Nonlinear evolution of unstable fluid interface

Transcription

1 Nonlinear evolution of unstable fluid interface S.I. Abarzhi Department of Applied Mathematics and Statistics State University of New-York at Stony Brook

2 LIGHT FLUID ACCELERATES HEAVY FLUID misalignment PRESSURE and DENSITY gradients INSTABILITY TURBULENT MIXING Rayleigh-Taylor instability Richtmyer-Meshkov instability sustained acceleration (gravity) impulsive acceleration (shock) thermonuclear flashes on surface of stars; supernova explosion inertial confinement fusion; interaction of laser with matter Basic objective: reliable description of turbulent mixing Fundamental issues: the cascades of energy the dynamics of small-scale structures the dynamics of the large-scale coherent structure Coherent structure an array of bubble and spikes periodic in the plane normal to the direction of acceleration (shock) ¾ Dynamics of 3D and 2D nonlinear structures in RMI ¾ Properties of the 3D-2D dimensional crossover in RMI

3 INTERFACE active regions passive regions Aref 989 small scales large scales intensive vorticity simple advection Abarzhi 996 (RTI) large-scale coherent motion spectral approach scalar fields group theory Group theory coherent structure periodicity group of invariance 7 plane crystallographic symmetry groups G translations in the plane + rotations + reflections The COHERENT STRUCTURE is OBSERVABLE A significant part of the fluid energy is concentrated in the coherent motion a DOMINANT mode K governs macroscopic dynamics The structure is stable under modulations K : ( ) ( K ) ( K ) 2 +ξ ϕ K +ξ ϕ + F ξ K K a scalar macroscopic function G is a symmorphic group with inversion in the plane 3D: p6mm, p4mm, p2mm, cmm, p2 2D: pm

4 LARGE-SCALE COHERENT MOTION * time t potential Φ( x, y, z, t) free surface z ( x, y, t) * z t + z * Φ Φ z Φ = 0, Φ = 0 * z=z = 0 Φ t z= * Φ + * z= z z= z, ( ) g( t) z 0 g 0 - instability: RT g > 0 RM g = 0 = * Initial conditions: z ( x, y, t ) v( x, y, t ) length scale ( ) ~ λ max 0 λ time scale τ ~ λ v0 Symmetry: periodic, symmorphic + inversion in the plane ( x, y) 0 difficulty SINGULARITY interplay of harmonics 2D RTI: Taylor 950, Fermi 952, Layzer 955, Garabedian 957, Birkgoff 957, Zuffiria 986, Inogamov 990, Tanveer 993, Hazak 997 3D RTI: Abarzhi 995 LOCAL EXPANSIONS ASYMPTOTIC SOLUTIONS 2D & 3D RMI: Shvarts 995, Inogamov 995, Mikaelian 998, Zhang 998, Abarzhi 2000 Layzer-type approach single-mode approximation

5 Expansion in terms of orthogonal functions: Φ = n= Φ n γ () t exp γ ( z z () t ) n n 0 + i k jrn j + c. c j irreducible representations of group G wave-vectors k, Gk project operators Fourier expansion Local expansion at a highly symmetric point of the interface x 0, y 0, z z0 * () t : ( x, y, t) = z () t + ζ () t 0 ij i+ j= 2i z x y, N = i + j =,2,... 2 j i+ j= Dynamical system of ordinary differential equations 2i 2 j D ( M 2i 2 j, M, ζ) x y = 0 K ( ζ, M, ζ) x y = 0 ij { } ζ = ; { } ζ ij M n i+ j= ij = Φ M = moments M n = ( km) m m n Local dynamics, any time t; the length scale(s) λ is invariable Multiple harmonics presentation 3D flows with general type of symmetry and 2D flows Desired accuracy, 0, y 0, z z () t, N = i + j =,2,... x 0

6 i+ j= REGULAR ASYMPTOTIC SOLUTIONS 2i 2 j D ( M 2i 2 j, M, ζ) x y = 0 K ( ζ, M, ζ) x y = 0 ij i+ j= regular asymptotic solutions t τ >> Richtmyer-Meshkov bubble: v() t ~ λ t, ζ() t ~ λ ij Layzer-type expansion: regular asymptotic solutions are absent in general case non-linearity is non-local singularities determine the interplay of harmonics At a fixed length scale(s) λ, shape of the regular bubble is free and is parameterised by the principal curvature(s) number of the parameters N p N p symmetry of the 3D (2D) flow 3 2D pm, 3D p4mm, p6mm N = 3D p2mm = 2 p N p 9 to capture the interplay of harmonics 9 to show existence and convergence for solutions in the family 9 to involve all bubbles allowed by symmetry of the flow 9 to choose the physically dominant (i.e. the fastest stable) solution

7 RICHTMYER-MESHKOV bubbles Curvature radius R (radii x y R, ) kr kr cr Velocity v = L( k, R) t surface variables ζ = ζ ( k R) Fourier amplitudes ( k R) t Φ n Φ Φ ~ exp = ϕ n n, ( pn) n n, β β Asymptotic stability v L( k R) t t, () ( ), ~ ζ t ζ k, R n n ~ β = β( kr) for stable solutions Re [] β < 0 t Properties. The physically dominant solution in the family corresponds to a bubble with a flattened surface, kr kr ~ 2. The bubble flattens in time as ( ) 3. For highly symmetric 3D flows: ( ) t τ β β kr3 D ~ t τ, v D ~ 4 k t 4. The local dynamics of 3D highly symmetric flows is universal; 2 2 near-circular contour z * ~ ζ ( x + y ) 5. 3D anisotropic bubbles tend to conserve a near-circular contour 6. 3D anisotropic bubbles are unstable 7. 3D Layzer-type square solution is the point of bifurcation 8. The dimensional crossover is discontinuous, 0 9. NO 2D flows β 3 3 D 2 D >

8 Family of regular asymptotic solutions in RMI v kt N=,2 2D 3D h 3D s /(kr) Velocity v as the function on the radius of curvature R Three-dimensional flows with hexagonal (3D h ) and square (3D s ) symmetry and two-dimensional flow (2D); k is the wave-vector, t is time, N is order of approximation. Black circles mark the Layzer-type solutions with R L = 4 k, v L = kt in 3D and R L = 3 k, v L 2 3kt = in 2D.

9 Family of regular asymptotic solutions in RMI lg[ Φ m / Φ max ] /(kr) Φ Φ 3 Φ2 N=2 3D h Exponential decay of the Fourier-amplitudes with an increase in their number. Three-dimensional flows with hexagonal symmetry p6mm (3D h ); Φ max bubble. = Φ ( kr ) ; black circle corresponds to the Layzer-type

10 Family of regular asymptotic solutions in RMI Re[β] N=,2 3D h /(kr) Stability analysis for the family of regular asymptotic solutions Real parts of exponents β as functions on the radius of curvature R Dashed lines correspond to N=, solid lines to N=2, black circle corresponds to the Layzer-type solution.

11 Evolution of the bubble front in RMI Highly symmetric 3D and 2D coherent structures time scale t << τ : curvature ζ () t ~ k t τ t : curvature () t k ~ τ τ ~ k v 0 v t v0 ~ v0 t, velocity () τ ζ ~ t >> τ: curvature ζ () t ~ k ( t τ), velocity v() t ~ C k t β Dynamic trajectories v / v 0 ζ /k Solid line corresponds to multiple harmonic solution, and black square - to the flattened bubble. Dashed line corresponds to Layzer-type single-mode solution, and black circle to the Layzertype bubble.

12 Evolution of the bubble front in RMI RM bubbles flatten RM bubbles decelerate Qualitative agreement with experiments Bubble velocity ζ v ~ C kt v L ~ C L kt C ~ 3 4 h ~ C ln( t τ) CL t ~ k β Bubble shape () ( ) t τ reliable parameter Existence of an exact analytical solution a rigid body curvature ~ R drag force = 2 R 2 ρv For a two-fluid system, Atwood number < the Layzer-type approach requires MASS FLUX through the interface Flattened RM bubble is a multiple-harmonic solution with NO MASS FLUX through the interface

13 Family of regular asymptotic solutions in RMI v k x t 3D s 3Dr 2D /(R x k x ) Dependence of velocity v L( R k ) t = on the bubble shape x, y, x, y Low-symmetric bubbles with rectangular symmetry 3D r, twoparameter family; various values of the aspect ratio; the highest curve 3D s is the family of solutions for 3D square bubbles with R = and k = y R x x k y ; the lowest curve 2D is the family of solutions for 2D bubbles flat in the y-direction with R y

14 Family of regular asymptotic solutions in RMI m x(y) /(R x k x ) m s m y m x Bifurcation of the Layzer-type square solution (black point) for nearly symmetric flows with k x ~ k y and R x ~ Ry

15 Family of regular asymptotic solutions in RMI Re[β] /kxrx N= 3Dr Stability analysis for low-symmetric RM bubbles Dashing lines corresponds to highly symmetric 3D square solutions with k x = k y and R x = Ry nearly symmetric solutions with. Nonsymmetric solutions are unstable.. Solid lines correspond to k x ~ k y and R x ~ Ry

16 Evolution of the bubble front in RMI Low-symmetric 3D coherent structures The dimensional 3D-2D crossover 2D bubbles under 3D modulations << τ time scale t ζ () τ >> τ ζ τ ~ v 0 kx x t ~ kx t ζy ~ k y t τ τ β t () ( ) t k t ( t ) 3D D x ~ x ζ ~ k x τ β y 2 the dimensional crossover is discontinuous, 0 β 3 D 2 D > Secondary instabilities Secondary instabilities in RMI are slow in contrast to RTI SINGULAR ASYMPTOTIC SOLUTIONS Richtmyer-Meshkov spikes small-scale structure dynamics Singular asymptotic solutions to dynamical system Zhang 998, Abarzhi ( ) z n, velocity v() t ~ v0 * t >> τ: shape () t ~ C exp( t τ) r n= Tanveer 993, Baker and Meiron 989, Pullin 200 For a two-fluid system, Atwood number <, the singular asymptotic solutions requires mass flux through the interface n

17 Conclusion 9 Large-scale coherent motion in RMI 9 Separation of scales active regions passive regions 9 Group symmetry large-scale coherent motion 9 Local dynamics of regular bubbles and singular spikes 9 Consideration of 3D flows with general type of symmetry 9 Singularity interplay of harmonics shape of the bubble 9 Family of regular asymptotic solutions symmetry of the flow 9 The physically dominant solution in the family 9 Multiple harmonic solution 9 Universality of local dynamics for 3D highly symmetric flows 9 Conservation of a near-circular contour of 3D bubbles 9 Discontinuous 3D-2D dimensional crossover 9 Singular asymptotes 9 Comparison between the local dynamics in RTI and RMI 9 Different types of the bubble front evolution in RTI and RMI 9 Layzer-type bubbles in RTI and RMI 9 New type of the evolution of the bubble front in RMI 9 Integral (velocity) and internal (shape) diagnostic parameters 9 Theory works effectively for a two-fluid system Discussion??? turbulent mixing in RMI and RTI