Numerical analysis of conservative unstructured discretisations for low Mach flows

Abstract

Summary Unstructured meshes allow easily representing complex geometries and to refine in regions of interest without adding control volumes in unnecessary regions. However, numerical schemes used on unstructured grids have to be properly defined in order to minimise numerical errors. An assessment of a low Mach algorithm for laminar and turbulent flows on unstructured meshes using collocated and staggered formulations is presented. For staggered formulations using cell‐centred velocity reconstructions, the standard first‐order method is shown to be inaccurate in low Mach flows on unstructured grids. A recently proposed least squares procedure for incompressible flows is extended to the low Mach regime and shown to significantly improve the behaviour of the algorithm. Regarding collocated discretisations, the odd–even pressure decoupling is handled through a kinetic energy conserving flux interpolation scheme. This approach is shown to efficiently handle variable‐density flows. Besides, different face interpolations schemes for unstructured meshes are analysed. A kinetic energy‐preserving scheme is applied to the momentum equations, namely, the symmetry‐preserving scheme. Furthermore, a new approach to define the far‐neighbouring nodes of the quadratic upstream interpolation for convective kinematics scheme is presented and analysed. The method is suitable for both structured and unstructured grids, either uniform or not. The proposed algorithm and the spatial schemes are assessed against a function reconstruction, a differentially heated cavity and a turbulent self‐igniting diffusion flame. It is shown that the proposed algorithm accurately represents unsteady variable‐density flows. Furthermore, the quadratic upstream interpolation for convective kinematics scheme shows close to second‐order behaviour on unstructured meshes, and the symmetry‐preserving is reliably used in all computations. Copyright © 2016 John Wiley & Sons, Ltd.

A pressure-based method for weakly compressible two-phase flows under a Baer-Nunziato type model with generic equations of state and pressure and velocity disequilibrium

“…The idea of multiple pressure variables can be implemented in several ways 38,46,73 , here we follow the strategy proposed by Bijl and Wesseling 38,68 , who defined the pressure scaling as…”

“…This strategy is called semi-implicit time integration 74,72 , and it is a common feature of pressure-based schemes, shared especially by the schemes that are able to represent all-Mach numbers-i.e., from very small to Mach numbers of order one. The concept can been easily implemented in a semi-discrete, fractional step projection method, in which the equations to be solved sequentially contain both implicit and explicit terms 67,53,75,55,73,51 . This is the strategy adopted in the present work, but, alternatively, a similar idea can be enforced by splitting the fluxes in two parts, advective and non-advective 47,52,76 .…”

A dynamic load balancing method for the evaluation of chemical reaction rates in parallel combustion simulations

“…The governing equations for the transported scalars are solved in non-conservative form due to the employed time-integration strategy, based on a predictor-corrector scheme for low-Mach number flows further detailed in the work of Ventosa et al [20]. At each step of the predictor-corrector scheme, first the transported scalars are integrated in non-conservative form in order to compute the density at the new time step, which is afterwards employed in the integration of the momentum equation.…”

“…Consul [29]. The splitting technique used in the present work has been developed for the predictor-corrector scheme detailed in previous works [20,30]. In the predictor step scalars are integrated using a second-order Adams-Bashforth time integration scheme.…”

A staggered semi-implicit hybrid FV/FE projection method for weakly compressible flows

“…In addition, a state equation is needed. Several finite volume, finite difference and finite element methods initially developed for incompressible flows have been extended to the weakly compressible and low Mach regimes (see [34,35,36,37,38,39,40]). Furthermore, some of the algorithms have been developed to solve all Mach number flows, which proves the potential of the extension of this family of schemes (see [41,42,43,44,45,46,47]).…”