Krista Nerinckx

A Mach-uniform algorithm: Coupled versus segregated approach

A Mach-uniform algorithm is an algorithm with a good convergence rate for any level of the Mach number. In this paper, the severe time step restriction for low speed flows is removed by treating the acoustic and diffusive terms implicitly. After identification of these terms in the conservative set, we end up with a semi-implicit system. The way to solve this system can be chosen. Three different solution techniques are presented: a fully coupled algorithm, the coupled pressure and temperature correction algorithm from [K. Nerinckx, J. Vierendeels and E. Dick. Mach-uniformity through the coupled pressure and temperature correction algorithm. Journal of Computational Physics, 206(2) (2005) 597-623], and a fully segregated pressure-correction algorithm. We analyse the convergence behavior of the considered algorithms in some typical flow problems. Moreover, a Fourier stability analysis is done for each of the algorithms. For inviscid flow, the fully segregated and the fully coupled algorithm need about as much time steps to reach steady state. Therefore, the more segregation is introduced, the cheaper the calculation can be done. In case of heat transfer, the fully segregated pressure-correction algorithm suffers from a diffusive time step limit. This is not the case for the semi-segregated coupled pressure and temperature correction algorithm. Finally, when the gravity terms play an important role, only the fully coupled algorithm can avoid an additional time step restriction.

A Mach‐uniform pressure‐correction algorithm with AUSM+ flux definitions

Purpose – To present conversion of the advection upwind splitting method (AUSM+) from the conventional density‐based and coupled formulation to the pressure‐based and segregated formulation.Design/methodology/approach – The spatial discretization is done by a finite volume method. A collocated grid cell‐center formulation is used. The pressure‐correction procedure is set up in the usual way for a compressible flow problem. The conventional Rhie‐Chow interpolation methodology for the determination of the transporting velocity, and the conventional central interpolation for the pressure at the control volume faces, are replaced by AUSM+ definitions.Findings – The AUSM+ flux definitions are spontaneously well suited for use in a collocated pressure‐correction formulation. The formulation does not require extensions to these flux definitions. As a consequence, the results of a density‐based fully coupled method, are identical to the results of a pressure‐based segregated formulation. The advantage of the pres...

A Mach-uniform pressure correction algorithm using AUSM flux definitions

A collocated finite-volume pressure correction procedure for the solution of inviscid compressible flow at all speeds is presented. Pressure correction methods usually adopt a so-called Rhie-Chow interpolation for the cell face velocities in order to provide pressure-velocity coupling. However, as is shown on the testcase of a one-dimensional transonic nozzle, this Rhie-Chow interpolation becomes highly diffusive in high Mach number flows, resulting in an extreme smearing of the shock. Therefore we replace the Rhie-Chow interpolation for velocity and the central interpolation for pressure by AUSM+ definitions. This results in a much sharper shock capturing, even with a first order scheme. However, the diffusive contributions of this flux scale badly when the Mach number diminishes. Furthermore, pressure-velocity coupling at low Mach numbers has to be provided. These two problems can be resolved by respectively introducing a preconditioned speed of sound and adding a pressure-diffusion component. The latter resembles the artificial dissipation introduced by the Rhie-Chow interpolation, but differently it is turned off when sonic values are reached.

Stability of Pressure-Correction Algorithms for Low-Speed Reacting and Non-Reacting Flow Simulations

A promising technique for reliable and accurate simulations of turbulent nonpremixed flames is large-eddy simulation (LES). With this technique timeaccurate flow field predictions are imperative and efforts for improvement of algorithms in terms of computational cost are of great value. One way of reducing computing time is to solve the Navier-Stokes equations in their low-Mach number formulation, and to apply a segregated solution method to the set of equations. Such methods are called pressure-correction methods, since the kinematic part of the pressure in low-Mach number flows acts as a variable whose magnitude is determined by imposing a constraint on the velocity field in a corrector step. In flows where the density remains constant in time and space, these methods have been extensively used [1] and do not suffer from inexpected stability problems. However, when these methods are applied to variable density flows in the context of non-premixed combustion simulations, instabilities arise.

An Efficient and Accurate Pressure-Correction Method for All Mach Numbers

We have presented a new type of algorithm: the coupled pressure and temperature correction algorithm. The essential idea is the separation of the convective phenomenon on the one side, and the acoustic/thermodynamic phenomenon on the other side. Based on a theoretical analysis, the algorithm was constructed so that Mach-uniform accuracy and efficiency are obtained, which was confirmed by the test results. Especially the removal of the acoustic and diffusive time step limits is an important feature. When the special case of a perfect gas without heat transfer is considered, the algorithm reduces to a fully segregated method with a pressure-correction equation based on the energy equation.