A high‐order accurate discontinuous Galerkin finite element method for laminar low Mach number flows

Summary In this paper, we present an efficient semi‐implicit scheme for the solution of the Reynolds‐averaged Navier‐Stokes equations for the simulation of hydrostatic and nonhydrostatic free surface flow problems. A staggered unstructured mesh composed by Voronoi polygons is used to pave the horizontal domain, whereas parallel layers are adopted along the vertical direction. Pressure, velocity, and vertical viscosity terms are taken implicitly, whereas the nonlinear convective terms as well as the horizontal viscous terms are discretized explicitly by using a semi‐Lagrangian approach, which requires an interpolation of the three‐dimensional velocity field to integrate the flow trajectories backward in time. To this purpose, a high‐order reconstruction technique is proposed, which is based on a constrained least squares operator that guarantees a globally and pointwise divergence‐free velocity field. A comparison with an analogous reconstruction, which is not divergence‐free preserving, is also presented to give evidence of the new strategy. This allows the continuity equation to be satisfied up to machine precision even for high‐order spatial discretizations. The reconstructed velocity field is then used for evaluating high‐order terms of a Taylor method that is here adopted as ODE integrator for the flow trajectories. The proposed semi‐implicit scheme is validated against a set of academic test problems, and proof of convergence up to fourth‐order of accuracy in space is shown.

Section: Introductionmentioning

confidence: 99%

High‐order divergence‐free velocity reconstruction for free surface flows on unstructured Voronoi meshes

Boscheri

Pisaturo

Righetti

2019

“…While in general the convergence is decreasing for lower Mach number, this behavior could be improved by suitable preconditioning. In the series of papers , a discontinuous Galerkin solver for low Mach numbers based on the compressible Navier–Stokes equations is developed. To decrease the stiffness of the system for low Mach numbers, preconditioning techniques are used.…”

Section: Introductionmentioning

confidence: 99%

A high‐order discontinuous Galerkin solver for low Mach number flows

Klein

Müller

Kummer

et al. 2015

Summary In this work, we present a high‐order discontinuous Galerkin method (DGM) for simulating variable density flows at low Mach numbers. The corresponding low Mach number equations are an approximation of the compressible Navier–Stokes equations in the limit of zero Mach number. To the best of the authors'y knowledge, it is the first time that the DGM is applied to the low Mach number equations. The mixed‐order formulation is applied for spatial discretization. For steady cases, we apply the semi‐implicit method for pressure‐linked equation (SIMPLE) algorithm to solve the non‐linear system in a segregated manner. For unsteady cases, the solver is implicit in time using backward differentiation formulae, and the SIMPLE algorithm is applied to solve the non‐linear system in each time step. Numerical results for the following three test cases are shown: Couette flow with a vertical temperature gradient, natural convection in a square cavity, and unsteady natural convection in a tall cavity. Considering a fixed number of degrees of freedom, the results demonstrate the benefits of using higher approximation orders. Copyright © 2015 John Wiley & Sons, Ltd.

“…The DG solution is extended to the incompressible limit by implementing a low Mach number preconditioning that affects both the time‐derivative terms of the governing equations and the numerical dissipation of Roe's Riemann solver with Harten's entropy fix through the action of a modified version of the Turkel preconditioning matrix . This approach, the so‐called full preconditioning, is particularly efficient for steady state computations . At sonic speed the local preconditioner smoothly reduces to the identity matrix, thus recovering the non‐preconditioned DG discretization and preserving the accuracy and performance of the method for solving high‐speed flows.…”

Section: Introductionmentioning

confidence: 99%

A high‐order discontinuous Galerkin method for all‐speed flows

Renda

Hartmann

Bartolo

et al. 2014

Self Cite

SUMMARYIn this article, we present a discontinuous Galerkin (DG) method designed to improve the accuracy and efficiency of steady solutions of the compressible fully coupled Reynolds-averaged Navier-Stokes and k ! turbulence model equations for solving all-speed flows. The system of equations is iterated to steady state by means of an implicit scheme. The DG solution is extended to the incompressible limit by implementing a low Mach number preconditioning technique. A full preconditioning approach is adopted, which modifies both the unsteady terms of the governing equations and the dissipative term of the numerical flux function by means of a new preconditioner, on the basis of a modified version of Turkel's preconditioning matrix. At sonic speed the preconditioner reduces to the identity matrix thus recovering the non-preconditioned DG discretization. An artificial viscosity term is added to the DG discretized equations to stabilize the solution in the presence of shocks when piecewise approximations of order of accuracy higher than 1 are used. Moreover, several rescaling techniques are implemented in order to overcome ill-conditioning problems that, in addition to the low Mach number stiffness, can limit the performance of the flow solver. These approaches, through a proper manipulation of the governing equations, reduce unbalances between residuals as a result of the dependence on the size of elements in the computational mesh and because of the inherent differences between turbulent and mean-flow variables, influencing both the evolution of the Courant Friedrichs Lewy (CFL) number and the inexact solution of the linear systems. The performance of the method is demonstrated by solving three turbulent aerodynamic test cases: the flat plate, the L1T2 high-lift configuration and the RAE2822 airfoil (Case 9). The computations are performed at different Mach numbers using various degrees of polynomial approximations to analyze the influence of the proposed numerical strategies on the accuracy, efficiency and robustness of a high-order DG solver at different flow regimes.