Segregated Runge–Kutta time integration of convection-stabilized mixed finite element schemes for wall-unresolved LES of incompressible flows

In this work, we develop a high-performance numerical framework for the large eddy simulation (LES) of incompressible flows. The spatial discretization of the nonlinear system is carried out using mixed finite element (FE) schemes supplemented with symmetric projection stabilization of the convective term and a penalty term for the divergence constraint. These additional terms introduced at the discrete level have been proved to act as implicit LES models. In order to perform meaningful wall-unresolved simulations, we consider a weak imposition of the boundary conditions using a Nitsche’s-type scheme, where the tangential component penalty term is designed to act as a wall law. Next, segregated Runge–Kutta (SRK) schemes (recently proposed by the authors for laminar flow problems) are applied to the LES simulation of turbulent flows. By the introduction of a penalty term on the trace of the acceleration, these methods exhibit excellent stability properties for both implicit and explicit treatment of the convective terms. SRK schemes are excellent for large-scale simulations, since they reduce the computational cost of the linear system solves by splitting velocity and pressure computations at the time integration level, leading to two uncoupled systems. The pressure system is a Darcy-type problem that can easily be preconditioned using a traditional block-preconditioning scheme that only requires a Poisson solver. At the end, only coercive systems have to be solved, which can be effectively preconditioned by multilevel domain decomposition schemes, which are both optimal and scalable. The framework is applied to the Taylor–Green and turbulent channel flow benchmarks in order to prove the accuracy of the convection-stabilized mixed FEs as LES models and SRK time integrators. The scalability of the preconditioning techniques (in space only) has also been proven for one step of the SRK scheme for the Taylor–Green flow using uniform meshes. Moreover, a turbulent flow around a NACA profile is solved to show the applicability of the proposed algorithms for a realistic problem.Peer ReviewedPostprint (author's final draft

Aspects of Unstructured Grids and Finite-Volume Solvers for the Euler and Navier-Stokes Equations

One of the major achievements in engineering science has been the development of computer algorithms for solving nonlinear differential equations such as the Navier-Stokes equations. In the past, limited computer resources have motivated the development of efficient numerical schemes in computational fluid dynamics (CFD) utilizing structured meshes. The use of structured meshes greatly simplifies the implementation of CFD algorithms on conventional computers. Unstructured grids on the other hand offer an alternative to modeling complex geometries. Unstructured meshes have irregular connectivity and usually contain combinations of triangles, quadrilaterals, tetrahedra, and hexahedra. The generation and use of unstructured grids poses new challenges in CFD. The purpose of this note is to present recent developments in the unstructured grid generation and flow solution technology

High order resolution and parallel implementation on unstructured grids

The numerical solution of the two-dimensional inviscid Euler flow equations is given. The unstructured mesh is generated by the advancing front technique. A cell-centred upwind finite volume method has been adopted to discretize the Euler equations. Both explicit and point implicit time stepping algorithms are derived. The flux calculation using Roe's and Osher's approximate Riemann solvers are studied. It is shown that both the Roe and Osher's schemes produce an accurate representation of discontinuities (e.g. shock wave). It is also shown that better convergence performance has been achieved by the point implicit scheme than that by the explicit scheme. Validations have been done for subsonic and transonic flow over airfoils, supersonic flow past a compression corner and hypersonic flow past cylinder and blunt body geometries. An adaptive remeshing procedure is also applied to the numerical solution with the objective of getting improved results. The issue of high order reconstruction on unstructured grids has been discussed. The methodology of the Taylor series expansion is adopted. The calculation of the gradient at a reference point is carried out by the use of either the Green-Gauss integral formula or the least-square methods. Some recently developed limiter construction methods have been used and their performance has been demonstrated using the test example of the transonic flow over a RAE 2822 airfoil. It has been shown that similar pressure distributions are obtained by all limiters except for shock wave regions where the limiter is active. The convergence problem is illustrated by the mid-mod type limiter. It seems only the Venkatakrishnan limiter provides improved convergence. Other limiters do not appear to work as well as that shown in their original publications. Also the convergence history given by the least-square method appears better than that by the Green-Gauss method in the test

Recent Advances in Graph Partitioning

We survey recent trends in practical algorithms for balanced graph partitioning together with applications and future research directions