Analysis and comparison of turbulence model coefficient uncertainty for canonical flow problems

Highlights

• Uncertainty quantification and sensitivity analysis of turbulence model closure coefficients.

• Point collocation nonintrusive polynomial chaos method uncertainty propagation.

• Canonical flow problems from the turbulence modeling resource website that span multiple flow physical realizations are investigated.

Abstract

The purpose of this paper is to present results of an uncertainty and sensitivity analysis study of commonly used turbulence models in Reynolds-Averaged Navier–Stokes codes due to the epistemic uncertainty in closure coefficients for a set of turbulence model validation cases that represent the structure of several canonical flow problems. The study focuses on the analysis of a 2D Zero Pressure Gradient Flat Plate, a 2D NASA Wall Mounted Hump, and an Axisymmetric Shock Wave Boundary Layer Interaction, all of which are well documented on the NASA Langley Research Center Turbulence Modeling Resource website. The Spalart–Allmaras (SA), the Wilcox (2006) $\kappa -\omega$ (W2006), and the Menter Shear-Stress Transport (SST) turbulence models are considered in the stochastic analyses of these flow problems, and the FUN3D code was utilized as the flow solver. The uncertainty quantification approach involves stochastic expansions based on non-intrusive polynomial chaos to efficiently propagate the uncertainty. Sensitivity analysis is performed with Sobol indices to rank the relative contribution of each closure coefficient to the total uncertainty for several output flow quantities. The results generalize a set of closure coefficients which have been identified as contributing most to the uncertainty in various output quantities of interest for the set of canonical flow problems considered in this study. Mainly, the SA turbulence model is most sensitive to the uncertainties in the diffusion constant ($\sigma$), the log layer calibration constant ($\kappa$), and the turbulent destruction constant ($c_{w2}$). The predictive capability of the W2006 model is most sensitive to the uncertainties in a dissipation rate constant (${\sigma }_w$), the shear stress limiter ($C_{lim}$), and a turbulence-kinetic energy constant (${\beta }^{*}$). Likewise, the SST turbulence model was found to be most sensitive to the diffusion constants (${\sigma }_{w1}$ and ${\sigma }_{w2}$), the log layer calibration constant ($\kappa$), and the shear stress limiter ($a_1$).

Introduction

Turbulence is still an unsolved problem in the study of fluid mechanics. The highly complex turbulent flow is uncertain in nature and thereby poses a difficult problem to solve. As such, comprehensive understanding of the phenomena has yet to be developed. Turbulence modelers have worked diligently in the creation of realizable predictive methods through the use of computational fluid dynamics (CFD); however, a lack of the complete understanding of turbulence has forced these modelers to use dimensional analysis in an effort to close this open problem. Resulting from the dimensional analysis, groups of constants, called closure coefficients, are introduced to balance the model equations. The values of these constants are gleaned from a combination of heuristic methods and empirical studies. Due to their formation, current turbulence models used in Reynolds-Averaged Navier–Stokes (RANS) simulations and used in sub-grid scale modeling of Large-Eddy Simulations are not guaranteed to perform well for any arbitrary flow and can often fail in flow regimes significantly dissimilar to the experiment used in their formulation.

To help facilitate the advancement of turbulence model development, implementation, application, and validation/verification, the Turbulence Modeling Resource (TMR) website [1] was developed to provide a centralized location to document RANS turbulence models. The objective of this website is to provide CFD developers accurate and current information regarding commonly used RANS turbulence models and a strategy to verify correct implementation of the models. Additionally, the TMR provides a validation process to compare CFD results against data in an effort to establish a model’s ability to reproduce important flow physics whereby a set of test cases are provided, including a series of refined grids, that incorporate fundamental fluid dynamics phenomena.

The purpose of this paper is to present uncertainty quantification (UQ) and sensitivity analyses of commonly used turbulence models in RANS codes due to the epistemic uncertainty (uncertainty due to lack of knowledge) in closure coefficients for a number of validation cases documented on the TMR website [1]. These cases include a 2D Zero Pressure Gradient Flat Plate (2DZP), a 2D NASA Wall-Mounted Hump Separated Flow (2DWMH), and an Axisymmetric Shock Wave Boundary Layer Interaction at $M=7$ (ASWBLI). Three turbulence models are considered in this study: the Spalart–Allmaras One Equation Model (SA) [29], the Wilcox (2006) $\kappa -\omega$ Two-Equation Model (W2006) [36], and the Menter Shear-Stress Transport Two-Equation Model (SST) [23]. This research also includes the refinement and implementation of stochastic expansion techniques based on polynomial chaos for efficient uncertainty propagation and sensitivity metrics derived from non-linear global sensitivity analysis based on Sobol indices.

It is well-known that RANS models are not designed for strongly separated flows including shock induced separation. In fact, turbulence models in RANS simulations are derived and calibrated mostly for low speed attached and mildly separated flows. Despite this innate deficiency, RANS simulations are still used as one of the main analysis and design tools in aerospace industry for various flow regimes and problems due to its relatively low computational cost compared to Large-Eddy Simulations (LES) and Direct Numerical Simulations (DNS). This study aims to support the validation and improvement of RANS turbulence models by identifying a set of closure coefficients for each model that contribute to the output uncertainty most for different flow problems so that the future validation and experimental efforts can be prioritized to focus on the improvement of the accuracy of these coefficients (i.e., reduction of the epistemic uncertainty of the closure coefficients).

Previous studies on turbulence model closure coefficient uncertainty focused on transonic wall-bounded flow problems and hypersonic internal and external flow. John Schaefer et al. [27] investigated turbulence model closure coefficient uncertainty for a transonic bump problem and an RAE 2822 airfoil. Di Stefano et al. [7] investigated turbulence model closure coefficients for a scramjet isolator and scramjet strut flow field. Erb and Hosder [9] performed an in-depth anaylsis of the Axisymmetric Shock Wave Boundary Layer Interaction problem where the flow field quantities of interest (QoIs) included density, Mach number, and pressure, while surface and point QoIs included pressure, heat flux, and skin friction distribution, separation bubble size, and drag coefficient. These previous works employed stochastic expansions to efficiently propagate the uncertainty. Similarly, some aerothermodynamic studies employed the use of stochastic expansions to perform an uncertainty quantification and sensitivity analyses. In particular, West et al. [34], [35] studied the uncertainty in convective and radiative heating in hypersonic entry flows. Brune et al. [5], [6] investigated the uncertainty in the hypersonic flow field, fluid structure interaction, and the thermal response of a flexible thermal protection system (TPS) due to uncertainties in flowfield modeling and TPS properties. Godfrey and Cliff [15] used the sensitivity-equation method to quantify the sensitivities of the Baldwin Lomax algebraic, the SA one-equation, and the Wilcox $k$-$\omega$ two-equation turbulence models due to closure coefficients, but they stopped short of quantifying the uncertainty in the results. Platteeuw et al. [25] used the probabilistic collocation method to quantify the uncertainty in the solution due to uncertainties in the standard $k$-$ϵ$ turbulence model closure coefficients for a flat plate test case. Han and Hosder [18] performed a mixed uncertainty quantification for the 2DWMH with flow control where they included a scaling factor on the turbulent eddy viscosity definition in the SA model. Xiao and Cinnella [38] recently published a review paper where they examine both the parametric and structural uncertainties in turbulence models.

The main contribution of the current work to the literature is that this is the first to study and generalize the impact of the uncertainty in turbulence model closure coefficients on various QoIs for a set of important canonical flow problems with different flow structures selected from the NASA TMR while employing the point collocation non-intrusive polynomial chaos and the global non-linear sensitivity with Sobol Indices in the uncertainty and sensitivity analysis, respectively. The objective is to investigate and identify a common set of coefficients for each turbulence model which contribute most to the uncertainty for all the flow problems studied. The results presented here will advance the understanding of turbulence model uncertainty and will enable the composition of general conclusions and suggestions for potential refinement and improvement of the turbulence models.

The paper is organized as follows: In Section 2, an overview of the test cases included in this study are presented. Section 3 provides a brief description of the flow solver used in this study as well as the three turbulence models; including a description of closure coefficients for each model. In Section 4, the UQ and sensitivity analysis methodologies are presented. In Section 5, the results of the UQ study is discussed along with a comparison to previous UQ work focusing on relevant turbulence model closure coefficient uncertainty problems to generalize the findings of the current study. Finally, in Section 6, major conclusions of the study are presented.