Turbulence Lecture 1 Non-lnear Dynamcs Strong non-lnearty s a key feature of turbulence. 1. Unstable, chaotc behavor.. Strongly vortcal (vortex stretchng) 3 s & 4 s Taylor s work on homogeneous turbulence Kolmogorov. 6 s & 7 s Krachnan and followers addressed non-lnear stochastc processes. Deductve theory hasn t been developed. 1. Learned much about non-lnear dynamcs.. Theores are based on ad hoc assumptons. Smplfy the problem and consder homogeneous turbulence. - Contans essental physcs of the non-lnear dynamcs. - Smple as possble. Homogeneous n all spatal drectons. If t s then: a. Energy exchange wth mean flow. b. Dffuson terms are not mportant. Lose some large structures pathologcal (?) flow. (Could be too much of a smplfcaton.) Ideas at least apply to small-scale n hgh Re no. flows. Can generate a homogeneous flow n a lab. 1
Screen; mesh sze, small compared to D. / U Possbly wth constant mean shear to provde energy to perturbatons downstream. Not homogeneous 1. Taylor ntroduced the concepts n the 3 s and developed the experment to study t.. Also can study homogeneous turbulence on the computer. Solve the N.-S. equatons drectly at a hgh enough Re. No. Re 1, λ realzable based on Taylor mcroscale. 3. If the Re No. of the flow s hgh enough, the small scale structures of the flow are homogeneous. A. Homogeneous turbulence can be produced n the presence of background shear. = λx (Corrsn, Champagne) u1 B. Background densty stratfcaton 1 ρ = constant. ρ z C. Magnetc feld. Consder the case where U =, P = Mean momentum equaton says = Choose a coordnate system attached to mean flow gves U =. There s an obvous non-homogenety n x.
L (Integral scale n x) Ut x T s decay tme of turbulence. T L x = u \ So Turbulence level u 1 Then approx. homogeneous n x. U x = Ut Relate lab, spatally decayng flud to computer, tme decayng turb. smulaton. For homogeneous turb. the 1-pt. energy equaton s: u t = µ ss = ε ε - dssppaton rate. It can only decay. The equaton hdes the reason for decay (.e., vortex stretchng) Need -pt statstcs, go to -pt measurements to understand the dynamcs. For hgh Re. No. homogeneous turbulence scalng s the same. ε = E u u tme u 3 u u K approxmately ndependent of R. The mean square vortcty (enstrophy) equaton reduces to ω t ω = ωω s v x / producton dsspaton long tme \ / scale short tme scales 3
Scalng leads to terms on RHS beng n balance. Note: A couple of references for homogeneous turbulence: Batchelor, Monn & Yaglom Vol., Hnze, Leseur. ω can actually grow although t must eventually decay. E( Κ) Κ E / dsspaton rate Κ Startng pont pt spatal correlaton. (, ) = (, ) ( + ) R rt u xt u x rt, or equvalently, the knetc energy spectrum. Homogeneous n x, no mean flow, want an equaton for R ( rt, ) Defne x = x+ r Prme means dealng wth a dfferent pont n space. u s u at x 4
x - equaton: u 1 p = uu + v u t x x at x : k k ρ u 1 p = + t x x uu v u ρ Multply by u and u then add, average. { } 1 p uu = u uu + u u u u u p v u u u + + + u t xk x ρ x x k (Page 61 of Matheu & Scott) Now prmed quanttes can be taken nsde of unprmed dervatve because they do not depend on x. u u = and we can use the notaton: e.g. uk uuu k xk xk ( ) ( uuu k = fk x x = fk r ) where r = x x uuu k r k p Smlarly u u u uu = u and u = u p x r x x u p = up x r u u = uu xk rk u u = uu x r and puttng t together 5
1 R ( rt, ) = { uuu k uuu k} + pu pu + v R rt t rk ρ r r rk I ( r) ( r) P Smlar to / \ eqn. 6.9 two pont pressure velocty 6.95 on pg. 61 cubc moments correlatons n Matheu & Scott (, ) N.B. We have an equaton for the Reynolds stresses (-pt). uu, but now we have ntroduced the closure problem at a hgher level through the trple moment and pressure-velocty correlaton terms that were not present n the R.A.N.S. equatons. Normally a relatonshp between pressure and velocty s found by takng the dvergence of the momentum equatons to form a Posson equaton, e.g. x 1 p = uu ρ x x for nfnte doman, homogeneous flud. 1 1 p( x) = ρ 4π uu x x x x x and the pressure-velocty correlatons. 1 1 x pu = uuku ρ 4πρ x x k x x Pressure velocty correlaton can be wrtten n terms of a trple velocty correlaton. For weak turbulence the trple correlatons are small. R = v R t r Κ 6
R ( rt, ) 11 1 R ( rt, ) 11 Dffuse n tme. 7