Transcript Simulations of Turbulent Flows with Strong Shocks and

Towards robust and accurate computations of
shock/turbulence interactions
Johan Larsson
Center for Turbulence Research
Stanford University
Queen’s University, Nov 13, 2007
Outline
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Flows of interest and potential applications
The SciDAC program and my work
Numerical challenges in shock/turbulence interaction
The proposed hybrid method
Evaluation and verification of the method
Summary, the next steps, and some ideas for the future
• Note -- I’m assuming some familiarity with:
• Numerical solution of PDEs -- finite difference methods
• Fluid mechanics -- shock waves and turbulence
• Feel free to interrupt!
Problems of interest
• Flows with interactions between turbulence, shock waves, and material
interfaces occur in a wide range of interesting applications
• Super/hypersonic flight, shock/boundary layer interaction, inertial
confinement fusion (ICF), supernovae explosions, scramjet
combustion, shock wave lithotripsy,…
Turbulent mixing of two fluids
with different densities
(Rayleigh-Taylor instability)
E.g. early stages of supernova
explosion, late stages of ICF
Source: Andy Cook, LLNL
Problems of interest
X-43 (Mach 9.6)
Source: www.dfrc.nasa.gov/Gallery
Scramjet
Source: www.tipmagazine.com
Problems of interest
Shock wave passing through a cylinder of heavier gas, generating
vorticity and mixing (Richtmeyer-Meshkov instability)
Source: Andy Cook, LLNL
E.g. scramjet combustion (shock interacting with injected fuel)
Problems of interest
Supernovae explosions -- natural
convection, shock waves,
combustion (fusion)
Source: flash.uchicago.edu
National Geographic, March 2007, on supernovae:
The turbulent infalling gas starts shaking the core, causing it to pulsate… the oscillations
are so intense they send out sound waves. The waves exert a pressure that expels
material, reinforcing the shock wave created by the star's collapse. They also amplify the
core's vibrations in a runaway reaction, says Burrows, "until the star finally explodes."
The SciDAC program
• Dept of Energy ‘Scientific Discovery through Advanced Computing’
• Multi-disciplinary program to advance ‘peta-scale’ computational
science
• Computer science -- MPI, filesystems, visualization, etc
• Applied mathematics -- scalable algorithms, etc
• Science -- climate, quantum mechanics, numerical relativity,
astrophysics, biology, etc
• Our project
• Stanford, NASA Ames, Lawrence Livermore, UCLA
• Initial focus -- numerical methods for shock/turbulence/material
interface interactions
• This talk is about my part of this project
Working roadmap
• Hybrid numerical method
• Existing methods capture shocks well, but sacrifice accuracy in
treating turbulence -- core problem is numerical dissipation
• Proposed method largely eliminates this numerical dissipation
• Verify method on a sequence of problems
• Increase complexity step by step
• Canonical shock/turbulence interaction study
• Unanswered questions of flow physics
• Basic problem for shock/turbulence modeling
• This talk will cover the first 2 items
isotropic turbulence
shock
Introduction -- governing equations
• Navier-Stokes equations for a perfect gas
• Convective terms (LHS) contain amazing range of physics
• Shock waves -- discontinuities in the flow field
• Vortex stretching etc -- energy transfer towards smaller eddies
• Convective terms also pose greatest numerical challenge
• Special ‘shock-capturing’ schemes needed for shocks
• Numerical energy transfer (aliasing errors) often cause blow-up
Introduction -- capturing of shock waves
• Shock thickness is roughly the molecular mean-free-path (1 nm in air)
• Unfeasible to resolve numerically
• Shock-capturing -- get the correct ‘jump’ on a realistic grid
• Need nonlinear (solution-dependent) dissipation
• Need conservative form of convective terms -- proven to give
correct weak solution
• Conservative and non-conservative
forms:
Introduction -- capturing broadband turbulence
• Stability affected by aliasing error:
• The energy is ‘aliased’ to some unphysical wavenumber
• Could lead to catastrophic energy growth and numerical instability
• Linear, ‘dealiasing’ dissipation
• Split form of convective terms -- reduces aliasing error
The Taylor-Green vortex (3D)
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Idealized vortex-stretching for nearly incompressible flow
No shock waves, but large aliasing errors
Note instability of conservative form, stability of split form
Note effect on bandwidth: 1/3, 1/2, 2/3 of maximum wavenumber
Kinetic energy evolution
Energy spectra at t=5
Introduction -- contradictory requirements
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Need conservative form and shock-capturing dissipation for shocks
Conservative form => need dealiasing dissipation for stability
Dissipation harms accuracy -- only 1/3 or 1/2 of wavenumbers accurate
‘Unified’ methods -- same scheme everywhere
• Conservative, both types of dissipation
• E.g. hyperviscosity (Cook, PoF 2007), WENO (Martin et al, JCP
2006)
• ‘Hybrid’ methods -- different schemes around and away from shocks
• Can use split form for ‘turbulence’ => non-dissipative
• Issues of conservation and stability at the interface, where to use
each scheme
• With upwinding (Adams and Shariff, JCP 1996), filtering (Rizzetta
et al, AIAA J 2001), central (Pantano et al, JCP 2007)
Hybrid method -- general approach
• Concept: different numerics for different physics
• Minimal dissipation through split form
• Novelties in present method:
• Conservative coupling for general split schemes
• Stability proof at the interface (JCP, under review)
Numerical grid
Hybrid method -- numerical flux framework
• Conservation at interfaces by numerical flux form
with defined
• Hybridize by
• Have reduced problem of interface conservation to finding
Hybrid method -- WENO scheme
• Adaptively chosen weighted combination of
candidate fluxes
• Weights
chosen based on smoothness
• Candidate stencils and sample weights
• State-of-the-art for shock-capturing, but
expensive and dissipative for
turbulence
Hybrid method -- central split scheme
• Split convective form by Ducros et al (JCP 2000)
• Derive bilinear interpolation stencils for flux form
such that
yields split form by Ducros et al
Hybrid method -- shock sensor
• Must find regions of shock waves robustly
• Many sensors possible, and area in need of improvement
• Currently based on comparing dilatation and vorticity
• Dilatation -- small in turbulence, large negative at shocks
• Vorticity -- large in turbulence, small at shocks
• Then set
Hybrid method -- final details
• Use 8th order central scheme / 5th order WENO scheme
• 4th order Runge-Kutta scheme in time
• 8th order scheme for viscous terms
• Compare results to
• ‘Pure’ WENO -- 7th order WENO everywhere
• Hybrid + 8th order ‘dealiasing’ dissipation of form
• Hybrid with 2nd order central scheme
Test cases -- verification and illustration of method
• 3D Taylor-Green problem
• Verify accuracy and stability for broadband ‘turbulence’
• Illustrate adverse effect of numerical dissipation
• 1D shock/entropy interaction
• Verify shock-capturing and hybrid concept
• 2D shock/vorticity/entropy interaction
• Verify method on idealized interaction with shockwave
• 3D isotropic decaying turbulence with shocklets
• Verify accuracy for compressible turbulence
• Provide inflow condition for full shock/turbulence case
The Shu-Osher shock/entropy interaction in 1D
• Mach 3 shock moves into entropy wave, interaction amplifies entropy
waves and creates acoustic waves
• WENO confined to shock waves
• No numerical noise at interfaces -- evidence of stability
• Hybrid method less dissipative than pure WENO
Entropy profiles
QuickTime™ and a
decompressor
are needed to see this picture.
2D shock/vorticity/entropy interaction at Mach 1.5
• Oblique vorticity and entropy waves interacting with a normal shock
• Amplification of vorticity and kinetic energy by shock
• Compared to linear theory -- 5th order convergence for amplification
ratio (order of the shock-capturing scheme)
Contours of vorticity
Vorticity amplification
Isotropic decaying turbulence
• Initial conditions:
• Large enough to spontaneously generate shocklets (weak shocks)
• Shock-sensor finds these appropriately
• Dilatation flatness much larger than 3 good measure of shocklets
Contours of dilatation
Isotropic decaying turbulence
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Compare on 64^3 grid with filtered DNS on 256^3
Shock-capturing dissipation (pure WENO) overly dissipative
Linear 8th order dissipation same thing, but less severely
Dissipation more important than order of accuracy
Kinetic energy decay
Vorticity decay
Isotropic decaying turbulence
• Accurate bandwidth: 1/4, 1/2, 2/3 of maximum wavenumber
• Similar to the inviscid Taylor-Green vortex
• Note: addition of explicit subgrid-scale model would only make this
worse
Velocity spectra
Dilatation spectra
Summary
• Numerical dissipation has negative effect on the accuracy for
broadband turbulence -- decreases accurate bandwidth by factor of 2
or more
• Would need 8 times more grid points for equivalent accuracy
• Dissipation larger than ideal subgrid-scale model -- should not
evaluate models with dissipative numerics
• Hybrid approach allows for minimal dissipation by use of split form
• Introduces additional complications, but increased accuracy worth
the ‘price’
• Side-benefit of speed -- central scheme 5 times faster than WENO
• Overall robust, accurate, and efficient -- good framework for future
DNS and LES studies
• Order of accuracy less important -- at least for turbulence statistics,
and for these problems
Future work
• Shock sensor
• How to find shocks robustly for more general flows?
• How to parallelize a hybrid method efficiently?
• The expensive WENO scheme is used locally, in ‘random’ portions
of the domain -- load balancing non-trivial
• Canonical shock/turbulence interaction at high Mach and Reynolds
numbers
• How to limit the size of the inflow database?
shock
isotropic turbulence
Some interesting research topics
• Modeling of shock/turbulence interaction in large eddy simulation
• Mathematically unresolved shock waves are no different than
unresolved turbulence -- can/should they be modeled
jointly/analogously?
• Why are split convective terms more nonlinearly robust?
• Two partial explanations exist, neither is complete
• 30 years of numerical evidence -- there must be a reason…
• Shock identification
• Structure of the velocity gradient tensor?
• Jumps in entropy?
Some interesting research topics
• NASA X-43 achieved Mach 9.6 with scramjet engine in 2004
• Better understanding and modeling of shock/turbulence interactions
and induced mixing of fuel and oxidizer needed to design better
scramjets
• Heat load on high-speed vehicles depend on the transition to
turbulence
• Discovery shuttle in 2005 -- ground control could not predict the
transition point (Annu. Rev. Fluid Mech. 2006)
• Experiments on inertial confinement fusion are ongoing
• Mixing induced by shock/turbulence interactions can severely
degrade the fusion process (indeed prevent it completely)
Acknowledgements
• Financial support
• NSERC Postdoctoral Fellowship
• US Department of Energy SciDAC program
• Center for Turbulence Research
• Stimulating discussions
• Many people, including Sanjiva Lele, Parviz Moin, Bertil
Gustafsson, Albert Honein, and Andy Cook