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UQ for Turbulent Fluid Mixing

James Glimm1,3

Stony Brook University

With thanks to:

Wurigen Bo1, Baolian Cheng2, Xiangmin Jiao1, T. Kaman1, Hyun-KyungLim1, Xaolin Li1, Roman Samulyak1,3, David H. Sharp2, Justin Iwerks1,Yan Yu1

1. SUNY at Stony Brook2. Los Alamos National Laboratory3. Brookhaven National Laboratory

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Some DefinitionsTurbulence

Active vortices on many length scales

MixingTwo or more fluids in (turbulent) flow

Hot-cold water; salt-fresh water; oil-water

Instabilities generated by acceleration or shearSteady acceleration: Rayleigh-TaylorImpulsive acceleration: Richtmyer-MeshkovShear: Kelvin-Helmholtz

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Multiscale, so a challenge for computation. This lecture: LES

Direct Numerical Simulation (DNS)Resolve all important scales

Large Eddy Simulation (LES)Resolve the large but not the small turbulent scalesModel scales not resolved

Reynolds Averaged Navier-Stokes (RANS)Model all turbulent scales

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LES regimeThe (laminar) viscosity and mass diffusion are small

Equations are Euler like

But the effects of the unresolved terms are diffusive, generating turbulent viscosity and turbulent mass diffusion, which are not small

Solutions need to be Navier-Stokes like

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Convergence in the LES regimeLES regime is inherently stochastic

Convergence of probability distribution functions (pdfs) Pdf indexed by space time = Young measureLimit = Pdf solution (Young measure) of Euler equation

Verification: fine grid agrees approximately with course grid?

Fine grid pdf should be sampled, not averaged, to agree (approximately) with coarse grid pdfSimpler: convergence of mean, variance, higher moments

Mean, variance etc. generated by variation of fine grid solution over the coarse solution resolution scale

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A mathematical theorem (G-Q Chen, JG; also others?)

Incompressible Euler equationsAssume Kolmogorov 1941 turbulence bounds

Fluctuations in velocity satisfy an integrable power law decayThus velocity belongs to a Sobolev spaceBounds and convergence (through a subsequence) to a classical weak solution follows

Couple to passive scalar (mixing)Volume fractions are bounded, thus w* convergence (subsequences), as a pdf to a pdf limit, i.e. a Young measure solution of the concentration equation coupled incompressible Euler solution

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Conceptual Picture conjectured(compressible or active scalars for incompressible)

Euler equation andLES simulation of Navier Stokes equation

Is inherently stochasticSolution is a pdf (Young measure)Convergence is convergence of pdfsConsistent with view of DNS and Navier-Stokes as having smooth solutions

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Numerical algorithm for LES turbulent mixing: Front Tracking + turbulence models

Capturing: Computation with steep gradients, rapid time scales

Tracking is an extreme version of this idea

Turbulence models: Good subgrid modelsBest of two ideas combined

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Central Features of Combined Algorithm

Tracking allows very steep gradient at a numerical level

Subgrid scale (SGS) models ensure proper choice of turbulent transport

Dynamic SGS models have no free parametersnumerical Schmidt number = physical Schmidt number?A good idea in general, or a lucky accident?

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Geometry preservation, accuracy and high (subgrid) resolution

Geometry preservation Subgrid resolution

Accuracy

64x64x64

128x128x128

2D

3D

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Robust geometry and topology functions

Robust topological bifurcation

Efficient mesh deformation

Robust mesh merging

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Subgrid models for turbulence, etc.

Typical equations have the formAveraged equations:

( )tU F U Uε+∇ = Δ

( )

( ) ( )

( ) ( ) ( )

t

SGS

U F U U

F U F U

F U F U F U

ε+∇ = Δ

≠

≈ +

is the subgrid scale model and corrects for grid errorsSGSF

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Subgrid models for turbulent flow

( ) ( ) ( )

( ) ( )( )

( ) ( )turbulent

turbulent

(key modeling step)

SGS

t SGS

SGS

t

F U F U F U

U F U U F U

F U U

U F U U

ε

ε

ε ε

≡ −

+∇ = Δ +∇

∇

+∇ = + Δ

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Dynamic Sub Grid Scale (SGS) ModelsNo Free Parameters!

xΔ2

Missing coefficient determined at xΔ2

xΔ

Transfer coefficient to  xΔ

Coarsening

SGS by model formula at xΔ2 Close SGS

model

xΔ2

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Tests of numerical algorithmsVerification

Does the numerics solve the mathematical equation? Requires proof of convergence

ValidationIs the physical problem solved?

Requires comparison to experiment or observational data

Uncertainty QuantificationWhat kind of bounds (error bars) can be established for the combined effect of errors of all types?

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Tests, continued: What to measure?

Macro variablesOuter boundary of mixing region

Meso variablesCoherent structures in flow

Micro variablesMeasures of molecular mixing

Pdf of species concentrationFirst, second moment of species concentration

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Tests in context of applications(we will know if we are wrong!)

Hydro instabilitiesRayleigh-Taylor and Richtmyer-Meshkov

ICF applications

Turbulent mixing and combustionScram jet applications

Mixing in Couette flowSpent nuclear fuel separation

Crystal growthNuclear fuel; porous media applications

Diesel sprayDroplet size in primary jet beakup

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Rayleigh-Taylor Instable MixingLight fluid accelerates heavy

Across a density contrast interface

Overall growth of mixing regionMolecular mixing: second moment of concentration

2

2 1

2 1

acceleration force

(1 )1

h Agt

A

g

f ff f

αρ ρρ ρ

θ

=−

=+

=

⟨ − ⟩=⟨ ⟩⟨ − ⟩

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Simulation study of RT alpha forSmeeton-Youngs experiment #112

Agreement with experiment (validation)Agreement under mesh refinement (verification)Agreement under statistical refinement (verification)Agreement for Andrews-Mueschke-Schilling alpha (code comparison; different experiment)Agreement within error bounds established for uncertain initial conditions (uncertainty quantification)

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Experiment : V. S. Smeeton and D. L. Youngs,  Experimental investigation of turbulent mixing byRayleigh‐Taylor instability (part 3). AWE Report Number 0 35/87, 1987Simulation : H. Lim, J. Iwerks, J. Glimm, and D. H. Sharp,  Nonideal Rayleigh‐Taylor Mixing

The simulations reported here were performed on New York Blue, the BG/L computer operated jointly by Stony Brook University and BNL.

Simulation-ExperimentComparison Movie

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Uncertainty quantification regarding possible long wave length perturbationsReconstruction oflong wave lengthinitial perturbationssimulated at +/- 100% to allowfor uncertaintyin reconstruction.Net effect: 10%for alpha.

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Molecular level mixing: second moment of concentration 2 experiments, 3 simulations (one DNS) compared

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3D Simulation for Primary Jet Breakup

Work done by W.Bo, Stony Brook UniversitySimulation is performed at NewYorkBlue (A 126Tf BGL/P at Stony Brook/BNL)

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Scram jet: H2 injection into M = 2.4 cross flow: Mixing Only

Black dots are the flame frontextracted from the experimentalOH-PLIF image.

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Scram jet: Mixing Only

H2 mass fraction contour plotted at the midline plane

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Crystal growth unstable regime Da = 285

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Two phase Couette flow in a fluid contactor.

Simulation of a smallangular sector

Used in chemical processingfor separation of spentfuel from nuclear powerreactors

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ConclusionsReacting, turbulent, mixing flows require

LES solutionsControl over numerical mass diffusion (front tracking)Subgrid scale turbulence modelsPdf convergenceV&V+UQTesting in realistic examples where “truth” is known

Partial results on all of the above was presented