Mathematical analysis of low Mach number limit of some two phase flow models

Transcription

1 Mathematical analysis of low Mach number limit of some two phase flow models B. DESJARDINS Fondation Mathématique Jacques Hadamard FMJH CMLA ENS Cachan 61, avenue du Président Wilson Cachan cedex France Paris, November 2015 Low Mach number flows workshop

2 Coworkers Physical Models for Two Phase Flows A series of joint works, and ongoing work... with: Didier Bresch (CNRS, Université de Savoie at Chambéry) Jean-Michel Ghidaglia (CMLA ENS Cachan) Emmanuel Grenier (UMPA, ENS Lyon) ANR DYFICOLTI managed by D. LANNES

3 Outline Physical Models for Two Phase Flows 1 Two phase one pressure models: some mathematical properties 2 Low Mach number limit for single phase models 3 Generalization to two phase flows.

4 Complex free surface flows Physical Models for Two Phase Flows Interfaces subject to Variable accelerations g(t), shocks, shear Mixing induced by instabilities 1 Rayleigh-Taylor 2 Richtmyer-Meshkov 3 Kelvin-Helmholtz... in the fully turbulent regime See I.W. Kokkinakis, Dimitris Drikakis, David Youngs, R.J.R. Williams, Two-equation and multi-fluid turbulence models for RayleighTaylor mixing International Journal of Heat and Fluid Flow 12/2015; 56:

5 Rayleigh Taylor Instabilities (1/3) Rayleigh-Taylor (acceleration induced) instabilities (LLNL,

6 Rayleigh Taylor Instabilities (2/3) B. Cabot, A. Cook, P. Miller (2006). 3 to 1 density ratio Atwood = 0.5, simulation

7 Rayleigh Taylor Instabilities (3/3) B. Cabot, A. Cook, P. Miller (2006). grid spacing η, Kolmogorov scale

8 Physical model: two phase barotropic flow model Two phase flow models are widely used in industrial framework (turbulent mixing in nuclear industry, reactive flows, propulsion, sprays... ) We consider here only barotropic flows (no temperature dependance of the e.o.s. P ± = P ± (ρ, θ)) 1 Each phase ± is characterized by its velocity u ±, density ρ ±, and equation of state P ± = P(ρ ± ). 2 The volume fraction of phase ± is denoted α ± : α + + α = 1. 3 Pressure equilibrium: pressure relaxation effects are assumed to be more rapid than any other physical phenomena: P + = P. 4 Turbulent diffusion of velocity: stress tensor S ± = 2α ± ρ ± ν ± D(u ± ), where ν ± > 0 are constant (zero bulk viscosity). 5 Surface tension expressed as diffuse interface models: div K ± = σ ± α ± ρ ± α ± ρ ± where σ ± 0 are constants.

9 Four equations model Physical Models for Two Phase Flows The four equations - one pressure two phase model writes as: α + + α = 1, t(α + ρ + ) + div (α + ρ + u + ) = 0, t(α ρ ) + div (α ρ u ) = 0, t(α + ρ + u + ) + div (α + ρ + u + u + ) + α + P = D + α + ρ + g + div S + + div K +, t(α ρ u ) + div (α ρ u u ) + α P = +D + α ρ g + div S + div K, P = P (ρ ) = P +(ρ +), with D = momentum exchange (u + u ) u + u β or interfacial pressure π α +, 0 α ± 1, S ± denotes the stress tensor of phase ±.

10 Difficulties (1/2) Physical Models for Two Phase Flows As a consequence of the pressure equilibrium assumption, the system (with right hand side = 0) is non-conservative, and non-hyperbolic if with 0 u + u < c m, (when ν± = 0 and σ ± = 0 ) c 2 m = c 2 c 2 + ((α + ρ + ) 1/3 + (α ρ ) 1/3 ) 3 α + ρ c 2 +. α ρ + c 2 + In general, c m is large compared to u + and u and thefore flow belongs to non-hyperbolic region. where c 2 ± = P ± (ρ ± ) denotes the sound speed of phase ±. See: H.B. STEWART, B. WENDROFF, J. Comp. Physics, , (1984) (Appendix I).

11 Difficulties (2/2) Physical Models for Two Phase Flows Properties of the model with zero diffusion, capillary and momentum exchange terms Non linear and non conservative equations: Existence of global in time weak solutions is open Regularizations Hyperbolization with additional physical effects such as interfacial pressure Viscosity and capillarity at high frequency What happens when one phase or the two are only slightly compressible? ρ + constant, or ρ ± constant?

12 Hyperbolization: interfacial pressure Interfacial pressure models π are often used as a low frequency stabilization effect α + + α = 1, t(α + ρ + ) + div (α + ρ + u + ) = 0, t(α ρ ) + div (α ρ u ) = 0, t(α + ρ + u + ) + div (α + ρ + u + u + ) + α + P + π α + = 0, t(α ρ u ) + div (α ρ u u )) + α P + π α = 0, P = P (ρ ) = P +(ρ +), with 0 α ± 1.

13 Hyperbolization: interfacial pressure In Cathare Thermohydraulic code, D. Bestion introduced the interfacial pressure π = δ α+ α ρ + ρ α + ρ + α ρ + (u+ u ) 2. with δ > 1 to get hyperbolicity for small relative velocity. If ρ + >> ρ and α << 1, hyperbolicity if u + u 2 < c 2. See paper by M. NDJINGA, A. KUMBARO, F. DE VUYST, P. LAURENT-GENGOUX, ISMF (2005) for geometric discussions: number of intersecting points of parabola and hyperbola in quarter plane. Extension of H.B. STEWART, B. WENDROFF s approach. For a brute force study: analytical calculations linked to resolvent cubic equations. D. BRESCH, B. D., J. M. GHIDAGLIA, E. GRENIER. Low Mach Number Limit and two phase Systems, In preparation.

14 Viscosity and surface tension With viscosity and capillarity effects, the two phase system rewrites: α + + α = 1, t(α ± ρ ± ) + div(α ± ρ ± u ± ) = 0, t(α ± ρ ± u ± ) + div(α ± ρ ± u ± u ± ) +α ± p = div(α ± S ± ) + σ ± α ± ρ ± ( α ± ρ ±). with S ± = 2µ ± D(u ± ) + λ ± div u ± Id p = p ±(ρ ± ) = A ± (ρ ± ) γ± where γ ± are given constants greater than 1 Assume µ ±(ρ ± ) = ν ± ρ ±, λ ±(ρ ± ) = 0.

15 Viscosity and surface tension Then, using the mathematical structure associated with the additional entropy and additional regularization due to capillary stress tensor, Theorem (D. Bresch, B.D., J.M. Ghidaglia, E. Grenier, 2009) Stability of weak solutions in T d in dimension d 1. D. BRESCH, B. D., J. M. GHIDAGLIA, E. GRENIER. On Global Weak Solutions to a Generic Two-Fluid Model. Arch. Rational Mech. Anal. Volume 196, Number 2, , (2009). In dimension d = 1, no capillarity effects are required Theorem (D. Bresch, X. Huang, J., 2011) Stability of weak solutions in T 1 without capillarity. D.Bresch, X. HUANG, J. LI. A Global Weak Solution to a One-Dimensional Non-Conservative Viscous Compressible Two-Phase System. Communications in Mathematical Physics, (2011).

16 Formal energy conservation Physical Models for Two Phase Flows Formally, total energy is conserved: ( ) d 1 dt ± Ω 2 α± ρ ± u ± 2 + α ± ρ ± e ± (ρ ± ) + σ± 2 (α± ρ ± ) 2 + 2ν ± α ± ρ ± D(u ± ) 2 = p tα ± = 0 ± Ω ± Ω where the internal energy is given by e ± (ρ ± ) = A ± (ρ ± ) γ ± 1 γ ± 1

17 Global weak solutions Physical Models for Two Phase Flows Definition of weak solutions. We shall say that (ρ ±, α ±, u ± ) is a weak solution on (0, T) if the following three conditions are fulfilled: The following regularity properties hold α ± ρ ± u ± 2 L (0, T; L 1 (Ω)), α ± ρ ± L (0, T; L 2 (Ω) 3 ), α± ρ ± u ± L 2 ((0, T) Ω) 3 3, σ± (α ± ρ ± ) L (0, T; L 2 (Ω) 3 ), σ± (α ± ρ ± ) L 2 (0, T; L 2 (Ω)), with following time continuity properties α ± ρ ± C([0, T]; H s (Ω)), for all s < 1/2, α ± ρ ± u ± C([0, T]; H s (Ω) 3 ) for some positive s,

18 Global weak solutions Physical Models for Two Phase Flows the initial conditions α ± ρ ± t=0 = R± 0 and α± ρ ± u ± t=0 = m± 0 hold in D (Ω). Mass equations hold in D ((0, T) Ω) and momentum equations multiplied by α ± ρ ± hold in D ((0, T) Ω) 3 : for all ψ ± C ([0, T] Ω) d and denoting R ± = α ± ρ ±, one has R ± 2 u ± (t, ) ψ ± (t, ) R ± 0 m ± 0 ψ(0, ) Ω t = 0 Ω Ω [ ( D(ψ ± ) : R ± u ± R ± u ± 2ν ± R ± 2 D(u ± ) +σ ± R ± R ± R ±) + σ ± R ± 2 ψ ± R ± α ± R ± ψ ± p R ± 2 div u ± (u ± ψ ± ) 2ν ± R ± ψ (D(u ± ) R ±) + R ± 2 u± tψ ± σ ± div ψ ± ( R ± 3 /3 R ± R ± 2)]

19 = div(ν ± R ± u ± i ) D i + σ ± R ± i R ± (1) Physical Models for Two Phase Flows Why a degenerate viscosity may help? In dimension d 1, the auxiliary velocity satisfies w ± = u ± g ±, where g ± = ν ± R ± R ± t(r ± w ± i ) + div(r ± u ± w ± i ) + R± ρ ± ip so that d dt + + Ω Ω Ω ) (R ± w ± 2 + R ± g ± 2 + σ± R± 2 (ν ± R ± u ± 2 + σ ± ν ± R ± 2) R ± ρ ± p w± = 0

20 Formal energy conservation Physical Models for Two Phase Flows Summing up the two phases ±, one gets ) ν d (R ± w ± 2 + R ± g ± 2 + R ± e ± (ρ ± ) + σ± dt ± Ω R± 2 +ν + ν ) (R ± u ± 2 + R ± c ± 2 ρ ± 2 + σ ± R ± 2 ± Ω ρ ± 2 = ν p tα ± ± Ω

21 Incompressible - compressible models When one of the two phases becomes incompressible ρ + constant and phase remain compressible Regular limit No acoustic waves p = p (ρ ), ρ + = ρ + 0 constant Invariant regions Definition A closed subset S R n is invariant for the local solution w over [0, T] if w(t, x) S for all (t, x) [0, T] Ω as soon as w(0, x) S. Theorem (J.M. Ghidaglia) For smooth solutions to inviscid equation, the region α 0 is invariant under the evolution if and only if phase + is compressible. Numerical schemes sometimes lead to negative values for α and therefore require clipping techniques which consists in modifying (increasing) variable α when it gets too small. However, this procedure destroys mass conservation and very often leads to nonphysical results.

22 Invariant regions: proof Physical Models for Two Phase Flows The two phase inviscid model may be rewritten in w = (u +, u, α, p), where α = α + α : tu + + u + u + + p/ρ + = 0, tu + u u + p/ρ = 0, tα + u α α + (1 α 2 )F(w, w) = 0, tp + u p p + A div(α + u + + α u ) = 0, where (a, b) F(a, b) is a smooth function of a and b, so that tw + where M i are smooth functions of w. 3 M i(w) iw = 0, i=1

23 Invariant regions: proof Physical Models for Two Phase Flows Boundary α = 0 corresponds to w 3 = α = 1 and matrices w M i(w) are smooth functions of w. Let [0, T] be a time interval for which local smooth solutions exist. From CHUEH, CONWAY, SMOLLER (Springer-Verlag (1983)), w 3 = 1 is invariant if and only if dw 3 = (0, 0, 0, 0, 0, 0, 1, 0) is a left eigenvector of M i(v +, v, 1, q) where v ± and q > 0 are arbitrary. In the case when both phases are compressible (0, 0, 0, 0, 0, 0, 1, 0) M i(v +, v, 1, q) = v i (0, 0, 0, 0, 0, 0, 1, 0). When phase + is incompressible, (0, 0, 0, 0, 0, 0, 1, 0) M i(v +, v, 1, q) = (2, 2, 2, 0, 0, 0, v + i, 0).

24 Incompressible - Incompressible Model In the case when the two phases are incompressible (constant densities ρ ± ) α + α + = 1, t(α + ) + div (α + u + ) = 0, t(α ) + div (α u ) = 0, ρ + ( t(α + u + ) + div (α + u + u + )) + α + P + π α + = 0, ρ ( t(α u ) + div (α u u )) + α P + π α = 0, with ρ and ρ + constants and P the Lagrangian multiplier associated with the constraint α + + α = 1.

25 Incompressible - Incompressible Model Hyperbolic with Bestion interfacial pressure: with δ > 1. π = δ α+ α ρ + ρ α + ρ + α ρ + (u+ u ) 2 Remark: The two-layers shallow-water system between rigid lids shares the same form. In this model, π = 0 and a term cst α + appears in the + momentum component. D. BRESCH, M. RENARDY. Well-Posedness of Two-Layer Shallow-Water Flow Between Two Horizontal Rigid Plates. Nonlinearity, 24, , (2011).

26 The model (SW) in Ω = T 2 or R 2 h t + div (hv 1) = 0, Model and Theorem Physical Models for Two Phase Flows h t + div ((1 h)v 2) = 0, (v 1) t + (v 1 )v 1 + ρ 1 ρ h + 1 p ρ = 0, (v 2) t + (v 2 )v 2 + p = 0. Remark. Indices 1 and 2 refer to the bottom and top layer respectively. Density of bottom layer ρ = ρ 1/ρ 2 > 1, the top one equals 1. The depth of the bottom layer is h 1 = h and top h 2 = 1 h. Gravity g is taken equal to 1. Theorem (D. Bresch, M. Renardy, 2011) Let ρ > 1 and s > 2. Assuming that (h 0, v 0 1, v 0 2) (H s ) 5 with 0 < h 0 < 1 are such that v 0 1 v < (ρ 1)(h 0 + ρ(1 h 0))/ρ. (2) is satisfied and, moreover, div (h 0v (1 h 0)v 0 2) = 0. Then, there exists T max > 0, and a unique maximal solution (h, v 1, v 2) C([0, T max); (H s ) 5 ) (and a corresponding pressure p) to the system (SW), which satisfies the initial condition (h, v 1, v 2) t=0 = (h 0, v 0 1, v 0 2).

27 Key points Physical Models for Two Phase Flows First result in the non-irrotational case Main result: Local well-posedness under optimal restrictions on the data by rewriting the system in an appropriate form which fits into the abstract theory of T.J.R. HUGHES, T. KATO and J.E. MARSDEN related to second order quasi-linear hyperbolic systems. Idea: Isolate the essential part, using the total derivative t + V operator with V the weighted average velocity V = (ρ(1 h)v 1 + hv 2)/(ρ(1 h) + h).

28 Single phase low Mach number limit Single phase low Mach number limit

29 Incompressible limit for single phase flows Incompressible limit of single phase barotropic flows can be expressed as: tρ ε + div(ρ ε u ε ) = 0 t(ρ ε u ε ) + div(ρ ε u ε u ε ) + Pε γε 2 = div S ε = µ u ε + (λ + µ) div u ε P = P(ρ) = Aρ γ, ε = U c denotes the Mach number, where c 2 = P (ρ). Formally when ε 0, P ε (and ρ ε ) tends to a constant value P (and ρ = 1), and div ū = 0. Introducing density fluctuations Ψ ε = (ρ ε 1)/ε, and momentum m ε = ρ ε u ε tψ ε + div mε ε tm ε + Ψε ε = 0 = div(u ε m ε ) + div S ε P(ρε ) P(1) P (1)(ρ ε 1) ε 2

30 Incompressible limit for single phase flows In the well prepared case: u ε 0 ū 0 where div ū 0 = 0, u ε converges strongly to ū solution of the incompressible Navier Stokes equations tū + div(ū ū) + π = ν ū, div ū = 0, ū t=0 = ū 0. Refs: S. Klainerman, A. Majda (1982), S. Ukai (1987), S. Schochet (1994), E. Grenier, P.L. Lions, N. Masmoudi ( ). Periodic Boundary Conditions In the ill prepared case u ε ū L ( t/ε) v 0 in L 2 ((0, T) T d ), where v solves the nonlinear acoustic equation where t L(t) denotes the linear wave operator where (ψ, m) = L(t)(ψ 0, m 0) solves tψ + div m = 0 tm + Ψ = 0 (Ψ, m) t=0 = (Ψ 0, m 0). i.e. ( 2 t )Ψ = 0. Rk: t L(t) is an isometry over H s s R.

31 Incompressible limit for single phase flows Whole space Acoustic waves disperse and tend to zero locally in (t, x): u ε ū in L 2 loc(r + R d ), due to Strichartz estimates: L(t/ε)φ L q (R + t ;L p (R d )) Cp,qε1/q φ 0 Ḣs (R d ) where (p, q) (2, ) (2, ) and σ (0, ) such that ( 2 1 = (d 1) q 2 1 ), σq = d + 1 p d 1. Bounded domain Most of acoustic waves are damped near the boundary u ε ū in L 2 ((0, T) R d ), unless there are acoustic modes ψ 0 such that ψ = λψ in Ω, ψ = Cst and nψ = 0 on Ω (3) (Schiffer conjecture: ψ 0 satisfying (3) = Ω is a ball).

32 Ongoing work: the incompressible limit in the two phase framework When one of the two phases becomes incompressible ρ + constant and phase remain compressible Regular limit No acoustic waves Double incompressible limit: sound speeds C + and C are of same order of magnitude. No relevant sound speed for two phase mixtures. Asymptotic behavior of the spectrum of linearized operator.

33 The double incompressible limit The scaled two phase model for the double incompressible limit can be written as α + + α = 1, t(α + ρ + ) + div (α + ρ + u + ) = 0, t(α ρ ) + div (α ρ u ) = 0, t(α + ρ + u + ) + div (α + ρ + u + u + ) + α+ ε 2 P = D + α+ ρ + g + div S +, t(α ρ u ) + div (α ρ u u ) + α ε P 2 = +D + α ρ g + div S, P = P (ρ ) = P +(ρ +),

34 Two phase low Mach limit : reformulation of the system The two phase system can be rewritten in terms of (α +, P, v r, U) variables ( ) tα + +V ρ α + +B ρ div v r + Bρ ρ + (ˆV P γ ρ ρ P+A γp div U) = 0 (4) + γ tp + ˆV γ P + PA γ div U = 0 (5) where V ρ = ρ + u + α + ρ u + α ρ + α + ρ + α, ˆV ρ = α + u + ρ + α u + ρ α + ρ + α + ρ 1, A ρ = α +, B ρ = ρ + α + ρ 1 ρ + α + ρ + α, V γ = γ + u + α + γ u + α γ + α + γ + α, ˆV γ = α + u + γ + α u + γ α + γ + α + γ 1, A γ = α +, B γ = γ + α + γ 1 γ + α + γ + α U = α + u + +α u, v r = ρ + u + ρ u, u r = u + u, and K = α + ρ +α ρ +.,

35 Reformulation of the two phase system and ( ) ρ + u + tv r + V ρ v r + A γ γ ρ u ( div U + A ρu + γ r U A ρu r ur α +) ( ) ρ + u + + γ ρ u (ˆV + γ γ V ρ) P P π( 1 α + 1 ) α + = 0. (6) + α tu + ˆV ρ U + 1 ( ρ + P + Bρur div vr + urbρ A ρε2 ( +B ρu r v r + u r(v ρ ˆV ρ) α + B 2 ρu r u r P ( 1 +π ρ 1 ρ + ( α + u + ρ + α u ρ + γ γ + ) ( α + α + u + ρ + U γ ρ + γ ) ( ρ + 2 P + α u ρ + γ γ + ) P PK ) A γ div U α + γ + + ρ 2 α γ ) U P = 0. (7) PK )

36 First step: uniform local existence Methodology à la Métivier Schochet to prove local existence over an interval [0, T] independent of ε a( tq + v q) + 1 divxu = 0, ε b( tm + v m) + 1 (8) ε ψ = 0 with a(t, x) and b(t, x) known and positive, and q = 1 Q(t, x, εψ), m = µ(t, x)u, v = V(t, x, u, q), (9) ε where Q, µ and V are smooth functions of their arguments with Q(t, x, 0) = 0, θ Q > 0 and µ > 0. Note that q is not singular in ε and satisfies q = Q 1(t, x, εψ)ψ, with Q 1 > 0. (10) a and b depend on functions satisfying a non singular equation (like α + and v r). Theorem T > 0, ε 0 > 0 such that sup t [0,T],ε<ε 0 (u ε (t), ψ ε (t)) H s C T (u ε 0, ψ ε 0 ) H s

37 Second step: analysis of the spectrum Asymptotic behavior of solutions depend on averaging phenomena in variable t/ε, close to what happens in systems like ε ϕ = ω(i) + εg(i, ϕ), ϕ(0) = ϕ İ = f (I, ϕ), I(0) = I (11) Under generic non resonance assumptions, for a.e. initial data (ϕ, I ), I ε converges to I such that İ = 1 f (I, ϕ)dϕ, I(0) = I. T d T d Reduction of (4)(5)(6)(7) to (11) requires constant multiplicity of the spectrum of the linearized operator.

38 Second step: analysis of the spectrum Detailed description of the spectrum of system (4)(5)(6)(7) is needed to complete the analysis. Eigenvalues are the roots of the polynomial in X P(X) = (X λ) 2 (X + λ) 2 K 1(X λ) 2 K 2(X + λ) 2 + K 3 with K 1 = α ρ + α + π/c 2, K 2 = α + ρ α π/c 2, K 3 = π/γ 2, λ = u r/2γ γ 2 = C 2 /(α ρ + + α + ρ ), Questions currently being investigated - Resonances? Eigenvalue crossings? - Local existence in [0, T] uniformly in ε, and convergence to the incompressible - incompressible system. - Wave dispersion / damping depending on domain geometry?