A third-order subcell finite volume gas-kinetic scheme for the Euler and Navier-Stokes equations on triangular meshes
Introduction
In the past decades, high-order CFD methods on unstructured meshes have attracted many attentions due to the advantage of high accuracy and efficiency, especially for complex geometries. A lot of high-order methods have been proposed for solving the Euler and Navier-Stokes (NS) equations, such as the Discontinuous Galerkin (DG) method [1], [2], [3], [4], [5], correction procedure via reconstruction (CPR) method [6], [7], [8] and [9], [10] method, etc. Based on the finite volume (FV) method, many high-order schemes have been developed and investigated extensively, typically including the k-exact method [11], [12], [13], ENO method [14], [15] and WENO [16], [17], [18], [19] method. The FV schemes are relatively simpler, more robust near strong discontinuities and may allow larger CFL numbers, when compared with DG methods [20], [21], [22]. However, since only cell-averaged values are available, a large stencil is usually required for high-order reconstruction, which leads to the difficulty of high-order boundary conditions, cache missing, etc [23]. It is more difficult for a non-compact stencil to choose and search for farther cells.
To avoid a large stencil in high-order FV methods, the conservation of spatial derivatives on the face-neighboring cells, or the minimization of the derivatives difference across cell interfaces can be additionally taken into account, resulting in the compact least-square reconstruction and the variational reconstruction [24], [25], [26]. A large linear equation system should be solved to obtain the derivatives, where an iterative method can be adopted and combined into the iteration of implicit schemes to effectively reduce the computational cost. By taking the point values at cell vertexes as additional variables, the multi-moment finite volume scheme has been developed with high-order compact reconstruction, in which the cell averaged values are solved via the finite volume formulation, while the point values are computed based on the differential form of the governing equations [27], [28], [29].
Another way to develop a compact scheme is to subdivide a cell into a set of subcells, based on which high-order reconstruction can be performed compactly. It was originally adopted by Wang et al. in the spectral volume (SV) method [30], [31], [32], [33]. Nevertheless, since the number of subcells is strictly equal to the number of unknowns on each main cell, how to subdivide main cells becomes another difficulty, especially for quadrilateral and hexahedral meshes.
The subcell finite volume (SCFV) method [34] proposed by Pan et al. is similar to SV method considering the use of subcells and the FV discretization. The main difference between them is that, SCFV method extends the stencil to face-neighboring main cells and adopts a weighted least-square reconstruction. In this way, the partition of subcells is easier and more flexible. Theoretically, by a suitable subdivision of subcells, SCFV method can achieve arbitrarily high-order accuracy with a compact stencil. So far, this method has been applied to solve the Euler equations on two-dimensional quadrilateral meshes [20]. Riemann solvers are adopted to compute the numerical flux, combined with the multi-stage Runge-Kutta (R-K) method for updating solutions. Nonetheless, the weighted least-square reconstruction is unable to conserve the averaged solutions on subcells directly. To fix this problem, a simple correction is applied to the solution polynomial on each subcell, which introduces jumps across interfaces.This correction can improve the stability and the resolution of discontinuities. However, it also reduces the efficiency of flux evaluation, compared to SV method.
Besides the reconstruction, the flux solver is also very important for a FV method. The Riemann solver based on the Euler equations is usually adopted, along with a central scheme for the viscous term when solving the NS equations. Based on the mesoscopic Bhatnagar-Gross-Krook (BGK) model, the gas-kinetic scheme (GKS) provides an alternative way to approach the NS solutions [35]. The NS equations can be recovered through the first-order Chapman-Enskog expansion of the gas distribution function, based on which, as well as a Taylor expansion in space and time, the time evolving solution can be obtained through the integral solution of BGK equation. GKS adopts this local solution to compute the flux at a cell interface, achieving the second order accuracy in both space and time through a single stage. Besides, the time evolving solution provides a dynamic adjustment of numerical dissipation inherently. Furthermore, the inviscid and viscous effects are coupled and computed simultaneously, which is also different from traditional Riemann solvers. They guarantee GKS a good balance between the accuracy and robustness, especially in solving hypersonic viscous flows [36].
With the help of a second-order Taylor expansion as well as the local integral solution of BGK equation, a third-order accurate time evolving solution can be explicitly obtained, and thus a third-order multi-dimensional GKS (HGKS or HBGK) can be developed [37], [38], [39]. The solution describes the flow evolution from a piecewise discontinuous initial state around a cell interface to the equilibrium state, in which the high-order spatial and temporal derivatives of the gas distribution function are coupled nonlinearly. Based on this high-order evolution model, a series of high-order GKS have been developed, such as the DG-GKS [40], SV-GKS [33], CPR-GKS [41], CLS-GKS [42], etc. With a third-order Taylor expansion, a fourth-order GKS can be constructed as well [43]. In particular, by using the time-dependent gas distribution function to update flow variables at cell interfaces which are borrowed in the reconstruction, the compact GKS offers a novel way to develop high-order compact FV methods [21], [44]. Recently, a two-stage fourth-order time-stepping method [45] has been proposed for the Lax-Wendroff type flux solvers. It provides a promising way to develop high-order GKS which is very efficient by adopting the second-order gas-kinetic flux solver [46], [47], [48]. Besides, for the time integration of the hyperbolic conservation law, a multi-stage multi-derivative method (MSMD) has been proposed. Then a family of MSMD-GKS has been developed [47], as well as the compact scheme with spectral-like resolution [48]. In viscous flows, it is still necessary to introduce high-order terms to the gas distribution function for better time accuracy, which are derived from the high-order gas evolution model.
The objective of the present study is to develop an efficient and robust high-order scheme on triangular meshes, denoted as SCFV-GKS, by combining the simple and compact reconstruction of the SCFV method with the high-order gas-kinetic flux solver. Different from the original SCFV method, a constrained least-square reconstruction [49] is adopted so that the averaged solutions on subcells can be conserved automatically. As a result, no correction is required and the solution polynomial can be continuous inside each main cell so that the flux for interfaces between subcells can be computed more efficiently. Furthermore, thanks to the high-order space-time evolution mechanism, the flux at a cell interface can be directly integrated along the tangential direction to avoid the Gauss integration, as well as temporally integrated within a time step so that the multistage R-K scheme is unnecessary. When simulating supersonic flows, a suitable limiting procedure is necessary, which is usually time consuming. Thus, it is more efficient to adopt the single stage time-stepping method.
This paper is organized as follows. In Section 2, the construction of the current method is introduced, including the high-order reconstruction based on the SCFV method, as well as the k-exact method, the hierarchical limiting procedure and the high-order gas-kinetic flux solver. Numerical tests are presented in Section 3 with several benchmark tests to demonstrate the performance of the current method in a wide range of flow problems. The last section draws the conclusions.
Section snippets
SCFV method
First of all, we briefly review the original SCFV method. By subdividing a set of subcells, each target cell and its face neighbors provide enough stencil cells for high-order reconstruction on the target cell. The weighted least-square technique is then adopted for computing unknown coefficients in the solution polynomial. The main difference between the SCFV method and the SV method, as well as method, is that the number of subcells in each cell can be less than the number of unknown
Numerical tests
Several benchmark flows are simulated to demonstrate the performance of the current scheme. The ratio of specific heats is set as . For viscous flows, the collision time is given by where μ is the dynamic viscous coefficient, p is the pressure at the cell interface. The first term in Eq. (38) represents the molecular viscosity, while the second term indicates the artificial viscosity, in which and are the pressure at the left and right sides of the cell
Conclusions
A third-order GKS based on the SCFV method is developed for the Euler and NS equations on triangular meshes. By subdividing each main cell into a set of subcells, the current scheme easily overcomes the difficulty of designing a compact stencil for high-order reconstruction, which is usually encountered by traditional finite volume schemes. To improve the efficiency of the SCFV method, a constrained least-square reconstruction is adopted so that the conservation in each subcell can be achieved
CRediT authorship contribution statement
Chao Zhang: Conceptualization, Formal analysis, Methodology, Software, Visualization, Writing – original draft. Qibing Li: Conceptualization, Resources, Supervision, Writing – review & editing.
Declaration of Competing Interest
The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.
Acknowledgements
This work is supported by the National Natural Science Foundation of China (11672158, 91852109) and National Key Basic Research and Development Program (2014CB744100). We also would like to acknowledge the technical support of PARATERA and the “Explorer 100” cluster system of Tsinghua National Laboratory for Information Science and Technology.
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2023, Journal of Computational PhysicsA new gas kinetic BGK scheme based on the characteristic solution of the BGK model equation for viscous flows
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2022, Journal of Computational PhysicsCitation Excerpt :Through a second-order Taylor expansion of the gas distribution function, a third-order multi-dimensional GKS (HGKS or HBGK) approach has been developed successfully [35–37]. Based on this high-order gas evolution model, a series of third-order gas-kinetic schemes have been developed within a single stage, such as the compact GKS [38,39], SV-GKS [40], DG-GKS [41] CLS-GKS [42], SCFV-GKS [43], etc. Recently, a single-stage third-order gas-kinetic CPR method on triangular meshes has also been developed [44].