Spectral and High-Order Methods Spectral and High-Order Methods for - - PowerPoint PPT Presentation

Spectral and High-Order Methods for Shock-Induced Mixing Spectral and High-Order Methods for Shock-Induced Mixing

Andrew W. Cook William Cabot Jeffrey A. Greenough Stephen V. Weber

This work was performed under the auspices of the U.S. Department of Energy by the University of California, Lawrence Livermore National Laboratory under Contract No. W-7405-Eng-48

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High-resolution numerical simulation data are needed to augment experimental data High-resolution numerical simulation data are needed to augment experimental data

• Data from high-energy shock experiments are difficult to obtain

and are usually in the form of integrated quantities.

• Direct numerical simulation (DNS) and large-eddy simulation

(LES) can be used to obtain highly detailed data from turbulent flows (e.g., velocity, correlations, pdf’s, energy budgets) which cannot be obtained from experiments.

• Our goal is to develop the capability to perform accurate, high-

resolution hydrodynamic simulations for shock-induced mixing problems.

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Which numerical scheme is best suited to perform high- resolution simulations of shock-induced turbulent flows? Which numerical scheme is best suited to perform high- resolution simulations of shock-induced turbulent flows? ! In astrophysical and Inertial Confinement Fusion (ICF) applications, shocks deposit vorticity at material interfaces, which subsequently evolve into turbulent mixing regions. ! Shocks and interfaces require robust numerical schemes which are typically monotonic and of low order. ! Accurate simulations of turbulent mixing requires high-resolution numerical methods to capture the large range of scales participating in the flow dynamics. Can spectral/compact methods be made sufficiently robust to handle shocks, while maintaining their high resolution properties for turbulent mixing?

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Two test problems were used for intercode comparisons Two test problems were used for intercode comparisons

• 2D Richtmyer-Meshkov instability (RMI)
• vorticity deposited by a shock on a thin

material interface

• 3D Taylor-Green vortex (TGV)
• vortex stretching; cascade to small scales
• similarities to turbulence

(Collins & Jacobs 1999) (The University of Edinburgh)

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Intercode comparisons were performed with five different hydrodynamic codes/schemes Intercode comparisons were performed with five different hydrodynamic codes/schemes

• Miranda - S/C: spectral/high-order compact scheme with

modifications: ♦ high-order filtering removes high frequencies globally for stabilization (but leaves ringing) ♦ artificial viscosity/diffusivity with high-order switches smooths shocks and interfaces locally (removes ringing)

• Raptor - HOG: High-order Godunov scheme (formally 2nd order) in

AMR framework

• HYDRA - ALE: Arbitrary Lagrangian-Eulerian scheme (2nd order)

with non-standard diffuse interface treatment

• Miranda - CENO: Kurganov & Tadmor central finite difference ENO

scheme (2nd order) (for RMI)

• WENO (Don, Gottlieb &

Shu): weighted ENO scheme (5th-order) (for TGV)

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Global filtering can degrade the resolution of the solution and generate additional ringing (Gibbs oscillations) Global filtering can degrade the resolution of the solution and generate additional ringing (Gibbs oscillations)

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Local artificial diffusivity/viscosity is used in Miranda to remove ringing (Gibbs oscillations) Local artificial diffusivity/viscosity is used in Miranda to remove ringing (Gibbs oscillations)

Shu-Osher 1D shock/density wave case

Compact scheme with global filtering (F) has ringing, which local artificial dissipation (AD) removes

0.5 1 1.5 2 2.5 3 3.5 4 4.5 5

• 2
• 1

1 2 3 4 density x Raptor: Godunov (3200 pts, converged) Miranda: 8th-order compact F (256 pts) Miranda: 8th-order compact AD+F (256 pts)

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Local Artificial Diffusion Terms in Miranda Local Artificial Diffusion Terms in Miranda

• Artificial transport coefficients are added to the molecular transport

coefficients in the Navier-Stokes equations for a multicomponent mixture of ideal gases:

µ ρ µ ρ

µ * * *

/ / / / / / / k C D k C D C x u c C x T T C x Y x t

p i p i s k D i n

          =           + ∇ ∇ ∇               ∆ ∆ ∆ ∆ ∆

2 2 2 2 2 2 2

Momentum Internal Energy Mass Fraction

• High-order switch localizes diffusion terms to regions with ringing

high-order switch

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Experimental Apparatus

• Slightly diffuse, membrane-

less interface is created with a stagnation flow

• Single, long-wavelength

perturbation is generated by a stepper motor

• M=1.1, 1.2, 1.3 shots
• PLIF laser diagnostics

Shock tube configuration

Pressure Transducers Stepper Motor SF

6

Driver Pivot Diaphragm Slot Test Section Interface Air + Acetone Vapor Lenses Laser Mirror

A A A A A A A A A A A A A A A A A A A A A A

Air

6

SF

Jacobs (U of Arizona) shock tube apparatus uses no membrane and produces high-quality laser diagnostics Jacobs (U of Arizona) shock tube apparatus uses no membrane and produces high-quality laser diagnostics

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Planar Laser Induced Fluorescence (PLIF) image from Collins-Jacobs shock tube experiment (2D) Planar Laser Induced Fluorescence (PLIF) image from Collins-Jacobs shock tube experiment (2D)

• Richtmyer-Meshkov

instability

• M=1.2 shock
• air-acetone (light) →

→ → → SF6 (heavy)

• single sinusoidal

perturbation

• diffuse interface (no

membrane) Air-acetone concentration (Collins & Jacobs 1999)

air + acetone vapor SF6

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Simulation setup for Collins-Jacobs 2D test case used by all codes/methods Simulation setup for Collins-Jacobs 2D test case used by all codes/methods

• Local domain near interface
• Periodic transverse boundary conditions
• Two perfect gases (Atwood number = 0.6)

♦ air+acetone vapor (γ =1.27) ♦ sulfur hexafluoride (γ =1.09)

• M=1.186 to match displacement speed
• Initial interface (ka0 = 0.2)

♦ thickness δ0 = 5 mm ♦ amplitude a0 = 2.2 mm ♦ wavelength λ0 = 59 mm

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N=128 N=256 N=512 CENO Godunov S/C

Density t = 6 ms

Results from all simulations show similar large-scale growth, but different small-scale phenomena Results from all simulations show similar large-scale growth, but different small-scale phenomena

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Large-scale structure and amplitude growth are insensitive to the numerical scheme and resolution Large-scale structure and amplitude growth are insensitive to the numerical scheme and resolution

• 5

10 15 20 25 30

• 1

1 2 3 4 5 6 7 amplitude [mm] time [ms]

• experiment

S/C 512 Godunov 512 ALE 512 CENO 512

M=1.2 single-mode Richtmyer-Meshkov instability

Jacobs-Collins Shock Tube: Amplitude

expansion wave reflected shock

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N=128 N=256 N=512 ALE S/C

Density at t = 6 ms

Greater differences are evident between numerical schemes at low resolution than at high resolution Greater differences are evident between numerical schemes at low resolution than at high resolution

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Fine-scale features of the vortex cores are very sensitive to the numerical scheme and resolution of the thin interface Fine-scale features of the vortex cores are very sensitive to the numerical scheme and resolution of the thin interface

S/C ALE HOG CENO Density N = 512 t = 6 ms

The S/C interface with artificial diffusion is about 6 points thick, retarding vortex breakdown and allowing more rollup.

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5 10 15 20 25 30

• 1

1 2 3 4 5 6 7 maximum vorticity (kHz) time (ms) S/C 128 S/C 256 S/C 512 5 10 15 20 25 30

• 1

1 2 3 4 5 6 7 time (ms) S/C 512 CENO 512 Godunov 512

The maximum vorticity in the interface/vortex core is sensitive to the effective resolution The maximum vorticity in the interface/vortex core is sensitive to the effective resolution Jacobs-Collins M=1.2 Shock Tube: Vorticity

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Despite intercode agreement for large-scale features, there is lack of agreement between simulation and experiment Despite intercode agreement for large-scale features, there is lack of agreement between simulation and experiment

Experiment Simulation

Differences in the setup of initial, boundary, and/or physical conditions between simulation and experiment may account for this.

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Features neglected in the simulations that may lead to the observed discrepancies: Features neglected in the simulations that may lead to the observed discrepancies:

• Boundary conditions

❏ side slots ❏ no-slip walls ❏ reflected expansion from top wall

• Initial conditions

❏ subharmonics in forcing ❏ non-uniform interface thickness

• Physical conditions

❏ gravity ❏ materials

❏3D

❏ DNS resolution

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New tests use no global filtering to improve resolving power New tests use no global filtering to improve resolving power

• High-order simulations of the previous 2D RMI case used both

local artificial dissipation and global filtering.

• There appeared to be no great benefit in using high-order

schemes for the 2D RMI case rather than some lower-order schemes.

• More recent tests are being conducted without global filtering

to reduce ringing and degradation of resolution, using only local artificial dissipation with an exponential switch.

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Shock passing through density wave Compact scheme with no global filtering gives most accurate result in 1D test case Compact scheme with no global filtering gives most accurate result in 1D test case

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3D Test Case: Taylor-Green Vortex (inviscid, mildly compressible) 3D Test Case: Taylor-Green Vortex (inviscid, mildly compressible) Initial conditions on 2π π π π3 domain:

ρ = 1 u x y z = sin( )cos( )cos( )

v x y z = −cos( )sin( )cos( ) w = 0 p z x y = + +

[ ]

+

[ ] −

{ }

100 16 2 2 2 2 2 ( / ) cos( ) cos( ) cos( ) ρ

Features

• Vortex stretching, cascade to small scales (similar to turbulence)
• Exact incompressible solution for t < 3.5
• Inviscid solution becomes singular at t ≈ 5
• For 643 grids, small scales become unresolved for t ≥ 3

(The University of Edinburgh)

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Compact scheme with no global filtering gives the most accurate results in 3D TGV case Compact scheme with no global filtering gives the most accurate results in 3D TGV case

1 2 3

time

0.9 0.92 0.94 0.96 0.98 1 1.02

Kinetic Energy

Taylor−Green Vortex

64 x 64 x 64 grid points Exact 10th−order compact (MIRANDA) 2nd−order Godunov (RAPTOR) 5th−order WENO (Brown) 1st/2nd−order ALE (HYDRA) 1 2 3

time

1 2 3

Enstrophy

Taylor−Green Vortex

64 x 64 x 64 grid points Theory 10th−order compact (MIRANDA) 2nd−order Godunov (RAPTOR) 5th−order WENO (Brown) 1st/2nd−order ALE (HYDRA)

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Conclusions Conclusions

• Spectral/compact methods can be made sufficiently

robust to capture shocks and contact discontinuities.

• The benefits of higher-order differencing are readily

apparent in certain cases (Shu, TGV), but less so in cases dominated solely by shocks and interfaces (2D RMI).

• Shear flow cases and fully turbulent test cases (late-time

RMI, reshock) still need to be examined to assess whether high-order or low-order methods are more effective for shock-induced turbulence.