Research Papers

Learning an Eddy Viscosity Model Using Shrinkage and Bayesian Calibration: A Jet-in-Crossflow Case Study

[+] Author and Article Information
Jaideep Ray

Sandia National Laboratories,
MS 9159, P.O. Box 969,
Livermore, CA 94550
e-mail: jairay@sandia.gov

Sophia Lefantzi

Sandia National Laboratories,
MS 9152, P.O. Box 969,
Livermore, CA 94550
e-mail: slefant@sandia.gov

Srinivasan Arunajatesan

Sandia National Laboratories,
MS 0825, P.O. Box 5800,
Albuquerque, NM 87185
e-mail: sarunaj@sandia.gov

Lawrence Dechant

Sandia National Laboratories,
MS 0825, P.O. Box 5800,
Albuquerque, NM 87185
e-mail: ljdecha@sandia.gov

1Corresponding author.

Manuscript received July 7, 2016; final manuscript received August 2, 2017; published online September 7, 2017. Assoc. Editor: Yan Wang. The United States Government retains, and by accepting the article for publication, the publisher acknowledges that the United States Government retains, a nonexclusive, paid-up, irrevocable, worldwide license to publish or reproduce the published form of this work, or allow others to do so, for United States Government purposes.

ASME J. Risk Uncertainty Part B 4(1), 011001 (Sep 07, 2017) (10 pages) Paper No: RISK-16-1100; doi: 10.1115/1.4037557 History: Received July 07, 2016; Revised August 02, 2017

We demonstrate a statistical procedure for learning a high-order eddy viscosity model (EVM) from experimental data and using it to improve the predictive skill of a Reynolds-averaged Navier–Stokes (RANS) simulator. The method is tested in a three-dimensional (3D), transonic jet-in-crossflow (JIC) configuration. The process starts with a cubic eddy viscosity model (CEVM) developed for incompressible flows. It is fitted to limited experimental JIC data using shrinkage regression. The shrinkage process removes all the terms from the model, except an intercept, a linear term, and a quadratic one involving the square of the vorticity. The shrunk eddy viscosity model is implemented in an RANS simulator and calibrated, using vorticity measurements, to infer three parameters. The calibration is Bayesian and is solved using a Markov chain Monte Carlo (MCMC) method. A 3D probability density distribution for the inferred parameters is constructed, thus quantifying the uncertainty in the estimate. The phenomenal cost of using a 3D flow simulator inside an MCMC loop is mitigated by using surrogate models (“curve-fits”). A support vector machine classifier (SVMC) is used to impose our prior belief regarding parameter values, specifically to exclude nonphysical parameter combinations. The calibrated model is compared, in terms of its predictive skill, to simulations using uncalibrated linear and CEVMs. We find that the calibrated model, with one quadratic term, is more accurate than the uncalibrated simulator. The model is also checked at a flow condition at which the model was not calibrated.

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Grahic Jump Location
Fig. 1

Top: Removal of CEVM coefficients ci as λ is increased. Bottom: Mean prediction error and ±2 standard deviation bounds of prediction error, as a function of log(λ). λmin (left vertical line) and λ1se (right vertical line) are also shown.

Grahic Jump Location
Fig. 2

Top: Plot of the physically realistic part of the parameter space, R (filled circles) along with the region where the RANS simulator runs without crashing C3 (crosses). Bottom: The experimental vorticity field as a flood plot, with locations with large vorticity (○) and the subset of probes with accurate surrogate models (+).

Grahic Jump Location
Fig. 8

Plots of streamwise velocity deficit (top row) and vertical velocity (bottom row) at three locations 200, 300, and 400 mm downstream of the jet. Experimental data are plotted using symbols, LEVM (nominal) using the dotted line, the CEVM (nominal) using the dashed line, and the ensemble mean of 100 samples from the posterior using the solid line. The + symbols are the predictions using Copt. The crossflow is M = 0.7.

Grahic Jump Location
Fig. 5

Results of the posterior predictive test using 100 samples. We plot the predicted vorticity normalized by the measured value at the probes with accurate surrogate models. The horizontal line indicates the experimental measurement. The error bars span the 5th–95th percentile range and the filled ○ are the median prediction.

Grahic Jump Location
Fig. 3

Marginalized PDFs of c3,Cε2,Cε1, and σ2. The dashed line is the prior density due to R and the solid line denotes the posterior density. The vertical lines are the nominal values of the parameters. The figure at the bottom right shows magnitude of the data—model misfit. As a comparison to σ, the 95th percentile of the (experimental) vorticity magnitude is 3.8×103 s−1.

Grahic Jump Location
Fig. 4

Plots of simulated streamwise vorticity field (as a flood plot) with contours of experimental vorticity overlaid. Left: Simulations using RANS-LEVM driven by nominal parameters. Middle: RANS–CEVM, driven by the nominal parameters in Ref. [5]. Right: Predictions using Copt. The improvement is stark. Note that the scales of the vertical and horizontal axes are different.

Grahic Jump Location
Fig. 6

Box-and-whisker plots of the posterior samples' runs with RANS–CEVM, for the PVM. The horizontal line denotes experimental results. The open ○ are prediction using nominal CEVM parameters [5], whereas the filled ⋄ are predictions using an LEVM with nominal parameters. Model predictions are normalized by their experimental counterparts.

Grahic Jump Location
Fig. 7

Plots of streamwise velocity deficit (top row) and vertical velocity (bottom row) at three locations 200, 300, and 400 mm downstream of the jet. Experimental data are plotted using symbols, LEVM (nominal) using the dotted line, the CEVM (nominal) using the dashed line, and the ensemble mean of 100 samples from the posterior using the solid line. The + symbols are the predictions using Copt. The crossflow is M = 0.8.




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