Direct Numerical Simulation of Rayleigh-Taylor ... - ASC at Livermore

Direct Numerical Simulation ofRayleigh-Taylor InstabilityBill Cabot,Andy Cook&Paul MillerThis work was performed under the auspices of the US Department of Energy by the University of California,Lawrence Livermore National Laboratory under Contract No. W-7405-ENG48.UCRL-PRES-222608CS and Viz Support: Mike Welcome (LBL), Brian Miller,Peter Williams, Hank ChildsIWPCTM10, Paris17–21 July 2006WC-IWPCTM10

Rayleigh-Taylor instability is an unsettled issue.• Does the flow forget its initial conditions?• Is the flow self-similar?• What is alpha?• How does mixing influence the growth rate?• When does the flow become fully turbulent?• What is the Reynolds number dependence?• How should subgrid-scale models beinitialized?Results from high-resolution numerical simulations can beused to elucidate these issues and guide modeling.WC-IWPCTM10-2

The Navier-Stokes equations have been aroundfor 185 years…∂ρ∂t + u j∂ρ= ρ ∂∂x j∂x j⎛⎜⎝D ∂ρ ⎞ρ ∂x⎟ =−ρ ∂u j,j ⎠ ∂x j∂ρu i∂t+ ∂ρu iu j∂x j=− ∂p∂x j+ ∂τ ij∂x j+ ρg i, g i= 0,0,−g ( )⎡ ⎛ ∂uτ ij= ρν i+ ∂u j⎢ ⎜⎣ ⎢ ⎝ ∂x j∂x i⎞⎟ − 2 ⎠ 3 δ ij∂u k∂x k⎤⎥⎥ ⎦…nevertheless, fluid mechanics is not a dry subject.WC-IWPCTM10-3

The governing equations are solved withhigh-resolution numerical methods.‣ 10 th -order compact/Padé spatial derivatives‣ 3 rd -order predictor-corrector timestepping‣ Direct Fourier/Padé Poisson solver‣ 8 th -order dealiasing filter‣ Vertically expandinggrid is matched to apotential-flow solutionin the far field.The Crab Nebula is RT unstable.WC-IWPCTM10-4

Paris is the birthplace of the Navier-Stokesequations as well as some key solution methods.Claude-Louis NavierGeorge StokesSiméon-Denis PoissonWho’snotFrench?Joseph FourierHenri PadéWC-IWPCTM10-5

Setup for Rayleigh-Taylor DNS:‣ 3:1 density ratio (Atwood number = 0.5)‣ Schmidt number = ν/D = 1‣ Grid spacing Δ ≈ η (Kolmogorov scale)‣ 3072 x 3072 x 3072 grid points‣ Error function for diffuse initial interface (5 grid pointsthick) with horizontally perturbed position‣ Periodic side boundaries‣ Potential-flow vertical boundaries at early times‣ Free-slip walls at top and bottom boundariesViscous and diffusion scales are resolved.WC-IWPCTM10-6

Direct Numerical Simulation serves as alow-Re “numerical experiment”• DNS resolves all relevant scales in turbulentflow (inertial, dissipation, and diffusion);there are no model approximations.• Turbulence is inherently three-dimensional.• DNS is limited to low to moderate Reynoldsnumbers, constrained by computer resources.• To reach a fully turbulent state (e.g., mixing transition), the outerscaleReynolds number must exceed Re > 10 4 (microscale Re λ> 10 2 ).• The range of scales (∝ number of grid points Ν) in any direction isΛ/η ∼ Re 3/4 ; hence the cost of DNS ∼ Ν 4 ∼ Re 3 (assuming perfectparallel scaling).• DNS needs large resources to reach higher Re.BG/L is the fastest computer in the world.WC-IWPCTM10-7

There is not much pure heavy fluid in the spikes.WC-IWPCTM10-8

There is not much pure light fluid in the bubbles.WC-IWPCTM10-9

Gravity and initial perturbations setcharacteristic length and time scales.Dominant wavelength:l = 2π∞∫0∞∫0E(k)kdkE(k)dk, l 0= l(t = 0)τ = l 1/2⎛ ⎞0Corresponding timescale: ⎜ ⎟ A = ρ 2− ρ 1,=1/2⎝ Ag⎠ρ 2+ ρ 1Initial spectrum peaked at mode 96.WC-IWPCTM10-10

Outer-scale Reynolds number reaches 32,000.Re=H H&/ vH is based on the 1%concentrationthreshold.Previous DNS at512 x 512 x 2040ended here.Re ~ 10 4 (Re λ~ 10 2 )marks the beginningof the turbulentregime and theformation of aninertial range in theenergy spectrum.Re crosses 10,000 around t/τ = 19.WC-IWPCTM10-11

Spectra develop scale separation andinertial ranges at late time.Separation > 10 for t/τ > 19. Kolmogorov spectrum for w.WC-IWPCTM10-12

Peak of energy spectrum follows a k -2 trajectory once theflow becomes self-similar. (see Olivier Poujade’s talk)WC-IWPCTM10-13

Growth and mixing are characterized in termsof a product function X p .Heavy-fluid mole fraction:X = ρ − ρ 1ρ 2− ρ 1X st=12Product (mixed fluid):X p( X) ={X / Xstif X ≤ Xst(1 − X ) /(1 − Xst) if X > XstProduct thickness:∞∫h = X p ( X )dz−∞Ξ=Mixedness:∞∫ X p−∞∞X p∫−∞( X) dz( X )dzIntegral measure of mix height is insensitiveto statistical fluctuations.WC-IWPCTM10-14

In the self-similar regime, (dh/dt) 2 = 4αAgh(Ristorcelli & Clark, JFM 2004 and Jacobs & Dalziel, JFM 2005).• h orelates to thespectrum of initialperturbations.• The linear term nevercompletely goesaway.• Using h/Agt 2 , largersimulations, run tolater times, givesmaller α.• There is a slight butsteady increase forRe>10,000.h = αAgt 2 + 2(αAgh o ) 1/2 t + h oWC-IWPCTM10-15

Many models assume a constant ratio of kineticenergy to released potential energy.Potential energy δPis converted tokinetic energy K,which cascadesdown to smallscales where it isremoved byheat dissipation Ψ.Alpha Group derivedK z/δP = 12αK/δP rises steadily for Re > 10,000.WC-IWPCTM10-16

The mixing rate lags the entrainment rate whenthe flow enters the turbulent regime.Ξ=∞∫−∞∞∫−∞X pX p( X) dz( X )dz1 homogenized0 segregatedEver bigger blobs have to be broken down to ever finer scales.WC-IWPCTM10-17

Surface area exhibits weak Re dependence forRe>10,000.Area of equimolar surface scales with Taylor microscale.WC-IWPCTM10-18

Self-similarity gives: λ∼h 1/4 ∼t 1/2 , η∼h -1/8 ∼t -1/4(Ristorcelli & Clark, JFM 2004).Taylor microscales (λ i )stay anisotropic.Kolmogorov microscales(η i ) become isotropic bylate time.DNS confirms moment similarity predictions except for λ z .WC-IWPCTM10-19

Enstrophy becomes isotropic near midplaneonly at very late time.isotropicanisotropicFlow near bubble and spike fronts is always highly anisotropic.WC-IWPCTM10-20

Similar trends were observed in a previous1152 3 LES (Cook, Cabot & Miller, JFM 2004).A = 0.53072 3 DNS1152 3 LESLES results suggestthat some quantitiesmay asymptote atlater times.WC-IWPCTM10-21

Shallower slope of density spectrum isobserved in both DNS and LES.3072 3 DNS, 1152 3 LESWC-IWPCTM10-22

DNS of R-T instability at Re up to 32,000yields some surprises.• α cannot be determined by plotting h vs. Agt 2 .• Kinetic/Potential energy ratio keeps rising.• Taylor microscales are always anisotropic.• Kolmogorov scales and enstrophy eventually become isotropic.• Flow is weakly Reynolds number dependent for Re>10,000.• How should growth and mixing curves be extrapolated to veryhigh Reynolds number regimes?Extremely large simulations are requiredto escape initial, boundary, and low-Re effectsand obtain good statistical samples.WC-IWPCTM10-23