On the force on a body moving in a fluid

Fluid Dynamics Research 38 (2006) 716 – 742

On the force on a body moving in a fluid Arie Biesheuvel∗ , Rob Hagmeijer J. M. Burgers Centre for Fluid Mechanics and Department of Mechanical Engineering, University of Twente, P.O. Box 217, 7500 AE Enschede, The Netherlands Received 22 August 2005; received in revised form 20 March 2006; accepted 7 June 2006 Communicated by Y. Fukumoto

Abstract It is well-known that freely falling or rising objects and self-propelling bodies shed vorticity. It is then a natural question to ask how to define the forces (drag and lift) experienced by the body in terms of the vorticity distribution in the surrounding fluid and the normal velocity of the body surface, since these define the velocity distribution uniquely. In this paper we outline the answer given by Burgers in an almost forgotten paper from 1920, and point at the close relationship of Burgers’s ideas in these matters with those of Sir James Lighthill. The connection with more recent work by Kambe and Howe is established and we briefly discuss related issues concerning “vortex methods” and “vortex sound”. © 2006 The Japan Society of Fluid Mechanics and Elsevier B.V. All rights reserved. Keywords: Rotational flow; Potential flow; Vorticity; Impulse; Lift; Drag; Vortex sound

1. Introduction A fundamental problem in fluid mechanics is the determination of the hydrodynamic force experienced by a body as it moves in a fluid. Of course, since the i-component of the force exerted across a material surface element with vector area n
S by an incompressible fluid on the side to which the normal n points is
 ju j ju i nj
S, + −pni
S +  jxj jxi ∗ Corresponding author: Tel.: +31 53 489 4068.

E-mail address: [email protected] (A. Biesheuvel). 0169-5983/$30.00 © 2006 The Japan Society of Fluid Mechanics and Elsevier B.V. All rights reserved. doi:10.1016/j.fluiddyn.2006.06.001

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with p denoting pressure, u fluid velocity and  fluid viscosity, it follows immediately that the i-component of the force exerted on the body by the surrounding fluid is given by   
juj jui Di = − pe ni dS +  + (1) nj dS − gi VB . jxi SB SB jxj Here SB and VB denote the surface and the volume of the body,  is the fluid density, pe is the excess pressure over the hydrostatic distribution, and gi is the i-component of the gravitational acceleration vector. However, since the velocity field u(x, t) in a fluid is uniquely specified by the vorticity distribution
(x, t) (defined as
= ∇ × u), together with the normal component of the velocity of the surface of the body Us · n, it would be useful to express the hydrodynamic force in terms of just these quantities. This has been the subject of numerous investigations, notably those of Sir James Lighthill, Kambe and Howe. When clearing his office after his retirement, L. van Wijngaarden gave one of us (A.B.) a series of offprints of publications by Burgers, one of which was entitled On the Resistance of Fluids and Vortex Motion and was communicated to the Koninklijke Akademie van Wetenschappen te Amsterdam on September 25, 1920. The paper seems to have been forgotten and its significance hardly recognized, as exemplified by its omission in the book Selected Papers of J. M. Burgers, which was published to commemorate the 100th anniversary of the birth of Burgers on January 13, 1895 (Nieuwstadt and Steketee, 1995). The paper begins with a derivation of an expression for the resistance in terms of the vorticity distribution in the fluid and then continues with an investigation of the physical processes that lead to a resistance, identifying it as minus the hydrodynamic impulse of the vorticity that must be produced per unit time at the surface of the body in order to meet the no-slip condition as the flow evolves. The conclusions of Burgers’ investigation are expressed succinctly in the Summary of his paper, which merits being cited in full (the meaning of the symbols is explained below Eq. (5)): When a body in a fluid is brought into motion a vortex layer is generated at its surface. This layer diffuses into the fluid by the friction and is carried on by the current, is “washed away”. At the surface new vorticity is generated, which diffuses again etc. The generation of each vortex layer demands a certain impulse and the sum of the impulses that must be produced per second, forms the resistance W experienced by the body. At a definite moment the total impulse of all vortices together is equal to the time integral of W:  t  W dt = I(t) =  Ci Ai − V; the impulse may be calculated from the products: (circulation) · (surface); of the separate vortex lines. Part of this impulse can be received back when the body is retarded; viz. the part given by classical hydrodynamics, for which may be put: (“ apparent mass” ) · (velocity of the body). Of the rest a small part can be received back; the greater part, however is lost. When we have to do with an ideal fluid (absolutely without friction) these considerations need not be changed, when only we say that the vortices always remain in an infinitely thin layer at the

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surface of the body. They do not diffuse and are not washed away. The impulse therefore is always seated in this layer and has the value (“ apparent mass” ) · (velocity of the body); this amount can be totally received back when the motion of the body is retarded. In order to obtain an “irreversible” resistance viz. to give an impulse to the fluid that cannot be received back, the vortex motion must come outside this layer, there must be diffusion of the vorticity, be it to a low degree. It is the purpose of this paper to point out the significance of this publication by Burgers by discussing its relationship to more recent studies concerned with “the resistance of fluids and vortex motion”. We propose to demonstrate this, first, by deriving in Section 2 a “formula for the resistance” in the manner of Burgers and, next, by showing how generalizations of alternative formulas due to Lighthill (in Section 3), Kambe (in Section 4) and Howe (in Section 5) follow from it. For simplicity, the discussion is limited to deformable solid bodies of constant volume, but it may easily be seen that these formulas are equally valid for droplets or gas bubbles. Section 6 relates Burgers’s formula for the resistance to the “theory of vortex sound”, which originated with the work of Powell in 1964. Throughout the text, square brackets around a capital A followed by a number, e.g. [A1], refers to a remark given in the Appendix.

2. Burgers’s formula 2.1. Two derivations of the formula for the resistance Burgers’s “elementary derivation of the formula for the resistance” [A1] uses (i) the classical result that the impulse of a distribution of vorticity
(x, t) in an unbounded fluid (without a body present) with density  is given by the volume integral (Lamb, 1932, art. 152)  1  x ×
dV , (2) 2 and (ii) that the force with which a moving body acts on a fluid (−D) is equal to the rate-of-change of the impulse I that has to be applied to generate the fluid motion instantaneously from rest, so that the force exerted on the body by the fluid is given by [A2] D=−

dI . dt

(3)

To calculate the impulse of the fluid I when there is a body is present, imagine that body is replaced by fluid with exactly the same motion as the body. The impulse of this new system is given by   1  x ×
dV = I +  u dV , (4) 2 V +VB VB with the integration on the left-hand side to be taken over all space, i.e. over the volume V of the actual flow field and over the volume VB of fluid replacing the body. Substituting this in (3), and using the

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definition of the velocity of the center of volume, viz.  1 u dV , U= VB VB then yields what we propose to call Burgers’s formula
  d 1 dU D=−  x ×
dV + VB . dt 2 V +VB dt

(5)

In Burgers’s original notation (see the concluding summary of Burgers (1920) cited here in the Introduction) the velocity of the body is V and its volume , while W is the “fluid resistance”, so that W = −D (cf. [A1]). The vorticity distribution is envisaged as an ensemble of closed vortex lines (which may extend into the body), each of which contributes to the impulse an amount CA, where C is the circulation around a vortex line and A the surface area enclosed by that line (cf. Lamb, 1932, art. 150 and Section 4 below). It is, perhaps, worth emphasizing that Burgers’s formula is applicable for rigid bodies as well as deformable bodies, and the surrounding fluid may be a “real” as well as “ideal” (without viscosity). Also, in any of these cases, the vorticity in the fluid need not all have been produced and shed by the body; for example, it also covers the case of the transient forces experienced by a body when vorticity generated by some means elsewhere in an ideal fluid is swept past the body in a mean flow. In recent literature Burgers’s formula for the resistance is often attributed to Wu (1981). Indeed, (5) is equation (42) of Wu (1981), a result that is actually obtained there by a line of reasoning in the spirit of Burgers’s “elementary derivation” but which employs, rather than the impulse, the momentum of the fluid, made definite by bounding the flow field by a large sphere. The equivalence may be illustrated by the words in which Wu describes the meaning of (42): “. . .the aerodynamic force exerted by a fluid on solid bodies . . . is equal to the inertia force due to the mass of fluid displaced by the solid bodies plus a term proportional to the time rate-of-change of the first moment of the vorticity field in the solid bodies and the fluid”. The vorticity field in the solid body that is meant here can be inferred from the remark at the bottom of p. 435: “. . .the proper extension of the fluid region is simply the solid regions in which the correct vorticity values to assign are the actual vorticity of the solid bodies”. Recent literature also sometimes mentions an alternative form of (5). This may be obtained by using the identity (which only holds in three dimensions)    2 a dV = x × (∇ × a) dV − x × (n × a) dS, (6) where it is understood that the normal n points away from the volume V [A3]. Taking a = u, applying the no-slip condition, and substituting the result in (5), the expression for the force is found to be given by

    d 1 d 1 D=−  x ×
dV −  x × (n × Us ) dS , (7) dt 2 V dt 2 SB where Us denotes the velocity of the surface of the body, and here and below, the normal n to the surface of the body points into the surrounding fluid. After this elementary derivation Burgers continues with what he considers “a proof of the formula for the resistance”. It begins with “in the same way as above the body is replaced by a fluid mass which has

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the same motion as the body and zero pressure (this means that the pressure has the same value as at infinity)”, and may be paraphrased further as follows: consider the fluid particles to be acted upon by a system of external forces F such that the resulting pressure distribution and fluid motion is the same as if the body were present. This system consists of a distribution du dt per unit volume acting on the fluid particles replacing the body and a system F2 acting “on a thin layer that is always there where the surface would have been, equal to the force exerted by an element of the surface of the body on the fluid”, and which is treated as a distribution of volume forces. Thus, the total external force acting on the fluid is given by    dU F dV = F1 dV + F2 dV = VB − D. (8) dt V +VB V +VB V +VB F1 = 

Direct evaluation then shows that the rate-of-change of the hydrodynamic impulse of the system equals this external force, i.e. [A4]  
 d 1  x ×
dV = F dV , dt 2 V +VB V +VB whence (5) follows immediately. Burgers’s paper continues with a series of remarks and footnotes that lead to the observations presented in the Summary cited at the beginning of our paper. We will now briefly consider some of these remarks. 2.2. Gedanken experiment Burgers’s Summary begins with: When a body in a fluid is brought into motion a vortex layer is generated at its surface. This layer diffuses into the fluid by the friction and is carried on by the current, is “washed away”. At the surface new vorticity is generated, which diffuses again, etc. The generation of each layer demands a certain impulse and the sum of the impulses that must be produced per second, forms the resistance experienced by the body. This is the outcome of a Gedanken experiment in which Burgers first imagines that a rigid body is replaced by fluid and that the motion of the system takes place under the action of a distribution of volume forces acting on the fluid particles that replace the body and on a very thin “transition layer” there where the surface of the body has been. Next, he envisages a situation in which over a very short time interval,
t say, a body in steady translational motion with velocity U does not offer resistance to the motion of the surrounding fluid. During this time interval the vorticity in the fluid will evolve “freely” by convection and diffusion, to become redistributed in space, while the hydrodynamic impulse is conserved. The result will be that the velocity field produced by the “new” distribution of vorticity,
f , will no longer satisfy the condition of vanishing tangential velocity at the body surface. To obtain the velocity distribution that would have existed if the body had offered resistance to the motion of the fluid during the short time interval
t, a thin layer of concentrated vorticity at the body surface must be added: a vortex layer with a surface concentration such that the no-slip condition is satisfied, and which produces an irrotational

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velocity field that can be superposed on the velocity field produced by the “new” vorticity distribution
f (ignoring the very small effect of diffusion during the short time interval
t). The surface concentration of this layer will be 
dxn = n × [∇  − (U − uf )], (9) in which xn measures distance from the surface [A5]. Here, the velocity uf is that induced by the freelyevolved “new” vorticity distribution
f and is given by the Biot–Savart formula 
f (y) dV (y), (10) uf (x) = ∇x × 4|x − y| the velocity potential  is defined by ∇ 2 = 0

with n · ∇  = n · (U − uf ) on SB ,

 → 0 as |x| → ∞.

(11)

The hydrodynamic impulse of the vortex layer,
Jst say, must equal the time integral of the resultant of the forces acting on the transition layer, viz.
J = −Dst
t,

where Dst is the force experienced by the body during this steady motion. As time continues, vorticity diffuses out of the transition layer and is subsequently “washed away”, a new layer forms, and so on . . . . “The generation of each vortex layer demands a certain impulse” and the resistance experienced by the body is formed by “the sum of the impulses that must be produced per second”. When the velocity of the body is not steady during the short time
t, for example changes by
U, a second vortex layer at the body surface must be added: a vortex layer that induces an additional irrotational motion with a velocity potential
 specified by the boundary condition at the body surface n · ∇(
) = n ·
U. The surface concentration of this second layer is 
dxn = n × [∇(
) −
U], and its hydrodynamic impulse,
Jacc , is given by
Jacc = −Dacc
t + VB
U,

in which Dacc is the “acceleration resistance”, the force experienced by the body because of its acceleration. This acceleration resistance does not depend on the presence of vorticity in the flow and can be determined “by the methods of classical hydrodynamics”. Indeed, the total impulse acquired by the fluid (including that replacing the body) may also be written as  1  x × {n × [∇(
) −
U]} dS, (12) 2 SB an expression that is easily shown to be equal to [A6]  VB
U −  (
)n dS = VB
U +
IB , SB

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where
IB is the change in the “virtual momentum of the body” (following the terminology of Saffman, 1992); hence, the acceleration resistance follows from the classical hydrodynamics result Dacc
t = −
IB . Finally, as Burgers points out, the second transition layer can be combined with the first by merely replacing U in (9) by U +
U, the body velocity attained at the end of the element of time
t, and combining the expressions for the impulses as
J =
Jacc +
Jst = −D
t + VB
U,

in agreement with the results in the previous paragraph, on observing that
J =
I + VB
U.

2.3. Ideal fluids Burgers points out in his Summary that his formula for the resistance applies equally to bodies moving in ideal fluids, i.e. fluids “absolutely without friction”. Clearly, he distinguishes the no-slip condition as the mechanism of generation of vorticity at the surface of the body, and diffusion by viscosity as the mechanism that brings this vorticity into the interior of the fluid. Whether the fluid is “ideal”, with zero viscosity, or whether it is “real”, with a viscosity that may be vanishingly small, the no-slip condition applies to both types of fluids (see Landau and Lifshitz, 1959, Section 9 for what seems to be a similar view) [A7]. This implies that because of the no-slip condition, a body moving in an ideal fluid (in the absence of vorticity in the interior of the fluid created by some external means) is surrounded by a thin layer of concentrated vorticity with surface concentration 
dxn = n × [∇  − Us ] (13) with  now defined by ∇ 2 = 0

with n · ∇  = n · Us on SB ,

 → 0 as |x| → ∞.

(14)

This vorticity cannot diffuse out of the layer and be washed away, since the fluid has no viscosity. Upon replacing the body by fluid, the hydrodynamic impulse of the unbounded flow is given by   1  x ×
dV = VB U −  n dS, (15) 2 V +VB SB in which the right-hand side follows by using the method outlined in [A6]. By inserting this in (5) the classical expression 
 dIB d D=−  n dS (16) = dt dt SB is obtained. It is now not a major step (but not made explicitly by Burgers) to perceive the vortex layer in Burgers’s Gedanken experiment described above, consists of two parts, one of which being a layer with a surface

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concentration given by (13), to be evaluated after the time increment
t. This part of the layer can be imagined to persist as long as the body moves, unchanged when the body has constant velocity, changing its surface concentration only when the body accelerates, slows down, rotates or deforms. The layer disappears when the body is brought to rest, and as found above, the force to be exerted on the fluid is the rate-of-change of the virtual momentum of the body. In the words of Burgers “part of this impulse can be received back when the motion of the body is retarded; viz. the part given by classical hydrodynamics”. The remaining part of the vortex layer has a surface concentration given by 
dxn = n × (∇ f + uf ) (17) with f defined by ∇ 2 f = 0

with n · ∇ f = −n · uf on SB ,

f → 0 as |x| → ∞,

(18)

again to be evaluated after the time increment
t. The fluid velocity field associated with the freely evolved new vorticity combined with the irrotational motion induced by this second layer is such that it vanishes at the body surface. The rate at which hydrodynamic impulse is to be supplied to this second layer, in other words, to maintain the condition of vanishing velocity at the body surface constitutes the “steady resistance” or “irreversible resistance” of the body. This part of the impulse of the fluid “cannot received back”. 3. Lighthill’s formula Burgers’s Gedanken experiment offers essentially a discrete version of a continuous process in which the body generates a “boundary vorticity flux” to meet the conditions on the velocity at its surface as the flow field evolves. It is reminiscent to the numerical method to solve the vorticity equation that was outlined in Lighthill’s famous introductory chapters to Laminar Boundary Layers of 1963 (Lighthill, 1963), and that laid the foundations for what is now known as “vortex methods” (see Howe, 2001 for a clear description of Lighthill’s method). The beauty of Sir James Lighthill’s derivation of a formula for the resistance (Lighthill, 1979, 1986a,b, 1991) is that the body is not replaced by fluid in identical motion. This requires, firstly, the definition of the hydrodynamic impulse I of a fluid in vortical motion when there is a moving body present in the flow; this he shows to be given by   1 I = − n dS +  x ×
a dV (19) 2 V SB and to be related to the far-field dipole strength G of the velocity field by G = I + VB U.

(20)

The first contribution to the right-hand side of (19) is the virtual momentum of the body, viz. the hydrodynamic impulse given to the fluid when it is imagined that the motion of the body is generated instantaneously from a state in which the fluid is completely at rest. This would result in a velocity field which in the interior of the fluid is continuous and irrotational and can be expressed by a velocity potential  as defined by (14); the velocity distribution follows from u = ∇ . At the surface, because of the no-slip

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condition, a thin layer with a concentrated distribution of vorticity is generated with a surface concentration given by (13). The second contribution to the right-hand side of (19) is the hydrodynamic impulse of the distribution of additional vorticity
a , defined as the actual distribution of vorticity in the fluid minus the above-mentioned distribution of vorticity in a thin layer at the body surface. The distribution of additional vorticity is very nearly the sum of the vorticity distribution
f of Burgers’s Gedanken experiment and the thin layer concentrated at the body surface with surface concentration (17). The velocity field it produces is given by the Biot–Savart formula (10) with
f replaced by
a , and vanishes at the body surface; in other words, it is the velocity field produced by a distribution of vorticity in the presence of a body at rest [A8]. Next, Lighthill proves that the relation between the force D experienced by a body in a fluid with vorticity and the hydrodynamic impulse of that fluid is given by (3), viz. dI , dt which Burgers seems to have taken as self-evident. Finally, upon substituting (19) in (3) it is found that

   d d 1 D=  n dS −  x ×
a dV , dt dt 2 V SB D=−

(21)

an expression that we propose to call Lighthill’s formula. Clearly, there is a close connection with Burgers’s formula. Indeed, upon considering the vorticity distribution in the fluid as a combination of a layer of concentrated vorticity at the body surface with surface concentration (13) and a distribution of additional vorticity
a , Burgers’s formula immediately gives

   d 1 d 1 D= −  x ×
dV −  x × [n × (∇  − Us )] dS dt 2 VB dt 2 SB
  dU d 1  x ×
a dV + VB , − dt 2 V dt which can be shown to reduce to Lighthill’s formula by the method described in [A6]. An alternative form of Lighthill’s formula is obtained on substituting relation (20) in (3), viz. dG dU + VB , (22) dt dt and is admirably used by Lighthill (1991) to explain aspects of fish swimming. To conclude, the similarity between Burgers’s and Lighthill’s ideas may be expounded further by citing Lighthill in connection with the loading on off-shore structures (Lighthill, 1986b): D = −

. . .the force on a body may be divided into (i) a potential-flow force that depends linearly on the body velocity, and can be accurately calculated; and (ii) a vortex-flow force that varies nonlinearly and is related in a definite way to vortex shedding and to the convection of shed vorticity. The potential-flow force is the first contribution to (21), viz.
  d  n dS , dt SB

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the vortex-flow force is the second contribution,
  d 1  x ×
a dV . − dt 2 V 4. Kambe’s formula We now make a connection with the theory of vortex sound by calculating the far-field dipole strength, Ga say, of the velocity field produced by a distribution of additional vorticity, as defined in the previous section. In other words, we derive an alternative expression for the impulse per unit mass  1 x ×
a dV , Ga = 2 V and in doing so, obtain an alternative expression for the vortex-flow force. It is helpful to follow Burgers, and view the distribution of additional vorticity as an ensemble of closed very thin vortex tubes of strength n lying within the volume elements
Vn , i.e.   
a dV =
a dV = n dln , (23) n


Vn

n

where dln denotes the vector element of length of member n of the ensemble. A closed line vortex of strength  in an unbounded flow is equivalent to a uniform distribution of dipoles with strength n over any surface bounded by the line vortex, in the sense that they produce the same irrotational motion [A9]. The velocity potential induced at x is 
 1 dA(y), (x) = n · ∇ x 4|x − y| which at large distances from the line vortex is approximately given by 
  1 . (x) ∼ n dA(y) · ∇x 4|x| Hence, in the hydrodynamic far-field the closed line vortex is equivalent to a point dipole of strength  n dA(y) at the origin. This classical result is a key element in the theory of vortex sound founded by Powell (1964). When there are surfaces bounding the flow, the free-space solution for the field at x due to a unit source at y, 0 (x, y) = −

1 , 4|x − y|

(24)

needs to be replaced by a solution, S say, for which the normal velocity vanishes on the surface of the body. The equivalence between a closed line vortex and a uniform distribution of dipoles remains however [A10].

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The idea now is to construct an approximate expression for S with the same asymptotic behavior as the exact solution, and which therefore yields an exact expression for the far-field dipole strength. The method by which this is done is essentially a simple version of that devised by Howe (1975) to construct what he calls “compact Green functions”, which are now commonly used in acoustics (see, for example, Howe, 2003). It relies on the same principle of reciprocity as mentioned in [A10], but in a form in which it states that the velocity in the direction n at point y due to point source of strength  at point x is equal to the potential at x due to a dipole of strength n at y, everything else being the same; a principle that allows one to treat the determination of S as a scattering problem. Without loss of generality, we assume that the center of volume of the body coincides with the origin. Let S (y, x) be the velocity potential at y produced by a unit source at x. In the absence of bodies in the flow this potential is given by (24), and in the presence of a body (at rest) it may be written as S (y, x) = 0 (y, x) + ∗ (y, x),

where ∗ can be regarded as the disturbance potential induced by the body. Now take the source point x in the hydrodynamic far-field of the body, while the observer point y is in the neighborhood of the body. In that case, near the body 
j 1 1 + yi , 0 (y, x) ∼ − 4|x| jxi 4|x| where the first term is a constant and the second term can be interpreted as the velocity potential of a uniform flow impinging on the body with velocity 
j 1 jxi 4|x| in the i-direction. This means that the disturbance potential may be approximated by 
j 1 ∗ ,  (y, x) = −i (y) jxi 4|x| where i (y) is a velocity potential determined uniquely by ∇ 2 i = 0

with nj

ji jyj

= ni on SB ,

Hence j 1 + Yi (y) S (y, x) ∼ − 4|x| jxi




1 4|x|

i → 0 as |y| → ∞.

(25)

 (26)

with Yi (y) = yi − i (y)

(27)

the i-component of what Howe calls the Kirchhoff vector, the velocity potential of a hypothetical irrotational flow with unit velocity in the i-direction at infinity around a body, instantaneously at rest at the origin. The velocity at y in the direction n associated with this potential is 
j 1 n · ∇y S = n · ∇y Yi . (28) jxi 4|x|

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It is readily shown that, up to the order of approximation, S as given by (26) is symmetrical with respect to the positions x and y. Thus, by the principle of reciprocity, the right-hand side of (28) is equal to the velocity potential at the source point x in the far-field due to a dipole of strength n at position y near the body. The velocity potential produced by a closed vortex-filament with strength  in the presence of a body at rest therefore behaves asymptotically as (suppressing for the moment explicit reference to the independent variable y)   
 j 1 n · ∇Yi dA  jxi 4|x| and this identifies the i-component of the far-field dipole strength as   n · ∇Yi dA. Generalizing this result to an ensemble of line vortices finally yields   Gai = n · ∇Yi dAn . n

(29)

n

To revert this last expression to a volume integral over the distribution of additional vorticity we use an idea due to Möhring (1978). Since the hypothetical velocity field ∇Yi is solenoidal, one may introduce for each direction i a vector function
i , defined by ∇Yi = ∇ ×
i ,

∇ ·
i = 0.

(30)

The components of each of these vector functions are harmonic functions since ∇ × (∇Yi ) = ∇ × (∇ ×
i ) = −∇ 2
i = 0, and so each
i is a vector potential, namely, as mentioned above, that associated with the hypothetical potential flow with unit velocity in the i-direction at infinity, around a body at rest at the origin. The problem of determining the vector functions
i is made unique by requiring that
i = 0 on the surface of the body. It now follows that     Gai = n · ∇Yi dAn = n · (∇ ×
i ) dAn n n n

=

 n

 n

n


i · dln =

V


i ·
a dV ,

(31)

which is the desired alternative expression for the far-field dipole strength Ga . On substituting (31) in Lighthill’s formula (21) the i-component of the force on a body moving in a fluid is found to be also given by  

 d d Di =  ni dS − 
i ·
a dV . (32) dt dt SB V Recalling that the difference between the actual vorticity distribution
surrounding a moving body and the distribution of additional vorticity
a is a thin layer of concentrated vorticity at the body surface with

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concentration (13), and observing that the vector function
i = 0 at the body surface, it becomes clear that this distribution of concentrated vorticity at the body surface may be added to the distribution of additional vorticity on the right-hand side of expression (31); this will not affect the value of far-field dipole strength Ga . In other words,
a may be replaced by
. Hence, an equivalent and more appealing expression for the resistance reads

   d d Di =  ni dS − 
i ·
dV . (33) dt dt SB V This is a generalization of the expression derived by Kambe (1986, 1987) when considering the force experienced by a rigid body at rest, as a consequence of the self-induced motion past the body of an isolated distribution of vorticity [A11]. In that case, of course, only the second term of (33) contributes to the force. As shown here, it is possible to generalize Kambe’s original expression and give it a much wider application; which is why we propose to call (33) Kambe’s formula. The practical significance of Kambe’s formula becomes apparent if ones looks more closely at the difference between Lighthill’s expression for the far-field dipole strength, viz.  1 Gai = {x ×
a }i dV , 2 V and Kambe’s expression 
i ·
a dV . Gai = V

Actually, the vorticity distribution envisaged originally by Kambe was not the distribution of additional vorticity as defined above, but rather some isolated “free-field” distribution of vorticity (a turbulent spot generated by some means elsewhere in the flow). In general, the velocity field produced by such a “freefield” vorticity distribution does not satisfy the condition of vanishing tangential velocity at the body surface, which means that in any “real” flow (be it of an ideal fluid or a fluid with viscosity) there must also be a very thin layer of concentrated vorticity present at the body surface through which the velocity field satisfies this no-slip condition. By definition, the distribution of additional vorticity
a includes both the “free-field” vorticity and this thin vorticity layer at the body surface. Now, from (28) it appears that the i-component of the far-field dipole strength associated with a dipole of strength n at position y near a body at rest may be written as the sum of two contributions, n · ∇y Yi = n · ∇y yi − n · ∇y i , the first contribution, equal to ni , arising from the “free-field” dipole, the second contribution obviously due to an “accompanying” distribution of dipoles on the surface of the body; in other words, to a thin layer of concentrated vorticity surrounding the body induced by the “free-field” dipole [A12]. Correspondingly, the function
i introduced in (30), may be divided in a part
1i = 21 [ei × y]

(34)

representing the uniform velocity ei , where ei is the unit vector in the i-direction, and a second part
2i , which is the solution of ∇ 2
2i = 0

with
2i = − 21 [ei × y] on SB ,


2i → 0 as |y| → ∞.

(35)

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729

Hence, the hydrodynamic impulse ( times the far-field dipole strength) associated with the distribution of additional vorticity consists of the sum of a contribution due to the “free-field” vorticity   1 [ei × y] ·
a dV , 
1i ·
a dV =  (36) 2 V and a contribution associated with the presence of a very thin layer of vorticity at the body surface  
2i ·
a dV . (37) V

In other words, if one chooses the vorticity distribution
a in (36) and (37) to consist of just “freefield” vorticity then the sum of expressions (36) and (37) constitutes the hydrodynamic impulse of the surrounding fluid. But, as the earlier derivation shows, that same sum also constitutes the hydrodynamic impulse if
a is the true distribution of additional vorticity (i.e. compatible with the no-slip condition). This means that the second contribution, viz. (36), must equal zero; a conclusion that is corroborated by the fact that the vector identity 1 2 [ei

× y] ·
a = 21 {y ×
a }i ,

(38)

shows that (36) is equivalent to Lighthill’s expression of the far-field dipole strength. The condition 
2i ·
a dV = 0, (39) V

or expressed differently,  
2i ·
dV = V

SB


2i · [n × (∇  − Us )] dS,

(40)

is in fact the projection condition that is often imposed in numerical computations of vortex flows, derived first by Quartapelle and Valz-Gris (1981) by a formal mathematical analysis. In the spirit of Burgers, one might say that as the vorticity field evolves by convection and diffusion, the surface of the body acts as a source of new vorticity in order to maintain a distribution of vorticity such that the integral condition (40) remains satisfied. Summarizing, Kambe’s formula for the resistance (33) may be obtained from Lighthill’s formula (21) by (i) rewriting the far-field dipole strength as   1 Ga = x ×
a dV =
1i ·
a dV , 2 V V using the vector identity (38) and upon introducing the vector potential (34), (ii) introducing a second vector potential specified by (35) and adding a term that is exactly zero, viz. 
2i ·
a dV , V

to obtain Ga =

 V


i ·
a dV

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A. Biesheuvel, R. Hagmeijer / Fluid Dynamics Research 38 (2006) 716 – 742

in which
i =
1i +
2i , and finally, (iii) exploiting that the sum
i of the two vector potentials vanishes at the body surface, giving  Ga =

V


i ·
dV .

The great improvement obtained by this alternative formulation, is that if one wishes to calculate the far-field dipole strength Ga of the velocity field produced by some observed distribution of vorticity near a body, and this vorticity distribution is such that the condition of vanishing tangential velocity at the body surface is not satisfied, Lighthill’s expression would lead to an error if in the specification of
a the thin layer of concentrated vorticity that must be present at the body in any real flow is not included. Kambe’s expression, on the other hand, would still give the correct result, for the error in the first contribution (36) is canceled exactly by the error in the second contribution (37), because
= 0 at the body surface. It should be noted that Kambe’s original derivation does not rely on envisaging the vorticity distribution as an ensemble of thin closed vortex tubes of constant strength surrounded by irrotational fluid; he describes that as a special application, of practical significance. Flows behind fishes and birds are often found to have that appearance, as do flows behind solid spheres held stationary, or behind bubbles, drops and particles rising or falling freely under the action of gravity. As Kambe points out, for each of the thin vortex tubes appearing in expression (29), the integral can be viewed as the volume flux through the surface bounded by that thin tube, as induced by the potential flow Yi ; i.e. for tube n  Jni (t) =

n · ∇Yi dAn .

This flux depends on time because Yi changes as the body rotates or changes shape and, more generally, because the position of the vortex tube changes. It follows that the i-component of the vortex-flow force on a body due to the presence of a closed thin vortex tube is minus the strength of the vortex tube times the rate-of-change of the mass flux through the surface area enclosed by the vortex, as induced by a hypothetical potential flow with unit velocity in the i-direction. Hence the general expression for the i-component of the vortex-flow force on the body (in other words, the resistance when the body is in steady motion), may be written as and calculated from Dsti

d = − dt







n Jni

.

(41)

n

Note that when the collection of vortex tubes encompasses the true distribution of additional vorticity this expression becomes simply Dsti

d = − dt





 n

ni dAn ,

(42)

n

in which ni dAn is the projected area in the i-direction of vortex tube n; an expression well-known, of course, from classical wing theory.

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5. Howe’s formula An important general expression for the resistance was derived by Howe (1989, 1995) following an earlier idea of Quartapelle and Napolitano (1983) to convert the pressure contribution to expression (1) into one involving the vorticity distribution in the flow. For what concerns the connection with Burgers’s formula, it should be noted Burgers’s formula is derived in Section 4.4 of Howe’s excellent textbook Theory of Vortex Sound (Howe, 2003), by a different method than that outlined in Section 2 above, and subsequently provides the starting point in Section 4.4.2 for a derivation of a Howe’s formula for the resistance of a rigid body in translational motion. Since that derivation uses generalized functions, readers may perhaps appreciate an alternative derivation of Howe’s formula, now for the general case. It is convenient to begin with the following alternative form of (7),

   d d Di = − 
1i ·
dV − 
1i · (n × u) dS , (43) dt dt V SB where, as before,
1i = 21 [ei × x] and we have used that for any vector a 1 2

{x × a}i =
1i · a.

The first term on the right-hand side may be evaluated by means of Reynolds’s transport theorem as   j
−
1i · [
1i ·
] (u · n) dS dV −  jt V SB and, since the local rate-of-change of the vorticity is given by j
jt

= ∇ × (u ×
) − ∇ × (∇ ×
),

this may also be written as   −
1i · ∇ × [(u ×
) − (∇ ×
)] dV −  [
1i ·
](u · n) dS. V

SB

On repeated use of the vector identity
1i · (∇ × a) = ai − ∇ · (
1i × a),

(44)

the divergence theorem, and the asymptotic behavior of u and
, this may be seen to be equivalent to    −  {u ×
}i dV −  {n ×
}i dS −  ui (u · n) dS V SB SB   + n · {
1i × [(u ×
) − (∇ ×
)]} dS +  ∇ · (
1i × u) [u · n] dS. (45) SB

SB

The second contribution to the right-hand side of (43) may be evaluated by using standard vector identities and Reynolds’s transport theorem as
   ju − n ·
1i × ∇ · (
1i × u) [u · n] dS. dS −  jt SB SB

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A. Biesheuvel, R. Hagmeijer / Fluid Dynamics Research 38 (2006) 716 – 742

On combining this result with (45), one obtains    Di = −  {u ×
}i dV −  {n ×
}i dS −  ui (u · n) dS V SB SB     ju − (u ×
) + (∇ ×
) dS, n ·
1i × − jt SB

(46)

where the term between square brackets in the last integral may be recognized from the Navier–Stokes equation to be −∇(p/ + 21 u2 ) = −∇H . Now, if i is chosen as defined by (25), then by (44) and Green’s theorem    n · (
1i × ∇H ) dS = {∇H }i dV = (∇H · ∇ i ) dV , SB

V

V

whence (46) may also be written as 
  ju Di = −  ui (u · n) dS · ∇ i dV +  jt V SB   ∇Yi · (n ×
) dS, −  ∇Yi · (u ×
) dV −  SB

in which Yi (x) = xi − i (x) is the i-component of the Kirchhoff vector. From
  
  
ju ju ju − ∇ · i i · ∇ i dV = −  dV =  · n dS jt jt jt V V SB
    d ji + u · ∇ i (u · n) dS, i u · n dS −  = dt jt SB SB one finally obtains
    d jYi Di =  + u · ∇Yi (u · n) dS i u · n dS +  dt jt SB SB   ∇Yi · (u ×
) dV −  ∇Yi · (n ×
) dS, − V

SB

(47)

(48)

which is the expression for the resistance found by Howe (1995, (2.11)), there under the assumption that the body does not change shape. The first term on the right-hand side of Howe’s formula (48) is just the rate-of-change of the hydrodynamic impulse of the irrotational motion determined uniquely by the instantaneous shape and velocity Us of the surface of the body; in other words, it is the potential-flow force
  d  ni dS . dt SB

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733

Indeed, by introducing  as the solution to the (hypothetical) potential-flow problem (14) and applying the kinematic boundary condition,      i u · n dS = i Us · n dS = i n · ∇  dS = n · ∇ i dS = ni dS. SB

SB

SB

SB

SB

The remaining three terms on the right-hand side of (48) constitute the vortex-flow force. This may seem a little puzzling in that the vorticity distribution does not appear in the first contribution. However, when the surface vorticity distribution (13) is substituted in the third term on the right-hand side of (48) one obtains, on using that n · ∇Yi = 0 at the body surface,    ∇Yi · {[n × (∇  − Us )] × u} dS =  [(∇  − Us ) · ∇Yi ](u · n) dS. SB

SB

This may be combined with the second term on the right-hand side of (47) to give   jYi  + ∇  · ∇Yi (∇  · n) dS, jt SB an expression that is shown to be zero in Appendix A of Howe (1991) [A14]. Thus Howe’s formula for the resistance may equally be written as
    d Di =  ni dS −  ∇Yi · (u ×
a ) dV −  ∇Yi · (n ×
) dS. (49) dt SB SB In presenting the vortex-flow force in terms of normal surface stresses produced by the vorticity in the flow, and in explicitly showing the contribution due to viscous diffusion, Howe’s formula provides an important alternative to that Burgers. Numerous applications are discussed in Howe (1995). Since that paper has been written from the viewpoint that the no-slip condition should not be imposed for flows of an ideal fluid, and also because it uses the concept of image vorticity, it is worthwhile to briefly reconsider some of the material in Sections 2.2–2.5 of that paper. 5.1. A rigid body in translational motion For a rigid body in translational motion with velocity U, the first term on the right-hand side of (48) may be written as −VB ij

dUj , dt

in which the tensor coefficient ij is defined by  1

ij = − i nj dS. VB SB The second term becomes   [(u − U) · ∇Yi ](u · n) dS, SB

(50)

(51)

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A. Biesheuvel, R. Hagmeijer / Fluid Dynamics Research 38 (2006) 716 – 742

and vanishes upon application of the no-slip condition. Hence, the resistance is given by   dUj Di = −VB ij − ∇Yi · (u ×
) dV −  ∇Yi · (n ×
) dS, dt V SB

(52)

an expression that was first derived in Howe (1989, equation (A4)), and later for example in Howe (1995, equation (2.27)), Howe (2001, equation (2.24)) and Howe (2003, equation (4.4.4)); albeit, in the form   dUj Di = −VB ij ∇Yi · [(u − U) ×
] dV −  ∇Yi · (n ×
) dS. (53) − dt V SB This is an equivalent formulation, for the kinematical boundary condition and the fact that n · ∇Yi = 0 at the body surface allows one to rewrite the vanishing contribution (51) as     [u · ∇Yi ] (U · n) dS −  [U · ∇Yi ] (u · n) dS +  [u · U] (n · ∇Yi ) dS. SB

SB

SB

On using the divergence theorem and the vector identity ∇ · [U(u · ∇Yi ) − u(U · ∇Yi )] = ∇ · [(U · u)∇Yi ] − ∇Yi · (U ×
), (easily proven by noting that each side is ∇Yi · [U · ∇u]) this becomes   ∇Yi · (U ×
) dV , V

and (53) follows. The advantage of (53) over (52) is that any distribution of vorticity at the body surface does not contribute to the volume integral in (53) and need not be specified accurately. This means, for example, that this contribution to the vortex-flow force is not affected by a specification of
that is inconsistent with the condition that at the body surface the induced velocity distribution equals U. The power of this result is illustrated beautifully in Howe et al. (2001). 5.2. General motion of a rigid body For a rigid body moving at constant angular velocity  and with velocity of its center of volume U, instantaneously at position x0 , say, the first term on the right-hand side of (48) becomes −VB

d ( ij Uj + ij j ), dt

with ij as defined above and  1 ij = − i {(x − x0 ) × n}j dS. VB SB

(54)

In the general case the rotation of the body causes a change in the potential Yi and the second term on the right-hand side of (48) now reads  − { × Y}i (u · n) dS, SB

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735

whence


  d Di = − VB ( ij Uj + ij j ) −  ij k j Yk (u · n) dS dt SB   ∇Yi · (u ×
) dV −  ∇Yi · (n ×
) dS, − V

SB

(55)

which is equivalent to expression (2.23) in Howe (1995); it is also equivalent to the third line of expression (2.19) of that paper if one assumes that the fluid is ideal. 6. Vortex sound Having seen that major contributions to hydrodynamics, viz. the formulas for the resistance of Kambe and Howe, are closely linked to developments in acoustics, it seems appropriate to briefly return to the subject of vortex sound. The general expression for the acoustic pressure field due to a dipole is (Lighthill, 1978, Section 1.5)
 Gac (t − |x|/c) p(x, t) = −∇ · , 4|x| with c the speed of sound. The relation between hydrodynamics and acoustics follows from the fact that for acoustically compact systems (a rough measure is that a length-scale measuring the size of the region with vorticity is much smaller than the wavelength of the sound) an acoustic dipole strength Gac can be defined by the replacement (Lighthill, 1978, Section 1.7) 

jG jt

→

Gac ,

(56)

by which dU . (57) dt The pressure in the acoustic near-field, which may be viewed as the hydrodynamic far-field, may be approximated by 
1 p(x, t) ∼ −Gac (t) · ∇ , 4|x| Gac = −D + VB

in agreement with the expressions given in [A11]. The pressure in the acoustic far-field, where differences between t and the retarded time t − |x|/c need to be taken into account, can be obtained by applying the rule (Howe, 2003, p. 22) j jxj

←→ −

1 xj j , c |x| jt

whence p(x, t) ∼

1 x jGac (t − |x|/c) · . 4c |x|2 jt

(58)

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Thus, Kambe’s formula defines the pressure in the acoustic far-field as p(x, t) ∼

x j2 · 4c |x|2 jt 2 



 SB

n dS(y) + VB U +

x j2 · 4c |x|2 jt 2 



 V


i ·
dV (y) ,

where, here and below, the quantities between square brackets are to be evaluated at the retarded time t − |x|/c. The expression is valid for viscous and inviscid fluids, and, as explained in Section 4, is of practical significance in that it does not lead to significant errors if the specified vorticity distribution fails to meet the conditions on the velocity at the surface of the body. Several interesting illustrations of its use are given in Kambe (1986). Similarly, Howe’s formula for the force experienced by a rigid body in translational motion defines the acoustic far-field pressure as  x j2 p(x, t) ∼ · n dS(y) + VB U 4c |x|2 jt 2 SB   x j − · ∇Yi · {(u − U) ×
} dV 4c |x|2 jt V   x j − · ∇Y · (n ×
) dS . i 4c |x|2 jt SB 

This expression has the further advantage that any concentrated distribution of vorticity at the body surface does not contribute to the second term on the right-hand side. Moreover, the (often negligible) contribution due to viscosity effects is shown explicitly. Examples of its use may be found in Acoustics of Fluid–Structure Interactions (Howe, 1998) and in Theory of Vortex Sound (Howe, 2003). For what concerns Burgers’s formula, this yields   x j2 1 p(x, t) ∼ · y ×
dV (y) , 4c |x|2 jt 2 2 V +VB 

(59)

as an attractive, but lesser known, alternative to Kambe’s and Howe’s expressions mentioned above. Actually, Burgers’s result is briefly mentioned by Howe (2003, (5.6.2)), who points out that the vorticity is the “generalized vorticity, including the bound vorticity on SB ”; by which must be meant the actual vorticity distribution, including the vorticity distribution that is obtained when the body is replaced by fluid which has the same motion as the body. Expression (59) is also similar to that derived by Möhring (1978, 5 lines below (8a)) for the dipole contribution to the sound produced by a system of compact vorticity spots in an unbounded inviscid fluid; here, the volume integral in (59) is to be taken as the sum of volume integrals over the vorticity spots. This inadvertently suggests that there is such a dipole contribution, while actually there is none. By itself vorticity is not a source of dipole sound, because the hydrodynamic impulse of a compact region of vorticity that evolves freely in an unbounded fluid, viscous or inviscid, is conserved. The rate-of-change of the hydrodynamic impulse of an isolated distribution (or spot) of vorticity with volume Vv is given by

A. Biesheuvel, R. Hagmeijer / Fluid Dynamics Research 38 (2006) 716 – 742

737

(Saffman, 1992, Section 3.7)  dIv = ue ×
dV , dt Vv where the integral on the right-hand side is the “vortex force” on the vorticity within the spot due to the “external” velocity field ue induced by the vorticity in the other spots; so Iv is not conserved. Each of the spots may now be considered as a source of dipole sound, in reaction to the forcing by the others; however, taken together the vorticity spots do not produce dipole sound. Thus, it is only when there is a fluctuating force acting on an unbounded homogeneous fluid of uniform density, for example the “resistance” due to the presence of a body in the flow, that dipole sound is produced. In that case the body must be considered as the source of the sound, and it is because of the fluctuations in the hydrodynamic impulse of the vorticity that it produces per unit time. When there are no external forces acting on a body moving in a fluid, the acoustic dipole strength is simply given by (57) as 
 dU Gac = − 1 − mB , B dt where mB = B VB denotes the mass of the body; whence, the vorticity distribution surrounding the body is of no concern [A15]. However, appreciable levels of sound intensity result from fluctuations that are only a small fraction of the total force exerted on the fluid; fluctuations that may be hard to determine accurately from experimental observations or numerical simulations. The theory of vortex sound, viz. the expressions due to Burgers, Kambe and Howe, then provides an excellent means of estimating the dipole radiation. Acknowledgments A large part of this work was carried out when A.B. was a guest at the Laboratoire de Mécanique des Fluides et d’Acoustique at the Ecole Centrale de Lyon and at the Department of Mechanical Engineering of the University of Tokyo. He gratefully acknowledges the warm hospitality shown by Professors Michel Lance and Richard Perkins in Ecully and by Professor Yoichiro Matsumoto and Dr Shu Takagi in Tokyo. Appendix A. Remarks A1. In Burgers’s words “the resistance is the force that needs to be exerted on the body to overcome the resistance by the presence of the fluid”, hence it is equal to minus the force that the fluid exerts on the body, which we denote D. A2. Burgers considers this as self-evident. Actually, the ingenious prove due to Sir Horace Lamb (see Lamb, 1932, art. 119), known to Burgers of course, concerns an inviscid fluid with no vorticity present in the interior of the flow. A proof along similar lines, but for a viscous fluid in rotational motion has been given first by Sir James Lighthill (1986a). A3. To prove this vector identity, note first that in three dimensions x × (∇ × a) = ∇(x · a) − ∇ · (xa) + 2a,

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A. Biesheuvel, R. Hagmeijer / Fluid Dynamics Research 38 (2006) 716 – 742

so that, using the divergence theorem,    2 a dV = x × (∇ × a) dV − [n(x · a) − n · (xa)] dS. Next, rewriting the surface integral using the identity x × (n × a) = n(x · a) − n · (xa), yields the required expression. A4. For example (cf. Saffman, 1992),
   d 1 j
1 dV  x ×
dV =  x× dt 2 V +VB 2 V +VB jt  1 =  x × [∇ × (u ×
)] dV 2 V +VB   1 1 +  x × ∇ 2
dV + x × (∇ × F) dV . 2 V +VB 2 V +VB By taking a = u ×
in the vector identity (6) and using that
vanishes at infinity the first integral on the right-hand side can be written as    1 2 u n − u(u · n) dS,  u ×
dV =  V +VB S∞ 2 where S∞ is a very large surface “at infinity”. The surface integral vanishes if asymptotically u=O(|x|−3 ), which we assume. Next, taking a =∇ 2 u in (6) and applying the divergence theorem shows that the second integral on the right-hand side can be written as   1  (n · ∇)u dS +  x × (n × ∇ 2 u) dS, 2 S∞ S∞ which can subsequently be shown to vanish, given the asymptotic behavior of u. The required result finally follows by taking a = F in (6), and using that F vanishes at infinity. A5. Note that “. . .this thin layer of vorticity at the body surface”, in the words of Burgers, “is not the boundary layer from the theory of Prandtl”. It is then compared with the latter, the vorticity in the boundary layer forms part of the actual distribution of vorticity in the flow. A6. For example (Saffman, 1992, Section 4.2), let a velocity potential  for the flow surrounding the body take the value B on the surface SB , and consider the potential P, defined as the solution of ∇ 2P = 0

inside SB ,

P = B on SB .

Since P and  are equal on SB , so are their tangential derivatives on SB ; hence n × ∇P = n × ∇ 

on SB .

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739

With the help of the divergence theorem and the vector identity (6), taking a = ∇P , it follows that   1 1 x × (n × ∇ ) dS = x × (n × ∇P ) dS 2 SB 2 SB    ∇P dV = − P n dS = − n dS. = − VB

SB

SB

By the same vector identity one obtains    1 1  x ×
dV −  x × (n × Us ) dS =  u dV = VB U. 2 VB 2 SB VB Setting  =
 and
= 0 inside SB yields the required result. A7. A different view is that for an ideal fluid the no-slip condition does not apply; fluid particles can slide freely past the surface of a solid body. The difference between the two views is well exemplified by the Klein’s 1910 Gedanken experiment, known as the Kaffeelöffel experiment, in which a very thin disc in an unbounded ideal fluid is impulsively set into motion in a direction normal to its plane and immediately after that is removed or dissolved (see Saffman, 1992, Section 6.1). The result of the experiment is a continuous unbounded fluid with a certain hydrodynamic impulse. In the Burgers’s view the fluid particles have acquired vorticity, and the hydrodynamic impulse resides in two material vortex sheets with equal strength “there where the body has been”. In the alternative view the fluid particles have not acquired vorticity and the fluid motion is irrotational; the hydrodynamic impulse is the virtual momentum of a non-material vortex sheet. A8. In view of what follows, note that the vortex lines of the distribution of additional vorticity form closed loops within the fluid outside the body surface. Vortex lines inside the body are, when they end at the body surface, continued in the thin layer of vorticity with surface concentration (13). A9. The velocity field induced by a closed line vortex in an unbounded fluid is given by the Biot–Savart law as  
1 u(x) = ∇x × dl. 4|x − y| By Stokes’s theorem this relation may be written as  
 1 × n dA(y) , ∇y u(x) = −∇x × 4|x − y| from which, by the use of standard vector identities,  
 1 dA(y) u(x) = −  n · ∇x ∇y 4|x − y|   
1 = ∇x dA(y) . n · ∇ x 4|x − y| This is the irrotational velocity field produced by a uniform distribution of dipoles of strength n over the open surface surrounded by the line vortex. A10. Replace in [A9] the free-space potential by S and use in the last step that S must be symmetrical in x and y according to the principle of reciprocity, which states that the velocity potential at position x

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A. Biesheuvel, R. Hagmeijer / Fluid Dynamics Research 38 (2006) 716 – 742

due to a unit source at position y is equal to the velocity potential y due to a unit source at x, everything else being the same (Rayleigh, 1896, art. 296). A11. Kambe (1987) first shows that two different methods of obtaining the hydrodynamic far-field pressure distribution produced by the vorticity around the stationary body (based on Howe, 1975) yield  
 j 1 p(x, t) ∼

ij nj dS jxi 4|x| SB and



d p(x, t) ∼ − dt



 V


i ·
dV

j




jxi

 1 , 4|x|

and subsequently proves the equivalence of the two terms between curly brackets directly. Note that the second expression agrees with what is found here because in the hydrodynamic far-field p ∼ −j/jt. A12. This means that for a dipole of strength m at position x = x0 , the modification of the far-field dipole strength due to the presence of a body near the origin correction is given by  1 x × [n × (∇ 0 + ∇ v )] dS = −m · ∇x i (x0 ), 2 SB where 0 is the velocity potential due to the “free-field” dipole and v is the velocity potential defined by ∇ 2 v = 0

with n · ∇ v = −n · ∇ 0 on SB ,

v → 0 as |x| → ∞.

To actually proof this, one may employ a method based on the Appendix of Wells (1996). Observe first that by a similar procedure as in [A6]   1 x × [n × (∇ 0 + ∇ v )] dS = − (0 + v ) n dS. 2 SB SB With ni = n · ∇ i on SB , it follows from Green’s theorem that     v ni dS = v n · ∇ i dS = i n · ∇ v dS = − SB

SB

SB

SB

i n · ∇ 0 dS.

Again using Green’s theorem, now to the volume outside the body, yields   (0 ni − i n · ∇ 0 ) dS = − (0 ni − i n · ∇ 0 ) dS, SB

S

where S is the surface of a small sphere with radius surrounding the “free-field” dipole at position x0 . Writing i (x) = i (x0 ) + (x − x0 ) · ∇x i (x0 ) on S one obtains   (0 ni − i n · ∇ 0 ) dS = −∇x i (x0 ) · (0 n − {x − x0 }n · ∇ 0 ) dS, SB

S

which on letting become vanishing small becomes  (0 ni − i n · ∇ 0 ) dS = m · ∇x i (x0 ). SB

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A13. This may be compared by the observation of Lamb (1932, art. 150) that the velocity potential at x due to a thin vortex ring with strength  is equal to the flux through the aperture of the vortex due to point-source of strength  at x; a result that follows immediately from the reciprocal theorem. A14. Briefly, rewrite the integral as
 
    jYi j∇  d + ∇  · ∇Yi (∇  · n) dS = · n dS. Yi ∇  · n dS − Yi jt dt jt SB SB SB Next, by the condition n · ∇Yi = 0 at SB this is also
  
  
d j∇  j∇Yi ·n − · n dS. Yi [Yi (∇  · n) − (∇Yi · n)] dS − dt jt jt SB SB By Green’s theorem the integrals may be transformed to integrals over a large surface at infinity S∞ , say, which yields
  
  
d j∇  j∇Yi − ·n − · n dS Yi [Yi (∇  · n) − (∇Yi · n)] dS + dt jt jt S∞ S∞ and subsequently can shown to vanish on account of the asymptotic properties of the potentials Yi and . A15. An amusing illustration of this simple result is given in Lighthill (1991): that fish swim so silently is because the mass of a fish is not very different from the mass of the fluid that it displaces. References Burgers, J.M., 1920. On the resistance of fluid and vortex motion. Proc. K. Akad. Wet. 23, 774–782. Howe, M.S., 1975. Contributions to the theory of aerodynamic sound, with application to excess jet noise and the theory of the flute. J. Fluid Mech. 71, 71625–71673. Howe, M.S., 1989. On unsteady surface forces, and sound produced by the normal chopping of a rectilinear vortex. J. Fluid Mech. 206, 131–153. Howe, M.S., 1991. On the estimation of sound produced by complex fluid–structure interactions, with application to a vortex interacting with a shrouded rotor. Proc. R. Soc. London Ser. A 433, 573–598. Howe, M.S., 1995. On the force and moment on a body in an incompressible fluid, with application to rigid bodies and bubbles at high and low Reynolds numbers. Q. J. Mech. Appl. Math. 48, 401–426. Howe, M.S., 1998. Acoustics of Fluid–Structure Interaction. Cambridge University Press, Cambridge. Howe, M.S., 2001. Vorticity and the theory of aerodynamic sound. J. Eng. Math. 41, 367–400. Howe, M.S., 2003. Theory of Vortex Sound. Cambridge University Press, Cambridge. Howe, M.S., Lauchle, G.C., Wang, J., 2001. Aerodynamic lift and drag fluctuations of a sphere. J. Fluid Mech. 436, 41–57. Kambe, T., 1986. Acoustics emissions by vortex motions. J. Fluid Mech. 173, 643–666. Kambe, T., 1987. A new expression of force on a body in viscous vortex flow and asymptotic pressure field. Fluid Dyn. Res. 2, 15–23. Lamb, H., 1932. Hydrodynamics. sixth ed. Cambridge University Press, Cambridge. Landau, L.D., Lifshitz, E.M., 1959. Fluid Mechanics. Pergamon Press, Oxford. Lighthill, M.J., 1963. Introduction boundary layer theory. In: Rosenhead, L. (Ed.), Laminar Boundary Layers. Oxford University Press, Oxford, pp. 46–113. Lighthill, J., 1978. Waves in Fluids. Cambridge University Press, Cambridge. Lighthill, J., 1979. Waves and hydrodynamic loading. In: Proceedings of the Second International Conference on Behaviour of Off-Shore Structures, BHRA Cranfield, vol. 1, pp. 1–40. Lighthill, J., 1986a. An Informal Introduction to Theoretical Fluid Mechanics. Oxford University Press, Oxford. Lighthill, J., 1986b. Fundamentals concerning wave loading on offshore structures. J. Fluid Mech. 173, 667–681.

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Lighthill, J., 1991. Hydrodynamic far fields. In: Miloh, T. (Ed.), Mathematical Approaches in Hydrodynamics. Society for Industrial and Applied Mechanics, Philadelphia, PA, pp. 3–20. Möhring, W., 1978. On vortex sound at low Mach number. J. Fluid Mech. 85, 685–691. Nieuwstadt, F.T.M., Steketee, J.A. (Eds.), 1995. Selected Papers of J. M. Burgers. Kluwer, Dordrecht. Powell, A., 1964. Theory of vortex sound. J. Acoust. Soc. Am. 36, 177–195. Quartapelle, L., Napolitano, M., 1983. Force and moment in incompressible flows. AIAA J. 21, 911–913. Quartapelle, L., Valz-Gris, F., 1981. Projection conditions on the vorticity in viscous incompressible flows. Int. J. Numer. Methods Fluids 1, 29–44. Rayleigh, Lord., 1896. The Theory of Sound. Macmillan, New York. (Reprinted by Dover, 1945). Saffman, P.G., 1992. Vortex Dynamics. Cambridge University Press, Cambridge. Wells, J.C., 1996. A geometrical interpretation of force on a translating body in rotational flow. Phys. Fluids 8, 442–450. Wu, J.C., 1981. Theory for aerodynamic force and moment in viscous flows. AIAA J. 19, 432–441.