Prediction of Oil Production With Confidence Intervals*

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1 UQ for Turbulent Fluid Mixing James Glimm 1,3 Stony Brook University With thanks to: Wurigen Bo 1, Baolian Cheng 2, Xiangmin Jiao 1,T.Kaman 1, Hyun-Kyung Lim 1, Xaolin Li 1, Roman Samulyak 1,3, David H. Sharp 2, Justin Iwerks 1, Yan Yu 1 1. SUNY at Stony Brook 2. Los Alamos National Laboratory 3. Brookhaven National Laboratory 1

2 Some Definitions Turbulence Active vortices on many length scales Mixing Two or more fluids in (turbulent) flow Hot-cold water; salt-fresh water; oil-water Instabilities generated by acceleration or shear Steady acceleration: Rayleigh-Taylor Impulsive acceleration: Richtmyer-Meshkov Shear: Kelvin-Helmholtz 2

3 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 scales Model scales not resolved Reynolds Averaged Navier-Stokes (RANS) Model all turbulent scales 3

4 LES regime The (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 4

5 Convergence in the LES regime LES regime is inherently stochastic Convergence of probability distribution functions (pdfs) Pdf indexed by space time = Young measure Limit = 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 pdf Simpler: convergence of mean, variance, higher moments Mean, variance etc. generated by variation of fine grid solution over the coarse solution resolution scale 5

6 A mathematical theorem (G-Q Chen, JG; also others?) Incompressible Euler equations Assume Kolmogorov 1941 turbulence bounds Fluctuations in velocity satisfy an integrable power law decay Thus velocity belongs to a Sobolev space Bounds 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 6

7 Conceptual Picture conjectured (compressible or active scalars for incompressible) Euler equation and LES simulation of Navier Stokes equation Is inherently stochastic Solution is a pdf (Young measure) Convergence is convergence of pdfs Consistent with view of DNS and Navier-Stokes as having smooth solutions 7

8 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 models Best of two ideas combined 8

9 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 parameters numerical Schmidt number = physical Schmidt number? A good idea in general, or a lucky accident? 9

10 Geometry preservation, accuracy and high (subgrid) resolution 64x64x64 2D 128x128x128 Accuracy 3D Geometry preservation Subgrid resolution 10

11 Robust geometry and topology functions Robust topological bifurcation Efficient mesh deformation Robust mesh merging 11

12 Subgrid models for turbulence, etc. Typical equations have the form Averaged equations: U + F( U) = εδu t U + F( U) = ε ΔU t FU ( ) FU ( ) FU ( ) FU ( ) + F ( U) SGS F SGS is the subgrid scale model and corrects for grid errors 12

13 Subgrid models for turbulent flow SGS t ( ) ( ) ( ) F U F U F U ( ) ε ( ) SGS U + F U = Δ U + F U SGS ( ) F U ε U turbulent (key modeling step) t ( ) ( ε ε ) turbulent U + F U = + ΔU 13

14 Dynamic Sub Grid Scale (SGS) Models 2Δx No Free Parameters! Coarsening Δx SGS by model formula at 2Δx Close SGS model Missing coefficient determined at 2Δx Transfer coefficient to Δx 2Δx 14

15 Tests of numerical algorithms Verification Does the numerics solve the mathematical equation? Requires proof of convergence Validation Is the physical problem solved? Requires comparison to experiment or observational data Uncertainty Quantification What kind of bounds (error bars) can be established for the combined effect of errors of all types? 15

16 Tests, continued: What to measure? Macro variables Outer boundary of mixing region Meso variables Coherent structures in flow Micro variables Measures of molecular mixing Pdf of species concentration First, second moment of species concentration 16

17 Tests in context of applications (we will know if we are wrong!) Hydro instabilities Rayleigh-Taylor and Richtmyer-Meshkov ICF applications Turbulent mixing and combustion Scram jet applications Mixing in Couette flow Spent nuclear fuel separation Crystal growth Nuclear fuel; porous media applications Diesel spray Droplet size in primary jet beakup 17

18 Rayleigh-Taylor Instable Mixing Light fluid accelerates heavy Across a density contrast interface Overall growth of mixing region Molecular mixing: second moment of concentration h A g = α Agt = = 2 ρ2 ρ1 ρ2 + ρ1 acceleration force f(1 f) θ = f 1 f 18

19 Simulation study of RT alpha for Smeeton-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) 19

20 Simulation-Experiment Comparison Movie Experiment : V. S. Smeeton and D. L. Youngs, Experimental investigation of turbulent mixing by Rayleigh Taylor instability (part 3). AWE Report Number 0 35/87, 1987 Simulation : 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. 20

21 Uncertainty quantification regarding possible long wave length perturbations Reconstruction of long wave length initial perturbations simulated at +/- 100% to allow for uncertainty in reconstruction. Net effect: 10% for alpha. 21

22 Molecular level mixing: second moment of concentration 2 experiments, 3 simulations (one DNS) compared 22

23 3D Simulation for Primary Jet Breakup Work done by W.Bo, Stony Brook University Simulation is performed at NewYorkBlue (A 126Tf BGL/P at Stony Brook/BNL) 23

24 Scram jet: H 2 injection into M = 2.4 cross flow: Mixing Only Black dots are the flame front extracted from the experimental OH-PLIF image. 24

$fraction contour plotted at the midline$

25 Scram jet: Mixing Only H 2 mass fraction contour plotted at the midline plane 25

26 Crystal growth unstable regime Da =

27 Two phase Couette flow in a fluid contactor. Simulation of a small angular sector Used in chemical processingfor separation of spent fuel from nuclear power reactors 27

28 Conclusions Reacting, turbulent, mixing flows require LES solutions Control over numerical mass diffusion (front tracking) Subgrid scale turbulence models Pdf convergence V&V+UQ Testing in realistic examples where truth is known Partial results on all of the above was presented 28

[1] Nielsen [2]. Richardson [3] Baldock [4] 0.22 mm 0.32 mm Richardson Zaki. [5-6] mm [7] 1 mm. [8] [9] 5 mm 50 mm [10] [11] [12] -- 40% 50%

[1] Nielsen [2]. Richardson [3] Baldock [4] 0.22 mm 0.32 mm Richardson Zaki. [5-6] mm [7] 1 mm. [8] [9] 5 mm 50 mm [10] [11] [12] -- 40% 50% 38 2 2016 4 -- 1,2, 100190, 100083 065007 -- 0.25 mm 2.0 mm d 10 = 0.044 mm 640 3 300. Richardson--Zaki,,, O359 A doi 10.6052/1000-0879-15-230 EXPERIMENTAL STUDY OF FLUID-SOLID TWO-PHASE FLOW IN A VERTICAL