Implicit Time Integration of Discontinuous Galerkin Approximations to the Navier-Stokes Equations

2020 ◽  
Author(s):  
Luigi Martinelli ◽  
Mark Lohry
Keyword(s):  
Discontinuous Galerkin ◽  
Stokes Equations ◽  
Time Integration ◽  
Navier Stokes ◽  
Implicit Time ◽  
Navier Stokes Equations ◽  
Implicit Time Integration ◽  
Galerkin Approximations

Simulation of the Transitional Flow in a Low Pressure Gas Turbine Cascade With a High-Order Discontinuous Galerkin Method

2013 ◽  
Vol 135 (7) ◽  
Author(s):  
A. Ghidoni ◽  
A. Colombo ◽  
S. Rebay ◽  
F. Bassi
Keyword(s):  
Discontinuous Galerkin ◽  
Turbulence Model ◽  
Stokes Equations ◽  
Time Integration ◽  
High Order ◽  
Transitional Flow ◽  
Navier Stokes ◽  
Implicit Time ◽  
Turbine Cascade ◽  
High Reynolds

In the last decade, discontinuous Galerkin (DG) methods have been the subject of extensive research efforts because of their excellent performance in the high-order accurate discretization of advection-diffusion problems on general unstructured grids, and are nowadays finding use in several different applications. In this paper, the potential offered by a high-order accurate DG space discretization method with implicit time integration for the solution of the Reynolds-averaged Navier–Stokes equations coupled with the k-ω turbulence model is investigated in the numerical simulation of the turbulent flow through the well-known T106A turbine cascade. The numerical results demonstrate that, by exploiting high order accurate DG schemes, it is possible to compute accurate simulations of this flow on very coarse grids, with both the high-Reynolds and low-Reynolds number versions of the k-ω turbulence model.

scholarly journals A Comparative Study of Rosenbrock-Type and Implicit Runge-Kutta Time Integration for Discontinuous Galerkin Method for Unsteady 3D Compressible Navier-Stokes equations

2016 ◽  
Vol 20 (4) ◽  
pp. 1016-1044 ◽  
Author(s):  
Xiaodong Liu ◽  
Yidong Xia ◽  
Hong Luo ◽  
Lijun Xuan
Keyword(s):  
Comparative Study ◽  
Discontinuous Galerkin ◽  
Stokes Equations ◽  
Time Integration ◽  
Differential Algebraic Equations ◽  
Navier Stokes ◽  
Third Order ◽  
Navier Stokes Equations ◽  
Index 2 ◽  
Runge Kutta

AbstractA comparative study of two classes of third-order implicit time integration schemes is presented for a third-order hierarchical WENO reconstructed discontinuous Galerkin (rDG) method to solve the 3D unsteady compressible Navier-Stokes equations: — 1) the explicit first stage, single diagonally implicit Runge-Kutta (ESDIRK3) scheme, and 2) the Rosenbrock-Wanner (ROW) schemes based on the differential algebraic equations (DAEs) of Index-2. Compared with the ESDIRK3 scheme, a remarkable feature of the ROW schemes is that, they only require one approximate Jacobian matrix calculation every time step, thus considerably reducing the overall computational cost. A variety of test cases, ranging from inviscid flows to DNS of turbulent flows, are presented to assess the performance of these schemes. Numerical experiments demonstrate that the third-order ROW scheme for the DAEs of index-2 can not only achieve the designed formal order of temporal convergence accuracy in a benchmark test, but also require significantly less computing time than its ESDIRK3 counterpart to converge to the same level of discretization errors in all of the flow simulations in this study, indicating that the ROW methods provide an attractive alternative for the higher-order time-accurate integration of the unsteady compressible Navier-Stokes equations.

Exponential convergence of mixed hp-DGFEM for the incompressible Navier–Stokes equations in ℝ2

2020 ◽  
Author(s):  
Dominik Schötzau ◽  
Carlo Marcati ◽  
Christoph Schwab
Keyword(s):  
Discontinuous Galerkin ◽  
Degrees Of Freedom ◽  
Stokes Equations ◽  
Rates Of Convergence ◽  
Navier Stokes ◽  
Small Data ◽  
Navier Stokes Equations ◽  
Slip Boundary ◽  
Hp Spaces ◽  
Galerkin Approximations

Abstract In a polygon $\varOmega \subset \mathbb{R}^2$ we consider mixed $hp$-discontinuous Galerkin approximations of the stationary, incompressible Navier–Stokes equations, subject to no-slip boundary conditions. We use geometrically corner-refined meshes and $hp$ spaces with linearly increasing polynomial degrees. Based on recent results on analytic regularity of velocity field and pressure of Leray solutions in $\varOmega$, we prove exponential rates of convergence of the mixed $hp$-discontinuous Galerkin finite element method, with respect to the number of degrees of freedom, for small data which is piecewise analytic.

Implicit numerical simulation of transonic flow through turbine cascades on unstructured grids

2005 ◽  
Vol 219 (1) ◽  
pp. 35-47 ◽  
Author(s):  
Y Mei ◽  
A Guha
Keyword(s):  
Numerical Simulation ◽  
Finite Volume ◽  
Turbulence Model ◽  
Stokes Equations ◽  
Time Integration ◽  
Navier Stokes ◽  
Implicit Time ◽  
Turbine Cascade ◽  
Navier Stokes Equations ◽  
Flow Through

Numerical simulation of the compressible flow through a turbine cascade is studied in the present paper. The numerical solution is performed on self-adaptive unstructured meshes by an implicit method. Computational codes have been developed for solving Euler as well as Navier-Stokes equations with various turbulence modelling. The Euler and Navier-Stokes codes have been applied on a standard turbine cascade, and the computed results are compared with experimental results. A hybrid scheme is used for spatial discretization, where the inviscid fluxes are discretized using a finite volume method while the viscous fluxes are calculated by central differences. A MUSCL-type approach is used for achieving higher-order accuracy. The effects of the turbulent stress terms in the Reynolds-averaged Navier-Stokes equations have been studied with two different models: an algebraic turbulence model (Baldwin-Lomax model) and a two-equation turbulence model ( k-ɛ model). The system of linear equations is solved by a Gauss-Seidel algorithm at each step of time integration. A new treatment of the non-reflection boundary condition is applied in the present study to make it consistent with the finite volume flux calculation and the implicit time discretization.

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