Engevik, Leif Eirik

University of Bergen

DepartmentOfAPPLIED MATHEMATICSOn the stability of plane inviscidCouette flowbyLeif Engevlk.November 1966.Report No. 12.UNIVERSITY OF BERGENBergen, NorwayOn the stability of plane inviscidCouette flowbyLeif Engevik.Report No. 12. November 1966.In the present paper we investigate the stability ofplane Couette flow of a homogeneous,, incompressible andInviscid fluid. We show how we can find an asymptotic seriesfor the stream function at large values of t (time), anddemonstrate the connection between Case's results [1 ] andours.This work has been supported by the Royal Norwegian Councilfor Scientific and Industrial Research.Abstract..X Introductlon.In a paper on inviscid plane Couette flow, K.M. Case [1]investigated the asymptotic behavior of the x-component ofthe velocity. His treatment of the question of stability isnot satisfactory. He asslgned certain initial vorticitiesand studied the asymptotic behavior at a flxed point in thefluid. Therefore he cannot be sure whether there is convectiveinstability or not.In this work an symptotic series for the stream functionis found_, showing how it depends on the initial vorticity.We flnd that the stream function vanishes as t within afluid partlcle mo ving wlth velocity z.At the end of sectlon VI the connection between Case r sresults and ours is shown.1yXI» Fprmulatlon of the problem.The system to be considered is the following: a horizon-tal flow of a homogeneous jandincompressibleVlnviscid fluidconfined between two rigidplanes, situated at z = ± 1 ,(see fig. 1). The equationsFig. 1governing this system., are the hydrodynamlc equations formotion in the gravity field, viz.:(2.1)where k is the unit-vector in the z-direction.With appropriate choice of velocity units, the basic motionin plane Couette flow is given by:The perturbation velocity can be written as:(2.3)where j_ is the unit vector in y-direction, and l(x, z,t)is the stream functlon.Llnearizing the first of eqs. (2.1), elimlnatlng thepressure, and using eqs. (2.2) and (2.3), we obtain:The boundary condltlons to be imposed on eq. (2.4), are2,ay ,p(st + -  = - vp - pgk ,V•v = 0 ,(2.2) V =z } V = 0ox oyV 1 = V x y(x,z,t)j_ ,(2 - 4) (å + z å)v2 '? = o3d¥/ . \=0 at z = ± 1(2.5)which shows that the vorticity is conserved within the fluidparticle moving with velocity Vqx = z .We assume that \{/ depends on x as:With this assumption eqs. (2.4), (2.5) and (2.6) become:(2.9)(2. 10)(2.11)(2.12)which is the equation solved by Case [1]. From his solutionhe tried to find the behavior of \J/(z,t) at large valuesof t. But his treatment is not satisfactory, because heLet the vorticity field be asslgned at t = 0( 2 -6) (xjZj O) = F(x,z)Integrating eq. {2.K), we obtain:(2.7) V27(x, z,t) = P(x - zt,z) ,(2-8) T(x,z,t) = \{/(z,t)e lkx ,where Is the k^n Fourier component.2( §1 + ikz) (•—• - k2 )t =0 ,dz\J/(z, t) = 0 at z = ± 1- W VZ > *where F(x,z) = Fk (z)e ikxFrom eq. (2.7), we obtain;- k2 H -P, (z)e‘ lkzt ,hz K4assigned certain initial vorticities and studied the timedependence at a fixed point (x,z) in the fluid. Thereforehe cannot be sure whether there is convective instability ornot. In the Appendix it is shown how we can find the asymptotic behavlor of \(/(z,t), using the theory in [2]. Herewe will demonstrate another method, which can be used in orderto find the asymptotic behavior of vj/(z_,t).In eq, (2.9) we use the Laplace transformation toobtain:(3.1)whereEq. (3.1) is easily solved. We find:(3.2)where W =W(sinhk(z + 1) , sinhk(z - 1)) = ksinh2k is theWronskian for sinhk(z + 1) and sinhk(z - 1).We observe thatIII. Solution to the problem.00= f ptdt ,0r= i £1 kj, ( z t) - slnhkfz -i) r Fk^ u^W Jik(u -TT Slnhk ( u + 1 )du- 1-]slnhkf z+ 1) rW J ”ik(u - n s^n-^( u " 0duz5(3.3) i(i,0 = = 0 ,consistent with our boundary conditions. When \j/(z,£)is found, (zj t) is easily obtained by inversion:In eq. (3.4) we can change the order of integration, andthen obtain the solution of eq. (2.12), given by eq. (A1).In this type of problems this is in general not possible.Generally \|/(z,t) will be written in the form of eq. (3.4).Here we will show how we can find the asymptotic behavlorof this function directly from the representation in eq. (3.4)In order to evaluate the integral in eq. (3.4) we willthe contour 0 , shown inperform an integration aroundfig. 2 (k > 0). TheUsing Caushy's residueFig. 2theorem, since theintegrand has no poles within the contour C, we obtain:Z F (u)(3.4) \|/(z,t) =— J | [sinhk(z-l) J ) sinhk ( u+1 )dulC “ 00 - 1or pk ( u )+ slnhk;{z + 1 ) J — (u^T)sinhk ( u" 1 )du]d?zintegrand has singularities at £ = -1.^Therefore we have tomake cuts in the com-plex Aplane, as shown.(3.5) = -f-J% Tj Tg A-BDE s<1 Sg(z) z_-S Qssumed éo be ~fo>r z_ £: 1 i^JU-1)1 “ip-j 1-iSanalogous expressions for theTherefore:(3.6)We have analogous expressions for/ and /72 7^1 1 ~iooLet us examine the integral1 - i»oo z ~ 1 +iT)+ \ _ _ik_ r-ikt-kT)t r .u, , , pF (v+1-iri)' 2rrWJ [sinhk(z-l) J — ±kv sinhk(v+2-lq)dv0 “2+irjThe first term in eq. (4.1) will be canceled by the6where /= J + / . We have7-j 1~iS 1 -ipintegrals / and / * S 3 f and / are the Integralsy2 y3 S 1 s 2 s3around the small clrcles and surroundingC - It is easy to show that / , / and / -* 0 whenS 1 S 2the radii (i = 1,2,3) of the circles tend to zero.Also / -> Oj when H, S -> coABDE'Kz ’ t) = -•7 1 721 1 “ioowhere now / = / + f7-j 1 - !00 1Æ. The asymptotlc behavior of the integrala along -y and7when t qq .We have f = f + f7 1 1"1°° 1J • Into this integral we introduce r], given by£ ~ 1 “ iTl j as a new variable. We then obtain:1T1. ,, , .r Pk (v+1-1T1)+ sinhk(z + 1) J sInhk(v - ln)dv]drj.Z-1+iTj1corresponding term in the integral f In the vicinity of(*.2)00The term ) ar "nn is of no interest for the same reason asabove. is easily found, by differentiating eq, (4.2) ,and letting p tend to zero. We find:(4.3)We are now in a positlon to find the first term in theasymptotic serles for / . Using Watson’3 lemma [3], we findrthat the first term is given by:(4.4)where we have used that InpOf T].a many-valued functionTaklng into account eq. (4.4), we can write71 -i°of} = 0 the last term can be written as:i n „ / . \ co oo Nr Fk (v+1-1T|) V1 n V n )J sinhk(v - lt))dv = > anT| + liji)z-1+1 ti n=0 n=0n=0b =0o< t,= - pk d)0j~i-j e“ lktb 1 sinhk(z +1) [ J e^SltVndn0000 / \+ / + 0 •0 K r*) f sWe assume that Fk (z} is an analytic function of ze[-1 j]jn P fl) , x(k r) / kl__ sinhkfz + 1) “ikt , ,J ~ ,2.2 sinh2k 6 + ... . when t-> -7 1 k tAn analogous expression is found for the integral along 7 ,We find:W* g'Symptotic behavior of the integral along vy whent -* 00 .Let us cxamine the integral72 z - ioo zz~iooJ . Into this Integral we introduce r], given by £ = z-irjzas a new variable. We obtain:(5.2)xiie 1 ii-tot terms in eqs. (5.2) are of no since they8r> F f- 1 ) / \/), c\ I _k sinhkfz -1) ikt(4 ' 6) J ~ k2 t2 sinhSk 6 + ••• ' when t7 3” Z - iooWe have J = / + fco irj(5.1) P( Z) t) = -|^e-ikzt /e-^tdT1 [sinh1c( 2 -l) / x0 -1-z+ir]1-z+i T)o P (v+z-irj)slnhk(v+z+1-iT])}dv+sinhk(z+l) J — sinhk(v+z-1-lr)) dv]IT]In the vicinity of rj = 0 we hovo:rnFk (v+z-lri) _ ” ”J ikv slnhk(v+z+1 -ir))dv = a lnA-' nil ) b^T]11-1-z+iT] n=0 n=01 -Z+iT)pPk (v+z-ir|) A “j ik7 slnhk(v+z-1 -ir|)dv = a2n^+ln^iTl n=0 n=02).will be canceled by the corresponding terms in the integralz/ . Let us find b, and b p , Differentlating eqs.(5-Z-ioo Inand letting q o, we find:(5.3)(5A)Using Watsonte lemma, and taking into account eqs. (5.3) and(5.4), we obtain:_ ik -ikztrIttU6 l(5.5)VVI. Some commentsTaking into account eqs. (3.6), (4.5), (4.6) and (5-5),we obtain;9r Vz)b 1Q = — sinhk(z +1)P (z)< = - i — sinhk(z + 1)]Fk (z)b 20 = ilT~ sinhM z " 0N P, (Z)bsi = 1 slnhk ( z - 1)]0 00 00/ e" kllt lnnC £ b 1nTin + £ b 2nnn }dr)00 n=0 n=0CO 00 co+J e kTit (lnr| + 2tt1) [ 2 b 1nnn + £ drl]0 n~0 n=0F, (z) ,7 ~ikzt~ p p ® when "fc —* 00k t3From eq. (o.l) we find that the velocity in the x-directioniis damped as — when t 00Differentiating eq. (6.1) twice with respect to z, weobtain:_ tp t„\ ~ikzt2 “ Fk (zje +oz >consistent with eq. (2.12).We observe that if Fk (z) is assigned in such a way thatFk () and all its derivatives are equal to zero at z =1 1then there is no contribution from the integrals along 7and 7^.As to the question of stability of the system, theIntegral J is of the greatest interest, since it is only72this integral which contributes to the leading term in theasymptotic series for andOZwhere*)This is analogous to the situation studied by Case•jf \We Interpret >|/(z,t) and Fk (z) as the Fourier transforms of and F(x, z).10F (z) p /i](6.1) il/(z,t) = - e~ lkzt + MQhk(z+i) -iktk 2 t 2 k 2 t 2 smh2kF sinhk ( z-1) iktsinh2k 0 + *• • •Let us assume - i) — 0 . Then fromeq. (6.1), we obtain by inversion:(6.2) if(x,z,t) - - G ~„ zt ’ z ) +3)s(6.3)Integrating eq. (6.3). we obtain:where and 0 o are functions of z only, andc/ 1 ) =0 3 so that the boundary conditions eq. (2.are satisfied. The Fourier transform of G(x,z) exlsts inthe meaning of Fourier transform of a generalisedsee [2].Let us in conclusion show the connectlon between Caseresults and ours. We consider the two cases:associated with the perturbation, isfinite.-j-00P(xjz)dx — 00 3 i.e. the total initial vorticityII 00associated with the perturbation* isinfinite.Case I.For the integral to exist |F(x,z)| must tend to zeroat least as fast as |x| a (a > 1 ) when |x| s Then(g(xjz)[ — when jxj °o . And this togetherwith eq. (6.2) shows us that the perturbation at any fixedpoint (x, z) in the fluid is damped as — when t-> ooU11å2P(x, z) =—- G(x,z)hx(6.4) G(x,z) =Jdx J P(x,z)dx + C 1 (z)x + C 2 (z) ,-fcoI• J' F(x_,z)dx <00 3 i.e. the total initial vorticity— 00It is assumed that |f(x.,z)| tends to zero asIntroducing thls into eq. {6.2), we find that at any fixedpoint in the fluid the perturhation will vanish as t awhen t -> °o .These are the results of Case.The financial support given by the Royal NorweglanCouncil for Scientific and Industrial Research is acknowledged.The author is grateful to the staffmembers of theDepartment of Applied Mathematics for their interest inthls work.12Case II.II """CXx l (0 < a < 1 ) when jxj -* <» . From eq. (6,4) we thenobtain:Gr(x.,z) ( Is of order Ixl when Ixi -> ooAcknowledgments.Appendix.¥e will show how we can use Theorem 19 in [2] to findthe asymptotic behavior of \|/(z,t) in thls simple case.The solution of eq. (2.12) is given by(Al)1slnhkfz + 1)¥ Fk (u)sinhk(u - l)e ikutduzThe integral in eq. (A1 ) can be written in the followingby rneans of the Heaviside unit function:m(A2)+ 00The integrand in the first integral in (A2) has singularitiesat u = - 1 and u= z. Supposing Fk (z) to be analyticat every point z e [-1,l],we may wrlte:13'Kz.t) = ' S ~ n ~ k t j Z ~ 1 f F k (u)sinhk(u + i)e lkut du-1zJ F k (u)Sinhk(u + l)e” lkut du-1+ 00= J F k (u)sinhk(u + l)H(u + l)H(z - u)e" lkut du ,— 001J F k (u)sinhk(u - l)e~ lkut duz= J F k (u)sinhk(u - 1 )H( 1 ~ u)H(u - z)e~ lkut du-00The integrand in the second integral in (A2) has singularitiesat u= z and u= 1. As above we may write:Let us14F k (u)sinhk(u + 1) = kF k (-l)(u + 1) +in vicinity of u = -1 .(A3) n + 1) = F^(z)slnhk(z + 1)+ k (u)si n hk(ui + l)](u - z) +u=zin vicinity of u = zsinhk(u - 1) = - i)+ -§^[F k ( u )si nh k(u - l)](u ~ z) +u=z(A4) in vicinity of u = zF k (u)sinhk(u - 1) = kF fc (l)(u - i) +in vicinity of u = 1Let us put:f*= kP k (-i)(u + 1) h(u + i) ,F 2 (u) = [P k (z)sinhk(z+1) +^(F k (z)sinhk(z+l))(u-z)]H(z-u),(A5) <,F 3 (u) = [F k (z)slnhk(z-1 ) + k (z)sinhk(z-1 )} (u-z)]H(u-z),= kP k (l)(u - 1)H(1 - u) .Using Theorem 19 in [2] with N = 2, we findwhere— 00In order to ohtain expressions (A7), we have used Table 1on page 43 in [2]. We are now in a position to determinethe asymptotic behavlor of t(z,t) for large values of t.Taking into account eqs. (A1), (A6) and (Aj), we flnd:15z 2J Fk (u)sinhk(u+1 )e' lkutdu =£F. T. (Fm (u) ) +0(—) ,-! m=1 l kt lwhen |kt( -4 00 J(A6)1 4f Fk (u)slnhk(u-1 )e' lkutdu = V P.T. (Fm (u)) + o(——) ,z m l’:t|when IktI —> 00+ooF. T. (u) }=J kPk (-l)(u +1 ) H(u + l)e~ lkutduIkt= “ (" 1 ) ~~2~2 s when |kt | -> ook tP . T . [Fp(u)) = e- ikztj - V-Z)Slnhk(Z +1 }d L iktk (z)sinhk(z+1)+ 2~2 I J when |kt | —> ook t J(A7). . . , - ik7+ r Pk (z)smhk(z - i)P.T.{F,(u 3 = e lKZt- —L iktk (z)sinhk(z-1 )' 2~3 f * when |kt | -> wk t J-ikt= kF (1 ) F' o v/hen |kt | -> ook t- 16-3which is equivalent to eq. (6.l) 0F. (z) .. , F f1 ) / Nvi/r z t) = p " lkz t k l ; slnhk{z +1 ) -iktk2t 2 + k2t2 slnh2kp /_1 \k y slnhkfz - 1) Iktsinh2k 6 +17References.[i] Case, K,M.: Stability of Inviscid Plane CouetteFlow.The Physics of Pluids, Vol. 3, {^^6o),(p. 1^3).J Llghthlllj M. J. ; Introduction "to Fourien analysis andgeneralised functions.Cambridge University Press (1962).[3] Jeffreys, H. : Asymptotic approximations.Oxford Clarendon Press (1962), (p. 14).

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On the stability of plane inviscid Couette flow

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On the stability of plane inviscid Couette flow

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