A direct discontinuous Galerkin method for the compressible Navier–Stokes equations on arbitrary grids

Cheng, Jian; Yang, Xiaoquan; Liu, Xiaodong; Liu, Tiegang; Luo, Hong

doi:10.1016/j.jcp.2016.09.049

Cited by 42 publications

(20 citation statements)

References 25 publications

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“…, where τ w is the local wall shear stress, on the airfoil surface are compared in Fig. 6 with the discontinuous Galerkin (DG) solution of the compressible NS equations [33]. Moreover, the comparison of the drag coefficient with other numerical results is shown in Table IV.…”

Section: A Accuracy Testmentioning

confidence: 99%

Semi-Lagrangian lattice Boltzmann model for compressible flows on unstructured meshes

2020

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Compressible lattice Boltzmann model on standard lattices [M. H. Saadat, F. Bösch, and I. V. Karlin, Phys. Rev. E 99, 013306 (2019).] is extended to deal with complex flows on unstructured grid. Semi-Lagrangian propagation [A. Krämer et al., Phys. Rev. E 95, 023305 (2017).] is performed on an unstructured second-order accurate finite-element mesh and a consistent wall boundary condition is implemented which makes it possible to simulate compressible flows over complex geometries. The model is validated through simulation of Sod shock tube, subsonic and supersonic flow over NACA0012 airfoil and shock-vortex interaction in Schardin's problem. Numerical results demonstrate that the present model on standard lattices is able to simulate compressible flows involving shock waves on unstructured meshes with good accuracy and without using any artificial dissipation or limiter.

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Section: A Accuracy Testmentioning

confidence: 99%

Semi-Lagrangian lattice Boltzmann model for compressible flows on unstructured meshes

2020

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“…In this section, we will give some numerical test cases to prove the convergence of the proposed asymptotic expansions by comparing them with the corresponding numerical results obtained by the direct discontinuous Galerkin (DDG) method detailed in Liu and Yan, Cheng et al, and Zhang et al And we will also illustrate the utility of the asymptotic expansion solution for verification purpose by determining the accuracy order for the DDG method. In all cases, the boundary values will be chosen to satisfy

S_{1} > S_{0}

so that the second law of thermodynamics will not be violated at least with respect to the inlet and outlet points.…”

Section: Verification and Discussionmentioning

confidence: 99%

An asymptotic expansion for 1D steady compressible Navier‐Stokes equations under nonuniform enthalpy

Zhang

Cheng

Liu

2020

Math Methods in App Sciences

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The solutions of the one-dimensional (1D) steady compressible Navier-Stokes equations have been thoroughly discussed before, but restrained for uniform total enthalpy, which leads to only a shock wave profile possible in an infinite domain. To date, very little progress has been made for the case with nonuniform total enthalpy. In this paper, we affirm that under nonuniform total enthalpy, there also exists steady solution for the 1D compressible Navier-Stokes equations, but the flow domain must be finite in the positive x-axis. The 1D steady compressible Navier-Stokes equations can be reduced to a singular perturbed nonlinear ordinary differential equation (ODE) for velocity with the assumptions of Pr = 3∕4 and a constant viscosity coefficient. By analyzing the mathematical property of the nonlinear ODE for velocity, we propose an asymptotic expansion for the solution of it as an exponential type sequence and also prove the convergence. Unlike the case of uniform total enthalpy, where the solutions for all variables keep monotone, we show that under nonuniform total enthalpy and some specific boundary conditions, there exists extreme inside the thin boundary layer. Numerical results verify the accuracy and convergence of the asymptotic expansion. This asymptotic expansion solution can serve as an important testing to demonstrate the efficiency of numerical methods developed for compressible Navier-Stokes equations at high Reynolds number.

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“…Generally, the basic principle in defining the characteristic lengthh on arbitrary grids is that the characteristic lengthh should be associated with the target interface where the viscous flux is evaluated and also be orthogonal to the common interface or boundary face. A variety of different definitions had been tested in our previous work [25], and the one gives the best performance is given as follows:…”

Section: Review Of Original Direct Dg Methodsmentioning

confidence: 99%

“…The most remarkable feature of the DDG method is its simplicity in implementation and its efficiency in computational cost. Very recently, the DDG method has been successfully extended and applied for solving the more complex compressible Navier-Stokes equations [25,26] on arbitrary grids. The DDG method shows its potential to deliver comparable accuracy as the widely used BR2 method at a significantly reduced computational cost, thus, becomes an attractive alternative to discretize the compressible Navier-Stokes equations on arbitrary grids.…”

Section: Introductionmentioning

confidence: 99%

A Direct Discontinuous Galerkin Method with Interface Correction for the Compressible Navier-Stokes Equations on Unstructured Grids

Cheng¹

2018

AAMM

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Since the original DDG method has been introduced by Liu et al. [8] in 2009, a variety of DDG type methods have been proposed and further developed. In this paper, we further investigate and develop a new DDG method with interface correction (DDG (IC)) as the discretization of viscous and heat fluxes for the compressible Navier-Stokes equations on unstructured grids. Compared to the original DDG method, the newly developed DDG (IC) method demonstrates its superior in delivering the optimal order of accuracy under demanding situations. Strategies in extension and application of this newly developed DDG (IC) method for solving the compressible Navier-Stokes equations and special treatments designed for handling boundary viscous fluxes are presented and examined in this work. The performance of the new DDG method with interface correction is carefully evaluated and assessed through a number of typical test cases. Numerical experiments show that the new DDG method with interface correction can achieve the optimal order of accuracy on both uniform structured grids and nonuniform unstructured grids, which clearly indicates its potential for further applications of real engineering practices.

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A direct discontinuous Galerkin method for the compressible Navier–Stokes equations on arbitrary grids

Cited by 42 publications

References 25 publications

Semi-Lagrangian lattice Boltzmann model for compressible flows on unstructured meshes

Semi-Lagrangian lattice Boltzmann model for compressible flows on unstructured meshes

An asymptotic expansion for 1D steady compressible Navier‐Stokes equations under nonuniform enthalpy

A Direct Discontinuous Galerkin Method with Interface Correction for the Compressible Navier-Stokes Equations on Unstructured Grids

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