A Direct Discontinuous Galerkin method for the compressible Navier-Stokes equations on arbitrary grids

46th AIAA Fluid Dynamics Conference

Yang

et al. 2016

Self Cite

A Direct Discontinuous Galerkin (DDG) method is developed for solving the compressible Reynolds-averaged Navier-Stokes (RANS) equations with a modified Spalart-Allmaras (SA) model on hybrid grids. The characteristic face length in the numerical flux functions defined by the DDG method is carefully examined and re-evaluated to take into consideration the fact that grids in near-wall regions for high Reynolds number flows are generally curved and highly stretched. The effect with respect to the different choices of penalty parameter in simulating turbulent flows is also discussed in this work. A number of typical test cases are presented to demonstrate the performance of applying DDG method for high Reynolds number turbulent flows. Numerical results obtained demonstrate that the developed DDG method is able to deliver satisfactory and reliable results for simulating high Reynolds number turbulent flows and indicates its potential for practical engineering applications.

Section: A Direct Dg Methods For Rans Equations With Sa Modelmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

A Direct Discontinuous Galerkin Method for Computation of Turbulent Flows on Hybrid Grids

46th AIAA Fluid Dynamics Conference

Yang

et al. 2016

Self Cite

“…However, how to define a characteristic face length on nonuniform grids is a critical issue in extending the DDG method to arbitrary or unstructured grids, as the numerical viscous flux function explicitly depends on it. In the work of Cheng et al, three choices for the characteristic face length are given as follows:

normalΔ = false| {boldI}_{e} false|,

normalΔ = false| {boldI}_{e} \cdot {boldn}_{e} false|,

normalΔ = \frac{false| {normalΩ}_{+} false| + false| {normalΩ}_{-} false|}{2 false| {normalΓ}_{e} false|},

where I e is the displacement vector from the centroid of the element K + to the centroid of the element K − , Γ e is the common internal face shared by both K + and K − , n e is the unit normal vector at the middle point of face Γ e , and |Ω ± | is the area of K ± , respectively (see Figure A).…”

Section: High‐order Dg Methods On Arbitrary Gridmentioning

confidence: 99%

“…It can be easily verified that all of the three expressions above are consistent with the original definition of the characteristic length if a uniform Cartesian grid is used. Extensive numerical experiments in the work of Cheng et al indicate that the second choice defined by Equation delivers the best results and, thus, is adopted in this work.…”

Section: High‐order Dg Methods On Arbitrary Gridmentioning

confidence: 99%

A high‐order discontinuous Galerkin method for the incompressible Navier‐Stokes equations on arbitrary grids

Zhong

Numerical Methods in Fluids

2019

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Summary A high‐order discontinuous Galerkin (DG) method is proposed in this work for solving the two‐dimensional steady and unsteady incompressible Navier‐Stokes (INS) equations written in conservative form on arbitrary grids. In order to construct the interface inviscid fluxes both in the continuity and in the momentum equations, an artificial compressibility term has been added to the continuity equation for relaxing the incompressibility constraint. Then, as the hyperbolic nature of the INS equations has been recovered, the local Lax‐Friedrichs (LLF) flux, which was previously developed in the context of hyperbolic conservation laws, is applied to discretize the inviscid term. Unlike the traditional artificial compressibility method, in this work, the artificial compressibility is introduced only for the construction of the inviscid numerical fluxes; therefore, a consistent discretization of the INS equations is obtained, irrespective of the amount of artificial compressibility used. What is more, as the LLF flux can be obtained directly and straightforward, no numerical iteration for solving an exact Riemann problem is entailed in our method. The viscous term is discretized by the direct DG method, which was developed based on the weak formulation of the scalar diffusion problems on structured grids. The performance and the accuracy of the method are demonstrated by computing a number of benchmark test cases, including both steady and unsteady incompressible flow problems. Due to its simplicity in implementation, our method provides an attractive alternative for solving the INS equations on arbitrary grids.

“…A variety of numerical methods have been proposed in the literature, such as symmetric interior penalty (SIP) 5-6 , local discontinuous Galerkin (LDG) 7 , compact discontinuous Galerkin (CDG) 8 , reconstructed DG (rDG) 9 , and the second Bassi-Rebay (BR2) 10 schemes. Recently, inspired by the formula of general Riemann solution for a purely diffusion equation, a direct discontinuous Galerkin (DDG) 11 method was introduced by Liu and Yan and extended to solve compressible Navier-Stokes equations 37 . Compared to the BR2 scheme, DDG method shows attractive protentialty for solving diffusion problems owning to its simplicity in implementation and efficiency in computational cost.…”

Section: Introductionmentioning

confidence: 99%

A Reconstructed Direct Discontinuous Galerkin Method for Compressible Turbulent Flows on Hybrid Grids

Yang

46th AIAA Fluid Dynamics Conference

et al. 2016

Self Cite

In this paper, a reconstructed direct discontinuous Galerkin (DDG) method is developed for solving RANS equations with the latest negative Spalart-Allamars (SA) turbulence model based on the framework of hybrid reconstructed DG (rDG) methods. The previous successful experience of applying DDG method for solving compressible laminar flows intrigues us to further unreveal the possibility and assess the capability of using DDG method for simulating high Reynolds number turbulent flows. Several typical test case are selected to assess the performance of applying DDG method for solving high Reynolds turbulent flows. Numerical results show that the grid curvature must be taken into consideration for the formulation of DDG numerical flux. Through the numerical experiments, the reconstructed DDG method clearly demonstrates its capability for solving high Reynolds turbulent flows with highly stretched and curved grids and indicates its further potential for real engineering applications.