Vortex dipole rebound from a wall

Vortex dipole rebound from a wall Paolo Orlandi Citation: Physics of Fluids A: Fluid Dynamics (1989-1993) 2, 1429 (1990); doi: 10.1063/1.857591 View online: http://dx.doi.org/10.1063/1.857591 View Table of Contents: http://scitation.aip.org/content/aip/journal/pofa/2/8?ver=pdfcov Published by the AIP Publishing Articles you may be interested in The effect of slip length on vortex rebound from a rigid boundary Phys. Fluids 25, 093104 (2013); 10.1063/1.4821774 Depinning of vortex domain walls from an asymmetric notch in a permalloy nanowire Appl. Phys. Lett. 101, 082402 (2012); 10.1063/1.4745788 The dynamics of a viscous vortex dipole Phys. Fluids 21, 073605 (2009); 10.1063/1.3183966 Formation of vortex dipoles Phys. Fluids 18, 037103 (2006); 10.1063/1.2182006 Dynamics of a nonlinear dipole vortex Phys. Fluids 7, 2220 (1995); 10.1063/1.868470

This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitationnew.aip.org/termsconditions. Downloaded to IP: 132.239.1.231 On: Thu, 21 May 2015 08:09:57

Vortex dipole rebound from a wall Paolo Orlandi Uniuersita di Roma, "La Sapienza," Dipartimento di Meccanica e Aeronautica, Via Eudossiana 18, 00184 Rama, Italy

(Received 6 July 1989; accepted 25 April 1990) Accurate numerical simulations of vortex dipoles impinging on flat boundaries have revealed interesting new features. In the case of free-slip boundaries the dipole does not rebound from the wall. In the case of nonslip walls rebounding occurs and complex interactions of secondary and tertiary vortices appear. The numerical simulation of the first dipole rebound from the wall agrees with experimental visualizations. Numerical experiments extending in time beyond the real experiments show multiple rebounding. Each rebound is associated with the detachment of a secondary vorticity layer from the wall, these layers merge, and at a value of Reynolds number Re = 1600, form a new dipole. This dipole has sufficient circulation to induce on itself a motion in the opposite direction to the motion of the initial dipole.

i. INTRODUCTION

The interaction of vortices with boundaries is an important problem in many areas of practical interest. For example, trailing vortices from aircraft interact with the ground and a knowledge of their time evolution is helpful in the landing and takeoff phases of flight. In geophysical applica~ tiOllS, large-scale vortices interact with solid boundaries, such as coastal or mountain ridges, influencing the micrometeoroiogy. The interaction can be with either free-slip boun~ daries or nonslip boundaries. In the first case, wall vorticity is zero for a straight boundary, while for curved boundaries it is inversely proportional to the radius of curvature of the boundary. In the second case, a very strong vorticity is created at the boundary while secondary and tertiary vortices interacting with the primary vortex give rise to very complex flow structures. A number of experimental studies of this flow have been carried out, starting with Harvey and Perry, 1 who observed how a single trailing vortex interacts with a moving floor. They observed that the vortex rebounds and travels far from the waH, and explained that this effect is due to the creation of a secondary vortex at the wall. Vortex rebounding was also observed by Barker and Crow 2 in water in which the vortex interacted with a free surface (zero shear stress). They attempted to explain the observation by an inviscid theory, which Saffman 3 later proved unacceptable. Van Heijst and Flof, 4 in a recent experiment performed in stratified flow to simulate two-dimensional flows, were able to clearly visualize the rebounding of a vortex dipole impinging on a solid wall. By numerical simulation, Peace and Riley5 described the rebounding effect in the presence of both nonslip and free-slip boundaries. They attributed the rebounding, in both cases, to viscous effects, but they did not give a satisfactory explanation for the case of free-slip boundaries. Our numerical simulation confirms the Harvey and Perry I explanations and investigates new features of the motion near the wan of the vortex dipole. In the case of free-slip boundaries the phenomenon called rebounding by Peace and Riley" can be explained by the fact that as the vorticity is diffused, the recirculating region increases and the vortices 1429

Phys. Fluids A 2 (8), August 1990

move apart. Since the effects of diffusion are also significant while the vortices approach the solid wall, it is not appropri~ ate to call this behavior rebounding. The numerical simulations of Peace and Riley 5 were done at rather low Reynolds numbers, Re = (J)maxa2/v, at which diffusion quickly destroys circulation. Although real situations occur in the presence of turbulent flows, the simulation oflaminar flows at high Re emphasizes very intricate and unexpected situations. In our calculations, done 011 a very fine grid, we have investigated the important role of secondary vortices and the creation of tertiary vortices. Tertiary vortices are created at the wall and their smaU scale is the reason for their rapid dissipation. Secondary vortices survive longer, and depending on the Re number, after multiple rebounding, pairing of vortices can form a new dipole that moves away from the wall. II. EQUATIONS AND NUMERICAL SCHEME

Numerical simulation of two-dimensional flows, involving interactions of vortices of different scales, requires a large number of grid points when the system of NavierStokes equations is discretized. The system of equations in vorticity-streamfunction formulation reduces the memory and decreases the CPU time requirement with respect to other formulations. Furthermore, this method has the advantage of using a discretization of the advective terms, which, for v = 0, conserves kinetic energy and enstrophy and maintains the skew symmetry property of the Jacobian (Arakawa 6 ). It must be stressed that these conservative properties are very important, since the failure to conserve both energy and enstrophy is often responsible for unphysical flow field behaviors. The Navier-Stokes equation in the vorticity-streamfunction formulation is .1. = -1V 2(u. -a(J) + J«(J),'I')

(1)

at Re The velocities are related to the streamfunction by VI = aifJ/ax z, V2 = - a¢/aXI' The streamfunction ¢tis related to the vorticity OJ by

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(2) Periodicity along the x I direction is assumed. The wall vorticity condition for the nonslip boundary is obtained by differentiating the vorticity definition

(i) = (av 2

ax,

_

au , )

(3)

aX 2

with respect to x 2 , whereas for slip boundaries, (J) = 0 has been assumed. The numerical scheme is given in Orlandi: 7 here we illustrate only its main features. The viscous terms are discretized by a centered, second-order scheme and the advective terms by the Arakawa 6 scheme. To avoid viscous stability restrictions, the viscous terms are treated by an implicit scheme, and to save CPU time the resulting pentadiagonal matrix has been factorized into two tridiagonal matrices. as suggested by Briley and McDonald. 8 The solution ofthe first tridiagonal matrix requires boundary conditions that do not coincide with the physical ones. The choice of periodicity along XI overcomes this inconvenience. The wall vorticity has been discretized by an expression similar to Woods' 9 formulation. The use of the delta form for the vorticity equation results in second-order time accurate values for the waH vorticity. A very accurate expression for the wall vorticity is necessary for high Re flows to represent the very thin layers of vorticity generated at the wall. The nonlinear terms were discretized in time by an explicit third-order Runge-Kutta scheme, with two level storage developed by Wray. IO This scheme allows Courant numbers larger than those permitted with the Adams-Bashforth scheme, further reducing CPU time requirements. The streamfunction was obtained bv using a fast Fourier transform (FFT) in the x I direction ~nd a tridiagonal solver along the X 2 direction. The CPU time required for the solution of the viscous calculation on a 256 X 256 grid for 2000 time steps was 850 sec on the eRAY

YMP. The numerical method was tested by calculating the time evolution of small perturbation superimposed on the plane PoiseuiHe flow at Re = 7500. The initial perturbation was derived from solutions of the Orr-Sommerfeld eigenvalue problem. The solutions obtained compared very wen with the solutions obtained by full spectral methods, given by Canuto et al. [[

III. INITIAL CONDiTIONS The vortex dipole has an initial distribution (Il = k 2¢ throughout the recirculation domain. In the inviscid case this dipole moves with a constant velocity Uc and the streamfunction distribution is given by

if; = -

PUJ, (kr)lkJ; (ka)] sin e (4)

in the region r

8

e

"

•" •

..,

0

0

9

~

0 ®

"

I' * 8

'"dI

~

0.0

4.0

a.o

12.0

16.0

:00.0

t

(8) Here Uo = VI (0, x z ), Vo = V2 (XI' 0). As expected, at Re = 800 a very good agreement is obtained [Figs. 3(a) and 3(b)] and this agreement persists even at Re = 40 [Figs. 4(a) and 4(b)].

FIG. 3. Velocities of vortex centroids in time at Re = 800. (a) Horizontal velocity, (b} vertical velocity. Present: e; Eqs, (8): O.

B. Nonslip wall

~ ~ Q,

:n

~

~ ~

§ 0.0

1.0

s.o

3.0

4.0

1I.a

XjD FIG. 2. Trajectory of vortex centroids scaled with D at 6., Re =. 40; 0, Re = 80; and 0, Re o.~ 800. 1431

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Harvey and Perry! showed through an experiment that when a single vortex approaches a nonslip wall it induces at the waH an intense thin vorticity layer that becomes unstable and, rolling up, forms a secondary vortex that causes the rebounding of the primary vortex. The design of a purely two-dimensional experiment with a vortex pair approaching a solid waH is very difficult, both in creating the initial dipole and because, immediately after the collision, the dipole becomes unstable and substantial three-dimensional effects occur. Experiments with axisymmetric rings are much simpler to set up but unstable effects occur above a certain Re number. 15 The main difference between a vortex pair and a vortex ring is that, in the motion of a vortex ring, vortex stretching plays a role absent in the motion of a vortex pair. In the experiment described in Ref. 15, a very accurate flow visualization revealed features very similar to those observed in the present simulation, mainly the creation of strong secondary vortices and weak tertiary vortices at the wall. For Rei' = r /v > 2400, after complex interactions between the primary and secondary vortex rings, a vortex ring moving far from the wall was observed. Paolo Orlandi

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9

{a}

~

~ i=l

~ til

10

--

~

g

~

laO

0.0

.34.0

3t:te1

4

r

=

I

a '"I a a

. . l' " "i

0

"l

~

*

0.0

lio

M.O

t FIG. 4. Velocities of vortex centroids in time at Re = 40. (a) Horizontal velocity, (b) vertical velocity. Present: e; Egs. (8): 0.

The intrinsic differences between a vortex ring and a vortex pair led us to compare our results, not with those obtained by Walker et ai., 15 but rather with the experiment of Van Reijst and Flor4 done in a stratified fluid by the turbulent horizontal injection of a small fluid volume in a strati.fied fluid. Initially, the turbulent jet resembles a horizontal jet in a nonstratified ambient fluid, with a three-dimensional core. In a next stage, the turbulent cloud collapses under gravity, and after the propagation ofinternal waves that collapse, the dipole structure is formed. This mechanism is described in a greater detail in Ref. 4, pp. 596-598. Although this experiment is very close to a pure two-dimensional flow, nevertheless, in the long term viscous dissipation introduces three-dimensional effects. The initial dipole created as described above has a vorticity distribution similar to that employed in the present calculation. Van Heijst and Flor 4 then let this structure interact with solid walls and were able to clearly visualize the evolution of dye concentration near the wall. They evaluated the Reynolds number, Red = Ued lv, based on the dipole characteristic, with a translation velocity of the dipole, Uc = 0.30 m/sec, and a dipole diameter d = 0.3 m at Red;::::; 900. The present numerical simulation at 1432

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Re = 800 predicts dimensions and positions of the primary and secondary vortices in good agreement with the real experiment, as is shown in Fig. 5. To show how well the numerical and experimental results agree the time scales of the laboratory and those of the numerical simulation have been compared. The dimensionless time differences between the four experimental pictures of Fig. 5 are !.i.tel = 2.30, Ate2 = 0.9, and Ate3 = 3.2. The nondimensional time differences for the numerical experiment are !.i.t n, = 2.25, Atn2 = 0.8, and !.i.t n3 = 2. L The first two values of At are quite close in the two cases. The difference between the values of the third At are mainly due to the fact that in the laboratory the dipoles become weaker, owing to the viscous dissipation in the direction normal to the plane of the motion. The vorticity diffusion in the normal direction changes the shape of the vortices, which from observation of the dye distribution resemble pancakes that are thin on the edges and thicker at the center. This effect, which with the inclusion of dye diffusion, are the reasons for the real experiment no longer being two dimensional. The numerical experiment does not suffer from any such limitations and allows observations over a long period of very complex vortex interactions that are strongly Re dependent. These are described in the remainder of the paper. The analysis of the numerical simulation after the first rebounding has shown situations different from those extrapolated by Van Heijst and Flor,4 who assumed that after the formation ofthe four structures in Fig. 5 the two couples interact, forming one smaller structure moving far from the wall and a larger structure that impinges with the wall for a second time. Whether or not what Van Reijst and Flor4 extrapolated occurs depends on the ratio between the circulation of the primary and secondary vortices. The last picture of Fig. 5 shows that the dimensions of the vortices in the numerical simulation are close to the dimensions in the real experiment. The circulations of the primary and secondary vortices were calculated at this time and the ratio lr = Ip/ls was found to be 2.1. In the numerical experiment this circulation ratio predicts a trajectory with a radius of curvature causing the secondary vortices to couple near the solid boundary and remain there. To verify whether the coupling on the line of symmetry actually occurs or not and whether the subsequent movement of the two couples in opposite directions occurs or not, a numerical experiment was devised. In this experiment, dipoles with different r r were located at a distance from a free-slip wall, equal to the distance from the line of symmetry shown in Fig. 5. Only when r r ;::::; 1 was the trajectory such that the small vortices couple on the line of symmetry and moved, as had been suppossed by Van Heijst and Flor,4 but in all these cases the dipole structure was very different from that shown in the last picture of Fig. 5. In the following paragraph the description of the numerical experiment emphasizes that the behavior extrapolated by Van Heijst and Flor4 never occurs, with the initial vorticity distribution given in Eq. (4). Figure 6 shows contour plots of the vorticity field obtained at Re = 800 on a 256 X 256 grid; the simulation lasting up to a nondimensional time, t * = tU */ a, of 40. As expected, the flow field maintains symmetry; thus in Paolo Orlandi

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FIG. 5. Flow visualization of experiment in Ref. 4 c{)mpared with the contour plot of vorticity of numerical simulation.

describing the flow behavior we will discuss the left side only. The nondimensional time is indicated by t. At t = 6 the primary vortex, during its approach toward the boundary, induces a very strong positive vorticity sheet at the wall that becomes unstable and detaches from the waH because of the effect of the primary vortex. The secondary vortex induces a weak and thin tertiary vortex at the wall that does not significantly influence the motion of the primary and secondary vortices. The secondary vortex, moving upwards, stops the horizontal motion of the primary vortex encountered in the presence of free-slip boundaries. This mechanism was experimentally observed at a much higher Re by Harvey and Perry. I The secondary vortex has a strong vorticity core connected to the vorticity layer at the wall. Together with the core of the secondary vortex the primary vortex forms a couple that moves away from the wall. During this movement the weak layer elongates until it breaks. The weaker part remains closer to the wall and continues to induce a thin layer of negative vorticity at the wall. In the new dipole, from t = 12 until t = 14 the circulation of the primary vortex is stronger than the circulation of the secondary vortex (r r:::::: 2.0), thus the dipole moves along a circular trajectory, concave in the direction of the stronger vortex. At t = 12 the two new couples of vortices begin to interact and at t = 14 the mutual interaction results in moving the couples again toward the wall. The secondary vortices are sheared by the primary vortices and are engulfed by the vorticity of the same sign that was attached to the wall. During this process of pairing, very small-scale tertiary vortices are induced near the line of symmetry. Between t = 14 and t = 18 viscosity dissipates the two tails of the primary vortex and leaves the primary vortex still very active. Once more, the primary vortex detaches the vorticity layer far from the 1433

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wall. At this stage, the new couple has the secondary vortex with a smaller circulation than the circulation it had in the previous, similar, event (r r;:::; 2.9). Thus the couple rotates faster and impinges on the wall again. During this second impingement of the couple against the wall, the secondary vortex does not pair with the stronger positive core of vorticity that was attached to the walL Instead, it penetrates between the primary vortex and the core of the secondary vortex, and it amalgamates with the weak flat vorticity layer below the primary vortex. The primary vortex still has sufficient strength to detach the positive vorticity from the wall and form a new dipole. This process repeats until viscosity dissipates the weak secondary vorticity layer, leaving only a circular vortex rotating around its center, with a circulation reduced to half its initial value. The couple formed near the line of symmetry is not sufficiently strong to detach itselffrom the wall. This couple is dissipated faster than the primary vortex because the scales of the couple are smaner than the scales of the primary vortex. To understand the effect of the viscosity on the main features of the interaction of dipoles with a flat boundary a second simulation was performed at Re = 1600. Until t = 16 the behavior is similar to the one described in Fig. 6, the only difference being the secondary vortex has a greater vorticity and vorticity tails more elongated (Fig. 7). At t = 22 during the second rebounding both vortices have a strength greater than in the case at Re = 800 and the couple rotates faster. At Re = 800, r r was equal to 2.85, whereas at Re = 1600, r r = 2.67; the reason being that at higher Re the smaHer secondary vortex is dissipated more slowly. At t = 26 the secondary vortex does not penetrate between the primary vortex and the previously generated secondary vorPaolo Orlandi

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tex located near the line of symmetry; instead, as shown at t = 28, the two secondary vortices pair and form a large structure with strong vorticity near the line of symmetry. The circulation of the vortex is of opposite sign to the initial vortex and has half its value. At this stage the primary vortex detaches for the third time the vorticity layer at the wall and together they form a couple that describes a circular trajectory. At t = 34 the secondary vortex penetrates between the primary and secondary vortices and contributes to move away from the waH the couple that was at the line of symmetry. This couple has a large dimension, so that the effect of the viscosity does not reduce the strength of the couple. As a final result the couple detaches from the wall and moves in the opposite direction to that of the initial vortex dipole. The 1434

Phys, Fluids A, Vol. 2, No, 8, August 1990

FIG, 6, Contour plot, of vorticity at Re= 800,

ratio between the circulation of one of the initial vortices and one of the ejected vortices is lr = - 2.80. The formation of this new dipole, moving in the opposite direction of the primary dipole, is a very interesting feature that has not yet been seen in real experiments with vortex pairs, but it has been observed at almost the same Re number in the experiment of a vortex ring impinging a solid wall. 15 The evolution of kinetic energy distribution is helpful in the understanding of the ejection ofthe dipole from the wall. During the first impingement, when the secondary vortex is created, maximum energy is found near the waH and concentrated at the location where zero vorticity occurs, that is, along the line connecting the vortex centers. The location of the maximum energy rotates with the couple and during the Paolo Orlandi

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FIG. 7. Contour plots of vorticity at Re = 1600.

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Phys. Fluids A. Vol. 2, No.6, August 1990

Paolo Orlandi

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second impingement a high level of energy is maintained in the region close to the line of symmetry, particularly between the secondary and the primary vortices. At this stage, from t = 18 until t = 22 the level of energy in the re gion of the line ofsymmetry is lower than the level of energy between the primary and secondary vortices. During the second impingement the level increases, but it only reaches a high value during the third impingement when the energy is high enough to anow the couple at the centerline to migrate from the wall. The formation at the centerline of the secondary dipole, strong enough to move in the opposite direction to the primary dipole, occurs only in a certain range of Reynolds numbers. A numerical experiment at Re = 3200 shows that the behavior ofthe multiple waH impingement persists with the structure of primary and secondary vortices very similar to that observed at Re = 1600. However, the smaller vortices near the line of symmetry reach a very high vorticity, thus creating a very elongated and strong tertiary vortex, which encircles them and prevents any vertical movement of these small vortices. The secondary vortices on the centerline rotate and form two small couples that rebound at a smaner scale than that of the primary vortex. This phenomenon has been observed on a 256 X 256 grid used for the simulation, but this grid was not completely adequate to capture the small structures influencing the behavior. As a result of truncation errors an experiment with a 128 X 128 grid produced a background small-scale vorticity field comparable to the vorticity of coherent structures. This numerical vorticity contaminated the calculation and the inverse energy cascade caused the formation of a couple similar to those observed at Re = 1600 moving away from the wall. The fact that the solution is not grid independent at Re = 3200 suggests that still higher resolution may be necessary. This will be described in a subsequent paper that will be based on simulations on finer grids than used here. Q

v. CONCLUSIONS The present accurate numerical simulation of dipoles rebounding from solid walls revealed main features that are in agreement with experimental investigations. 4 • 15 Numerical experiments can be pursued for a longer time than the real experiments because the former are not affected by the diffusion of the dye used for visualization. Furthermore, there is no contamination of the two-dimensional flow field by three-dimensional effects. Three-dimensional effects are significant in experiments with two-dimensional vortex pairs because these vortices are very unstable to small perturbations. Also, in the case of an axisymmetric ring azimuthal instabilities are observed after the collision if the Reynolds number is sufficiently high. The longer time evolution of the present simulation revealed very interesting features, such as mUltiple rebounding and pairing of vortices forming new dipoles that detach from the wall. The observation of an eruption of fluid from the wall in this two-dimensional simulation may give some insight into the bursting phenomenon in the near waH region of turbulent boundary

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layers. A similar event has been observed in the experiment of Walker et al.,15 in which a vortex ring was used; the authors claim that the fluid eruption is caused by azimuthal instabilities. The present method can be very easily extended to the simulation of axisymmetric flows. Then the numerical simulation is a very useful tool to clarify whether the fluid eruption i" caused by vortex ring interactions similar to those described in this paper or by three-dimensional instabilities. The numerical experiment produces accurate visualizations of vortices of very different scales and how long these survive. For this purpose an animation of vorticity fields emphasizes aspects difficult to see in the sequences of Figs. 6 and 7. A videotape of the numerical experiments described in this paper has been presented in the picture gallery of the 1989 meeting of the APS held in Palo Alto, and a copy can be obtained from the author. Here we have dealt only with flat boundaries. In a parallel study we have described dipoles impinging en cylinders of different radii. Rebounding was observed for all cylinder radii. The features of the flow field are different from those described in this paper and these will be published elsewhere. ACKNOWLEDGMENTS

The author wishes to express his gratitude to Dr. G. F. Carnevale for many very fruitful discussions and for comments on a draft of this manuscript. We also gratefully acknowledge Dr. Van Heijst and Dr. Flor for permitting the use of the pictures from their experimental work and for their useful comments on the paper. The computational time of this research was supported by Istituto di Fisica dell' Atmosfera of Consiglio Nazionale delle Ricerche.

'J. K. Harvey and F. J. Perry, AIAAJ. 9,1659 (1971). 2S. J. Barker and S. C. Crow, J. Fluid Mech. 82, 659 (1977). 3P. G. Saffman, J. Fluid Mech. 92, 497 (1979). 40. J. F. Van Heijst and J. B. Flor, in Mesoscale/Synoptic Coherent Structures in Geophysical Turbulence, edited by J. C. J. Nihoul and B. M. Jouart (Elsevier, Amsterdam, 1989), pp. 591·-608. SA. J. Peace and N. Riley, J. Fluid Mech.129, 409 (1983). 6A. Arakawa, J. Comput. Phys. 1,119 (1966). 'P. Orlandi, in Proceedings of the 8th GAMM Conference on Numerical Methods in Fluid Mechanics, edited by P. Wesserling (Vieweg, Braunschweig, 1989), pp. 436--441. "W. R. Briley and H. McDonald, Lecture Notes in Physics (Springer, New York, 1975), Vol. 35, p. 105. "L. C. Woods, Aero. Q. 5, 176 (1954). lOA. A. Wray, submitted to J. Comput. Phys. IIC. Canuto, M. Y. Hussaini, A. Quarteroni and T. A. Zang, Spectral Methods in Fluid Dynamics, Springer Series in Computational Physics (Springer, New York, 1987). 12H. Lamb, Hydrodynamics (Cambridge U. P., London, 1932). "G. K. Batchelor, An Introduction to Huid Dynamics (Cambridge U. P., London, 1967). I..S. Staneway, K. Sharili', and F. Hussain, in Proceedings of the 1988 Summer Program, Center ofTurbulencc Research, Report No. CTR-S88, pp. 287-309. ISJ. D. A. Walker, C. R. Smith, A. W. Cerra, and J. L. Doligaski, J. Fluid Mech. 181,99 (1987).

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