Instability criterion for oblique modes in stratified circular Couette flow

a r X i v :0807.0702v 1 [p h y s i c s .f l u -d y n ] 4 J u l 2008Instability criterion for oblique modes in stratified circular Couette flow C.Normand Institut de Physique Th´e orique,CNRS,URA 2306CEA,IPhT,91191Gif-sur-Yvette Cedex,France.contact:christiane.normand@cea.fr July 4,2008Abstract An analytical approach is carried out that provides an inviscid stability criterion for the strato-rotational instability (in short SRI)occurring in a Taylor-Couette system.The control parameters of the problem are the rotation ratio µand the radius ratio η.The study is motivated by recent numerical [1]and experimental [2]results reporting the existence of unstable modes beyond the Rayleigh line for centrifugal instability (µ=η2).While the modified Rayleigh criterion for stably stratified flows provides the instability condition,µ<1,unstable modes were never found beyond the line µ=η.Taking into account finite gap effects and considering non axisymmetric perturbations in the small azimuthal wavenumber limit,we derive the instability condition :µ<2η2/(1+η2).For a thin gap,η=1−ε,this is in agreement with the condition µ<ηfound numerically for η=0.78and experimentally for η=0.8.For a wide gap and ηsmall the instability condition is approximated by µ<2η2.Whatever the gap size,instability is predicted for values of µlarger than the critical one,µc =η2,corresponding to centrifugal instability.1IntroductionDespite its long-lasting study since the work of Taylor[3],the stability of cylindrical Couette flow remains a vivid research area attracting many investigators.A survey of the literature on the topic can be found in[4].Theflow occurs in the annular gap between two concen-tric cylinders of radii R1<R2rotating independently at the angular velocitiesΩ1andΩ2, respectively.The control parameters are the radius ratioη=R1/R2and the rotation ratio µ=Ω2/Ω1.Inside the gap,the angular velocity profile of the laminar steadyflow isΩ(r).Its linear stability with respect to axisymmetric perturbations is governed,in the inviscid limit, by the Rayleigh criterion[5]:d(r2Ω)2/dr>0.A generalized Rayleigh criterion for non-axisymmetric centrifugal instabilities has been derived for a free axisymmetric vortex using a large axial wavenumber WKB approximation[6].The instability takes the form of a spatially oscillating mode localized between two turning points where it matches with the exponentially decaying solutions outside.The generalization to take into account a background rotation and a stable stratification did not reveal fundamental changes in the Rayleigh criterion.This contrasts with what was found for boundedflows[7,8,9].In these studies,it was shown that a stable stratification drastically changes the stability criterion,showing a strong analogy with the modified Rayleigh criterion obtained when a magneticfield parallel to the cylinders’axis is present.In the hydromagnetic case[10,11]the condition for stability with respect to axisymmetric perturbations is:d(Ω2)/dr>0.The presence of a stable stratification has a similar effect on the Rayleigh criterion though its validity is in that case restricted to non axisymmetric perturbations[7,8,9].In the astrophysical context,the magneto-rotational instability(MRI)was recognized as a potential source of turbulence in accretion disks[12],especially Keplerian disks,with angular velocityΩ(r)∼r−3/2,unable to sustain centrifugal instability according to the standard Rayleigh criterion.The MRI mechanism is robust,it occurs in bounded as well as unbounded flows and for compressible or incompressiblefluid.The capacity of the strato-rotational instability(SRI)to trigger turbulence in Keplerian disks raises the open question of what are the appropriate radial boundary conditions at the edges of the disk.The existence of SRI has been demonstrated in the inviscid limit for perturbations that satisfy no normal flow conditions on the channel walls.In the viscous case,both the no-slip[1]and stress-free [8]conditions give rise to instability.Mixed boundary conditions in which no normalflowis imposed on one side of the channel and zero pressure on the other side were not able to sustain growing modes[9].The effect of a vertically varying stratification was also examined [9]showing that SRI persists in that case.Beside possible applications in geophysics and astrophysics,SRI was studied in the labo-ratory.Until very recently the conclusion of experimental studies was that a stable vertical stratification stabilizes theflow[13,14,15].The experimental evidence of the strato-rotational instability(SRI)in a Taylor-Couette system was definitely assessed in[2].Moreover,the val-ues of the control parameters for which instability occurs were found in good agreement with numerical predictions[1]that yields the condition:µ<η,for instability.The aim of the present contribution is to derive an instability criterion by an entirely ana-lytical analysis that improves previous studies achieved in the small gap limit[7,8,9].When curvature effects are neglected the problem under consideration reduces to the stability of a stratified plane Couetteflow rotating at constant angular velocityΩ.In cartesian coordinates (x,y,z)theflow velocity is V=(0,Sx,0)where S is the constant shear.The condition for instability in the stratified case is S/2Ω<0while the standard Rayleigh-Pedley criterion gives:S/2Ω<−1,when there is no stratification.The substitution S→rΩ′,is often used to deduce the modified Rayleigh criterion for stratifiedflows with curved streamlines :d(Ω2)/dr<0.When applied to the circular Couetteflow,the instability criterion simply yields:µ<1.However,in numerical and experimental studies[1,2]achieved forfinite values of the gap,unstables modes were never found beyond the lineµ=η.The narrowing of the instability range is possibly due to curvature effects that were not taken into account with sufficient accuracy in the transposition S→rΩ′.In the present contribution a more appropriate treatement of curvature effects is carried out leading to an instability criterion that involves the two parametersµandη.Our stability results will be compared to the experimental ones[2]obtained for a value of the radius ratioη=0.8,and to the numerical ones[1]forη=0.78andη=0.3corresponding to a wider gap.2Stratified circular CouetteflowIn cylindrical coordinates(r,ϕ,z)the velocityfield in the basic state is V=(0,rΩ(r),0)with the angular velocity given by[1]Ω(r)=A1−η2,and B=Ω1µ−η2rZu=−iGrΩp=−iσDp(3)where D≡∂/∂r and D∗=D+1/r.Here,σ=ω−lΩ,and Z=2Ω+rΩ′.The quantities K and G are given belowK=σ2−N2,and G=σ2m2+l2withσ1=ω1−l1Ω.It must be stressed that the generalization of the Rayleigh criterion[6] performed in the limit m>>1,with a different scaling for the frequency assumed of order unity,is not equivalent to our approach.This could explain why SRI was not found in[6].The perturbations u and p are expanded according top=p0+εp1+...and u=εu1+ (5)and substituted in(2)-(3)to gives at the leading order inεthe set of equations:l1σ1D∗u1+Ωp0=iσ1Dp0(7)rWe have introduced the Rayleigh discriminantΦ=2ΩZ and the function g(r)=ˆm2σ21−l21/r2 withˆm=m/N.Eqs.(6)-(7)are the circular version of those obtained for a plane Couette flow[9],they constitute a set of coupled equations providedΦ=0.It was shown in[9] that elimination of u1leads to a simple equation for p0,its two independent solutions being exponential functions.Then substituting p0in(7)gives immediately the expression for u1. Finally,satisfaction of the conditions u1=0,on the two boundaries confining theflow leads to the instability criterion.The same procedure is applied here to the cylindrical case.After some calculations,thefirst step gives the equation satisfied by p0σ1 D2p0+ 1Φ Dp0−ˆm2Φp0 +2l1Ω∂r Log ΦΩ′r−Φx)found in[9]for the stratified rotating plane Couetteflow when the Rayleigh criterion for stability,Φ>0,is√satisfied.For circular Couetteflow andΦ=4BΩ,we shall introduceˆp=p0/Ωr4 2−3Awith the differential operator L0=D∗D accounting for curvature effects.We shall assume thatˆm is large enough so that the term inˆm2dominates in Eq.(10).Our analysis differs from previous studies that linearized the angular velocity:Ω(r)=Ω(r0)+(r−r0)Ω′+···around a mean radius r0as in the thin gap limit.Here,the exact expression ofΩ(r)given in (1)is substituted in(10)that becomesL0ˆp−4ˆm2B AAB andα=2ˆmB.The expression forˆp isˆp=A0I n(αr)+B0K n(αr)(12) where the unknown coefficients A0and B0will be determined by the satisfaction of the boundary conditions on u1.Substituting(12)in(7)and using the relation between the√Bessel functions and their derivatives,one gets u1=ˆu/Ψn+σ1 n2Ω Ψn∓αΨn−1 (14)rIn the following,we shall neglect the terms inΩ′/Ωthat appear in factor ofσ1,which are small compared to the terms proportional to n orαwhich behave likeˆm.Satisfaction of the boundary conditions u1(R1)=u1(R2)=0provides the algebraic systemMA=0(15) with the vector A=(A0,B0)and the elements of the matrix M given by M ij=U j(R i). Their expressions are easily deduced fromλiU(R i)=2.3Dispersion relationThe vanishing of the determinant associated to(15)leads to the dispersion relationλ1R2S n−α2ˆσ1ˆσ2S n−1+αˆσ2λ1R2C21=0(17)where we have introduced the quantitiesS n=I(1)n K(2)n−I(2)n K(1)n and C ij=I(i)n K(j)n−1+I(j)n−1K(i)n for(i,j)∈(1,2)(18) To simplify the calculations we shall use the asymptotic expansions at large arguments of the modified Bessels functions I and K.At the lowest order,the asymptotic behaviors of expressions(18)are:S n=S n−1≈sinh(α1−α2)α1α2and C12=C21≈cosh(α1−α2)α1α2(19)With q=α(R2−R1),the dispersion relation(17)readsλ1λ2R1−ˆσ1λ2R1R2−X+Ω1Ω2l21 X−4(n−1)R1(1−η) nω21−nl1(Ω1+Ω2)ω1+(n−2)l21Ω1Ω2 (22) where a term of order of magnitude smaller than n and n−2has been neglected in T2. The quantity X involved in expression(21)for T1can be expressed in terms of the control parameters,µandη,as followsX=n21−η(23)2.4Instability criterionIt is worth while noticing that the sign of the quantityη−µwhich appears in expression(23) for X seems determinant for the stability of the system according to what has been observed both in experiments[2]and in numerical computations[1].To check whether our calculations support thisfinding we shall determine the roots of the dispersion relation(20).Instabilitycould occur if there is a pair of complex conjugate roots,one with negative imaginary part corresponding to instability growth,the other to decay.The calculation of the discriminant and the determination of its sign is facilitated by introducing the quantities Y,Z and Q= Q0coth q defined as follows:Y=2nR1R2and Q0=nα1−η(1−µ)(µ−η2)(R1R2)2(Ω1+Ω2)2−4(X−Q)1+η1+η2(28)For small gap size we can write:η=1−ε.Expansion of(28)tofirst order inεgives the approximated conditionµ<η,in agreement with numerical[1]and experimental[2]results found forη=0.78andη=0.8respectively.For these two values of the gap the exact condition (28)predicts instability forµ<0.756andµ<0.78,well beyond the corresponding Rayleigh linesµ=0.6andµ=0.64.For larger gap size,consideringηas a small expansion parameter and neglecting terms of orderη4,the condition(28)becomesµ<2η2.This confirms that the frontier for instability is beyond the Rayleigh line which is exceeded by a factor two.However,the instability condition(28)do notfit with accuracy the numerical results found in[1]for η=0.3.The discrepancy could be explained by the behavior of the numerical neutral curves (Reynolds number versusµ)that strongly depend on the azimuthal wavenumber[1].The instability condition(28)which is independent of l cannot reproduce this feature.However, the instability boundary:µ=0.165,deduced from(28)forη=0.3,fits surprisingly well with the numerical results obtained for the azimuthal mode with wavenumber l=3,while it is not the case for the smaller values l=1,2.3ConclusionWe performed an inviscid stability analysis of SRI in a Taylor-Couette system.Finite gap effects were taken into account more appropriately than in previous inviscid approaches.The assumptions that have been made never concerned the angular velocity profile which is kept equal toΩ=A/r2+B.Thus,gap size effects manifest themselves through the quantities A and B which depend onµandη,the control parameters of the system.For intermediate values of the gap(η≈0.8),the stability boundary:µ=ηwas recovered with a good accuracy in agreement with numerical[1]and experimental[2]results.For a wider gap(η=0.3)the agreement with numericalfindings is better for the mode l=3than for the modes l=1,2, though the reason of such a behavior is not yet clearly understood.The angular velocity profile of circular Couetteflow is peculiar since it allows an analytical resolution for the pressure perturbations in terms of Bessel functions.This is an essential step in the above derivation of the stability criterion for SRI in incompressiblefluid.A slight change inΩ(r)can lead to completely different stability results.Aflow with constant angular momentum(Ω∼r−2)obtained when B=0in(1),was considered in a thin cylindrical shell [16].This type offlow is generally assumed centrifugally stable,its Rayleigh discriminant being equal to zero.WhenΦ=0,Eqs.(6)and(7)are decoupled and the above analysis for SRI cannot be applied.In that case,the existence of nonaxisymmetric unstable modes was proved for unstratifiedflow in a compressiblefluid[16]with an equation of state of the type p∼ργ.It will be interesting in future work to examine if the present analytical approach could be extended to compressiblefluids for appropriate angular velocity profiles.References[1]D.Shalybkov and G.Rudiger,Stability of density-stratified viscous Taylor-Couetteflows, Astronom and Astrophys.438,411-417(2005).[2]M.Le Bars and P.Le Gal,Experimental analysis of the Strato-Rotational Instability ina cylindrical Couetteflow,Phys.Rev.Lett.99,064502(2007).[3]G.I.Taylor,Stability of viscousfluid contained between two rotating cylinders,Phil. 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