On the stability of projection methods for the incompressible Navier-Stokes equations based on high-order discontinuous Galerkin discretizations

The present paper deals with the numerical solution of the incompressible Navier-Stokes equations using high-order discontinuous Galerkin (DG) methods for discretization in space. For DG methods applied to the dual splitting projection method, instabilities have recently been reported that occur for coarse spatial resolutions and small time step sizes. By means of numerical investigation we give evidence that these instabilities are related to the discontinuous Galerkin formulation of the velocity divergence term and the pressure gradient term that couple velocity and pressure. Integration by parts of these terms with a suitable definition of boundary conditions is required in order to obtain a stable and robust method. Since the intermediate velocity field does not fulfill the boundary conditions prescribed for the velocity, a consistent boundary condition is derived from the convective step of the dual splitting scheme to ensure high-order accuracy with respect to the temporal discretization. This new formulation is stable in the limit of small time steps for both equal-order and mixed-order polynomial approximations. Although the dual splitting scheme itself includes inf-sup stabilizing contributions, we demonstrate that spurious pressure oscillations appear for equal-order polynomials and small time steps highlighting the necessity to consider inf-sup stability explicitly.Comment: 31 page

High-order DG solvers for under-resolved turbulent incompressible flows: A comparison of L2L^2 and HH(div) methods

The accurate numerical simulation of turbulent incompressible flows is a challenging topic in computational fluid dynamics. For discretisation methods to be robust in the under-resolved regime, mass conservation as well as energy stability are key ingredients to obtain robust and accurate discretisations. Recently, two approaches have been proposed in the context of high-order discontinuous Galerkin (DG) discretisations that address these aspects differently. On the one hand, standard $L^2$-based DG discretisations enforce mass conservation and energy stability weakly by the use of additional stabilisation terms. On the other hand, pointwise divergence-free $H(\operatorname{div})$-conforming approaches ensure exact mass conservation and energy stability by the use of tailored finite element function spaces. The present work raises the question whether and to which extent these two approaches are equivalent when applied to under-resolved turbulent flows. This comparative study highlights similarities and differences of these two approaches. The numerical results emphasise that both discretisation strategies are promising for under-resolved simulations of turbulent flows due to their inherent dissipation mechanisms.Comment: 24 pages, 13 figure

Flux Approximation Scheme for the Incompressible Navier-Stokes Equations Using Local Boundary Value Problems

We present a flux approximation scheme for the incompressible Navier- Stokes equations, that is based on a flux approximation scheme for the scalar advection-diffusion-reaction equation that we developed earlier. The flux is computed from local boundary value problems (BVPs) and is expressed as a sum of a homogeneous and an inhomogeneous part. The homogeneous part depends on the balance of the convective and viscous forces and the inhomogeneous part depends on source terms included in the local BVP.</p

An upwind-differencing scheme for the incompressible Navier-Stokes equations

The steady state incompressible Navier-Stokes equations in 2-D are solved numerically using the artificial compressibility formulation. The convective terms are upwind-differenced using a flux difference split approach that has uniformly high accuracy throughout the interior grid points. The viscous fluxes are differenced using second order accurate central differences. The numerical system of equations is solved using an implicit line relaxation scheme. Although the current study is limited to steady state problems, it is shown that this entire formulation can be used for solving unsteady problems. Characteristic boundary conditions are formulated and used in the solution procedure. The overall scheme is capable of being run at extremely large pseudotime steps, leading to fast convergence. Three test cases are presented to demonstrate the accuracy and robustness of the code. These are the flow in a square-driven cavity, flow over a backward facing step, and flow around a 2-D circular cylinder

A "poor man's" approach for high-resolution three-dimensional topology optimization of natural convection problems

This paper treats topology optimization of natural convection problems. A simplified model is suggested to describe the flow of an incompressible fluid in steady state conditions, similar to Darcy's law for fluid flow in porous media. The equations for the fluid flow are coupled to the thermal convection-diffusion equation through the Boussinesq approximation. The coupled non-linear system of equations is discretized with stabilized finite elements and solved in a parallel framework that allows for the optimization of high resolution three-dimensional problems. A density-based topology optimization approach is used, where a two-material interpolation scheme is applied to both the permeability and conductivity of the distributed material. Due to the simplified model, the proposed methodology allows for a significant reduction of the computational effort required in the optimization. At the same time, it is significantly more accurate than even simpler models that rely on convection boundary conditions based on Newton's law of cooling. The methodology discussed herein is applied to the optimization-based design of three-dimensional heat sinks. The final designs are formally compared with results of previous work obtained from solving the full set of Navier-Stokes equations. The results are compared in terms of performance of the optimized designs and computational cost. The computational time is shown to be decreased to around 5-20% in terms of core-hours, allowing for the possibility of generating an optimized design during the workday on a small computational cluster and overnight on a high-end desktop