Bubble effect on Kelvin-Helmholtz’ instability 8 0 0 S. L. Gavrilyuk∗, H. Gouin∗ and V.M.Teshukov† 2 n February 2, 2008 a J 6 1 Abstract ] n We derive boundary conditions at interfaces (contact discontinuities) y for a class of Lagrangian models describing, in particular, bubbly flows. d Weusetheseconditions tostudyKelvin-Helmholtz’instability which de- - velops in theflow of two superposed layers of a pureincompressible fluid u andafluidcontaininggasbubbles,co-flowingwithdifferentvelocities. We l f show thatthepresenceofbubblesin onelayerstabilizes the flowin some . s intervals of wavelengths. c i s y 1 Introduction h p Many mathematical models of fluid mechanics are derivedthrough the approx- [ imation of the solution of boundary value problems for Euler equations. For 1 example, equations for bubbly flows are derived as an approximation of the v solution of a complex free-boundary problem describing the motion of the mix- 7 ture of water and gas bubbles (Iordanski (1960), Kogarko(1961), Wijngaarden 5 4 (1968)). Green-Naghdi’s model describing wave motions of a liquid layer of fi- 2 nite depth in a shallow water approximation with account of dispersion effects 1. is another example (Green et al (1974), Green & Naghdi (1976)). Due to the 0 fact that averaging procedures and asymptotic expansions have been used in 8 thederivation,itisnotobvioustodecidewhatboundaryconditionsarenatural 0 for these systems of equations. In this paper, a Hamiltonian formulation of the : v problemin(t,x)-spaceisproposedwhichallowsonetofindboundaryconditions i for a general class of models. This class includes a model of bubbly fluid and X dispersiveshallowwater. Weusetheboundaryconditionstostudythestability r a of two co-flowing layers of bubbly and pure fluids. In absence of gravity and capillarity, it is well known that Kelvin-Helmholtz’ instability develops for any wave lengths in the flow of two superposed layers of pure incompressible fluids ∗LaboratoiredeModélisationenMécaniqueetThermodynamique,FacultédesScienceset TechniquesdeSaint-Jéroˆme,Case322,AvenueEscadrilleNormandie-Niemen,13397Marseille Cedex20,France,e-mail: sergey.gavrilyuk@univ.u-3mrs.fr;henri.gouin@univ.u-3mrs.fr †LavrentyevInstituteofHydrodynamics,Lavrentyevprospect15,Novosibirsk630090,Rus- sia, also INRIA, SMASH, 2004 route de Lucioles, 06902 Sophia Antipolis, France, e-mail: teshukov@hydro.nsc.ru 1 (see, for example, Drazin & Reid (1981)). We prove that the presence of bubbles in one layer can produce a stabilizing effect: the flow becomes stable with respect to perturbations of some wave lengths. 2 Governing equations and conditions on mov- ing interfaces 2.1 Variation of Hamilton’s action Here we calculate the variation of Hamilton’s action for a special class of La- grangians. Usually this procedureisusedonlyfor derivationofgoverningequa- tions. We obtain below not only governing equations but also boundary conditions at inner contact surfaces. Let us consider the Lagrangianof the form: ∂J L=L(J, ,z) (2.1) ∂z t where z= zi , (i = 0,1,2,3), t is the time, x is the space variable, x ≡ (cid:18) (cid:19) ρ (cid:0) (cid:1) J= , ρ is the fluid density and u is the velocity field. Let us calculate ρu (cid:18) (cid:19) thevariationofHamilton’sactioninthecasewhenthe4-DmomentumJverifies the equation of continuity: ∂ρ Div J +div(ρu)=0 (2.2) ≡ ∂t Hamilton’s action is defined by a= Ldz (2.3) Ω Z where Ω is a material domain. Considering a smooth one-parameter family of virtual motions z=Φ(Z,ε), Φ(Z,0)=ϕ(Z) (Z stands forthe Lagrangiancoordinates,ε is a smallparameterat the vicinity of zero and z = ϕ(Z) is the real motion), we define the virtual displacements ζ(Z) and the Lagrangianvariations δJ(Z) by the formulae: ∂Φ(Z,ε) ∂J(Z,ε) ζ(Z)= , δJ(Z)= (2.4) ∂ε ∂ε (cid:12)ε=0 (cid:12)ε=0 (cid:12) (cid:12) Due to the fact that Z = ϕ−1(z(cid:12)(cid:12)), we consider the variat(cid:12)(cid:12)ions as functions of Eulerian coordinates and use the same notations ζ(z) and δJ(z) in Eulerian variables. Hamilton’s principle assumes that ζ(z) = 0 on the boundary ∂Ω of Ω. 2 In the following, the transposition is denoted by T. For any vectors a and b we use the notation aTb for their scalar product a b and abT for their tensor · producta b. ThedivergenceofthesecondordertensorAisacovectordefined ⊗ by Div(Ah)=Div(A)h whereh is anyconstantvectorfield. Div anddiv, Grad and arerespectively ∇ divergence and gradient operators in the 4-D and 3-D space. The identity matrix and the zero matrix of dimension n are denoted by I and O . n n In calculations we shall use the equality ∂ζ δJ= (Div ζ) I4 J (2.5) ∂z − (cid:18) (cid:19) which was proved in Gavrilyuk & Gouin (1999). Variation of the Hamilton action is da ∂ζ δa= = δL+L tr dz (2.6) dε(cid:12)ε=0 ZΩ(cid:18) (cid:18)∂z(cid:19)(cid:19) (cid:12) Since (cid:12) (cid:12) ∂L ∂L ∂J ∂L δL= δJ+tr δ + ζ ∂J ∂J ∂z ∂z ∂ (cid:18) (cid:19)  ∂z   (cid:18) (cid:19)    and ∂J ∂δJ ∂J∂ζ δ = ∂z ∂z − ∂z ∂z (cid:18) (cid:19) we get from (2.5) and (2.6): ∂L ∂ζ ∂δJ ∂J∂ζ ∂L δa= (Div ζ)I4 J+tr AT + ζ+L Div ζ dz Ω ∂J ∂z − ∂z − ∂z ∂z ∂z Z (cid:18) (cid:18) (cid:19) (cid:18) (cid:18) (cid:19)(cid:19) (cid:19) with AT = ∂L or AT j = ∂L (2.7) ∂J i ∂Ji ∂ ∂z (cid:0) (cid:1) ∂zj (cid:18) (cid:19) (cid:18) (cid:19) Forthesakeofsimplicitythemeasureofintegrationwillnotbeindicated. Since for any linear transformation A and vector field v ∂v Div(Av)=(Div A)v+tr(A ) ∂z we get ∂L ∂J ∂ζ ∂L ∂δJ ∂L δa= tr J AT + L J Divζ+tr AT + ζ = Ω ∂J − ∂z ∂z − ∂J ∂z ∂z Z (cid:18)(cid:18) (cid:19) (cid:19) (cid:18) (cid:19) (cid:18) (cid:19) ∂L ∂J ∂ζ ∂L = tr J AT + L J Div ζ+ Ω ∂J − ∂z ∂z − ∂J Z (cid:18)(cid:18) (cid:19) (cid:19) (cid:18) (cid:19) 3 ∂L +Div ATδJ Div AT δJ+ ζ , − ∂z and finally (cid:0) (cid:1) (cid:0) (cid:1) ∂L ∂J ∂ζ ∂L δa= tr J AT + L J Div ζ+Div ATδJ Ω ∂J − ∂z ∂z − ∂J − Z (cid:18)(cid:18) (cid:19) (cid:19) (cid:18) (cid:19) (cid:0) (cid:1) ∂ζ ∂L Div AT (Div ζ)I4 J+ ζ = − ∂z − ∂z (cid:18) (cid:19) (cid:0) (cid:1) ∂L ∂J ∂ζ = tr J AT JDiv AT + Ω ∂J − ∂z − ∂z Z (cid:18)(cid:18) (cid:19) (cid:19) (cid:0) (cid:1) ∂L ∂L + L J+ DivAT J Div ζ+Div ATδJ + ζ = − ∂J ∂z (cid:18) (cid:19) (cid:0) (cid:1) (cid:0) (cid:1) ∂L ∂J = Div J AT J Div AT ζ+ATδJ Ω ∂J − ∂z − − Z (cid:18)(cid:18) (cid:19) (cid:19) (cid:0) (cid:1) ∂L ∂J Div J AT J Div AT ζ+ − ∂J − ∂z − (cid:18) (cid:19) (cid:0) (cid:1) ∂L ∂L T ∂L +Div L J+ DivAT J ζ Grad L J+ DivAT J ζ+ ζ − ∂J − − ∂J ∂z (cid:18)(cid:18) (cid:19) (cid:19) (cid:18) (cid:19) (cid:0) (cid:1) (cid:0) (cid:1) Let us denote δL ∂L Div AT =KT. δJ ≡ ∂J − Then (cid:0) (cid:1) ∂L ∂J δa= Div J KT AT + L KT J I4 ζ+ (2.8) Ω ∂z − − ∂z − Z (cid:18) (cid:18) (cid:19)(cid:19) (cid:0) (cid:1) ∂J + NT J KT AT + L KT J I4 ζ+NTATδJ Z∂Ω (cid:18) − ∂z − (cid:19) (cid:0) (cid:1) with NT = ( D ,nT), D denotes the surface velocity and n is the space n n − unit normal vector. Virtual displacements vanish at the boundary ∂Ω and the surface integral is zero. The volume integral yields the equations of motion in conservative form as in Gavrilyuk & Gouin (1999). Inthecasewhenfluidtensorialquantitiesarediscontinuousattheinnerinterface Σ, expression (2.8) becomes ∂L ∂J δa= Div J KT AT + L KT J I4 ζ+ (2.9) Ω ∂z − − ∂z − Z (cid:18) (cid:18) (cid:19)(cid:19) (cid:0) (cid:1) ∂J + NT J KT AT + L KT J I4 ζ+NTATδJ Σ − ∂z − Z (cid:20) (cid:18) (cid:19) (cid:21) (cid:0) (cid:1) where the jump through Σ is denoted by [ ]. 4 2.2 Governing equations and boundary conditions We explicit the governing equations and the inner boundary conditions for the Lagrangian L= 1ρ u2 W ρ,ρ· , where (·) = d = ∂() +uT∇() (2.10) 2 | | − dt ∂t (cid:16) (cid:17) Such a Lagrangian appears in the study of wave propagation in both shallow water flows with dispersion and bubbly flows (a complete discussion of these models is given in Gavrilyuk & Teshukov (2001)). We get ∂ρ ∂ρ ∂J ∂t ∂x ∂L u2 ∂W 1∂W 1∂W = , = | | + ( ρ)T u, uT ( ρ)T ∂z ∂j ∂j ∂J − 2 − ∂ρ ρ ∂ρ· ∇ − ρ ∂ρ· ∇ !    ∂t ∂x      Matrix (2.7) becomes ∂W 0T · − ∂ρ ∂L   AT = = . ∂J ∂  ∂W  ∂z  · u O3  (cid:18) (cid:19)  − ∂ρ      We get ∂ ∂W ∂W Div(AT)= div u , 0T , (2.11) · · −∂t ∂ρ !− ∂ρ ! ! 2 u δW 1∂W 1∂W KT= | | + ( ρ)T u, uT ( ρ)T , · · − 2 − δρ ρ ∂ρ ∇ − ρ ∂ρ ∇ ! ∂W NT J=ρ nT u D , NTAT = nT u D ,0T . n · n − − ∂ρ − ! (cid:0) (cid:1) (cid:0) (cid:1) We study the case when Σ is a contact surface and, consequently, NT J = 0, NTAT =0T. We see that the surface integralin (2.9) vanishes if the boundary conditions : L KT J [p]=0 (2.12) − ≡ are fulfilled with p L K(cid:2)T J. Vani(cid:3)shing the volume integral in (2.9) and ≡ − using relations (2.11), we obtain the governing equations in the form ∂ρ +div(ρu)=0, (2.13) ∂t ∂ρu +div ρuuT+pI =0, ∂t (cid:0) (cid:1) 5 ∂W ∂ ∂W ∂W δW p ρ div u W =ρ W. · · ≡ ∂ρ − ∂t ∂ρ !− ∂ρ !!− δρ − A complete set of boundary conditions at the contact interfaces for the class of models considered is: nTu D =0, [p]=0 (2.14) n − Equations (2.13) have been obtained earlier in Gavrilyuk & Shugrin (1996) by meansofthe classicalmethod ofLagrangemultipliers. Notice thatthis method did not give jump conditions (2.12). System (2.13) is reminiscent of Euler equations for compressible fluids. The term p in equations of motion (2.13) andboundaryconditions(2.14)standsforthe pressure. Nevertheless,pisnota functionofdensityasinthecaseofbarotropicfluids: itdependsalsoonmaterial time derivatives of the density. Thus, p is not usual thermodynamic pressure. In general, it implies that the boundary conditions are not just consequences of conservation laws. For example, for Korteweg-type fluids where the stress tensordependsonthedensitygradient,theboundaryconditionscontainnormal derivatives of the density at interfaces (Seppecher (1989), Gouin & Gavrilyuk (1999)). As we have shownhere, the boundary conditions at inner interfaces in bubbly fluids do not depend on normal derivatives of the density: they involve only tangential derivatives. If the flow domain is sharedbetween two domains consisting of pure ideal fluid andbubblyfluid,allpreviouscalculationsarestillvalidandboundaryconditions (2.14) can be extended to such a case. 2.3 Exact statement of the problem Now,weproposetouseconditions(2.14)tostudyKelvin-Helmholtz’instability of a parallel flow between two rigid walls of a layer of bubbly fluid in contact with a layer of pure incompressible fluid. Considerthe 2-Dflowoftwosuperposedfinite layersofinviscidfluids inthe channel: < x < + , 0 < y < H0. The first one is governed by the Euler −∞ ∞ equations : div(u1)=0, ∂u1 + uT1 u1+ ∇p1 =0, N 1 (3.17) − 16N2 3N−2 ◦ 2 2 2 3 N >1, d > 8N 6 1 +8N 9 (3.18) 27 − r − 4 − ! (cid:0) (cid:1) 10

Bubble effect on Kelvin-Helmholtz' instability

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