A parallel adaptive numerical method with generalized curvilinear coordinate transformation for compressible Navier–Stokes equations

Abstract

SUMMARYA fourth-order finite-volume method for solving the Navier-Stokes equations on a mapped grid with adaptive mesh refinement is proposed, implemented, and demonstrated for the prediction of unsteady compressible viscous flows. The method employs fourth-order quadrature rules for evaluating face-averaged fluxes. Our approach is freestream preserving, guaranteed by the way of computing the averages of the metric terms on the faces of cells. The standard Runge-Kutta marching method is used for time discretization. Solutions of a smooth flow are obtained in order to verify that the method is formally fourth-order accurate when applying the nonlinear viscous operators on mapped grids. Solutions of a shock tube problem are obtained to demonstrate the effectiveness of adaptive mesh refinement in resolving discontinuities. A Mach reflection problem is solved to demonstrate the mapped algorithm on a non-rectangular physical domain. The simulation is compared against experimental results. Future work will consider mapped multiblock grids for practical engineering geometries.

A Fourth-Order Finite-Volume Method with Adaptive Mesh Refinement for the Multifluid Plasma Model

“…This reformulation mathematically satisfies the divergence constraints (Eqs. (9) and (10)). By assuming the Coulomb gauge approximation, where…”

“…Additionally, Chord supports MPI type parallelization. [5][6][7][8][9][10] We propose implementing a new approach to solving the governing equations of multi-fluid, continuum plasmas, by leveraging the AMR functionality and highly parallelizable framework within Chord. This proposed solution can be 4th-order accurate for most equations, improving on the numerical error compared to existing plasma codes.…”

A high-performance finite-volume algorithm for solving partial differential equations governing compressible viscous flows on structured grids

“…This work focuses on the development of new strategies to enhance the high performance of a CFD algorithm, namely Chord [3,4,5,6,7,8], solving PDEs 20 governing compressible fluids on Cartesian grids. The novelty of the present study is to introduce the loop chaining concept [9,10] to a modern CFD code for the first time in order to achieve significant improvement in parallel performance on modern computers.…”

“…To generate Cartesian grids for complex geometries, one can use embedded boundary (cut-cell) techniques, or mapping a structured grid 60 in physical space to a Cartesian grid in computational space. The latter approach effectively recovers Cartesian methods with the cost of some additional complexity in terms of grid metrics, and we have performed a significant amount of work on mapped grids [5,8] in which the necessary mathematical and numerical details are documented. The present study concerns the computational 65…”

“…8, the spatial terms on the right hand side must be evaluated, and then the ODEs will be evolved in siderations for minimizing the size of stencil without compromising the fourthorder accuracy. The sequence of transformations between the conservative and primitive variables is detailed in previous works [16,5,8].…”