Barry Koren

Defect correction and multigrid for an efficient and accurate computation of airfoil flows

Results are presented for an efficient solution method for second-order accurate discretizations of the 2D steady Euler equations. The solution method is based on iterative defect correction. Several schemes are considered for the computation of the second-order defect. In each defect correction cycle, the solution is computed by non-linear multigrid iteration, in which collective symmetric Gauss-Seidel relaxation is used as the smoothing procedure. A finite volume Osher discretization is applied throughout. The computational method does not require any tuning of parameters. The airfoil flow solutions obtained show a good resolution of all flow phenomena and are obtained at low computational costs. The rate of convergence is grid-independent. The method contributes to the state of the art in efficiently computing flows with discontinuities.

A new formulation of Kapila's five-equation model for compressible two-fluid flow, and its numerical treatment

A new formulation of Kapilas five-equation model for inviscid, non-heat-conducting, compressible two-fluid flow is derived, together with an appropriate numerical method. The new formulation uses flow equations based on conservation laws and exchange laws only. The two fluids exchange momentum and energy, for which exchange terms are derived from physical laws. All equations are written as a single system of equations in integral form. No equation is used to describe the topology of the two-fluid flow. Relations for the Riemann invariants of the governing equations are derived, and used in the construction of an Osher-type approximate Riemann solver. A consistent finite-volume discretization of the exchange terms is proposed. The exchange terms have distinct contributions in the cell interior and at the cell faces. For the exchange-term evaluation at the cell faces, the same Riemann solver as used for the flux evaluation is exploited. Numerical results are presented for two-fluid shock-tube and shock-bubble-interaction problems, the former also for a two-fluid mixture case. All results show good resemblance with reference results.

Upwind schemes, multigrid and defect correction for the steady Navier-Stokes equations

2.1. Evaluation of diffusive fluxes For the evaluation of the diffusive fluxes at a volume wall, it is necessary to compute grad(u), grad(v) and grad(c 2) at that wail. For this we use the standard technique as outlined in [11]. The technique applied is central, the directional dependence coming from the cross derivative terms is neglected. For sufficiently smooth grids this flux computation is second-order accurate.

A pressure-invariant conservative Godunov-type method for barotropic two-fluid flows

Discretizations of two-fluid flow problems in conservative formulation generally exhibit pressure oscillations. In this work we show that these pressure oscillations are induced by the loss of a pressure-invariance property under discretization, and we introduce a non-oscillatory conservative method for barotropic two-fluid flows. The conservative formulation renders the two-fluid flow problem suitable to treatment by a Godunov-type method. We present a modified Osher scheme for the two-fluid flow problem. Numerical results are presented for a translating-interface test case and a shock/interface-collision test case.

Accuracy analysis of explicit Runge-Kutta methods applied to the incompressible Navier-Stokes equations

This paper investigates the temporal accuracy of the velocity and pressure when explicit Runge-Kutta methods are applied to the incompressible Navier-Stokes equations. It is shown that, at least up to and including fourth order, the velocity attains the classical order of accuracy without further constraints. However, in case of a time-dependent gradient operator, which can appear in case of time-varying meshes, additional order conditions need to be satisfied to ensure the correct order of accuracy. Furthermore, the pressure is only first-order accurate unless additional order conditions are satisfied. Two new methods that lead to a second-order accurate pressure are proposed, which are applicable to a certain class of three- and four-stage methods. A special case appears when the boundary conditions for the continuity equation are independent of time, since in that case the pressure can be computed to the same accuracy as the velocity field, without additional cost. Relevant computations of decaying vortices and of an actuator disk in a time-dependent inflow support the analysis and the proposed methods.

Adjoint-based aerodynamic shape optimization on unstructured meshes

In this paper, the exact discrete adjoint of an unstructured finite-volume formulation of the Euler equations in two dimensions is derived and implemented. The adjoint equations are solved with the same implicit scheme as used for the flow equations. The scheme is modified to efficiently account for multiple functionals simultaneously. An optimization framework, which couples an analytical shape parameterization to the flow/adjoint solver and to algorithms for constrained optimization, is tested on airfoil design cases involving transonic as well as supersonic flows. The effect of some approximations in the discrete adjoint, which aim at reducing the complexity of the implementation, is shown in terms of optimization results rather than only in terms of gradient accuracy. The shape-optimization method appears to be very efficient and robust.

Using coordination to parallelize sparse-grid methods for 3-D CFD problems

The good parallel computing properties of sparse-grid solution techniques are investigated. For this, an existing sequential CFD code for a standard 3D problem from computational aerodynamics is restructured into a parallel application. The restructuring is organized according to a master/slave protocol. The coordinator modules developed thereby are implemented in the coordination language Manifold and are generally applicable. Performance results are given for both the sequential and parallel version of the code. The results are promising, the paper contributes to the state-of-the-art in improving the efficiency of large-scale computations. Also a theoretical analysis is made of speed-up through parallelization in a multi-user single-machine environment.

Finite-difference schemes for anisotropic diffusion

In fusion plasmas diffusion tensors are extremely anisotropic due to the high temperature and large magnetic field strength. This causes diffusion, heat conduction, and viscous momentum loss, to effectively be aligned with the magnetic field lines. This alignment leads to different values for the respective diffusive coefficients in the magnetic field direction and in the perpendicular direction, to the extent that heat diffusion coefficients can be up to 1012 times larger in the parallel direction than in the perpendicular direction. This anisotropy puts stringent requirements on the numerical methods used to approximate the MHD-equations since any misalignment of the grid may cause the perpendicular diffusion to be polluted by the numerical error in approximating the parallel diffusion. Currently the common approach is to apply magnetic field-aligned coordinates, an approach that automatically takes care of the directionality of the diffusive coefficients. This approach runs into problems at x-points and at points where there is magnetic re-connection, since this causes local non-alignment. It is therefore useful to consider numerical schemes that are tolerant to the misalignment of the grid with the magnetic field lines, both to improve existing methods and to help open the possibility of applying regular non-aligned grids. To investigate this, in this paper several discretisation schemes are developed and applied to the anisotropic heat diffusion equation on a non-aligned grid.

Development of the Discrete Adjoint for a Three-Dimensional Unstructured Euler Solver

The discrete adjoint of a reconstruction-based unstructured finite volume formulation for the Euler equations is derived and implemented. The matrix-vector products required to solve the adjoint equations are computed on-the-fly by means of an efficient two-pass assembly. The adjoint equations are solved with the same solution scheme adopted for the flow equations. The scheme is modified to efficiently account for the simultaneous solution of several adjoint equations. The implementation is demonstrated on wing and wing-body configurations.

Multigrid solution method for the steady RANS equations

A novel multigrid method for the solution of the steady Reynolds-averaged Navier-Stokes equations is presented, that gives convergence speeds similar to laminar flow multigrid solvers. The method is applied to Menters one-equation turbulence model. New aspects of the method are the combination of nonlinear Gauss-Seidel smoothing on the finest grid with linear coarse-grid corrections, and local damping in the initial stages of the computation, to keep the solution stable; the damping needed is estimated with the nonlinear smoother. Efficiency on the finest grid is increased with full multigrid, second-order accuracy is obtained with defect correction. Tests on boundary layers and airfoil flows show the efficiency of the method.