Structure of Turbulence in Boundary Layers

Structure of Turbulence in Boundary Layers W . W . WILLMARTH Department of Aerospace Engineering The University of Michigan. Ann Arbor. Michigan

1. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . I1. Development of Turbulent Boundary Layer Flow . . . . . . . . . . . 111. Background Knowledge of the Structure of Turbulence Prior to 1955 . . . A . Turbulence-Intensity Profiles and the Effect of the Wall . . . . . . . B. Production and Dissipation of Turbulent Energy . . . . . . . . . . C. Comments on Classical Measurements of the Structure of Turbulence . IV. Recent Developments in Research on the Structure of Turbulence . . . . A . The Various Regions of the Boundary Layer and Their Time and Length Scales . . . . . . . . . . . . . . . . . . . . . . . . . . . . B. Modern Experimental Techniques Used in Measurements . . . . . . C . Space-Time Correlation Measurements and Convection Effects . . . . D. Measurements in the Intermittent Region . . . . . . . . . . . . . E . Measurements of the Structure of the Viscous Sublayer . . . . . . . F. The Occurrence of Bursts . . . . . . . . . . . . . . . . . . . . G. Measurements of Statistical Properties of Turbulence . . . . . . . . V. Discussion of Coherent Structures . . . . . . . . . . . . . . . . . . A . The Burst Sequence . . . . . . . . . . . . . . . . . . . . . . . B. Cyclical Occurrence of Bursts . . . . . . . . . . . . . . . . . . List of Symbols . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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I Introduction In this review of the status and extent of our present knowledge of the structure of turbulence in boundary layers. only the simplest case of incompressible flow over a smooth. plane surface with zero pressure gradient is 159

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considered. Rapid advances in our knowledge of turbulent structure have been made in the past ten years, but our understanding of it is far from complete. In this chapter we will present those fundamental topics that we think are necessary for better understanding of the subject and take stock of our present knowledge of turbulent structure. In the absence of a viable theory for turbulent structure this chapter will necessarily.concentrate upon a summary of the results of recent experimental research. Modern experimental investigations employing sophisticated flow visualization and computer-aided data processing techniques have revealed a large amount of new information. We shall discuss these new results and briefly mention the nature of existing theoretical investigations when appropriate. Many excellent reviews of our knowledge of the turbulent boundary layer have already appeared. An excellent summary of the knowledge of the mean flow field in constant pressure, incompressible boundary layers was written by Clauser (1956). The papers by Coles (1955, 1956) also contain a careful analysis and survey of knowledge of mean profiles. Rotta (1962) has produced a monograph on turbulent boundary layers in incompressibleflow containing many of the significant results that had been obtained at that time. The book on turbulence by Hinze (1959) also summarizes numerous results of research on turbulent boundary layers. Recently Kovasznay (1970, 1972) and Laufer (1972) have written reviews which describe the status of research on turbulent boundary layers and treat some aspects of research on turbulent structure. Mollo-Christensen (1971) has also presented an excellent review of the physics of turbulent flow and includes a detailed discussion of the transition process. The present chapter deals exclusively with the structure of the fluctuating turbulent flow field. 11. Development of Turbulent Boundary Layer Flow In boundary layer flows the nature and structure of the flow are controlled by the vorticity which is produced by the passage of initially irrotational fluid over the wall. Let us review the concepts of vorticity production, diffusion, and convection as they occur in boundary layers developed on rigid walls. In the following discussion we rely on Lighthill (1963) for motivation, guidance, and stimulation. In Lighthill's view (1963),vorticity considerations are needed to place the boundary layer correctly in the flow as a whole. Vorticity considerations illuminate the detailed development of the boundary layer just as clearly as do momentum considerations and allow a compact description of the dynamics of turbulence. To start our discussion we recall that an inviscid fluid

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initially devoid of vorticity will continue to have zero vorticity, but a viscous fluid, which is initially irrotational, will acquire vorticity when a solid obstacle is passed through the fluid. Vorticity is imparted to the fluid through the agency of the viscous boundary condition that the fluid in contact with the surface adheres to the surface. Thus, a reasonably flat solid boundary in a flow acts as a distributed source of vorticity tangential to the boundary. If one considers a flow in the x direction over a plane surface (the x,z plane), the flow of z vorticity out of it is (Lighthill, 1963) am, - v-= aY

-v-

= v

where y is the distance normal to the surface. In (2.1) the vanishing of the velocity components at the wall requires that at every instant the surface forces acting upon a fluid element in contact with the wall be in balance. The meaning of (2.1) is that the pressure gradient along a wall creates vorticity tangential to the surface in the direction of the surface isobars. The sense of rotation is that of a ball rolling down the line of steepest pressure fall, and the magnitude of the vorticity source strength per unit area [ - v dcoz/ay in Eq. (2.1)] is of magnitude l/p times the pressure gradient. Now consider the development of the flow in the boundary layer about a solid body (Lighthill, 1963). Let us imagine that the flow development is to be computed with a large, high-speed digital computer. The computation is carried out in steps, and the computer is used to monitor and keep account of the process of vorticity development. Before the initial step, the flow is stationary but during the initial step, when the body begins to move, an inviscid flow is produced which satisfies the condition of no flow through the surface. At the end of the first small time increment (or step) one computes the potential flow field which satisfies the conditions that the velocity vanish (sufficiently rapidly) far from the body and that the flow component normal to the surface also vanishes. Having satisfied the condition of zero flow component normal to the boundary, the vanishing of the tangential velocity component is ensured by the introduction at the surface of a sheet of vorticity tangent to the surface whose strength at any point is such that the velocity component tangent to the boundary becomes zero. This completes the initial step of the flow development and computation. Notice that in this computation of flow development using vorticity, the pressure and its gradient do not enter explicitly, although physically it is the pressure gradient that causes the fluid at the boundary to move in such a way that a sheet of surface vorticity is produced.

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The presence of the sheet of surface vorticity makes the next and succeeding steps in the computation considerably more complicated. First the normal component of vorticity is deduced from the previously introduced tangential vorticity distribution at the surface using the fact that the vorticity field is solenoidal. Then the time interval is advanced one small step. At the end of the small time step the new position of the solid surface is computed. The evolution of the vorticity during the time step caused by convection, stretching, and diffusion is now computed. Here one uses the equation

a0

-=

at

-(q.V)u,

+ (0 - V)q + v v 2 0 :

obtained by taking the curl of the momentum equation. In (2.2)the first term on the right-hand side is the convection term, the next is the stretching (or compression) term, and the last is the diffusion term. (An excellent discussion of the terms in this equation has been given by Whitham, 1963.)At this point the velocity field produced only by the vorticity field is computed by the Biot-Savart law

where r is a vector giving the position at which q relative to the volume element dV in the vorticity field is to be determined. Then the remainder of the velocity field can be computed. This portion is a potential flow field which satisfies the condition that when added to the previously determined velocity field produced by vorticity the flow component normal to the body surface is zero. (Part of the potential flow field is produced by “image” vorticity within the solid body.) Finally, the condition of no flow tangential to the surface is satisfied by the introduction at the surface of a new distribution of vorticity tangent to the surface. At this point the computational process is complete. The time can be advanced another step and the above computation repeated. According to Lighthill (1963) there are definite advantages in computing or thinking about flow fields in terms of vorticity. For example, in computing or thinking about a given problem one can confine one’s attention mostly to the region containing vorticity. In addition, the vorticity distribution varies smoothly and is not subject to large peaks as is the pressure during impulsive motion for example. The above computational process has been used successfully for the flow about a circular cylinder (for example, see Payne, 1958). In this chapter the description of the process of flow development in terms of vorticity is emphasized because it sheds light upon the nature of the turbulent structure in the boundary layer. At relatively high Reynolds

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number the boundary layer is confined to a thin layer near the wall. It is a thin region because high Reynolds number means that the ratio of convection rate of vorticity (of order Uo)to diffusion rate of vorticity normal to the flow (of order vo/6) is large. Since the boundary layer is thin, the process of flow development described above makes it clear that there is a strong interaction between the convection, diffusion, and creation or annihilation of vorticity. The role of the wall as a source of both mean and fluctuating vorticity is of primary importance for an understanding of the structure of turbulence in the boundary layer. Consider, for example, the idealized, classical case of a laminar boundary layer with zero pressure gradient. In this idealized case the singular point at the leading edge of a half-infinite flat surface must be excluded. The excluded region is, however, very important because it contains a steady source of tangential vorticity whose diffusion and convection result in the boundary layer flow downstream. The fact that there is no streamwise pressure gradient downstream of the singular point at the leading edge means that with the exception of the leading edge the tangential-vorticity source strength is zero all along the surface; see (2.1). When the Reynolds number is increased, the laminar boundary layer is no longer stable. At this time the laminar flow becomes unsteady and on a smooth flat plate initially two-dimensional waves (called TollmienSchlichting waves) develop. The waves do not remain two-dimensional for long, and soon small spanwise variations occur within the two-dimensional wave system. A little later spanwise variations in the instantaneous profiles of streamwise velocity are observed which rapidly intensify at certain locations to develop localized regions of concentrated vorticity. Kovasznay et al. (1962) have suggested that the rapid development of concentrated vorticity is caused by the stretching of streamwise vorticity produced when the initial spanwise velocity variations interact in a nonlinear fashion (Benney and Lin, 1960; also see Benney, 1961, 1964) with the two-dimensional TollmienSchlichting waves. The theoretical and experimental papers which lead to this description of the beginning of the transition process can be found in two excellent reviews (from a theoretical point of view) by Stuart (1965)and (from an experimental point of view) by Tani (1967). Hama and Nutant (1963) and Knapp et al. (1966) have made visual studies of the transition process in which the flow structures responsible for the flow development during transition may be observed. Quantitative and detailed measurements of the flow field during the transition process have been made by Klebanoff et al. (1962) and by Kovasznay et al. (1962).Figure 1 is a photograph of the transition process and flow structure that has been made visible by a layer of smoke on the surface of the cylinder. This remarkable picture shows the entire process of transition initiated by acoustic

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FIG.1. Flow visualization of transition process on a cylinder using smoke. Transition produced by acoustic disturbances in the flow. Photograph by C. W. Ingram.

disturbances. It was made by C. W. Ingram using techniques of F. N. M. Brown (see Knapp et al., 1966). In this picture even the small-scale disintegration (or confusion) of the concentrated vorticity pattern which marks the initial stage of the development of a random turbulent flow may be observed. After development of localized regions of concentrated vorticity, a poorly understood process occurs within these regions in which intense random velocity fluctuations are produced. The region of random chaotic motions then enlarges as the flow carries the diffusing and evolving vorticity concentrations downstream. In naturally occurring transition on a smooth flat plate the localized chaotic motions (turbulent spots) are observed to occur at random locations in space and time once the Reynolds number is large enough to produce an unstable laminar layer. Emmons (1951) was the first to observe and describe the process of the formation and growth of the turbulent spots. He produced a laminar flow of water on an inclined sheet of glass. At sufficiently high velocity, the turbulent spots could be detected by the change in the light reflected from the initially smooth free surface of the water. The turbulent spots were observed to grow as they were convected downstream, but a sharp interface was maintained between the turbulent fluid containing random vorticity within the growing spot and the surrounding laminar flow. The trajectory of the edges of the spots were observed to form an included angle of the order of 20" to the flow direction. Explanations for the occurrence of the sharp interface between the turbulent and nonturbulent fluid and for the so-called turbulent contamination angle between the stream and the edge of the enlarging turbulent spot (of the order of 10") are still lacking. Fully developed turbulent flow is finally observed far downstream of the transition region. The fully developed turbulent flow appears to occur after the merging of numerous turbulent spots initially produced at random upstream locations. There appears to be no systematic study of when the structure of the turbulent fluctuations attain a statistically stationary value

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after natural transition occurs. At present the attainment of a universal mean flow distribution may be ascertained by measuring mean velocity profiles and comparing them with previously well-documented results (Clauser, 1956; Coles, 1955, 1956). This provides a necessary but perhaps not a sufficient condition for the attainment of a statistically stationary and universal structure of turbulent fluctuations. Owing to the dominance of the effect of streamwise convection over the transverse diffusion of turbulence, most investigators have used tripping devices to hasten the transition process. The use of a tripping device is necessary if the flow facility is of limited streamwise extent. However, the effect of the tripping device upon the evolution and eventual attainment of a universal statistical structure of turbulent fluctuations (especially the larger eddies) has not been reported. A few observations were made by the author of the effect of discrete tripping devices several feet upstream of the transition region upon the large eddy development 30 ft downstream of the region of transition in a thick turbulent boundary layer at Re, N 38,000. In this case at the 30-ft station the eddy structure was noticeably larger when the tripping devices were present. It appeared from the smoke traces (viewed along the edge of the boundary layer in the downstream direction) that large streamwise vortices were present when discrete tripping devices were used. It is the author’s opinion that there may well be important variations in the turbulent structure owing to differences in boundary layer initiation in different experiments. One must view experimental results, especially of measurements involving large eddy structure, with some caution. Our basic experimental knowledge of the structure of turbulence is only rudimentary at present. Thus, we are forced to study mainly its gross features. It is probably not yet necessary to be extremely careful with regard to the more subtle features of turbulence structure that may not be strictly reproducible between one experiment and the next. However, the time will soon come when it will be necessary to bring together and document the existing measurements of the structure of turbulence in the same fashion that Clauser (1956) and Coles (1955, 1956), for example, have done for the mean flow. 111. Background Knowledge of the Structure of Turbulence prior to 1955

Hot wire anemometry [see Kovasznay (1954) and Melnik and Weske (1968) for modern and historical descriptions] is the basic technique used for most of the quantitative measurements of the structure of turbulence. The results of the early measurements (until approximately 1955) have been

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collected and are available in books or review papers (see Townsend, 1951; Hinze, 1959; Rotta, 1962). There were very many papers produced during this early period. It is not possible to provide a complete coverage of them in this review. However, the framework of knowledge gained from the early measurements will be mentioned in order to provide background for the present review of modern experimental results (after 1955 approximately).

A. TURBULENCE-INTENSITY PROFILES AND THE EFFECTOF THE WALL Many of the studies of the structure of turbulence made during this early period consisted of measurements of the root-mean-square and spectra of the turbulent velocity fluctuations. Fluctuations of streamwise velocity and the two velocity components transverse to the stream were measured. The hot wire array with the X configuration was used for the measurements of the transverse velocity components. The measurements were not simple (in fact most measurements of turbulent quantities are usually very difficult). The difficulties are of two kinds: those pertaining to obtaining faithful response at high frequencies or for small-scale fluctuations; and those (already mentioned in Section 11)connected with obtaining or setting up a boundary layer flow whose structure of turbulence is acceptably uniform and repeatable. As a matter of fact, the author made a single, unpublished attempt, in 1960, to bring together the then existing results of turbulence-intensity profiles of the boundary layer on a single plot. The curves of u'/uI,v'/u,, and w ' / q , as functions of y/S (or of y') from various published measurements (including our own) did not agree very well (not within -t 50%). Part of the reason for the lack of agreement may have been caused by free-stream disturbances [see Bradshaw (1965) for an example of curious disturbances, caused by turbulence damping screens] or by differences in the methods used to trip the boundary layers. Among the measurements considered were those of Townsend (195l),Klebanoff (1954),Klebanoff and Diehl(1952),and Willmarth and Wooldridge (1963), and for flow near the wall (in a pipe) Laufer (1954). In spite of the differences between various measurements of turbulence intensity it is definitely established (in all the measurements) that within the boundary layer, u'/u, > d / u , > u'/U,. These differences between the root-mean-square velocity fluctuations become larger as one approaches the wall. Finally, very near the wall, the fluctuations begin to damp out through viscous action within the viscous sublayer. A plausible explanation for the difference in magnitude of the fluctuating velocity components is the following. The presence of the wall makes it necessary that the normal component of velocity vanish. The vanishing of

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the normal velocity fluctuations is ensured by replacing the wall with an exactly identical fluctuating field of image vorticity beneath the wall, thus canceling the normal component of velocity at the location of the wall. If a distribution of sources or higher singularities were present in the flow (in a compressible boundary layer for example), one would also add their images beneath the wall. The image field of vorticity and other singularities will reduce normal velocity fluctuations but does nothing to suppress tangential velocity fluctuations (in fact the effect is generally to enhance them) therefore u’ < u’, w’. The other effect of the wall is to create, through the nonslip boundary condition, a mean shear field in which the streamwise velocity gradient becomes large as one approaches the wall. This means that whenever u fluctuations do occur (even though they are suppressed by the presence of the wall), they will bring fluid parcels with high streamwise velocity nearer to the wall or move lower speed fluid parcels farther from the wall. This will cause u’ to be greater than w‘ and is incidentally the same mechanism responsible for the Reynolds stress which will be discussed later in detail. Uzkan and Reynolds (1967) have reported an interesting experiment devised to study a shear free turbulent boundary layer. In their experiment they created a turbulent flow behind a grid adjacent to a wall and then, as the turbulence was carried downstream, caused the wall to move at speeds near the stream speed. New turbulence was produced only when the wall velocity did not exactly match the stream speed. Uzkan and Reynolds found that even when the wall moved at the stream speed the wall suppressed the streamwise velocity fluctuations. They also state that they believe that the wall would have suppressed the normal velocity fluctuations to a greater extent than the u and w fluctuations (unfortunately they did not measure v or

4-

To emphasize further the effect of the wall let us consider an initially laminar boundary layer flow that undergoes transition and becomes turbulent. In the boundary layer the turbulent mixing or stirring acts generally to bring high-speed fluid closer to the wall and carries low-speed fluid from the wall region far out from the wall. The result is that the mean velocity near the wall is higher and far from the wall the mean velocity is lower than in a laminar boundary layer at the same Reynolds number. Figure 2 is a sketch of the two profiles. Note that the turbulent mixing causes the profile of the turbulent velocity to intersect that of the laminar velocity at one point near the wall and that the turbulent boundary layer is thicker than the laminar one. It turns out that considerable quantitative documentation for this description of the effect of turbulent mixing on the development of the turbulent profile has been obtained from recent experimental studies using methods of

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I

y

1 I

I

I I I

II

L-L I

___---U/U-

FIG.2. Sketch of laminar flow profile and turbulent flow profile immediately after transilaminar; - - -, turbulent. tion to turbulence. ---,

conditional sampling. Kovasznay et al. (1970) have studied the flow in the intermittent region far from the wall, and Zaric (1972)has studied the region very near the wall. Both papers, which will be discussed later, illustrate the power of new experimental techniques developed in the past ten years. Another fundamental property of the turbulent mixing described above is that, as pointed out by Lighthill (1963), it concentrates most of the mean vorticity much closer to the wall than in a laminar boundary layer. Figure 3, from Lighthill (1963), shows results taken from the measurements of Schubauer and Klebanoff (1955) before and after transition. Curve (i) is the laminar distribution of mean vorticity just before transition at Re, = 2.3 x lo6. Curve (ii) is the distribution of mean vorticity in the turbulent boundary layer just after transition at Re, = 3.3 x lo6.The turbulence redistributes the mean vorticity so that most of it is very near the wall and the mean /,u) is eight times the laminar value. A small portion vorticity at the wall (q,, (some 5% of the total) of the mean vorticity is now found much farther from the wall. The turbulent mixing has caused the mean vorticity to migrate out farther from the wall (in a region of intermittently turbulent flow) than would have occurred in the normal development of the laminar layer. Curve (iii) in Fig. 3 gives rough values of the root-mean-square fluctuation of vorticity in the fully turbulent region after transition. The vorticity fluctuations attain a high maximum very close to the wall near the edge of the sublayer and extend to the outer edge of the boundary layer.

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FIG.3. Distribution of mean vorticity in a boundary layer with uniform external flow: (i) at beginning, (ii) at end, of transition. Curve (iii) gives rough values of the root-mean-square vorticity at end of transition. From Lighthill (1963).

It is indeed remarkable that the turbulence in the boundary layer is able to maintain large gradients of mean and fluctuating vorticity near the wall despite the large viscous diffusion down the gradient. The processes that accomplish this are central to an understanding of the structure of turbulence in the boundary layer. It is our aim in this chapter to bring together what is currently known about this structure so that we can obtain a better understanding of it.

B. PRODUCTION AND DISSIPATION OF TURBULENT ENERGY In a turbulent flow of an incompressible fluid the energy of the turbulence may be expressed as the kinetic energy per unit mass (u2)/2. Using the momentum equation and introducing the fluctuating velocity we can write the energy balance equation for the average of the fluctuating turbulent energy at a given point in the flow as

For the derivation of this equation, see Hinze (1959), for example.

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One can obtain an equation for balance of the total turbulent energy by integrating each term in (3.1) over the entire volume containing turbulence. If the turbulence is confined to a finite region so that the pressure and velocity fluctuations vanish on the boundaries of this region, then the integral over this region of the divergence term on the right-hand side vanishes. For an incompressible fluid the integral of the second term on the left-hand side will also vanish because it can also be written as a divergence. The remaining terms are simply

This equation expresses the time rate of change of the total turbulent energy as the sum of a production term (the first term on the right-hand side) and a dissipation term (the other term on the right-hand side) proportional to the viscosity. Let us consider the production and dissipation term in (3.1) at a given location within the boundary layer. If at a given point turbulent energy is produced, it is necessary that there be a mean velocity gradient dUj/axi and a Reynolds stress uiuj at the point in question. As we have already discussed, the turbulent mixing transports high-speed fluid toward the wall and moves away from the wall region. Both processes create Reynolds low-speed fluid__ stress so that uI u2 becomes negative, and at the same time the mixing of fluid causes a large increase in the mean velocity gradient 8U ldx, near the wall. The result is that the production term is positive and of large magnitude near the wall. It turns out that the gradients of the velocity fluctuations are very large near the wall with the result that the dissipation term is also large near the wall just where the production term is largest. Measurements of the production term and approximate measurements of the dissipation term and many of the other terms in Eq. (3.1) have been reported by a number of investigators. Many of these measurements were carried out some 15 or more years ago. The first measurements were made by Townsend (1951). Other measurements by many other people are described by Hinze (1959)or Rotta (1962), among others. In addition to measuring the terms in (3.1),Townsend (1956) also measured the terms in the energy balance equation for the mean motion. The result of his investigation was the discovery that the loss of mean flow energy to turbulence becomes large as the wall is approached. The conclusion is that energy from the mean flow is transferred to the turbulence primarily near the wall where the energy dissipation by turbulence is also largest. ~

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C. COMMENTS ON CLASSICAL MEASUREMENTS OF THE STRUCTURE OF TURBULENCE At the time when most of the energy balance measurements were made there was very little understanding of the precise nature of the flow field and of the turbulent structure responsible for the energy transfer and dissipation. This lack of understanding is caused by the extreme complexity of turbulent flow and by the very great difficulty of the experimental measurements. A major problem with the early measurements is the fact that use of hot wire anemometry to obtain quantitative measurements had the effect of concentrating attention on large amounts of detailed data. Less attention was paid to the structural features of the boundary layer until recently, when new results using flow visualization methods have stimulated interest in the structure of turbulence. In addition, prior to 1955 one usually averaged the signal from a single hot wire, and this further obscures the physical phenomena of the process one is attempting to study. Mollo-Christensen (1971) has commented upon the fact that averages hide rather than reveal the physics of a process. He presented an extreme example in which a blind man used a single road bed sensor in an attempt to find out what motor vehicles looked like. “Happening to use a road traveled only by airport limousines and motorcycles, he concludes that the average vehicle is a compact car with 2.4 wheels.” The example is appropriate for turbulent flows since the signal from a sensor placed in the flow is caused by passage of randomly occurring and ever evolving entities containing vorticity which are commonly called eddies. In this chapter we shall not attempt to define an eddy further. We shall assume that the reader is familiar with the concepts of two-point correlation and spectrum measurements. Townsend (1956) has discussed some of the problems implicit in deducing structural features from correlation measurements.

IV. Recent Developments in Research on the Structure of Turbulence There has been a gradual but accelerating increase in the pace of publication of research papers on the structure of turbulence. The number of papers on the subject is already very large. Unfortunately, there are a few that the author has not been able to obtain and others that owing to time and space limitations could not be included. The discussion here is separated into a number of topics upon which the research has been focused. We will discuss

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numerous existing results of recent research and attempt to reveal the areas in which it appears to us that current understanding is good and those areas in which fundamental problems exist.

A. THEVARIOUS REGIONS OF THE BOUNDARY LAYER AND THEIR TIME AND LENGTH SCALES Kovasznay (1967) has discussed the boundary layer in terms of four main regions: (1) First is the wall region in which the flow properties are dominated by the presence of the wall. This is often referred to as the region of the law of the wall. (2) Within the wall region adjacent to the wall there is a viscous sublayer. In this region the effect of viscosity is dominant but turbulent fluctuations are still large relative to the mean velocity within the region. (3) Far from the wall there is a large region of nearly homogeneous turbulent flow bounded by the potential flow outside the boundary layer. The flow in this region is intermittent and wakelike. It is known as the region of the law of the wake or simply the wake region. (4) At the outer edge of the wake region there is a thin interfacial region between the turbulent and nonturbulent fluid which Corrsin and Kistler (1955) called the superlayer. Across this relatively thin corrugated region the outer potential flow acquires randomly oriented vorticity through fluctuations in viscous shear stress which are driven by the turbulence within the wake region. Within this region vorticity acquired by the originally potential fluid is stretched and deformed so that the fluid is incorporated into the turbulent flow in the wake region. Note that the flow in the outer potential region is fluctuating and not quiescent, although the fluctuations are devoid of vorticity. Each of these four regions has its own characteristic length scale. In the wall dominated layer, region 1, the length is proportional to distance from the wall. One may remark, however, that the thickness of this region, i.e., the distance from the wall at which the wake region joins it, is a function of Reynolds number. In the viscous sublayer, region 2, the length scale is measured in units of the viscous length v/u,. The thickness of the viscous sublayer is of the order of 5v/u,. The wake region, region 3, has a length scale of the order of the boundary layer thickness 6. The superlayer, region 4, has a length scale set by the smallest turbulent eddies. Kovasznay (1967) states that his estimate of the thickness of the superlayer is 10v/Vo where V, is the entrainment velocity given by Vo = U,(d/dX)(G - 6*).

(44 Phillips (1972) prefers the length scale proposed by Corrsin and Kistler (1955) in which the thickness of the superlayer is assumed to be governed by

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the balance between viscous diffusion of vorticity and mean rate of straining induced by the turbulence. The viscous diffusion rate is proportional to v, while the mean rate of straining is proportional to E~ /v so that the thickness of the superlayer must be measured in terms of the Kolmogoroff microscale (V3/EO)1’4.

The velocity scales in the various regions can also be mentioned. In the wall region, region 1, and the viscous sublayer, region 2, the velocity scale is the friction velocity u, . In the wake region, region 3, the velocity scale is the free stream velocity U , . In the superlayer, region 4, the velocity scale is the average entrainment velocity, i.e., the relative speed of advance of the turbulent interface with respect to the nonturbulent fluid. According to Kovasznay (1967) this is given by (4.1), while Phillips (1972) and Corrsin and Kistler (1955) use the Kolmogoroff velocity scale ( t ov)’I4. Although the boundary layer has been neatly partitioned into four reasonably definite regions, one must not become complacent when considering the structure of turbulence. It has become apparent that there is a strong interaction between these regions. For example, the latest flow visualization studies, Nychas et al. (1973) and Offen and Kline (1973), which we shall discuss, show that eruptions of low-speed fluid from deep within the wall region emerge outward as they are carried downstream and contribute to the corrugations of the superlayer. At the same time Offen and Kline (1973) and Falco (1974) have observed large-scale irrotational fluid parcels within the boundary layer that extend deep into the wall region. The results of recent research are not at all conclusive as regards the nature of the interaction between the various regions of the boundary layer. Indeed, only qualitative visual methods have been able to show that definite interactions caused by the motion of the large-scale eddies do occur. The nature of this interaction and its quantitative assessment are without doubt the central problem in the study of the structure of turbulence in boundary layers.

B. MODERNEXPERIMENTAL TECHNIQUES USEDIN MEASUREMENTS The modern era of experimental research on turbulence has been, to a great extent, dependent upon the development of a new technologyprimarily of electronic devices and computers. In one recently exploited area newly developed electronic techniques (conditional sampling) have allowed us to obtain a glimpse of the rapidly changing intermittent flow in the outer regions of the boundary layer. The first observations of intermittently turbulent flow were made in 1943 by Corrsin (1946) along the boundary of a turbulent jet. Townsend (1949) reported measurements of the intermittency

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and average values in the turbulent portions of a two-dimensional wake. In the boundary layer the on-off nature of hot wire signals was first studied by Corrsin and Kistler (1955). In these studies, electronic gates controlled by the intermittently turbulent signal from a hot wire array sensitive to streamwise vorticity were used to study the boundaries of the intermittent flow. Measurements of averages in the turbulent portions of the flow were limited to those quantities that were zero or very small in the nonturbulent portions. It was not until the development of very fast switching circuitry for analog or digital computation that it became possible to obtain detailed statistical measurements of the flow phenomena within the laminar or turbulent regions of the flow. Detailed measurements in the intermittent region of the boundary layer during turbulent or nonturbulent periods (using conditional sampling) have been made using analog methods by Kovasznay et al. (1970) or using digital computer methods by Kaplan and Laufer (1969), Antonia (1972b), and Hedley and Keffer (1974). Another line of development that has produced many new results was initiated by Favre (1946) who used an analog wire recorder (he later used a tape recorder) to produce a time-delayed turbulent signal. By recording two signals simultaneously and reproducing them using a movable reproduce head on one channel, a variable time delay between the two recorded signals was produced. If the two signals originate from different positions in the turbulent flow, then the space-time correlation of the turbulent fluctuations can be measured. Favre et al. (1957, 1958) have used their method to measure space-time correlations of velocity fluctuations in a turbulent boundary layer. The same method was used by Willmarth and Wooldridge (1962) and Bull (1967) to study convection and decay of wall pressure fluctuations beneath the turbulent boundary layer. Owing to rapid advances in technology it is now possible to perform similar space-time correlation measurements using digital computers (indeed self-contained devices for this purpose using analog and/or digital computing methods are commercially available).The virtue of space-time correlation measurements is that they allow one to obtain quantitative measurements of the convection and decay of turbulent fluctuations that could not previously be obtained using simultaneous spatial correlation measurements. Another intriguing development has been the use of new flow visualization techniques. The study of the sublayer flow and of the transition process using dyed fluid injected at the wall has generated much controversy and interest in flow visualization techniques within the community of research workers in turbulence. Interesting flow structures were observed during transition that were caused by dyed fluid collecting in certain regions. Hama and Nutant (1963) have interpreted these patterns as being caused by con-

Structure of Turbulence in Boundary Luyers

175

centrations of vorticity. Klebanoff et al. (1962) maintained that the dyed fluid particles which formed lines of concentration have nothing to d o with the concentrated pattern of vorticity. In response to this controversy and interest, Hama and Nutant (1963) initiated an important improvement in boundary layer flow visualization methods. They introduced small hydrogen bubbles into the flow that were produced on a fine wire by electrolysis. The bubbles were carried off the wire by the flow and were so small that they were almost completely passive and served as markers that could easily and efficiently be introduced in almost any region of interest in the flow field without producing appreciable disturbances. The method, which was originally developed by Clutter et al. (1959), was successfully used by Hama and Nutant (1963) for visual studies of transition. At about the same time S. J. Kline and his colleagues at Stanford University began the development of the hydrogen bubble method for use in a fully turbulent boundary layer flow. The research work on the visualization of boundary layer flow at Stanford University is summarized in three papers; Kline et al. (1967), Kim et al. (1971), and recently Offen and Kline (1974). As will become apparent later, the research at Stanford University has very considerably increased our understanding of the structure of turbulent boundary layers. A related flow visualization technique is the method of flash photolysis used by Popovich and Hummel (1967) to study the viscous sublayer in a turbulent flow in a pipe. Marker particles in the specially prepared flowing fluid in the form of a dye are produced (without disturbing the flow) by flashing a narrow beam of light into the fluid. Some details of the flow at the wall that could not be obtained in any other way were measured by Popovich and Hummel (1967). However, the method has not yet been fully exploited. The use of a flow visualization method, which was developed many years ago by Fage and Townsend (1932), has been reported by Corino and Brodkey (1969). They and later Nychas et ul. (1973) used a motion picture camera moving with the flow to obtain high-speed magnified photographs of the motion of preexisting tracer particles in the fluid. A detailed analysis of their photographs has revealed new information about frequently occurring coherent flow structures in the turbulent boundary layer.

C. SPACE-TIME CORRELATION MEASUREMENTS AND CONVECTION EFFECTS The first measurements of two-point space-time correlations in boundary layers were made by A. Favre and his colleagues (1957, 1958).The quantity measured was the streamwise velocity component u at two locations of hot

176

W . W . Willmarth

wire probes. Assuming that the turbulent fluctuations are statistically stationary in time, the correlation coefficient R(x’, X, Z) = u(x’,t)u(x’

+ X, t + z)/(u’(x’,

t)u’(x’

+ X, t + T))”’

(4.2)

is a function only of the time delay z between the two velocity signals. If the turbulent flow is statistically homogeneous in space, the correlation coefficient (hereafter called the correlation for convenience) is a function only of the relative separation vector x between the two positions of the velocity sensors. In a boundary layer of slowly increasing thickness the flow is approximately homogeneous in planes parallel to the wall, so that the correlation becomes a function of the distance from the wall of one wire and the separation vector between that wire and the other:

R(x’, x, 7 ) = R(y’, x, z).

(4.3) Favre et al. (1957, 1958) have reported a rather complete set of measurements of the correlation (4.3) for many arrangements y’ and x of the two wires. In their measurements the hot wires were always well outside the sublayer-we estimate that the hot wire probes were never closer to the wall than approximately 40 wall lengths, yu, /v,N 40. Their measurements revealed several new and interesting features about the larger length scales of turbulence well outside the sublayer. These features were: (1) In general the fluctuations in the large-scale streamwise velocity are convected with the local mean speed so that Taylor’s hypothesis may be applied to the boundary layer at distances from the wall greater than 3% of the layer thickness. (2) The isocorrelation surfaces obtained with the optimum time delay have a large aspect ratio in the streamwise direction even when one velocity sensor is relatively close to the wall at 3% of the layer thickness. The optimum time delay is the sum of two time increments that depend upon the relative transverse and streamwise displacement of the two hot wire probes.

For exact details one should consult the original papers, but a qualitative understanding of the optimum time delay is not difficult. The space-time correlation between two points located on a line normal to the wall was a maximum at a certain small time delay which was a function of the distance y separating the two points. When the separation distance y increased, the time delay required to obtain maximum correlation increased. To obtain the optimum time delay, for points separated in both the x and y directions, the above time increment was added to the average convection time (computed under Taylor’s hypothesis) by dividing the x separation distance between the two probes by the average of the mean velocity at the two probes.

Structure of Turbulence in Boundary Layers

177

Figures 4 and 5 from Favre et al. (1957) show the large aspect ratio of the isocorrelation surfaces caused by streamwise fluctuations within the boundary layer. It is extremely significant in view of recent modern work on the occurrence of bursting that in Fig. 4 the signals from one probe at a fixed location near the wall at y' = 0.036 are strongly correlated with signals from the other movable probe even when the movable probe is at y N 0.36. This result illustrates the influence of the larger scale fluctuations upon the flow

X

FIG.4. Space-time isocorrelation surfaces with optimum time delay in the boundary layer on a flat plate; 6 = 33 mm, Re, Y 1600, y'/6 = 0.03. From Favre et al. (1958).

near the wall. Note that in Fig. 5 when the fixed probe is far from the wall, y = 0.776, the isocorrelation contours are still of large aspect ratio but are not strongly correlated with velocity fluctuations near the wall. A study of transverse velocity fluctuations as complete as the study that Favre et al. (1957, 1958) have made of streamwise fluctuations has not yet appeared. Such a study might prove fruitful. The investigation of pressure fluctuations beneath turbulent boundary layers was initiated at about the time (1953) that Favres' boundary layer measurements became known. The first measurements of space-time correlations of the wall pressure were reported by Willmarth (1958) and Harrison (1958). See Willmarth (1975) for a review of the subject. The initial

178

W . W . Willmarth

showed that important convection effects also occur for the wall pressure fluctuations. The overall convection speed of the pressure fluctuations was of the order of 0.8 U , . This result is not inconsistent with the results shown in Fig. 4 in which velocity fluctuations a t distances of the order of 0.2 < y/6 < 0.3 from the wall where U N 0.8U, have considerable correlation with the velocity fluctuations near the wall. These early measurements were soon refined by Willmarth and Wooldridge (1962), Hodgson (1962), and Bull (1967). The convection and decay measurements of the wall pressure for various streamwise spatial separations will be discussed in terms of the variable time delay z between the pressures measured at the transducers, and of measurements of pressure correlations in narrow frequency bands. One should consult the results of Bull (1967) if detailed information is desired in terms of narrow-band correlations [which in the limit of vanishing bandwidth become the temporal Fourier transform of R,,(x, z)]. Figure 6 from Willmarth and Wooldridge (1962) shows the wall pressure correlation as a function of dimensionless time delay and streamwise spatial separation. There is a ridge of large pressure correlation running out and decaying in the first quadrant of the x1/6*,

Structure of Turbulence in Boundary Layers

179

FIG. 6. Longitudinal space- time correlation of the wall pressure. From Willmarth and Wooldridge (1962).

zCJ,/6* plane. The presence of the ridge of high correlation in the first quadrant indicates that coherent pressure-producing eddies are carried downstream between pressure transducers separated by a distance x1 and a time z. As x1 increases, one must wait for a long time z for an eddy to arrive. Moreover, as x1 increases, the eddies gradually lose their identity (i.e., become incoherent) so that the height of the ridge becomes smaller. It turns out that the trace of the ridge crest in the (xl /6*, T U , /6*) plane is not a straight line but is curved. The slope of the ridge trace d(x, /d*)/d(zU, /a*) in the x1plane increases as the distance from the origin along the ridge increases. One can regard the slope of the ridge trace as a measure of the convection velocity of the wall-pressure fluctuations. The convection becomes larger as x1 increases. This is interpreted as a spatial filtering effect in which for large separations between measuring stations the pressure fluctuations produced by small-scale eddies become incoherent during the travel time between the two measuring stations. Only the largescale pressure disturbances retain their identity (coherence) during the passage between measuring points. Since the effective center of the larger scale eddies is farther from the wall where the mean velocity is greater, the convection velocity is increased for space-time correlations measured with widely separated transducers. To illustrate the variation of convection velocity with size of the convected pressure fluctuations, let us consider the results of longitudinal space-time correlations measurements in narrow frequency bands, shown in Fig. 7. These measurements show that for a band centered at low frequencies the convection velocity is higher than for a band centered at high frequencies.

180

W . W . Willmarth Experiments Theory, Re = 5,000 Theory, Re = 10,000 -. .- ..- Theory, Re = 40,000

0

2

4

6

w8*/U,

FIG.7. Convection velocities of the wall pressure in narrow frequency bands. Experiments by Willmarth and Wooldridge (1962) as analyzed by Corcos (1964). Theory and figure from Landahl (1967).

The reason is that the larger convected eddies are responsible for the majority of the low-frequency contributions to the space-time correlation and move more rapidly because they extend to larger distances from the wall where the mean velocity is higher. One can also estimate the decay of pressure producing eddies of various sizes from measurements of space-time correlations in narrow frequency bands Figure 8 shows the results of early measurements of this type in which the decay is scaled by use of the convection velocity U,(w) and logitudinal separation xl. An important interpretation of the results in Fig. 8 is that since U,(o) is approximately constant, an eddy of a given size decays, as it is convected, in a distance proportional to its size. A crude quantitative estimate is that an eddy of a given length scale 1 has decayed after traveling with the flow a distance of the order of five times its length scale (i.e., decay has occurred when ax1/ U , = 30 and if U , 10 = 2111, for frozen convection, x1 N 51). For more exact results, Bull (1967) should be consulted. Figures 7 and 8 are from Landahl's (1972)paper describing his theory and calculation of the convection and decay of wall-pressure fluctuations. In his calculation the nonhomogeneous Orr-Sommerfeld equation is used to describe the behavior of linear disturbances in a shear flow. The shear-flow velocity profile used in the analysis is the mean turbulent boundary layer profile which admits only stable disturbances. In order to obtain convection velocities, Landahl assumed that the cross-spectral density of the pressure obtains its largest contribution from the least attenuated mode of a disturb-

Structure of Turbulence in Boundary Layers

20

10 4

W

18 1

30

C

FIG.8. Streamwise decay of cross-spectral density of the wall pressure. Experiments by Willmarth and Wooldridge (1962) as analyzed by Corcos (1964). Theory and figure from Landahl (1967).

ance and that these modes propagate normal to the stream. The convection velocity and decay of these modes were obtained numerically. There is reasonably good agreement with the older experimental results shown in Figs. 7 and 8. Notice that no explanation of Emmerling's (1973) results discussed below were obtained from these computations. It has only recently been definitely determined by Blake (1970) and Emmerling (1973) that the above early measurements of wall-pressure fluctuations are seriously in error with regard to the very smallest spatial scales of pressure fluctuations. The error is caused by the inability of the relatively large flush-mounted pressure transducers that were used in the early experimental work to resolve adequately the smallest spatial scales of the pressure fluctuations. The measurements of Blake (1970) and Emmerling (1973) were made with a small pinhole in the surface that communicates with the sensitive membrane of a miniature condensor microphone. It has only recently been possible to obtain small condensor microphones which ensure that the Helmholtz resonator formed by the pinhole and microphone combination has a natural frequency above the frequency range required for the measurements (i.e., above approximately 15 KHz). The results of these and other measurements of the root-mean-square wall pressure as a function of transducer size were collected by Emmerling (1973). Figure 9 shows conclusively, for the first time, that the wall-pressure intensity scales with the wall length scale. We cannot learn very much more about the spatial dependence of the small-scale pressure field from existing measurements which show only the dependence of the root-mean-square (rms) pressure upon the diameter of the transducer. Further measurements are required. The dramatic increase

W . W . Willmarth

182

-.-.

0

4-

.=

-

7-.-

, 2

I

I

700

8 3

900

1000

ur.( diameter ) / v

FIG.9. Dependence of the measured root-mean-square wall pressure upon the pressure Blake (1970); x , Emmerling (1973); 0, Schloemer (1967); A, Bull transducer diameter; 0, (1967); 0 , Willmarth and Roos (1965); V, Bull and Willis (1961); and 0, Harrison (1958). From Emmerling (1973).

(by a factor of two) in the intensity of the rms wall-pressure fluctuations when the small-scale fluctuations are included cannot be predicted by correction theories taking into account the transducer size. See Corcos (1963) or Willmarth and Roos (1965). The correction theories require an accurate set of somewhat attenuated measurements of the pressure fluctuations sensed by finite size transducers which, as a function of transducer size and relative orientation of transducers, can be used in the correction theories to determine the unattenuated pressure fluctuations. An accurate set of measured data was not available because the relatively intense small-scale pressure could not be detected with the rather large transducers at necessarily large spatial separations used in the early work. We have mentioned these details about the experimental difficulties with the hope that this example will serve as a warning that experiments in turbulent flow can be difficult and misleading. One must use great care in selecting experimental results for the construction or verification of theories. The relatively large portion of the intensity of the rms wall pressure that can be attributed to the smallest scale of pressure-producing eddies suggests that in or near the sublayer there are important structural details and convection effects that are not included in the measurements discussed to this point. Evidence that this is true has been obtained by Emmerling at the Max-Planck Institut fur Stromungsforschung in Gottingen. In addition to

Structure of Turbulence in Boundary Layers

183

pressute measurements with a pinhole microphone system, Emmerling (1973) performed a remarkable experiment. A section of the wall of an acoustically quiet vibration-isolated wind tunnel was used as one of the reflecting mirrors of a Michelson interferometer. The boundary layer developed on this wall deflected a thin reflecting membrane (35 pm thick made from silicone rubber) that covered an array of 650, closely spaced, 2.5-mm diameter holes drilled in the wall. The wall-pressure fluctuations caused a deflection of the membranes observable in the optically recombined fringe pattern produced by the interferometer. The fringe-shift patterns for each hole were analyzed by Emmerling, at the expense of considerable labor, for a few frames from high speed (7000 frames/sec) motion pictures of the entire wall. Figure 10 shows four frames of data from Emmerling’s work that display the constant pressure contours. Each frame contains 650 data points analyzed by hand. The detail is astonishing and shows a convected pressure contribution that is at first intense and roughly circular, Figs. 10a and lob, and then moves downstream (to the right) in Figs. 1Oc and 10d. Emmerling has obtained a large quantity of photographs of the wall-pressure fluctuations ; these, unfortunately, have not yet been analyzed. When Emmerling’s data are analyzed, it is likely that much new quantitative knowledge will be obtained about the convection and decay of smallscale pressure fluctuations. For example, Emmerling’s (1973) report contains plots of pressure as a function of time and displacement along a line of 17 circular holes lying one behind the other in the streamwise direction. The plots are made from frames of his motion pictures which were taken 1/7000 sec apart. At random locations in the sequence of data small intense individual.pressure fluctuations can be observed that travel downstream at speeds as low as O.39Um.In general the individual lifetimes of these small pressure fluctuations are considerably longer than the values for larger eddies measured with the larger transducers. See Fig. 8. Emmerling states that small individual contributions to the pressure fluctuations with a streamwise extent of S*/2(50v/u,) could be followed for distances of the order of 96*(900v/t4). In other words, near the wall small pressure-producing eddies traveled a distance approximately 18 times their streamwise extent before decaying. This is at least three times farther than the average distance (five times the streamwise extent) that the larger eddies travel before decaying, as shown in Fig. 8. Further analysis of this data would be very valuable since the results of measurements of the rms pressure show that these smallscale fluctuations are very intense relative to the intensity of the larger scale pressure fluctuations. Note that the rms pressure increases by a factor of two when the transducer diameter is reduced from d = lOOv/u, to d = lOv/u,. If one assumes that the small-scale pressure fluctuations are uncorrelated with

184

W . W . Willmarth

b

- IUnm

-

FIG. 10. Contours of instantaneous pressure fluctuations. The darker shading indicates large pressure changes. Positive fluctuations are outlined with solid lines and negative fluctuations with dashed lines. Stream velocity is from left to right, time increases from (a) to (d): first frame (a) time = 17.57 msec; (b) time = 18 msec; (c) time = 19.14 msec; (d) time = 20.85 msec. From Emmerling (1973).

the larger scale fluctuations, this means that the rms intensity of the small, times larger than that of the larger-scale fluctuascale fluctuations is b tions. This large increase in intensity is probably reasonable because one would expect that the small pressure-producing eddies are very close to the wall since their convection speed is low, and hence very close to the transducer where their measured intensity will naturally be large. Let us now consider how the pressure field is related to the velocity field. For an incompressible fluid, the fluctuating velocity is related to the fluctuat-

Structure of Turbulence in Boundary Layers

185

ing pressure through Poisson's equation (obtained from the divergence of the momentum equation)

P(P

+ p)/axf = - p

f3"(Ui

+ Ui)(Uj+ U j ) ] / d X i d X j .

(4.5) From the integral representation of the solution of (4.5), in which terms on the right-hand side are regarded as source terms, it is clear that the pressure at one point is produced by velocity contributions at many other points. Therefore, the pressure fluctuations at any given point will not be highly correlated with the velocity fluctuations at any neighboring point. The fluctuating terms can be extracted from (4.5) by Reynolds' decomposition in which the actual pressure and velocity are sums of a mean plus a fluctuating portion and the boundary layer approximation is made (Kraichnan, 1956). When this is done, two of the source terms on the right are linear in the spatial derivatives of the velocity fluctuations, and the many remaining terms are quadratic in the spatial derivatives of the velocity fluctuations. The two linear source terms are simply - 2p(dU/8y)(av/dx) and represent the interaction of turbulence with the mean shear. Most investigations in which calculation of the wall pressure is attempted begin with the assumption that the contribution of the quadratic terms to the wall pressure can be neglected. See Willmarth (1975) for a list of references. Corcos (1964), however, is of the opinion that the nonlinear quadratic terms also are important. This appears to be an open question especially in view of Emmerling's (1973) results, discussed above, in which the smallest scale of the wallpressure fluctuations are quite intense. In any case (4.5) shows the relationship between pressure and velocity fluctuations. Measurements of correlations between large-scale wallpressure and velocity fluctuations in the flow have been made by Willmarth and Wooldridge (1963), Serafini (1963), and Tu and Willmarth (1966). As stated above one should not expect the correlations to be large. In fact they are not, since the measured correlation coefficients R,, ,R,, , and R,, rarely exceed magnitudes of 0.15. R,, is defined by

+ ~ ) l ( P * ( O ,t)U2(X,t + 2 ) p 2 ,

(44 the origin of x being at the center of the wall-pressure transducer. R,, and R,, are similarly defined. The convection velocity of pressure-producing disturbances has been estimated from space-time correlation measurements of R,, and R,, (Willmarth and Wooldridge, 1963). It was found that the convection velocity is approximately the local mean velocity at the position of the hot wire. In other words when the hot wire is at a given distance from the wall, the space-time correlation curve is displaced in time an amount approximately equal to the downstream displacement of the hot wire from the pressure Rpu(X,z> = P(0, t)u(x, t

186

W. W . Willmarth

transducer divided by the local mean speed at the hot wire location. More elaborate corrections to account for any possible change in the time delay as a function of distance of the hot wire normal to the wall were not attempted; for example, the transverse dependence of time delay found by Favre et al. (1957, 1958) in their streamwise space-time correlation measurements was not considered. Convection at the local speed refers only to the larger scale pressure-producing fluctuations because the flush-mounted pressure transducer used could not resolve the smaller scale pressure changes studied by Emmerling (1973). Consider next the decay of the pressure-velocity correlations during convection. Unfortunately, a quantitative analysis of the decay rate of pressure-velocity correlations does not exist. Examination of the measurements of Willmarth and Wooldridge (1963) shows that as the hot wire which measures u or u is moved downstream the pressure-velocity correlation decays much more rapidly when the hot wire is near the wall than it does when it is far from the wall. This agrees with the previous result that smaller pressure-producing eddies near the wall decay more rapidly than the larger eddies farther from the wall. It can also be observed from the measurements of Willmarth and Wooldridge that the R,, correlation does not decay as rapidly as the R,, correlation even though the initial (at z = 0) magnitudes of R,, and R,, are about the same. This may simply be an indication of the effect of the boundary coqdition at the wall in suppressing motions with a velocity component u normal to the wall. Although the pressure-velocity correlations are not large, the contours of constant correlation with zero time delay (obtained by moving the hot wire about) were readily determined. Figures 11-16 from Willmarth and Wooldridge (1963) show the isocorrelation surfaces of R,, and R,, for zero time delay (note that the coordinate x1 in these figures is positive in the stream direction). Both the R,, and R,, surfaces have approximately the same spatial extent and an aspect ratio of the order of 4. The dimension transverse to the stream is of the order of 6/2 and the streamwise extent is 26. Note especially that the correlations R , and R,, are primarily antisymmetric with respect to the stream direction across a plane normal to the wall and the stream that passes through the pressure transducer. The important exception for R,, near the wall will be discussed later. The striking antisymmetric property of the correlations suggested to Willmarth and Wooldridge (1963) the construction of the vector field of pressure velocity correlation, shown in Fig. 17, in a plane normal to the wall passing through the pressure transducer and containing the free stream velocity vector. The figure is constructed upon the assumptions that: ( 1 ) The orientation of each vector, measured from the positive x1 axis, is given by arc tan(R,, IR,,).

I

Structure of Turbulence in Boundary Layers

187

B L Edpi

FIG. 11. Correlation contours of R,,, in the x1-x2plane. Correlation normalized on the rms value of the velocity fluctuation at x2/6* = 0.51. Origin of coordinate system at wall pressure transducer. From Willmdrth and Wooldridge (1963).

f

L Edge

A--I.25

'0

t8

Rk

-

tOOlO

+0025

+0090 +0090 0

- 2 - 1

I

2

3

FIG. 12. Correlation contours of R,, in the xz-x3plane. Correlation normalized on the rms value of the velocity fluctuation at xz/S* = 0.51. Origin of coordinate system at wall-pressure transducer. From Willmarth and Wooldridge (1963).

188

W . W . Willmarth

FIG. 13. Correlation contours of R,, in the x1-x3plane. Correlation normalized on the rms value of the velocity fluctuation at xz/S* = 0.51. Origin of coordinate system at wall-pressure transducer. From Willmarth and Wooldridge (1963).

/I

i." 1 1; i te

-12 I

-10

L

Eaqe

ic

I

. - L A

10

FIG.14. Correlation contours of R p cin , the x,-x2 plane. Correlation normalized on the rms value of the velocity fluctuation at x2/6* = 0.51. Origin of coordinate system at wall-pressure transducer. From Willmarth and Wooldridge (1963).

12

Xi -

f

P

189

Structure of Turbulence in Boundary Layers

E.L. Edge

B.L. Edge

1.25

(I

FIG. 15. Correlation contours of R,, in the x2-x3plane. Correlation normalized on the rms value of the velocity fluctuation at x z / 6 * = 0.51. Origin of coordinate system at wall-pressure transducer. From Willmarth and Wooldridge (1963).

15 a*

FIG.16. Correlation contours of R,, in the x1-x3plane. Correlation normalized on the rms value of the velocity fluctuation at x2/S* = 0.51. Origin of coordinate system at wall-pressure transducer. From Willmarth and Wooldridge (1963).

190

W . W . Willmarth

-

d(zi,

+.ki,). FIG.17. Vector field of correlation. Magnitude of the vector at any point is Direction of the vector at any point as measured from the positive x 1 axis is given by From Willmarth and Wooldridge (1963). tan- '(Rp,,/Rpu).

Jw.

(2) The length of a vector at any point is equal to (3) The direction in which the arrow points on each vector is obtained by assuming that on the average the wall pressure is negative when the principal contributions to both R,, and R,, occur. Then from the value of R,, and R,, an approximate average value of v/u is R,, IR,,, .

The vector field of correlations of Fig. 17 was the result of an interesting observation by this author. Some 17 years ago on the occasion of a meeting of the Institute of the Aeronautical Sciences in Los Angeles a field trip to observe aircraft flights aboard the USS Enterprise was announced. This author attended, but spent most of his time peering over the rail observing the boundary layer developed on the side of the ship. The boundary layer was about 5 ft thick, at the three-quarter point of the ship aft of the bow. The birth, convection, and decay of large swirling eddies was readily observable upon the ocean surface. Occasionally, large rotating masses of fluid with a length scale of the order of the boundary-layer thickness were observed which began growing at the one-quarter point of the ship aft of the bow and often retained their identity for a long enough time to allow the ship to move forward half its length. These entities or eddies always rotated in the sense of a ball rubbing against the side of the ship and urged onward by the outer stream. To the shipboard observer it appeared that large individual eddies were born and decayed after traveling a distance of the order of 50-100 times

Structure of Turbulence in Boundary Layers

191

their length scale. They could readily be observed upon the surface both because they caused smaller eddies to be swirled around their center and because the ocean surface was depressed near the center of these rotating eddies. These observations led me to suggest that the passage of a swirling rotating group of fluid particles in the boundary layer will produce a reduced pressure at the wall. As this entity or eddy approaches a given point in the boundary layer there will be a flow toward the wall (u < 0) of higher streamwise momentum fluid ( u > 0) from the outer part of the boundary layer. After the eddy passes it will cause a flow away from the wall ( u > 0) of lower streamwise momentum fluid ( u < 0).This picture is entirely consistent with the recent visual observation of Falco (1974) and with the recent computerized pattern recognition measurements of Wallace et al. (1974), to be discussed below. Some time ago W. W. Willmarth (unpublished) proposed a crude model for the pressure-velocity correlation in which a two-dimensional vortex moves past a wall-pressure transducer and a hot wire above the wall. A correlation coefficient was then defined as in (4.6) (with u replaced by v) and with an arbitrary displacement x of the hot wire probe (measuring u ) with respect to the pressure probe. F. W. Roos (unpublished) computed the correlation by integrating the contributions of the wall pressure and u to the correlation during passage of the vortex (with solid-body core) past the hot wire and wall-pressure transducer, holding x = constant. The result of the computation is shown in Fig. 18 along with an actual measurement of R,, from Willmarth and Wooldridge (1963). This crude two-dimensional model produces correlations which are qualitatively comparable to the actual measurements. In reality the isocorrelation surfaces must be highly three dimensional so that the transverse extent and magnitude of actual pressure-producing eddies will be limited. Recently Falco (1974) has made a visual study, using smoke, of the typical eddies observed in a boundary layer. The evolution of a typical eddy, viewed from the side, consists of the sequence of spurting, bending over, and rolling up. As a matter of fact, Falco’s typical eddies look very much like Townsend’s (1957) mixing jets. At low values of Re, (Re, = 600) the typical eddies arc the large eddies in the boundary layer and are observed to evolve across most of the layer. At high Re, the typical eddies are still present, but they are smaller and are no longer the large eddies. The new large eddy family found at high Re, did not appear to Falco to involve streamwise overturning but no other features of these large eddies could be determined from the visual observations. When viewed from the side in the x-y plane the typical eddies had a generally elliptical boundary and a cochlear spiral was often apparent within

192

W . W. Willmarth

0“-

SOLID BODY” ‘iORTEX CORE

FIG.18. Qualitative model for the correlation R,, computed by F. W. Roos and compared measurement; - - -, with a measurement by Willmarth and Wooldridge (1963): --, qualitative model.

this boundary. The spanwise view in the y-z plane showed that typical eddies have a double cochlear or mushroom appearance. Figure 19 is a sketch of these two views of the typical eddy. Falco also noted that the interaction of a typical eddy with other features of the boundary layer often resulted in the typical eddy being rotated. The mushroom-shaped spanwise view was sometimes seen in the streamwise view, or occasionally a typical eddy which apparently had the opposite sign of rotation could be observed. Another aspect of typical eddy evolution observed by Falco was that for 600 < Re, < 1500 the largest features of the boundary layer structure often

F

U,

-

Section A-A

Section 6-B

(a)

(b)

FIG. 19. Typical eddy shape: ( a ) streamwise view, (b) spanwise view. From Falco (1974).

Structure of Turbulence in Boundary Layers

193

resulted from the coalescence of two or more typical eddies. The lifetimes of typical eddies as they completed their evolutionary cycle corresponded to movement over 8- 15 boundary layer thicknesses. The movement across the layer was generally one or two times their scale c, or c, (Fig. 19). We shall consider Falco’s (1974) work and the relationship of his typical eddy scales c, and c, to other measurements when we consider turbulent bursts in a later section. We will also consider the behavior of the isocorrelation contours of R,, near the wall (Fig. 16) and discuss the correlation R,, at that time.

D. MEASUREMENTS IN THE INTERMITTENT REGION Since the pioneering studies of Corrsin and Kistler (1955) experimental work in the outer intermittent region of the boundary layer remained at a virtual standstill, with the exception of the visual studies of intermittency by Fiedler and Head (1966). Within the past five years many new experimental measurements in the intermittent region have been reported, notably by Kibens (1968), Kaplan and Laufer (1963), Kovasznay et al. (1970), Antonia (1972b), and Hedley and Keffer (1974). All the above authors made their measurements in boundary layers with Re, of the order of 3000 with the exception of Hedley and Keffer who made their measurements at Re, N 9700. In these investigations the technique of selective or conditional sampling was employed to allow statistical measurements of a turbulent signal to be made only when an identifiable isolated event occurred. The complicated question of how one identifies or detects an isolated event in the intermittent region is a subject of current research. At present there is no generally accepted method, and investigations in future years will doubtless be concerned with this question for some time. Each of the investigations mentioned above used a detection method based upon different attributes of the turbulent portion of the flow. Corrsin and Kistler (1955) detected the presence of the streamwise component of fluctuating vorticity. Kibens (1968), whose results are summarized in Kovasznay et al. (1970), based his detection scheme on the presence of large-amplitude fluctuations of the derivative du/dy which is one term in the spanwise vorticity component. Smoke injected into the boundary layer far upstream was observed by Fiedler and Head (1966). The presence of large-amplitude streamwise velocity fluctuations was used as a burst detector by Kaplan and Laufer (1969). The presence of large-amplitude contributions to Reynolds stress uu served as a burst detector by Antonia (1972b) and large amplitudes of the signal was used as a detector by Hedley and Keffer (1974). + The intermittent region is of interest because it is the interface for energy transfer from the mean flow to the turbulence. We will discuss the existing

194

W . W . Willmarth

experimental studies of the intermittent region and attempt to indicate common features among the various existing investigations. All investigators mentioned above agree upon the general shape of y(y), the profile of intermittency as a function of y (Fig. 20), even though they used different attributes to detect the presence of turbulence. Once turbulence is detected the conditional sampling technique allows one to determine statistical properties of a flow variable at a point or in a zone related in some way to the location of the point of turbulence detection. The results of Kovasznay et al. (1970) show typical results that indicate the nature of the intermittent flow region. Included in Fig. 20 are profiles from the work of Kovasznay et aE. (1970) of the streamwise velocity component within the turbulent zone ZJU and within the nonturbulent zone GJU.Within the nonturbulent zones the fluid moves a few percent faster than it does in the turbulent zone, probably because within the turbulent region the fluid has originated at a lower level and still retains a small streamwise velocity deficit relative to its new location. Within the turbulent regions the velocity fluctuations are greater than in the nonturbulent regions. Figure 2 1 shows zone-averaged measurements by Kovasznay et al. (1970) of the streamwise velocity components in the two regions. In this case the fluctuations in the nonturbulent region are of smaller magnitude than in the turbulent region but are by no means negligible. In fact, the fluctuations in the nonturbulent region are potential flow fluctuations driven by the fluctuations within the turbulent region. Kovasznay et a!. (1970) reported good agreement of their measurements with Phillips’ (1955) theory for the dependence of the intensity of potential-flow fluctuations upon the distance from the interface. Phillips (1955) obtained the theoretical result that the mean-square fluctuations decay as the inverse fourth power of the distance from the interface. Another feature discovered by Kovasznay et al. (1970) was that the zone-averaged velocity normal to the wall was always positive in the turbulent and negative in the laminar regions. This is consistent with the concept that on the average the low-speed turbulent fluid emerges from deeper in the boundary layer and is replaced by nonturbulent fluid at a higher speed (Fig. 20). Some features of the shape of the interface in the streamwise plane can be deduced from the hot wire measurements. Kaplan and Laufer (1969) display a computer-generated representation of the extent of the turbulent zone all across the boundary layer that is obtained from their experimental measurements using an array of 10 hot wires (sensitive to streamwise fluctuations) which spanned the boundary layer. The turbulent front is deemed to arrive at a given hot wire when the streamwise fluctuation level relative to the streamwise velocity averaged over a small time interval increased appreciably. Using this scheme it was found that the nonturbulent regions of

195

Structure of Turbulence in Boundary Layers

1.0 0.8

Y

0.6 0.4

0.2

0.6

0.4

0 1.2

1.0

0.8

YIS

FIG.20. Zone averages of the streamwise velocity component. (The interrnittency factor y is given for reference.) From Kovasznay et a/. (1970). 0.10

1.0

0.09

0.5 y

0.08 0

>

0.07

0.06

d 0.05

L

.s”- 0.04

I2$

0.03

0.02 0.01 0

0

0.2

0.4

0.6

0.8

1.0

1.2

4

YlS

FIG.21. Zone averages of the intensity of the streamwise velocity fluctuations. From Kovasznay et al. (1970).

196

W . W . Willmarth

fluid did not appear to be completely surrounded by turbulent fluid but were connected to the free stream fluid. The interface appeared highly corrugated. Conditional sampling measurements also allow one to determine point averages so that one can measure the average fluid velocity at the interface by summing up the velocity measured at the interface for many arrivals of the interface. Kovasznay et al. (1970) found that the point averaged streamwise velocity at a given distance from the wall on the “front ’’ of a turbulent bulge was higher than on the “back.” Here the “back” faces upstream. Thus, the turbulent bulges are continually enlarging. The “ front ” velocity was a few percent lower than the stream velocity and the “back” velocity was 5% below that. This gave an average bump velocity of the order 0.93Um and suggests that the outer fluid rides over the turbulent bulges. Kovasznay et al. (1970) also proposed a model for the formation of the turbulent bulges. The model was based upon the above measurements and upon extensive measurements of double space-time correlations of the velocity components measured at two well-separated points. Their correlation measurements are too extensive to include here. The model for the turbulent bulge resulting from their measurements is in good correspondence with the recent visual observations of Falco (1974). In fact, their correlation maps appear to contain the rotary motion implied by the spurt, bend-over, and roll-up sequence observed by Falco (Fig. 19). Their correlation contours in planes parallel to the wall also appear to be in agreement with the spanwise view of a double cochlear spiral of mushroom shape observed by Falco in his typical eddies (Fig. 19). Blackwelder and Kovasznay (1972b) have recently reported an interesting, more detailed series of measurements in the intermittent region that complement their previous results (Kovasznay et al., 1970).Using the same experimental setup they have, for example, measured conditional averages of the Reynolds stress in the intermittent region. Figure 22 shows the results of these measurements. Here the detector function is again one term of the spanwise fluctuating vorticity &lay, and the sampled function is the Reynolds stress. The data of Fig. 22 show that in the nonturbulent regions the Reynolds stress is very small (note that the vertical axis is logarithmic). Measurements of the space-time correlation of u and v for various large streamwise separations between the u and v probe were also reported. Large streamwise spacing effectively removes contributions of smaller eddies to the correlation. Their measurements showed that with the proper time delay strong correlations between u and v exist even for streamwise separations as large as 166 when the distance from the wall is 0.456. From the decay of the correlation between u and v Blackwelder and Kovasznay estimated that the large eddies contribute as much as 80% of the Reynolds stress for y > 0.26. In addition Blackwelder and Kovasznay also measured point averages of

Structure of Turbulence in Boundary Layers

197

1.0

Y 0.5

0

-

0

0.2

0.4

0.6

0.8

1.0

1.2

Y/s FIG.22. Conditional averages of the Reynolds stress. From Blackwelder and Kovasmay

(1972b).

streamwise velocity at various locations relative to the detector probe during passage of the front and back of the turbulent bulge over the detector probe. Their results were then combined with similar point averages of velocity normal to the wall, from Kovasznay et al. (1970), to construct the average flow pattern within and around a turbulent bulge in the outer region. The results are displayed in Fig. 23 which shows a circulatory flow within the turbulent bulge in agreement with the “cochlear spiral” found within the typical eddies in visual studies of the smoke-filled boundary layer (Falco, 1974). The outer flow is apparently “riding over” the turbulent fluid within the bulge. One must, however, view this picture with caution because it is an average constructed from a large number of events. The actual interface observed in photographs of the smoke-filled boundary layer (Fiedler and Head, 1966; or Falco, 1974; or Kaplan and Laufer, 1969) appears highly irregular and corrugated. Hedley and Keffer (1974) have made studies very similar to those of Kovasznay et al. (1970) but at larger values of Reo = 9700. Their gross results were similar, in general, to those of Kovasznay et al. although they noticed a change in the sign of the difference between the velocities of the “ fronts ” and backs ” of the bulges deep within the boundary layer y < 0.5. “

198

0.2

W . W . Willmarth

1

FIG.23. Compositc velocity distribution in the outer region of the boundary layer. From Blackwelder and Kovasznay (1972 b).

Hedley and Keffer also found that the turbulent Reynolds stress sharply increased across the upstream face (back) of the bulges and that there was hardly any change across the downstream face (front). These results are in agreement with Falco’s (1974) visual observations a t higher Re,. See the discussion at the end of this section. Antonia (1972b) has reported conditionally sampled measurements in the intermittent region in which the detection of turbulence is accomplished by observing the Reynolds stress fluctuations uu. Turbulence is presumed to be present if (duz~/dt)’ exceeds a certain level originally determined by visual comparison between the pulse train indicating the presence of turbulence and traces of the Reynolds stress fluctuations. The results of Antonia’s investigation are in substantial agreement with those of Kovasznay et al. (1970) and Kaplan and Laufer (1969). Some differences with regard to the shape of the interface and the point-averaged streamwise velocity were observed. A possible explanation for the differences based upon Falco’s (1974) recent measurements is discussed at the end of this section. One interesting result from Antonia’s work is that the average of the Reynolds stress in the turbulent zones is of the order of half the wall shear stress, which provides support for the idea that the strength of the large eddy motion is closely related to the wall shear stress. This result is in approximate agreement with the measurements of Blackwelder and Kovasznay (1972b). The process of entrainment of nonturbulent fluid by turbulent fluid is of great interest for many problems in turbulence. Phillips (1972) has presented a theory to describe the evoIution and corrugations of the interface. In his

Structirre of' Turbulence in Boundar-J.Layer.\

199

theory the large-scale eddies of the motion produce convolutions of the interface. At the same time and independently the small-scale motions cause the interface to advance relative to the fluid in which it is imbedded, as a result of a microscale entrainment process. In his paper he presents an interesting analysis of the experimental results of Kovasznay et ul. (1970). He finds a surprisingly large entrainment speed of the order of 15 times the Komogorov velocity, which he attributes to augmentation by microconvolutions of the interface that are caused by the small-scale and mesoscale part of the turbulence. It appears that further progress in this area will require more accurate information about the interfacial shape and convolutions. Threedimensional information has not yet been obtained using conditional sampling methods. Laufer (1972) has also discussed the entrainment problem. From the experimental results already mentioned he suggests that the flow of the free stream over the backs of the bulges results in a mixing and diffusion region at the top and along the front of the bulges. However, as he states, no information is yet available on the extent of this mixing and diffusion region. Also, a note of caution is necessary, for Falco (1974) has visually observed a surprising decrease in the spatial scale and behavior of his typical eddies as the Reynolds number is increased. Figure 24, redrawn from Falco's paper, shows that the scales c, and c, of his typical eddies (Fig. 19) decrease by almost a factor of ten relative to 6 when Re, increases from 600 to 10,000. The change in spatial scale is accompanied by a visually observable change in the structure of the smoke-filled regions of the boundary layer. Specifically, at low Reo the typical eddies are the large eddies in the boundary layer and are observed to evolve across most of the layer. At high Refla

FIG.24. Typical eddy scales as a function of Re,. From Falco (1974).

200

W . W . Willmarth

new large-eddy family not involving streamwise overturning is observed. Falco stated that he found engulfment at the turbulent-nonturbulent interface to occur at typical eddy scales in any case. Therefore, one must proceed carefully because at high Re, the typical eddies of reduced scale were then observed (Falco, 1974) to evolve on the “backs” (i.e. the upstream side of larger scale features). This is not to say that Laufer and Falco are in disagreement, but it may simply be that there is a change in the phenomena with Reynolds number.

E. MEASUREMENTS OF THE STRUCTURE OF THE VISCOUS SUBLAYER For many years the region very near a smooth wall, where the flow is mainly viscous, was called the laminar sublayer. Investigations of the flow in the sublayer beginning as early as the ultramicroscope observations of particles in the flow very near the wall (Fage and Townsend, 1932) have only emphasized the fact that the sublayer was not truly laminar. In fact, within the sublayer the rms streamwise velocity fluctuations relative to the local mean velocity are higher, of the order of u‘/U = 0.3, than at any other place in the boundary layer (Fage and Townsend, 1932; Laufer, 1954; Eckelmann, 1974). During the past 20 years the viscous sublayer has been the subject of a number of experimental investigations, but there are many questions that remain to be answered. The experiments are extremely difficult primarily because the viscous sublayer in a boundary layer developed in a flow of air is very thin unless the stream speed is very low. On a conventional aircraft wing the sublayer is of the order of 0.1 mm thick or less. For this reason, almost all our knowledge of the viscous sublayer structure has been obtained at low Reynolds number either in rather viscous liquids or in air at low speeds. The effect of wall roughness on the sublayer structure can be expected to be profound when the roughness height is greater than the sublayer t.hickness, which is generally accepted to be of the order of 5v/u,. In this chapter the very practical and important problem of wall roughness effects will not be considered. Research on the sublayer structure on smooth walls was stimulated by Hama’s observation in 1953 (Corrsin, 1957; Hama et al., 1957) that a film of dye injected tangentially into the sublayer, formed extended streamwise streaks of high dye concentration within the sublayer. The streaks appeared to have a more or less regular spacing and traveled slowly downstream waving randomly in the spanwise direction. The streaky sublayer structure was also reported and investigated by Kline and Runstadler (1959) and Runstadler el al. (1963). Their visual investigations using either dye or hydrogen bubble tracers showed that within

Structure of Turbulence in Boundary Layers

20 1

the sublayer relatively regularly spaced streaks appeared at random locations and times with a characteristic wavelength A, determined from limited ensemble averages, of the order of 20 sublayer thicknesses, i.e., II N lOOv/U,. Their observations showed that at any instant the streamwise velocity varied almost periodically in the spanwise direction within the sublayer. The streaks moved slowly away from the wall as they progressed downstream, undergoing transverse oscillations. Some of the streaks were observed to interact with the outer flow in a process consisting of (Kline et al., 1967) gradual liftup, sudden oscillation, bursting, and ejection. We will delay discussion of this process until the next section, Section IV,F. In a recent paper Gupta et al. (1971) have reported studies of the sublayer streaks using a spanwise array of 10 hot wires within the sublayer. The spacing was such that a number of spanwise spatial correlations could be measured from various pairs of wires. The two-point spatial correlations of streamwise velocity, averaged over a long time interval, showed no evidence of alternating high and low speeds (Fig. 25). However, upon measuring the correlations for relatively short time intervals regions of alternating high and low speeds could be detected and had a characteristic spacing of the order of lOOv/U,. Figure 26, from their report, shows a sequence of correlations measured over short time intervals. This is an example which shows, as pointed out by Mollo-Christensen (1971), that long time averages can hide rather than reveal the physics of a random process. Gupta et al. (1971) statistically analyzed an ensemble of short time correlations to locate the first maximum and minimum of the correlations. Their results for the average streak wavelength were that below Re, = 4700, X N lOOv/U,, but at Re, N 6500, 2 N 150v/U,. Some of the models proposed for the flow structure in the sublayer involve streamwise vortices. Bakewell and Lumley (1967) have studied the spacetime correlations of streamwise velocity in a thick sublayer beneath a fully developed turbulent flow in a tube using glycerine as the working fluid. They used Lumley’s method of proper orthogonal decomposition to show that the dominant large-scale structure of their sublayer flow is a randomly distributed pair of counterrotating eddies aligned in the stream direction. Morrison et al. (1971) have reported an extensive analysis of Morrison’s (1969) measurements of the two-dimensional frequency-wave-number spectra and narrow-band shear stress correlations in turbulent pipe flow. Figure 27 shows a plot of contours of constant spectral density as a function of frequency and spanwise wave number at y + = 5.92, within the sublayer. The spectral density contours at yf = 1.56 and 2.96 are similar. The maximum spectral density occurs at spanwise wave number k,‘ N 0.047 and in fact a characteristic transverse wave number is relatively well defined since the “ridge line” on these plots tends to be almost parallel to the w + axis.

202

W . W . Willmar t h

0

0

7.5

50 Z+

25

0 0

-.

%

so

7

150

I00

Z+

c;l

0

0

0

I

0

So

I00

IM)

0.2

IS0

z+

260

-

z+

200

0.4

3M)

0.6

I

0.8

2 (in.)

FIG.25. Two-point spanwise long time average correlations of u fluctuations. R,,(O, 0, z') at various velocities. y = 0.014 in. (a) Re, = 2200, U , = 11.3 ft/sec, y' = 3.4; (b) Re, = 3300, U, = 18.8 ft/sec, J.' = 5.4; (c) Re, = 4700, U , = 20.0 ft/sec, y + = 7.8; (d) Re, = 6500, U , = 39.5 ftisec, 1'' = 10.8. From Gupta et al. (1971).

The lines of w+/kt = constant indicate the convection velocity as discussed by Wills (1964). A transverse wave number of 0.047 yields a characteristic 1 2: 2n/0.047 P 134, with small convection velocity transverse wavelength, ' of the sublayer streaks, a result in relatively good agreement with other observations of the transverse streak spacing mentioned above. Figure 28, on the other hand, shows a plot of contours of constant spectral density as a function of frequency and streamwise wave number at y f = 5.92. In this case the ridge line is well defined and indicates a definite effect of streamwise convection. What is surprising is that in Fig. 28 and in similar plots at y + = 1.52 and 2.96 the ridge line indicates convection at higher velocities than the local mean velocity. In fact, Morrison et al. (1971) found that the convection velocity was the same throughout the sublayer for

Structure of Turbulence in Boundary Layers

0

203

0

FIG.26. Typical short-time average two-point correlations of u fluctuations R,,(O, 0, z'). Averaging over 0.375 msec. Re, = 3300, Lix = 18.8 ft/sec, y + = 5.4. From Gupta et ul. (1971).

y+ I 5.93, and that the slope of the ridge line could be approximated by

w+/k,f = 8, which indicates a streamwise convection velocity of 8U, (i.e., the

local mean streamwise velocity at y + = 9) over the wave-number range 0.001 < k,f < 0.1. It is interesting that for their flow, which was apipeflow,

8 U , is approximately 0.4 of the centerline velocity. We recall that Emmerling (1973) found that his smallest scale pressure fluctuations, which could only be measured with transducers of diameter less than lOOv/U,, were convected at speeds as low as O.39Umin a boundary layer. Further analysis of Emmerling's data should prove to be very valuable. Morrison et al. (1971) have proposed a wave model to explain the observed sublayer structure. From their extensive measurements (Figs. 27 and 28) they deduce that the spectral power in their wave model is spread out over a range of speeds between 6 and 12 times the shear velocity. The wave size is of the order of 271/k, and the inclinations of the waves vary by a factor

W . W . Willmarth

204 1.0

+ 0.1 3

h

B

8

& 0.01 0.001

0.01

0.1

Wave number k: FIG. 27. Two-dimensional spectrum of streamwise velocity fluctuations in the sublayer. Fully developed turbulent flow in a tube, Re = 17,100, y+ = 5.92. From Morrison et al. (1971).

1.0

0.1

+

9

1

0.o

0.0001

0.001

0.01

0.1

Wave number k:

FIG.28. Two-dimensional spectrum of streamwise velocity fluctuations in the sublayer. Fully developed turbulent flow in a tube, Re = 17,100, y + = 5.93. From Morrison et al. (1971).

Structure of Turbulence in Boundary Layers

205

of 10. Morrison et al. have discussed their wave model in considerable detail in their paper. It appears that by proper combination of waves a plausible argument for the existence of waves in the sublayer has been proposed. It is the author’s opinion that one should also consider the existence of small-scale convected vortices to explain the observed sublayer structure. The pressure field produced by a convected vortical structure would cause the occurrence of a similar structure of disturbances throughout the sublayer which is necessarily convected at the same speed at all layers. One must not lose sight of the fact that the flow disturbances in and just above the sublayer are very intense relative to the mean sublayer flow and that the mean vorticity in this region is large. This means that when large disturbances occur in the rapidly rotating fluid near the wall, strong nonlinear interactions will occur within this rotating fluid. Sternberg (1965) and Schubert and Corcos (1967) have independently attempted to construct a linear theory for the structure of the viscous sublayer. In these theories the sublayer fluctuations are supposed to be driven by the convected pressure field developed in the logarithmic region above the sublayer. In other words, the sublayer acts in a passive manner in its response to the turbulence above it. Sternberg has stated that the sublayer records the footprints of the turbulence above. Morrison et al. (1971)object to both theories because the analysis is only valid for wave speeds well in excess of the local fluid velocity. Certainly the convection speed observed by Morrison et al. was much lower than the wall-pressure convection speeds of the order of 0.8U , , measured before Emmerling’s (1973)paper appeared. In addition, Morrison et al. state that the turbulent velocity component normal to the wall computed by Schubert and Corcos is two orders of magnitude below the experimentally observed value. Again, it is difficult to believe that [in view of Emmerling’s (1973) measurements, and the very high fluctuations in streamwise velocity] there are not strong nonlinear processes which occur near the edge of the sublayer, which would invalidate the linear theories. We should also mention the measurements of fluctuating shear stress at the wall by Hanratty (1967) who used an electrochemical technique. Hanratty found that on occasion the flow fluctuations near the wall are so large that the flow is stagnant. Eckelmann (1974) on the contrary found that in his hot-film study of sublayer fluctuations in oil flowing in a channel stagnation did not occur. Eckelmann’s measurements of the fluctuating velocity gradient at the wall using a calibrated heat-transfer element always gave positive gradients. The velocity gradient fluctuated between 0.4 and 1.7 times the mean. We conclude this section with a few comments about the effect of Reynolds number. Morrison et al. (1971) note that some of their measurements just outside the sublayer indicate that the character of the sublayer will

206

W . W . Willmarth

change radically at Reynolds numbers above 30,000 in their pipe flow (i.e., about Re, 2 1500 if we assume a boundary layer with 0 2 6/10 and that 6 is approximately half the pipe diameter). Their evidence for this statement was the finding that at higher Reynolds number when their sublayer had become too thin to allow accurate measurements within it the amount of energy at low frequency and low transverse wave number (k:) outside the sublayer had increased considerably relative to the energy outside the sublayer at lower Reynolds number and presumably had increased within the sublayer as well. The convection velocity of this new low-frequency energy was 16U, which is of the order of twice the previously measured convection velocity at lower Reynolds numbers. Recall also that Falco (1974) observed a marked change in his typical eddy structure when the Reynolds number was increased. At this date detailed investigations of the sublayer flow field have not been possible at large Reynolds number owing to the difficulties caused by spatial averaging of sublayer-flow quantities that occurs with finite-size probes. This author has attempted double correlation measurements of streamwise vorticity with two side-by-side identical probes just outside the sublayer at Re, = 38,000. The sublayer was 0.05 mm thick, and the characteristic probe dimension was 1.5 mm. Even when the probes were 1.5 mm apart (as close as possible) and placed on the wall, the correlation coefficient between the two vorticity signals was zero. Vorticity fluctuations of high frequency could be measured by a single probe, but doubtless these vorticity signals were severely attenuated, as has been discussed by Wyngaard ( 1969).

F. THEOCCURRENCE OF BURSTS Fifteen years ago, at Stanford University, improved flow visualization methods were developed and used to study coherent flow structures in the turbulent boundary layer. See Kline et al. (1967), Kim et al. (1971), and Offen and Kline (1973, 1974) for summaries of the work done at Stanford University. Their visual observations of coherent structures stimulated further visual observations by other investigators (Corino and Bodkey, 1969; Grass, 1971; Nychas et al., 1973).The new information obtained from visual studies initiated renewed interest in research on turbulence and has created a need for more quantitative information about the nature of the coherent bursting structure. During the past few years there has been a veritable explosion in turbulence research using hot wires or films in which the data on coherent structures are processed by conditional sampling methods and/or a digital computer. In this area we shall mention work reported in at least ten papers published since 1971. There is now also feedback from these hot wire measurements of coherent structure which have raised new

Structure of Turbulence in Boundary Layers

207

questions that only visual investigations of the coherent burst structure can answer. For this reason, Offen and Kline (1973) consider the relationship between a number of burst-detection methods in which hot wire signals a h used and the visually observed bursting structure. We shall examine in this and the next section the very considerable progress that has been made toward a better understanding of the turbulence-production mechanism in the boundary layer. 1. Visual Observations of Bursts As mentioned above, visual observations of coherent structures in the fully developed turbulent boundary layer were first reported at Stanford University by Kline and his colleagues. The first phase of their work (covering the period 1963-1967) was reported in Kline et al. (1967). This paper contained a description of the streaky sublayer structure discussed in Section IV,E. It also presented a description of an identifiable randomly occurring process (that we now call a burst) in which sublayer streaks were observed to gradually " lift up," then suddenly oscillate, followed by bursting and ejection. Unfortunately, there is not space to describe many of the details of their observations in this review. Perhaps a summary of the randomly occurring process that they identified can be obtained from a sketch. Figure 29, from Kline et al. (1967), is a sequence depicting the above process, from typical side views of a dye streak as seen in motion pictures. The arrow follows a prominent portion of the ejected streak. The oscillation occurs in the third sketch, and bursting and ejection with considerable contortion of the dye streak in the fourth and fifth sketch. Kline et al. (1967) also report statistical studies of the path of the ejected dye in planes normal to the wall and parallel to the stream. Figure 30 shows the distribution and average trajectories of many occurrences obtained from motion pictures in the boundary layer on a flat plate parallel to the general direction of flow. We remark that these results cannot show the burst structure transverse to the flow since the depth of field of the motion pictures was large compared to the transverse extent of the bursts. It is interesting to note that Kline et al. found that the ejected fluid had a streamwise velocity of roughly 80% of the local mean velocity as it moves across the outer part of the boundary layer. This is an indication that the ejection process is responsible for some portion of the Reynolds stress since the ejected fluid, for which u > 0, has a local momentum deficit (u c 0), so that uu < 0. In their next paper, Kim et al. (1971) studied the process of turbulent production (and Reynolds stress contributions) during bursting. Motion pictures showing the trajectories of successive lines of bubbles (obtained by pulsing the bubble-generating current) were evaluated to determine velocity

W . W . Willmarth

208

t=6r

.+'"L -

t = 26t

0

100

Y

I =

36t

/

Y+ n

FIG.29. Dye streak breakup during bursting; illustration as seen in side view. From Klii et a!. (1967).

components u and u. Note again that there is no way to be certain tht bubble lines observed at fixed x and y on successive motion picture frame have the same spanwise location (i.e., z coordinate). One can minimize t h source of error by measuring u and u at points near the bubble-generatin wire so that the measured velocities approximate those at a fixed location ; y, z = constant. From their pictures Kim et al. concluded that in the zon 0I y+ I 100 essentially all the turbulence production occurs durin bursting. They also thought it likely that this would also be true fc y + > 100.

Structure of Turbulence in Boundary Layers

209

1.0

-9 9

a

0.5

0

1.5

1.0

0.5

2.0

t (sec)

(4

0

0.5

’

1.0

1.5

2.0

2.5

t (sec)

(b)

FIG.30. Trajectories of ejected eddies during bursting-flat plate flow, zero pressure gradient. From Kline et al. (1967).

Owing to the laborious process of data reduction from films the sample size was limited, and their results for average production and Reynolds stress were only accurate to f25%. Kim et al. also obtained data showing instantaneous streamwise velocity profiles during bursting. They were able to observe that during the gradual lift up of low-speed streaks from the sublayer unstable (inflectional) instantaneous velocity profiles were formed. Also, they observed that after ejection described above there was a return to the wall of the low-speed streak and a more quiescent flow which completed the bursting cycle. This probably means that another low-speed streak reappeared (not the same one); this matter will receive further discussion in Section V. At approximately the same time, at Ohio State University, Brodkey and his colleagues (Corino and Brodkey, 1969)were also making visual observations of the turbulent boundary layer during bursting in the region near the wall. Their results were obtained from high-speed motion pictures of the

2 10

W . W.Willmarth

trajectories of very small particles near the wall suspended in a flow of liquid in a tube at a Reynolds number (based on diameter) of 20,000, i.e., Re, N 900.The depth of field of their photographs was relatively shallow, of the order of 20v/u,, so that their observations show (approximately) a slice through the bursting structure. This should be kept in mind as we describe their observations because their method provides information that, when properly interpreted, can give some idea of the transverse scale of the burst structure. The camera was mounted on a traversing mechanism so that the motions responsible for the bursting phenomena could be kept in view as the pattern of the burst was swept downstream. The observations of the burst phenomena reported by Corino and Brodkey are in essential agreement with those reported by Kim et al. The use of numerous tracer particles for flow visualization allows the observation of all the fluid particles passing through the field of view of the camera. Corino and Brodkey were able to identify additional features of the breakup process and the flow after breakup that could not be observed by marking only the fluid elements that passed over the upstream bubble-generating wire used by Kim et al. The sequence of events before and after chaotic breakdown during the bursting process reported by Corino and Brodkey began with the formation of a low-speed parcel of fluid near the wall in 0 I y + I 30. The velocity of the low-speed region was often as low as 50% of the local mean velocity with a very small streamwise velocity gradient within the low-speed region. After a low-speed region had formed the next step occurred, and was called acceleration by Corino and Brodkey. During acceleration a much larger high-speed parcel of fluid came into view and by “interaction” began to accelerate the fluid. At various times the entering high-speed fluid appeared to occupy the same region on the photograph as the low-speed fluid. The explanation is that the high-speed region was within the field of view but at a different spanwise station to one side or the other of the low-speed parcel of fluid. It appears to this author that the spanwise variation revealed by the above observation may be related to the observation of sudden oscillations of the low-speed streaks just before bursting and ejection reported by Kline et al. (1967). This is supported by the fact that the depth of field was of the right order of magnitude, Z + N 20, to allow observation of a single transverse shear layer formed by adjacent high- and low-speed regions near the wall where the streaks have a typical spanwise spacing of z+ = 100. In the acceleration phase, if the high- and low-speed fluid met at the same spanwise station, the interaction was often immediate; the low-speed fluid above a particular y f location was accelerated, and a very sharp interface or shear layer between accelerated and retarded fluid was formed. The next step

Structure of Turbulence in Boundary Layers

21 1

in the process was called ejection by Corino and Brodkey. During ejection one or more eruptions of low-speed fluid occurred immediately or shortly after the start of the acceleration process. Once ejection began, the process proceeded rapidly to a fully developed stage during which ejection of lowspeed fluid persisted for varying periods of time and then gradually ceased. The length scale of ejected fluid elements was small, of the order of 7 < z+ < 20 and 20 < x+ < 40.Most of the ejections occurred at distances from the wall in the range 5 < y + < 15. When the ejected low-speed fluid encountered the interface between high- and low-speed fluid, at the high shear layer, a violent interaction occurred with intense, abrupt, and chaotic movements. The intense interaction continued as more fluid was ejected. The end result was the creation of a relatively large-scale region of turbulent motion reaching into the sublayer as the violent interaction region spread out in all directions. The ejection or bursting phase ended with the entry from further upstream of fluid directed primarily in the stream direction with a velocity approximating the normal mean velocity profile. The entering high-speed fluid carried away the retarded fluid remaining from the ejection process; this was called the sweep event by Corino and Brodkey. Both Corino and Brodkey and Kim et al. agree that the bursting phenomena is an important process for turbulent-energy production. Corino and Brodkey conclude that “ the results do indicate that the ejections are very energetic and well correlated so as to be a major contributor to the Reynolds stress and thus the production of turbulent energy.” Their rough estimates of the Reynolds-stress contribution during bursting from a small sample of bursting events indicated that 70% of the Reynolds stress was produced during ejections. Corino and Brodkey’s flow visualization studies were confined to the region very near the wall. In a later study, Nychas et al. (1973), using a similar apparatus and techniques, studied larger scale motions throughout the boundary layer by means of tracer particles. Again the field of view represented essentially a two-dimensional slice of the flow and, as in the previous investigation, some interpretation and inferrence is required in order to correlate results obtained in this way with those observed by other visual methods in which the depth of field is large. The observations of Falco (1974), discussed in the previous section, which were made of an illuminated slice of the smoke-filled boundary layer, allow one to draw the same major conclusions as those reached by Nychas et al., who observed that, at Re, N 900, the single most important event in the outer region was a largescale fluid motion that appeared as a transverse vortex transported downstream with a velocity slightly less than the mean. It appears that Falco’s observations are completely consistent with those of Nychas et al. This

212

W . W. Willmarth

includes the important observation that at low Re, the observed large-scale motions were the result of an instability-producing interaction between accelerated and decelerated fluid that is closely associated with wall-layer ejections. The motions associated with these events extended all across the layer at these low values of Re, and made substantial contributions to the Reynolds stress. Grass (1971) has also reported visual studies of the structure of turbulent boundary layers developed on smooth and rough surfaces. Motion pictures of hydrogen bubbles were used to observe instantaneous longitudinal and vertical velocity profiles. Mean and fluctuating velocities u and u and also contributions to the Reynolds stress were computed from these profiles. No discussion is given in the paper about the fact that the z coordinate for the hydrogen bubble traces is not known. A computer was used to select from a large number of profile pairs of the u and v velocity those profiles with either very large or small streamwise velocity at a particular distance from the wall. Reynolds stress contributions that were computed for the sampled profiles showed that Reynolds stress contributions uv were dominated by both ejection events, for which zi > 0, and in-rush events, for which v < 0. However, the ejection events appeared to make appreciable contributions to Reynolds stress throughout the boundary layer while the in-rush events were more important near the wall. The results of the investigation were in essential agreement with all that has been discussed above. 2. Quantitative Measurements of Bursts Perhaps the most difficult problem that is encountered in making quantitative measurements of bursts is the detection problem. Unlike the intermittent region near the outer part of the boundary layer, the burst near the wall is immersed in the background turbulence. It is not enough simply to detect the presence or absence of turbulence as one does when detecting intermittency in the outer regions. Whether one uses a visual method or a measurement from a probe, or probes, immersed in the flow, there are two not unrelated aspects of the burst-detection problem : what attribute (or attributes) of the burst should be used for detection and how does one decide when the selected attribute indicates that a burst is present. The latter problem is really a problem of detecting a signal (or signals) buried in noise. At present it appears to this author that the most reliable detection scheme is a visual method. In fact Offen and Kline (1973) have (as we shall discuss in Section V) compared their method of visual detection of bursts with other detection schemes based upon measurements with a single probe at a point in the flow. As we describe the existing studies of bursts we will discuss the detection schemes used to obtain the measurements.

Structure of Turbulence in Boundary Layers

213

a. Mean Burst Period. Consider first the paper by Rao et al. (1971) in which the mean time interval between bursts and the mean burst duration were investigated. Rao et al. detected bursts using a complicated scheme to process the fluctuating streamwise velocity signal from a single hot wire placed at various points in the wall region of the boundary layer at low Reynolds numbers (Re, N 620). The signal u was apparently (although this is not definitely stated in the paper) differentiated with respect to time. The signal was then filtered using a narrow bandpass filter. Traces of the filtered signal showing intermittent periods of relatively large-amplitude oscillations were recorded and the frequency and duration of bursts were counted manually. In order to count the bursts, Rao et al. devised a complicated scheme to determine what signal level (above the background noise) would indicate that a burst had occurred. Their scheme involved the arbitrary rule that individual bursts had to be separated in time by more than twice the period of the center frequency of the bandpass filter. Next, the time between bursts was determined as a function of detection level. The burst interval was found to pass through a minimum (where it was relatively insensitive to detection level) and this was the deflection level used in their measurements of mean burst frequency and duration. One could change the detection scheme so that a different minimum time interval between periods of signal activity would be required before individual bursts would be counted ; this would change the mean burst interval. Rao et al. (1971) did not report the effect of changing this minimum time interval; however, they did determine that the mean burst frequency was relatively independent of the center frequency of the bandpass filter. Rao et al. found that the mean burst frequency was independent of y in the region between the wall and the intermittent region, a result that is in agreement with the observation (Falco, 1974) that at low Re, the typical eddies are observed to evolve across most of the layer. Rao et al. also summarized the results for mean burst period T obtained from the visual studies of Kim et al. (1971), Schraub and Kline (1965),and Runstadter et al. (1963), and they also included results from hot wire correlation measurements by Laufer and Badri Narayanan (1971)and Tu and Willmarth (1966). Their summary showed that the mean burst period T scales with outer variables U, and 6 or 6" and that TU,/6* N 30 or TU,/6 21 5 in the Reynolds number range 500 < Re, < 9000. As mentioned above, Rao et a!. (1971) included a data point obtained from autocorrelation measurements of streamwise velocity fluctuations in the sublayer at Re, = 38,000 (Tu and Willmarth, 1966). However, in a recent note, Lu and Willmarth (1973b) demonstrated that the Tu and Willmarth (1966) autocorrelation measurements do not indicate the burst frequency. It turned out that the second mild maximum of the autocorrelation curve from Tu and Willmarth that was

214

W . W . Willmarth

used by Rao et aE. to determine mean burst period was produced by a low-pass filter used to remove low frequency fluctuations. In fact, Johnson and Saylor (1971) have reported studies of the effect of high-pass filtering on the determination of the mean burst period from single hot wire signals and conclude that serious errors can occur. Despite these difficulties it appears that the scaling with outer variables and the approximate magnitude of the mean burst period is established for Re, < lo4. Further supporting results will be briefly mentioned later.

b. Burst Structure and Relationship to Reynolds Stress. Let us turn now to quantitative measurements of the structure of bursts. We will attempt to bring together evidence from a number of investigations and at the same time indicate the detection scheme that was used. Consider first the instantaneous velocity profiles during the burst. Blackwelder and Kaplan (1972) reported measurements of instantaneous profiles of the streamwise velocity obtained by the use of a rake of ten hot wires in the wall region. Samples of the ten simultaneously recorded signals were selected by a digital computer at various times before and after a burst was deemed to occur. The occurrence of the burst was inferred from a digital processing scheme devised by Kaplan and Laufer (1969). Using a sequence of digitized values of the u signal, the variance was computed over a short time interval centered about the digitized value of the u signal at the current time. A burst was deemed to occur if the short time variance was greater than a predetermined level. If the short time variance was less than the predetermined level, the process was repeated for the next digitized value of the u signal. The method (which is roughly equivalent to computing the variance over a short time after filtering out the low-frequency fluctuations) is sensitive to large fluctuations about the short time average of the signal. The detection scheme was applied to the u signal at y + = 16 and many samples of the u signals at ten different distances from the wall were obtained and stored at times from -16 to 36 msec before and after detection. Figure 31 shows the results that Blackwelder and Kaplan obtained at Re, = 2550. Note that in agreement with the visual results of Kline et al. and Corino and Brodkey the velocity profile is inflectional near the wall just before the detection of the burst. Willmarth and Lu (1971) used a different scheme to detect the occurrence of bursts. The scheme was based upon the visual observations of Kline et al. (1967) and Corino and Brodkey (1969) who found that fluid ejections near the wall are preceded by a region of fluid with low streamwise velocity very near the wall, within which they originate. A single hot wire, at y + = 16.2, was used for burst detection. This location was chosen in accordance with Corino and Brodkey’s (1969) observation that the approximate center of the low-speed region near the wall before a burst occurred was y + = 15. It was

+

Structure of Turbulence in Boundary Layers

0

0

0

0

0

0

0

0 - 0 O

0

0

215

0 01 0.20.3040.50.60.7

u, u. FIG.31. Conditionally sampled velocity profiles before, T < 0, and after, T > 0, burstdetecmean velocity profiles, - - -. Re, = 2550, U m = 14.0 ftisec, y = 0.52 ft/sec. tion, -; From Blackwelder and Kaplan (1972).

necessary to filter out the high-frequency components of the detector signal before applying the detection criterion. The criterion was that when the velocity first became lower than the mean by a certain amount, called a trigger level, a burst was presumed to occur. Note that there appears to be some correspondence between this method and the method of Blackwelder and Kaplan since Blackwelder and Kaplan found that at y+ = 15 the velocity was considerably below the mean value shortly before their detection scheme indicated burst occurrence (Fig. 31). When we review the work of Offen and Kline (1973) in Section V, we shall be in a better position to compare these and other methods of detection. Willmarth and Lu (1971) made conditionally sampled measurements of contributions uu to Reynolds stress during bursting at y + = 30, directly above the detector wire located at y+ = 15. Corino and Brodkey (1969) have observed that y+ = 30 is the approximate center of the violent interaction region during bursting. The conditional samples of uo were obtained with a digital computer and indicated that the lower the trigger level for the burst detector velocity, at y + = 15, the greater and less frequent were the contributions to Reynolds stress at y+ = 30. In a more detailed and complete study Lu and Willmarth (1973a) used the same detection method to

216

W . W . Willmarth

obtain conditional samples of the uv product that were measured at different points relative to the detection point and were also sorted into four categories represented by the four quadrants of the uv plane in which the sampled uu products were found to occur. Their results showed that large contributions to Reynolds stress occurred in the second quadrant (which represents ejection of low speed fluid, for which u < 0, u > 0), when the large-scale velocity fluctuations near the wall became lower than the mean. On the other hand, the remainder of the contributions to a negative uv occurred in the fourth quadrant (which represents sweeps or the in-rush of high-speed fluid (u > 0, v < 0), when the large-scale velocity fluctuations near the wall became larger than the mean. In addition to these results Lu and Willmarth studied the downstream convection of the bursts and sweeps by detecting the burst or sweep at an upstream station at y + = 15 and sampling the uu signal at various downstream locations. Figure 32 shows the results of the measurements obtained when the detector signal u, became equal to - u k . The results are plotted in a space-time format. As can be seen in the figure there is a time lag required for the occurrence of a peak in each plot. The origin of each plot in the figure is displaced vertically in proportion to the downstream distance between the detector wire and the x wire used to measure uv. The dashed line in the figure represents the space-time trajectory of the burst. From the slope of the dashed line the burst convection velocity is of the order of 0.8 times the local mean velocity at the x probe. The x probe was approximately 0.156* from the wall. In terms of wall variables this distance from the wall is 39v/u,. Figure 33 shows similar results for the sweep event detected when the upstream detection signal u, was first equal to ul, . Note that the convection speed is the same as for the burst, but the magnitude of the contributions to the Reynolds stress from the sweep event are less. The ratio of burst to sweep contributions was approximately 1.7 : 1. Lu and Willmarth (1973a) also measured the contributions to uv from bursts as a function of the y and z coordinates of the position of the x-wire probe with respect to the position of the detector probe. Their results showed that the spanwise extent of the bursts was confined to a narrow swept-back region with an included angle of approximately 20" centered upon the free-stream direction. In the x-y plane (normal to the wall and parallel to the stream) their measurements show that the region of Reynolds stress contributions emanates from the wall region as it travels outward and is carried downstream. As the region of Reynolds stress contributions travels outward, it is also sheared and distorted because the convection velocities, which are somewhat less than the local mean velocities, increase as one moves away from the wall. The trajectory of the bursts in the x-yplane is in agreement with the results of Kline et al. (1967). See Fig. 30. Detailed sur-

Structure of Turbulence in Boundary Layers

217

I

I

I

I

I

, -+,

I

(4

FIG.32. Convection and decay of sampled sorted Reynolds stress ( u c 2 ) / & with sampling conditions u,/& = - 1, negative slope; y/6* % 0.169, z/6* = 0 and U, 'Y 20 ft/sec. (a) x/6* = 0, (b) x/6* = 0.34, ( c ) x/6* = 0.84,(d) x/6* = 1.69, (e) x/6* = 2.53. From Lu and Willmarth (1973 a).

veys of the burst trajectory have not yet been performed and probably should not be attempted until more reliable detection schemes are developed. c. Comments on the Comparison between Methods of Burst Detection. The recent results of Blackwelder and Kaplan (1974) show that measurements based upon their detection method gave a result for the

W . W . Willmarth

218

I._i/uy

-4

7

2

i

FIG.33. Convection and decay of sampled sorted Reynolds stress (uc4)/ui, with sampling conditions u,/ul, = + 1, positive slope; y/6* N 0.169, z/6* = 0 and U , N 20 ft/sec. (a)-(e) same as in Fig. 32. From Lu and Willmarth (1973a).

conditionally sampled Reynolds stress that was different from the corresponding result based on the measurements of Willmarth and Lu. Blackwelder and Kaplan sampled uv as a function of time measured from the time of detection of a burst at y + = 15, using their short time variance scheme described previously. Near the wall the sampled uv values obtained by Blackwelder and Kaplan have two maxima, one before and the other after detection. At the detection time the sampled uv product was zero. This is a

Structure of Turbulence in Boundary Layers

2 19

result quite different from that obtained by Willmarth and Lu. See Fig. 32. Further understanding of these recent results must await the development and adoption of a reliable method of burst detection. In a recent paper Offen and Kline (1973) have compared the burst detection techniques of Willmarth and Lu (1971), Blackwelder and Kaplan (1972), and three others devised by Offen and Kline (1973); based upon normal velocity, velocity-profile slope, and the im signal, with their own visual observations of bursts. The principal result of the comparison quoted from their summary was none of the detection schemes correlated very well with the visual indications of bursting or with any other scheme. Hence, there remain serious questions about what events are measured by each technique. Despite the poor correlation, the various schemes rarely detect ejections that do not pass the probe in the plane parallel to the wall, they agree with each other to a certain extent in their relationships to the visual data, they generally produce conditional averages and velocity signatures which are similar and agree qualitatively with the expected results (i.e., streamwise velocity defect, outward motion of the fluid, and Reynolds stresses greater than the mean), and many of them are as effective as the visual data at detecting periods of high uu.

It is clear that a significant improvement in burst-detection methods would be valuable.

G. MEASUREMENTS OF STATISTICAL PROPERTIES OF TURBULENCE The development of accurate analog to digital conversion devices and the speed and versatility of the digital computer has allowed, at last, the detailed study of statistical properties of turbulence signals. Early attempts (Liepmann and Robinson, 1953) to measure probability distributions were successful but were cumbersome and tedious requiring great care to prevent drift during on line measurements of long time averages. It was not until the rapid recording and storage of large quantities of digital data was perfected that one could efficiently and reliably measure the statistical properties of turbulence. An example is the recent boundary layer measurements of Frenkiel and Klebanoff (1973), in which a digital computer was used to measure some of these statistical properties. Digital techniques have only recently been applied to measurements of various statistical properties of the Reynolds stress fluctuations uv(t). The average value of uv(t) (ie., the Reynolds stress itself) was first measured more than two decades ago by Townsend (1951) and Schubauer and Klebanoff (1951). We have had to wait nearly 20 years for the development of electronic techniques (either analog or digital) which can be used to produce accurately instantaneous values of uu(t) so that other statistical measurements can be performed. We will consider some of the recent results that have been obtained with the aid of

220

W . W . Willmarth

modern data-acquisition and data-processing equipment. The rapid advances in this area in the past few years indicate the direction of the progress that is now occurring. We consider below the measurements of Antonia (1972 ), Gupta and Kaplan (1972), Willmarth and Lu (1971), Wallace et al. (1972), Lu and Willmarth (1973a), Kaplan and Blackwelder (1973), and Brodkey et al. (1974). Gupta and Kaplan (1972) have reported measurements of the first four moments of Reynolds stress fluctuations uu(t). Their results, as a function of distance from the wall, are displayed in Fig. 34. The peak value of the rms Reynolds stress fluctuations (Fig. 34b) occurs near the edge of the sublayer where the turbulence-production term is the largest. The third moment of Reynolds stress fluctuation (Fig. 34c) shows that the fluctuations are negatively skewed across most of the boundary layer and are most strongly skewed in the viscous sublayer and the wake region. Antonia (1972a) also reported measurements of the root-mean-square Reynolds stress fluctuations in the boundary layer on a smooth wall. He

4c

FIG.34. Distribution of the dimensionless moments of the instantaneous Reynolds stress; = 6500; 0 ,R , = 1900; 0, Laufer (1954): (a) mean values of uu product; (b) root mean square of fluctuations; (c) skewness factor of fluctuations; (d) flatness factor of fluctuations. From Gupta and Kaplan (1972).

f , R,

Structure of Turbulence in Boundary Layers

22 1

reported that the rms level is approximately twice the shear stress in the region close to the wall and equal to about three times the local shear stress throughout most of the layer. This behavior as a function of distance from the wall is different from that reported by Gupta and Kaplan (Fig. 34b). Kaplan and Blackwelder (1973) have recently reported measurements of the two-dimensional probability density of the Reynolds stress. Their results for measurements near the edge of the sublayer were displayed on a threedimensional plot constructed by a digital computer. The results have not yet been published. Willmarth and Lu (1971) and Wallace et al. (1972) independently conceived the idea of sorting the contributions to the uzi product into the four quadrants of the u-u plane. The reason for this is to obtain quantitative measurements of the relative importance of bursts and sweeps. The visual studies show that during bursts, the ejection phase should occur in the second quadrant (in which u > 0 and u < 0) and on the other hand, the sweep phase, representing an inflow of high-speed fluid, should occur in the fourth quadrant (u > 0, u < 0). After digital sorting of the uu contributions into four quadrants of the u-zi plane the magnitude of the contributions in each quadrant was determined. See Fig. 35 from Brodkey et al. (1974).

.,

Y+

FIG.35. The sorted Reynolds stresses normalized with the local average Reynolds stress, 0 , results of Willmarth and Lu (197 1):. . . . ',sweep; - . . -, ejection; - . - i,, outward interaction; - - - iw, wallward interaction. From Brodkey et a/. (1974).

222

W . W . Willmarth

One could obtain similar results from appropriate integrals of the twodimensional probability density of the uu product, but the computing cost should be considered. The measurements of Brodkey et al. were made in a channel flow at low Reynolds number while the initial results of Willmarth and Lu (1971)shown in Fig. 35 were obtained at a low Reynolds number in a boundary layer in air. Figure 35 shows that the four distinct types of motion make different contributions to the Reynolds stress in the wall region. The ejection phase contributes more to uu than the sweep phase (for y+ > 15), the ratio of the two contributions being as large as 1.6 : 1. Lu and Willmarth (1973a) found that the ratio of ejection to sweep contributions was of the order of 1.4 : 1 for y + > 100, but increased for points closer to the wall and was largest, 1.9 : 1, near the wall at y + = 25. This is not the behavior found by Brodkey et al. since, in Fig. 35, the ratio of ejection to sweep contribution is less than one for y + < 15. The results of Lu and Willmarth (1973a) for small y f should be regarded with caution because the dimensions of the x-wire array used in the measurements are not small compared to either the sublayer thickness or the distance of the probe from the wall. An extension of the technique of sorting uu contributions into quadrants was reported by Lu and Willmarth (1973a). They introduced a further classification of the uu contributions in each quadrant depending upon the magnitude of the contribution in the quadrant. (Again these results could also be obtained from appropriate integrals over the two-dimensional probability density of uu.) Contributions to & from different regions in the u-u plane were measured with an x wire at various distances from the wall. The u-u plane was divided into five regions as shown in Fig. 36. In the figure, the cross-hatched region is called the “hole,” which is bounded by the curves 1 uu I = constant. The four quadrants excluding the “hole” are the other four regions. The size of the “hole” is decided by the curves 1 uu 1 = constant. The parameter H is introduced, where I uu I = H u‘u‘. The parameter H is called the hole size. With this scheme, we can extract large contributors to relative to the local rms values of u and v from each quadrant, leaving the smaller, fluctuating uu(t) signal in the “ hole.” The contributions to the Reynolds stress from the “hole ” would be those during the more quiescent periods, while the second and fourth quadrants represent the more intense burst or sweep events. The contributions to & from the four quadrants were computed from the following equations:

-

u”u,(H) Uv

1 . uv T

- - hm -~

+

1

Jb

~-

T~

uv(t)Si(t,H ) dt,

i = 1, 2, 3, 4,

(4.7)

Structure of Turbuknce in Boundary Layers

223

V

t

1

FIG. 36. Sketch of the “hole” region in the u,u plaile. From Lu and Willmarth (19733).

where the subscript i refers to the ith quadrant and if

1

Sj(t>H ) =

io

1 uu(t) 1 > N . U’D’ and the poht (u, t,) in the u-v plane is in the ith quadrant, otherwise.

(4.8)

Contributions to % from the “hole” region were obtained from (4.7) but with Sireplaced by S h , where 1 sh

=

{D

if { uo(t) { < Hu‘v‘, otherwise.

(4.9)

These five contributions uvi and uvh are all functions of the hole size H , and (4.10)

224

W . W . Willmarth

Typical results of the measurements at low speed at various distances from the wall are shown in Figs. 37-39. There was only one measurement for the case of high-speed flow, made at a distance of y + = 265 from the wall. This result is shown in Fig. 40. The results shown in these figures appear similar for measurements at both high and low Reynolds numbers, regardless of the location of the x-wire probe in the turbulent boundary layer. In Figs. 37-40 curves representing the fraction of total time that the uu signal lies in the “hole ” region are also included.

FIG.37. y/6 = 0.024. Measurements of the contributions to uu from different events at various distances from the wall. .“u,/uu: H, measured; - -, computed. .“u2/G: b, measured; - - - -, computed. G,/i: p, measured; - - -, computed. iiZ,/G: g , measured; - - -, computed. &/i: 0,measured; - -, computed. Fraction of time in “hole”: ---, measured; - - -, computed. U , u 20 ft/sec, Re, N 4,230. From Lu and Willmarth (1973a). -

~

~

~

I I

For a large portion of the time, uu is very small. Stated in another way, uu has an intermittency factor of the order of 0.55 since, in Figs. 37-40,99%

of the contribution to occurs during 55% of the time. The intermittency of the uu(t) signal is striking, especially when compared to the nonintermittent fluctuations of the u ( t ) and u ( t ) signals [whose product is uu(t)]. Gupta and Kaplan (1972) present an example in their paper in which traces of the three signals as a function of time may be compared. The intermittent behavior of uu(t) is probably a good example of what Townsend hypothesized as the active motion in turbulence. Bradshaw (1967) has discussed Townsend’s concept that turbulent motion in the wall region consists of an active and inactive part. The active part is identifiable because it is responsible for the

225

Structure of Turbulence in Boundary Layers 1.0

1% 0.8 0

Y

ICn

2

P

0.6

Y

0

0.4

s

L

.

< + I -,/---.

C

.-c

-------- - _ _ _ _ _ _

0.2

0

4

Hole Size, H -0.2 .L

FIG.38. y/6

=

0.052; for legend see Fig. 37.

/.

/

I -

/-

Hole Size, H

-0.4

FIG. 39. y/6

= 0.823; for

legend see Fig. 37.

226

W . W . Willmarth

Hole Size, H

from different events. I/, = 200 ft/sec, FIG.40. Measurements of the contribution to Re, z 38,000, y/6 = 0.014 (y’ N 265). Notation as in Fig. 37. From Lu and Willmarth (1973a).

shear stress. This aspect clearly presents an interesting problem for further study using the conditional sampling technique. Although the signals u and u are not Gaussian, the fraction of the total time in the “hole and the contribution to & from the “hole” region can be derived from the assumption of joint normality of u and u signals. For details, see Lu (1972). The predicted curves are included in Figs. 37-40 for comparison. The assumption of joint normality of u and u signals implies that the contribution to from the second quadrant Z2should equal that = Z3. The predicted curves from the fourth quadrant G4.Similarly, 6, for GI and G, are also shown on the figures. The deviation from joint normality is apparent since Gz# iZ4and El # G 3 regardless , of the flow speed and the location in the turbulent boundary layer. As can be seen, the largest contribution to comes from the second quadrant. The second largest contribution is iZ4. The contributions from uvl and uu3 are negative and relatively small. When the hole size H becomes large, there are only two contributors. One is iZ2 and the other one comes from the “hole” region. Thus the importance of the burstlike events in the turbulent boundary layer is obvious. At H = 4.5, which amounts to I uu I > 10 . I 1, there is still a 15-30 % contribution to Uu from the second quadrant, i.e., Z 2/Uu z 0.15-0.30. At this level there are almost no contributions from the other three quadrants. ”

227

Structure of Turbulence in Boundary Layers

Lu and Willmarth (1973a) attempted to estimate, with a technique different from those already mentioned in Section IV,F,2, the characteristic times related to bursts and sweeps and their durations. Similar difficulties, mainly definitive identification of bursts and sweeps, were encountered. Extensive measurements were made, in a consistent manner, for the low-speed flow across the turbulent boundary layer. A single high-speed measurement was also made to study the Reynolds number effect on the burst and sweep rates. In their method they assumed that, if the uu signal reached a certain specific level relative to the local u' and u' (i.e., hole size H ) or larger in the second quadrant, a burst had occurred. By counting, with the aid of the digital computer, the number of times the uu signal exceeded a given hole size in a given time interval, the meantime interval between burst contributions at a given hole size could be found. The nondimensional mean time interval U , T/S* between bursts is shown in Fig. 41 as a function of the hole size H 500

50

t t t

UfxT b*

20

10

5

0

5

10

Hole Size, H

FIG.41. Mean time interval between bursts as a function of hole size H and distance from y/6 = 0.021; A, y / S = 0.052; 0, y/6 = 0.103; V, y/6 = 0.206; 0, y/6 = wall; Re, 2 4230. 0, 0.412; @, y/6 = 0.618; ,. y/6 = 0.823. From Lu and Willmarth (1973a).

228

W . W . Willmarth

with the distance from the wall y/6 as a parameter. These data were obtained from low-speed ( U , M 20 ft/sec) measurements. The mean time interval between bursts exceeding a given H is nearly independent of the distance from the wall throughout the turbulent boundary layer. On the other hand, the mean time interval between bursts T exceeding a given value of H increases rapidly as H is increased (Fig. 41). A satisfactory criterion for determining T should have the property that the value of T determined from the criterion is independent of small changes in the criterion. In Fig. 41, the absence of a plateau in the variation of T as a function of H indicates that the value of H alone is not an acceptable criterion for determining the actual value of the mean burst rate. However, upon close examination of the plots of the contributions to from different events at different distances from the wall (Figs. 37-40) a unique and consistent feature is observed. As the hole size becomes large, the contributions to Uv from quadrants one, three, and four vanish more rapidly than contributions from the second quadrant. It is observed that, when H reaches a value of between 4 and 4.5, only fi2/G is not zero regardless of the distance from the wall. Contributions to above this value of H must have come from the large spikes in the uu signal related to the bursts. For a hole size of H P 4.5, Iuu I is about ten times the absolute value of the local mean Reynolds stress. These bursts certainly are very violent, since uu is large compared to the value of at a given distance from the wall. Using this unique feature, applied consistently throughout the boundary layer, one can obtain a consistent measure of the characteristic time interval Tc between relatively large contributions to (which are larger than contributions to Uv from any other quadrant at a given distance from the wall) by setting the specified level of H at 4 to 4.5. Using this scheme, a consistent estimate of the characteristic time interval between relatively large bursts is shown in Fig. 42, which was obtained from Fig. 41 by setting a level of H N 4-4.5. A value of U , T,/6* z 32 is found for most of the boundary layer. Measurements from the single highReynolds-number run are also included in Fig. 42. The fact that the value of Tc determined as described above scales with the outer flow variables is in accord with the scaling of the mean period between bursts reported by Rao et al. (1971). It must be regarded as a coincidence that the actual value of U , Tc/6* = 32 (determined with H N 4-4.5) is almost the same as the value of the mean burst period determined by Rao et al. (1971). Lu and Willmarth (1973a) used a similar scheme to measure the mean time interval between sweep contributions. A sweep was assumed to occur if the uz) signal in the fourth quadrant reaches a specified value or larger. Thus, as in the case of bursts, the mean time interval between sweeps became also a function of the hole size H . As in the case of bursts, there was no plateau in the mean time between sweep contributions as H increases. Therefore, H

229

Structure of Turbulence in Boundary Layers

40

2o

0

H

t

p ’

0

a a. 2

4

1

4

Um(ft/sec)

H

0

200

0 A

200

4.0

20

4.5

v

20

4.0

4.5

I

0.4

Yb

0.8

0.8

1.0

FIG.42. Characteristic mean time intervals between large bursts. Re, N 38,000: 0 , 4.5; 0, H = 4.0. Re, Y 4230: A, H = 4.5; V, H = 4.0. From Lu and Willmarth (1973a).

=

alone could not be used to determine the actual mean time between sweep contributions. There was, however, another unique feature in the plots of the contributions to from different events (Figs. 37-40). At a hole size of H N 2.25-2.75, at any distance from the wall in the boundary layer, GI .1. and u”u3 /G vanish. Thus, the characteristic time interval between relatively large sweeps can be obtained by setting the level at H = 2.25-2.75. In this fashion, a consistent estimate of the mean time between sweeps which are larger than the largest positive contributions to k at any given distance from the wall was obtained. A value of about 306*/U, was obtained for the mean time between sweeps throughout most of the boundary layer. The data were more scattered than the data for the mean burst time. A similar result for the mean time between sweeps at Re, 2: 38,000, namely 306*/U,, was also obtained. It appears that the mean time between both bursts and sweeps scales with outer flow variables. Further studies are obviously required. Note also that apother important feature revealed by the measurements of Lu and Willmarth (Figs. 37-40) is the remarkable uniformity (with regard to the relative magnitude and frequency) of Reynolds stress contributions at various distances from the wall. This is consistent with Falco’s (1974)observation that the typical eddies are much the same at various distances from the wall. These observations are also supported by the recent work of Wallace et al. (1974) in which a repetitive pattern in the streamwise velocity fluctuations could be recognized. Upon recognition of this pattern (a rapid increase followed by a slower decay of streamwise velocity), Wallace et al. sampled the u and u signals and averaged them after normalizing each

230

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sample to unit duration. A repetitive pattern was obtained in which antisymmetric u and u fluctuations occur which make important contributions to Reynolds stress. This pattern must be related to Falco’s typical eddies (i.e., streamwise sections through the spurt, bendover, roll-up sequence he observed). This is also consistent with the results of Fig. 17 showing the vector field of pressure-velocity correlations. Zaric (1972) has reported an interesting study of fluid temperature and streamwise velocity fluctuations within and outside the sublayer in the boundary layer developed on a heated surface. He used a digital computer to evaluate the probability densities of the velocity and temperature fluctuations at various distances from the wall, within 2.1 < y+ < 45.1. The probability densities of the streamwise velocity and temperature fluctuations inside the sublayer was highly skewed toward high velocity or low temperature, and both probability densities had large flatness factors. In the region 15 < y + < 45.1, Zaric found that both skewness factors had changed sign from the values measured within the sublayer. He investigated these phenomena using a conditional sampling technique in which the sampling criterion used short-time averages, reminescent of the method used by Gupta et al. (1971). Zaric divided his data consisting of N digitized velocities into M small groups of N / M digitized values. The small-group average and rms fluctuation about this average within each of the M groups of N / M values were then computed. Two sampling criteria were defined in which the product of the small-group average and the rms fluctuation about this average were required to have large positive or negative values. When either of these criteria were met, the data at that point became members of one of two subensembles whose statistical properties (mean, probability density, etc.) were separately computed. Zaric then discussed the statistical properties of the two subensembles and the remaining data. He found that one subensemble with low group average of u represented fluid from the wall and the other represented fluid originating far from the wall. He found that the high skewness and flatness factors of the probability density inside the viscous sublayer result from the dual structure of the flow in this region, i.e., the probability densities were highly skewed toward high velocities because of the presence of intermittent high-momentum in-rushes. Similar results were found for the conditionally sampled temperature fluctuations except that in the sublayer the probability density of the temperature was skewed toward low temperatures indicating a substantial amount of in-rush of highvelocity, low-temperature fluid parcels from the regions farther away from the wall. Zaric’s technique, like the other burst-detection methods needs further development. It is, however, an indication that the further development of sampling methods will surely provide new and interesting quantitative information.

Structure of Turbulence in Boundary Layers

23 1

Another interesting development has been the recent progress in measuring the triple space-time correlation of turbulent fluctuations. Dumas et al. ( 1973) have measured the space-time correlation between the streamwise velocity at a fixed point y’ near the wall and the product of two components of velocity at a downstream point in the x-y plane passing through the upstream point. The quantity measured is the triple correlation coefficient defined as

where, it is important to note, the normalization factor

is a function of the separation vector x between the upstream and downstream points. The values of the triple correlation coefficient that were measured are not large (of the order of 0.1 at most), but the scatter of the measurements is small enough to allow interesting comparisons to be made with our previous discussions of Reynolds stress measurements during bursting, Section IV,F,2. Figure 43 from Dumas et al. (1973) shows contours of constant triple correlation rl,12 and rl,2 2 . In the figure the streamwise separation distance between the measuringpoints, 1.416,was held constant while the downstream probe measuring Reynolds stress contributions u1 u2 or normal velocity squared u2 u2 was traversed in a direction normal to the wall. The upstream probe measuring streamwise velocity was held at a constant distance y’ = 0.0566 from the wall. We estimate that for the upstream probe y+ N 15. This allows a comparison to be made between these results of Dumas (1973) and the burst-structure measurements of Lu and Willmarth (1973a), discussed in Section IV,F,2,b. Recall that y + N 15 is the approximate center of the low-speed fluid near the wall along whose upper boundary outward eruptions of low-speed fluid were observed by Corino and Brodkey (1969). The results shown in Fig. 43a indicate a positive triple correlation between upstream streamwise velocity near the wall and the downstream Reynolds stress contributions u1 u2 in the outer half of the boundary layer. Note that if upstream u1 < 0 and downstream u1 u2 < 0, a positive triple correlation is obtained. The dashed line on Fig. 43a connects

232

W . W . Willmarth

FIG.43. (a) rl,,2; (b) l-l,22.Triple space-time isocorrelation contours in a turbulent boundary layer. Notation as in (4.10) and (4.11), upstream wire at y'/6 = 0.056, x , / 6 = 1.41. From Dumas et al. (1973).

points of maximal space-time correlation. These correlation contours represent contributions from convected disturbances since Dumas et al. state that the time delay indicated by this line is consistent with the previous scheme used to define an optimum time delay (Favre et al., 1957, 1958), which we discussed in Section IV,C. Lu and Willmarth (1973a) made their measurements in a boundary layer with Re, = 4230 and were not able to trace individual Reynolds stress contributions as far from the wall as Dumas et al. (1973) did. However, the Reynolds number in the boundary layer used by'Dumas et al. was much lower, Re, N 500. Falco (1974) observed that his typical eddies often extended all across the layer at Re, N 600 (Fig. 24). It would be interesting to dissect these triple correlation measurements by the technique of Lu and Willmarth (1973a), for sorting the upstream u1 signals to find bursts and sweeps and the downstream u1 u2 signals into quadrants to select the more important contributions. The results of Dumas et al. in Fig. 43b also show that the mean-square normal velocity fluctuations at the downstream point are correlated with the upstream disturbances near the wall. They have also measured rl, l(x,T). Figure 44 shows contours of constant correlation between the upstream streamwise velocity and the mean-square streamwise velocity at the downstream location. Note that the downstream probe contributions to the triple correlation are always positive since they are squared. Thus, the interesting sign change in this plot reflects a change in sign of the upstream probe signal. Clearly very interesting phenomena are revealed in these measurements

233

Structure of Turbulence in Boundary Layers

FIG. 44.

rl,ll.

'

. \,..'~ ,

A0

Remainder of legend as in Fig. 43.

connected with the intensity of the sweep relative to the burst signals, since the long time correlation presents a composite picture in which cancellation effects between the different effects can make interpretation extremely difficult. V. Discussion of Coherent Structures

A. THEBURSTSEQUENCE Lighthill (1963) has emphasized (see Section 11) that in the turbulent boundary layer the spanwise mean vorticity is concentrated very near the wall. The concentration of mean vorticity is accomplished by the turbulent fluctuations which in some fashion create large Reynolds stress contributions in which the low-speed fluid near the wall is exchanged or replaced with high-speed fluid from regions farther from the wall. Lighthill emphasized that the rms fluctuating vorticity is also concentrated in the same region near the wall (Fig. 3). Furthermore the recently acquired evidence from both visual studies and quantitative measurements of bursts, presented in Sections IV,F,l and 2, indicate that in this same region near the wall the turbulent fluctuations responsible for most of the contributions to Reynolds stress are large and highly intermittent. The conclusion from this is that near

234

W . W . Willmarth

the wall the vorticity fluctuations are also intermittent and accompany the large intermittent contributions to Reynolds stress. Evidence for this is contained in the visual observations and quantitative measurements of bursts. Kline et al. (1967) and Corino and Brodkey (1969) observed a high shear layer containing large spanwise vorticity that is involved in the burst while Blackwelder and Kaplan (1972), in their conditionally sampled velocity profile measurements, find a high shear layer with large spanwise vorticity just before their burst detection criterion is satisfied (Fig. 31). Since the Blackwelder-Kaplan detection criterion is sensitive to large fluctuations relative to the short time mean velocity, it seems clear that the large spanwise vorticity which is present in this region just before they detect a burst and disappears after burst detection is involved with and is a part of the burst itself. Lighthill (1963) also emphasized that in the boundary layer the strong concentration of fluctuating vorticity near the wall means that only the stretching of vortex lines can explain how the gradient of mean vorticity is maintained in spite of viscous diffusion down it. We may add that the same conclusion must then hold for the fluctuating vorticity since if one stretches vortex lines in the presence of large turbulent fluctuations near the wall the vorticity fluctuations will also be large. Lighthill also discussed a possible vortex stretching mechanism in which the effect of a solid surface on the turbulent vorticity close to it was to correlate inflow toward the surface with lateral stretching in the spanwise plane. He also discussed the observations of Townsend (1956) that large-scale motions are elongated in the stream direction, as if their vortex lines had been stretched longitudinally by the mean shear. He proposed a cascade process in which movements of smaller and smaller scale bring vorticity closer and closer to the wall and continually stretched the vortex lines. In considering this aspect of the problem the visual observations of Kline and his colleagues at Stanford University and the other visual observations mentioned in Section IV,F, 1 provide valuable new evidence indicating that the important flow processes that produce most of the turbulent energy and are responsible for the major portion of the Reynolds stress near the wall are highly intermittent. The visual observations also show that these flow processes consist of a reoccurring coherent pattern of events. The pattern as described by Kline et al. (1967) consists of the gradual " liftup" of low-speed streaks from the sublayer which then suddenly oscillate, followed by bursting and ejection (Section IV,F,l). There appears to be no evidence of the cascade process of vorticity of smaller and smaller scale being brought closer and closer to the wall, which was suggested by Lighthill (1963). The visual observations and the conclusion discussed above (that only the stretching of vorticity can explain the strong concentration of both mean and fluctuating vorticity near the wall) leads inescapably to the conclusion

Structure of Turbulence in Boundary Layers

235

that the visually observed sequence of recognizable events (i.e., gradual “ liftup ” sudden oscillation, followed by bursting and ejection) is nothing other than the vortex stretching process itself. The series of visual observations of Kline and his colleagues which are summarized in their latest work (Offen and Kline, 1973) have led them to the same conclusion. They made the following statement, quoted from the abstract of their report: The dye and bubble patterns observed in the pictures from the visual study, when combined with the conditionally processed results from the anemometer/dye experiment, strongly suggest that the three kinds of oscillatory growth reported by Kim et al. (1971) are associated with just one type of flow structure-the stretched and lifted vortex described by Kline et al. (1967). The streamwise and transverse vortical patterns are conceived as the passage of different locations of the stretched and lifted vortex; the wavy perturbation occasionally seen in the time-lines prior to one of the vortical patterns may indicate the presence of velocity pulsations that emanate from a bursting structure which is still farther upstream.

A sketch from Kline et al. (1967) of the stretched and lifted vortex is displayed in Fig. 45. We note that in the movies showing bursts one can often observe the swirling of fluid about axes parallel to the stream. The other visual observations are not contradicted by the model (see Fig. 45) proposed by Kline et al., since (for example) the Corino and Brodkey (1969) observations are then to be regarded as randomly occurring (in space and time) streamwise views of cross-sectional cuts through the burst pattern. We recall that the depth of field was very small in Corino and Brodkey’s work. Tu and Willmarth (1966), after quantitative measurements with hot wire probes, also proposed a model involving stretched streamwise vorticity that is essentially the same as that of Kline et al. (Fig. 45). The configuration of the vortex structure shown in Fig. 45 is hardly a new idea since Theodorsen

Lifted and Stretched vortex element

FIG.45. The model of streak breakup that precedes bursting. From Kline et al. (1967).

236

W . W . Willmarth

(1952,1955) proposed that " horseshoe " vortices were the mechanism for the turbulent transfer of momentum and heat. Direct quantitative evidence in support of the model of Fig. 45 was obtained by B. J. Tu and W. W. Willmarth (unpublished) who measured the correlation between the streamwise velocity at a fixed point near the wall just outside the sublayer and the streamwise vorticity at various points above and downstream of the fixed point just outside the sublayer. The results from these measurements which had not been published before are shown in Fig. 46. The measurements of streamwise vorticity were made with a small probe consisting of four hot wires arranged in the configuration described by Kovasznay (1954). This probe was carefully constructed by Dr. Tu; and although it was as small as possible (wire spacing of approximately 1.5 mm), the thickness of the sublayer of the boundary layer at Re, N 38,000, in which it was used, was of the order of 0.05 mm. This (as already discussed in Section IV,E) means that the vorticity probe cannot resolve the smaller scale streamwise vorticity. This does not invalidate the main conclusions to be drawn from the measurements shown in Fig. 46; it simply means that the correlations that were measured are probably larger. 6 -

x/s*

6

x/g4 4

uW.,JI'~=

-0.025

A

2

0

-2

FIG.46. Isocorrelation contours of %,/u'w:. The velocity u is measured at a fixed point just outside the sublayer at y + = 6.8. The w, probe is moved about in the plane; (a) y + 2 263, (b) y+ = 524. Re, N 38,000, U , = 204 ft/sec, u , / U , e 0.0326. Measurements by Dr. Bo Jang Tu and the author.

Structure of Turbulence in Boundary Layers

231

The main conclusion to be drawn from Fig. 46 (the notation is sketched in Fig. 47) is that downstream on either side of the fixed point where the streamwise velocity is measured there is an antisymmetric pattern of highly sweptback streamwise vorticity emanating outward from the wall in the downstream direction. J. Sternberg made the suggestion (private communication) that it would be of interest to know whether the velocity just outside the sublayer was lower or higher than the mean when contributions to the correlation occurred, This question led us to reinvestigate the original analog-tape recorded measurements with the technique of sorting applied to both the u and oxsignals. These results (Fig. 47), published by Willmarth and Lu (1971), show that the contribution of largest magnitude to the correlation occurs when u c 0 and w, > 0. This connects the pattern of Fig. 46 to the bursts and to the pattern of Fig. 45 since we know (Section IV,F,2,b) that bursts with ejection of fluid occur when the velocity near the edge of the sublayer is lower than the mean. The sign of the vorticity w, is on the average positive when this occurs, and so the streamwise rotation of the fluid indicated in Fig. 45 is consistent with the measurements of Fig. 47. Further evidence supporting a sweptback pattern of vorticity near the wall was also obtained by Tu and Willmarth (1966)T who reported spacetime correlation measurements of wall pressure and spanwise velocity. From measurements in which the probe measuring the spanwise velocity was R ~uw,/u'w'~=-O.O95

Z/6" FIG.47. Contributions to correlation between streamwise vorticity and velocity in the sublayer from four quadrants in the u, oxplane. U, = 204 ft/sec. Note, R , , = 1/2n = 0.159 for two uncorrelated Gaussian random variables. From Willmarth and Lu (1971).

t Willmarth

and Tu (1967) is a short summary.

238

W . W . Willmarth

moved about they constructed isocorrelation contours of the correlation between spanwise velocity and wall pressure with zero time delay in planes normal to the stream. In Fig. 48 the isocorrelation contours in four crosssectional planes show the liftup with downstream distance of what must be streamwise vorticity which acts to produce a reversal in the spanwise velocity and hence in the sign of across the plane of zero correlation. Directly related to the correlation measurements of Fig. 48 are the wall-pressure-normal-velocity correlation measurements of Willmarth and Wooldridge (1962) that were discussed in Section IV,C. Very near the wall the contours of constant correlation in a plane parallel to the wall show a sweptback structure of isocorrelation contours. See Fig. 37 at x2/6* = 0.10.

FIG.48. Three-dimenslonal diagram of isocorrelation contours of R p w , .Re, = 38,000. = 204 ftlsec, itr!U7 = 0.0326. From Tu and Willmarth (1966).

U,

It is not possible to make a quantitative relationship between these measurements and the correlation measurements of Fig. 48. However, the important qualitative result is that again near the wall a highly sweptback pattern of disturbances becomes evident. Additional fragmentary evidence that is consistent with'the presence of strong streamwise vorticity just above the sublayer (specifically unexpected changes in the sign of the correlations % and -~ zw as a function of the location of the probes) was also reported by Tu and Willmarth (1966). The conclusion from the measurements is that the evidence indicates that the burst mechanism consists initially of a pair of counterrotating vortices with primarily a streamwise vorticity component that are stretched during

Structure of Turbulence in Boundary Layers

239

the liftup phase of the bursting sequence.This leads us to a possible explanation for the rapid eruption phase of the burst sequence. Consider the flow in the cross-sectional plane normal to the wall and stream which contains the streamwise vortices near the wall. Figure 49 is a sketch of a vortex pair near the wall and shows the image pair beneath the wall. Note that the direction of rotation is that indicated in the sketch of Fig. 49. This sketch is reminiscent of the flow in the cross-sectional plane through the trailing vortex pair behind an aircraft near the ground in a uniform flow, except that the direction of rotation of the flow around the vortices behind an aircraft is opposite to that shown in Fig. 49. Bleviss (1954) has analytically studied the motion of the trailing vortices near the ground behind an aircraft. The mutual

I '

FIG.49. Sketch of vortex pair near the wall. Image ofpair below the wall. Dashed lines are hyperbolas giving trajectory of the vortex centers.

induction effects for discrete vortices of infinite extent behind an aircraft are such that the centers of the vortex pair above the ground move farther apart and toward the ground along hyperbolic paths (if the flow is uniform) which are ultimately parallel to the wall. If the direction of vortex rotation is reversed, the motion computed by Bleviss (1954) will reverse. Then the motion (if in uniform flow) will be along the hyperbolic paths (dashed lines in Fig. 49) but in the direction of the arrows so that the vortex pair above the wall will move together and away from the wall. As the vortex pair moves away from the wall it will be convected downstream more and more rapidly as the distance from the wall increases. This will cause severe stretching of the vortex pair with rapid increase in vorticity

240

W. W . Willmarth

which will cause more rapid motion away from the wall. This, we suggest, is the fundamental process involved in the last stage of the burst sequence, i.e., the eruption or ejection phase. This stretching process and the motion of the pair away from the wall occurs in a background of relatively intense turbulent fluctuations. One can expect that large random motions and contortions of the vortices will occur during the intense stretching of the vortex pair embedded in the turbulence. It seems likely to us that this is the fundamental nonlinear mechanism responsible for the intense random motion developed in the last stage of the burst sequence. It is tempting to compare the burst sequence in a fully turbulent flow with the rather thoroughly studied transition process. We have outlined the transition process in Section 11. It is not possible at the present time to make quantitative comparisons between the stages of transition and the sequence of events during the burst. We suggest that the last stages of transition in which violent eruptions and breakdown occur are caused by the same processes we have just outlined for the burst sequence. An analysis along similar lines for the amplifier effect of stretched pairs of vortices in a shear flow was performed by Hama (1963). He made proposals of a similar nature for the random breakdown process during the transition to turbulence and studied the deformation of perturbed vortex lines (Hama, 1962, 1963). The deformations were computed numerically and found to propagate along the vortex lines. His results seem to us to be qualitatively consistent with the concept that severe instabilities in the stretching vortex pair are responsible for the random breakdown process which produces turbulence. It is unfortunate that the proper instrumentation for the measurement of small-scale vorticity components in turbulent boundary layer flows has not yet been developed. As already emphasized in Sections IV,E and V,A, the present hot wire probes are far too large to resolve the small-scale vorticity fluctuations. It appears that turbulence structure cannot easily be understood without the concept of vorticity. It is apparent that the experimental possibilities in this problem are still very great and the theory for it is almost nonextant.

B. CYCLICAL OCCURRENCE OF BURSTS

To complete our discussion we will now consider the evidence for the cyclical regeneration of bursts. The concept of cyclical regeneration arises naturally from the fact, as discussed in Section IV,F,2,a, that the mean burst period scales with the outer flow variables. As we have discussed, the burst originates within the sublayer, then becomes very intense and grows emerg-

Structure of Turbulence in Boundary Layers

24 1

ing into the outer wake region. This has led to the suggestion that the bursting process is part of a cycle in which some type of interaction between the outer regions of the boundary layer and the wall region is important. One suggestion that we shall discuss is that new bursts are created as a result of an alteration or disturbance of the flow field in the sublayer caused by the debris produced by an old burst. The concept of cyclical regeneration is not new. Einstein and Li (1956) proposed that the sublayer was in a state of unsteady laminar twodimensional flow. In their theory the sublayer thickness periodically increases and decreases. The periodicity is supposed to be caused by the instability and breakdown of the supposedly laminar sublayer after its thickness has grown to the point that the critical Reynolds number is exceeded. Then the cycle begins again. Loeffler (1974)has recently proposed a similar three-dimensional approximation to this theory. Naturally, these theories cannot account for the interaction with the turbulence and do not appear to be useful for understanding turbulence structure. Recent visual observations of intermittent bursting were made by Offen and Kline (1973, 1974) designed to investigate the interaction between the outer and the wall regions. They used a combination of dyed fluid at the wall of one color, hydrogen bubbles shed from a wire normal to the wall, and a dye filament of another color injected into the flow above the wall to observe simultaneously the flow disturbances and interactions between the inner and outer parts of the boundary layer. Their very detailed report should be perused for better understanding of the interesting interaction phenomena. The interaction process is explained in two ways; one from the viewpoint of vorticity (which we prefer) and the other from the point of view of pressurevelocity interactions. The result of their very fine observations is that the bursting process is cyclical beyond reasonable doubt. The gist of their work is that the vorticity produced during the bursting sequence is observed to emerge from the vicinity of the wall as it is carried downstream and leaves the place of its origin. Often this vorticity was observed to interact and/or combine with other similar accumulations of vorticity to make a larger accumulation of vorticity. This, according to Offen and Kline (1973), was akin to the twodimensional vortex pairing process studied for some time by Browand and reported by Winant and Browand (1974). As these larger scale accumulations of vorticity, which are generally in the outer wall region but not the wake region, pass over the dye at the wall, wallward-moving disturbances are observed in the outer dye filament and then the wall dye indicates the burst sequence of lift up, then sudden oscillation followed by bursting and ejection. The sequence after sudden oscillation occurred was not always completed but would on occasion subside. The sequence would begin again

242

W . W . Willmarth

when another accumulation of vorticity in the outer region passed over the wall. Offen and Kline (1974) positively conclude that, (i) each lift up in the wall region is associated with a disturbance which originates in the logarithmic region and is characterized by a mean motion towards the wall and that (ii) such disturbances are generated by the interaction of an earlier burst from further upstream with the fluid motion in the logarithmic region.” Figure 50 is a pair of photographs from their (1974) paper and shows what “

FIG.50. Two photographs ofsmooth sweep event at low Re,. Intersection of lines through pairs of long arrows gives location of smooth sweep. Short arrow indicates subsequent liftup of wall dye (dark area just above the wall extending to the righ-i.e. upstream from the short arrow. From Offen and Kline (1974).

they call a typical smooth sweep of outer dye which accompanies the larger downstream outer disturbances which in turn trigger a liftup of the wall dye in the lower photograph. The two photographs are displaced relative to each other by the distance the free stream has traveled during the time interval between the photographs. One observes that the outer vortical disturbances are traveling at a somewhat lower speed than the stream speed. Offen and Kline (1974) have proposed a model for the cyclic process. Quoting from their paper: “This model is based on the hypothesis that the slow-speed wall streak behaves as a boundary layer within the conventionally-defined turbulent boundary layer. Due to a temporary, local, adverse pressure gradient, this inner boundary layer separates, or lifts up from the wall. The pressure pulsation is probably imposed upon the slow-

Structure of Turbulence in Boundary Layers

243

speed streak by a wallward-moving disturbance that originated in the logarithmic region of the turbulent boundary layer. Such disturbances have been called ‘sweeps’ in the earlier discussion.” The concept that the cyclical process of bursting is initiated by pressure disturbances in the wall region was also proposed by Laufer (1972).Kovasznay (1967) speculated about the interaction process and Kovasznay in 1971 had privately suggested to the author that the pressure should correlate with the bursts.? The author, with the aid of V. Kibens, R. Winkel, and D. Christians, has performed a conditionally sampled measurement of wall pressure and Reynolds stress contributions during bursting. The experimental setup was similar to that described in Lu and Willmarth (1973a) but with the addition of a wall-pressure transducer beneath the detector wire and the Reynolds stress probe. The experimental measurement of wall pressure fluctuations at low speeds required that a sensitive rather large ($in. diameter) condenser-microphone be used which was mounted flush with the wall. The low-frequency background pressure fluctuations in the potential flow were severe. It was necessary to subtract another electrical signal representing these pressure disturbances from the wall-pressure signal. This was accomplished by using an operational amplifier to subtract from the wall-pressure signal the signal from another microphone installed in the potential flow far from the wall at the stagnation point of a streamlined body of revolution. The effectiveness of the subtraction scheme was checked at low speeds when transition occurred at the measuring station. It was verified that both the hot wire and the corrected wall-pressure signals showed similar intermittent signals during active and inactive periods in the intermittently turbulent transition region. The measurements that are important for understanding the effect of wall pressure upon the cyclical burst-regeneration process consist of conditional sampling of the Reynolds stress and wall pressure during eruptions of lowspeed fluid from the wall region. The detection criterion for the burst was similar to that used by Lu and Willmarth (1973a) (i.e., that the streamwise velocity at y = 15v/u, after filtering to remove high frequencies shall have decreased to a level twice the rms velocity below the mean). The results are displayed in Figs. 51 and 52. Note that in Fig. 51 the Reynolds stress contribution is of short duration compared to the sampled wall pressure during bursting. The result that the conditionally sampled wall pressure is low at the time of burst detection indicates that before burst detection the pressure gradient experienced by the fluid near the wall that is later involved in the burst was adverse (i.e., the pressure downstream was higher). This, coupled with the fact that the region of adverse pressure gradient is of relatively large scale, is t Published in Kovasznay (1972).

244

W . W . Willmarth

8L

’40.00

4S.W

-30.00

-15.00

TIME

15.W

(DI&%IONLESS)

30.W

U5.W

I

6O.W

f U J V

FIG.51. Sampled Reynolds stress contributions using detection method of Lu and Willmarth (1973a) with u, = - 2 4 at y + N 15. Re,, = 6830, U , = 33.0 ft/sec, y / U , = 0.037.

in agreement with Offen and Kline’s (1974) proposed mechanism for lowspeed streak liftup in the beginning of the burst sequence. In addition to Offen and Kline’s proposed mechanism we add the fact that near the wall the inertial forces are small so that in the sublayer the important terms in the momentum equation are the pressure gradient and stress

fl

‘-60.00

I

Y5.W

-30.00

-15.W

TIME

.W

1s.w

(DIMENSIONLESS1

3o.w

KW

6o.w

tU,/S*

FIG.52. Sampled wall-pressure contributions using same detection method and boundary layer as in Fig. 51.

Structure of Turbulence in Boundary Layers

245

terms. Therefore, the fluid near the wall is prepared for the burst sequence by the convected large-scale vorticity in the outer flow which creates a moving field of adverse and favorable pressure gradients. These moving pressure fields act on the sublayer flow and actually push the moving fluid parcels in the sublayer about. Offen and Kline (1973) have discussed a possible mechanism for small-scale low-speed streak generation which also may be a part of the process we are describing. However, we are now proposing a gross large-scale effect. To continue, the action of the convected adverse pressure gradient upon the sublayer near the wall will generate new vorticity at the wall [see (2.1)] with sign opposite to the mean vorticity. If the adverse gradient is large enough and lasts long enough, a low-speed region is developed near the wall which contains reduced spanwise vorticity and is bounded from above by a high shear layer. Note that this is what Corino and Brodkey (1969) observed near the wall just before a burst occurs. They stated that in the low-speed region near the wall deficiencies as great as 50 % of the local mean velocity were observed. Our contribution here is that since both the fluid near the wall and the adverse pressure field produced by the vorticity from previous bursts in the outer fluid are moving downstream, there is more time for the fluid near the wall to be affected (i.e., deaccelerated) than would normally be the case, thus the vorticity produced at the wall (with opposite sign to mean vorticity) by the adverse gradient will accumulate in this region as time goes by. The high shear layer that is produced above the low-speed fluid is unstable and is the source of the vorticity that is stretched and incorporated in the ejection and chaotic motion in the bursting sequence. To conclude: We believe that the initiation of a burst is caused by a convected “massaging” action that acts on the low-speed sublayer fluid. This creates an unstable high shear layer near the wall. The massaging action is produced by the adverse gradient portions of the wall pressure that accompany the convected large-scale vorticity from previous bursts, as observed by Offen and Kline (1974). It is significant to note that Elliot’s (1972) recent paper contains measurements of the coherence and phase between pressure and velocity near the wall in an atmospheric boundary layer. Elliott found that for large-scale pressure fluctuations the streamwise velocity near the wall was in phase with the pressure at the wall. That is to say that the eddies with a scale as large or larger than their distance from the wall “feel” the wall so that a positive wall pressure will occur when downward moving fluid, typically of higher-thanaverage u, is decelerated upon contact with the boundary. If one looks carefully at the flow ahead of the low-speed liftup in Offen and Kline’s (1974) photographs (see Fig. 50, for example), one finds a large downward motion just downstream of the lift up. This produces a high pressure downstream of liftup so that a convected adverse gradient just upstream of the convected

246

W . W . Willmarth

high-pressure region acts on the fluid in the sublayer and creates near the wall an unstable high shear layer which then lifts up and starts the burst sequence. The momentary retardation of the fluid near the wall over a finite largescale region (which we believe leads to the formation of a low-speed region bounded by the high shear layer) will also produce some type of spanwise variations in the sublayer. Mollo-Christensen (1971) has emphasized this aspect in his review, but there is as yet little concrete information about the nature of the three dimensionality that must occur. It is clear from the results of Fig. 52 that the scale of the regions of convected adverse gradient are very large compared to the sublayer thickness. Furthermore, the threedimensional effects must be confined mainly to the sublayer since that is where the fluid velocity is affected by the large-scale pressure gradient. When one considers the scale of the sublayer relative to that of the large-scale wall-pressure disturbance causing streamwise and spanwise motions, it seems quite possible that smaller scale (possibly unstable) spanwise variations containing warped vortex lines may occur. This may be the origin of the streaky structure observed in the sublayer before bursting. This might present an interesting theoretical problem if the proper simplifications can be made so that a mathematically tractable model could be obtained. On the other hand, Offen and Kline (1973)have proposed a different mechanism for streak formation that the interested reader should peruse. Obviously, these suggestions must be documented by further experiments and by theoretical work. The present suggestion for the " massaging " action of the adverse gradients, if correct, may also explain the observation (see Blackwelder and Kovasznay, 1972a for recent experiments) that a boundary layer subjected to a favorable pressure gradient becomes less turbulent. Blackwelder and Kovasznay found that in a flow with a strong favorable pressure gradient the boundary layer turbulence level could be reduced to a negligible value. We would explain this by observing that the outer portion of the vorticity from the burst sequence was accelerated rapidly downstream in the region of favorable external-pressure gradient. There would then be little time for oppositely directed (relative to the mean) spanwise vorticity to be created and accumulate near the wall [see Eq. (2.1)]and strongly unstable high shear layers would not be produced. Thus, the burst sequence would be inhibited. On the other hand, in a flow with adverse external-pressure gradient, the outer vorticity accumulations from previous bursts would travel more and more slowly over the sublayer fluid as they were carried downstream. This would produce a long duration massaging action of the sublayer fluid and in the adverse gradient regions strong production [see Eq. (2.1)] and accumulations of vorticity directed oppositely to the mean

Structure of Turbulence in Boundary Layers

247

vorticity would occur. This would in turn intensify the unstable high shear layer above the low-speed region and lead to the generation of the intense turbulence which is observed in an adverse gradient, as has been explained. Landahl(l972, 1973) has considered both the transition process in which turbulent spots are produced and the burst phenomenon in a fully turbulent flow from a theoretical point of view. The essential new idea in his first paper is the concept that small-scale secondary waves riding on a large-scale primary instability wave may accumulate at a certain point on the primary wave owing to nonlinear trapping” of the small-scale secondary waves. Landahl uses the kinematical-wave theory to explain his concepts and then applies the ideas to the experimental profiles measured by Klebanoff et al. (1962) during artificially induced transition. In the case of the fully turbulent flow considered by Landahl (1973), calculations along these lines were not attempted because the problem is highly three dimensional and therefore mathematically too complicated. Instead, Landahl(l973)discussed the bursting process observed in experiments as we have done in this review and then proposed a different burst-regeneration mechanism, for which we refer the reader to his paper. We have now taken stock of the current situation with regard to the problem of the generation and structure of turbulence. It is clear that much more experimental work is needed before a viable theory can be produced. The prospects are good for many advances in the experimental work by means of recently developed techniques. Future progress will benefit greatly from development of better methods to measure small-scale vorticity and to detect bursts. “

ACKNOWLEDGMENTS The support of the Office of Naval Research, Contract N00014-67-A-0181-0015 and the National Science Foundation, Grant GK-30888 during the preparation of this review is acknowledged. The author is grateful to the Office of Naval Research for their continued support for many years and to Valdis Kibens for his helpful comments after reading a draft of the paper.

LIST OF SYMBOLS cx

f

3

c,

H k, , k, , k, P

longitudinal and lateral scales of typical eddies; see Fig. 19 frequency, Hz hole size, H = I uu 1 /u’u’, see Section IV,G wave number vector components in x, y, or z direction where k = 27~11 mean pressure

W . W . Willmarth fluctuating pressure. p = 0 fluid velocity vector distance between two points Reynolds number based on x distance Reynolds number based on momentum thickness time difference mean period between bursts characteristic time interval betweer, large contributions to uu(t), see Section IV,G time mean velocity component in x, y , or z direction average contribution to uu in ith quadrant of u-v plane mean velocity component in ith direction fluctuating velocity component in x, y, or z direction fluctuating velocity component in ith direction shear velocity T ~ / P mean convection velocity component in x direction mean free stream velocity component in x direction volume entrainment velocity at the superlayer, as defined by Kovasznay (1967) orthogonal coordinate in stream direction, i = 1 orthogonal coordinate normal to wall and stream, i = 2 orthogonal coordinate parallel to wall and normal to stream, i = 3 intermittency factor, fraction of the time the flow is turbulent at a given point boundary layer thickness displacement thickness total dissipation in turbulent flow momentum thickness wavelength coefficient of viscosity kinematic viscosity p / p mass density of fluid time difference shear stress at the wall vorticity vector vorticity component in x, y, or z direction vorticity component in ith direction 0 = 2nf

(4

root-mean-square value of fluctuating flow quantity sampled quantity, a quantity in terms of wall variables, ( )uJv average value of quantity, a, in the nonturbulent zone average value of quantity, a, in the turbulent zone average value of quantity, a

Structure of Turbulence in Boundary Layers

249

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