Lawrence Dechant

Bayesian calibration of a k-ε turbulence model for predictive jet-in-crossflow simulations

‡§ We propose a Bayesian method to calibrate parameters of a RANS model to improve its predictive skill in jet-in-crossflow simulations. The method is based on the hypotheses that (1) informative parameters can be estimated from experiments of flow configurations that display the same, strongly vortical features of jet-in-crossflow interactions and (2) one can construct surrogates of RANS models for certain judiciously chosen RANS outputs which serve as calibration variables (alternatively, experimental observables). We estimate three ke parameters (C∝, C 2 , C 1 ) from Reynolds stress measurements from an incompressible flowover-a-square-cylinder experiment. The k-e parameters are estimated as a joint probability density function. Jet-in-crossflow simulations performed with (C∝, C 2 , C 1 ) samples drawn from this distribution are seen to provide far better predictions than those obtained with nominal parameter values. We also find a (C∝, C 2 , C 1 ) combination which provides < 15% error in a number of performance metrics; in contrast, the errors obtained with nominal parameter values may exceed 60%.

Bayesian Calibration of a RANS Model with a Complex Response Surface - A Case Study with Jet-in-Crossflow Configuration

We demonstrate a Bayesian method that can be used to calibrate computationally expensive 3D RANS models with complex response surfaces. Such calibrations, conditioned on experimental data, can yield turbulence model parameters as probability density functions (PDF), concisely capturing the uncertainty in the estimation. Methods such as Markov chain Monte Carlo construct the PDF by sampling, and consequently a quickrunning surrogate is used instead of the RANS simulator. The surrogate can be very difficult to design if the model’s response i.e., the dependence of the calibration variable (the observable) on the parameters being estimated is complex. We show how the training data used to construct the surrogate models can also be employed to isolate a promising and physically realistic part of the parameter space, within which the response is wellbehaved and easily modeled. We design a classifier, based on treed linear models, to model the “well-behaved region”. This classifier serves as a prior in a Bayesian calibration study aimed at estimating 3 k − parameters C = (Cμ, C 2, C 1) from experimental data of a transonic jet-in-crossflow interaction. The robustness of the calibration is investigated by checking its predictions of variables not included in the calibration data. We also check the limit of applicability of the calibration by testing at an off-calibration point.

Learning an eddy viscosity model using shrinkage and Bayesian calibration: A jet-in-crossflow case study

We demonstrate a statistical procedure for learning a highorder eddy viscosity model from experimental data and using it to improve the predictive skill of a Reynolds-Averaged Navier Stokes simulator. The method is tested in a 3D, transonic jet-in-crossflow configuration. The process starts with a cubic eddy viscosity model developed for incompressible flows. It is fitted to limited experimental jet-in-crossflow data using shrinkage regression. The shrinkage process removes all 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 a Reynolds Averaged Navier-Stokes simulator and calibrated, using vorticity measurements, to infer three parameters. The calibration is Bayesian and is solved using a Markov chain Monte Carlo method. A three-dimensional 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 a Markov chain Monte Carlo loop is mitigated by using surrogate models (“curve-fits”). A support vector machine classifier is used to impose our prior belief regarding parameter values, specifically to exclude non-physical parameter combinations. The

K-ε Turbulence Model Parameter Estimates Using an Approximate Self-similar Jet-in-Crossflow Solution.

The k-ε turbulence model has been described as perhaps “the most widely used complete turbulence model.” This family of heuristic Reynolds Averaged Navier-Stokes (RANS) turbulence closures is supported by a suite of model parameters that have been estimated by demanding the satisfaction of well-established canonical flows such as homogeneous shear flow, log-law behavior, etc. While this procedure does yield a set of so-called nominal parameters, it is abundantly clear that they do not provide a universally satisfactory turbulence model that is capable of simulating complex flows. Recent work on the Bayesian calibration of the k-e model using jet-in-crossflow wind tunnel data has yielded parameter estimates that are far more predictive than nominal parameter values. Here we develop a self-similar asymptotic solution for axisymmetric jet-incrossflow interactions and derive analytical estimates of the parameters that were inferred using Bayesian calibration. The self-similar method utilizes a near field approach to estimate the turbulence model parameters while retaining the classical far-field scaling to model flow field quantities. Our parameter values are seen to be far more predictive than the nominal values, as checked using RANS simulations and experimental measurements. They are also closer to the Bayesian estimates than the nominal parameters. A traditional simplified jet trajectory model is explicitly related to the turbulence model parameters and is shown to yield good agreement with measurement when utilizing the analytical derived turbulence model coefficients. The close agreement between the turbulence model coefficients obtained via Bayesian calibration and the analytically estimated coefficients derived in this paper is consistent with the contention that the Bayesian calibration approach is firmly rooted in the underlying physical description

Robust Bayesian Calibration of a RANS Model for Jet-in-Crossflow Simulations

Compressible jet-in-crossflow interactions are poorly simulated using Reynolds-Averaged Navier Stokes (RANS) equations. This is due to model-form errors (physical approximations) in RANS as well as the use of parameter values simply picked from literature (henceforth, the nominal values of the parameters). Previous work on the Bayesian calibration of RANS models has yielded joint probability densities of C = (Cμ, C 2, C 1), the most influential parameters of the RANS equations. The calibrated values were far more predictive than the nominal parameter values and the advantage held across a range of freestream Mach numbers and jet strengths. In this work we perform Bayesian calibration across a range of Mach numbers and jet strengths and compare the joint densities, with a view of determining whether compressible jet-in-crossflow could be simulated with either a single joint probability density or a point estimate for C. We find that probability densities for C 2 agree and also indicate that the range typically used in aerodynamic simulations should be extended. The densities for C 1 agree, approximately, with the nominal value. The densities for Cμ do not show any clear trend, indicating that they are not strongly constrained by the calibration observables, and in turn, do not affect them much. We also compare the calibrated results to a recently developed analytical model of a jet-in-crossflow interaction. We find that the values of C estimated by the analytical model delivers prediction accuracies comparable to the calibrated joint densities of the parameters across a range of Mach numbers and jet strengths.

An Approximate Expression for Base Pressure Fluctuation Spectra for Bluff Bodies

[Abstract] The goal of this research is to derive approximate expressions for the pressure fluctuation spectra in the base region of blunt bodies. Though theory-based algebraic models are known for vortex shedding frequency, no such model is available to provide spectral behavior of the pressure fluctuation field in the lee of a body with a blunt base. Here we extend the algebraic model of Ahlborn (2003) to be an unsteady partial differential equation (wave equation) in terms of a scalar potential that is valid in the lee of the blunt body. The existence of a scalar potential is justified via the use of a Cauchy pair which relates vorticity and pressure near a bounding wall. The wave equation, which exhibits a mixed time space behavior is rotated to canonical wave equation form and solved using normal modes. The wave is dispersive where the dispersion relationship captures one of Ahlborn’s conservation expressions. The resulting solution (which is based upon near wall/wake assumptions and is therefore only locally valid) is extended to a full space-time domain via a heuristic damping term. The pressure field follows immediately via the Cauchy equations. Remarkably, because both fluctuation and mean pressure are governed by a linear equation, Reynolds decomposition is trivial and both mean and fluctuation are governed by the same model. With access to a space-time pressure field we can readily compute a fluctuation correlation function and power-spectral density (PSD) through the associated cosine transform pair. Comparison to data sets from Eldred (1961) and Shvets (1978) suggest that frequency associated with maximum PSD magnitude is predicted by the model. Moreover, rate of decay of high frequency fluctuations decays at the observed rate as St. Thus, we suggest that the closed form expressions derived here may be a useful physics based approach for base pressure fluctuation with direct application to fluid structure loading problems.

Estimating Turbulent Wall Shear and Boundary Layer Thickness for Hydro-dynamically Rough Surfaces by Perturbing Known Smooth Results

[Abstract] It is not uncommon when utilizing hydrodynamically smooth experimental or computational results that one would like to estimate the additional effect of wall roughness without performing either new experiments or computations. Here we consider a simple analytical model based on inner law methods which extends smooth wall skin friction and boundary layer thickness to be valid for rough wall flows. The approach described here uses a formal perturbation model of the compressible (adiabatic) skin friction, the rough wall equivalent (Van-Driest) log-law boundary layer thickness. Though perturbation-based approaches provide a correction expression, they are not valid for many physically interesting problems where the roughness is more significant. For these flows we solve the appropriate roughness expressions using an approximate procedure that is valid for a wider range of roughness. Though useful to demonstrate behaviors associated with roughness and providing a connection to smooth expressions, this approximate method is of less value when one requires compressibility corrections, whereby it is perhaps more appropriate to dispense with approximation and simplly solve the full expression numerically. We also note that the empirically based expression described by Fang et. al. (2003) 1 is in excellent agreement with the inner law based model described here.