The onset of filamentation for compact vortex structures in two-dimensional incompressible flows is elucidated. An estimate is presented for the filamentation time of an unstably perturbed Kirchhoff ellipse, obtained from a linear analysis of the geometry of the instantaneous corotating streamfunction

Flierl, G. R.

Polvani, Lorenzo M.

Zabusky, N. J.

English

Columbia University Academic Commons

Filamentation of unstable vortex structures via separatrix crossing: A quantitativeestimate of onset timeL. M. Polvani, G. R. Flierl, and N. J. Zabusky  Citation: Physics of Fluids A: Fluid Dynamics (1989-1993) 1, 181 (1989); doi: 10.1063/1.857485 View online: http://dx.doi.org/10.1063/1.857485 View Table of Contents: http://scitation.aip.org/content/aip/journal/pofa/1/2?ver=pdfcov Published by the AIP Publishing   This article is copyrighted as indicated in the abstract. Reuse of AIP content is subject to the terms at: http://scitationnew.aip.org/termsconditions. Downloaded to  IP:128.59.154.119 On: Thu, 07 Nov 2013 15:45:11scribed pressure gradient (e.g., atl = 40 Hz and A = 0.1 %; see Fig. 2), the wake component could not be fitted with the universal function W, while the "law-of-the-wall" still ap-plied. The wake component is, therefore, more strongly af-fected by the imposed oscillations than is the wall compo-nent. Ensemble-averaged velocities phase locked to the oscil-lations indicate that a large fraction of the fluctuations are correlated to the motion of the flap and are coherent across the boundary layer. The amplitudes and the relative phases of the fundamental (forced) frequency suggest that the ex-ternal excitation generates an array of span wise vortices ex-tending across the entire boundary layer. Fourier analysis of the phase-locked data reveals that the maximum value of the fundamental frequency and its first harmonic at a given loca-tion decreases rapidly with increasing X. The maximum val-ue of the subharmonic content of the signal remains approxi-mately constant with increasing X, and dominates the flow at large distances from the flap. The emergence of the sub-harmonic as the dominant frequency is correlated with dH / dX being positive. One may conclude by restating that the introduction of harmonic, two-dimensional oscillations results in a reattach-ment of the flow and changes the proportions between the "wake" and the "wall" functions whose linear combination represents the streamwise velocity distribution in a turbulent boundary layer. It does not, however, alter the universal form of these functions. ACKNOWLEDGMENT This work is supported in part by AFOSR Grant No. 86-0323. aj Present address: Department of Aerospace and Mechanical Engineering, University of Ariwna, Tucson, Arizona 85721. 10. Neuburger and I. Wygnanski, in Proceedings o/the AFOSR Workshop on Unsteady Separated Flow, F. J. Seiler Research Labs-TR88-0004, 1987. 20. Oster, I. Wygnanski, B. Dziomba, and H. Fiedler, in Lecture Notes in Physics, edited by H. Fiedler (Springer, Berlin, 1978), Vol. 75, p. 48. 'G. L. Brown and A. Roshko, J. Fluid Mech. 64, 775 (1974). 4M. Gaster, E. Kit, and I. Wygnanski, J. Fluid Mech. 150, 23 (1985). 50. Coles, J. Fluid Mech. 1, 191 (1956). Filamentation of unstable vortex structures via separatrix crossing: A quantitative estimate of onset time L. M. Polvania) and G. R. Flierl Center for Meteorology and Physical Oceanography, Massachusetts Institute of Technology, Cambridge, Massachusetts 02139 N. J. Zabuskyb) Department of Mathematics and Statistics, University of Pittsburgh, Pittsburgh, Pennsylvania 15260 (Received 25 August 1988; accepted 25 October 1988) The onset of filamentation for compact vortex structures in two-dimensional incompressible flows is elucidated. An estimate is presented for the filamentation time of an unstably perturbed Kirchhoff ellipse, obtained from a linear analysis of the geometry of the instantaneous corotating streamfunction. The filamentation of compact vortex structures in two-dimensional (2-D) incompressible flows is being recognized as a dominant mechanism for enstrophy cascade and is asso-ciated with their chaotic evolutions. Melander et al. I showed that the geometrical patterns of critical (stagnation) points and separatrices of a proper, comoving streamfunction yield insight into the relative flow of vorticity. Deem and Za-busky2 were the first to observe very fine filaments emerging from a strongly perturbed piecewise constant stable vortex (Kirchhoff) ellipse and DritscheP found a remarkable re-peated filamentation of a weakly perturbed Rankine (circu-lar) vortex at long times. In this Letter, we present concepts and quantitative re-sults for simple cases that relate the onset offilamentation to the time, t c , when perturbed vortex contours cross the sep-aratrices emanating from hyperbolic points in the comoving streamfunction. In contrast with previous work,3-5 where amplification was due to nonlinear interactions, we examine in detail linearly unstable harmonic perturbations6 and the filamentation of Kirchhoff ellipses. Calculating the geome-try of the comoving streamfunction and the contour, we ob-tain an analytic estimate for tc based on linear theory. At that time the hyperbolic points of the comoving streamfunc-tion penetrate the vortex contour, the onset offilamentation. We also show that the suppression offilamentation observed for strongly perturbed equivalent barotropic evolutions, re-ported by Polvani et al., 7,8 can also be understood in terms of the geometry of the streamfunction in the comoving frame. In two dimensions, with V·u = 0, the streamfunction 7/J can be calculated from the vorticity distribution. With piecewise constant vorticity and V27/J; = q, for rED, V27/Jo = 0, for rEiD, ( 1) where q is a constant and the velocities are functionals of the 181 Phys. Fluids A 1 (2), February 1989 0899-8213/89/020181-04$01.90 @ 1989 American Institute of Physics 181  This article is copyrighted as indicated in the abstract. Reuse of AIP content is subject to the terms at: http://scitationnew.aip.org/termsconditions. Downloaded to  IP:128.59.154.119 On: Thu, 07 Nov 2013 15:45:11shape of the boundary aD of the region D. The points on aD are convected with the velocities (u,v) = (x,y) = ( - tPY ' tPx). This fact is the source for recent work with contour dynamics9 and contour surgery. 10 In this Letter, D will be, initially, an unstably perturbed Kirchhoff ellipse. First, consider finding the critical points for the corotat-ing streamfunction of the unperturbed ellipse, whose angu-larvelocityll isfi = qab/(a + b)2. For convenience we take q = 1, ab = 1, and A=b /a to be the ratio ofthe minor-to-major axis (0 < A < 1 ). The elliptic-Cartesian coordinate system conversion is (x,y) = c(coshp cos O,sinhp sin 0) , wherec = (a2 b 2) 1/2 (l - A 2)/ A. The ellipse is defined by the equation p = Po tanh - 1 A and the inner and outer streamfunctions are, respectively, tPi =!(1 +A)-I(AX2 +r), for P<Po, tPo =!p+ !e- 2Pcos 20, for P>Po· The proper outer corotating streamfunction (see Fig. 1) is tPc = tPo -! fi(x2 + r) tPo -! fic2 (cosh 2p + cos 20) , and the hyperbolic critical points (SI and S2 in Fig. 1) are obtained by solving ap tPc = 0 and a{J tPc = 0 , which yield, respectively, Pc = pog[ (1 + 3A)/(1 - A)] and Oc 0,17'. Thus the ratio of the distance of the critical point from the origin along the x axis (xc) to the major axis of the ellipse (a) is given by (xJa) = (1 +A)3/2/(1 +3A)1I2.Foracir-cle (A-d), (xc/a) .... .J2, and for a sheet (A .... O), (xJa) .... 1. One can conjecture that, in general, critical points may al-ways be very close to contours on stationary states which have very high curvature. Near a hyperbolic point the curva-ture of the corotating streamline is very high, and for a sta-tionary state the contour itself must be a streamline. Incidentally, for a circle of radius R, it is easy to show that a harmonic perturbation of mode m has m critical points equispaced along a circle of radius 'c.m = [ml(m - 1) p/2R . y t--F'----x FIG. 1. The geometry of the corotating streamfunction of a Kirchhoff el-lipse. The dotted lines are the separatrices. S, and S2 are the hyperbolic (saddles) critical points. and C, and C2 are the elliptic ones (centers). 182 Phys. Fluids A. Vol. 1. No.2. February 1989 Thus, for increasing m, they approach the vortex boundary. This simple result explains why the limiting rotating Euler V statesl2 decrease their aspect ratio as m increases. Namely, one class of limiting V states arises when the critical points on the corotating streamfunction touch the vortex bound-ary, so that the boundary has a comer. This concept general-izes to the multilayer geostrophic context. 13 The above result also indicates that for localized Gaussian perturbations on a Rankine vortex, higher initial steepness leads to filamenta-tion at shorter times, in accordance with the findings of Dritschet,3 Finally, these considerations lead us to conjec-ture that, due to the inevitable presence of critical points, monopolar 2-D nonaxisymmetric vortex equilibria with a smooth (i.e., infinitely differentiable) vorticity distribution cannot exist. Next we examine the linear stability and corresponding critical points of a perturbed ellipse. Let the total stream-function be IIJ = tP + tp, where tp is the perturbation. Follow-ing Love,6 we expand the outer and inner perturbation streamfunction in harmonics, tpi = (amlm)cosh mp sin mO + (Pm/m)sinh mp sin mO, ({Jo = (amlm)cosh mpo e- mp cos m{) + (Pm1m)sinh mpo e- mp sin m{), with am and Pm functions of time alone and the perturbed boundary given by P=Pv(O,t) =Po -h~(am cosm{}+Pm sinmO), where h 0- 2 = A -I cos2 0 2 + A sin2 {}. Love6 showed that am and Pm are related by a coupled harmonic system, for which the growth rates s are found from the dispersion rela-tion ( 2mA )2] (1 +A)2 . With the initial conditions am = € and Pm = 0, we obtain (.Q t) _ + cosh st cos m{} + !f sinh st sin m{} Pv v, -Po € . ' A cos 20 + (l/A)sm 20 (2) <=> .C7 tJ () 0 ~ ~ ~ ~ <=> 0' tJ () 0 ~ ~ +~ +~ +~ FIG. 2. The evolution of a A. 0.3 Kirchhoff ellipse. initially perturbed with an unstable m = 3 Love mode of amplitUde £ = 0.02. The crosses indi-cate the positions of the hyperbolic critical points of the instantaneous coro-tating streamfunction. Time advances to the right and downward. and the frames shown correspond to t 0.2,4.6 ..... 36. Letters 182  This article is copyrighted as indicated in the abstract. Reuse of AIP content is subject to the terms at: http://scitationnew.aip.org/termsconditions. Downloaded to  IP:128.59.154.119 On: Thu, 07 Nov 2013 15:45:1111=0.3 m=3 0:=0.02 2.4 2.3 dmax, I 2.2 2.1 2.0 dcp 1.9 J 10 15 20 25 30 35 FIG .. 3. The maximum distance from the contour to the origin (dma• ) and the dIstance from the critical point S, to the origin (dcp ) versus time for the run of Fig. 2. where .Y = e- mpo cosh mpo - (mA)/(1 + A)2. The per-turbed critical points, located at (Pc (t), e c (t»), are found by solving ap I{I c (Pc,ec ) = 0 and a{} I{I c (Pc,ee) = 0 , where I{I e = tPe + tPo is the perturbed outer corotating streamfunction. If Pe = P~o) + EP~l) + O(c), ee = e~O) + E0~1) + O(c) , then we find that P~O) = Pc> e~O) = {fe' and h {f -mp P(I) _ cos mPe cos m ce C e - 2 cosh st , ( 3a) (A - e - PC) eO) _ .Y sinh mPe cos m{fee - mpc e - 2 cosh st , (3b) (A - e- PC) where A = (1 - A)/(1 + A). The linear estimate tCL ' for the time when the critical point coincides with the perturbed vortex boundary, is found by substituting (3a) and (3b) into (2): tCL ~+[ log (Pc ~ Po ) I ( 1 coshmpee-2PC)] - og -- . U 2(A cosh 2pc - e - 2pc) (4) The approximation in (4) arises from mixing the orders of the asymptotic expansion and omitting terms proportional toe-st. Figure 2 shows the contour dynamical evolution of a perturbed A = 0.3 ellipse (m = 3,E = 0.02). The angular ve-locity necessary to determine a corotating streamfunction is obtained by fitting the entire contour to an ellipse at each time step13 (this is not a unique procedure): the hyperbolic points are shown by crosses ( + ). The crossing is shown more precisely in the plot of dcp and dmax vs t (Fig. 3), where dcp is the distance of the relevant hyperbolic point (SI) to the origin and dmax is the distance from the origin to the extreme point on the ellipse. The time of crossing, in Fig. 3, is tc ~28.5, whereas from (4) we obtain tCL ~30.4. The mini-183 Phys. Fluids A, Vol. 1 , No.2, February 1989 'c=::>' ,c?' (/ 0 '~, '~, ,c:::;, ,0' ?5 U ~ '~ , c:::::::l, ,c/' ,!l 0 ~ '~, '~, ,c;:J' ,t/ 8 ~ '~, • c:::::::>. FIG. 4. The evolution of an initially unperturbed equivalent barotropic A = 0.3 elliptical vortex at r = 3. The time frames are t = 0,40,80, ... ,960. mum value of the curvature along the contour becomes neg-ative at t~27.8; this corresponds to the appearance of an inflection point on the contour and, shortly afterward, the filament is seen to emerge. It should be noted that the in-crease of dcp vs t for t < 28 is a property of the linear theory [cf. Eq. (3a) 1 and is due to the outward motion of the con-tour. A similar comparison for a A = 0.2 perturbed ellipse (m = 4,E = 0.02) yields a crossoverattc ~ 14.3, an estimate from (4) of tCL ~ 17.5, and the appearance of negative cur-vatures at t~ 14.0. An examination of equivalent barotropic dynamics 7,8 shows a similar phenomenon. If ( 1) is replaced by (V2 - r)tP; = n, for rED and with n constant and (at + u'V) (V2 - r)tP;.o = 0, we ob-tain the' evolution and critical points shown in Fig. 4 for an initially unperturbed A = 0.3 ellipse at r = 3. Although the vortex undergoes nonlinear oscillations, no filamentation or crossings are observed, and the critical points never come into contact with the contour. ACKNOWLEDGMENTS The authors wish to express their gratitude to Anne-Marie Michael who graciously expended much effort to pro-duce a legible manuscript from a virtually unreadable origi-nal, and to Dr. D. G. Dritschel for providing them with his latest results before publication. This work was funded in part by the Office of Naval Research under Contracts No. NOOO14-85-K-0024 to NJZ and No. NOOO14-86-K-0325 to GRF. a) Present address: Department of Mathematics, Room 2-339, Massachu-setts Institute of Technology, Cambridge, Massachusetts 02139. b) Permanent address: Department of Mechanical and Aerospace Engineer-ing, Rutgers University, Piscataway, New Jersey 08855-0909. Letters 183  This article is copyrighted as indicated in the abstract. Reuse of AIP content is subject to the terms at: http://scitationnew.aip.org/termsconditions. Downloaded to  IP:128.59.154.119 On: Thu, 07 Nov 2013 15:45:11'M. V. Melander, J. C. McWilliams, and N. J. Zabusky, J. Fluid Mech. 178,137 (1987). 2G. S. Deem and N. J. Zabusky, Phys. Rev. Lett. 40,859 (1978). 3D. Dritschel, J. Fluid Mech. 194, 511 (1988). 'D. I. Pullin, J. Fluid Mech. 108,401 (1981). 5D. I. Pullin, P. A. Jacobs, R. H. J. Grimshaw, and P. G. Saffman (private communication) . 6 A. E. H. Love, Proc. London Math. Soc. 35, 18 (1893). 7L. M. Polvani, N. J. Zabusky, and G. R. Flierl, Fluid Dyn. Res. 3, 422 (1988). "L. M. Polvani, N. J. Zabusky, and G. R. Flierl, submitted to J. Fluid Mech. "N. J. Zabusky, M. H. Hughes, and K. V. Roberts, J. Comput. Phys. 30, 96 ( 1979). IOD. G. Dritschel, J. Comput. Phys. 77, 240 (1988). "Sir H. Lamb, Hydrodynamics (Dover, New York, 1945), 6th ed., p. 159. 12H. M. Wu, E. A. Overman, and N. J. Zabusky, J. Comput. Phys. 53, 42 (1984). 13L. M. Polvani, Ph.D. thesis, Massachusetts Institute of Technology, 1988. Space-time correlations in turbulence: Kinematical versus dynamical effects Victor Yakhot, Steven A. Orszag, and Zhen-Su She Applied and Computational Mathematics Program, Princeton University. Princeton. New Jersey 08544 (Received 6 October 1988; 17 November 1988) Space and time correlation functions in a randomly stirred turbulent fluid are evaluated to lowest order of the E expansion in the renormalization group theory of turbulence. It is shown that wavenumber and frequency energy spectra differ substantially since random sweeping effects on the small eddies by large-scale eddies do not contribute to the decorrelation in the RNG theory. The strong nonlinearity of the Navier-Stokes equations at high Reynolds numbers makes theoretical analysis offully developed hydrodynamic turbulence difficult and excludes the usual perturbative treatments. From the physical point of view, the difficulty stems from the fact that, at large Reyn-olds numbers, a very large range of scales is excited; the interaction between these scales is still poorly understood physically. One of the important questions is how the most energetically excited scales (at the so-called integral scale) interact with much smaller inertial-range scales, and how the interaction influences Eulerian experimental measure-ments. The random Galilean invariance principle intro-duced by Kraichnan 1 postulates that the small-scale motions undergo purely kinematic advection by large-scale eddies, which causes decorrelation by a sweeping effect directly measurable in the Eulerian frame. Since the large-scale ed-dies have the greatest excitation, this sweeping effect leads to a shorter correlation time in an Eulerian frame than that in a Lagrangian frame, where the correlation time is determined mainly by the local eddy turnover time. However, this rela-tionship between Eulerian and Lagrangian correlation times does not hold uniformly. In fact, it is easy to construct exam-ples of flow systems2-4 in which the Lagrangian correlation time is smaller than the Eulerian one. To illustrate the decay of velocity correlations through sweeping effects, consider the Fourier-space correlation, G(k,t) = (v(k,O)v(k,t» = (v2(k,0)eikVt-.akZt)v,v, (1) where V is the random sweeping velocity introduced by the large scales on the small ones. The term proportional to k Z takes into account the dynamics of interactions of the modes v(k) and v(q) with k=q. The exponent z must be deter-mined by appropriate dynamical theory. The average in (1) is done over both small- and large-scale fields, v(k,O) and V, respectively. If it is assumed that the small and large scales are decorrelated and that the large-scale field V is Gaussian, then G-t/J(k)e- akZt (eikVt ) v-t/J(k)e- k2 V2 t2 - ak't, (2) where t/J(k) = (v2 (k,0»v' As t_ 00, the decay ofthe corre-lation function G is dominated by dephasing caused by ran-dom sweeping effects. This kinematic sweeping effect also appears in the form of infrared divergences in the field-theoretical diagrammatic perturbation description of turbulence. Partial resumma-tions like the direct interaction approximation (OIA) suffer from this divergence. The primary motivation for the La-grangian modifications of OIA was the elimination of this infrared divergence.5,6 In this Letter, we use the renormalization group (RNG) approach developed by US7- 9 for the analysis of space-time correlations in homogeneous turbulence. The RNG analysis has already produced good quantitative pre-dictions for inertial-range dynamics. The RNG analysis is based on the correspondence principle7 that general, fully developed turbulent flow subject to initial and boundary conditions is equivalent in the inertial range to flow governed by the Navier-Stokes equations driven by the Gaussian ran-dom force, defined by the three-dimensional correlation function (..t; (k,w )fj (k',o/» = (21T)42Dok 1- 'P(;(k)8(k + k')8(w + w'), (3) where p(; (k) = 8(; - kik/k 2, and E = 4 - 1"/2 with 0<1" < 10. At the present time, the correspondence principle is a 184 Phys. Fluids A 1 (2), February 1989 0899-8213/89/020184-03$01.90 @) 1989 American Institute of Physics 184  This article is copyrighted as indicated in the abstract. Reuse of AIP content is subject to the terms at: http://scitationnew.aip.org/termsconditions. Downloaded to  IP:128.59.154.119 On: Thu, 07 Nov 2013 15:45:11

Filamentation of Unstable Vortex Structures via Separatrix Crossing: A Quantitative Estimate of Onset Time

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Filamentation of Unstable Vortex Structures via Separatrix Crossing: A Quantitative Estimate of Onset Time

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