Implementation of the transition model for high order discontinuous Galerkin method with hybrid discretization strategy

Highlights

• The γ-$\stackrel{⌢}{R{e}_{\theta t}}$ model is implemented within high-order DG context.

• Develop hybrid implementation strategy to reduce computation challenge.

• Improve accuracy for steady and unsteady transitional simulations.

Abstract

The γ-$\stackrel{⌢}{R{e}_{\theta t}}$ transition model has been applied within the framework of high order discontinuous Galerkin (DG) discretization of Reynolds-averaged Navier-Stokes (RANS) equations. A hybrid discretization strategy that takes advantage of all available information from DG method is suggested for the implementation of the transition model. Some techniques for the spatial and temporal discretization are utilized to enhance the effectiveness of the method. The resulting approaches achieve high accuracy of transition prediction and meanwhile do not increase notable computational challenge when compared to DG discretization of fully coupled RANS/transition system. We carry out both steady and unsteady transitional simulations to examine the performance of the designed method. Numerical results have demonstrated that the present approaches are capable of capturing the transition flow characteristics as well as improving transition prediction accuracy with high order DG schemes.

Introduction

Laminar-turbulent transition of boundary layers is of practical relevance in various aerodynamic flows. In the past few decades there has been a large body of publications on the subject of numerical modeling of laminar-turbulent transition. However, the transition modeling does not offer the same wide spectrum of computational fluid dynamics (CFD)-compatible model formulations that is currently available for turbulent flows [1]. While various methods are able to predict flow transition, the last generation of transitional flow models for unstructured parallel CFD Reynolds-averaged Navier-Stokes (RANS) codes can be typically divided into two classes: local correlation based transition models introduced by Menter et al. [2,3] and Langtry and Menter [4], and eddy viscosity based phenomenological transition models such as the k-k_L-ω model introduced by Walters and Cokljat [5].

For the first class of methods, the γ-$\stackrel{⌢}{R{e}_{\theta t}}$ transition model [4] maybe currently the most successful approach due to its careful and elegant formulation that avoids non-local operations and predicts multiple modes of transition through simple correlations. It should be noted that the γ-$\stackrel{⌢}{R{e}_{\theta t}}$ model was initially coupled with the SST k-ω turbulence model by its developers, but the model could be applied to other models, too. More recently, the authors of [6] have demonstrated the application of the γ-$\stackrel{⌢}{R{e}_{\theta t}}$ model coupled with one-equation Spalart-Allmaras (SA) turbulence model. Moreover, various implementation and assessment of the transition model coupled with different solvers are carried out in the following work [7], [8], [9]. However, currently the local correlation based transition models have been mainly developed in the context of low order finite volume (FV) or finite element (FE) frameworks (see for example [4], [5],8]). Recently, the adaption of transition models has been developed for high order methods such as the DG method, see for example [10,11].

On the other hand, during the last decade, the DG [12,13] methods have become popular to solve nonlinear hyperbolic conservation laws to arbitrary order of accuracy. In the DG methods, the uniform high-order accuracy is obtained by using high degree polynomial approximation within elements as in the classical finite element methods, and the physics of wave propagation is accounted for by solving the Riemann problems at element interfaces as in upwind FV methods. The DG methods have many desirable features, such as the capacity to handle complicated geometries, the compact stencils, the flexibility for the h/p adaptation, and the nice mathematical properties with respect to conservation, stability, and convergence.

As for the extension of the DG method to the predication of laminar-turbulent transition of the boundary layer, the implementation can be still subject to many limitations. Firstly, the application of high-order DG methods in solving the RANS equations with the turbulence/transition model is not commonplace, especially for high order DG methods. For fully turbulent simulations, various strategies of robustness enhancement [14], [15], [16], [17], [18] need to be carried out to eliminate non-smooth behavior of the turbulence model variable (or variables) due to the stiffness induced by the source terms. Furthermore, a great deal of effort has also been devoted to increase the efficiency of higher-order DG RANS solvers [19], [20], [21], [22]. Secondly, the inclusion of the transition model equations will introduce more model variables and thus even more degrees of freedom. As a result, the high order DG discretization of RANS and transition equations will deal with the issues such as the lack of computation robustness and the increment of computation effort. Even with the reduced order for transition model equations, it is still quite challenging for the DG method to solve fully coupled RANS/transition system. In recent years, there has been some effort in bringing other transition prediction methods into the DG discretization framework, which explores either the apparent transition of the turbulence model [23] or the non-local formulation of the transition model [24].

Therefore, the objective of this work is to develop effective method to extend the transition model to high order DG method. Firstly, the SA model is chosen as the underlying turbulence model related to the large SA model diffusion in commercial, open-source and research software. Secondly, instead of direct exploration of the method where governing equations are treated in a fully coupled fashion, here we propose a hybrid discretization strategy for implementing the transition model. At last, some techniques are utilized in the spatial and temporal discretization for current hybrid implementation strategy. We note that the resulting approaches are able to avoid issues such as the lack of computation robustness and the increment of computation effort, when comparing to solving fully coupled RANS/transition system with the DG method. At the meantime, the approaches are also able to fully explore the advantage of high order DG methods. It should be mentioned that because the primary work is to develop hybrid implementation strategy for extending the transition model to the high-order DG method, thus the simple γ-$\stackrel{⌢}{R{e}_{\theta t}}$ model is chosen instead of other advanced models. However the designed method is believed to be easily extended to these models when one considers more challenging problems such as the crossflow transition for transition models nowadays.

The rest of the paper is organized as follows. Section 2 briefly introduces the governing equations and the DG formulation. Section 3 describes the details of hybrid discretization strategy for implementing the transition model in the DG method. Numerical tests are carried out in Section 4 and some conclusions are drawn in Section 5.