Vorticity Dynamics on Boundaries

ADVANCES IN APPLIED MECHANICS. VOLUME 32

Vorticity Dynamics on Boundaries J . Z . WU and J . M . WU

.

The UniL'errity of Tennesscv Space Institute. Tullahoma Tennessee

I . Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . A . Development of Boundary Vorticity Dynamics . . . . . . . . . . . . . . . B. Plan of Prescntation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

120

I1. Splitting and Coupling of Fundarncntal Dynamic Processes . . . . . . . . . A . Dynamic Processes and Boundary Conditions . . . . . . . . . . . . . . . .

127

B. The Splitting and Coupling of Dynamic Processes . . . . . . . . . . . . . C . Splitting and Coupling inside thc Fluid: The Helical-Wave Decomposition Approach . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . D . Splitting and Coupling on Boundaries: A Model Problem . . . . . . . .

121 125

128 133

138 142

I11. General Theory of Vorticity Creation at Boundaries . . . . . . . . . . . . . A . Boundary Vorticity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . B. Boundary Fluxes of Vorticity and Enstrophy . . . . . . . . . . . . . . . . C . Creation of Boundary Vortex Sheets . . . . . . . . . . . . . . . . . . . . .

148 148

IV . Vorticity Creation from a Solid Wall and Its Control . . . . . . . . . . . A . The Effect of the Pressure Gradient . . . . . . . . . . . . . . . . . . . . . . B . The Effect of Wall Acceleration . . . . . . . . . . . . . . . . . . . . . . . . C . Three-Dimensional Effects . . . . . . . . . . . . . . . . . . . . . . . . . . . D . Vorticity-Creation Control . . . . . . . . . . . . . . . . . . . . . . . . . . . .

168 169

151 161

..

175 181

187

V . Vorticity Creation from an Interface . . . . . . . . . . . . . . . . . . . . . . . . 198 A . Dimensionless Parameters on a Viscous Interface . . . . . . . . . . . . . 199 B. Flat Interface and Free Surface . . . . . . . . . . . . . . . . . . . . . . . . . 203 C . Free-Surface Boundary Layers . . . . . . . . . . . . . . . . . . . . . . . . . 207 D . Complex Vortex-Interface Interaction and Surfactant Effect . . . . . 219 VI . Total Force and Moment Acted on Closed Boundaries by Created Vorticity Fields . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . A . The Vorticity Moment and Kutta-Joukowski Formula . . . . . . . . . . B. Total Force and Convective Vorticity Flux on a Wake Plane . . . . . . C. Force and Moment in Terms of Boundary Vorticity Flux . . . . . . . . . VII . Application to Vorticity Based Numerical Methods . . . . . . . . . . . . . . A . A n Anatomy of Vorticity Based Methods . . . . . . . . . . . . . . . . . . B. Numerical Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 119

224 226 231 239 247

248 257

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Copyright 0 1996 hy Acadcmic Press Inc . All rights of reproduction in any form reserved. ISBN 0-12-002032-7

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J. Z. Wu and J. M. Wu

VIII. Concluding Remarks. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

264

...................................

267

Refcrcnccs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

267

Acknowledgrncnts

I. Introduction One of the central problems in vorticity and vortex dynamics is the interaction of vorticity field and boundaries. The boundary can be a rigid wall, a flexible solid wall, or an interface of two different fluids. For example, an aerodynamicist may be concerned with the formation of the three-dimensional boundary layer on a wing surface, its interaction with shock waves, transition to turbulence, separation and control, as well as the effect of these on the wing lift and drag. A biomechanicist may be concerned with how a fish generates a Khrmhn vortex street with signs opposite to those behind a cylinder to gain thrust most efficiently. And a naval hydrodynamicist may be concerned with ship-wake vortices and how they interact with a wavy water surface. These problems, although apparently belonging to different fields, fall into the same category of vortex-boundary interaction at a fundamental level. In any case, boundaries are the most basic source of vorticity (in particular, the unique source for incompressible flow of a homogeneous fluid in a conservative external body-force field), and the whole life of a vortex usually begins at a boundary. Due to the continuous creation of vorticity from boundaries, a bounded vortical flow is much more complicated than an unbounded flow, in particular in the regions near the boundary. The complexity of vorticity dynamics caused by a boundary is further seen in turbulent boundary layer, where extremely abundant coherent vortical structures occur (c.g., Robinson, 1991), and in various vorticity-based numerical methods, where a difficult issue is how to formulate proper boundary conditions for the vorticity (Majda, 1987; Hald, 1991; Gresho, 1991, 1992). In fact, in most practical cases, unbounded flows are merely idealized approximations, and ultimately dealing with a boundary is inevitable. It is perhaps too ambitious to give a comprehensive review on vorticity-boundary interactions in a single chapter. However, the problem can be decomposed at least into two subjects. The first is ~ o v t e xdynamics inside a flow field, which is dominated by the highly nonlinear advection as

Vorticity Dynrrmics on Boundaries

121

well as diffusion, such as vortex stretching, instability, and breakdown; interacting vortex systems; and so forth; as well as their complicated consequences, including turbulent coherent structures. On that subject many excellent reviews are available. The second subject is the orti ti city dynamics on boundaries (or “boundary vorticity dynamics,” for short), which concentrates on the vorticity creation from a boundary and the reaction of the created vorticity to the boundary. It is this subject that makes a bounded vortical flow differ from an unbounded one and hence is an indispensable fundamental constituent and necessary prerequest in dealing with any vorticity-boundary interactions. This important subject has never been systematically reviewed before. In this chapter, we first present a unified general theory of the vorticity dynamics on various boundaries for viscous compressible flows and then review its applications to specific problems, which at this writing are mainly confined to incompressible flows. We shall consider two types of boundaries: a solid boundary, either rigid or flexible; and an interface of two immiscible fluids. Both are regarded as sharp material surfaces. Some other boundaries, such as a porous wall, regrettably are omitted, though they are also important in practice. In what follows we briefly review the development of boundary vorticity dynamics, then introduce the contents of the chapter.

A. DEVELOPMENT OF BOUNDARY VORTICITY DYNAMICS The study of boundary vorticity dynamics was first concentrated on the solid-wall type of boundary pioneered by Lighthill (1963). He called the normal diffusion flux on a solid wall (denoted by dB), d o u = u n - O o = udn

on

dB

(1.1)

as the iwficity source strength (per unit of time and per unit of area). Here o = V X u is the vorticity, v the kinematic viscosity, and n is the unit normal vector pointing out of the fluid domain. With this flux (referred to as the bounduly iwticity flux hereafter ), Lighthill expounded the physics of two-dimensional vorticity creation from a stationary wall, in particular its direct dependence on the tangential gradient of pressure, which resulted

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J. Z. Wu andJ. M. Wu

from applying the tangent component of the Navier-Stokes equation to dB: dw 1 dp (T= v-on d B (1.2) dn p dx Lighthill went on to present the whole boundary layer theory in terms of vorticity.’ Then, Batchelor (1967) devoted the main body of his classic book to incompressible rotational flows, and he gave a detailed but descriptive explanation of the vorticity creation from a solid wall, mainly in terms the formation of boundary vorticity rather than its flux. The importance of the vorticity creation and these pioneer works had not been fully recognized until the late 1970s. Lighthill (19791, in a survey paper on the waves and hydrodynamic loading, restressed the idea he raised 16 years before, which now stands at the center of his book (Lighthill, 1986). Then, Lighthill’s analysis was adopted by Panton’s (1984) textbook, which emphasizes both boundary vorticity and its flux. Morton (1984) criticized the common lack of fundamental understanding of vorticity generation mechanism-in the paper he still cited only Lighthill (1963) and Batchelor (1967). Morton tried to compromise the views of Lighthill and Batchelor and noted that for a moving wall its tangential acceleration should be added to (1.2) as another constituent of vorticity source. Then, in a review of unsteady, driven separated flows, Reynolds and Carr (1985) qualitatively explained the physics of many different forced, unsteady, separated flows in a unified way, based on the boundary vorticity flux, and added one more term due to the boundary transpiration to (1.2). They stressed the importance of a good understanding of vorticity production and transport in designing effective mechanisms for separation control. On the other hand, owing to the need for developing numerical vortex methods, eq. (1.2) and its normal counterpart,’

1 dp

do

-- - -uP dn dX I

.

on

dB

(1.3)

Lighthill (1963) found that “although momentum considerations suffice to explain the local behavior in a boundary layer, vorticity considerations are needed to place the boundary layer correctly in the flow as a whole.” He showed that the vorticity considerations “illuminate the detailed development of the boundary layer just as clearly as do momentum considerations.” Therefore, Lighthill has in fact placed the whole boundary layer theory into vorticity dynamics. ’Note that (1.2) and (1.3) are a pair of Cauchy-Riernann equations. Thus, on a hvo-dimensional stationary boundary, p and vu constitute the real and imaginary parts of a n analytical function. For a Stokes flow with advection being completely ignored, that complex function is also analytic inside the flow field.

Vorticity @nrrmics on Boundaries

123

have been repeatedly rederived, widely cited, slightly extended as state earlier, or partly utilized by Pearson (1969, Roache (1972), Leonard (1980, 198.9, J. C. Wu et al. (1984), Leconinte and Piquet (1984), among others. Even though this development had been confined basically to twodimensional flow over a flat platc, progress appeared at the fundamental level as J. Z. Wu (1986) extended (1.2) to three-dimensional flows over an arbitrarily curved stationary wall, and found extra contributors to u, such as the skin friction (or equivalently, boundary vorticity) and surface curvature, which exist in three dimensions only. Lighthill (1963) has mentioned that the effect of the third dimension and wall curvature is of smaller order, but exceptions exists near a sharp edge or a spiral point of the skin-friction lines. More important, Wu’s extension enabled him to study the u distribution on a closed surface, which leads to a novel total force formula exclusively in terms of the vectorial moment of the boundary vorticity fluxes due to these effects as well as pressure gradient (J. Z. Wu, 1987). Therefore, the creation of vorticity, that is, the action of a solid body to the vorticity field, and the reaction of created vorticity to the body reached an intrinsic unity on the basis of boundary vorticity flux. These results were soon extended to moving wall and compressible flows by J. Z. Wu et al. (1987, 1988a). A himilar extension, based o n a novel approach, applicable to any continuous media, was made by Hornung (1989, 19901, independently and almost simultaneously. But a detailed presentation of the whole general theory, along with an in-depth physical discussion, appeared in monograph and journals only recently (J. Z. Wu et al., 1993a; J. Z. Wu and Wu, 1993). During this period, the general theory was applied to various specific problems as well, of which some will be reviewed in this chapter. Owing to these efforts, the theoretical foundation of vorticity dynamics on solid boundaries has now been well established. Parallel to the preceding development, the vorticity dynamics on an immiscible interface has its own history. Although it has long been known that, on a free surface, there is a viscous boundary layer (e.g., Lamb, 1932; Wehausen and Laitone, 1960; Moore, 1959, 1963; Lundgren 1989), works from the point of view of vorticity dynamics were relatively rare. The physical interpretation of vorticity formation on an interface started from Longuet-Higgins (1953; see also Longuet-Higgins, 1992) as cited by Batchelor (1967). Longuet-Higgins showed how the vorticity appears on a free surface as a direct consequence of the continuity of tangent stress across the surface and expressed this boundary vorticity in terms of the surface

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J. Z. Wu and J. M. Wu

motion and geometry. For example, in steady two-dimensional flows, the vorticity on a free surface S simply reads

where U is the tangent velocity of S and K twice of thc mean curvature. But once again, this type of boundary vorticity dynamics had not received sufficient attention until the 1980s. Due to the great interest in the interaction between vortices and a free surface that causes the surface to significantly deform (see, e.g., the review of Sarpkaya, 1992a, b), rapidly growing works in publications have appeared. Like the solid-wall case, one of the key mechanisms involved in the interaction is the creation of new vorticity or the loss of existing vorticity (a negative creation) from the surface. This vorticity creation is inherently a viscous process and in most cases highly three-dimensional. How to understand the process and identify its role in observed complicated surface-deformation patterns during the interaction became utmost important. This practical need motivated corresponding theoretical studies at fundamental level. In a study of local flow properties on a viscous interface, Lugt (1987, 1989a) made a distinction between the roles of the vorticity on the surface and its diffusion flux across the surface. He pointed out that the surface vorticity does not provide information on the rate of production of vorticity or on the diffusion of vorticity into the interior of the fluid; this information should be furnished by the boundary vorticity flux. On a two-dimensional free surface, if the flow is steady, Lugt showed that the flux is (in our notation, s is the arc length along the surface) (1.5)

being the total head. Equation (1.2) was thereby extended to an interface for the first time, and theories for two different boundaries started to merge. We shall see in Section 111 that a combination of (1.4) and (1.5) is indeed the simplest prototype of the entire vorticity dynamics on interface. Later, Rood (1991) stressed the importance of u without giving details. H e then extended Lugt’s approach to unsteady flow and attempted to explain a series of observed phenomena (Rood, 1994a, b). In fact, extending the vorticity dynamics on a solid wall to include an interface is

Vorticity Dynamics on Boundaries

12s

straightforward; a general three-dimensional theory was presented by J. Z. Wu (1995). Hence, now the development of a theoretical foundation of vorticity dynamics on an interface is almost as mature as its solid-wall counterpart.

B.

PLAN OF PRESENTATION

This chapter consists of eight parts. In Section 11, an overall observation is made on three fundamental fluid dynamic processes driven by surface forces: the longitudinal compressing-expanding process, the transverse shearing process, and the surface-deformation process. The shearing process represented by vorticity is our main concern, but is coupled with the other two. In particular, the coupling on a boundary, as revealed by the Cauchy-Riemann equations (1.2) and (1.3) on a solid wall, or as (1.4) and (1.5) on a free surface, crucially affects the vorticity formation on the boundary. The coupling of the three processes as well as their splitting is the foundation of the entire body of boundary vorticity dynamics. We use a novel triple decomposition of stress tensor for Newtonian fluid to analyze the stress constituents, followed by a unified treatment of boundary conditions. These allow one to study the splitting and coupling of dynamic processes in both momentum balance inside the fluid and surface stress balance on a boundary. The former is further explored in terms of helical-wave decomposition, and the latter, exemplified by a unidirectional flow with both rigid and interfacial boundaries. Section 111 gives the general theory of vorticity dynamics on any material boundaries of viscous compressible flows. We first present the general formulas for the jump of normal stress and tangent vorticity across a boundary surface. Then, we present Hornung’s (1989, 1990) definition of vorticity flux and the derivation of its general formula. For Newtonian fluid with constant dynamic viscosity, definition (1.1) is recovered. Formulas for boundary vorticity and its flux are decomposed into tangent and normal directions to show the specific physical implication of each term. The difference between a rigid or flexible wall and an interface is addressed. As a complement to the boundary vorticity flux, the boundary enstrophy flux is introduced, and the central role of viscosity in the formation of both boundary vorticity and its flux is stressed. As the Reynolds number approaches infinity (the Euler limit of viscous flow), the vorticity creation

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J. 2. Wu and J. M . Wu

from boundaries manifests itself as the boundary-vortex sheet creation, whose three-dimensional dynamic equation and velocity in a general circumstance are derived. In Sections IV and V we enter various specific physical mechanisms responsible to vorticity creation from a solid wall and an interface, respectively. Some easily misleading conceptual issues are clarified. In Section IV, many model problems and applications are reviewed, either briefly or in detail. Several guiding principles for controlling vorticity creation from solid walls, of great interest in applications, are reviewed and exemplified. In Section V, we start with a general dimensionless formulation of the interfacial vorticity and its creation rate, followed by simplified approaches, along with a few worked out examples. These include the flat interface or free surface and the free-surface boundary layer at large Reynolds numbers. The section ends with a brief observation of vorticity creation in complex vortex-interface interactions and the surfactant effect on the creation. We turn to the reaction of created vorticity field to boundaries in Section VI. It is shown that, due to the coupling of shearing and compressing processes, the total force and moment acted on a closed boundary can always be expressed as vorticity-based formulas, even if the flow is dominated by a compressing process, as long as the Mach number is not in hypersonic regime. This is done systematically as an observer moves from the very remote far field to near field, until to the closed-boundary surface. Correspondingly, the resulting formulas arc increasingly accurate and general: from the classic Kutta-Joukowski formula applicable to the Euler limit of steady viscous flows, to the accurate near-wake plane analysis for arbitrary steady viscous flows, to the total force and moment exclusively in terms of boundary vorticity fluxes. The unique characters and great potential in application of these formulas, compared with conventional force and moment formulas, are addressed. Examples are given to illustrate the implication of new formulas in aerodynamic diagnostics and optimization. One of the main applications of boundary vorticity dynamics is to provide proper boundary conditions for vorticity-based numerical methods. The relevant theoretical analysis and numerical examples are given in Section VII. The success of computations further confirms the power of the theory and the importance of correct physical understanding. We conclude the chapter with Section VIII. According to the preceding definition of the boundary vorticity dynamics, in the main body of this chapter we shall not go deeply into the interior

Vorticity Djn~inzicson Boundaries

127

of the flow field, and hence in most cases the advection effect is avoided. This confinement makes a unified general theory possible. However, this confinement also makes the theory alone insufficient to give a complete answer to any vorticity-boundary interaction problem where nonlinear advection is involved. Eventually, one has to rely on experiments or computations for every specific problem. The chapter includes some fully resolved vorticity-boundary interaction problems merely to illustrate the general theory. More examples of this type of interactions can be found in the review of Doligalski et 01. (1994) for rigid boundary and that of Sarpkaya (1996) for interfacial boundary. The material selection of this chapter inevitably reflects the authors’ personal experience. Some topics seldom mentioned in the chapter are by no means less important. On the contrary, they may precisely be the place where the boundary vorticity dynamics has great potential to apply. For example, the application to flow over a flexible wall is illustrated only once, but almost the entire field of biofluiddynamics, such as animal flight and swimming and blood flow (e.g., T. Y. Wu, 1971; Fung, 1971; Lighthill, 1975; T. Y. Wu et al., 1975; Childress, I981), falls into this category. Clearly, the vorticity creation and reaction on such a flexible boundary, either active or compliant, is of vital importance and hence the theory has a room to grow including its extension to non-Newtonian fluids. We believe that a combination of the boundary vorticity dynamics with existing relevant theories, or a re-examination of these theories from the viewpoint of vorticity dynamics, will open a very fruitful new avenue.

11. Splitting and Coupling of Fundamental Dynamic Processes

A vorticity field d x , t ) characterizes the most abundant dynamic process in a flow field-the transverse shearing process. This process coexists and is coupled with other fundamental dynamic processes as well as the thermodynamic process, both inside the flow field and on its boundaries. In particular, the specific behavior of vorticity dynamics on a boundary depends on the type of boundary, which to a large extent determines with what process the shearing is coupled. To provide a general background for the theory of boundary vorticity dynamics, therefore, in this section we make a systematic examination of the coupling and splitting of these processes.

128

J. Z. Wu and J. M. Wu A. DYNAMIC PROCESSES A N D BOUNDARY CONDITIONS

We first clarify how many fundamental dynamic processes coexist in a Newtonian fluid and what kinematic and dynamic conditions are imposed on different boundaries. 1. The Triple Decomposition of Stress Tensor For a Newtonian fluid, the viscous stress tensor is proportional to the strain-rate tensor D = D', where the superscript means transpose. Thus, to identify all possible dynamic processes driven by surface forces explici t l ~one , ~ needs to first decompose D into its corresponding constituents, in particular on boundaries. This, for a rigid stationary wall dB, was first studied by Caswell (1967), who obtained an elegant formula that can be easily extended to an arbitrarily moving wall with angular velocity W ( t ) (J. Z. Wu and Wu, 1993): 2D

=

26nn

+ n(w, X n> + ( w , X n)n

on

dB

(2.1)

where 6 = V . u is the dilatation and w, = o - 2W, the relutiL!e Liorticity. However, for an arbitrary surface, the following novel but much simpler decomposition of D is most appropriate. Let R = - R T be the antisymmetric spin tensor such that V u = D + R . Then, because VuT = D - R , there is D = 61

+R

-

(2.2)

B

where I is the unit tensor and

B = 61 - Vu'

with V . B

=

0

(2.3a, b)

is called the su~ace-strain-ratetensor (Dishington, 1969, (where the definition of B differs from (2.3a) by a transpose) since for any surface element dS = n dS there is (for component form see, e.g., Truesdell, 1954; Batchelor, 1967) D -dS Dt

=

n . B dS

(2.3~)

'There are also dynamic processes driven by body forces and thermodynamic processes. The conservative forces like gravity can be absorbed into pressure force as we shall do later, and nonconservative forces like those in magnctohydroclynamics are beyond our present concern. The thermodynamic process introduces additional complications that will be touched upon but not our main concern here.

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Vorticity Dynumics on Boundaries

Now, let p and A be the first and second dynamic viscosities. The implication of (2.2) in the dynamics of a Newtonian fluid is immediately evident by substituting it into the Cauchy-Poisson constitutive equation: T

=

+ A6)I + 2 p D

(-p

which yields an intrinsic triple decomposition of the stress tensor T (J. Z . Wu and Wu, 1992, 1993; J. Z. Wu, 1999:

+ 2pcL.n - 2 p B

T = -111

where

n

-p

-

(A

(2.4)

+ 2p)6

(2.5)

is the isotropic part of T, which characterizes the compressing-expanding process (compressing process, for short) and will be referred to as the compression itanable? Then, on any surface S with normal n, either inside the fluid or on a boundary, the surface stress also has an intrinsic triple decomposition (2.6a)

t - n . T = -nn+.r+tt,

where T = p.cr, X

n,

t,

= -2pn.B

=

1 D

-2p--dS dS Dt

(2.6b,c)

Therefore, the surface stress of a Newtonian fluid consists of three parts: the normal stress -Hn, which i s dominated by the pressure p, plus a viscous compressible correction; t h e shear stress T , which is proportional to the tangent vorticity; and the surface-deformation stress t , , which represents the viscous resistance of a surface element of unit area to its strain rate. Each stress drives a fundamental dynamic process. Thus, through the stress balance, the shearing process may couple with other two dynamic processes as well as the thermodynamic process. Note that usually t , is neither parallel nor perpendicular to d S , see Figure 1. Among the three stresses the surface-deformation stress t, is less familiar and deserves a further analysis. From its physical meaning, we expect that t , should depend exclusively on quantities defined on the surface S. Indeed, due to the vector identity,’ (b

X

V)

X

c

=

(VC). b

-

b ( V . C)

=

b . { ( V C ) ~- ( V . c)I}

(2.7)

‘The compression variablc does not have a unique definition and specific name (this explains why the hydroacoustic variable is not uniquely defined either), hut the shearing i~uriahleis always the vorticity. 5 W e thank Professor T. Y. Wu’s suggestion on using this identity, which simplified relevant mathematic manipulation.

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J. Z. Wu and J. M. Wid

FIG.I . The triple decomposition of the surface stress.

it immediately follows that

where the right-hand side contains only tangent derivatives. Hereafter, we specifically denote the surface velocity by a capital U, which is feasible only if no normal derivative of u is involved. In studying the boundary vorticity dynamics, we often need to decompose a vector, including the gradient operator, into normal and tangent components on a surface S. We use suffix rr to denote the tangent components of a vector; thus, for instance, n x V = n X V,. Then, let K = -V,n

=

-(Vnn)

7

and

K

= -V;n

be the (symmetric) cunlature tensor of surface S, of which only four tangent components are nonzero, and twice the mean curcature, respectively (for a neat and convenient technique of calculating various tangent derivatives of an arbitrary tensor defined on a surface, see an appendix of J. Z. Wu, 1995). Correspondingly, we write U = U, + n u , . Then, we may further split the normal and tangent components of (2.8a) explicitly (J. Z. Wu, 1995):

Note that from the generalized Stokes theorem

13 1

VorticityL?ynamics on Boundaries

for any tensor 9and any admissible tensor product 0 , eq. (2.8a) implies that the integral of n . B over a closed S (of which dS vanishes) must be zero, as is directly seen from (2.3b) and the Gauss theorem.

2. Primary and Deriiwl Boundury Conditions The kinematic and dynamic conditions on different boundaries can be stated in a unified way. We denote a material boundary of a viscous fluid by ~ 8which , can be either a rigid or flexible wall, or an immiscible interface of two different fluids. When there is a need to distinguish between a solid wall and an interface, we set : != dB for the former and .8 = S for the latter. Let us agree that the unit normal vector n points out of t h e flow domain on dB and from fluid 1 to fluid 2 on S. Let [ Y ] denote the jump of any quantity 9right across .%', so that on a solid boundary [u] = u - b, where b is the solid velocity, whereas on an interface between fluid 1 and fluid 2, say, [u] = u , - u 2 . Then, the boundary conditions should ensure the continuity of velocity and that of stress with allowance of surface tension (Wehausen and Laitone, 1960; Batchelor, 1967). Therefore, the velocity continuity implies [u] = 0 on :2,or, in decomposed form, the no-through and no-slip conditions: n . [ul

=

0,

n x [u] = 0 on

(2.10a, b)

A?

Similarly, let T be the surface tension that vanishes on dB, then the stress continuity across ~2? implies n . [t] + TK = 0,

n x [t] = 0 on

9

(2.1 l a , b)

where K is the mean curvature of 9 defined before. We call (2.10) and (2.11) the primary boundary conditions. Note that, with surface tension T = 0, they apply equally well to any material surface inside a viscous flow. Now, because the vorticity transport equation is one order higher than the momentum equation, in addition to these primary conditions we need some deriued conditions for vorticity dynamics, which are corollaries of (2.10). First, eq. (2.10b) directly implies the well-known continuity of normal uorticity : n.[co]=Vo,x[u,l=O

on

9

(2.12)

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1. Z. Wu and J. M. Wu

Thus, on a nonrotating M there must be n . ~ r = ) 0; and a viscous vortex tube with ~ r ) n. f 0 cannot terminate at LB but will penetrate 94. The latter occurs if a solid wall is rotating (so the vorticity goes into the solid body as twice its angular velocity), or if a viscous tornadolike vortex intersects an interface. The second derived condition concerns the surface acceleration. If, at an initial time r = 0, a fluid particle sticks to a point of a solid wall or a particle of fluid 1 sticks to a particle of fluid 2 at an interface, by (2.10) the stickiness will continue as time goes on. Therefore, there must be the continuity of acceleration:

[a] = 0

forall

and x €9. a =

t

Du ~

Dt

Inversely, if this condition holds and if in addition

[u,,] = 0 at

t

=

0 and all

x

E&‘

then (2.10) is guaranteed. Therefore, the adherence condition (2.1 Oa, b) can be equivalently stated as n . [a] = 0,

n . [u,,] = 0

n x [a3 = 0,

n x [u,)]= 0

1

on

9

(2.13a, b)

respectively (J. Z. Wu et al., 1990). We shall see the key role of this acceleration condition in boundary vorticity dynamics. Obviously, (2.12) and (2.13) will be redundant if one confines oneself to primitive variables. In a recent article on interface dynamics, Yeh (1995) attempted to derive the continuation of velocity u, stress t (confined to the case without surface tension T ) , and acceleration a across %‘ from dynamic equations. While some of Yeh’s argument is incomplete, it may be easily improved to an extent, as briefly reviewed below. At this stage, we do not consider Yeh’s approach as a superior alternative to the conventional one; what makes it interesting is that it may further confirm the consistency between each boundary condition and a corresponding dynamic aspect of the fluid motion. In particular, the no-slip condition (2.10b) has been a hypothesis based on physical observation, but it now appears to be consistent with the energy balance.

133

Vorticity L?ynamics on Boundaries

Suppose that on both sides of a material boundary 9’a dynamic equation of the following general form holds:

dF

-

Dt

+ FV.U+ V .S + V

X

A

+G

=

0

where F, S, A, and G can be any tensor provided that all terms are of the same rank and are smooth functions of (x,t ) on each side. Consider a domain 9 =8, + g 2 across c(8,where 9,and 9i2are subdomains on sides 1 and 2 of S‘, respectively. Take the integral of the above equation over 23.Then, using the weak-solution technique to handle the possible discontinuity across 9, Yeh ( 1995) arrived at a general jump condition: n . [Sl

+n X

[A]

=

0 on

9

Now, we first specify the above dynamic equation as the Cauchy motion equation (see (2.14) below), i.c., taking S = T and A = 0. Then the jump condition immediately gives (2.11) with T = 0. Second, by taking the divergence and curl of the motion equation per unit mass, a similar argument may lead to n . [a] = 0 and n X [a] = 0, respectively. Finally, we specify F as the kinetic energy, of which the equation has a single divergence term V * ( u . t), leading to [ ~ . t =] [ u ] . t

=

o

on

LB

due to the continuity o f t . This result implies that the mechanical energy cannot be stored on a surface clement without volume. Writing t = nt,i + t,, we see that (2.10a) follows as long as t,, # 0, and (2.1%) follows as long as the tangent stress t, is nonzero and not perpendicular to the tangent velocity u, on at least one side. Although a rigorous proof of this last condition is not yet available and exception at some special isolated boundary points cannot be excluded, physically it should be generically true on a material boundary. Note that for inviscid flow with t, = 0 there is no restriction to [u,], again consistent with the common result.

B. THE SPLITTING AND

C O U P L I N G OF

DYNAMIC PROCESSES

The shearing process or vorticity field is not always coupled with all other dynamic and thermodynamic processes. Quite often, one or more processes are unimportant or decoupled from the shearing. Then the

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J. Z. Wu and J. M. Wu

relevant physics and analysis can be simpler. We now examine when and in what sense this situation happens. 1. Splitting and Coupling in Momentum Balance The most important use of the triple decomposition (2.4) is its combination with the Cauchy motion equation (where f is any external body force per unit mass) pa=pf+V.T

(2.14)

yielding a corresponding triple decomposition of the Navier-Stokes equation (J. Z. Wu and Wu, 1993) p(a

-

f)

=

- ~ nv x ( p a ) -

-

2 v p . ~

(2.154

where the right-hand side represents three surface forces that balance the body forces (inertial and external). The third term on the right has a simple physical interpretation: because p is a function of temperature, eq. (2.3~)indicates that this term is the viscous resistance (per unit volume) of isothermal sugaces to their deformation caused by dynamic-thermodynamic interaction. This point can be made clearer by writing

where dS, is an isothermal-surface element with normal n,, and S, = d(log p)/d(log T ) , usually of 0(1), is the dimensionless sensiticity (here T denotes temperature only in this single context). This effect is rather weak in comparison with other surface forces, except when heat transfer is extremely strong. So, from now on we shall always assume p = constant for simplicity. Then the surface-deformation process is absent from the momentum balance, and (2.15a) implies a natural Stokes-Helmholtz decomposition of the body force (Truesdell, 1954): p(a - f) =

-VIT

-

V x (pa>

(2.1%)

with II and pw being the scalar and vector potentials. Only the compressing and shearing processes are involved, which contain only three independent components because puw is solenoidal. Therefore, as first noticed by

Vorticity Dynamics on Boundaries

135

J. Z. Wu and Wu (1992), so f a r (is the momentum balance with constant Liscosity is concerned, the six-component stress tensor T can always be replaced by a three-component tensor

T

=

-n1+ 2 p a

(2.16)

which consists of only the isotropic and antisymmetric parts of T and will be referred to as the reduced stress tensor. This replacement is feasible in most applications and implies a big simplification.6 In fact, in (2.14), T can be replaced by any T’ as long as V . (T - T’) = 0, and 5. is the simplest among these infinitely many T’. We digress to observe that T is nothing but a tensor expression of the Stokes-Helmholtz potentials I1 and p w . Mathematically, for any tensor S (symmetric or not), one can always find a three-component tensor S such that S and S have the same divergence, which amounts to finding the Stokes-Helmholtz decomposition of that divergence. In the preceding case, T automatically emerged owing to the intrinsic decomposition (2.4) or the physically natural Stokes-Helmholtz decomposition (2.15). In other situations, the reduced tensor will be not so simple, although it still exists. For example, for an incompressible homogeneous turbulence, in the wave-number space the conventional six-component turbulent stress tensor T I j ( k ) can be replaced by a threc-component one (J. Z. Wu et al., 199Sb):

We return to (2.1Sb) and ask whether and when the remaining two surface forces can be further decoupled, at least approximately. Mathematically, the transverse and longitudinal vector fields V X w and V n in (2.13 are said to be orthogonal in a functional space, if (2.18) over the flow domain 53; in this case they can be solved independently (assuming a is not further split-the coupling in nonlinear advection is of ‘Some saving in computational fluid dynamics has been gained by implicitly using the reduced stress tensor: In the stress computation o f three-dimensional viscous incompressible flows by a finite-volume method, the number of grid points in a cubic element can be reduced from 27 t o 7 (Eraslan el al., 1983). The CPU time of overall computation can thereby be reduced by 40%.

136

J. 2. Wu andJ. M. Wu

a different nature, see Section 1 1 . 0 . Then, by the Gauss theorem, we have

where = d 9 is the closed boundary of 53.Hence, eq. (2.18) holds if one of the following conditions is satisfied:

n=O,

n ~ w = 0 , (nxV).w=0,

n x V n = O , on

.B

(2.19)

Therefore, in an unbounded flow the ( w , I I ) decoupling is always possible. For a bounded flow, then, the (w, II) coupling inside the flow amounts to that on the boundary, clearly indicating the crucial importance of boundary vorticity dynamics. The third and fourth conditions of (2.19) are special cases of (1.3) and (1.2) and represent a homogeneous Neumann condition for pressure and vorticity equations, respectively. An obvious example where the fourth condition exactly holds is the Stokes's first and second problems, to be examined in Section II.D, including their generalization to rotating circular cylinder. A further example of practical interest is the incompressible Blasius boundary layer, in which d p / d x = 0 for sufficiently large x, and all the vorticity inside the whole layer must be created near the leading edge, where the full Navier-Stokes equation has to be applied. More generally, any attached boundary-layer approximation with known external main flow is a decoupled approximation. On the other hand, the coupling becomes very strong at a small Reynolds number, where, as noted in a note 2 to Section I, p and pu are simply the real and imaginary parts of an analytical function. This observation suggests that the strength of the (a, p ) coupling depends on the Reynolds number; which will be confirmed more unambiguously in Section VII. For an incompressible flow with a rigid boundary, the dynamics in the interior of the flow is further reduced to the shearing process alone (an incompressible potential flow belongs to kinematics), whose coupling with the compressing process (the pressure force) occurs merely at a boundary dB as indicated by (1.2) and (1.3). In other words, to compute an incompressible flow based on vorticity only, the tangent pressure force on dB needs to be solved (in numerical computation even this boundary ( w , p ) coupling can be bypassed, see Section VI1.A).

Vorticity Dytianiics on Boundaries

137

It should be stressed that we are not proposing a new “constitutive equation” (2.16) to replace (2.4). Obviously, the surface-deformation process may appear once we go beyond the momentum balance. One example is the angular momentum balance, but there the effect of t still amounts to shearing. In fact, in a fluid element of volume V bounded by dV, it can be shown that

where x is the position vector. This effect of t , again reduces to vorticity and vanishes if the boundary is not rotating. Another example is the dissipation rate a.From an identity (Truesdell, 1954) 1

D,,D,, = (I2 + --w? - (B,,u,),, 2 it follows that

and hence

Thus, all three dynamic processes contribute to the dissipation. However, substituting (2.21) into the energy equation immediately leads to the cancellation of t h e part due to t , . 2. Splitting and Coupling in Stress Balance Contrary to (2.20) and (2.20, where t , adds merely a small part to the total effect, this stress may play an role in the stress t = n . T on boundaries, because no divergence of T is taken. However, an exception still occurs on a rigid wall dB with angular velocity W(t); in that case, the boundary velocity can be written as u = U,,(t> W ( t >X x, and from (2.8) it follows that t , = - 2 p W x n, which can be absorbed into the shear stress. This yields the familiar formula, which also directly follows from (2.1):

+

t = -nn+pW,Xn

on

rlB

(2.22)

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J. Z. Wu and J. M. Wid

where the relative vorticity or is nothing but [m], satisfying n . m, = 0 due to (2.12). Therefore, along with (2.15b), we see that except for strongly heat-conducting fiows, in the entire rorticity dynamics with rigid boundary, only the coupling of shearing and compressing processes is important (J. Z. Wu and Wu, 1993). Note that on a rigid wall the stress balance (2.11b) occurs between solid and fluid; should the solid stress be known, then so would be the boundary vorticity. But the real situation is, of course, precisely the opposite. In contrast, on a flexible boundary, either solid or fluid, the role of surface-deformation stress t , becomes very active. A n extreme case opposite to rigid wall is free surface, where by (2.11b) there must be n x t = 0, and hence right on a free surface the tangent r!orticityis balanced solely by the surface deformation, of which (1.4) is a simple example. We leave a full exploration of this issue to Sections I11 and V. So far we have met two types of couplings of shearing process and other processes o n a boundary: one is due to momentum balance, which leads to a coupling like (1.2); and the other is duc to stress balance, which leads to that like (1.4). Further decouplings may happen in both types, if (2.19) holds for the former and if the free surface is flat for the latter (where, by (1.41, the surface vorticity vanishes). It should be stressed that these two types represent different physics. The former is of one order higher than the latter ( V . T versus n . T ) but, as will be shown later, the former determines the latter. C. SPLITTINGAND COUPLING INSIDE THE FLUID: THE HELICAL-WAVE DECOMPOSITION APPROACH We now further examine the splitting and coupling inside a flow field. Consider a slightly compressible, isentropic but viscous flow (the incompressible flow model is oversimplified for compressing process). In this case we assume the variation of the kinematic viscosity v = p / p is negligible and so are the external body force f and the viscous dilatation term ( A + 2 p ) 6 in II. Let h = / d p / p be the enthalpy. Then (2.1%) reduces to a = -Vh - vV x o (2.23) If the flow is unbounded or the material boundary of the flow domain is at rest, it is more convenient to write (2.23) as (2.24)

VorticifyLlynamics on Boundaries where L

5

139

o X u is the Lamb vector and

(2.25)

is the total enthalpy that now takes the place of compression variable. It is well known that to split the shearing and compressing processes one simply takes the curl and divergence of (2.23) or (2.24). This gives the vorticity equation and the “compression equation.” It will then be clear that the shearing and compressing processes are governed by the Reynolds number and Mach number, respectively; and as the range of these parameters changes, a hierarchy of approximations of this pair of equations can be constructed (J. Z. Wu and Wu, 1989a). Although some new results of these well-known curl and divergence operations will be presented later in Section VILA, here we introduce a less familiar but physically very appealing approach to the splitting. Let the Stokes-Helmholtz decomposition be also applied to the velocity and the Lamb vector and their respective vector potentials be made divergenceless:

Then (2.24) yields

)

+ G X [ Z + J + v w

(2.26)

For the present purpose, it is beneficial to work in the wave-number space, because then the spatial derivatives are simplified to multiplications by a wave vector k. Denoting the Fourier transform of any vectorial function f(x) by F{f(x)) = f(k), there is

Note that this pair of operations clearly indicates that dilatation waves are always longitudinal and vorticity waves are always transverse. Therefore, just by taking the inner and vector products of the Fourier transform of (2.26) with k/k, we immediately obtain the longitudinal and transverse

140

J. Z. Wu and J. M. Wu

"Bernoulli integrals" in the wave-number space: d

-4(k,t) at

d

-+(k,t) dt

+ X(k,t) + H ( k , t ) = 0

+ J(k,t) + u o ( k , t )

=

along k

(2.28a)

0 normal to k

(2.28b)

which govern the compressing and shearing processes, respectively. The decomposition (2.28), however, has not yet reached the finest fundamental building blocks of fluid dynamic interactions. Moses (1971) proved that the vector potential in a general Stokes-Helmholtz decomposition can be further intrinsically split into two, representing the righthanded and left-handed helical states. This is possible because the curl operator and the vector potential are both axial or pseudo vectors, which have handedness or polurity as an intrinsic property or dimension.' Thus, in (2.24) and similarly in (2.28), the vorticity and velocity can also be further split. Mathematically, this is achieved by the so-called helical-wave decomposition (HWD for short), first studied by Moses (1971) and Lesieur (1972; see also Lesieur, 1990). The decomposition uses the eigenvectors of the curl operator as the basis. In the wave number space the HWD basis becomes eigenvectors of the operator i k X , denoted by QA(k):

The eigenvalues for each k are -t k and 0. To see the structure of this basis explicitly, let e,(k) be an arbitrary real unit vector perpendicular to k, and e,(k) = k X e , / k , such that ( e , , e , , k/k) form an intrinsic Cartesian basis. Then a simple representation of Q*(k) is 1 QA(k) = -[e,(k)

fi

Q"(k) =

k k

for

+ iAe2(k)] A

=

0

for

A

=

I

(2.30)

'The relation between handedness and polarity is that a circularly polarized wave has only a single handedness, and a linearly polarized wavc has no handedness: that is, the right-handed and left-handed components are the same. A superposition of sufficiently many randomly polarized waves may have zero averaged polarity and zero net handedness.

VorficityDynanzics on Boundaries

141

These QA(k)also form an orthonormal and complete basis in the complex wave-number space, and hence any u(k) can be decomposed to u(k)

uA(k)QA(k), with

=

uA(k)= u ( k ) . QA*(k) (2.31~1)

A = f 1,O

where * means complex conjugate. The components with A = -t 1 and 0 clearly describe the transverse and longitudinal parts, respectively. More remarkably, the transverse components now represent a right-handed or left-handed circularly polarized state, depending on A taking 1 or - 1; and the HWD vorticity components are simply given by wA(k) = AkuA(k),

A

=

+1

(2.31b)

Consequently, by (2.27) and (2.29) and using the continuity equation, for weakly compressible flow, (2.28) can be cast to a very neat form:

where c is the sound speed. Equation (2.32a) is nothing but the Fouriertransformed Liorfex-sound equution at low Mach numbers (Howe, 1975; J. Z. Wu and Wu, 1989a). On the other hand, eq. (2.32b) is both the HWD transverse Navier-Stokes equation and the HWD vorticity equation due to (2.31b). Note that the nonlinear Lamb vector LAin (2.32b) is a convolution integral, which contains the self-coupling among different transverse modes and between the right- and left-handed velocity components. The cross-coupling between the two processes inside a flow field is also evident from (2.32). First, even though an acoustic wave is a longitudinal wave, its fluid dynamic source is the potential part of the Lamb vector, L", which vanishes without shearing process. Second, in the Lamb vector o x u, there can be a contribution of the potential velocity u"(k,t ) . Thus, although a sound wave is not the source of vorticity inside a fluid due to the absence of H in (2.32b), it will affect the advection of a vorticity field. The most typical and important example of such a coupling is the wellknown vortex stretching due to a background irrotational straining flow U = (ax,By, y z ) , which is implicitly contained in the transverse Lamb vector LA(k,t ) of (2.32b) as the counterpart of V x L(x, t) in the physical

142

J. Z. Wu and J. M. WLI

space. This coupling has been the object of extensive studies; for a recent analysis and review of previous works, see Moffatt et al. (1994). We stress that these cross- and self-couplings inside a flow field are essentially due to nonlinear kinematic adcection, which should be distinguished from dynamic couplings discussed before. The polarity of a vorticity field has recently attracted much attention due to its significant effect on vortex evolution and turbulent cascade process (e.g., Waleffe, 1992; Melander and Hussain, 1993; Virk et al., 1994). In a sense, theories on vorticity and vortex dynamics would be incomplete if the polarity effect was ignored. Because this new property contains two independent real scalars (HWD splits a divergenceless vector potential into two), it can be conveniently characterized by the relative amplitude and phase of the right-handed and left-handed components, of which the role in nonlinear interactions should be further explored. D. SPLITTING

BOUNDARIES: A MODELPROBLEM AND COUPLING ON

Although the kinematic coupling of shearing and compressing inside a flow field due to advection is inherently highly nonlinear, the dynamic coupling on a given boundary 9? is apparently of a linear nature, as exemplified by the Cauchy-Riemann equations (1.2) and (1.3). This apparent linear character makes it possible to develop a general formal theory, which is to be reviewed in this chapter. Here we first clarify some basic facts and concepts. First, eq. (1.2) indicates that the ( w , p ) coupling is necessary for producing vorticity on a rigid wall. The coupling would disappear if the flow were strictly inviscid and if the no-slip condition on .D were removed. Therefore, the dynamic boundary coupling is inherently a iiscous phenomenon. J. Z. Wu and Wu (1993) have stressed that under a pressure gradient the particles of a strictly ideal fluid on a solid wall will only slide over it but never rotate; there is no mechanism to give these particles an angular velocity, and hence no vorticity can be created. This argument applies equally to an interface. Therefore, we shall be confined to viscous fluid exclusively. Even if the Reynolds number approaches infinity, the flow will behave as the Euler limit of a Nauier-Stokes flow (Euler limit, for short) rather than an ideal flow. In this case, the created vortex layer due to the

143

Vorticity Dynmzics on Boundaries

no-slip condition will degenerate to a vortex sheet, which is still essentially different from the pure sliding on a mathematical contact discontinuity. For an elegant exposition of the difference between the Euler limit of a Navier-Stokes flow and the Euler solution of an ideal fluid, see Lagerstrom (1973). Second, in the analysis o f the boundary behavior of vorticity, such as that by Lighthill (1963) and Batchelor (19671, we meet two quantities: boundary vorticity (sometimes denoted o Bfor clarity) and its flux u. As we explained before, o B and u appear in the two types of couplings through the balance of momentum and surface stress, respectively. J. Z. Wu and Wu (1993) showed that on a solid wall the implications and roles of these quantities are very different. This is also true on an interface, as Lugt (1987) stressed. For a fluid element sticking to a material boundary, its vorticity represents twice of the angular velocity of the principal axes of its strain-rate tensor-a physical interpretation of vorticity by Boussinesq (see Truesdell, 1954), and in fact the only consistent interpretation applicable to fluid elements both i n the interior of fluid and on a boundary. Figure 2(a) sketches such a fluid element sticking on a boundary and shows how the principal-axis rotation leads to an o B. Note that, once a material it will vortex line sticks to a boundary (of which every point has an oB), remain there and never go into the interior of fluid; but o B can be diffused into the fluid as shown in the figure. In contrast, as indicated by its definition (1.1), the boundary vorticity flux represents a mechanism that sends vorticity into the fluid from a boundary. Figure 2(b), reproduced from Lighthill (1979), clearly indicates how a fluid

High Pressure Low Pressure (a)

@)

FIG. 2. The physics of boundary vorticity w, and its flux u. (a) The rotation of the principal axes of the strain-rate tensor of a fluid element sticking on a boundary gives w B, which can be diffused into the fluid. (Reproduced from J. Z. Wu and Wu, 1993. Reprinted with the permission of Cambridge University Press.) (b) A fluid element neighboring a stationary boundary is set to rotate due to a tangent pressure gradient and the no-slip condition. (Reproduced from Lighthill, 1991, with permission.)

144

J. Z. Wu and J. M. Wu

element neighboring to a boundary (but not right on the boundary) is forced into rotation by a tangent pressure gradient. As Morton (1984) pointed out, this d p / d x can be replaced by a boundary acceleration; so Figure 203) is a pictorial interpretation of (1.2) and (2.33) later; that is, that of u. An external body force may cause similar effect; for instance, the gravity acted on an inclined boundary (Section I1I.B). Note that, although without a no-slip condition and viscosity the fluid ball in Figure 2(b) would not rotate, the amount of vorticity being sent into the fluid due to (1.2) or (2.33) as well as a body force is independent of the magnitude of v (Lighthill, 1963). We shall also see the explicit viscous effects on u, but except at some special local regions, they are much weaker if v is small. The magnitude of v determines only how deep the vorticity can be diffused into the fluid. In the Euler limit, therefore, the same amount of vorticity is still going into the fluid under the same d p / d x , say, but all confined within a vortex sheet. This sheet exists inside the fluid and is conceptually different from the boundary surface. For example, as a material sheet, it has a definite velocity differing from, say, a stationary solid wall from which it is created (this fact is well known as one approximates a boundary layer by a vortex sheet). To illustrate these basic concepts, we consider a simple model problem taken from J. Z. Wu and Wu (1993) and J. Z. Wu et al. (lYY4b): a unidirectional, incompressible, viscous flow over a flat plate. The fully general case will be treated in the next section. Assume the flow occurs on the half plane y > 0 with u = [ d y ,t > , 0,0], w = [O,O, w ( y , t)] and p = 1. The fluid and boundary are at rest for f < 0, and at t = 0 there suddenly appears a tangent motion of the plate with speed h ( t ) and a uniform, time-dependent pressure gradient d p / d x = P(t). Then, applying (2.23) to y = 0 and imposing the no-slip condition yields the boundary vorticity flux cr(t) =

dh -

dt

+ P(t)

at

y

=

0

(2.33)

which represents the force balance on the plate and in which u is the viscous force. Mathematically, (2.33) gives a Neumann condition for d y , t) and leads to the solution

Vorficit.yDynamics on Boundaries

145

Here, the flux u can be regular or singular. If, at t = 0, there is an impulsive P ( t ) and db/dt, they must cause a suddenly appeared uniform fluid velocity U = ( U , 0,O) and wall velocity b,,, respectively, such that

P ( t ) = -U8(t),

db

-

dt

=

b , , 8 ( t ) for 0 - 1 t I 0’

Thus, the vorticity flux will also be singular (still due to a no-slip condition at t = 0):

a ( t >= - ( U

-

h , , ) 6 ( t )= y , , 8 ( t ) for 0 - 1 t 5 0’

(2.35)

where y o = - ( U - 6,) is the strength of the initial vortex sheet. Separating this singular part from (2.34) yields

(2.36) Obviously, the boundary vorticity is given by

indicating clearly that wR is exclusively from u. Moreover, the total amount of the vorticity being scnt into the fluid is (2.38) which is indeed independent of v , as asserted before. In particular, if the pressure gradient and wall acceleration vanish from t = O’, eqs. (2.36) and (2.37) reduce to the Stokes’s first problem (or the Rayleigh problem):

On the other hand, if P ( t ) = 0 for t > 0’ but the wall has a sinusoidal oscillation, say b ( t ) = cos nt, then we have the transient Stokes’s second problem, which has been thoroughly studied by Panton (1968) and of which the boundary vorticity can be integrated from (2.37) analytically:

146

J. Z. Wu and J. M. Wu

where S(x) and C(x) are Fresnel's functions. As t classic result

4

=, eq. (2.40) gives the

Stokes's first and second problems have been generalized to include different boundary shapes (J. C. Wu and Wu, 1967). For the flow caused by an impulsively started rotating circular cylinder with constant angular velocity and that by a circular cylinder with rotatory oscillation, the analytical solutions (the latter is confined to the asymptotic steady state t + );. can be found in Lu (1987) and PCpin (19901, respectively. J. Z. Wu et al. (1994b) also gave the computed time evolutions of the vorticity field for these two cases, including a numerical result in the transient period for the generalized Stokes's second problem. The behavior of boundary vorticity and its flux is qualitatively the same as the flat-plate case. From the previous unidirectional-flow solution (2.33)-(2.38), two observations can be made. First, the no-slip condition is the key in deriving (2.33) or (1.21, including the possible singular part of (r, and hence in the creation of vorticity. Therefore, any inviscid interpretation for this creation mechanism should be rejected. Second, it is the boundary vorticity flux u that is directly (locally and simultaneously) coupled with the compressing process ( d p / d x here) through the force balance on the boundary. In contrast, the boundary vorticity o Ris a time-accumulated effect of a.For more general case, the space-time integrated effect of advection and diffusion also contributes to c o n , see Section VILA. Therefore, the boundaly r>orticityflux u, rather than the boundary isorticily c o R , measures the creation rate of iwrticity from a bounduiy. Now, if the rigid boundary becomes an interface S of two immiscible viscous fluids on which the velocity adherence still holds, one would immediately ask if the earlier basic assertion is still true, because (1.4) seems already gives a local and instantaneous relation on the boundary vorticity (which in the present case is zero). The answer is yes. First, eq. (1.4) applies to only a free surface where the fluid motion on one side of S is negligible; otherwise one can obtain a condition for only the vorticity jump (see Section III.A), which is insufficient to determine the vorticities of both sides. Second, as will be seen in Section V, even for a free surface, its geometry and motion are themselves a space-time accumulated effect of the force balance on it, and hence are still an accumulated effect of the boundary vorticity flux.

147

Vorticity @waniics on Boundaries

To support this assertion, J. Z. Wu (1995) extended the above Stokes's first problem to a two-fluid system. Assume that, in addition to an impulsively started rigid wall at y = 0 with U = 1, there is a horizontal flat interface S at y = 1. Thus, the wall drives fluid 1, which then drives fluid 2, which in turn reacts to fluid I . Along S the pressure is constant. Then the dimensionless governing equations and initial-boundary conditions at y = 0 and y = are

db,

dU, - =

V?

dt

~

d y -?

u 2 ( y , 0 ) = 0,

1
for

uz(=,t)

=

0 for

t >0

I

(2.41b)

By applying (2.10) and (2.111, the initial-boundary conditions on S are

Denote a , ( [ )= pressed integral

u 2 at y = 1 by r 4 t ) , and note that, from (2.41a,b,c), there = -dr!/dt. Then, u , ( t ) and u 2 ( t )can be analytically exin terms of the unknown ~ ( t or) cr,(t), for which an additional equation due to (2.41d) holds to close the system:

uI

=

-u,(t)

rs(t)

=

F(t)

+ I' 0

i

2G(t

-

7)

+

k

k

=

sE PI

(2.42)

with F ( t ) and G ( t )being known functions. The second integrand of (2.42) represents the reaction of fluid 2 to fluid 1 that makes the latter rotational on S with an O ( k )vorticity. I t is remarkable that, by comparing (2.42) and (2.37), the vorticity flux on an interface causes a rotutionaZ rdocity there in a way exactly the same as the rwrticity on a rigid wall, but with factor k . As will be addressed later, this is an indication of the basic difference in the mechanisms of vorticity creation from a solid wall and an interface. But on the interface the vorticity creation rate is still u .

148

J. Z. Wu and J. M. Wu

For the water-air interface with k = 3.87 x lop3, the solution of system (2.41) and (2.42) is potted in Figure 3. Figure 3(a) shows the velocity profiles at different times. The shear “layer” of the air is much thicker than that of the water. Figures 3(b) and 3(c) give the computed evolution of o1 and g1 at S, respectively. Indeed, the peak value of lol(l, t)l is about 4.5 X lop3 = O(k). If we replace the interface by a flat free surface with k = 0 in (2.42), then the surface vorticity vanishes (shear-free). From the preceding analysis, it is clear that the shear-j?ee behavior on a flat Pee su$ace comes from the vanishing of k alone, entirely independent of the flow Reynolds number.8 Then, u ( y , t ) and d t ) on S can be easily solved from (2.41a) (Rood, 1994b; J. Z. Wu, 19951, which for the water case are also plotted in Figure 3. Regardless of the missing of the residual interface vorticity, this solution is very close to that of the full water-air system. It was found that, for the generalized Stokes’s second problem of the water-air system, the effect of k # 0 is even smaller than in this example. 111. General Theory of Vorticity Creation at Boundaries

In this section we develop the general theory of vorticity creation from an arbitrary material boundary. This is the major aspect of the boundary vorticity dynamics. As mentioned before, two quantities, the vorticity and its flux on the boundary, are involved in this creation aspect. The other aspect, the dynamic reaction of the created vorticity to the boundary, will be discussed in Section VI. A. BOUNDARY VORTICITY

We first infer the vorticity behavior on a boundary from the surface stress balance (2.11). From now on, we specifically denote the tangential and normal vorticities on 9 by 6 = o, and 5 = o,,= n J , respectively. On a rigid wall one can go no further than (2.22), because the stress experienced by the wall is unknown and, in practice, is to be inferred from the fluid stress-still based on (2.11) with T = 0, a simple use of Newton’s third law. Therefore, we concentrate on an interface S of two Newtonian ‘In particular, we did not invoke a free-slip condition at all; on the contrary, the no-slip condition (2.41~)was used in deriving (2.42).

b

U

0.0045

0.004 0.0035 -

.. .. .,.._..., .. .... -0.0005

c

0

o

50

60

70

80

cwpled

free-surface

-

-0.006

5

-0.012-

>

40

-

.e

.-0

30

-0.002 -

-0.01-

z

20

Time

-0.004

2

10

...

-0.008

-0.014. -0.016 -0.018 -

-0.02c

0

10

20

30

40

50

60

70

80

Time

FIG.3. The generalized Stokes’s first problem for the interacting water-air system (from J. Z. Wu, 1995): (a) velocity profiles at different times, (b) the evolution of the water vorticity on the interface, (c) the evolution of the vorticity flux on the interface. Also shown in the figures are the approximate solution with k = 0 (dotted lines). (Reproduced with the permission of the American Institute of Physics.)

150

J. Z. Wu and J. M. Wu

fluids. Except in Section V.D.2, where an elementary discussion is given to the surfactant effect, throughout this chapter we consider only a clean interface with constant surface tension. Thus, the jump of compression variable and tangent vorticity across S directly follows from (2.11), (2.6) and (2.8):

Equation (3.lb) indicates that, although across S the normal vorticity is continuous, the tangent vorticity is generally not (also true for solid wall); the jump of E. j is exclushiely balanced by that of the tangent components of sui$ace-deformation stress. In particular, when t , = 0, as in the case of a flat interface with uniform U , (3.2)

For example, on a flat water-air interface, the tangent vorticity of the water will be about 1.8 X 10- * times smaller than that of the air. The continuity of 5 and discontinuity of US across S implies that a vortex line must be refracted by S, first studied by Lugt (1989b) for the case where (3.2) applies. The general “refraction law” on a curved S can be easily inferred from (3.lb) and (2.12): (3.3) Thus, the refraction occurs in a plane determined by n and 5 , - 2n X ( n . B). Even if 6 , = 0, there is still a refraction, with US2 being 2[ p ] / p 2 times ( n . B), rotating 90” around n. J. Z. Wu (1995) remarked that if a well organized vortex in fluid 1 hits a wavy moving S , its vortex-line refraction is very likely a source of turbulence in fluid 2. Then, what is still continuous across S is the tangent vector (3.4) Note that a shearing near S must cause a rotation of n

Dn ~

Dt

=

1 D -(-dS) dS Dt

= 71

(B. n),

=

W,

X

n

=

-(V,U,

+ U;

K) (3.5)

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Vorticity Dynamics on Boundaries

Thus, W, is exactly the familiar angular velocity of n. This rotation of n is caused by a nonuniform normal motion and a tangent advection that turns n to a new direction due to the curvature. Although the present theory is developed on a unified basis, the preceding results reveal some interesting differences between a solid wall and an interface regarding the behavior of vorticity on boundaries. First, for an attached flow over a solid wall, the vorticity usually arrives at maximum on the boundary, which is of O ( R e t ) for large Re, as is clear from comparing (2.36) and (2.37). But (3.1) indicates that the interfacial vorticity is of 0(1), of which the physics is explained by Batchelor (1967). Therefore, the maximal vorticity usually occurs in the interior of the fluid. Next, on a solid wall, when its motion and geometry are given, the only thing we know immediately is the normal vorticity, all the rest remains to be solved. In contrast, on an interface the situation seems to be entirely opposite: once the fluid motion on one side of S is given or its effect is negligible, and if the geometry and motion of S are known, the tangent vorticity and compression variable on the other side can be immediately obtained. What remains unknown is the normal vorticity, which does not enter into the stress balance at all. However, the main difficulty is now shifted to the calculation of the geometry and motion of S , which ultimately relies on the space-time accumulated effect of force balance on S. Moreover, because the nonlinear advection is also involved in this force balance, problems with an interface are factually harder to solve than those with a solid wall.

B. BOUNDARY FLUXES OF

VORTTCITY AND

ENSTROPHY

While the velocity and stress conditions are sufficient for dealing with primitive variables, for vorticity dynamics one has to explore further the boundary behavior of w and n. This leads us to the theory of boundary vorticity flux, which stands at the center of entire boundary vorticity dynamics. In short, eq. (1.1) indicates that, if at a boundary 9 the shearing is stronger than that away from .'8 then , the vorticity will be diffused from 9 into interior of the flow, and vice versa (Lighthill, 1963, 1986). In what follows we give a sufficiently general formula, similar to (3.1), that reveals all physical mechanisms causing the boundary vorticity flux. This type of formula, of which the simplest form is (1.2), is obtained by taking the vector product of n and the Navier-Stokes equation applied to 28, along

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J. Z. Wu and J. M. Wu

with the acceleration adherence condition (2.13). Just as a combination of (2.10) and the boundary stress balance leads to the jump of boundary vorticity and compression variable, a combination of (2.13) and the force balance per unit mass will lead to their respective normal gradients. Once again, the approaches to a rigid wall, a flexible wall, and an interface are unified. 1. The Boundary Vorticig Flux in Arbitrary Continuous Media Since the flux u represents a vorticity source, it is natural to inquire the role of u in the vorticity transport equation; and because in (l.l), u is stated for Newtonian fluids of constant shear viscosity, it is also natural to ask if u can be defined for any continuous media without specifying a constitutive structure. These two questions were addressed by Hornung (1989, 19901, who gave a general definition of u and derived its formula. Hornung’s approach allows for a deeper physical understanding of a,of which a similar version is presented here. Assume that the divergence of the stress tensor T in (2.14) has a formal Stokes-Helmholtz decomposition

V.T

=

p ( a - f)

=

-Vq - Q

x A,

V.A

=

0

(3.6)

where cp and A now represent the compression variable and shear variable, respectively, which become n and ~ D in J (2.1%) for Newtonian fluid with a constant p. Taking the curl of (3.6) and using the continuity equation, we obtain the vorticity equation

Thus, the total vorticity variation in an arbitrary material subdomain B bounded by r ? 9 is given by

D

1

0 . 0 +~ T V p

Dt

P-

+

X

d 1

L23

nXfdS-

L3

i

Vcp dV (3.8)

-nX(VXA)dS

Here, on the right, the volume integral includes the contribution of vorticity stretching and turning and the baroclinicity (note that the latter is due to a self-interaction of compressing process, as it should be: the density

Vorticity Dynatnics on Boundaries

153

gradient does not align to the compressing force), and the first surface integral is due to nonconservative force. The second surface integral should contain our boundary vorticity flux, which is now defined as the normal gradient of A divided by p (Hornung, 1989): 1

(3 3)

u - -n.VA P

as a generalization of (1.1). Indeed, by using identity (2.7) and noticing that V . A = 0, it easily follows that -

1 - n X ( V X A) P

=

u

-

1 -(n P

X

0) X A

(3.10)

This observation also indicates how to obtain the general formula of u for any continuous media on any boundary 9: by (3.6) and (3.10), we simply have u

=

n

X

(a

-

f)

1

+ -{n P

X

Vq

+ (n x

V ) x A)

on boundary 9 (3.11)

of which the right-hand side only contains quantities right on 9. It is worth noticing that in the above procedure u is identified via the vorticity equation (3.7) or (3.8) but its formula is obtained via the momentum equation (2.14). We stress that, if the arbitrary subdomain 9 is entirely inside a homogeneous fluid, then the u on 1?9l becomes an interior flux of A across d 9 , which must be accompanied by an equal but opposite flux on the other side of d i B and hence no net vorticity is created. The flux u represents a vorticity source only when it is on a solid boundary dB or an interface S across which some flow quantities have a jump (see (3.20) below).' Moreover, using (2.8a) as an identity, there is (n X 0 ) X A = VA . n, which disappears in two dimensions. Related to this is the fact that (3.8) seems to suggest that one could take - n x ( V x A)/p, instead of (3.9), as the measure of vorticity-creation rate from 9; since then the explicit dependence of u on (n x V ) x A would disappear in three dimensions as well. That this alterative is physically unacceptable was argued by J. Z. Wu and Wu (1993) for Newtonian fluid, which holds true for the general case as well. Let 9 be a small material volume adjacent to our boundary .i-%9 (a "In other words, inside a homogeneous Navier-Stokes flow, the torque due to surface force causes only vorticity diffusion and dissipation but not creation.

154

J. Z. Wu and J. M. Wu

solid wall dB or an interface S ) . Then, the last term of (3.8) represents the integrated torque over the closed d 9 , of which only a part belongs to 9. In contrast, as a vorticity source strength, u must be defined on any open boundary element of 3, and its formula (3.11) was indeed obtained by applying the tangent components of (2.14) on such an open element. Therefore, using - n X ( V X A)/p to measure the vorticity source would lead to a missing of part of its constituents due to the cancellation during integration. This part is precisely the purely three-dimensional effect (n x 0)X A/p, as is evident by comparing (3.10) and (3.11). The physical correctness of using (3.9) to define the boundary vorticity-creation rate will be further verified when we consider thc reaction of the created vorticity field to the boundary 5 3 ' (Section VI). In fact, as a general rule, defining a boundary source on an open surface based on a close-surface integration always has a risk of some effect being cancelled. This kind of cancellation can be seen more clearly from the integrated effect of u over a closed B' ((3.21) below). Therefore, we use the integrated equation (3.8) only as a clue to identifying the boundary vorticity source, but not as a basis of the entire analysis. Finally, similar to (2.16), we may define a reduced stress tensor (3.12a)

T = -pI+S where S - A x I =

or

-ST

S,, =

E,,A ~ , =

-S,,

(3.12b)

is the skew-symmetric tensor associated with the axial vector A. Then by (3.6) there is V . T = V . T. Meanwhile, (3.1 1) can be cast to a neater form u

=

n x (a

-

f)

-

1 - ( n x v > . T on P

9.

(3.13)

2. Boundary VorticityFlux in Viscous Incompressible Flows We return to Newtonian fluid. For a general compressible viscous flow, the vector potential A in (3.9) will be very complicated due to the variable dynamic viscosity p, as implied by the last term of (2.15a). But, for a

155

Vorticity Dynamics on Boundaries

constant p, (2.1%) indicates that (1.1) is still the proper statement of u, and to obtain its general formula one simply replaces A in (3.13) by po.'" We shall be concerned with this situation exclusively. For simplicity, in most of the rest of the chapter, we further confine ourselves to incompressible flow, governed by a = -gk-Vh-vVxo,

(3.14a, b)

V-u=O

where k is the unit vector vertical up, g is the gravitational acceleration, and h = p / p is the incompressible enthalpy. The gravity effect can be absorbed into a modified enthalpy h = h gz, such that (3.13) is specified as

+

u

= =

+ n x vt; + v(n x V ) x o n x a + (n x V ) . ( h ~ 2 v ~ )on nx a

-

(3.15) c%

where R is the spin tensor first appearing in (2.2). The explicit viscous term exists only in three-dimensional flows. According to the discussion about (2.18) and (2.19), eq. (3.15) clearly shows that the (o, h, coupling is in general inevitable. Like (2.22) and (3.lb), eq. (3.15) can be viewed as a derived boundary condition for vorticity as well. We shall return to this issue in Section VII. Note that, even though on an interface we have only one (vectorial) jump condition for the stress, now (3.15) provides two vector conditions, each for one fluid, but they are related by the continuity of a, gk, and (. We now compare different types of boundaries. For a rigid wall with angular velocity W(t), the flux exists only on one side of dB. Write o = o,+ 2W(t) = 5, 2W(f) due to (2.121, with 5,. being the relative tangential vorticity, in (3.15) we may replace w by 6,. Along with the easily derived normal component of (3.14a), this leads to, on dB (J. Z. Wu, 1986; J. Z. Wu and Wu, 1993).

+

ah _ - -n.(a +gk) dn u I0

=

n x (a

-

v(n x V ) . o

+ vL) + v(n x

V) x

5,

(3.16)

(3.17)

J. Z . Wu and Wu (1993) studied the interaction between a rigid surface and a compressible viscous flow, where the definition of boundary vorticity flux is in terms per unit volume, because in that paper only the Navier-Stokes equation was involved and equations per unit volume are convenient for examining surface forces per unit area (see Section V1.B later) and their relevant integrals. Thc rcsults can be easily re-expressed in terms of per unit mass. But, Wu and Wu defined the normal gradient of pc~w as the flux even for variable viscosity, which is inconsistent with the general definition (3.9).

156

J. Z. Wu and J. M. Wu

For vorticity dynamics, this pair of equations complement the boundary information given by (2.22). Similar to the derivation of (2.8b), eq. (3.17) can be further decomposed to

Thus, the curvature will cause a flux if there exists a relative tangent boundary vorticity or shear stress. The creation of normal vorticity from a rigid wall, which cannot be revealed by the stress balance, is now recognized as due to a nonuniform distribution of relative tangent vorticity. The vorticity fluxes on an interface S can be similarly studied. The fluxes on sides 1 and 2 are a , = v1-

dW I Jn1

=

vl-

I

dn '

(n2 = - n l

a 1 =

=

vz- dw2

an,

-

JW?

-v1-

dn '

-n)

Then, on each side of S, we may decompose (3.15) into (J. Z. Wu, 1995)

This result was also obtained by Yeh (1995) in a different context. Compared with (3.18), both a,, and a,, on S are complicated by an extra term due to the unknown calying normal boundary vorticity. Moreover, the net vorticity-creation rate reads

Although a and g k are part of the vorticity source on a rigid wall or each fluid side, they are no longer so for the net creation rate on S. In all of the previous cases, the tangent and normal components of the boundary vorticity flux, a, and (T,,, represent two different physical patterns by which the vortex lines neighboring a boundary A? enter into the interior of the flow (J. Z. Wu and Wu, 1993). As sketched in Figure 4(a), a, implies a ilortex-line ascending pattern. Due to a a,, vortex lines

157

Vorticit?,Dynumics on Boundaries

(a)

(b)

FIG.4. Two basic patterns for the vorticity to leave a boundary: (a) the ascending of vortex lines close to a boundary due to diffusion, and (b) the turning of tangential vortex lines to the normal direction. The gray vortex lines represent boundary vorticity and always stick to the boundary. (Reproduced from J. 2. Wu and Wu, 1993. Reprinted with the permission of Cambridge University Press.)

infinitesimally close to a wall (but not the boundary vortex lines, which cannot leave 95’) spread from and parallel to the wall (but not necessarily parallel to the boundary vorticity at the foot of n). In two dimensions, ascending is the only possible pattern. But CT, is entirely different. The first equality of (3.19b) clearly shows that the solenoidal feature of the vorticity forces some near-wall vortex lines to turn to the normal direction, provided Vv. 6.1 # 0 (a two-dimensional source or sink). This Llorfex-line turning pattern is shown in Figure 4(b) and is the only mechanism that can form normal vorticity near a nonrotating boundary. 3. Bounduq Enstrophy Flux If we integrate (3.17) over a closed material boundary S’,by (2.9), (3.21 ) with all the rest of the vorticity sources vanishing. This happens because the vorticity lines sent into fluid by those sources are all closed, and their vorticities cancel each other during integration. In fact, eq. (3.21) is a dynamic counterpart of the Fiippl total-vorticity conservation law (e.g., Truesdell, 1954). This observation also tells us that, in defining the boundary vorticity flux or examining its physical implication, one cannot start from this type of integral relation, such as the integral of (3.6) over the fluid domain. To avoid this cancellation, J. Z. Wu (1986; see also J. Z. Wu and Wu, 1993; J. Z. Wu, 1995) invoked the enstrophy consideration to

158

J. Z. Wu and J. M. Wu

analyze how much vortical flow, regardless the direction of vorticity, is created from a boundary. For incompressible flow, over the flow domain 9,

in which

and @,=vo:020

in

9

(3.23b)

are the boundary enstrophy flux and the enstrophy dissipation rate, respectively. The first term on the right of (3.22) represents the enstrophy variation due to vortex stretching or shrinking, and the second term is its creation rate on 9. Some remarks are needed here. First, because 77 = 5 . u, + lo;,, on a nonrotating 9 with ( = 0, only uT contributes to the creation of new enstrophy. In other words, a nonrotating boundary does not create new normal covticity dynamically. This is consistent with the kinematic background of a,. Second, eq. (3.23a) implies that the enstrophy, and hence Llorticity, can be diffused across a boundary only if both LIorticity and its flux are nonzero on the boundary. This reflects a fundamental difference between the diffusion of a vector field and a scalar field. Physically, as Lighthill's (1963) pointed out, the vorticity diffusion cannot be viewed as diffusion of angular momentum; rather, it is related to the diffusion of momentum through a momentum gradient, which appears whenever there is a shearing. Therefore, if u # 0 but o = 0 on one side of a boundary 9, say side 1, then it would be imprecise to say that the vorticity is diffused across A?. The correct explanation is that some new vorticity is created on ,Y$ and diffused into fluid 1. We shall return to this point later. Third, in two-dimensional flows whether a piece of boundary is a source or sink of vorticity can be easily identified by the sign of u (Lighthill, 1963, 1986; Morton, 1984; Sarpkaya, 1994). But evidently this criterion is not applicable to three-dimensional flows. The proper source-sink criterion should be the sign of boundary enstrophy flux. The second equality of (3.23a) shows that 77 > 0 on a region of 9 implies that the newly created

159

Vorticity Dytiumics on Boundaries

vorticity by u enhances the existing one adjacent to the boundary, and hence this region is a vorticity source; inversely, 7 < 0 implies a vorticity sink. For example, in Stokes’s second problem mentioned in Section II.D, as t + M, (3.24) Therefore, during a time period, the flat plate serves as a vorticity source and sink alternatively but not equally: in time averaging, it is always a source. Namely, the wall oscillation makes the near-wall flow field more and more vortical. 4. The Role of Viscosity in Vorticity Creationfront Boundaries In (3.18) and (3.19) the vorticity flux is decomposed with respect to spatial directions. We may alternatively split the terms in u according to whether the viscosity v occurs explicitly, denoted by u,,, and uyIF, respectively. Thus, by (3.19, u,,, = n x (a

I

+~i;>

u,,, = v(n x V ) x

ct)

(3.25a, b)

Obviously, at a large Reynolds number, the apparently inviscid vorticity source (inuiscid, for short) uillV is much stronger than the explicit viscous source uVis. In the Euler limit we simply have u = u We restress that, as argucd in Section II.D, even in the “inviscid” source, the viscosity still plays a key role. More precisely, eq. (3.25a) is a consequence of the no-slip condition, because (2.13) has to be imposed; and if its right-hand side is fixed, the normal gradient of corticity is iniwsely proportional to the kinematic 1-iscosity. But, in the literature, there exists some confusion on this almost trivial fact. For the solid-wall case, some authors favored a completely inviscid interpretation, which has been clarified by J. Z . Wu and Wu (1993). Similarly, for the interface case, some authors attributed the creation mechanism to the baroclinic effect alone, because S is a limiting case o f a density-stratified layer if the surface tension is negligible. We now show that the appropriate explanation for the vorticity creation right on an interface should be a combined baroclinic-L:iscous cffect. Indeed, it was emphasized in Section 1II.A that 5 on S is from the viscous tangential stress balance, and without viscosity, there can by no

160

J. Z. Wu and J. M. Wu

means be a boundary vorticity. But the baroclinicity is also indispensable, because otherwise (3.1) would trivially become 0 = 0. On the other hand, after the preceding discussion, there should be no more doubt about the viscous root for the boundary vorticity flux; but the baroclinic effect does not show up in the momentum equation nor can it be seen in the one-sided u formula (3.13) or (3.17). However, its effect appears in the total flux formula (3.20), where without the density jump (and the jump of u ) u , ( r 2 would be zero, as in the case of inside a homogeneous fluid. The baroclinic effect enters the vorticity equation explicitly; for example, the third term on the right of (3.7). Thus, to further understand how this effect combines with viscosity in the vorticity creation from an interface, we apply (3.7) to a Newtonian fluid with density stratification. The resulting equation has been given by Dahm et al. (1989) in a study of vortexinterface interaction. They considered a finite layer of thickness 6, across which the density changes from p 1 to p z , and then took the limit 6 + 0 to obtain a density interface. The circulation and characteristic length scale were assumed to be r and a , respectively, and the dynamic viscosity p was assumed constant. Using a , a’/T, T/u, I’/u’ and p , + p z to rescale x, t , u, ct), and p , respectively, and writing the density gradient as

+

Dahm et al. obtained the rescaled dimensionless vorticity equation:

where A is the Atwood ratio, and g* = a 3 g / T and E = v / r = R e - ’ are the inverse Froude number and inverse Reynolds number, respectively. The surface tension T is rescaled to yield a Weber number W = ( p 1 + p 2 ) r 2 / a T . Obviously, the second term of (3.26) represents the baroclinic generation of vorticity. Dahm et al. (1989) went on to obtain some further simplifications. Assume W is sufficiently large and 6 / a is sufficiently small, that the surface tension can be ignored and the density gradient can be expressed in terms of a delta function 6 ( n ) . Then, eq. (3.26) reduces to

of which the viscous term was also dropped by Dahm

el

al. (1989) in view

Vorticity Dyriciniics O H Boundaries

161

of E < 1. This deletion has led one to explain the vorticity creation on an interface solely by baroclinicity. However, the density-stratified layer is not an immiscible interface; in a strict sense, this baroclinic effect is a body source of vorticity rather than a surface source (hence it is absent from a). Moreover, as remarked before (also will be addressed later), for a fixed circulation density, the thinner is the vortex layer, the stronger would be the normal vorticity gradient and hence the viscous effect. Therefore, the viscous term in (3.27) should be retained in the limiting process. This term contains the mechanism leading to a vorticity-creation rate on the interface, to which the baroclinicity is an indispensable contributor. C. CREATION OF BOUNDARY VORTEXSHEETS

So far the theory applies to flow$ at any Reynolds number, Re. When Re >> 1, the theory can be simplified to boundary-layer approximation. The solid-wall boundary layer has been addressed by Lighthill (1963) in terms of vorticity dynamics, and the interface boundary layer has some special features and will be treated in Section V.C. Here, we jump to a further simplification, to look at the asymptotic form of vorticity creation from boundaries at the Euler limit. In this case, the vortex layer created from a boundary 97 reduces to a vortex sheet of thickness 6 + 0. The jump across 9’ and the attached vortex sheet is different from that only across 9; so we denote the former by a double square bracket . , According to our definition of the direction of n, the vortex sheet strength is given by

(in literature one often writes y = c1 x 1 1 ~ 1 , with i3 = - n pointing from side 2 to side 1). Because the velocity jump ‘[u must be tangent to S, lu = n x y

(3.28b)

On a solid boundary ,jlB we have a single-layer sheet, whereas on an interface S the sheet consists of two shear layers of vanishing thickness with strength y 1 and y 2 , each on one side of S. None of these sheets should be identified as 9 itself, which is a surface in geometric sense.

162

J. Z. Wu and J. M. Wu

FIG.5. The sandwich structure of vortex sheets on both sides of an interface.

Thus, for the interface case we have a sandwichlike structure as sketched in Figure 5. The strength of the two sheets are

(3.29)

The total vortex sheet strength is y = y + y ? . For flow over a solid wall the vortex sheet model is certainly oversimplified in modern studies of boundary layer and its separation, but it nevertheless provides overall physical insight and may serve as the basis of perturbation expansion at a finite Re. On the other hand, for simulating an interface the vortex sheet model is still in use. From the viewpoint of three-dimensional boundary vorticity dynamics, two issues addressed by J. Z. Wu (1995) for an interfacial vortex sheet are of interest: the dynamic equation and velocity of a three-dimensional boundary vortex sheet. His results can be applied to a vortex sheet created on a solid wall as well and are reviewed in the following. 1. The Transport Equatiorz of Boundary Vortex Sheets

Unlike a vortex sheet inside an otherwise irrotational incompressible homogeneous fluid, for which all relevant studies are built exclusively on the kinematic Biot-Savart law, the dynamics of vorticity creation plays a key role in the evolution of a boundary i’ortex sheet. To derive the general dynamic equation for an incompressible boundary vortex sheet, it is convenient to start from the Lagrangian description. Let T = t be the time

Vorticity Dynurnics on Boundaries

163

variable in the Lagrangian description, X be the Lagrangian coordinate of the fluid particles on the sheet. We define fi = w . VX as the incompressible Lugrungiun Llorticity (Casey and Naghdi, 1991; here fl is not to be confused with the spin tensor). Correspondingly, let 6 c = 6 x . 6s = 6 n 6 S ( 6 n = 6x . n 0) be a thin material volume element surrounding a piece of the sheet, then

-

r=y.VX=fl6n

(3.30)

is the Lagrangian vortex sheet strength. Therefore, because lim ( V x a ) 6 n

6n-0

=

- n x ,:a!

(3.31)

from the incompressible vorticity equation (Truesdell, 1954)

an

-dT

the dynamic equation for

r

-

(V x a ) * VX

reads (3.32a)

or

a-

dr

+ rv,. u

=

- ( n x l L a J .VX

(3.32b)

The corresponding dynamic equation in the Euler description is

where D Dt

=

d

at

+u.v

(3 33b)

The right-hand side of (3.33a) obviously represents the dynamic source of y,which appears only on a boundary <9. Inside a homogeneous fluid (3.33) falls back into pure kinematics. We now show that, on both a solid wall and an interface, the vector - n X a1 is nothing but the “inviscid” part of boundary vorticity flux: -n

X

llal = alnv

(3.34)

which explicitly reveals the dynamic essence of a boundary vortex sheet. Indeed, for a sheet attached to a solid wall dB, there is -all = a,,, - a R ,

164

J. Z. Wu and J. M. Wu

the difference between the acceleration of the inviscid outer flow and that of d B ; but because alnv= - V h , where h has the same value as that on dB, we immediately obtain - n x lla

=

n x (aB + ~i;)

which is precisely the “inviscid” part of (3.18). On the other hand, for an interfacial vortex sheet, the strength y in (3.33) should be the total value y , + y 2 , and Lal means the difference of inviscid accelerations at the outer boundaries of the “sandwich.” Clearly, this simply equals n X V [ h l , and (3.34) becomes the “inviscid” part of (3.20). In practice, on a solid wall, the strength y of the boundary vortex sheet can be directly inferred from the velocity jump l,uI between the outer inviscid flow and the wall. Thus, eq. (3.33) could be used reversely to obtain (T,”~without solving the pressure field. In contrast, because in general the location and shape of an interfacial vortex sheet has to be solved, eq. (3.33) must be included in its governing equations. More specifically, let

be the Atwood ratio and mean acceleration, respectively, then (3.33a) gives (J. Z. Wu, 1995)

DY Dt

--

y.VU

+ yV;U

=

2n

X

i

Aa,, +

TVK ~

PI +

P2

+ Agk]

(3.35)

The two-dimensional version of (3.35) and its Lagrangian counterpart have been derived by Tryggvason (1989) based on the consideration of a pure tangent discontinuity. For instance, eq. (3.32a) reduces to

(3 3 6 ) 2. Bifurcation and Velocity o f a Vortex Sheet in Three Dimensions It was stressed in Section 1I.D that a material vortex sheet is conceptually different from a mathematical contact discontinuity of ideal fluid. The former consists of fluid particles with very high angular velocity, whereas

Vorticity Dynaniics on Boundaries

165

the latter consists no particles at all (particles o n both sides of the discontinuity only slide over each other). But, in spite of these differences, we saw that the two models lead to the same equation. The reason was explained by J. Z. Wu (199s). He noticed that, however, one difference between these two models may lead to some identifiable effect: the sheet velocity U in (3.33). A material sheet must have a definite velocity, but for a mathematic discontinuity talking about its tangent velocity is meaningless. Wu found that some new aspect of this sheet velocity is closely related to the vortex sheet bifurcation phenomena, which deserve to be briefly mentioned first. It is well known that a boundary layer may bifurcate into several vortex layers when the flow separates from 97, and so too a boundary vortex sheet. For an incompressible and steady flow over a solid boundary d B , by using the Bernoulli integral, it is straightforward to show that a steady, two- or three-dimensional separating vortex sheet must leave the surface tangentially, either from a sharp edge or a smooth surface (Batchelor, 1956; Mangler and Smith, 1970; Smith, 1977; J. Z. Wu et al., 1993a). Following the flow separation theory on a solid wall (Oswatitsch, 1958; see also J. Z. Wu et al., 1988b, and Section I V . 0 , Lugt (1987, 1989a) studied steady separation (or attachment) patterns on an interface by using local Taylor expansions of the Navier-Stokes equation. The separation occurs at stagnation points of S, where U = 0, which, by (3.1), are also the critical points of [ & I (note that Vnq, = 0 for steady flows). The results are in sharp contrast to that of the solid wall case. For example, Lugt found that, in two dimensions, the dividing strcamlines originating from such a critical point can be single, double, or even triple, see Figure 6. Recently, B r m s (1994) revisited Lugt’s work in a more general setting, including a variable curvature and surface tension on S , and obtained a complete description of the bifurcation patterns for two-dimensional steady interfacial flow. Note that, like three-dimensional vortex line refraction (Section IILA), during the bifurcation process dividing streamlines are also refracted (Figure 6(b)). Parallel to the solid wall case, in the Euler limit the appearance of dividing streamlines implies bifurcation of the interfacial vortex sheet. Unfortunately, most of interfacial flows cannot be assumed steady, but the theory of unsteady separation is still not well established even for the solid wall case. For unsteady flows, when and how an interfacial vortex sheet will bifurcate remains unclear. J. Z. Wu (1995) stressed that, in addition to the preceding separationinduced vortex sheet bifurcation, yet another type of bifurcation can

166

J. Z. Wu and J. M. Wu

(3) FIG. 6. (a) The single, double, and triple dividing streamlines near an interface. (b) The refraction of dividing streamline across an interface. (Reproduced from Lugt (1987, 1989a), with the permission of the American Institute of Physics.)

happen only in three dimensions: Some cortex lines in a sheet may turn away from the sheet sufuce to form a normal uorticity field. Quantitatively,

in the Euler limit. Here, a nonzero [[‘I does not conflict (2.12); rather, it means that the turning process of Figure 4 occurs inside the vortex sheet. Therefore, in three dimensions, one should not impose an ad hoc condition that the flow is irrotational away from the sheet. In particular, a nonnegligible concentrated “horn vortex” could form by this mechanism, as will be seen in Section 1V.C for a solid boundary and Section V.C for an interface. Equation (3.37) has led J. Z. Wu (1995) to a new vortex sheet velocity formula. It is well known that the normal component of U is always the same as that of u , or u 2 , and the problem is its tangent component. Traditionally, U has been assumed to be the mean velocity 1 U=ii=-(u 2

l

+u,)

(3.38)

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Vorticity Qytiamics on Boundaries

which is consistent with (2.10a). To support this choice, Milne-Thomson (1967) argued that if S is a mid-surface of a fluid layer of thickness 6, then ul

=

u

-

6 - n - ~+ u 0(6’), 2

u2 =

6

u + -n.Vu + O ( s 2 ) 2

and hence (3.38). But this argument is invalid, because it has nothing to do with the Euler limit of a vortex layer nor with any discontinuity ( S can even be a surface normal to a vortex sheet). In fact, as mentioned before, to a nonmaterial tangent discontinuity no definite tangent velocity n X U can be assigned; choices other than U = ii are allowable; for example, U = ii + ( Y I ’ u I I with (Y being a free parameter (Baker et al., 1982). However, a material vortex sheet must have a definite n X U,which has to be derived in the Euler-limit consideration. We write

U

=

U

+ U’,

U ’ - n= 0

(3.39a)

and determine U’.Based on thc kinematic Kelvin circulation equation

D -Dt $u.dx

=$a.dx

J. Z. Wu (1995) showed that in three dimensions U’ is related to the normal vorticities 6, and t2associated with y :

n x V(U’.’U =~ ) l lull

+ i t ,u’

Let e , and e2 be the unit vectors along the tangent directions of y and u l , respectively, such that U’ = t i ; e , + t i z e , , and s1 and s2 be the corresponding arc lengths measured from a point on S where 2 = 0. Then the solution of the preceding equation reads

An example of using (3.39) to determine U will be given in Section V.C. If =

0 (this includes all two-dimensional sheets and three-dimensional ones

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J. 2. Wu and J. M. Wir

in a potential background flow), we still have (3.38) (Friedrichs, 1966; Saffman, 1992). J. Z . Wu (1995) also noticed a simple relation between U ' and the sandwich structure of an interfacial vortex sheet: 1

U' = - ( y , 2

Hence, if strength.

(3.40)

y2) x n

-

l =0, then the vortex sheets on two sides of S always halie the same

IV. Vorticity Creation from a Solid Wall and Its Control We now take a closer look at various vorticity sources on a solid wall. When the flow is incompressible (or weakly compressible), we take the density p (or the unperturbed density) as unity, so that p = h and p = v. The global Reynolds number is assumed large. We shall consider four constituents of the boundary vorticity flux u, denoted by (see (3.18a, b)) u,] = n x V p , uv13?i = vg,*

K

= -

uvl,,, = -vn(V;Sr)

uo = n

(n

X

(4.la, b)

a

(4.1~)

X 7, ) . K

=

-n(n.(V x

7,))

(4.ld)

where T,+= - T = n x v S r is the wall skin friction. Thus, there are contributions from the pressure gradient, wall acceleration, wall curvature (combined with T, or boundary vorticity g r ) , and a rotational T,, distribution. The effect of gravity can be easily added to the pressure force if necessary. The enstrophy fluxes corresponding to (4.la-d) are 1

7p

= -

-T,,

v

.a

(4.2a, b)

where W,,(t,n) is the normal angular velocity of the wall. Note that the first two enstrophy fluxes are of O(Re;), the third is O(1), and the last one is only O(Re-+). Obviously, all these sources are not independent of each other, for instance wall acceleration will certainly affect the ( p , T ) distribution on the wall. We examine them separately merely for convenience.

169

Vorticity Dynamics on Boundaries A. THE EFFECTOF

TI-IE PRESSURE

GRADIENT

As seen from (1.21, up is the most basic vorticity source on a solid wall drB and is the only source in an incompressible flow over a twodimensional, stationary surface. Figure 7 sketches a typical variation of q, along a flat plate, from negative to positive. Also shown are some velocity and vorticity profiles at different x stations, as well as the corresponding distribution of v,] (not to scale). Note that the enstrophy sink 77 < 0 is a sign that the flow may tend to separate downstream, and before and after that we have enstrophy sources. Usually, across a separation point, a;, has a pulse behavior. If multiple separation occurs behind a bluff body, the amplitude of IT, will have violent oscillation, as shown by Koumoutsakos

Y

.................................. (a) Velocity Profile

*

(b) Vonicity Profile

v

(c) Boundary vonicity flux

(d) Boundary enstmphy flux

Flc;. 7. Vorticity source and sink due to pressure gradient (subscript p is omitted): (a) velocity profile, (b) vorticity profile, (c) boundary vorticity flux, (d) boundary enstrophy flux.

170

J. Z. Wu and J. M. Wu

and Leonard’s (1995) computation of an impulsively started flow over a circular cylinder. With this general picture in mind, we now consider a special class of pressure-created vorticity on a solid wall: acoustically generated riorticity waues. The pressure wave is assumed to be harmonic: (4.3) where n is the circular frequency and u u the velocity amplitude. This problem is a slight extension of Stokes’s second problem of Section ILD, simply with u = db,/dt = -nb, sin nf being replaced by (T

Jh

= -=

dX

Re(inu,, e l k ( r - i r} )= - nu,, sin(& - n t )

(4.4)

Let 6 = ( 2 v / n ) i << 1 be the thickness of the Stokes layer excited by (4.4), and [ = y / 6 be the rescaled normal distance. Then the velocity field inside the layer is known as =

u,(l

-

e(’-1)
(4.5a)

indicating that the flow is no longer unidirectional when k vorticity wave produced by the pressure wave is =

(i

-

UO

6

-
+

O(k’6)

f

0. The (4.6)

where the ignored part is the contribution of d c / d x . The associated enstrophy wave is (here one has to take the real part of (T and wB first then multiply)

which extends (3.24) and also has a positive average. Lin (1956) showed that, even with a mean flow, if the frequency is so high that the Stokes-layer thickness is much smaller than the boundary-layer thickness, then inside the Stokes layer we still have (4.5) and (4.6), with u g being understood as the corresponding inviscid value at y + 0.

Vorticity mnamics on Boundaries

171

More generally, we add a quasi-parallel local mean velocity U ( y ) and assume the flow is weakly compressible with disturbance dilatation 6 = V . u. Then, the disturbance shearing and compressing processes are governed by a pair of linear equations:

The basic boundary conditions are (1.2) and (1.3). Equations (4.8a, b) can be applied to different situations, and we present two examples. 1. Receptiriity of Boundary Layer to Pressure Waivs The first example concerns the amplification of the Tollmien-Schlichting (TS) instability waves by a forcing wave. For spatially developing TS waves we have

where A,, is their wavelength and k , < 0 means growing. Since the classic 1943 experiments of Schubauer and Skramstad (19481, it has been repeatedly found that an acoustic wave can excite TS waves. But A , , is much shorter than the acoustic wavelength A',' under the same frequency; so the question is how such excitation can happen. This is a typical receptkity problem. A heuristic argument was presented by Nishioka and Morkovin (1986), which directly reveals thc creation of vorticity and enstrophy from a solid wall by waves. Assume 6 = 0 and take p = 1. From (4.8a), the unsteady enstrophy equation follows, which, upon being averaged over a time period and integrated from the wall to any y , gives

172

J. Z. Wu and J. M. Wu

where

are the mean dissipation and boundary flux of the unsteady enstrophy, respectively. Setting y = 0 in (4.9) gives a mean of (4.2a) and the streaming part of (4.7). Now, the unsteady field contains a TS part and a forcing part, each satisfies (4.8a). The coupling between wTs and wf occurs on the wall only, through the adherence condition. But, in the enstrophy equation, the coupling occurs inside the flow as well. Thus, Nishioka and Morkovin (1986) - identified two inhomogeneous sources in (4.9) for the growing of w & : an interior source ~‘l.wTsU”,which transfers the vorticity of the mean flow to TS waves due to forcing and a boundary source w - , , ( d p / d x ) f , which directly relates the TS-wave vorticity to the forcing pressure gradient. Although the first source belongs to inviscid advection, the second is essentially viscous. More specifically, assume a long-wave ( k + 0) disturbance (T = A ( x ) e ” “ , where the amplitude is x dependent and has a Fourier transform A , ( k ) . Then, if A ( x ) makes a nonzero distribution hA ( k ) at the TS wave length between x - A/2 and x A/2 (quantities without suffix are referred to TS waves), Nishioka and Morkovin found that the mean boundary enstrophy source will yield a net local input into TS enstrophy, with a rate [I,;

+

where w(0) is the value at y = 0. Similarly, the continuity equation implies that uf is proportional to A ’ ( x ) , say, Vf(y){A’(x) ik,A(x)}. This gives a positive contribution to the TS enstrophy from the interior source:

+

where

* stands for complex conjugate.

Vorticity L?ynamics on Boundaries

173

Based on this heuristic argument, Nishioka and Morkovin proposed that the variation of A is a likely effective mechanism for the boundary layer receptivity. The increment A A , ( k ) reflects the commensurability of ATs and the local characteristic length at x introduced by the variableamplitude forcing waves. They carefully analyzed all available experiments, including their own and found that the proposed mechanism works well. 2. Sound- Vortex Irzteractiorz in Duct Shear Flow

In the receptivity problem the pressure wave is given and only (4.8a) was used. Another interesting application of (4.8a, b) is the sound-vortex interaction on a background mean flow U ( y ) in a duct (from y = 0 to y = 2). Because U ( y ) satisfies the no-slip condition and is nonuniform, a sound wave having a plane front at x = 0, say, must be refracted toward the walls, and only some of modes can reach far downstream. In the acoustic community this problem has been known for more than three decades (e.g., Pridmore-Brown, 1958; Shankar, 1971), and the completely inviscid sound-field character has been well clarified by solving (4.8b) alone under homogeneous boundary conditions (an eigenvalue problem). However, through (1.21, the sound must produce a vorticity wave of the same magnitude of order; and once this vorticity wave is formed it will in turn emit sound (Section 1I.C). Consequently, to obtain a full answer one has to solve the coupled system (4.8a, b) under the coupled boundary conditions (1.2) and (1.31, which becomes nonlinear. A simplified approach to this coupling was given by J. Z. Wu et a f . (1994a), who split the process into two subprocesses and solved them sequentially. First, because the viscous effect is ignored in the acoustic equation (4.8b), an inviscid refracted pressure field was computed as Shankar (1971) did. Thus, the vorticity wave followed from (4.8a) and (1.2). Second, eq. (4%) was recasted to a linearized vortex-sound equation in terms of the total enthalpy H. With M = U / c as the Mach number, the dimensionless H-equation reads

from which the pressure field due to the acoustically created vorticity wave can be obtained and added to the initial inviscid solution. Note that, in solving (4.11), a rriscous boundary condition, derived from (1.3), has to be imposed even though the equation is inviscid.

174

J. Z. Wu and J. M. Wu

1.0

1.0

08

08

06

06

0.4

04

02

02

00

00 -20

-I0

0

10

lOloSlO(p')

FIG.8. The acoustically created vorticity waves and its effect on the sound pressure level in the duct. M = 0.3. Left: The amplitude of vorticity waves (real part, solid line; imaginary part, dash line) at x = 20 with Re = 1000; top: h = 5, bottom: k = 10. Right: The viscous (solid line) and inviscid (dashed line) sound pressure level at k = 10 and different Y stations; 1994a). top: Re = 1000, bottom: Re = SO00 (from .I. Z. Wu ei d.,

Some results of this approach with a parabolic mean flow M ( y ) = M,(2y - y 2 > are shown in Figure 8. Traditionally, the viscous effect on sound propagation had been thought always as an attenuation, and was approximately treated by an equivalent admittance derived from the high frequency limit (4.5) of the Stokes layer. However, Figure 8 indicates that the viscosity has a nonmonotonic effect on the sound pressure level (SPL). This can be seen more clearly from Figure 9, which gives thc wall SPL variation as k and the Reynolds number Re. When a sound wave excites a vorticity wave through the no-slip condition, it loses some energy. But, as seen in the receptivity example, there is an interior unsteady source U"r' for disturbance vorticity, which makes the acoustically created vorticity

175

VorticityDyrzarnics on Bounduries

I .o

, 0.5

I

35

3.0

.

.

.

40

log io (Re) FIG.9. The wall SPL at x J. 2. Wu el (11.. 1994a).

=

20 for different wavenumbers and Reynolds numbers (from

wave able to absorb enstrophy from the mean flow and become a selfenhanced source of sound.

B. THEEFFECTOF WALLACCELERATION We now turn to (4.1b), the wall acceleration as a vorticity source. Its general feature is similar to cr,' ; here we concentrate on some less familiar issues. A key condition leading to (4.1) is the acceleration adherence (2.13), which was obtained by following fluid particles on a boundary 9. Therefore, the independent spatial variable in (4.1) is in fact the Lagrangian coordinates of these particles, say, X, . Correspondingly, the physics of vorticity creation from 97 should be unambiguously understood in terms of a(x(X,, t ) , t ) . This observation implies that special caution is necessary in dealing with moving boundaries if the Eulerian point of view is to be used. Indeed, once 9 has a normal motion U , , it is meaningless to break a on

176

J. Z. Wu andJ. M. Wu

9 into a local acceleration d u / d t and a convective acceleration u . Vu, because at t + dt each of these Eulerian accelerations will no longer refer to any particle on 93.For example, on a rigid wall d B with velocity b and angular velocity W,

Some confusion would easily occur if i)B has tangential motion because then a h e d spatial point x B , once belonging to JB at t , will always do so. However, this xN is factually sliding along the wall. Consequently, talking about a(x,,t) does not reflect the same physics as a(x(X,, t ) , t >does (J. Z. Wu et al., 1993b). Fortunately, the unidirectional flow model (2.33-2.381, including two Stokes’s problems and their extension to rotating circular cylinder, as well as the examples used by Morton (1984) to illustrate the vorticity creation from moving walls, happen to be free from trouble, because in these situations cr is independent of the location along the wall. Two examples lack of such symmetry are reviewed next. 1. Streaming Effect of Flow oiw a Cylinder with Rotary Oscillation The preceding remark becomes crucially important if wc consider the time-averaged vorticity creation from a periodically moving boundary 9 lack of symmetry. The conventional Eulerian mean (EM) over the period T , C(x)

=

1a ( x , t ) dt T o I

-

T

for fixed x

is in general either meaningless (if has a normal motion) or easily misleading (if 9moves only tangentially). Instead, one should either use a coordinate system comoving with &7 (in that system there is no boundary acceleration) or invoke a Lagrangian-type description, for instance the generalized Lagrangian mean (GLM) developed by Andrews and McIntyre (1978). As illustration, J. Z. Wu et al. (1993b) considered the timeaveraged (T for a uniform flow over a circular cylinder with rotatory oscillation of finite angular amplitude 6. The Reynolds number and Strouhal number were assumed sufficiently high; thus, the pressure gradient on the cylinder can be approximated by that at the outer edge of the boundary layer, and the Stokes layer is well inside the boundary layer so

177

Vorticity Dynuniics on Boundaries

that the outer inviscid flow does not feel the cylinder's rotation. Then, if the flow is attached, it was proven that the E M incorrectly predicts a zero mean vorticity creation. In contrast, for the dimensionless Lagrangian mean G I - , 3'- = 2 ~ ( i ) s i n 2 0

(4.12a)

where 8 is the polar angle and

which has a series of zeros as 6 increases from 0 (the first zero occurs at H^ = 68.9"). This is a typical nonlinear streuming effect (e.g., Stuart, 1963), which in the present case is measured by o,=

cL

-

c r I ~ == ~ ,-2{1 - ~ ( ~ ) ] s i n 2 0

and is strongest when A(G) = 0. At these zeros no mean vorticity is created over the cylinder; thus, thc flow behaves like a mean iiscous potential flow with zero mean pressure drag." Moreover, if the cylinder has a pair of attached wake vortices, IGLI become very small over most of the cylinder surface at 6 = 45", see Figure 10. In this way, Wu et ul. provided the first (though preliminary) explanation for an interesting experiment of Taneda (1978) shown in Figure 11.12 Through a conformal mapping, the flow over a circular cylinder can be cast to that over a flat plate. Thus, J. Z. Wu et al. (1993b) also made a preliminary study on the effect of wing oscillation on the lift generated by a trapped vortex, based on thc inviscid analysis of Saffman and Sheffeld (1977). It was proven that in a certain range of parameters a mean lift increase is possible due to the streaming vorticity flux. 2. Vunishing Mean Drug in Viscous Flow ouer a Flexible Waiy Wall Another interesting example showing the effect of boundary acceleration on vorticity creation can be found from J. M. Wu et ul. (1990) and its "A viscous potential flow is diffusion free due to the absence o f the viscous term in (2.24), hut not dissipation free. Lagerstrom (l9h4) pointed out that this is possible because the dissipation depends on the symmetric part of the velocity gradient only. The Rankine vortex is a simple example of such flow, where h e dissipation occurs in the irrotational flow region outside the vortex core, but the total dissipation can be equally inferred from the core vorticity (J. Z. Wu et al., 1993a). I? So far Taneda's experiment has never been rcproduced numerically for some unknown reason.

178

3.0

J. Z. Wu and J. M. Wid

-

6=0

2.0 1.0 -

0.0 -

-1.0 -

I

-4.0

0

60

30

I20

90

150

8

I80

FIG. 10. The distribution of boundary vorticity around a circular cylinder that is in a uniform flow, carrying a pair of FGppl vortices, and performing rotatory oscillation with different angular amplitudes of 0. (Reproduced from J. 2. Wu et ul., 1993b, with the permission of the American Institute of Physics.)

subsequent exploration by the present authors and A. H. Eraslan and K. J. Moore (unpublished, 1992). The problem involves a uniform flow U over a flexible wavy wall y'

=

af(E

-

n t ) = af(k.i

-

nt

+ h),

If1 I 1

(4.13)

which in the laboratory coordinates (2, j ) has an up-down oscillation and thereby forms a train of traveling waves. Here A is the wavelength, k = 27r/h is the wave number, and a the amplitude. In the coordinates

x=P-ct,

y = j

moving with the wave speed c = n / k , the wall motion becomes steady and along its tangent direction, with a variable velocity magnitude

Vorticity Dynamics on Boundaries

179

FIG. 11. Streamlines and streaklines around a circular cylinder performing a rotatory yil l at i o n in a uniform flow: d = 1 cni, U = 0.33 cm/sec, Re, = 35, N = 2 Hz, N d / U = 6 , 0 = 45”, x = U I .(a) x/d = 0; (b) 1.5; (c) 3.4; (d) 12. (Reprinted from Prog. Aerospace Sci. 17, S . Taneda, Visual study of unsteady separated flows around bodies. Pages 287-348. Copyright (19781, with kind permission from Elsevier Science Ltd, The Boulevard, Langford Lane, Kidlington OX5 lGB, UK.)

and direction toward upstream if c > 0. Therefore, we have a tangentially accelerating boundary. The value of a / A was taken large enough to allow for the generation of separated vortices at the troughs of the wavy wall. A similar problem had been studied by Caponi et al. (1982) with both linearized analysis and numerical computation, the difference being that Caponi et al. imposed a periodic boundary condition to the flow, which was absent in the work of J. M. Wu et al. In this way, they discovered that, for a given wave pattern f(hx),there exists a unique critical waue speed c, with 0 < c,/U < 1, such that at c = c, the r1ortafEow becomes periodic in nature, with both averaged fnction drag and averaged pressure drag being zero. Physically, with c = c, the separated vortices can be well captured and do not grow or break away as one goes downstream. They form a fluid “sheath” to isolate the near-wall shear layer from the main stream. Therefore, for a remote observer who can see only the averaged effect, there is again a viscous potential flow. This type of wavy-wall flow with captured vortices was observed experimentally by Taneda (1978) and

180

J. 2. Wu and J. M. Wu

181

Vorticity Dynumics on Boundaries

Savchenko (1980), and our full Navier-Stokes computation indeed confirmed the existence of a stable zero-mean drag state. Figure 12 shows one with the computed flow pattcrn at the nth and ( n 11th waves from leading edge ( n can be arbitrarily large within the computational ability), with c = c,.. In the computation no periodic boundary condition was imposed, so the flow is naturally periodic at c,. Computations also showed that, under off-design conditions ( c is not close to c,), the flow strongly fluctuated and never reached a steady state or, even worse, vortices broke away. For the present purpose we just remark that, because near the critical state the flow behaves almost inviscid as well as steadily, a simple inviscid analysis similar to that in the earlier cylinder example can predict the critical wave speed c, quite satisfactorily (for one such analysis, see J. M. Wu et al. 19901, and this analysis can well be made based on the requirement that the averaged boundary vorticity flux is zero, exactly the same as in the cylinder case. In fact, to reach the desired critical state one needs two conditions: first, the flow has to be periodic; and second, because the inviscid vortex-induced velocity on the wall is in general different from the wall velocity given by (4.14) (thus, there is a near-wall vortex sheet), the averaged sheet strength has to be zero. This requirement precisely predicts c,/U = 0.414 for the case of Figure 12.” Although the first condition is satisfied by requiring a zero mean u/), the second will be satisfied by requiring a zero mean u 0 .

+

C. THREE-DIMENSIONAL EFFECTS In three dimensions therc are two more vorticity sources uviSii and

u V i FCompared n. with cril and wi7, at large Reynolds numbers, they are

small, except for some highly localized regions of the surface where flow separates, which we now focus on. The pressure gradient effect is also involved. The behavior of steady separation in two and three dimensions has been well u n d e r ~ t o o d . ’For ~ three-dimensional steady Navier-Stokes flow, fluid ”In fact, the Navier-Stokes computation was made under the guidance of this inviscid prediction. 14 One The concept “separation” can be underhtood in two ways (J. Z. Wu et ul., I%%). may consider either the boundary layer separation under the boundary layer approximation or the fluid particle separation that may or may not cause the whole boundary layer to separate. The study of the latter is based on the Taylor expansion of the full Navier-Stokes equation and relevant critical point theory. Hcrc we confine ourselves to the latter.

182

J. 2. Wu and J. M. Wu

\ Gc

tedl 1

FIG.13. The separation stream surface and its leading-edge and trailing-edge streamlines.

particle separation initiates from a critical point of the T~ field, say, C , , and terminates at another critical point, say, C,. C, can be a saddle or a saddle node, corresponding to the “closed” or “open” separations, respectively; and C, must be a node, including focus (Zhang, 1985; J. Z. Wu et ul., 1988b). Like a two-dimensional separation streamline, a separation stream suiface grows from the skin friction line between C, and C,, which is the dividing surface of the free vortex layer. This surface has a leadingedge streumline initiated from C, and a trailing-edge streamline initiated from C, (Hornung and Perry, 19841, see Figure 13. To study the local vorticity behavior near these critical points, one may introduce an orthonormal basis (el ,e 2 ,e,) with origin at a C , , say, such that e 3 = ii along the normal and toward the fluid. The directions of e , and e 2 can be chosen along T~ and g , respectively, as one approaches C, from upstream (Figure 13).

1. Vorticity Flux at Wall Critical Points J. Z . Wu et al. (1988b) showed that at C j or C, with T, = 0 there must be vV,. g = - n . ( V x 7,) = 0; so by (4.1), we simply have u

=

dP

u,, = -e l dX,

-

dP

-e2

Jx,

(4.15)

with q,= 0 there. Away from these points, other sources, cryisnand uvisn, may play some role, as will be treated later. Then, on the (el ,e,) and

183

Vorticity Dynarnics on Boundaries

(e, ,e,) planes, the leading- or trailing-edge streamline has inclination angles's

On the other hand, near critical points we have expansion w(x)

=

x . (V&)O

+ O(lxl')

= -"3U UJO)

+",[:],e*

+ O(lx12)

Here, by (3.18a) and (2.6b), there is d p / d x , = - V T . I,, which at critical points is simply - d ~ , / i l x , in our local frame. Thus, in particular, at a point P on the leading-edge streamline, w(x> =

-

x3 v

2 a,,,(O)e, + -Ir,,(O)e,j 3

+ 0(1xi2)

(4.17)

Similarly, it can be shown that the velocity at P(x) is

(4.18) Therefore, the vorticity at the separation stream surface is fed by a;, at x = 0 and is convected away from the wall by a velocity (4.18), which to O(Ix1) is along the x direction. This vorticity will join those from the boundary layers of both sides of the separation stream surface to form a free vortex layer. For two-dimensional separation, these results reduce to a more familiar form. The separation region has a small streamwise scale of O(Re- ;),thus a,,has a narrow peak of O ( R ef ) .This peak causes the boundary layer to separate; that is, the attached vortex layer bifurcates into a free vortex layer and an attached inverse layer. With the preceding coordinates, we now have a;, = a,,, O 2 = ~ / 2 and , (4.16a) reduces to the result of Oswatitsh (1958). A peak a,, > 0 implies that the separation streamline must be inclined to downstream, and, by (4.171, at least for 1x1 << 1, the [,orticityon separation streamline is dominated by the upstream shear flow. This analysis can be carried to higher orders by a set of recurrence formulas (J. Z . Wu et al., 1088b), but the results will soon become extremely complicated. IF

In J. 2. Wu ef al. (1988b) some relevant formulas were not reduced t o the simplest form.

184

J. Z. Wu and J. M. Wu 2. The Effect of Wall Curi!atureand Rotational r, Field

Because right at the critical points of the T,, field both uvIyn and uv15,, vanish, to examine their effects we need to shift our focus from the neighborhood of these points to the narrow strip neighboring a separation line, which is a T, line connecting the initial and terminal critical points, C, and C,. First, if I K I = O(v-'),the vorticity flux uv1\71 due to the surface curvature is comparable to up.This happens at a sharp edge of d B . Each small piece of the edge can be represented by a straight line, and the T, line must be along the edge. Thus, only the large curvature in the cross section counts. Let e, be the unit vector along the edge, and e, tangent to the sectional curve that has a curvature radius E = O(vi)or even smaller. Then (J. Z . Wu and Wu, 1993), (4.19) indicating a strong vorticity creation along the edge. Note that, by (4.2~1, whether a curved surface is a source or sink of enstrophy depends only on the sign of the sectional curvature of dB in the (e2,n) plane: in terms of our basis, on a nonrotating wall, TV,,,

= V5%2

Therefore, a sharp convex edge is always an enstrophy source. As the edge sweeping angle increases, T~~ will be larger and hence the edge-vorticity creation will be stronger. The total creation from an edge cannot be overestimated, though, because to the leading order the integration of uv,s71 over the edge area is independent of the curvature radius E and hence is still of O(v4).However, this relatively weak vortex layer is a seed of flow separation from edges. In fact, although (4.19) indicates that the created edge vorticity is along the e2 direction (hence, an ascending pattern), after traveling an O ( E )distance it will soon turn to the normal direction simply due to the sharp turning of e2 itself. Then, this vortex layer may become a kernel surface, around which the vortex layers from the upstream of both sides of the edge develop into a strong separating free vortex layer. Therefore, the vortex layer shedding from a sharp edge should have a sandwichlike structure, and further study of the effect of such a structure on the layer's evolution would be desirable. Second, to have an appreciable crvl\,r we need sufficiently large n . ( V x T ~ ~Its ) . peak value appears near a separation line toward which a family of T, lines converge or near an attachment line from which a family of 7,

VorticityDynamics on Boundaries

185

lines diverge. With the local orthonormal frame introduced before, on a separation line (4.ld) reduces to (T3=

1 dr,.

-~

h , dx,

+--

T~

dh,

h , h , ax,

(4.20)

where h , ( a = 1,2) are the Lam6 coefficients on the surface. Thus, the faster e , and e 2 change their directions, the stronger will be the normal vorticity. In particular, the magnitude of normal vorticity flux reaches a maximum near a focus of the T~ field, where the separation line rolls up into spirals and brings the created normal vorticity to a local area (a focus of the T, field is always a concentrated sink or source of the 6 field and a source or sink of [ field; see Figure 4(b)). Unlike the vortex layer generated right from a sharp edge, therefore, the spiral structure provides a mechanism to form a concentrated normal vortex, often called tomadolike ilortex (or horn uortex). Such vortices have been observed in many separating flows. As an example of these three-dimensional vorticity sources, Figure 14 sketches a complex vortex system observed in a jet in crossflow (Shi et al., 1991). The jet was issued normally out of an orifice of a flat plate and interacted with the oncoming boundary layer flow. The flow was incompressible, and the vorticity of this vortex system was created entirely from the solid surface, including the flat plate, the inner wall of orifice, and their juncture. From the figure we first see the vorticity in the attached boundary layer (on the left), which was the result of a u,>.Approaching the jet, an inverse pressure gradient similar to that in Figure 7 caused the boundary layer to separate and form a horseshoe vortex. The separated flow reattached in front of the jet and there was a secondary separation and another horseshoe vortex. Similarly, inside the jet pipe a u,] provided the vorticity in the pipe shear flow, that eventually left t h e orifice and became a free vortex layer. Then, because along the side edge of the orifice T~ has a nonzero component, a pair of weak vortex layers was formed near the edge as predicted by (4.191, which was experimentally distinguishable from the orifice vortex layer. Finally, as the horseshoe vortices went downstream of the jet, some horn vortices appeared as the result of u,,,,,.These “microtornados” are somewhat similar to the KBrmrin vortices, as can be seen from Figure 15 (see color plates), a similar experiment on a jet in crossflow (J. M. Wu et a!., 1988. Copyright 0 1988 AIAA. Figure 15 reprinted with permission). However, the fundamental difference is that, unlike a solid

186

J. Z. Wu and J. M. Wu

U

a

U

+

t U-1703

b Z

,Vortex-layer fromJet-pipe Boundary Layer

Tomado-Like Wake Vortices

'Horse-Shoe Vortex due to Primary Separation Oncoming Crossflow Boundary Layer Vonicity C

x

-L

U-16%

FIG.14. The near-field vortex system in jet in crossflow. Sketched based on an experiment: (a) the side view (central plane), (b) the top view, ( c ) a perspective view (from Shi et ul., 1991).

Vorticiry Dynamics on Boundaries

187

cylinder, no new uorticity curl be created from the jet plume boundary, as stressed by Fric and Roshko (1989) and Shi et al. (1991). Unfortunately, to the authors’ knowledge, so far no quantitative results of u distribution came from a three-dimensional flow over a curved surface, and this discussion is of only qualitative feature. The reader may also revisit other available experimental and computational ( p , 7 , ) distribution to obtain a qualitative estimate of the u behavior. Many such examples, among the others, can be found in Doligalski et al. (1994). Finally, it is worth noticing that, although globally u,,,, is much smaller than u p ,they seem to become comparable in the near-wall region of turbulent boundary layer, where the flow is dominated by a much smaller local Reynolds number. Therefore, we may expect that the normal vorticity source, along with the pressure gradient source, could play an important role in understanding the near-wall coherent structures of turbulence, such as those reviewed by Robinson (1991). D. VORIICITY-CREATION CONTROL Vortex control stands at the ccnter of various flow controls (for an extensive review, see Gad-el-Hak, 1989), including the aeroacoustic noise control, because at low Mach numbers vortices are the only “voice of flow” (Muller and Obermeier, 1988; see the vortex-sound equation (2.32a)). At present time, our knowledge of various vortex controls is going through a transition from an empirical art toward a rational science. Although the process is not yet complete, the general theory of boundary vorticity dynamics nevertheless suggests several guiding principles for achieving a successful uorticity-creation control. Thc principles will be exemplified here by a few typical quantitative results, experimental, analytical, or numerical. Theoretically (and partially practically), vortex control can be achieved at any stage of its life. For a vortex generated from a solid wall, these stages include (1) vorticity creation from the wall; (2) boundary layer evolution; (3) flow separation and the formation of free vortex layer; (4) rolling up into a concentrated vortex, its instability, receptivity, and breakdown; and ( 5 ) dissipation and transition to turbulent eddies (for a systematic exposition of this event sequence, see J. Z . Wu et al., 1993a). However, as stressed by J. Z. Wii and Wu (1991), to control a vortex at different stages is by no means equally easy and effective. The basic guiding principle is, the earlier, the better. For a desired favorable vortex, it is much easier to form an excellent creation circumstance, so that it is

188

J. Z. WuandJ. M. Wu

strong and stable at very beginning, than to enhance it after it has been poorly born. Similarly, it is always less effective to eliminate or alleviate an already formed unfavorable vortex than to prevent its formation. Therefore, whenever possible, the uorticity-creation control is of fundamental importance in various controls of boundary layer and its separation.I6 Typically, we are given a baseline-configuration geometry and a set of flow conditions, and hence also the baseline distribution of various vorticity sources. Therefore, the main objectiiie of vorticity-creation control is to manage a local change of the u distribution to improve the global distribution of vorticity as much as possible. 1. Steady Separation Control by Monitoring Local a,, Usually, the goal of most steady separation controls is to eliminate or alleviate smooth-surface separation, which is almost always unfavorable and uncontrollable. The key local regions for control are the neighborhoods of critical points of the 7,. field; that is, those C j and C, of Section 1V.C. This is because, first, as shown by J. Z. Wu et al. (1988b), the separation stream surface (dividing stream surface of the free vortex layer) consists of only those streamlines initiating from these points, thus, monitoring C, or C, can gain the best effect; and second, due to the topological rules of critical points that the number of nodes minus the number of saddles must equal 2 on a closed, single-connected surface (e.g., Tobak and Peake, 1982; Chapman and Yates, 19911, if one can remove a saddle, say, a node some distance away must disappear simultaneously and hence a local control will lead to a global topological change. In fact, these critical points and their connections are the skeleton of the entire near-surface flow (Hornung and Perry, 1984). Therefore, another guiding principle applied to steady flows is that the local Liorticity-creation control should be focused on critical-point control. In two dimensions, critical points degenerate to straight lines, and as is well known, a control applied to a neighborhood of the separation “point” is indeed most effective. Moreover, it is also well known that more critical points imply more complicated separation pattern and worse flow quality. Thus, the next principle is, for gicen baseline configuration and flow condition, the number of critical points of the T, field should be minimized, the fewer the better. Two remarks are appropriate here. First, controlling the critical points of the T,, field is by no means merely modifying those explicitly viscous vorticity fluxes uvisT and uvin. Rather, the T ,field, and hence these fluxes, Ih

This does not exclude the necessity of later stage control, because the effect of early stage control may not be able to cover the whole working range.

Vorticity Dynunzics on Boundaries

189

is a consequence of the leading-order “inviscid” fluxes ulJand ua (J. Z. Wu and Wu, 1993). Therefore, the focus should be on the latter. Second, for unsteady flow, the separation criteria and a thorough critical-point analysis are not yet available; in that case the principles should be understood from only a time-averaged point of view. An example of truly unsteady control will be reviewed at the end of this section. On a stationary wall, the smooth-surface separation is caused by a local unfavorable up or a sink of enstrophy flux ql, (Figure 71, and any means that can fill up this sink will work. For example, a local suction is an effective means for two-dimensional separation control known since Prandtl. Essentially, associated with sucking out the low-energy retarded fluid in the boundary layer is a local change of pressure gradient, which causes a local change of rlr, from sink to source, see Figure 16. Note that, as pointed out by Reynolds and Carr (19851, the vorticity advection due to a suction with uniform injection velocity is exactly canceled by a corresponding advection term in the vorticity transport equation. Thus, there is

FIG. 16. Enstrophy flux control by local suction near the separation point: (a) suction changes the enstrophy flux by changing the local pressure gradient, (b) sufficient suction removes the separation, (c) insufficient suction moves the separation point to downstream.

190

J. Z. Wu and J. M. Wu

no net gain in removing vorticity from the fluid by suction and the mechanism of suppressing separation by suction is exclusively due to the change of vorticity creation. In three dimensions, the critical points of the 7, field are isolated, which can be recognized from the oil flow pattern with the aid of the aforementioned topological rules and the stability theory of topological structures (e.g., J. Z. Wu et al., 1993a). A local suction is again applicable. This is especially so for removing a horn vortex, because its “root” is highly localized, centered at a focus (a C,). As mentioned already, removing a horn vortex means automatically eliminating a distant saddle point, so a change of the flow behavior in a larger region is achievable. For example, at a high incidence a strong swirling flow may be formed in a S-shaped inlet, resulting in either a sharp reduction of the engine efficiency or a stall. By flow visualization, Guo and Lin (see Lin, 1986) found that the swirl was a horn vortex initiated from a focus at the lip of inlet. Thus, a local suction was applied to an S-duct at an incidence about 60°, which indeed effectively prevented the formation of this swirling flow. The effect of blowing at the same location, in contrast, was found to enhance the horn vortex by increasing its energy, which further deteriorated the inlet behavior.” The same method can be applied to external flow, as confirmed by our water tunnel experiment with a swept-forward wing flow at (Y = 40” and a cylinder-plate juncture flow (J. Z. Wu and Wu, 1991). A single-hole suction with cp = 0.016 could appreciably alleviate an unfavorable horn vortex on the upper surface of the wing and make most part of the outboard flow attached (the inboard flow was massively reversed and could not be controlled by a local suction). Similarly, three suction holes in front of the cylinder (near C , ) with cp = 0.077 greatly suppressed the necklace (horseshoe) vortices. It should be stressed that suction is not the only way to suppress a separation and, in fact, often not a practically adoptable way. Modifying the local configuration within an allowable range, if possible, may be the simplest and most reliable passive control method to achieve a desirable local up distribution. Gupta (1987) reported that, to eliminate the necklace vortex at the juncture of a vertical cylinder and a flat plate, one can install a small delta-wing-like device at a negative angle of attack in front of the juncture. The device produces two effects: first, the ramp formed by the delta wing acts as a barrier to the rolling up of the necklace vortex; 17 Therefore, if one needs to enhance a horn vortex in a duct, blowing near the focus is effective.

Vorticity Dytiamics on Boundaries

191

second, the wing produces a pair of counterrotating streamwise vortices from the wing tips, of which the sense of rotation is opposite to that of the original necklace vortex and so cancels it. The essence of this device is, in fact, still a local pressure control.18 Even though a qualitative diagnosis by oil-flow patterns, say, may reveal the nature of the problem and what kind of change is desired, a quantitative analysis is necessary to tell the strength of control. To do the latter, one needs a sufficiently accurate computation or measurement of the T,$,field along with the pressure distribution, especially near critical points. This allows one to estimate the force acted on the relevant local region, which is roughly proportional to the strength of control. As illustrated in Figure 15, an insufficient control strength cannot reach the goal. Later we shall see that the theory of boundary vorticity dynamics can provide both a much improved method of computing the T, field and a unique estimate of the control strength directly in terms of boundary vorticity flux. 2. Vorticity-Creation Control by Unsteady Forcing As a further guiding principle, most continuous steady controls that work by imposing a forcing can be replaced by a pulsating control with much less power input yet achieile the same or eivn better effects. Therefore, unsteady controls have recently been a subject of intensive study. The flow to be controlled can be basically stcady or inherently unsteady. The relevant physics, along with many examples, was systematically reviewed by J. Z. Wu et al. (1991). Here we address some topics most directly relevant to the vorticity-creation control. For the unsteady control of a basically steady flow, one’s concern is the mean effect, which is nonzero even if the forcing wave is harmonic. Thc nonlinear interaction and resonance will lead to a net streaming effect. But, unlike the receptivity problem of Section IV.A.l, where one needs to know only if the mean fluctuating enstrophy becomes stronger, for proper control the mean change of iwtorial vorticity field is important. What we require is a favorable vorticity field with certain direction being enhanced. The mean enstrophy-flux consideration cannot distinguish the alternative change of the vorticity direction and hence is not appropriate in studying unsteady controls. 18

We remark that, according to our guiding principles, some configuration modification is better than installing an extra device, which can directly prevent the juncturc separation rather than cancel the separated vortex.

192

J. Z. Wu and J. M . Wu

The unsteady means for control can be an acoustic wave, an oscillating flap, periodic blowing or suction, or other devices. The objective of control can be the suppression of separation from a smooth surface and, when the separation is inevitable or desired, achieving a well-organized attachment at a prescribed location (where the flow is usually turbulent even if the separation is laminar). In the latter case, not only vorticity creation but also its advection and instability are involved." So we confine ourselves to the former. Collins and Zelenevitz (1975) first found that an acoustic wave may delay separation. Since then, many experiments have been carried out to clarify relevant mechanisms and improve the control efficiency (see the review of J. Z. Wu et al., 1991). Several mechanisms may be involved in the change of mean profiles by forcing, such as the receptivity of the separated vortex layer and forced transition to turbulence, so for our present purpose of vorticity-creation control, the basic physics should be similar to that discussed in Section 1V.A. But, now the attention must be paid to the time-averaged effect on the pressure gradient and skin friction (or velocity profile). The mean velocity profile, and hence the separation status, can indeed be altered by forcing waves, as seen from Figure 17 from the experiment of Nishioka et ul. (1990). The streaming effect of forcing can be further understood from a two-dimensional perturbation analysis of X. H. Wu et al. (1991). For a steady laminar separation, around the separation point the flow is governed by an interacting triple-deck structure (e.g., Smith, 1982). Let Re = U,L/v >> 1 be the Reynolds number based on the distance L from the leading edge to the separation point, then the basic length scale near the separation point is q,= Re-; << 1. The order of the streamwise length of these decks is E : ; and the normal scales of lower, main, and upper decks are c ( : ,6: (the attached boundary layer thickness), and c i , respectively. Now, upon the basic steady flow we impose a high-frequency forcing wave with circular frequency n , so that a reduced frequency or Strouhal number St = n L / U , >> 1 enters the interaction. Let E = St-f << 1 be the "To understand such a complicated process, Reynolds and Carr (1985) combined the boundary-vorticity-flux analysis with a qualitative cartoon of unsteady vortex advection driven by oscillating control devices. In this way, they explained some basic physics of many experimental examples of separated flow control.

193

Vorticity Dynunzics on Boundaries

1s

1s

II

x 10

10

5

0

0 0

0

U&

U&

1.0

1.o

0

u/v,

1.o

FIG. 17. Mean velocity profiles with and without acoustic excitation on a flat plate at an angle of attack of 12" and Re,- = 4 x 10'. Curves with triangles show the case of no forcing and thosc with open and black circles show forcing of 200 and 600 Hz, respectively. (Reproduced from Nishioka et ul. Copyright 0 1990 AIAA. Reprinted with pcrmission.)

194

J. Z. Wu and J. M. Wu

normal scale of the viscous oscillating Stokes layer, then the asymptotic structure of the disturbed separating flow naturally depends on the ratio

(4.21) When a >> 1, the Stokes layer is much thinner than the lower deck; when a << 1, the viscous oscillating flow in the Stokes layer enters the main deck and is convected into the upper deck as well due to the up-welling. The frequency between these two opposite extremes, a = 0(1), is known as the Tollmien-Schlichting frequency. This parameter a and the up-welling velocity, along with the x derivative of t h e streamwise velocity (which may be large near separation), jointly determine the flow character and suggest how to introduce an appropriate control for the desired streaming effect. Quantitatively, let (x, y ) be the global coordinates along and normal to the wall, with origin (0,O) at the steady separation point. The steady vorticity w,) vanishes as the skin friction does at the separation point ( x , y ) = (0,O) and becomes positive for x > 0 (Figure 7). Therefore, the goal of control is to create a vorticity wave such that its streaming effect is sufficiently negative to cancel the positive w,):

(4.22)

To this end, X. H. Wu et al. (1991) carried out a perturbation analysis in the triple-deck region. The forcing was assumed to be a tangential oscillation of a small piece of the wall rather than an acoustic wave, because the former directly serves as a vorticity source ua, which creates a transverse vortical wave highly localized in the vicinity of separation point (Figure 1S(a)>.2"Thus, the wall boundary condition is

zu The small oscillating wall could be replaced by a slot from which a forcing sound wave emits.

195

Vorticity qyniimics on Boundaries

I

a

Amplitude

5

b

r

0.50

Numerical

.~

...... Analytical

0.40

-s

..-u

0.30

L

.-c

6

0.20

0.10

0.00

-40

- 30

-20

-10

0

10

20

FIG. 18. Breakaway separation control by a small piece of tangentially oscillating wall: (a) the forcing device, of which the amplitude function is idealized by (4.25) in computations, (b) streaming effect on the skin friction tor Re = 2.0 X lo5 and different forcing frequencies. X is the normalized triple-deck streamwise coordinate. The “original” curve corresponds to no forcing, the dash lines are the closcd-torm asymptotic solution with a >-> I , and the solid lines are numerical results (from X. H. Wu, 1991).

196

J. Z. Wu and J. M . Wu

Then, the perturbation expansion shows that, when a >> 1, the classic Rayleigh streaming law (e.g., Stuart, 1963) for an attached Stokes layer,

where 5 = e-leO4y/ is the rescaled normal distance, still holds even for separating flow. This suggests that, to create a negative w,$ on the wall, we need A ’ ( x ) < 0. Figure 1Na) indicates that this corresponds to the right half of the small oscillating wall. The left half is inevitably unfavorable; but, if it is located properly upstream of the strong up-welling region where w o is not too close to zero, it may only reduce the negative vorticity without changing its sign. Thus, not only the amplitude profile 4 x 1 but also its location is important for a successful control. The computation of X. H. Wu (1991) confirmed this observation. In terms of the normalized triple-deck streamwise coordinate X = A - $ e i 3 x with A = 0.332 (Smith, 19821, the forcing amplitude was idealized by the function (4.25) Here, the free parameters L , and B determine the central location of the oscillating piece of wall and the sharpness of the amplitude slope, respectively. This forcing was applied to both breakaway and bubble-type separating flows. In Figure 18, the basic velocity profile was taken from the triple-deck solution of Smith (1977) for breakaway separation, where the upstream attached boundary layer was assumed to be the Blasius solution. Figure 18(b) clearly shows that a proper forcing of Figure 18(a)’s type may completely eliminate the separation. In Figure 19, the basic flow is a separated bubble similar to that given by Carter (1975). From Figure 19(a) we see that the forcing effect is improved as St decreases for fixed Re. More precisely, the parameter a dominates the control efficiency, as shown in Figure 19(b) for different X stations. Because a = 0 means no forcing and hence w , = ~ 0, there must be an optimal value of a between 0 and O(1). Finally, let us briefly look at the unsteady control of a truly unsteady flow. A typical example is the transition control, where a forcing wave with

.

~Numerical

.....

Analytical

I

-2

0

2

4

X

6

I

I

10

0

I

I

I

12

0.40

0.30

a

\

7. 0.20

X =

X = 0.10

0.00 0.00

0.50

1.00

cL

1.50

2.00

FIG. 19. Bubble-separation control by a small piece of oscillating wall: (a) the streaming effect on the skin friction for Re = 2 x 10' and different frequencies, (b) the effect of a defined by (4.21) at Re = 2 X los (from X. H. Wu, 1991).

198

J. Z. Wu and J. M. Wu

opposite phase and proper frequency and orientation is imposed to cancel the Tollmien-Schlichting wave and thereby retain the laminar state of a boundary layer. The concept of using waue cancellation to eliminate or reduce the TS-wave started with Schilz (1965- 1966), and became popular due to the works of Milling (19811, Liepmann et al. (1982), and Liepmann and Nosenchuck (1982). The control means used by Schilz, Milling, and Liepmann were flexible wall, vibrating wires, and hot films, respectively. The technique can obviously be used to promote transition as well. Liepmann and Nosenchuck (1982) showed that a comparative stabilization by steady heating would require 2 X lo3 times more power than the unsteady wave cancellation. This is a vivid confirmation of our guiding principle about the advantage of using unsteady control. For relevant works see the review of Gad-el-Hak (1989) and a recent paper by J o s h et al. (1994a, b). Our interest here is that the hot films used by Liepmann and Nosenchuck are again a tool of controlling the vorticity creation rate u, but this time through the change of viscosity. A more satisfactory result was obtained by J o s h et al. (1994a,b), who used the direct numerical simulation (DNS) to confirm for the first time that an active periodic blowing and suction may lead to nearly exact cancellation for both smallamplitude (linear) and large-amplitude (nonlinear) disturbances. The computation used two sensors, one actuator, and a spectral analyzer to automate the wave cancellation process. Sensors measured wall pressure for the spectral analyzer, which told the actuator what frequency was evident and at what amplitude to cancel instability. Their results are shown in Figures 20 and 21, where the contours of u velocity, wall pressure, and skin friction with and without control are in a sharp contrast. As in the steady control case, this unsteady blowing-suction control of TS-waves is essentially a up control.

V. Vorticity Creation from an Interface

The general theory of vorticity creation on an interface has been presented in Section 111. In this section we further explore some of its aspects of wide interest, including several examples to illustrate its basic features, especially those that occur in three dimensions only. The flow is assumed incompressible, under a gravitational field.

VorticityDynamics on Boundaries

199

FIG.20. The contours of u velocity without (a) and with (b) control. The TS-wave is in the unstable regime and is intensified downstream for the no-control case. The unsteady control makes the wave approach a steady laminar flow with only slight modulations downstrcam of the controller (from J o s h et a/., 1994. 1995. Copyright 0 1995 AIAA. Reprinted with permission).

A. DIMENSIONLESS PARAMETERS ON

A

VISCOUS INTERFACE

We first cast the general results of Section 111 to dimensionless form as a conventional starting point of our analysis. Let U* be a typical interface velocity and L be a length scale, both common to two fluids. Denote a mean value by an overbar and use U *, L , and 2 p to nondimensionalize all quantities. The Atwood ratio A, Weber number W, and Froude number Fr are defined as

200

J. Z. Wu and J. M. Wu

t

-0.5

"-0

0

Uncontrolled

-1.0

300

1.5

1.0

-1.0

-1.5

350

400

500

450

550

x ~ o - ~

1

I.

C

0

5

Controlled Uncontrolled

0

;

0.8

0.9

1.0

1.1

1.2 x103

X

FIG. 21. The wall pressure (a) and skin friction (b) without and with el a[., 1994, 1995. Copyright 0 1995 A I M . Reprinted with permission).

control (from J d i n

20 1

Vorticity Dynamics on Boundaries respectively. For a viscous interface we have two Reynolds numbers:

both of which are assumed large and from which a single nominal Reynolds number can be introduced: Re

=

=

U*L

+ v2

~

vl

R e , Re,

-

Re,

+ Re,

>> 1

(5.3)

Then, following J. Z. Wu (l995), from A, e l and e2 we construct two constants of O(1) to characterize the jump of dynamic viscosity:

Thus, the dimensionless form of surface stress conditions (3.la, b) become (using the same notations as dimensional ones for relevant variables)

[ p ]= WK

[ 5 ]+ 2Ag =

2EpV

~

-4An

X

'

(V,U,,

Moreover, eqs. (3.19a, b) become, for i

=

u

+ U . K)

€,-

ai; dn

=

€,Il((K

-

In particular, if [ p ] = 0 and hence p = A density jump, eqs. (5.5a, b) degenerate to [PI

=

WK,

(5.5a,b)

S

1,2,

+ E,V,{ + e f t f .K (-l)f-'crrn =

on

o;g,> =

1

on

s

(5.hb)

0, no matter how great the

[g] = 0

(5.7a, b)

That is, the inviscid normal stress balance is recovered and the tangent vorticity is continuous across S, independent of the interface motion and geometry. Note that (5.5a) (along with the motion equation) determines the sueice elevation; thus, for a viscous interface with [ p ] = 0 , the surface

202

J. Z. Wu and J. M. Wu

elevation will remains exactly the same as the corresponding inviscid solution. On the other hand, if S is a free sugace of fluid 1, there must be A=l,

(5.8a, b)

P=A=l

and (5.5a, b) reduce to (we drop the suffix 1)

where

E

is defined by (5.2).For two-dimensional flow, (5.9b) becomes

where s is the arc length along the interface. If, in addition, the flow is steady so that U, = 0, then (1.4) is recovered. The formulas for a are similarly simplified; in particular, because n x (cl, x U) = wU,, - U l , for two-dimensional steady flow we return to (1.5). J. Z. Wu (1995) stressed that, although for inviscid flow the condition A = 1 is sufficient for identifying an interface as a free surface, for viscous flow one should check (5.8b) as well. On a water-air interface A = 0.988, A = 0.964, both being quite close to 1; but p = 0.062 << 1. Therefore, in the free-surface model of a water-air interface, the viscous effect on the normal stress balance is overestimated. This simply favorably ensures that the water-surface elevation is much closer to the inviscid result than the free-surface model predicts. However, if on an interface of two viscous fluids other than pure water and air it happens that A 0.5, say, then the free-surface model has to be given up even if A = 1. Wu also noticed the striking similarity between (5.9b) and the vorticity formed behind a curved shock wave (Hayes, 1957):

-

although their mechanisms are so different. Here the flow in front of the shock is assumed to have a uniform density po and a uniform velocity U and that behind the shock has a variable density p l . This similarity seems to be a hint that extending the present theory to include normai discontinuity might be possible.

203

Vorticity Dynamics on Boundaries B. FLATINTERFACE

AND

FREESURFACE

Based on the preceding dimensionless formulas, we first consider the simplest situation, where Fr << 1 or g" >> 1, so that the interface can be approximated by a flat horizontal surface. 1. General Obsen>ations On a flat interface, the surface tension disappears and so must be V n q l . Thus, because a uniform U,, can be made zero by a coordinate transformation, eqs. (S.Sa, b) and (S.6a, b) reduce to 1 - A (S.lla, b) PI = P2 - 2ePV,. u,, 5, = x 5 *

( - 1)'- 'a,,=

-

e , n ~ , .5,

respectively, where Hi = h , boundary enstrophy flux is

+ U 2 / 2 is the

total enthalpy. Moreover, the

1 , 2 (5.13) If a flat S is a viscous free surface with A = A = P = 1, then 6 , u,,,and q all vanish. This implies that a flat free sugace is free from shearing, where the uorticify can haile only a normal component that must be formed away from the surface and no uorticity clifises across the suface (recall the remark following (3.23)). What remain are a possible pressure variation of O( E ) and a tangent vorticity flux. This flux still creates new tangent vorticity, which, however, entirely enters the interior of the fluid. Moreover, like the case of an interface with [ p ] = 0, there will be no boundary layer near a flat free surface (see Section V.C later). Some further conceptual issues relevant to a flat interface and a flat free surface have been discussed by J. Z. Wu (19951, who stressed that the shear-free condition is not equivalent to and does not need the free-slip condition, as exemplified in Section 1I.D. In addition, when treating a flat water-air interface, say, as a free surface, a residual g on the water side (no matter how small) is necessary to avoid unrealistic physics if the air side is also to be of concern. Nevertheless, the preceding simplifications have been utilized in various

i

=

204

J. Z. Wu and J. M. Wu

approximate analyses and numerical simulations, of which a couple examples are reviewed here. 2. Vortex Pair Rebound from a Flat Interface

As a typical example of using the flat interface or free-surface model in theoretical studies, we consider the vortex pair rebound phenomenon from such a surface. A pair of inviscid vortices of equal and opposite circulation at the same height above a flat boundary, either a solid wall or a free surface, may approach the boundary under their mutual induction. In this process the vortices will separate from each other and never rebound from the boundary (Lamb, 1932; Saffman, 1979, 1991). The observed vortex pair rebounding from a ground has been attributed to the viscous separation induced by the vortex pair and the formation of secondary vortices (Harvey and Perry, 1971; Peace and Riley, 1983; Orlandi, 1990). However, whether or not a viscous vortex pair will rebound from a free surface is a more delicate problem. The experiment of Baker and Crow (1977) and lowReynolds number computation of Peace and Riley (1983) confirmed the rebounding phenomenon; but Orlandi (1990) and Tryggvason et al. (1992) showed numerically that, on a flat, “free-slip’’ (in fact, shear-free) surface, the rebound does not occur at high Reynolds numbers, and the latter attributed the rebounding to the effect of surface contamination. However, it seems that most of these discussions can be settled by the elegant and simple analysis of Saffman (19911, who provided a mathematic proof that on a flat free surface the vorticity centroid does not approach the surface monotonically. We now show that this conclusion can be strengthened by allowing for a residual surface vorticity. Following Saffman (1991), assume the vortex pair initially moves down toward a flat free surface y = 0, with the vorticity antisymmetrical about x = 0. Let

be the total strength and the height of the centroid of the vorticity in the first quadrant, respectively. The boundary conditions are u=w=O

u=0,

on

x=O

o = -

Vorticity Dynamics on Boundaries

205

Note that no free-slip condition is imposed; for a shear-free surface, we simply replace (5.14~)by w = 0 on y = 0. Then, Saffman’s results

(5.16) remain effective no matter if w = 0 on y surface vorticity is that we now have

=

0; the only effect of a residual

where the second term is due to (5.14c), which is always positive. Then, as Saffman argued, as t + m, eq. (5.16) indicates that the vorticity moves asymptotically away from x = 0 and hence the first term of (5.17) will tend to zero. Therefore, without the second term of (5.17) there is Tj

-

const

as

t

.j~0

which, along with (5.151, shows that eventually j will increase, implying a rebound. But, with the extra term we now have

-

d lim - ( T j ) > 0 1 x dt and hence the rebounding is slightly enhanced (J. Z. Wu, 1995). Even though this analysis proves that the rebounding will happen after a sufficiently long time, it does not tell whether the vorticity will be completely dissipated during this time. If this happens, then the rebounding might not get a chance to occur. Note that the enstrophy dissipation is especially important for a two-dimensional vortical flow bounded only by a flat free surface, because then (3.22) reduces to

Thus, 101 in 9 must monotonically decrease until its distribution is uniform; but = 0 on S implies that the final result can be only a potential flow.

206

J. 2. Wu and J. M. Wu

3. Turbulent Vortices under a Flat Free Sugace Because a flat free surface S can be simply taken as a known shear-free “wall,” using this model in numerical simulations entirely removes the most complicated task of determining the shape and location of S. Therefore, many recent direct numerical simulations (DNS) of free-surface turbulence have adopted this simplified model (e.g., Handler et al., 1993, and references therein). Figure 22 from Handler et al. is a typical DNS result of vortex structures in an open-channel flow with flat free surface, having a Reynolds number of 2340 based on the channel height and mean surface velocity. As expected, the figure clearly shows that the only coherent structure right on S is vertical vortices centered at the spiral

FIG. 22. Turbulent vortex structures in open-channel flow computed by DNS (from Handler et al., 1993): (a) particle paths on flat free surface convected by a frozen velocity field, (b) vortex structure associated with converging spiral in the small rectangle in (a).

VorticityDynumics on Boundaries

207

points of surface pathlines. This kind of pattern has been observed in experiments (e.g., Utami and Ueno, 19871, where the free surface is not strictly flat. Although the numerical picture reminds one of those horn vortices created on a solid wall (Section IV.C), a basic difference is that now since un = 0 the vortices are not creutedfrom S. They must have been turned to the normal direction before reaching S (Figure 21(b)), either by the mutual induction of turbulent eddies or due to a nonzero a, on the bottom wall. Obviously, the flat free-surface model suffers from severe limitation in practical applications, because most rich and colorful vortical structures unique to free-surface flows are missing. These structures appear once the Froude number is no longer small, among which is the free-surface boundary layer to be treated next.

C. FREE-SURFACE BOUNDARY LAYERS As mentioned in Section I, the existence of a boundary layer near a free surface S has been well known. Although the inviscid solution of a free-surface flow satisfies the normal stress balance, in general it does not satisfy the tangent stress balance (5.9b), which forces a vortex layer to form as a correction of inviscid solution. Unlike a solid wall boundary layer, the pressure also needs to be corrected, as first pointed out by Batchelor (Moore, 1963). A theoretical formulation for three-dimensional boundary layer near a free surface has been given by Lundgren (19891, which can be applied to any large Reynolds number free-surface problem with an irrotational global inviscid flow. J. Z. Wu (1995) generalized Lundgren's theory to include rotational outer flow and sharpened its form by using (2.8). This theory and some applications are reviewed here. Extension to an arbitrary interface is straightforward. 1. Linearized Roundary Layer Equation

Denote the inviscid velocity and modified enthalpy by un and respectively. They satisfy the equation

h,,,

Dnun Dn - -v&, +u,.v dt Dt Dt and on the free surface S there is h n = - W K . Thus, in contrast to the solid wall case, the viscous correction of velocity and enthalpy, denoted by u' and h e , must be of o(1). More specifically, we introduce the vector --

208

J. Z. Wu and J. M. Wu

potential A for u', such that u' = V x A and o' = -V2A. Then we find ), that at large Re = 6 - I the boundary layer thickness is 6 = O ( E ~and u;

=

n X dAT - O(S), dn

o; = ~ ( i ) ,

u:,= O W ,

u; = (n

A,

x V).A,

=

=

o(sz>,

O(S2) A,

=0

~

3

\J

(5.18) )

Thus, as remarked before, a free-surface boundary layer is much weaker than its solid wall counterpart. This implies that the boundary layer equation can be linearized. The result is (J. Z . Wu, 1995)

We see that the inviscid rotation enters the equation only through its normal vorticity, consistent with the three-dimensional vortex-sheet theory of Section 1II.C. Note that (5.19a) is a homogeneous equation; it has nontrivial solution only if there exists a forcing mechanism provided by the free-surface condition

d2A,

- - 5 ' = 2n X (V,UO, + U , , - K , , ) + O ( 6 ) on S (5.19b) dn2 That is, the fi-ee-suface boundary layer is drillen by the leading-order sufacedeformation stress. On a flat free surface, this forcing mechanism disappears and hence so does the boundary layer. Moreover, the excess enthalpy he reads --

he = - g ' . A ,

+ O(6')

on

S

(5.20)

which implies only an O(S2) correction to the surface shape. Finally, for (J. Z. Wu, 19951, the boundary vorticity flux u = uoa+ uh,+ u,,,, + uVisn

(5.21a, b, c, d) Clearly, except the first two O(6) terms in u Q sall, the rest are of O( 6 '1. Therefore, the leading-ordersource of vorticity on a free-surj5ace is the viscous correction of the surface acceleration. The weakness of u compared with

209

Vorticity Dynamics on Boundaries

solid wall case has been revealed by the smallness of (T on the flat free surface discussed in Section 1I.D (see Figure 3(c)), where the boundaly layer thickness is 6 = O(lOpl). Note that from a known inviscid solution and uvisn, and the other quantities one may immediately infer uviSr amount to solving the linear problem (5.19a,b) for the tangent vector potential A,. 2. Vorticity Creation Due to Surface Waves The preceding theory covers a wide range of applications. As the first and most classic example, we revisit the viscous two-dimensional linear water wave (Figure 23). This is a slight modification of the flexible wavy-wall case treated in Section IV.A, the main change being the constant-pressure condition on the free water surface S. The problem was solved by Lamb (1932, Section 349) for a freely decaying wave, to which we impose an applied stress to maintain a constant amplitude. Lamb’s result has been extended to three-dimensional flow by Lundgren (1989). Assume the free-surface elevation is y

=f

( x , t ) = a cos(krc - y t ) ,

where T‘

y2

=

gk

+ T‘k3,

ak

2rra

= -< <

h

1

=

T / p . The Cartesian components of the potential velocity are

u

=

ayekY cos(kx - y t ) ,

I)

=

ayeky sin(&

-

yt)

In this problem, (5.19) reduces to a linear equation for the scalar stream function, and the inviscid potential flow satisfies the Bernoulli equation. The resulting linearized vorticity solution is indeed weak: w =

2akyeB’ cos{ku - ( y t + B y ) }

FIG.23. A water wave.

=

O(1)

(5.22)

210

J. 2. Wu and J. M. Wu

where /3 = (y/2v)i. The boundary vorticity flux and enstrophy flux are

respectively, both being of O(vf). The latter has a nonzero positive average, showing that the overall enstrophy in a forced water wave is increasing. As indicated by (5.1c), the Froude number will be larger if the wavelength is smaller. The gravity-capillary wave belongs to this case, where we anticipate a stronger vorticity creation in a thin boundary layer. Figure 24 (see color plates) is an experimental result of a steady near-breaking gravity-capillary wave formed behind a hydrofoil, due to Lin and Rockwell (1995). The measurements were performed using digital particle image velocimeter (DPIV). As a comparison, a DNS of a similar but unsteady wave under slightly different conditions due to Dommermuth and Mui (1995) is shown in Figure 25 (see color plates). The Reynolds number based on wavelength (5 cm) is 3.5 x lo4, and the boundary layer is about 1 mm thick, which was resolved numerically using 4096 X 1025 grid points. The vortical structures can be clearly seen from both experimental and numerical results. Physically, the vorticity inside the flow must come from the u effect governed by the viscous correction of the surface acceleration, a', which creates new vorticity on S and then sends it into the flow. Thus, the location of strongest vorticity should be a mark of the largest la'l on S . Unfortunately, so far no computation has been -made to infer u from experimental or numerical data. Here we note that eqs. (5.5a,b) should be imposed as a primary boundary condition in solving any viscous free-surface flow. In fact, it is a sharpened and physically more appealing version of the stress condition (2.1 11, which usually involves a complicated calculation of all components of the velocity gradient Vu. If (2.11) has been directly used in the common way in a numerical scheme, then (5.5b) can be reversely used to test a posteriori the accuracy of the scheme. Figure 26, again from the DNS by Dommermuth and Mui (19951, gives such an example. The computation used 6.7 X lo7 grid points. The DNS vorticity on S is shown in the figure and compared with (5.10). The numerical error is less than 1%, indicating a very good accuracy of the scheme. Considering the huge number of grid

21 1

Vorticity Dynamics on Boundaries Surface Vorticity Check 50.0

l

"

'

1

"

'

I

"

'

l

"

'

0.0 .-x

.u 5

I

-50.0

>

g a

-100.0

vl

-150.0

1 -200.0 0.0

1

0.20

,

/

,

1

,

1

/

1

1

1

0.60

0.40

1

1

0.80

1

1

1

1

X-position

FIG. 26. Surface vorticity check of DNS for a two-dimensional unsteady near-breaking gravity-capillary wave with a 16384 x 4097 grid along the length and depth of the wave (from D. G. Dommermuth). At this particular time, the maximum wave slope is 1.22. The solid line is the computed free-surface vorticity, and the dotted line is 10 times the difference of DNS and (5.10).

points, however, this comparison also indicates the difficulty of reaching a high accuracy. It is remarkable to note that, although in general on a free surface 6 = 0(1), Figure 25 indicates that its peak value can be as large as O(Ref), like the solid wall case but confined to very narrow regions of troughs. This is a sign that to capture the detailed structure of stiff short waves the traditional potential wave theory is insufficient, because the created strong vorticity must inversely affect the surface motion.

3. Boundary Layer on u Bubble Surface Moore (1963) studied the boundary layer of a raising spherical gas bubble of radius u in a liquid with constant velocity U at a large Reynolds number Re = U u / v = E - ' = K 2 . The inviscid velocity on S uses the potential-flow solution U, = ( 3 / 2 ) U sin Oe,, where spherical coordinates ( r , 8, 4 ) are assumed, with 0 = 0 along the moving direction. Then, by

.o

212

J. Z. Wu and J. M. Wu

(5.19) and (5.21c, d), we immediately obtain the dimensionless surface vorticity

6’ = 3sin 8e, + O ( 6 )

(5.24)

and its explicit viscous flux:

From the viewpoint of boundary vorticity dynamics, what remain to be solved are the flux due to viscous correction of surface acceleration, u a , , and that due to the excess enthalpy, u h cAlthough . these can be obtained by solving (5.19a), which reduces to a scalar problem for the Stokes stream function, Moore directly solved the velocity components u’ = ( 6 ’ u ’ , 6 u ’ , 0) and excess enthalpy he from the original boundary layer equation. It is evident from (5.18) and (5.21a7b) that for our purpose we need to know only the value of u’,, on S, which is 6 u ‘ ( l , 0)e, and from which A,

=

1” -8

u’,,

X

n dn

m

=

fi2e,/, u‘dy, on

S,

y

=

F 1 ( 1- r )

=

O(1)

Therefore, from Moore’s (1963) result u’(1,e)

=

-61hsin

0$(8),

2 ~ ( 8 =) -csc4 8(2 9

-

3cos 8 + cos2 0 )

effective for 0 # T (near the rear stagnation point the boundary layer approximation blows up), it follows that (J. Z. Wu, 1995)

the latter diverging as 0 + T . A complete solution of he and u h over the whole bubble surface would be possible if a further singular peiturbation could be introduced near 0 = T as an analogy of the triple-deck structure near the separation from a smooth solid wall. Nevertheless, it can be anticipated that uhewill have a high peak at this stagnation point. A delicate application of the free-surface boundary layer theory was made by Lundgren and Mansour (1988), who considered the oscillating

Vorticity Dynamics on Boundaries

213

drops in zero gravity. After obtaining the inviscid axisymmetric solution where the drop surface was treated as a closed vortex sheet, the authors introduced a high Reynolds number correction to study the viscous damping effect on the oscillation. Because of the complexity of the solution structure, here we shall not go into detail but merely mention that the formulation of this problem, based on Lundgren (19891, can now be sharpened as (5.19a7b) and (5.20); and from the inviscid solution the vorticity creation from the drop surface can be likewise studied. 4. Interaction between a Vertical Vortex and Free Suface

The preceding examples all assume an irrotational outer flow. J. Z. Wu et al. (1995a), considered a situation where the free-surface boundary layer has a rotational outer flow: the interaction between a free surface and a vertical vortex. Assume the vortex is axisymmetric as sketched in Figure 27. Here, we need both the cylindrical coordinates ( r , 0, z ) for describing the basic vortex flow and the coordinates along and normal to S. Let e, and s be the unit tangent vector and arc length along the section curve of S in the ( r , z ) plane, say, z = f ( r ) , and n be the unit normal vector pointing toward air and n the normal distance. Then, as shown in Figure

FIG.27. A vertical vortex interacts with a water-air interface S . In the water, the basic vortex is inviscid and hvo-dimensional (the boundary layer is not shown). In the air, the viscosity is just turned on so that there appears a vortex shcet S + above S. As r + 0, the sheet bifurcates into a thin axial vortex. Both cylindrical and surface coordinates are shown.

214

J. Z. Wu and J. M. Wu

27, the basic geometric relations are de,dS

1 de, r do

dn dS

1

-

K,

4,

- e , cos

r

4 is the angle between

where

-_

K,n,

-

I

- Kse,

1 dn

(5.26a)

--- - ~ # e ~ r do

e, and e , , and 3

=f”

COS3

4 = f ” ( l +f”)-’,

(5.26b)

are the principal curvatures along e , and e H . The global inviscid flow, ug, allows for a discontinuity of tangent velocity and normal vorticity on the interface S. The vertical inviscid vortex can be two-dimensional, that is, ug = (0, V,, ,01, and any V,(r) is a solution of the steady Euler equations V:

_ = -

r

dh

dh o=-+g*

dr’

d2

For a given V,,(r) and assuming the surface tension is negligible, the dimensionless shape of S , z = f ( r ) , is determined by f(r)

=

r V,f F r i l -dr

(5.27)

r

-

-

The prescribed V J r ) should ensure a solid-core behavior V(, r as r + 0, and r - ’ as r + 30 (irrotational). Our interest is the boundary layer structure near S, which amounts to solving (5.19a, b). Following J. Z. Wu et al. (1995a) and in accordance with (5.18), the viscous correction to the velocity can be written as u‘ = ( S U , SV, S 2W ) , with U , V , W = 0(1), and the tangent vector potential is A, = e, A , + e , A , ] . Then (5.19a,b) give a pair of coupled homogeneous equations d2Ao dN’

--

V,I 2-A, r

cos 4

=

0,

dA,

~

I dr,,

+ -r- Adr, dN2

cos 4 = 0,

To = rV, (5.28a, b)

Vorticity Dynamics on Boundaries

215

with N = 6-'n being the stretched normal coordinate, subject to the free-surface boundary conditions

Obviously, eq. (5.29b) is the axisymmetric counterpart of (1.4). Note that V&r) has an associated w O s= w , , sin ~ 4, which would be consistent with (5.29b) only if woz= 2V,/r; that is, the vertical vortex behaves like a solid rotation for all r . But in reality this is not the case, and hence generically there is a viscous correction to l,',,(r), implying the appearance of a boundary layer. Alternatively, one can start from the familiar rotationally symmetric boundary layer equation; then, after subtracting the inviscid solution, the equations for V and the Stokes stream function I) = rA, = 6-'W are factually equivalent to (5.28a,b). J. Z. Wu et al. (1995a) found their analytical solution

C(r)Il -

are two parametric functions. Physically, L ( r ) characterizes the effect of inviscid vortex-core structure, and C ( r ) includes both the effect of L ( r ) and the forcing mechanism (5.29b). The latter dominates the r dependence of the layer. Note that to ensure the effectiveness of the solution, there must be C ( r ) = O(1) for 0 I r <

(5.31)

Two findings of J. Z . Wu et ul. (1995a) based on this solution are of fundamental interest. First, for two most frequently used steady vortex

216

J. Z. Wu and J. M. Wu

models, that is, a normalized “frozen” Taylor vortex

V,= ref(1-r’) and a normalized q vortex

V,(r)

=

4 -(1 - e-”’) r

with

q

=

1.398, s

=

1.256

(the normalization is made to have a unit maximum velocity at r = 11, the C ( r ) behavior does not satisfy (5.31). For the former, d r , / d r changes sign at r = fi and hence C ( r ) has a singularity, but for the latter, C ( r ) grows exponentially. This implies that separation will occur, and the attached boundary layer model, as assumed in deriving (5.191, blows up. Therefore, at least in the case of normal vortices (which often happens in turbulent flows as seen in Section V.B.3 or if a tornadolike vortex hits the surface), thefree-surface boundary layer is very susceptible to separation. In fact, the jet flow shown in Figure 25(b) (see color plates) is nothing but the result of such a separation, even though there is no vertical vortex in that case. J. Z. Wu et al. (1995a) found that the simplest form of C ( r ) satisfying (5.31) and ensuring the correct behavior of V,(r) near r = 0 and at infinity is ar2 which leads to an algebraic vortex model

They set a = 3.5 and r,, = 1.05 times of that of the normalized q vortex, and fixed the constants a and b by normalization. It is remarkable that, as Figure 28 indicates, this model represents merely a very slight modification of the q vortex, but their C ( r ) , and hence their boundary layer structure, are so different. The second finding of J. Z. Wu et al. (1995a) is, on using (5.32) to obtain an attached boundary layer, that away from the layer the axial velocity w = S 2 W does not return to zero. Rather, there is a persistent axial flow

w(r>lN+x=

6’ d C

-

--

r dr

217

VorficityDynamics on Boundaries a

Vo(r)

1.2

,

Algebraic Vertex Frozen Taylor Vertex ---Q-Vertex - - -

1

0.8

0.6

0.4

0.2

C

1

2

r

3

4

5

I , I I $ , , , /

, , 8 I I II

Algebraic Vertex Frozen Taylor Vertex - - - Q-Vertex

! I

1.5

1 :

-

1 ;

,

I

I ,

,,'

I

,--

/

-0.5-

1

0

1

I

,

2

3

r

1

4

5

FIG.28. The velocity profile V , ( r ) and function C ( r ) of a two-dimensional inviscid vortex, defined by (5.32) with (Y = 3.5. Also shown are the corresponding curves of the normalized q vortex and frozen Taylor vortex. See J . Z . Wu ei al. (1995a).

J. Z. Wu and J. M. Wu

218

with a scale of

Fr Re

-=--

vU* gL2

-

viscous force gravitational force

Therefore, the interaction of a vertical vortex with a free surface not only causes a boundary layer, but also alters the vortex structure itself a vertical vortex interacting a free-surface is inherently three dimensional. This phenomenon has been observed in some experiments. Physically, the r dependent axial velocity is induced by a circumferential vorticity component w, that is created from S and sent deeply into the fluid by a a, of O ( S 2 )caused by the interaction, see Figure 29. In this interaction problem, the free surface S can be taken as a water-air interface with small effect on the water motion. If we turn on the viscosity of the air at t = 0, say, to extend the preceding solution to the air side, an unsteady full Navier-Stokes equation has to be solved, of which the initial condition at t = 0' is an air vortex sheet S ' , say, adjacent to S with a nonzero mean normal vorticity 5 = w,,z cos + / 2 . Its

:::: t

-0.3

-0.4

FIG. 29. The circumferential vorticity and associated axial velocity of a vertical vortex interacting a free surface (from J. Z . Wu et al., 1995a). Without interaction, the vortex is two dimensional and inviscid, given by (5.32). In the figure, w 0 ( r ) and w ( r ) are values away from the interacting region (the boundary layer) and hence represent a persistent structural change along the whole vortex.

219

Vorticity Dynamics on Boundaries

u,,

velocity has to be determined by (3.39). Except the mean velocity the normal vorticity causes an additional circumferential velocity (J. Z. Wu, 1995)

which is always an increment to if 2 is single signed. Then, as time goes on, the air vortex sheet evolves to an air vortical flow. D.

COMPLEX VORTEX-INTERFACE

INTERACTION

SURFACTANT EFFECT

AND

The examples considered thus far have been limited to highly idealized simple circumstances, all assuming a clean interface or free surface. For completeness, before ending this section we briefly exemplify the vorticity creation from a free surface that has a complex interaction with nearby vortices and make some preliminary observations on the effect of surface contamination. 1. Interaction of a Vortex Pair with a Free Sur$ace

A typical complicated vortex-interface interaction occurs when a pair of submerged vortices or a vortex ring moves up to a free surface S , as revealed by the well-known experiments of Sarpkaya and coworkers (for reviews, see Sarpkaya (1992a, b). The experiments showed that as a vortex pair approaches the free surface S under mutual induction, the surface will be humped up to form a Kelvin oval, and at mean time a series of lateral vortices appears, riding on the quasi-cylindrical oval (“striations”), bounded by two rows of “scars” and whirls digging into the water at the roots of the oval (Figure 30). This interesting finding has excited many numerical simulations, such as those based on two-dimensional vortex sheet model for the free surface (e.g., Tryggvason, 1989; Yu and Tryggvason, 1990) and Navier-Stokes solver (Ohring and Lugt, 1991; Lugt and Ohring, 19921, as well as full three-dimensional Navier-Stokes simulation (Dommermuth, 1993). These computations enable us to outline the physics relevant to the vorticity creation in this interacting process. Initially, the interaction of the rising vortex pair with S is apparently a two-dimensional inviscid process and can be mimicked by taking the pair as point vortices and S as a weak boundary vortex sheet. As the Kelvin

220

J. Z. Wu and J. M. Wu striations

c7

/

scars

FIG.30. Schematic of striations and scars (excerpted from Sarpkaya and Suthon, 1991).

oval is formed, the surface tangent vorticity 5 increases to O(1) as indicated by (5.101, or equivalently, the sheet strength y is of O ( 6 ) , 6 = R e - f . Note that the variation of y already contains the vorticity creation process as seen from (3.33) and (3.34). Between the vortex pair and S, the flow can still be irrotational. Then, at a finite Re, new vorticity produced from S will eventually be sent into the fluid; and at a certain stage of the early interaction, the vortex sheet needs to be refined as a free-surface boundary layer. By (5.211, then, the vorticity flux is dominated by the boundary layer correction of surface acceleration; that is, uat= O(6). In two dimensions (5.21a) reduces to

where u: can be solved from (5.18) and (5.19), provided that the elevation of S and its velocity induced by the primary vortices have been known from inviscid calculation. Qualitatively, a, concentrates in the local region of high curvature, where separation may happen at a sufficiently large Froude number (about 0.5 and larger), so that a pair of secondary vortices of opposite sign is formed below S and toward the end of this stage the boundary layer approximation is no longer applicable. This newly produced secondary vortex pair is responsible to the observed scars and possible rebounding of the primary vortices. The preceding two-dimensional picture cannot explain the observed striations, which are related to the vortex instability along the axis. In a I

Vorticity Dynamics on Boundaries

221

three-dimensional Navier-Stokes simulation, Dommermuth (1993) introduced an initial disturbance of the location and vorticity distribution of the primary vortices to observe the effect of instability. It was found that, as a vortex tube interacts itself and its neighbors, sheets of helical vorticity are spiraled off. Due to shortwave inviscid instability, these sheets manifest themselves as braids of cross-axis vorticity, a structure independent of the presence of S . But, as they rotate around and translate with the primary vortices, some braids will approach S and their open ends become normal to S to form the observed whirls as the outer boundary of the scars. This complicated three-dimensional interaction, however, seems not to be accompanied by a strong viscous dynamic process of vorticity creation, although the surface vorticity keeps changing. The main event occurring on S is the formation of normal vortices due to the turning pattern of Figure 4(b), which is essentially a kinematic process.”

2. The Effect of a Suifactunt on Vorticity Creation from a Free Suface In reality, a clean interface can rarely happen. Even a slight surface contamination may significantly alter the interfacial vorticity distribution and hence the surface motion as well as the vorticity-creation rate. The effect of an oil film on calming the interfacial wave has been known for long time. In a broad sense, this calming can be viewed as an early example of intefacial uorticity-creation control, and perhaps introducing proper contamination could be a major means of such control in the future. The appearance of a surface-active material, or a sufactant, will reduce the local surface tension T from its equilibrium value T o , say, and a concentration of surfactant will thereby cause a tangent gradient of T , which in turn drives a motion of both areal and volumetric fluids (the Marangoni effect 1. In addition, the surfactant may have various rheological and chemical properties that can cause additional interactions with the bulk fluids (e.g., Edwards et al., 1991). All these will affect the interfacial vorticity dynamics. Ideally, if the surfactant is also a Newtonian fluid, so that a water-oil-air system, say, forms a sandwich structure with two interfaces, then the clean interface theory developed thus far can still be applied to each interface. However, in many cases, surfactants are non-Newtonian or even not fluids, 21

Viscosity and dynamics would enter if the air vortices were to be studied as well, as illustrated in the previous subsection.

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J. 2. Wu and J. M. Wu

and their interface with bulk fluids may not be immiscible. On the other hand, it often suffices to take the surfactant as a surface fluid with negligible thickness, density, and bulk motion. As a preliminary discussion, therefore, we confine ourselves to the simplest model. First, we assume the surfactant is Newtonian, of which the motion is governed by a twodimensional analogy of the Navier-Stokes equation (2.15) on a curved surface (Scriven, 1960). This includes introducing a surface shearing viscosity ps and a surface dilatational viscosity A,, the latter being nonzero even if the bulk surfactant is incompressible.22 Second, we assume that the surfactant density is well negligible compared with that of bulk fluids and so is its body force (inertial and external). Consequently, the surfactant simply moves with the interface velocity U. Moreover, the force acted by the surfactant on bulk fluids, say, f , = nf,, f,,, can be only a surface force and hence directly balanced by the surface stress of the bulk fluids. This leads to an extension of the classic interfacial stress condition (2.11):

+

n * [t]

=

T,K + f,,l,

n x [t]

=

n x f,,

(5.33a, b)

Here, f,, and f,, are given by (Scriven, 1960; Edwards et al., 1991) f

,, =

f,,

where

=

- K(AA

+ p,)v,.U

-

2psk:V,U

+ p.,)V,(V,.U) - P , ( ~ K U , + n x V,{~K.v,u,,) -V,T

-

(A,

-

K= -nXKXn=K-KI,,

I

K E -K:K

(5.34a, b)

(5.34c)

with I, = I - nn being the two-dimensional unit tensor on S. For twodimensional flows K = KI, so that K = 0 and K = 0. Note that f, is derived from the two-dimensional analogy of (2.15) in which the divergence of stress tensor, rather than the surface stress itself, is involved. Thus, f, contains derivatives one order higher than those in t. Although (5.34a) implies a modification of the elevation of S, the tangent-vorticity jump [ p.51 across S is modified to

Lp.51= - n x =

([t,,] - f,,)

[ &lo - n X V,T

-(,is

+ p,)n

+ p,V,l

x v,(v,.u)

-

2,411 x (KU,

+ K.v,u,)

(5.35)

"This situation is similar to the two-dimensional divergence of vortex sheet strength; see

(3.37).

FIG. 15. Horn vortices in the wake of a jet in crossflow (from the water tunnel viwalization of J. M. Wu c’t d.. 1988).

FIG.24. Experimental result of a wave behind a hydrofoil (from Lin and Rockwell, 1995). The wavelength is about 8.8 cm. The figure shows vorticity contours (upper) and velocity vectors (lower). Red and yellow denote counterclockwise and clockwise vorticities, respectively. (Reprinted with the permission of Cambridge Univer\ity Press.)

FIG.25. DNS result of a 5-cni wave (from Domrncrniuth and Mui, 1995). The vertical scale is stretched to highlight the boundary layer structure. At thi\ particular time tlie maximum wave \lope is 1.07: (a) vorticity contours (the scale is saturated because tlie w in the trough.; are so extreme): ( b ) rotational part of the velocity field. which highlights the newsurface jet that mixes the subcurface flow (a clockwise vortex is clearly seen in the crest of wave where separation occurs): (c) total velocity field (vortical plus potential) viewed in a moving fiame of reference to illustrate the qualitative agreement with experiments (Figure 24). (Reproduced from Dominerwutli and Mui. 1995. with permission from ASME.)

FIG.32. The experimental result of velocity components (a), total pressure (b), and static pressure (c) on a near-wake plane downstream a 76" swept delta wing at a = 20". The wing tip is at y/c = 0.25. Two velocity components in the wake plane are indicated by small arrows, and the chordwise velocity by color contours. M = 0.05, Re = 5 x lo5. The wake plane is normal to the central chord (from Visser and Wdshburn, unpublished).

FIG.33. The vorticity components iii body axe5 a s computed from Figure 32 (from Wu, Ondrusek, and Wu, 1996). In the main vortex core region wv and 0): are roughly antisymmetric with respect to the horizontal and vertical lines thorough the vortex center, rehpectively. Thus, they both I-educe lo zero at the center and then have a negative peak (as one moves upwei-tl and inboard. respectively). which is not clearly shown in the plots.

Vorticity Dynamics on Boundaries

223

where [ pgIOis the value on corresponding clean interface, given by (3.lb). In particular, on a two-dimensional free surface with arc element ds, the dimensionless form of (5.35) reduces to

E = lo+ ReW-

dT* dS

+

(5.36)

where to is given by (5.10), W is the Weber number defined by (5.lb) for equilibrium surface tension T o ,T * = T / T , , and Bo = (As + p s ) / (p L ) is the Boussinesq number, a new parameter. Therefore, the surfactant influences the interfacial tangent vorticity by (1) the gradient of surface tension and (2) the viscous resistance to the strain rate of the surfactant. The first effect is quite strong at a large Re if d T * / d s = 0(1), but not as strong as R e itself; because ReW = T(,/( p U * ) is inversely proportional to the reference momentum. Consequently, a contaminated inteqace may locally behave somewhat in between a clean inteflace and a rigid wall, as confirmed by recent experimental and numerical studies (for a brief review see, e.g., Tsai and Yue, 1995). On the other hand, compared with (3.lb) or (5.10), the additional viscous resistance also tends to increase boundary vorticity. Although usually Bo = 0(1), on a wavy surface the appearance of higher order derivatives may result in a significant local boundary vorticity as well. Interestingly, this resistance includes the gradient of normal vorticity 5, which is exactly of the same form as that in (3.19a) or (3.20a) but now affects [ pg] instead of u. We stress that, under the preceding assumptions, no volumetric force is exerted to the bulk fluids by the surfactant; thus, the force balance that leads to the net-u formulas (3.20a, b) is unaffected. However, the specific level and distribution of u will be indirectly affected by the surfactant, too, mainly via the change of surface acceleration. We return to the interaction of rising vortex pair and free surface, but now let the surface S be contaminated. The experimental measurement and numerical computation with flat free surface by Hirsa et al. (1990; see also Hirsa and Willmarth, 1994; Tryggvason et al., 1992) showed that the presence of surfactant greatly strengthens the formation of secondary vortices and rebounding of primary vortices, so that the contaminated free surface is indeed more like a solid wall. This finding was further confirmed and extended by Tsai and Yue (19951, who made a two-dimensional viscous simulation on the effect of soluble and insoluble surfactant on the

224

J. Z. Wu and J. M. Wu

interaction process. In this study, not only the surface surfactant was introduced as stress conditions (5.33), but also the bulk surfactant with variable concentration and its transport, not reviewed previously, was considered. It was found that the interaction between the surfactant and underlying vortex flows forms a close loop. The primary vortex pair induces gradients of surfactant concentration that leads to Marangoni stresses, strong surface vorticity, a boundary layer, and even separation. These in turn significantly alter the underlying vortex flows. For example, when ReW = O(10) and Bo = 0 0 1 , at a mild Froude number, Fr = 0.15, a free-surface vorticity 6 of O(1) may occur due to two comparable effects in (5.361, ReW-

dT* 6JS

and

Bo-

d’U, ds

*

Owing to the practical importance of surfactant effect on ocean waves and ship wakes, research along this line will certainly be further pursued.

VI. Total Force and Moment Acted on Closed Boundaries by Created Vorticity Fields

So far we have considered various theoretical and applied aspects of vorticity creation from boundaries. This creation process can be viewed as an action of the boundary to the flow field, which is one aspect of the boundary vorticity dynamics. The other aspect is the reaction of created vorticity to the boundary. Locally, the action and reaction have been fully reflected by the stress balance (2.11), where t is a reflection of compressing process, shearing process, and surface deformation process. However, if our concern is only the total force F and moment L acted on a closed boundary, the reaction aspect can be greatly simplified as evidenced by a series of vorticity based formulas of great interest in both fundamental theory of vortex dynamics and applications. For later convenience, we first list the original formulas for F and L. In these formulas, 7 is the total material volume of the fluid surrounding a body with boundary 9, of which the unit normal n points into the body, and I$ is a control volume bounded by 9 from inside and by a control surface Z from outside. The moment is taken about the origin of the coordinates. The flow is assumed incompressible, and the gravitational

Vorticity Dynamics on Boundaries

225

+

pgz. Then, in terms force is absorbed into pressure by denoting p - p of the momentum change or inertial force of the material fluid body, we have

D F = --/ppudV= Dt c y D x x p u d ~ - / x x p a d ~ L = -Dt Y

ly

(6.la) (6.lb)

which require knowing the whole flow field and is referred to as the global uiew. Or, in terms of control volume and control surface (only force formula will be considered here for simplicity), F=

--1 pudV+ dt v/ d

/Z(t - p u u . n ) d S

(6.2)

where t is given by (2.6a). For steady flow over a stationary body this requires knowing only the flow on an arbitrary wake plane, say, Y, and is referred to as the near view. When Ymoves sufficiently far downstream (a Trefftz plane), we return to the global view. Finally, in terms of the surface if 9 = dB is rigid, by (2.22) we have stress on 9,

Thus, as noted in Section 11, only the compressing and shearing processes are involved in the rigid-body force and moment analysis. Similarly, if 9 is deformable, say, a flexible solid surface, the surface of an air bubble in water or a water drop in air, we only need to add (2.20) to the moment formula; thus

Equations (6.3) and (6.3') require knowing the stress status on the body surface and are referred to as the close uiew. All these formulas are

226

J. 2. Wu and J. M. Wu

obtained in the framework of primary variables, although the vorticity has entered some of them. A basic observation of (6.11H6.3’) is that the inherent coupling of the fundamental processes, so important in boundary vorticity dynamics, has no reflection at all. In this sense, these formulas are not physically optimized and most appealing. J. Z. Wu and Wu (1993) systematically showed that, for general viscous compressible flow, the total force and moment acted on a closed rigid boundary by the flow field can be attributed exclusively to that by the vorticity field created from the boundary. This result has been extended by J. Z. Wu (1995) to closed fluid interface. Physically, this type of vorticity-based formulas are made possible by the viscous coupling between compressing and shearing processes via the no-slip condition on the boundary, which always enables one to express the force due to the former in terms of the latter even if the former is dominating (e.g., in a supersonic flow). In contrast, J. Z. Wu and Wu (1993) proved that the scalar compressing process alone does not own this nice ability, and hence it is impossible to obtain a complete set of dilatation-based force and moment formulas. In this section, the general vorticity-based force and moment theory are discussed in the order from (6.11, to (6.2), to (6.3), just like following an observer who moves closer and closer to the body from a remote distance. Thus, the observer will get a global view first, then a near view, and finally a close view. The application of the resulting vorticity based formulas will be illustrated by a practical problem, the aerodynamic diagnostics and optimization.

A. THE VORTICITV MOMENTAND KUTTA-JOUKOWSIUFORMULA We start from revisiting the earliest total-force formula in terms of vorticity w or circulation r: the Kutta-Joukowski formula F

=

U x re,

(6.4)

and its three-dimensional counterpart, where e y is the unit vector along the vortex axis. Traditionally, this formula is obtained by using strictly inviscid and irrotational flow models, in which for having a lateral force, a singularity (a point vortex in two-dimensional case) has to be artificially introduced inside the body (Joukowski’s derivation is cited in Batchelor, 1967, pp. 404-406). Although this result was a significant and ingenious

Vorticify Dynamics on Boundaries

227

achievement in early theoretical aerodynamics, today its inviscid derivation is easily misleading and can by no means reveal the physical source of the circulation, which exists only in viscous interaction. To obtain a physically consistent understanding, therefore, we rederive this type of formulas based on viscous consideration. In so doing we shall clarify the conditions for (6.4) to hold. We then make some general remarks on this type of vorticity based formulas. 1. Kutta-Joukowski Formula: Material Volume Derivation It is well known that the total momentum of an incompressible fluid body can be cast to the integral of the vorticity moment x x (I) by integration by parts, via the vector identity (d

-

1)f

=

x x ( V x f)

-

V ( X . f)

+ V . (xf)

in d-dimensional space for any f (Lamb, 1932; Batchelor, 1967). Therefore, by (6.la), we immediately obtain the first general incompressible force formula exclusively in terms of vorticity:

P

F =

d

-

D x x ( I ) ~ V -P D x x (n X d d S (6.5) 1 Dt -(y d-1DEk

+

Lighthill (1979) mentioned that (6.5) has been extensively used in estimating the hydrodynamic loading of offshore structures due to its ability of connecting vorticity and force directly. A systematic exposition of (6.5) and the corresponding moment formula for viscous flow was given by J. C. Wu (1981). Note that this formula gives the force on a fluid bubble or drop 7 as well, as long as the distribution of vorticity in 77 and velocity on 9 are known. Now, assume the viscous fluid is unbounded from outside, with a uniform oncoming velocity U = U e , , and the body surface dB is stationary. Then, the second term of (6.5) vanishes due to a no-slip condition. We consider the Euler limit of the first integral. In this case the boundary layer over dB reduces to a boundary vortex sheet, as discussed in Section III.C, which rolls up as leaving the body and becomes a pair of concentrated trailing vortices. Along with the starting vortex, we have a closed vortex loop, which in the Euler limit does not diffuse and can be represented by a closed vortex filament as viewed by a remote observer. In this case the integrated vorticity moment can be simply expressed as twice the vectorial area S spanned by the loop times its circulation (Batchelor, 1967);

228

J. 2. Wu and J. M. Wu

the direction of dS and the vorticity in the loop are defined according to the right-handed rule. Thus,

F=

DS

-pr-Dt

(6.6)

Moreover, if we assume that the wake vortex pair are straight, with constant separation b, and advected downstream by U everywhere, then the increase of S is due simply to the elongation of the trailing vortices as the starting vortex moves downstream with a rate U. Therefore, eq. (6.6) reduces to the simplest lift formula for a finite-span wing: (6.7)

F, = p b U r

which has been known since the time of Lanchester and Prandtl. Here we used = instead of equality because the trailing vortices have a downward induction, which makes the vortex loop nonplanar and is the source of induced drag. The quantitative determination of downwash depends on the circulation distribution along the span and is not our concern here. However, if we assume that b -+ x such that the downwash approaches zero, and that in any cross-section (x, z ) the flow is the same, then the force per unit span exactly recovers (6.4). 2. Kutta-Joukowski Formula: Control-Volume Denvation

Alternatively, we may obtain (6.6) and (6.7) from (6.2) in a way more closely parallel to the classic approach. Assume C is far away from the body (still the global view), so that we may set u = U + u’ with /u’I2 negligible on C. Note that C must exclude the starting vortices, which is even farther as t + 00, so that the flow inside C can be considered steady. The only deviation from the classic approach, where U‘ = Vcp and is singular inside the body, is that we now consider a uiscous, rotational but regular perturbation u’, with o = V X u‘. The viscous force is still negligible on C; hence, ox

u+v

(

U.U’

3

+-

=

O(IU’I*> on

c

Then, for a two-dimensional steady flow, it is known that in the Euler limit no vortex sheet is shed off into the wake, and in three dimensions, we may again assume the wake vortices are approximately along the direction of U.

229

Vorticity Dynamics on Boundaries

Consequently, in both cases w X U = 0 and the Bernoulli integral for steady potential flows still holds on X. This casts (6.2) to

F

=

-

p / {(U u’)n - (U . n)u’ - U(n . u’)}dS

c

5+

by the Gauss theorem and incompressibility, where V = V, is the fluid volume plus solid volume. Because V, is stationary, we finally obtain

F=pUX/wdV=pUX

v,

L

(n.o)xdS

(6.8)

where the second equality of (2.20) was used. We stress that in deriving (6.8) the no-slip condition must be imposed as well, for otherwise a solid sphere, say, can have arbitrary rotation and a nonzero o in V, without affecting the fluid motion. Then, in three dimensions, let x = xe, ye,, we obtain

+

which reduces to (6.7) if wx is concentrated at two points on Yseparated by b. On the other hand, in two dimensions, n * w occurs at the side boundaries of unit separation located at y = k 1/2. Thus, if x = xe, + ye, is the position vector of the centroid of the closed boundary vortex sheet surrounding the body, eq. (6.4) is recovered. Clearly, the conditions for (6.4), (6.6) and (6.9) to hold are (1) the Euler limit of incompressibleviscous flow; and (2) the starting vortex is sufficiently far from the body such that the flow in C is asymptotically steady. We digress to mention that, partially due to the misleading of the inviscid derivation of (6.41, some authors believed that this formula can be used to compute the force on a fluid vortex with circulation r. True, a side force will trivially appear if a vortex can be held by an external force; but it is always incorrect to apply (6.4) to the interaction of any free vortex and cross flow, because its velocity can in no way differ from that of the background flow, which must have included the velocity induced by the vortex. The only nontrivial case is that of a vortex held at t = 0 and then released. Again due to the viscosity and no-slip condition, the sudden appearance of a cross flow at t = 0 will create a vortex sheet “shell”

J. Z. Wu and J. M. Wu

surrounding the “bare” vortex, such that at t = 0’ the sheet induces a velocity inside the bare vortex different from the approaching velocity and therefore hold it in place for 0 < t << 1. However, as Caruthers et al. (1992) showed, in the Euler limit and at t = O+, the total force acted on such an enveloped Rankine vortex is exactly half of the value given by (6.4). This result is also true for a solid rotating cylinder of the same density if it is released at t = 0, as can be inferred from Ting and Klein (1991, Section 2.1). As time goes on, the vortex sheet is rapidly advected downstream of the remaining distorted core with an ever-decreasing influence upon the core motion. The vanishing of the sheet is associated with an acceleration of the core to match the background velocity, so that finally the side force reduces from its maximum value of pU I 7 2 to zero. Therefore, the Kutta-Joukowski formula can n e r w be applied to a free vortex, euen if it is initially bound. This revisit of the so-called vortexcylinder analogy is also helpful for understanding the viscous background of (6.4). 3. General Characters of Vorticity Based Force and Moment Formulas Inspecting the derivation of the preceding results and their structure, some remarks of general significance for the following development can be made. First, comparing (6.5) and (6.la) shows that replacing u by w in the force formula must introduce a position vector x. This is easily understood from the dimensional difference of velocity and vorticity. But the force must be independent of the choice of the origin of x; so there should be a compatibility condition to ensure this physical fact (J. Z. Wu and Wu, 1993). Let I be an integral operator (volume, surface, or line integral or their linear combination), xo be the origin of x, .B any tensor, and 0 any admissible tensor product. Then the general form of the compatibility condition reads I{(x + x , ) o F }

=

Z{xoF}

if and only if

Z{F}

=

0

(6.10)

For example, if we remove the operation x X from the integrand of (6.51, the result is obviously an identity. Similarly, taking off vector x from (6.8) simply tells the solenoidal feature of vorticity. This type of compatibility conditions holds for all vorticity based integral formulas that follow. Second, after integration by parts the new integrand no longer equals the original one; only the integrated result remains the same, as is evident from (6.la) and (6.5). In fact, this difference in integrand gives vorticity

Vorticity Dynuniics on Boundaries

231

based force and moment formulas new but clear physical meaning and significant practical value. The most remarkable point is that the regions in ’27, V,, or on 7with ct) # 0 are much narrower than that with u’ # 0, allowing one to focus on highly localized key regions in force and moment analysis. In addition, a few more features of these derivations are worth mentioning, as they also common to the formulas that follow: 1. Equations (6.4)-(6.9) contain various proofs of the D’Alembert paradox; 2. Unlike the inviscid derivation, where a “phantom vortex” has to be put inside the body, now the vorticity in ‘Y or Vf is continuously created from the body surface due to the no-slip condition; 3. What really counts in F (and L as well) is only the net circulation, part of vorticity with opposite directions having been automatically canceled during integration; and 4. In this type of vorticity based formula, the pressure force is absent, but its effect has been automatically included due to its coupling with vorticity, as explained earlier. B. TOTALFORCE AND CONVECTIVE VORTICITY FLUX O N A WAKEPLANE Equation (6.9), applied as 7 is far downstream, is the prototype of a series of subsequent studies that used vorticity based wake plane data to infer the information of the force status. These include, among the others, Betz (19251, Maskell (19721, J. C. Wu et a/. (19791, Hackett and Wu (19821, Hackett and Sugavanam (19841, Onorato et af. (1984), Yates and Donaldson (1986), Chometon and Laurent (1990), and Brune (1994). The interest in this type of analysis increased recently due to the significant progress of experimental techniques, which enables quantitative survey of flow field over one or multi-wake plane(s) very close to the downstream end of the body, or even cutting across the body. Thus, the wake plane analysis has become a powerful tool for relevant flow diagnostics and optimization. However, most of existing theoretical analyses are confined to leading-order approximation, unable to count thc detailed wake flow structure as one approaches the near wake. The following results (J. Z. Wu et a/., 1987a; J. Z. Wu and Wu, 1989b), again based on recasting (6.2) via an integration by parts, give an accurate version of (6.9) and can be used as the rational

232

J. Z. Wu and J. M. Wu

basis for making any desired approximations regarding the force components acting on the body. 1. Lift and Drag Constituents from the Near- Wake Plane Suruey

We take a uniform flow U over a finite-span wing as illustration. The theory equally applies to analyzing other moving bodies, like automobiles, trains, and ships, as long as the flow can be assumed steady. Let the lateral size of the control surface 2 be sufficiently large, so that except its arbitrary downstream plane 9 ( w e call it the near-wake plane) the flow is undisturbed, see Figure 31. Assume that the flow in the control volume I/ is steady. This implies that the starting vortex is far downstream of F a n d its motion has no effect on the near field. As the mathematic basis of this and next subsections, we introduce a general tensor formula of integration by parts. Let P be any second-rank tensor, x be the position vector, and dS = ndS be a surface element. Then, because in the component form

we have identity

x x (n x V ) . P = -(n x v ) . ( P x

XI'+

~

- - nntrace P

IL-1701

FIG.31. The control volume and near-wake plane ZJ. Z. Wu and Wu, 1989. (Reprinted with permission from SAE paper no. 892346 0 1989 Society of Automotive Engineers, Inc.)

233

Vorticity Dynamics on Boundaries

Therefore, from the generalized Stokes theorem (2.9), for any surface S we find X

(n X V ) . P d S = / ( P e n S

-

ntrace P)dS

-

k

dx.(P x

XIT

(6.11)

Then, following the notation of (3.121, we decompose any vector f into normal and tangent directions, so that formally f = ncp n X A = n . ( P I - S). In (6.10, we set P = cpI and S in turn. This gives a pair of surface integral identities:

+

n X A dS

= -

L

x x ((n x 0 ) x A} dS

+ $ x x (A x dx), dS

d

=

3 only (6.12a, b)

We now use (6.12) to transform (6.2). Assume that the integrand of the surface integral of (6.2) has been decomposed as cpn + n X A,, so that we may apply (6.12) with vanishing line integrals. Then, on the right-hand side cp and A, are under differential operators, so only the near-wake plane 7with n = e, needs to be considered. Denote u = ( u , u , w ) and w = ( w, , m y , wz), one finds

where 0, is the tangent gradient on Zand terms that can be cast to line integrals along the boundary of 9 ( w h e r e the flow is undisturbed) have no contribution. Hence, cp causes only a streamwise force and A, causes only a lateral force. Moreover, the viscous effect in has been accumulated before the fluid arrives at Z which manifests on 7as the variation of stagnation pressure F o = @ + pluI2/2. This effect is much stronger than the surface integral of the viscous term in (2.6a), so only the inviscid stress -pn needs to be retained in (6.2). Thus, cp = @

+ pu2,

A,

=

pu(u X e x )

=

pu(we,

-

ue,)

234

J. Z. Wu and J. M. Wu

Therefore, by using (6.13), for the lift and side forces in three-dimensional flow,

and, for the drag in d dimensions,

Now the physical causes of the total force on the wing can be easily identified. For example, for a typical steady attached wing flow with oncoming velocity and root-chord length as reference scales,

Thus, if in (6.14a) u = U + u' with u ' = O ( E ) ,the leading effect is precisely (6.9). But, for a steady separated flow, on a near-wake plane, there can be u' = O(1) and (6.9) is unreliable. Nevertheless, the lift is dominated by the spanwise moment of the adcective streamwise-vorticity flux through a near-wakeplane. Then, the O ( c 2 )term, - p y ( u w , + wwz) ,represents a change of lift due to downwash or upwash and sidewash. Finally, the last term in (6.14a) is O ( e 3 ) ,which reflects a modification due to the curvature of near-wake vortices; it is in general necessary to ensure that the computed F, is independent of the location 5 The side force Fy has exactly the same structure and naturally vanishes if the flow is symmetric with respect to the ( x , 2) plane. Similar identification can be made for the drag F,. But, unlike the lift, all constituents are of the same order of magnitude. The first term of (6.14~1, 1

.

(6.15a)

is the pressure drag, which exists as long as f , is not uniformly distributed on X Then, the second term of (6.14~)can be re-expressed as (zwy -yw,)

-

-(x.u,) dX

u,

=

i'eL+ w e ,

235

Vorticity Dynamics on Boundaries

Here, because the viscosity on Y has been ignored, the steady Euler equation V& = p(u X w) applies, yielding

Therefore, we identify a viscous drag m

due to the loss of kinetic energy in boundary layer and wake, which is always associated with the vorticity creation; and an induced drag

yw) dS for d

=

3

only

(6.15~)

caused by the lifting vortex associated with downwash, upwash, and sidewash. The remaining term, 2 FXC",

=

~

d - l b

d

pu-(x. dx

u,) dS

(6.15d)

corresponds to the third term of (6.14a, b) and represents a near-field curvature modification. For example, the two-dimensional viscous drag is directly measured by the vertical moment of advective vorticity flux, 2puzo,, through a near-wake plane, which is equivalent to the well-known wake velocity defect. Note that, through the Bernoulli equation, the pressure drag (6.15a) can also be expressed as the integral of p x * V,lu12/2 over where the tangent variation of the kinetic energy is entirely induced by the wake vortices via the Biot-Savart law. Thus, in three dimensions, even a perfect streamlined wing will experience a pressure drag caused by low pressure in the wake vortex cores, not counted in the induced drag. In fact, all FX,), F,,,,, and Fxind, and even F,,,,, have their roots at the wake vorticity. Compared with those works cited in the beginning of this subsection, this approach has some unique advantages: (1) It is accurate and can cover compressible flows (J. Z. Wu and Wu, 1989b; J. M. Wu et al., 1996); ( 2 ) the plane 7can be shifted upstream and cuts the wing, yielding the force constituents on any desired front portion of the wing to further assess the

236

J. 2. Wu and J. M. Wu

configuration design; and (3) each constituent comes from a physical mechanism, closely related to vorticity and vortex dynamics. Note that the appearance of the x-derivatives of u and w in (6.14a, b) (through wy and w z )and (6.15d) does not necessarily mean that one needs data on at least hvo adjacent near-wake planes to compute the forces, since by using the Euler equation they can be reexpressed in terms of quantities on a single % Alternatively, if the ( u , u, w >data on two adjacent planes are available (e.g., by three-dimensional particle image velocimetry that is being developed), then V T p and V w p o can be expressed in terms of these velocity components and their ( x , y , z ) derivatives, so that without pressure measurement the forces can also be computed.

2. Wake Plane Diagnostics of a Delta Wing As an illustration of the aerodynamic force diagnostics based on the wake plane survey data treated by (6.14) and (6.151, we mention a recent analysis of the experimental data on near-wake planes of a slender delta wing (J. M. Wu et al., 1996). The experimental survey was performed by K. D. Visser and A. E. Washburn (unpublished) at NASA Langley Research Center, with a = 20", M = 0.05, and Re = 5 x lo5. The swept angle was x = 76". The data include velocity components and static and total pressure on a near-wake plane perpendicular to the central chord (using the body axes), as shown in Figure 32 (see color plates). The irregular shape of the total pressure contour around the main vortex indicates some error (the maximum error is about 9%). The data were fitted by spline technique, from which the corresponding vorticity components were computed as shown in Figure 33 (see color plates). The various constituents of the integrand in (6.14a), now the normal force, and that of the chordwise force, (6.15a-d), are plotted in Figure 34, which clearly reveal the key contributors to the normal and chordwise forces and their sources. The survey was made mainly on the upper part of the wing flow, but the effect of model support can be partly seen near the centerline y = 0, which was not excluded in data processing, though this can be easily done if so desired. Another wake-plane data set was used in J. M. Wu et al. (1996), which gives a better result. The normal force is clearly dominated by the spanwise moment of the chordwise vorticity advective flux, the first term of (6.14a). Two peaks of this term appear in Figure 34(a). The higher peak is due to the main leading edge vortex, where the local axial velocity is about 1.8 times of the

VorticifyDynamics on Boundaries a

237

018 0 16

0 14 0 12 01

F

008

E

006

m0

0 04 0 02 0

-0 02 0 04

b

0

0 05

01

0 15

0.05

0.1

0.15

02

0 25

03

0 35

0.2

0.25

0.3

0.35

Y/C

005 0 04 0 03 0 02

0 01

F m

P E

O -001

-0.02 -0.03 -0.04

-0.05 -0.06 0

Y/C

FIG.34. The constituents of the integrand of (6.14a) and (6.15a-d) for normal and axial forces, (a) FN and (b) F A , respectively, and (c) the resulting integrand of lift and drag (from J. M. Wu et al., 1996).

238

J. Z. Wu and J. M. Wu

C

oncoming velocity and hence yields a favorable contribution. This also indicates that beyond small angles of attack applying the classic formula (6.9) to a near-wake plane is indeed inaccurate. The second peak occurs near the edge, a contribution of the feeding vortex layer as it just leaves the edge and before it rolls into the main vortex (behaving as a tip vortex), where the axial velocity is, however, only about half of the undisturbed value [Figures 32(a) and 33(a) (see color plates)]. Then, in the second term of (6.14a), the sidewash and downwash of t h e main vortex causes a downward normal force below its core, but an upwash outside the tip vortex is a favorable contribution (note that the tip vortex has larger y ) . After transforming to wind axes, the total lift coefficient is 0.69, lower than the experimental value 0.744 (no calibration was made in the wake-plane survey). The interference effect from the model support is automatically minimized without additional treatment, because its y = 0. Compared to the normal force F N , the constituents of axial force F A , Figure 34(b), are more complicated. One of the major contributors is the lateral motion of main vortex, which changes sign as crossing the vortex

VorticityDynamics on Boundaries

239

core from inboard to outboard, the latter being larger and beneficial. The same happens for the tip vortex. Note that to a certain extent the contribution of main vortex to the FA is canceled by the static pressure constituent, of which the physics is the balance between centrifugal force and pressure force. Then, a viscous force is manifested by the moment of the tangent gradient of p,, , which has both positive and negative values but the overall effect is a drag. The effect of (6.15d) is not negligible. Adding these together yields the total integrand of (6.15). After transforming the forces to wind axes (Figure 34(c)), we found that the total drag coefficient is 0.333, very close to the experimental value 0.34. The drag is mainly from the tip region, even though the high peak in the FN integrand, due to the main vortex, has a strong streamwise projection-it is largely canceled by the streamwise projection of the beneficial contribution to FA of the same vortex. Therefore, the main vortex has much higher aerodynamic efficiency than the tip vortex. This is a clear indication that the wing-tip shape needs to be modified for higher aerodynamic efficiency. In fact, the main purpose of wake plane survey is precisely discovering what spanwise portion of the wing should be improved. Note that improving the chordwise configuration can be achieved by wake plane analysis as well, provided that the flow is also surveyed on many cutting-in planes. But, in general and particularly for unsteady flows, one has to move to the close view as follows here.

C. FORCEAND MOMENTIN TERMS OF BOUNDARY VORTICITY FLUX We have transferred (6.1) and (6.2) to obtain the vorticity based force formulas on a solid or fluid body, viewed by far-field and near-field observers, respectively. In these results we see that F (L is similar) is related to a continuous shedding of the vorticity from the body. This vorticity can come only from a continuous creation from the body surface 9. This observation brings us back to the boundary vorticity dynamics and suggests that one must be able to express the total force and moment in terms of the boundary vorticity fluxes on 9. Such is achieved by transferring the close-view formulas, (6.3a, b) and (6.3'a, b). The result was first obtained by J. Z. Wu (1987) for incompressible flow over a rigid body, which has been extended to compressible flow (steady or unsteady) and deformable solid body or closed fluid interface by J. Z. Wu et al. (1987a, b),

240

J. 2. Wu and J. M. Wu

J. Z. Wu and Wu (19931, and J. Z . Wu (1995). Here we confine ourselves to incompressible flow.

1. Integrated Moments of Boundary Vorticity Fluxes For recasting (6.3a) and (6.3'a), we invoke the identities (6.12a, b) again. Consider a closed boundary .D in a homogeneous fluid first so that the line integrals are absent. Comparing the left side of (6.12a, b) with (6.3a) and (6.3a') and the right side with (3.17) and (3.15), respectively, it immediately follows that, in d dimensions, j?lnpdS

Lpt

X

n dS

P

x

X

a,;dS,

=

--

=

p / x x u,,,dS,

d-ljhn

A?

d

d =

=

3 only

2,3

(6.16a. b)

Here, as in (4.1) and (3.251, a h = n X Vh is the vorticity flux due to modified enthalpy (including the gravity effect); and u,,, is the explicit viscous part of vorticity flux, given by the last term of (3.17) and (3.25b) for rigid and deformable boundaries, respectively. Therefore, for threewe find an elegant force dimensional flow over either type of closed 9, formula (6.17) Similarly, for the total moment on a rigid body, J. Z. Wu and Wu (1993) showed that the corresponding three-dimensional formula is (6.18) Then, on a deformable body, except for the contribution of a nonuniform normal vorticity 5 , say, uc = v(n x V ) x 5, the remaining part of u is the same as that on a nonrotating rigid surface. But J. Z. Wu (1995) showed that

which is just the last term of (6.3'b). Consequently, eq. (6.18) can also be applied to both rigid and deformable bodies.

Vorticin,Dynamics on Boundaries

24 1

Equations (6.17) and (6.18) are the central result of this section. They reveal that the total force and moment acted on a closed boundary can be solely expressed by the proper moment of i1orticityJiuxesdue to su$ace stresses, which are inherently of viscous origin. In other words, F and L are directly related to the rate of work required to create new vorticity from the wall, that is, that for raising and turning t h e near-boundary vortex lines into the J. Z. Wu and Wu (1993) and J. Z. Wu (1995) took the Stokes flow over a rigid and fluid sphere as a simple example to show how the pressure force on the sphere can be clearly interpreted in terms of up,which is essentially a viscous process, but was explained misleadingly as “the viscous component of the normal stress” (Illingworth, 1963) that should have been referred to ( A 2pM in (2.5). Note that the flux due to acceleration does not appear; as in (6.3), it has been implicitly reflected by the stress status through the Navier-Stokes equation. As noted by J. Z. Wu and Wu (1993), a part of the stress, although exerting a force to the local surface clement, may have no contribution to the total force and moment if it does not send vorticity into the flow. This includes, for example, the hydrostatic pressure and the pressure associated with a potential flow. Even somc viscous stress may have no contribution to the total friction force; this situation occurs on a rigid boundary dB, as seen from (4.1c,d), if on a portion of dB, the boundary vorticity is two-dimensionally divergence free or along its direction dB has no curvature. In this sense (6.17) and (6.18) can be said theoretically irreducible: All local stresses that could cancel each other during integration are automatically excluded. Thus, if only the total force and moment status are to be studied, one can well focus on those local regions where there is a strong boundary vorticity flux-a point of great practical value to be exemplified later. We may also gain a deeper physical understanding of various vorticity based formulas like (6.61, (6.81, (6.14), and (6.15): They all “converge” to (6.17) and (6.18) and become more and more accurate and general. That is, moving toward 9, an observer will pass (6.6) or (6.8) first for the Euler limit of steady flow, then (6.14) and (6.15) for viscous steady flow, and finally, tracing the root of the vorticity or circulation appearing in those formulas, the observer will see the boundary vorticity fluxes and find the most general result (6.17) and (6.18). In short, we arrive at an inherent

+

23

This fact also provides another insightful physical interpretation for the d’Alembert paradox.

242

J. 2. Wu and J. M. Wu

unification of the action and reaction phases between a closed boundary and the vorticity field created thereon, which closes the fundamental theory of boundary vorticity dynamics. 2. Diagnostics and Optimization of an Airfoil

As a simple illustration and complement to the wake plane diagnostics, consider an attached flow over a two-dimensional airfoil in the (x,z ) plane to which (6.16a) can be applied to compute the pressure force. As remarked before, the origin of a Cartesian coordinate system (x,y, z >can be arbitrarily chosen without changing the result of (6.16a); so for convenience let it be so located that for x < 0 there is a favorable pressure gradient, and for x > 0, an adverse gradient, see Figure 35. By (6.16a), the contribution of a unit-length are element ds as x to the lift and pressure drag is given by

where a;, = e y . ur,= - d p / d s . We stress again that, due to integration by parts, these dF, and dF, do not represent the local lift and drag elements at all, but their integrations do give correct Fpz and q,,r. Now, on the upper surface of the airfoil, because up is positive when x < 0 and negative when x > 0, both portions give rise to a lift. The situation is opposite on the lower surface. The regions near the leading and trailing edges (in particular leading edge) are critical, because there we have both a large 1x1 and a sharp pressure gradient. Thus, to increase lift, the airfoil should be shaped so that, on the upper surface, the high-l up/ regions are located as close to the edges as possible, and on the lower surface, they are as close to the origin as possible, provided that the flow remains attached and the off-design performance is not quickly deterio-

FIG.35. The sign and location of vorticity flux due to pressure on airfoils: (a) a traditional airfoil, (b) a Piercey-Whitcomb type of airfoil with higher lift/drag ratio at low speed.

Vorticity Dynamics on Boundaries

243

rated. On the other hand, to reduce drag, \?,I should be minimized near the maximum thickness point. Remarkably, this simple observation leads to a configuration very like the low-drag airfoil previously proposed by Piercey in 1930s, which evolved to the supercritical airfoil of Whitcomb at a higher speed, with a wide flat midportion on upper surface (Figure 30(b)). The preceding discussion can be quantitizcd by computing the Euler limit of an attached flow (J. Z. Wu et al., 1996). A comparison between a symmetric NACA-0012 airfoil and a lift-optimized (and hence cambered) airfoil of the same maximum thickness is given in Figures 36 and 37 regarding the distribution of pressure and the normal force integrand (6.19a). The Euler limit was obtained by solving the full potential equation, because then imposing the no-slip condition immediately gives the desired u,, and its x moment. The figures indicate clearly that the moment -xu,, is indeed highly localized in the leading- and trailing-edge regions. This is even more so for the optimized airfoil, which by (6.194 explains why it has a higher lift. Note that the optimized airfoil is similar to that of Figure 36(b). Figure 37(b) shows t h e normal force benefit of the optimized airfoil at small angles of attack. The normal force of an airfoil with the same camber distribution as the optimized one but with the thickness distribution of NACA-0012 is also shown (as is well known, changing thickness distribution affects the lift only very slightly). The viscosity of a finite Re flow may cause an early separation if the trailing-edge angle is larger than a critical value, which should be imposed as a constraint that cannot be made by an inviscid code. As indicated by the previous example, the unique advantage of (6.17) and (6.18) over the common formulas (6.3a, b) is that the new formulas automatically focus one’s attention to highly localized key regions in diagnostics and optimization. Further exploration along this line would be of great interest, in particular for the fluid-dynamic optimization of threedimensional configurations.

3. Force on a Body Piercing an Interface The previous formulas can be further generalized to the case of a closed boundary 9 piercing an interface S between fluids 1 and 2. For simplicity we derive only the total force formula, of which the proper basis is obviously the full form of (6.12a, b). Let 9, and be immersed in fluids 1 and 2, respectively, with 9 = .3,+
244 a

J. 2. Wu and J. M. Wu 04

Optimized airfoil NACA0012 - - - ~

03

0.2

01

0

-0 1

-0 2

-0.3

-0 4

b

0

02 I

04

06

I

1

08 I

Optimized airfoil NACA0012 - ~ - ~

-3

-2

-1

0

1

0

0.2

0.4

0.6

0.8

1

FIG. 36. The pressure distribution over a symmetric NACA-0012 airfoil and a liftoptimized and cambered airfoil at M = 0.2 and a = 6". computed by a full potential code. (from J. Z . Wu et al., 1996).

245

Vorticity Llynamics on Boundaries a

150

Upper surface, Optimized airfoil Lpper surface. Optimized airfoil - - - Upper surface, NACAOOI 2 Lower surlace, NACAOOI 2 100

50

-0 5

b

-0.3

-0.4

-0.2

-0.1

0

0.1

03

0.2

0.4

0.5

14

Optimized airfoil -+ Optimized camber. NACA0012 thickness t NACA0012 0 1.2

1

0.8

z

0

0.6

0.4

0.2

0

I

I

1

1

1

1

1

ALPHA (degrees)

FIG. 37. The distribution o f -xu,, on thc two airfoils of Figure 31 and corresponding normal force coefficients at M = 0.2 and small angles of attack. Also shown are the normal force Coefficients of an airfoil with the optimized camber distribution but the NACA-0012 thickness distribution (from J . Z. Wu et d., 1996).

246

J. Z. Wu and J. M. Wu

directions as viewed from two sides of the line. As the intersection of 9 and S, this boundary line is a contacl fine. The total force follows from the procedure leading to (6.19) but retaining the line integrals of (6.12a, b). We denote the contributions to F of the surface and line integrals by F, and F,, respectively. Then,

where [PI and [ p s ] are the jump of stresses given by (3.1a, b). The force F/ can no longer be attributed to boundary vorticity fluxes. Rather, if e is the unit vector along the contact line and dx = eds, then because n . e = 0, by (3.lb), [ pg] X dx = n(e . [t,,]) d ~ ; therefore, ’~ we finally have

Note that U, K, and K measure the motion and shape of interface, of which only the values right of the contact line enter (6.20b). Therefore, F/ strongly depends on the contact line condition, which itself is a subject of much research and will not be pursued here. We remark only that on the contact line some quantities might become singular; but if this does not happen or the singularity is not sufficiently strong, then the line integral (6.20b) is obviously negligible compared with the surface integral (6.20a). Further extension of (6.20) to multi-interface flow over 9 is straightforward. A typical application of (6.20) with great practical interest is the hydrodynamic force of a surface ship, of which a significant part is the resistance to the creation of ship waves. If F, is negligible, it immediately follows that the wave force can come from only the water part of (6.20a). Therefore, associated with the ship-war:e creation there must be a corticity creation. This is another manifestation of the coupling between compressing and shearing processes, of which the mechanism is still the no-slip condition for viscous flow. The vorticity created thereby does not follow the wave 24

If the interface is contaminated, additional terms as in (5.35) should be included.

Vorticity mnamics on Boundaries

247

propagation but is advected downstream and forms a portion of wake vortices, which in turn affect the wave pattern and are the whole source of ship-wake turbulence. Theoretically, the ship wave has been studied under the potential flow assumption, in which the inherent coupling of ship wave and vorticity creation is cut off. However, as in the previous subsection, the coupling can be easily recovered by taking the potential solution as the Euler limit and turning on the no-slip condition. Then, the wave resistance must equal its associated drag component of the water part of (6.20a1, from which the total vorticity flux moment due to the wave can be estimated. It should be stressed that the vorticity source associated with wave creation, although still coming from the no-slip condition, is different from other well-known sources such as those due to flow separation from the hull or a rotating propeller. In fact, the wave resistance and associated u can be clearly identified by its highly nonlinear dependence on the Froude number (e.g., Wehausen, 1973; Newman, 1977). This situation is closely analogous to a supersonic flow over a body surface with shock wave. Although the inviscid theory can well predict the wave pattern and associated wave drag, as the viscosity is turned on the foot of a shock wave on the body surface immediately becomes a singular source of vorticity, because across the shock the pressure gradient is a delta function. For a finite Re, the shock-created vorticity is responsive to the interaction of shock wave and boundary layer. Then, from the compressible counterpart of (6.19) (J. Z . Wu and Wu, 19931, one can equivalently predict the wave drag but in terms of the vorticity creation rate. Further exploration of this interesting aspect of wave-vortex coupling would also be highly desirable.

MI. Application to Vorticity Based Numerical Methods From the discussions in previous sections it is clear that, except for some simple analytical or semianalytical solutions, any further quantitative applications of the theory of boundary vorticity dynamics have to rely on experiment or computation, because the boundary values of those key quantities such as the vorticity and its flux can be only the outcome of the solution. This situation naturally calls for proper numerical methods. Although any existing Navier-Stokes codes may serve the purpose, those vorticity based schemes that have inherent consistency with the theory are

J. Z. Wu and J. M. Wu most favorable. Inversely, the theory also sheds new light onto the resolution of some difficult problems of these methods and hence contributes to their development. In particular, the toughest difficulty is how to impose a proper boundary condition for vorticity on a solid wall, of which extensive studies have been conducted in the past two decades (see Gresho, 1991, 1992; Koumoutsakos et al., 1994; Koumoutsakos and Leonard, 1995; Sarpkaya, 1994). Here, boundary vorticity dynamics can make very unique contribution, which leads to an almost complete resolution of the difficulty. This is the topic to be reviewed here, basically following the recent work of J. Z. Wu et al. (1994b). We shall confine ourselves to the case of an incompressible flow with unit density over a rigid boundary dB, which is simplest and relatively mature. Even in this case, extra complexity will appear if dB has a normal acceleration, as can be felt from the discussion at the beginning of Section 1V.B. Thus, to make the key points stand out, we further assume that the wall has at most a motion along its tangent direction. A. AN ANATOMY OF VORTICITY BASEDMETHODS 1. Kinematic iiersus Dynamic Vorticity Condition on a Rigid Wall Vorticity based methods solve the vorticity equation, say (3.7), under the uelocity adherence condition (2.10) on d B . For an incompressible flow, such a formulation avoids resolving pressure by keeping the velocity solution divergence free. However, in implementing these methods one has to infer a boundary condition for w from (2.101, either by applying the Biot-Savart law to the solid surface or through a projection theorem (Gresho, 1991). This implies that in its strict nature the vorticity boundary condition is of global type. That is, eq. (2.10) imposes a kinematic integul constraint to the possible w distribution, rather than a local condition for the boundary behavior of w. Such an integral constraint cannot be easily included in a local algorithm (finite difference or finite element). Alternatively, this constraint can be represented as the differential relation between u and w , either by the Poisson equation or a Cauchy-Riemann type of equation. Then w and u must be solved together, which results in a larger coupled system. To avoid this basic difficulty, some well-known fractional-step approaches were proposed, where the near-wall w field is simplified as a

Vorticity Dynatnics on Boundaries

249

vortex sheet y such that the volume integral constraint is reduced to a boundary integral equation or even a local condition for y. These include, among the others, the integral formulation of J. C. Wu and coworkers (e.g., Wang and Wu, 1986, and references cited therein) and the vortex methods of Chorin (1978). A further analysis was recently made by Koumoutsakos et al. (19941, who noted that the boundary vortex-sheet strength y can be manipulated so that a Dirichlet or Neumann type of condition could be modeled. They pointed out that the Neumann type is better suited for vortex methods using the particle strength exchange (PSE) scheme. The error of all these approximations, however, had never been closely analyzed. We stress that any approximation of the near-wall c1) field by a vortex sheet could deteriorate the accuracy of computed boundary vorticity and its flux, which as we saw are of crucial importance in the entire boundary vorticity dynamics. This difficulty reflects a basic fact that neither the local boundary vorticity nor its normal gradient can be rigorously inferred from any kinematic constraint derived from (2. 10). Thus, kinematically, it is impossible to obtain a strict Dirichlet or Neumann condition for the Llorticity equation. Moreover, the vorticity equation is one order higher than the Navier-Stokes equation; hence an additional compatibility condition is necessary to exclude possible spurious solutions due to raising the equation’s order. But again, this condition is not derivable within kinematics. Essentially, a thorough resolution of this boundary conditioning problem lies in the following fundamental observation stated by J. Z. Wu et al. (1990). The Navier-Stokes equation, with primitive variables (u,p ) as unknowns, naturally matches the velocity adherence condition (2.10) (yet special consideration is necessary for the pressure boundary condition). In contrast, the vorticity equation, as a one-order higher equivalence of the Navier-Stokes equation, does not. This mismatch is the root of entire difficulty. It is then clear that the natural boundary condition for (I) should also be one order higher than (2.10); which is nothing but the acceleration adherence (2.131, which in turn leads to the dynamic Neumann conditions (3.17) and (3.16) for second-order vorticity and pressure equations.2s 25Althoughsuch a “derivative argument” is well known in other contexts, Anderson (1989) first applied it to the vorticity boundary conditioning problem. See also Anderson and Reider (I 994).

250

J. Z. W u and J. M. Wu

Symbolically, the incompressible Navier-Stokes and continuity equations can be written as %u)

+ Vp = 0

and

V .u

=

0

(7.la, b)

Then the vorticity and pressure equations, along with their respective dynamic Neumann conditions, are

V

X ~ ( U = )

0,

n

X ~ ( U ) B=

-n x Vp,

(7.2a, b)

where suffix B means values on dB. Note that (7.2b) and (733) are just (3.17) and (3.16), respectively. Suppose the proper initial condition u = uo at t = 0 is always satisfied. Then, guided by the above fundamental observation, J. Z. Wu et al. (1994b) proved the following theorem:

THEOREM 1. Assume u satisfies (7.2a) and ( 7 . 3 ~ )Then . it is the solution of (7.la), ij'either (7.2b) or (7.3b) holds. This theorem can be restated as

COROLLARY 1. Let u be a solution of problem (7.2a, b ) or (7.3a, b). Then it is also a solution of (7.la), if and only if it satisfies (7.3~1,or (7.2a), simultaneously. We have seen the equivalence of (2.10) and (2.13). Now the theorem proves that the compatibility condition mentioned earlier is simultaneously satisfied by the dynamic conditions. Thus, the Neumann problem for a coupled ( w , p ) field is well posed. Note that, once the u field has been solved by whatever method, V p can be inferred from (7.la); but this involves an inconvenient computation of d u/ d t . Solving the second-order pressure equation (7.3a, b) avoids this problem. However, the second form of the theorem shows that this approach needs to satisfy the vorticity equation as a prerequisite. Similarly, solving the vorticity equation (7.2a, b) also requires the second-order pressure equation. The pressure part of this problem has been extensively studied by Gresho and Sani (1987); but the ( w , p ) coupling on dB, and hence an interior ( c o , p ) coupling over the whole flow domain V , should not be ignored or oversimplified. In fact, this

VorticityDynamics on Boundaries

25 1

coupling is precisely the dynamic manifestation of the global kinematic constraint on the o field, and the basic difficulty of vorticity boundary condition would still exist if the effect of this coupling was not further clarified and bypassed. Fortunately, in contrast to the kinematic constraint, the order of magnitude of boundary (w, p ) coupling can be clearly identified and becomes negligible at high Reynolds numbers. This basic estimate leads to a very simple decoupled approximation of the dynamic Neumann condition, which will be addressed next. The most significant advantage of dynamic Neumann condition is that, because it amounts to computing (r (including up and u,,,),an accurate computation of the skin friction 7, on dB becomes possible. For ordinary schemes in terms of primary variables (u,p), as well as some vorticity based methods with kinematic boundary conditions, this is not an easy task because T,, has to be obtained by one-sided differencing along the normal, but the normal derivatives of u and w on dB are extremely large at high Reynolds numbers, O(Re i) and O(Re), respectively. In contrast, because (rl) and cr, are both computed from much milder tangent derivatives of p and T , ,with the knowledge of interior vorticity field and its slope u on ilB, one can use centered diference to obtain a much better estimate of w on dB, and hence 7 , . As will be exemplified in Section VII.B, this unique advantage derived from boundary vorticity dynamics is of great value in practical applications. The use of dynamic condition for vorticity does not eliminate the need for kinematic conditions in solving velocity from vorticity and ensuring the divergence-free property of computed vorticity field. On the contrary, Theorem 1 also sheds new light on the optimal form of such kinematic conditions. Although this chapter focuses on dynamics, this kinematic problem is worth examining briefly. For an incompressible flow with given vorticity w in domain V , the velocity is determined by V x u = w , or more conveniently, by the Poisson equation V2u = - V x w , along with (7.lb) and the normal condition (2.10a). The solution should ensures

V.w=0

in

V

( 7.4)

But, because the scalar condition (2.10) is insufficient for the vector u, additional condition is required that should simultaneously guarantee (7.lb) and (7.4) both in the interior of V and on the boundary d V. Inspired by Theorem 1, the optimal kinematic conditions was found by X. H. Wu

J. 2. Wu and J. M. Wu

252

et al. (1995), who proved the following theorem as a kinematic counterpart of Theorem 1:

THEOREM 2. For any gioen smooth ilector field o,there is a unique solution u of the problem V2u= - V x o

in

(7.5a)

with n x (V

x u)

=

n x o,

n*u

=

n . b on

dB

(7.5b,c)

that satisfies (7.1b) and V X u = cod, where adis an orthogonalprojection of o onto the divergence-free space. Note that (7.5b) is exactly the kinematic counterpart of (7.2b) or (3.17). The significance of this theorem will be mentioned in Section VII.B.2. 2. Solution Structure of the Coupled ( w , p ) Field

To analyze the order of magnitude of the (o, p ) coupling on dB, we first need to understand how this coupling affects the vorticity field in the flow domain V . This can be done by examining the structure of the ( o , p ) solutions without really solving them numerically. Thus, we recast the (o, p ) equations into integral representation by using Green’s functions. Here and below we take p = 1. Let r = x - y, r = IrI, and

be the fundamental solution of the heat equation in free space, with 2 being the step function and d the spatial dimensionality. Then, the integral form of incompressible vorticity equation follows from Green’s second identity:

- c d ~ $ G*n x L, dS dB

+ /:d.r/vVG* x

L dV

Here the subscript 0 implies values at T = 0. and u is given by (3.17). The first volume integral represents the effect of initial o distribution, and the

253

Vorticity Dynamics on Boundaries

last two, the nonlinear advection effect. As shown in Section II.D, if in an idealized model the flow is started by an impulsive pressure gradient, the boundary vorticity flux u must contain a 6 function at t = 0 and creates an infinitely thin vortex sheet with strength y o , which can be easily separated from its regular part:

c

dr

/

dB

G*u dS

=

[?,G; y o d S

+ /‘ d r / Of

dB

G*u dS

(7.7)

The initial velocity no-slip condition has been implied through (7.7), without which y o would be uncertain and has to be prescribed in advance. Now, a key step is to transfer this integral form of w(x, t ) by using (3.17) and the Stokes theorem (2.9). This finally yields

+cdr/VG* V

X

where

t,

=

(7.8)

LdV

np,

+nX

VW,

(7.9)

is the wall stress. Similarly, let

be the fundamental solution of the Laplace equation in free space. Then, as the counterpart of (7.81, for stagnation enthalpy H = h + lu12/2 (or stagnation pressure, as we have taken p = 11, H(x)

=

-/T,nVG.t,dS-/VG.LdV V

(7.11)

Equations (7.8) and (7.11) are a consequence of boundary vorticity dynamics, which have three remarkable features. First, although only the free-space Green’s functions (7.7) and (7.10) are used, the normal derivatives of w and p are absent in the surface integral due to the application of (3.16) and (3.17); that is, the Navier-Stokes equation on dB. Thus, we

254

J. 2. Wu and J. M. Wu

arrive at an innovative integral formulation of a Dirichlet problem with free-space Green’s function. However, as before, this problem is for the coupled (w, p ) . Second, because by (7.7) and (7.10), VG* and VG are along the direction of r = x - y, the boundary stress t, at a point y, E B affects the w field at a point x E V only by its components perpendicular to r, through diffusion; similarly, t, at y, affects the H field at x only by its component along r.26 Third, applying (7.8) to dB implies a global and implicit Dirichlet condition for w in V . Unlike the boundary vorticity flux a,there exists no local equation for w, . This is a general mathematical manifestation of the underlying physics stressed several times before: u rather than w R reflects the vorticity source directly, and w B itself arises through a space-time accumulated effect of u as well as advection and diffusion. The same is true for stagnation enthalpy. We mention that (7.8) and (7.11) are also valuable in various integral formulations of vorticity based methods. These methods are usually more time consuming than finite difference; but, owing to the desire of removing the less accurate random-walk approach from Chorin’s (1978) methods, some authors have recently turned to deterministic diffusion algorithms that are mostly of the integral type, because then a smooth connection between the Lagrangian advection substep and the Eulerian diffusion substep can be achieved (in contrast, to connect a Lagrangian convection and a finite difference diffusion one has to use techniques like vortex in cell, at the expense of artificial viscosity) (see, e.g., Degond and Mas-Gallic, 1989; Cottet, 1990; PCpin, 1990; Lu and Ross, 1991; Koumoutsakos and Leonard 1995; Koumoutsakos et al., 1994). We believe that (7.8) and (7.11) provide a common theoretical foundation to all grid-free deterministic diffusion schemes. 3. The Vorticity-PressureDecoupled Approxima tion

An inspection of (7.8) and (7.11) indicates once again that the boundary ( w , p ) coupling, through the appearance of t, in both equations, is a viscous effect. It does not enter the advection process represented by the volume integrals of the Lamb vector L. Thus, in numerical computations, using fractional-step (or operator splitting) methods can at least greatly 26This is closely similar to the transverse-longitudinal decomposition of shearing and compressing processes made in the wave number space (Section 1I.C).

255

VorticifyDynamics on Boundaries

reduce the ( o , p ) coupling. That is, one solves the Euler and Stokes equations

+ u , . d u , + vpl = 0,

dU1 ~

dt

du2 -

dt

+ Vp2

=

v V 2 u 2 (7.12a,b)

successively for each time step. Here, scalars p1 and p 2 are necessary to guarantee (7.lb). Symbolically, the solution of the Navier-Stokes equation (7.la) can then be written as

where uo is the initial velocity for a time step, and At),9 ( t ) , and E ( t ) stand for the solution operators for the Navier-Stokes, Stokes, and Euler equations, respectively. Ying and coworkers (e.g., Ying, 1987; Zheng and Huang, 1992; Zhang, 1993; Ying and Zhang, 1994) and Beale and Greengard (1992) proved that this scheme is convergent and it is first-order accurate in time (higher order schemes can be designed; e.g., Strang splitting scheme). Corresponding to (7.12a, b), we have

-+ v x L , dm1

dt

=

0,

L,

=

(7.13a)

w 1 x u,

(7.13b)

vV20, = 0

as the vorticity advection and diffusion equations. Indeed, o1and p1 are fully decoupled in (7.13a) because only (2.10a) is required on dB. In this substep, there is no vorticity boundary condition at all. Here, Lagrangian vortex methods exhibit most of their strength. The coupling between o 2 and p 2 persists in (7.13b) due to (2.10b), which, however, implies a much simpler linearized version of coupled equations (7.8) and (7.11): w,(x, t )

=

/

dB

Ggy,, dS

+ /GXw2,) dV + /‘ d7/ v

0’

dB

VG*

X t,,

dS (7.14)

Here, we have assumed that d B is smooth and the integral in (7.15) should be understood as taking the Cauchy principal value. The velocity does not appear in the coupling; it can be solved separately after o is obtained.

256

J. Z. Wu and J. M. Wu

Our purpose is to obtain a Neumann condition for the vorticity. So we estimate the boundary vorticity flux a after one time step At + 0. First, after solving the substep of inviscid convection (7.13a), a slip velocity us appears on dB, which implies a singular vortex sheet with strength y = n X (b - u,). This vortex sheet is a result of the driving force V p , = O(1) and serves as the initial condition of (7.13b). On the other hand, applying (7.12b) to dB yields du2B

u = n x - dt

+

u 2 p

+

U2"l\

where u2, = n X V p 2 Band uzvl, = v(n x V > x w Z Aare the fluxes due to p I R and respectively. Note that p 2 is the Stokes pressure that, unlike p , , has a viscous origin (Section VI.C.l) and should be classified as an explicit viscous part of a,the same as u2?vlc. Integrating this equation from t = 0 to t = A t , say, and requiring that the resulting u must ensure a no-slip condition at t = A t , there is

cr =

Y

-

L At-

+ a,,)+ crzvis

(7.16)

at each x B E dB. Here, the overline means time-averaged values; the first term is the singular part of u caused by advection, and the rest is the regular part caused by pure diffusion that contains the boundary ( w , p ) coupling. and y / A t are now easily The orders of magnitude of a,,, iT2v15, identified. Because ug should satisfy the no-slip condition, lyl = O ( A t ) and hence IyI/At = O(1). On the other hand, by (7.15) and the property , have p Z B= O(vlwAI).For a high of the linear operator acting on p Z B we Reynolds number attached boundary layer lwRl = O(vf), hence lG2pl

-

IG2vlal

-

"llV~GBll

'v

(7.17)

This estimate may have a local change in the neighborhood of separation lines or sharp edges, but because in (7.14) p Z Raffects w through a surface integral, the Re- 4 dependence should be a correct overall estimate. Therefore, for sufficiently high Re, the error due to dropping the last two terms of (7.16) is of O(Re-f), much smaller than the first term. This leads to the decoupled approximation W E - -

Y At

(7.18)

Vorticity Dynamics on Boundaries

257

having been used by several authors based on some plausible arguments (e.g., Hung and Kinney, 1988; P&pin, 1990; Koumoutsakos and Leonard, 1995; Koumoutsakos et al., 1994). Numerical tests have shown that (7.18) is much better than the corresponding local decoupled approximations of the Dirichlet condition, of which one commonly proposed form is

. is where h is a chosen small normal diffusive distance, say d z ~ This not surprising, because no local approximation for o Bcan be rational. Although in the preceding the decoupled approximation was obtained in an operator splitting scheme, a similar approximation can be derived without splitting. X. H. Wu et al. (1994) used a second-order Runge-Kutta method to integrate the vorticity equation in time, where the effect of p H , and hence C T , is updated without really solving it, but is approximated by means of the residual slip velocity u, according to (7.18). It then turned out that at the end of each time step the approximation produces a u, of O ( A t 2 ) (the actual value is much smaller than A t 2 ) . Nevertheless, if necessary, the global (0,p ) coupling can be easily recovered by a simple iteration procedure, which may effectively eliminate the residual slip velocity that appears due to dropping the O(Re-t)-order regular part of (7.16). The analyses of this subsection constitute some basic building blocks in constructing more accurate and efficient schemes. Two typical numerical tests applying these analyses follow. B. NUMERICAL EXAMPLES 1. Impulsively Started Flow over a Circular Cylinder A well-known benchmark test for various numerical schemes is the two-dimensional flow over an impulsively started circular cylinder. Except for the availability of careful experimental results at a Reynolds number up to 9500 (Bouard and Coutaneau, 19801, very accurate numerical data, usually based on vorticity-stream function (w-i,!~) formulation, have also been produced as a standard of comparison. Ta Phuoc LOCand Bouard (1985) used a fourth-order scheme to resolve the Poisson equation for II, and a second-order scheme for w . Anderson and Reider’s (1994) finitedifference scheme is of fourth order in both space and time, which uses

258

J. Z. Wu and J. M. Wu

(2.10) as boundary condition and was performed on a supercomputer with a uniform grid of 2048 x 256 = 0.5 X l o 6 and time step A t = 0.00033. More recently, Koumoutsakos and Leonard (1995) used a grid-free deterministic vortex method to reach about the same accuracy as Anderson and Reider, also performed on a supercomputer. In this scheme the dynamic Neumann condition for vorticity was employed, the time step for Re = 9500 was 0.01, and the particle number increases as time, up to 2 X 10' at t = 3 and more than lo6 at t = 6. All these computations retained the global nature of vorticity boundary condition. J. Z. Wu et al. (1994b) computed the same flow on a workstation by using the w-fi formulation and a finite-difference scheme, of only second order in space and first order splitting in time, with a grid of 301 X 256 (stretched in the radial direction so that it was much denser adjacent to the wall) and time step 0.0025 for Re = 9500. The ( w , p > decoupled approximation of dynamic Neumann condition, eq. (7.18), was imposed. Figure 38 shows the computed flow field using the first order splitting and no iteration, compared with experiment (Bouard and Coutaneau, 1980) and the computation by Ta Phuoc LOCand Bouard (1985). However, this comparison is not critical; many schemes can produce the same flow patterns but not all of them are able to predict the boundary vorticity or skin friction accurately. Figure 39 shows a comparison of computed boundary vorticity w B by J. Z. Wu et al. (1994b) and Anderson and Reider (1994). The violent oscillations of w Roccur due to separated vortices. It is remarkable that the low-order differencing and a simple use of (7.18) can already catch the w B distribution very well, and the small residual difference can be effectively eliminated by only a couple of iterations. Figure 40 shows the computed drag coefficient C, of the cylinder at Re = 1000 by different methods. In this case, the flow does not separate, and hence C, is dominated by skin friction T , ~ Note . that one-sided difference methods cannot predict the asymptotic steady C , satisfactorily, and the third-order method is no better than the second-order one. This test convincingly confirms the ability of boundary vorticity dynamics in improving the skin friction prediction. To check the estimate (7.17) numerically, J. Z. Wu et al. (1994b) carried out a group of tests on the dependence of the residual slip, and hence u Z pon , Re. They define the reduction factor of the slip by

Vorticity L?yrzamics on Boundaries

259

FIG. 38. The flow patterns of an impulsively started flow over a circular cylinder at Re = 9500. Top: The fractional-step scheme with (7.18) as dynamic boundary condition, J. Z. Wu et a / . (1994b). Middle: Flow visualization of Bouard and Coutaneau (1980). Bottom: Fourth-order simulation of Ta Phuoc Loc and Bouard (1985). (Reproduced from “Dynamic vorticity conditions: Theoretical analysis and numerical implementation,” J. Z. Wu et ul., Copyright 0 1994 (Inter. J . Numericul Mefhods in Fluids). Reprinted by permission of John Wiley & Sons, Ltd.)

150

-

-50

-

-100

-

3

fourth order no iteration iteration

----

-150 -

;

.7nn _”” 0

20

40

60

80

Angle

100

120

140

160

lourth order no iteration

--- I

180

-

t

?

._ ._

8

0.

5 ;-loo

.

0

m

-200 ’ -300 400 -500

I

I

I

I

I

Angle

FIG. 39. The boundary vorticity of impulsively started Row over a circular cylinder at R e = 9500. Comparison between fourth-order result (Anderson and Reider, 1994) and second-order (J. 2. Wu et al., 1994b) computation with and without iteration. (Reproduced from “Dynamic vorticity conditions: Theoretical analysis and numerical implementation,” J. Z. Wu et al., Copyright 0 1994 (Inter. J . Numerical Meihods in Fluids). Reprinted by permission of John Wiley & Sons, Ltd.)

26 1

VorticityDynumics on Boundaries I

I

I

I

I

1.4

1.2 1 c

c

.g

E

0.8

0)

$ 0.6 ol

2

0.4

2nd-order one-side 3rd-order one-side

0.2 0

-0.2

0

1.5

1

0.5

Time

2

2.5

3

FIG.40. Comparison of computed transient drag coefficient for flow over an impulsively started circular cylinder at Re = 1000 (from X. H. Wu, 1994).

where u : ’ ; and u:+l are the slip after the convection and diffusion substeps, respectively. Evidently fR also represents the ratio between p , and p 2 , or the inverse of the strength of boundary ( w , p ) coupling. To determine the variation of fR with Re, the index of fR was introduced as

I,

=

In ( fR /fR ,) ln(Re,/Re, 1

TABLE 1 THEREYNOLDS NUMBER DEPENDENCE OF THE ( 0 ,p ) COUPLING STRENGTH AND THE ERROR or: DECOUPLED APPROXIMATION. RESULTS AREOBTAINED WITH At = 0.01 A1 t = 1; j K IS THE REDUCTION FACTOR OF THE AVERAGED SLIP, zR IS ITSINDEX, E AND 1, ARE THE L 2 - N 0 RELATIVE ~~ ERROROF BOUNDARY VORTICITY AND ITS INDEX, RESPECTIVELY. RE 10 100 1000 10000

1.552 4.226 12.94 37.49

0.359 0.475 0.497 0.508

5.093e-3 1.992e-3 7.104e-4 2.659~-4

0.398

- 0.423 - 0.459 - 0.466

(Reproduced from “Dynamic vorticity conditions: Theoretical analysis and numerical Numerical Methods in FluidsJ implementation,” J. Z. Wu et al., Copyright 01994 (Inter. .I. Reprinted by permission of John Wiley & Sons, Ltd.)

262

J. 2. Wu and J. M. Wu

According to (7.17), I , should be about 0.5. Table 1 clearly confirms this especially as R e becomes higher. The table also presents the L,-norm relative error in oB of the decoupled scheme compared to the fully coupled one using iteration. The index of this error, I,, was defined similar to I , and also had a similar trend. Another indication of the strength of (0,p ) coupling is the number of iterations required to achieve convergence in u , . As expected, more iterations were needed for lower R e in the tests. A test with R e = 9500 showed that (7.17) is valid for smoothly separated flow. As the flow becomes more complicated, the estimate becomes invalid and the error of decoupling grows larger. Amazingly, even at Re = 10, about the lowest Reynolds number of most practical interest, the decoupling error might be considered as still reasonably small. 2. Three-Dimensional Lid-Driven Cavity Flow

In three dimensions, extending the dynamic Neumann condition for vorticity is straightforward. The key issue turns to ensuring a divergenceless incompressible (o,u) field. To this problem Theorem 2 provides a theoretical basis. First, owing to the theorem, there is no need to project o prior to solving u; in fact, the projection can be done at no extra cost by taking the curl of u numerically. Second, due to the orthogonality of the projection, the projected od is the optimum approximation of w in the least square sense. Third, because u satisfies (7.lb), in a d-dimensional space one need only solve d - 1 Poisson equations out of (7.5a) for u, inferring the other component from (7. lb). This enables solving velocity 50% and 30% faster than common o-u methods for d = 2 and 3 , respectively. Note that for d = 2 this saving makes o-u methods identical to the vorticity-stream function methods. Based on Theorem 2, as well as a careful treatment of discretization to preserve the kinematic properties of relevant variables inherent in their differential and continuous counterparts, X. H. Wu et al. (1995) arrived at very efficient finite-difference o-u schemes for two- or three-dimensional flows. The schemes were tested by computing the lid-driven flow in a cavity of unit cube. For the case of steady flow at Re = 1000, the result was in good agreement with that by pseudo-spectral computation. For more complicated unsteady flow at a higher Reynolds number, the result with different schemes are compared in Figure 41. It is seen that the nondivergence-free scheme, Figure 41(b), is not acceptable due to the large error of V . o,although it did not bring appreciable difference at lower Reynolds numbers.

263

Vorticity Dynamics on Boundaries

b

a

C

FIG.41. Three-dimensional unsteady lid-driven cavity flow at Re = 3200, T = 51.2: velocity field projected on the symmetry ( x - y ) plane. In the figure, different schemes for computing fluxes due to convection and stretching, respectively, are compared: QUICK and centered difference, denoted Q-C, which is not divergence free; QUICK and QUICK, denoted Q-Q, which is divergence frcc in the interior of the domain but not on the boundary; and Q-C-P, denoting the application of projection to Q-C. (a) Q-C-P, (b) Q-C, (c) Q-Q (from X. H. Wu et al., 1995).

This section did not intend to review various aspects of vorticity based methods. Rather, we have focused on topics relevant to boundary vorticity dynamics and used the examples to show evidence of its significance. In particular, the robustness of the low-order scheme of J. Z . Wu et al. (1994b) indicates that the key issue in vorticity based methods is indeed the proper boundary conditioning, whereas the fast convergence of itera-

264

J. Z. Wu and J. M . Wu

tion confirms the weakness of boundary ( q p ) coupling at high Re. To both of these, the theory of boundary vorticity dynamics made contribution. We may also mention that, because the prediction or measurement of T , at high Re is very often a troublesome engineering task, the excellent ability of computing T , by vorticity based methods with proper boundary conditions has great potential application. For example, based on given data of a pressure distribution over a complicated surface, obtained by either experiment or some numerical scheme (even an Euler code), one could continue the computation with this method and a little additional effort to obtain the best estimate of 7, within the error range of the pressure data.

VIII. Concluding Remarks We have discussed various aspects of the boundary vorticity dynamics, from fundamental theory to applications, from solid boundary to interface, and from the vorticity creation on a boundary to its reaction to the boundary. The main conclusions can be summarized as follows. 1. The boundary vorticity dynamics focuses on the vorticity creation from, and its reaction to, various fluid boundaries. The theory is derived by applying the tangent surface stress balance and force balance (tangent components of the Navier-Stokes equation) to a rigid or deformable boundary 9, along with the no-slip condition. The former gives the whereas the latter gives the creation tangent vorticity jump [ p.53 across 9, rate of vorticity, or boundary vorticity flux u = v d o / d n , from each side In any case, the boundary vorticity is always a space-time accumuof 9. lated effect of the boundary vorticity flux, along with that of advection and diffusion inside the flow domain. Therefore, the boundary vorticity flux u is the primary mechanism responsible to the creation of new vorticity from 9. 2. The specific dominating mechanisms of vorticity creation on 9 depend on the coupling situation of shearing process with other surfaceforce driven dynamic processes. Therefore, the splitting and coupling of these dynamic processes on 3 are the key physical basis for understanding the vorticity creation and its reaction. For a Newtonian fluid, the surface force drives three dynamic processes: shearing, compressing, and surface deformation. In general, these processes appear in both stress balance and force balance and are coupled via viscosity and adherence condition. In

Vorticily Dynarriics on Boundaries

265

the force balance one also needs to include the effect of external and inertial body forces. A unified theory of boundary vorticity dynamics is thereby developed for any kind of immiscible material boundaries. 3. On a rigid wall, dB, the surface-deformation process is absent and the coupling occurs between the shearing and compressing processes. The local stress balance on dB gives nothing but the ( p , 7 , ) distribution on d B that has to be obtained by solving the entire flow field. Similarly, the corresponding force balance shows that the vorticity creation is dominated by the pressure gradient and tangent wall acceleration, both being of O(1). The explicit viscous effect of O(Re- i ) exists only in three dimensions, which concentrates on highly local regions of dB, where there is a large surface curvature or strongly rotational 7, field. 4. In contrast, a free-surface S always adjusts its motion and shape, which to the leading order is governed by the inviscid normal-stress balance (pressure and surface tension) without coupling with shearing process. Consequently, unlike the solid wall, where the vorticity is of O(Re+),the tangent vorticity on S is only of O(1) and solely balanced by the tangent components of surface-deformation stress t ~,which in turn is dominated by the inviscidly determined velocity and curvature of S. The normal vorticity is free from the stress-balance condition; rather, it may come from the intersection of the free surface and an external vortex or from the kinematic turning mechanism of internal vorticity to the normal direction. Hence, at low Froude numbers the normal vorticity could be the dominating vortex structure near S. The boundary vorticity flux u, then, appears only in the viscous correction to the inviscid motion and hence is of O(Re- i), dominated by the viscous correction of surface acceleration. The coupling between shearing and compressing processes remains merely at a level of O(Re-’). Other types of boundaries, including flexible solid wall, fluid-fluid interface, and contaminated free surfaces, behave in between the two extremes, t h e rigid wall and clean free surface. 5. Because the coupling of shearing processes and other dynamic processes is Reynolds number dependent and weaker as Re increases, at large Re the theory of boundary vorticity dynamics can be simplified. On a solid wall with an attached boundary layer, the pressure is approximately decoupled from and u , in the sense that p can be determined first from inviscid solution and then the latter determined for a given p distribution. On a free surface, the attached boundary layer is weaker and its equation can be linearized. As Re m, that is, in the Euler limit of viscous flows, these attached boundary layers are further simplified to boundary vortex --f

266

J. Z. Wu and J. M. Wu

sheets, of which a basic difference from a free vortex sheet is that they are still continuously created dynamically by a boundary vorticity flux u, manifesting itself as a jump of tangent acceleration across the sheets. 6. The integrated reaction of the created vorticity from a closed boundary 9 to 99 amounts to various vorticity based total force and moment formulas. This is possible again owing to the viscous coupling of the dynamic processes and the no-slip condition, no matter how weak it is at a large Re. In particular, the total force and moment can be expressed solely in terms of proper vector moments of boundary vorticity fluxes, indicating that the rate of work done in creating the vorticity from B' is Compared with conventional faithfully reflected as the reaction to 9. force and moment formulas based on primary variables, the unique feature of vorticity based formulas is the high concentration of their integrand in local regions of S' for typical configurations of engineering interest. The formulas are, therefore, theoretically irreducible, in which those local stresses that could cancel each other during integration are automatically excluded. Hence, this type of formula is of great potential in applications, especially in hydrodynamic-aerodynamic diagnostiq and optimization of the configurations once combined with experimental measurements or numerical computations. Moreover, because the formulas can clearly identify the key regions for vorticity creation, they also provide a clue for optimizing the configurations via various local means of vorticity-creation control. 7. The theory of boundary vorticity dynamics alone is insufficient for solving a bounded vortical flow problem. Rather, it provides natural and optimal boundary conditions for vorticity based numerical methods. On a solid wall, this condition is of a Neumann type, in terms of the boundary vorticity flux that can be efficiently localized at high Reynolds numbers. On a free surface, the Dirichlet condition seems natural for tangent vorticity, if the surface shape and motion are known from inviscid approximation. 8. Although the basic theory of vorticity dynamics is almbst complete, it is highly desired to further explore its various aspects. First, the theory needs to be extended to flows with strong heat conducting with variable shearing viscosity. Essentially, this implies a close study of the boundary coupling between shearing and thermodynamic processes. Next, the potential applications of the theory have never been fully addressed, in particular those relevant to diagnostics and optimization of configurations, to

Vorticity Dynrrmics on Boundaries

267

near-boundary turbulence structures and modeling, and to developing vorticity based methods for compressible flows and free-surface flows. Acknowledgments

We owe much to many of our colleagues, friends, and students in preparing this article. Our thanks first go to Professor H. K. Cheng, who recommended writing the chapter for this series. The editor of the series, Professor T. Y. Wu, has been a very strong source of continuous support, understanding, and encouragement; he also carefully read the early drafts of the manuscript and made many valuable suggestions that greatly improved the content of t h e chapter. The helpful discussions with Professors H. Hornung, H. Y. Ma, H. Yeh, and M. Gharib, and Drs. E. Rood and D. G. Dommermuth, are very appreciated. W. L. Sellers, A. E. Washburn, Professor D. Rockwell, Drs. D. G. Domrnermuth, R. D. Juslin, and R. Handler kindly permitted us to cite their work and provided us their experimental and computational results, unpublished or published, to whom we are also very grateful. We also thank our able graduate assistants B. Ondrusek and J. S. Liu, as well as T. G. Zheng, for their help in preparing the manuscript. Our own work reviewed in this article were supported in part by NASA Langley Research Center under the Grant NAG-1-844 J. M. Wu Research Fund, and by National Science Foundation of China during the first author's visit of the Graduate School of Academia Sinica in the summer of 1994. In particular, we are deeply indebted to P. J. Bobbitt and Dr. R. W. Barnwell of NASA Langley Research Center, without whose support and great insight the exploration of relevant theory and applications would have been impossible. References Anderson, C. R. (1989). Vorticity boundary conditions and boundary vorticity generation for two-dimensional viscous incompressible Hows. J . Comput. Phys. 80, 72-97. Anderson, C. R., and Reider, M. B. (1994). A high order explicit method for the computation of flow about a circular cylinder. J . Cornput. Phys., in press. Andrews, D. G., and Mclntyre, M. E. (1978). An exact theory of nonlinear W A V ~ Son a Lagrangian-mean How. J . Fluid Mech. 89, 609-646. Baker, G. R., Meiron, D. I., and Orszag, S. A. (1982). Generalized vortex methods for free-surface flow problems. J . Fluid Mech. 123, 477-501.

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