High-order compact finite volume schemes for solving the Reynolds averaged Navier-Stokes equations on the unstructured mixed grids with a large aspect ratio

Highlights

• The high-order compact finite volume scheme is extended to solve the SA model on unstructured grids.

• The variational reconstruction algorithm is optimized for grids with large aspect ratio.

• An exponential decay procedure is proposed to cure the negative turbulent viscosity in the solution of the SA model.

Abstract

In this paper, high-order compact finite volume schemes on the unstructured grids based on the variational reconstruction are developed to solve the Reynolds averaged Navier-Stokes equations closed by the Spalart-Allmaras one-equation turbulence model. Encouraging progress is made in addressing the following two challenging problems: reducing the numerical errors on the large aspect ratio grids and avoiding the negative turbulent viscosity associated with the high-order methods. On grids with large aspect ratios, a three-step procedure is designed to optimize the functional parameters of variational reconstruction. In addition, an exponential decay procedure is proposed to cure the negative turbulent viscosity problem of the Spalart-Allmaras model. The exponential decay procedure has the advantage of being able to be used with any spatial discretization method and with the implicit temporal discretization. Numerical tests show significant benefits of the high-order schemes in predicting the skin frictions, capturing some important flow structures, and achieving grid-independent solutions. The numerical tests also show that the proposed schemes are sufficiently robust for practical applications.

Introduction

High-order methods have shown great capability in simulating multi-scale flow problems such as turbulences [1]. For flows with complex geometries, high-order numerical methods on the unstructured grids are preferred. Recent advances have been made in the finite volume (FV) method [2], [3], [4], [5], [6], [7], [8], [9], [10], [11], [12], discontinuous Galerkin (DG) methods [13], [14], [15], [16], spectral volume (SV) [17]/spectral difference (SD) methods [18], PnPm procedures [19], FR [20]/CPR [21] methods, hybrid FV/DG methods [22], [23], [24], and multi-moment methods [25] for solving multi-scale flow problems on the unstructured grids.

An active area of the study of the high-order schemes on the unstructured grids is the simulation of the turbulent flows governed by the Reynolds averaged Navier-Stokes (RANS) equations. The current design tools in aircraft industries are primarily based on the RANS equations [26]. RANS equations are also necessary for the wall modeling of the large eddy simulation (LES) [27], as well as the hybrid RANS/LES modeling such as the detached eddy simulation (DES) [28]. Various turbulent models have been proposed [29] to meet the needs of industrial applications. Since it is widely accepted that high-order schemes outperform their second-order counterparts in achieving grid-independent solutions and in resolving important flow structures such as the turbulent boundary layers, it is necessary to develop high-order accurate RANS solvers. To this end, high-order FV [31], [32], DG [33], [34], [35], [36], [37], [38], [39], [40], [41], [42], SD [43], FR/CPR [44], [45], and the hybrid FV/DG methods [46] have been applied to solve RANS equations in recent studies. The Spalart Allmaras (SA) one-equation model [30] is widely used among the turbulent models due to its good performance and easy implementation. In this paper, we will focus our discussion on the SA model.

It is still challenging to solve the turbulence transport equations such as the SA model using high-order numerical schemes. The first issue arises from the large aspect ratio of the near-wall grids to resolve turbulent boundary layers [35]. The time-step of the explicit time-marching scheme is significantly reduced due to the small grid size in the wall normal direction. The implicit temporal discretization scheme is preferred to improve the solution efficiency, as suggested in [35]. When using high-order FV schemes, it has also been reported that the grid aspect ratios affect the reconstruction accuracy [32]. In [32], a reconstruction scheme based on local curvilinear normal-tangential coordinates was proposed to enhance the reconstruction accuracy on grids with large aspect ratios. The second issue is the stiff source terms in the turbulence modeling equations. Negative turbulent viscosity in the outer edge region of the boundary layer can be predicted [32], [33] due to the stiffness of the turbulence model. To suppress the appearance of negative turbulent viscosity, the standard SA model is suggested to be replaced by the negative SA model [54], which has been proven to be effective [32].

In the present paper, we will study the numerical simulation of the RANS equations closed by the SA model using the recently proposed high-order compact FV schemes based on the variational reconstructions [4] on the unstructured mixed grids. A distinctive advantage of these schemes is that all operations of the algorithms are based on a compact stencil, and the large stencil problem associated with traditional high-order finite volume schemes is resolved entirely. Another advantage is that the variational reconstructions are non-singular with unique solutions.

Two outstanding problems for applying the variational reconstruction FV schemes to solve the RANS equations closed by the SA model are addressed in the present paper. The first one is to design an effective reconstruction algorithm for the quadrilateral grids with a large aspect ratio. The present FV scheme can be applied on the mixed grids with triangular and quadrilateral control volumes. However, near the solid walls of the turbulent boundary layers, the preferred meshes are quadrilateral. Therefore, optimization of the FV scheme for the quadrilateral grids with a large aspect ratio will be studied. The variational reconstruction relies on minimizing a particular functional with adjustable parameters. It is therefore possible to optimize these parameters to improve the performance of the FV scheme. For high-order accurate reconstructions, it turns out that the optimization procedure is challenging. To simplify the analysis, we propose a three-step optimization procedure, including 1) the derivative weights optimization, 2) the geometric weights optimization, and 3) the generalization to general unstructured grids. According to the tests based on a manufactured solution of physical importance, the optimized scheme significantly improves the reconstruction accuracy on the quadrilateral grids with large aspect ratios.

The second one is the avoidance of the possible negative turbulent viscosity in the solutions of the SA model. Different from other methods for eliminating the negative turbulent viscosity mentioned previously in this section, a new procedure called the exponential decay procedure (EDP) is proposed in the present paper. By analyzing the asymptotic behavior of the SA model when the turbulent viscosity is close to zero, we find that the turbulent viscosity will undergo an approximately exponential decay. This physics-based reasoning leads to EDP, a numerical positivity preserving procedure. The EDP is straightforward and effective and can be easily coupled with high-order spatial discretization and implicit time integration schemes. Furthermore, through EDP, the standard SA model is sufficient to achieve the positivity preserving solution, and no other modification such as the use of the negative SA model is needed.

Several test cases are solved numerically with the proposed FV schemes and the EDP. Numerical tests show significant advantages of the high-order schemes in predicting the skin frictions, capturing some important flow structures, and achieving grid-independent solutions. The numerical simulations also indicate that the proposed schemes are sufficiently robust for practical applications.