ON THE NONLINEAR STABILITY BEHAVIOUR OF DISTORTED PLANE COUETTE FLOW. Zhou Zhe-wei (~]~)

Transcription

1 Applied Mamatics and Mechanics (English Edition, Vol. 12, No. 5, May 1991) Published by SUT, Shanghai, China ON THE NONLNEAR STABLTY BEHAVOUR OF DSTORTED PLANE COUETTE FLOW Zhou Zhe-wei (~]~) (Shanghai University Technology, Shanghai nstitute Applied Mamatics and Mechanics, Shanghai) (Received June 10, 1990; Communicated by Chien Wei-zang) This paper discusses nonlinear stability behaviour distorted plane Couette flow to 2-dimensional disturbances, and compares it with that distorted plane Poiseuille flow. The one-dimensional The results show problem that plane Couette motion flow is more a rigid unstable flying than plate plane under Poiseuille explosive flow attack to has an analytic finite,amplitude solution only disturbances. when polytropic index products equals to three. n behavior Key reflection words shock distorted in plane explosive Couette products, flow, distorted and applying plane Poiseuille small parameter flow, purterbation method, an analytic, nonlinear first-order stability approximate behaviour solution is obtained for problem flying plate driven by various high explosives with polytropic indices or than but nearly equal to three. Final. ntroduction velocities flying plate obtained agree very well with numerical results by computers. Thus an analytic formula with two parameters high explosive (i.e. velocity and polytropic index) n for estimation experimental observations, velocity flying both plate plane is established. Poiseuille flow and plane Couette flow become unstable when Reynolds number is about 1000, but in stability analysis, se two flows belong in different classifications, instability plane Poiseuille flow belongs in subcritical range, and plane Couette flow, as well as pipe Poiseuille flow, belongs in bifurcation from infinity[q,pl. The failure Explosive to obtain driven flying-plate critical Reynolds technique number ffmds for its important latter use two in kinds study flow behavior brings much materials difficulties under to nonlinear intense impulsive stability loading, analysis. shock synsis diamonds, and explosive welding and cladding n reference metals. [1], The method author estimation suggested a hydrodynamic flyor velocity and stability way ory raising distorted it are questions laminar flow, common and presented interest. a kind distortion priles reflecting nonlinear interaction between various disturbances. Under By assumptions using such distortions, one-dimensional th paper plane discussed stability and rigid problem flying plate, parallel normal shear approach solving problem motion flyor is to solve following system equations flows. n linear stability analysis, stability behav.iour plane Poiseuille flow, plane Couette flow governing flow field products behind flyor (Fig. ): and pipe Poiseuille flow are investigated by a uniform method, results indicated that se flows are all linear unstable under circumstances background perturbations. The nonlinear stability behaviours plane Poiseuille flow ap and +u_~_xp pipe + Poiseuille au flow are also discussed, numerical calculations showed some differences au between au se 1 two flows. This paper discusses nonlinear stabifity behaviour plane Couette flow, and finds out that we can make some comparison between as its nonlinear as stability behaviour and that plane Poiseuille flow in spite many.differences. n a--t this paper author discusses in detail such similarities and differences. where. The p, p, Nonlinear S, u are pressure, Evolution density, Equation specific entropy Disturbance and particle velocity Amplitude products respectively, By using with artificial trajectory neutrality R reflected method shock suggested by Pr. wave Zhou D as a Heng, boundary nonlinear and trajectory evolution F equation flyor as anor disturbance boundary. amplitude Both can are unknown; be obtained as position follows: R and state parameters on it are governed by flow field central rarefaction wave behind wave D and by initial stage motion flyor also; 439 position F and state parameters products

2 440 Zhou Zhe-wei or da = (Alle + A,le 2 + A31e s + A,le s + A,le*)a dt + ( A,eZ + Asse3 + A,3e')aS + A,Be'a5 (2.1) d (ca) = (A,e + A~xe ~ + A~le 3 + A,le') (ca) dt + (A,3 + As3e + A,se ~) (ca) 8 + A,6(ea)5 ----Cl (ca) + C~(ea) s + Cs(ea) ~ = f (ca) (2.2) where C t, C 2, C 3 are constants determined by numerical calculations, C L is just linear growth rate dc~ disturbance amplitude. When d(ea)/dt=o, we obtain critical value (ca)c, 9 When 5th order term (ca) is involved, /(ca) =0 becomes a quadratic algebraic equation (ca) z, so in general situation, it has two real roots (or two equivalent real roots), we do not consider complex roots and negative real roots. The one-dimensional problem motion a rigid flying plate under explosive attack has an analytic f(ea) solution may have only two when situations polytropic as in figure index 1, arrow indicates products equals direction to three. n development disturbance amplitude (ca) with time, behavior reflection shock in explosive products, and applying small parameter pur- i(~a) /(ca) terbation method, an analytic, first-order approximate solution is obtained for problem flying plate driven by various high explosives with polytropic indices or than but nearly equal to three. Final velocities flying plate obtained agree very well with numerical results by computers. Thus an analytic formula with two parameters high explosive (i.e. velocity and polytropic index) for estimation velocity flying plate is established. a/(~a) d(ea) Fig. 1 Situation /(ca) Explosive driven flying-plate technique ffmds its important use in study behavior When 0, ~>0, critical amplitude obtained is threshold amplitude, above which materials under intense impulsive loading, shock synsis diamonds, and explosive welding and cladding metals. The method a/(ea) estimation [ flyor velocity and way raising it are questions disturbance grows; when d(ea) or <0, critical amplitude obtained is equilibrium common interest. Under assumptions one-dimensional plane and rigid flying plate, normal amplitude, disturbance amplitude will approach this value. We denote threshold amplitude by approach solving problem motion flyor is to solve following system equations governing flow field products behind flyor (Fig. ): Because ap +u_~_xp + au d( ea ) =C, + 3Cz( ea )~ + scs( ea )' (2.3) When two unequal positive real roots are obtained, if C3>0, larger root is threshold amplitude, smaller root is equilibrium a--t amplitude; ifc 3 <0, situation is opposite. When only one positive real root is obtained, we have to calculate (2.3).. Nonlinear Development Linear Neutral Disturbances where p, p, S, u are pressure, density, specific entropy and particle velocity products respectively, n reference with [], trajectory we changed R reflected amplitude shock distortion priles wave D to as make a boundary flow and into trajectory subcritical F or supercriticai flyor as anor range, boundary. and see Both variation are unknown; coefficients position in R nonlinear and state evolution parameters equation on (2.3). it are Here governed we use by this method flow field to calculate central plane rarefaction Couette wave flow, behind and compare results wave D with and those by initial plane stage Poiseuille motion flow flyor under also; same position conditions. F and state parameters products cr

3 Nonlinear Stability Distorted Plane Couette Flow 441 At first stage, we select an amplitude some distortion prile to make linear growth rate disturbance in plane Couette flow and plane Poiseuille flow vanish, namely C z = 0. The numerical results se two flows under various parameters are showed in Table!. Table Nonlinear stability behaviour linear neutral perturbations R now 6 C2 Cs lea) o, (eo)%r Couette X X10 -a Poiseuihe ~ 4.804X 101 Couette x X 10 -a Poiseuille ~ Coutttc x X 10 -a Poiseuille X 10 -'3 Couette o X 109 ~ X 10 -s The one-dimensional problem motion a rigid flying plate under explosive attack has Poiseuil/e o x 10 -z an analytic solution only when polytropic index products equals to three. n We can see that plane Couette flow requires Stronger modification to make disturbance behavior reflection shock in explosive products, and applying small parameter purterbation linear neutral, method, indicating an analytic, that first-order plane Couette approximate flow is solution more stable is obtained than for plane problem PoiseuiUe flying flow to plate infinitesimal driven by disturbances. various high explosives After such with modification, polytropic although indices or than disturbances but nearly are equal linear to three. neutral, Final nonlinear velocities development flying plate are obtained different. agree n plane very Couette well with flow, numerical we can obtain results a by threshold computers. amplitude Thus an analytic order formula magnitude with two parameters l0-3 under several high explosive Reynolds (i.e. numbers. But velocity in plane and Poiseuille polytropic flow, index) when for R=1500, estimation modification velocity flying plate mean is established. velocity is strong, we can obtain a threshold amplitude order magnitude 10-2, when Reynolds number is higher, modification mean velocity is weaker, we can only obtain an equilibrium amplitude or even a decaying amplitude, which shows that nonlinear development disturbances depends more Explosive driven flying-plate technique ffmds its important use in study behavior upon intensity modification on mean velocity than upon influence viscosity. materials under intense impulsive loading, shock synsis diamonds, and explosive welding and cladding V. Nonlinear metals. The Development method estimation Linear flyor Non-Neutral velocity and Disturbances way raising it are questions common interest. The Under results assumptions in last section one-dimensional show a tendency plane that and nonlinear rigid flying development plate, normal approach disturbances solving se two problem flows relate motion much to flyor amplitude is to solve distortion following priles. system Here we equations compare governing results flow se field two flows under products same behind Reynolds flyor numbers (Fig. ): and same modification intensity. We first calculate plane Couette flow, change amplitude distortion priles to make it in subcritical or supercritical range, n calculate plane Poiseuitle flow under same ap +u_~_xp + au parameters: From results we can see, in lower Reynolds y number =0, (R = 1500), both linear growth rate and nonlinear development disturbance in plane Poiseuille flow are more unstable than those in plane Couette flow. When Reyfiolds a--t number increases (R~ 1800), things are different. Although linear growth rate disturbance in plane Poiseuille is more unstable than that in plane Couette flow, threshold amplitude disturbance in plane Poiseuille flow is larger than that in plane where Couette p, p, flow, S, u which are pressure, is because density, in plane specific Poiseuille entropy flow, and particle coefficient velocity C 3 5th order products term respectively, nonlinear evolution with equation trajectory R disturbance reflected shock amplitude is less than wave zero, D as a but boundary in plane and Couette trajectory flow, C 3 is F greater flyor than as anor zero. So boundary. plane Couette Both are flow unknown; is more unstable position than plane R and Poiseuille state flow para- to meters finite-amplitude on it are governed disturbances. by flow field central rarefaction wave behind wave D and by initial stage motion flyor also; position F and state parameters products

4 442 Zhou Zhe-wei V. Discussions Through numerical calculation on nonlinear stability behaviour plane Couette flow, we obtain result that plane Couette flow is more unstable to finite-amplitude disturbances than plane Poiseuille flow. The result shows that growth rate infinitesimal disturbance would not decide ultimate stability behaviour flow, behaviour depends furr on wher nonlinear action will destabilize (C 2, C 3 > 0) or stabilize (C v C 3 < 0) fow. n se two different flows, we find some results that linear growth rate is large but threshold amplitude finite- amplitude disturbance is also-large. t is worth investigating wher in same flow such results can be found. So far mode whose decaying rate is minimum in linear stability analysis has been selected as fundamental mode in nonlinear stability analysis. Perhaps range mode selecting could be enlarged when searching most unstable situations. n reference [1], we suggested that instability flow results from occurrence reflections on distortion priles, namely, instability is non-viscous. From numerical results this paper, we can also see that when Reynolds number is rar small (R = 1500), linear growth rate is low, and threshold amplitude is large, linear and nonlinear action have same stabilizing The one-dimensional tendency; when problem Reynolds number motion is rar a rigid large, flying namely, plate under influence explosive attack viscosity has is an analytic solution only when polytropic index products equals to three. n small, action distortions dominates, The curvature mean velocity in plane Poiseuille flow is ~" = 2 and in plane Couette flow is ~" = 0, thus we can assume it is strong variation behavior reflection shock in explosive products, and applying small parameter purterbation curvature method, and an analytic, occurrence first-order reflections approximate on solution distortion is obtained priles that for make problem non-viscous flying plate instability driven behaviour by various in high plane explosives Couette with flow polytropic more remarkable indices and or than threshold but nearly amplitude equal to three. finite- Final amplitude velocities disturbance flying smaller. plate obtained agree very well with numerical results by computers. Thus an Table analytic 2 Nonlinear formula stability with two behaviour parameters linear high non-neutral explosive perturbations (i.e. velocity and polytropic index) for estimation velocity flying plate is established. cz~l, 1 R 6 flow C, C2 C, (ca) c~ (ca)%,. 1 Couette -- t, 1470X X10 to ~ 1.350)<10-3 Explosive O. driven 0098 flying-plate technique ffmds its important use in study behavior Poiseuilc , materials under intense impulsive loading, shock synsis diamonds, and explosive welding and 1500 cladding metals. The method Couelt~. estimation X 10 --~ flyor velocity and X way 10 lo 1.697X raising 0 it -4 are questions. 318X common interest. Poiseuille Under assumptions ~ _ one-dimensional plane and rigid flying plate, normal approach solving problem Couette --0,1495X motion 10-: flyor is to solve X following 109 system equations X governing flow field products behind flyor (Fig. ): 1800 T O Poiseui] r x x io -~ 2.671Xi0 1 Couettc O. 1549X lo-a -27t X 10 ~ 2.406X X i0-3 ap +u_~_xp + Poiseui]]e "X X X 0 -z Couette X 0 --~ X X Poiseuillc X X X a--t Couette o. 1262x x O g 2.336X lo x Poiseuitle x 10 -z X X 0 -z Couettc ] X 0 -~ x x 0-~ respectively, with trajectory Poiscuillr R X reflected 0 -z shock wave D 1.730X as a boundary 10 -z 1.685X and 0-2 trajectory 2500 F flyor as anor boundary. Both are unknown; position R and state parameters on it are governed Couette by flow u.3300x10-3 field central rarefaction x wave r v behind 3.959x x10 wave i au where p, p, S, u are pressure, density, specific entropy and particle velocity products D and by initial stage motion flyor also; position F and state parameters products Poiseuile x 0 -j 9, X 10 -z 1.795X lo -=

5 Nonlinear Stability Distorted Plane Couette Flow 443 This paper is under financial support Science Foundation Shanghai University Technology. References [ 1 ] Zhou Zhe-wei, On stability distorted laminar flow, Ph.D. Disertation Shanghai University Technology, July (1987), partly published in Applied Mamatics and Mechanics, 10, 2, 3, 4 (1989). [ 2 ] Kao, T.W. and C. Park, Experimental investigations stability channel flow: Part 1. Flow a single liquid in a rectangular channel, J. Fluid Mech., 43 (1970), 145. [3] Reichardt, H., Gesetzm~ssigkeiten der geradlinigen turbulenten Couette Str~Smung, Mitteilungen aus den Max-Planck lnstitute fiir Str~mungsforschung und der Aerodynamischen Versuchsanstalt, 22 (1959). [4 ] Rosenblat, S. and S.H. Davis, Bifurcation from infinity, SAM J. Appl. Math., 37, 1 (1979), 1. [ 5 ] Zhou, H., On nonlinear ory plane Poiseuille flow in subcritical range, Proc. R. Soc. The London, one-dimensional A381 (1982), problem 407. motion a rigid flying plate under explosive attack has an analytic solution only when polytropic index products equals to three. n behavior reflection shock in explosive products, and applying small parameter purterbation method, an analytic, first-order approximate solution is obtained for problem flying plate driven by various high explosives with polytropic indices or than but nearly equal to three. Final velocities flying plate obtained agree very well with numerical results by computers. Thus an analytic formula with two parameters high explosive (i.e. velocity and polytropic index) for estimation velocity flying plate is established. Explosive driven flying-plate technique ffmds its important use in study behavior materials under intense impulsive loading, shock synsis diamonds, and explosive welding and cladding metals. The method estimation flyor velocity and way raising it are questions common interest. Under assumptions one-dimensional plane and rigid flying plate, normal approach solving problem motion flyor is to solve following system equations governing flow field products behind flyor (Fig. ): ap +u_~_xp + au as a--t as where p, p, S, u are pressure, density, specific entropy and particle velocity products respectively, with trajectory R reflected shock wave D as a boundary and trajectory F flyor as anor boundary. Both are unknown; position R and state parameters on it are governed by flow field central rarefaction wave behind wave D and by initial stage motion flyor also; position F and state parameters products