A reconstructed direct discontinuous Galerkin method for simulating the compressible laminar and turbulent flows on hybrid grids

Yang, Xiaoquan; Cheng, Jian; Luo, Hong; Zhao, Qijun

doi:10.1016/j.compfluid.2018.04.011

“…Direct DG [33] 0.05543 Reconstructed DG [35] 0.05534 Spectral difference [36] 0 Figure 9 shows the temperature and Mach contours. It is observed that a bow shock is formed in front of the airfoil and oblique shocks appear from the trailing edge.…”

Section: Methods C Dmentioning

confidence: 99%

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

Saadat

¹

,

Bösch

²

,

Karlin

³

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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“…Then, it is combined with the simple numerical flux formula of the direct DG method to approximate the gradient of conserved variables. In summary, the new DDGIC method defines multiple individual diffusion processes, each of which is combined to calculate the viscous numerical fluxes for 2-D compressible NS equations as (21) + ∇(ρu) h • ξ (22) + ∇(ρv) h • ξ (23) ∇ρ h • ξ (31) + ∇(ρu) h • ξ (32) + ∇(ρv) h • ξ (33) ∇ρ h • ξ (41) + ∇(ρu) h • ξ (42) + ∇(ρv) h • ξ (43)…”

Section: The New Ddgic Scheme Formulation For 2-d Compressible Navier...mentioning

confidence: 99%

“…Similarly, the interface correction term can be simplified and computed as follows (21) + (ρu) h ξ (22) + (ρv) h ξ (23) • ∇φ j (x, y) ρ h ξ (31) + (ρu) h ξ (32) + (ρv) h ξ (33) • ∇φ j (x, y) ρ h ξ (41) + (ρu) h ξ (42) + (ρv) h ξ (43) + E h ξ (44) • ∇φ j (x, y)…”

Section: The New Ddgic Scheme Formulation For 2-d Compressible Navier...mentioning

confidence: 99%

“…In these studies, due to the simplicity of the DDG method, efficient implicit solvers were developed by computing the viscous part of the Jacobian matrix exactly. DDG methods were also employed to solve Reynolds-averaged NS equations for turbulent flows with high Reynolds numbers [37,41]. In the context of compressible NS equations, DDG methods were further studied in [42,43].…”

Section: Introductionmentioning

confidence: 99%

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A new direct discontinuous Galerkin method with interface correction for two-dimensional compressible Navier-Stokes equations

Danis,

Yan

2021

Preprint

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We propose a new formula for the nonlinear viscous numerical flux and extend the direct discontinuous Galerkin method with interface correction (DDGIC) of Liu and Yan [1] to compressible Navier-Stokes equations. The new DDGIC framework is based on the observation that the nonlinear diffusion can be represented as a sum of multiple individual diffusion processes corresponding to each conserved variable. A set of direction vectors corresponding to each individual diffusion process is defined and approximated by the average value of the numerical solution at the cell interfaces. The new framework only requires the computation of conserved variables' gradient, which is linear and approximated by the original direct DG numerical flux formula. The proposed method greatly simplifies the implementation, and thus, can be easily extended to general equations and turbulence models. Numerical experiments with P 1 , P 2 , P 3 and P 4 polynomial approximations are performed to verify the optimal (k + 1) th high-order accuracy of the method. The new DDGIC method is shown to be able to accurately calculate physical quantities such as lift, drag, and friction coefficients as well as separation angle and Strouhal number.

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“…The family of reconstructed DG schemes, which is similar to the P N P M framework, was forwarded independently by Luo et al in a series of papers, see e.g. [68,87,88,116,117,119] and references therein. At this point we also highlight that the use of reconstruction and filtering operators as a post-processor for improving the accuracy of DG schemes goes back to work of Ryan et al, see [72,77,90,99,100].…”

Section: Introductionmentioning

confidence: 99%

A Posteriori Subcell Finite Volume Limiter for General $$P_NP_M$$ Schemes: Applications from Gasdynamics to Relativistic Magnetohydrodynamics

Gaburro

¹

,

Dumbser

²

2021

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In this work, we consider the general family of the so called ADER $$P_NP_M$$ P N P M schemes for the numerical solution of hyperbolic partial differential equations with arbitrary high order of accuracy in space and time. The family of one-step $$P_NP_M$$ P N P M schemes was introduced in Dumbser (J Comput Phys 227:8209–8253, 2008) and represents a unified framework for classical high order Finite Volume (FV) schemes ($$N=0$$ N = 0 ), the usual Discontinuous Galerkin (DG) methods ($$N=M$$ N = M ), as well as a new class of intermediate hybrid schemes for which a reconstruction operator of degree M is applied over piecewise polynomial data of degree N with $$M>N$$ M > N . In all cases with $$M \ge N > 0 $$ M ≥ N > 0 the $$P_NP_M$$ P N P M schemes are linear in the sense of Godunov (Math. USSR Sbornik 47:271–306, 1959), thus when considering phenomena characterized by discontinuities, spurious oscillations may appear and even destroy the simulation. Therefore, in this paper we present a new simple, robust and accurate a posteriori subcell finite volume limiting strategy that is valid for the entire class of $$P_NP_M$$ P N P M schemes. The subcell FV limiter is activated only where it is needed, i.e. in the neighborhood of shocks or other discontinuities, and is able to maintain the resolution of the underlying high order $$P_NP_M$$ P N P M schemes, due to the use of a rather fine subgrid of $$2N+1$$ 2 N + 1 subcells per space dimension. The paper contains a wide set of test cases for different hyperbolic PDE systems, solved on adaptive Cartesian meshes that show the capabilities of the proposed method both on smooth and discontinuous problems, as well as the broad range of its applicability. The tests range from compressible gasdynamics over classical MHD to relativistic magnetohydrodynamics.

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A reconstructed direct discontinuous Galerkin method for simulating the compressible laminar and turbulent flows on hybrid grids

Cited by 14 publications

References 36 publications

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

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

A new direct discontinuous Galerkin method with interface correction for two-dimensional compressible Navier-Stokes equations

A Posteriori Subcell Finite Volume Limiter for General $$P_NP_M$$ Schemes: Applications from Gasdynamics to Relativistic Magnetohydrodynamics

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