An Investigation of Continuous and Discontinuous Finite-Element Discretizations on Benchmark 3D Turbulent Flows (Invited)

Abstract

Two high-order finite-element solvers are used to simulate two 3D benchmark problems provided by the NASA Turbulence Modeling Resource (TMR) website. The first problem is a subsonic turbulent flow over a hemisphere-cylinder body at angles of attack of 5 and 19 degrees. The second problem is a transonic turbulent flow over an ONERA M6 wing at angles of attack of 3.06 and 6.06 degrees. The first finite-element solver is based on the streamline-upwind Petrov-Galerkin (SUPG) discretization and the second one is based on the discontinuous Galerkin (DG) method. For both problems, second-and higher-order solutions are provided and a detailed mesh convergence study is presented. Also, the linear and nonlinear convergence behavior of the utilized solvers is investigated. The SUPG solver shows a very consistent non-linear convergence behavior across all grids while the DG solver struggles on the coarse grids. Both finite-element solvers demonstrate rapid convergence for high-order simulations when initialized with a second-order accurate solution. For all hemisphere-cylinder cases and the M6 wing at angle of attack of 3.06 degrees, the finiteelement solvers are outperforming the FUN3D finite-volume solver in terms of accuracy per degree of freedom. For the M6 wing at angle of attack of 6.06 degrees, although no reference solutions were provided by the TMR website, simulations were attempted for the finiteelement solvers. For this case, no conclusions can be made about asymptotic convergence of the integrated forces due to lack of consistency between results. This was initially thought to be caused by the provided grids. However, further investigation using custom unstructured grids leads us to believe that the flow conditions, not the grids, make this case challenging.

Comparison of ROW, ESDIRK, and BDF2 for Unsteady Flows with the High-Order Flux Reconstruction Formulation

“…High-order computational fluid dynamics (CFD) methods have been attracting much research attention in past decades due to their high-resolution and low-dissipation properties that enable high-fidelity simulation of intricate flows. High-order spatial discretization methods, such as discontinuous Galerkin (DG) methods [1,2,3] and FR/CPR methods [4,5,6,7], have shown their capabilities of dealing with turbulent flows [8,9,10,11,12,13,14,15]. Usually, the high-order explicit strong stability preserving Runge-Kutta (SSPRK) methods [16] are used to integrate the semidiscretized governing equations for unsteady flow simulation.…”

A comparative study of implicit Jacobian-free Rosenbrock-Wanner, ESDIRK and BDF methods for unsteady flow simulation with high-order flux reconstruction formulations

“…High-order computational fluid dynamics (CFD) methods have been attracting much research attention in past decades due to their superior numerical properties that enable high-fidelity simulation of intricate flows. High-order spatial discretization methods, such as discontinuous Galerkin (DG) methods [1,2,3], and flux reconstruction/correction procedure via reconstruction (FR/CPR) methods [4,5,6,7], have shown their capabilities of dealing with turbulent flows [8,9,10,11,12,13,14,15]. Usually, the high-order explicit strong stability preserving Runge-Kutta (SSPRK) methods [16] are used to integrate the semi-discretized governing equations for unsteady flow simulation.…”

A Residual Smoothing Strategy for Accelerating Newton Method Continuation

“…Newton methods have become popular strategies for solving large-systems of non-linear equations. In the field of computational fluid dynamics (CFD) for aerodynamics, Newton methods have enjoyed a resurgence in popularity, largely due to their ability to provide deep convergence levels for stiff systems of equations such as those resulting from emerging continuous and discontinuous Galerkin discretizations particularly for highly resolved steady-state problems [1,2,3,4]. In the final stages of convergence, when the iterative solution state is close to the exact nonlinear solution, Newton methods converge in a small number of nonlinear steps, and each step can often be solved effectively using preconditioned Krylov methods which are generally robust [5,6,7].…”

“…However, for most cases, a continuation strategy must be employed to iteratively approach the nonlinear state where fast convergence is obtained. A typical strategy consists of employing a pseudo-transient approach where a pseudo-time term is added to the diagonal of the Jacobian matrix with a variable pseudo-time step [8,9,4]. In the initial phases of convergence, when the pseudo-time step is small, the method approximates a pseudo-time explicit scheme, and in the final stages, when the pseudo-step becomes large, the exact Newton method is recovered.…”

“…In Section 2.2, this formulation is generalized and two possible interpretations are discussed. In Section 3 example solutions are given for a steady-state and a time-dependent aerodynamic Reynolds-averaged Navier-Stokes (RANS) CFD problem on unstructured meshes, demonstrating a more robust and efficient solution strategy compared to the traditional pseudo-transient continuation Newton-Krylov approach [8,9,4]. In the conclusion section, prospects for further improvements to this approach are discussed.…”