M. Darwish

TVD schemes for unstructured grids

Abstract A number of approaches have evolved over the last decade for the implementation of total variational diminishing (TVD) schemes within an unstructured grid finite volume method framework. Unfortunately none of these approaches has been comprehensive enough to permit the general implementation of TVD-based schemes in unstructured grids, and/or accurate enough to recover the exact TVD formulation in structured grids. In this paper we propose a simple method that allows the implementation of the full spectrum of TVD schemes in unstructured grids, while recovering their exact formulation on structured grids. Four schemes implemented using this approach, TVD-MINMOD, TVD-MUSCL (monotonic upstream-centered scheme for conservation laws, MUSCL), TVD-SUPERBEE, TVD-OSHER, are tested and compared to Bruner’s TVD formulation [Parallelization of the Euler equations on unstructured grids, AIAA paper 97-1894, 1995], and to the Barth and Jesperson linear reconstruction scheme [The design and application of upwind schemes on unstructured meshes, AIAA paper 89-0366, 1989] by solving four pure advection problems. Results indicate that the Bruner formulation yields, for the same original TVD scheme, overly diffusive results when compared to the current method. The BJ-MUSCL and TVD-MUSCL are shown to be comparable and more accurate than the OSHER scheme. The SUPERBEE performs best though showing tendency for stepping the modeled profile. In all tests the current method is found to retain the behavior of the structured grid TVD formulation.

Convective Schemes for Capturing Interfaces of Free-Surface Flows on Unstructured Grids

ABSTRACT In this article, the general methodology used in constructing interface-capturing schemes is clarified and concisely described. Moreover, a new interface-capturing scheme, denoted STACS and based on a switching strategy, is developed. The accuracy of the new scheme is compared to the well-known CICSAM and HRIC schemes by solving four pure advection test problems. Results, displayed in the form of interface contours for the various schemes, reveal deterioration in the accuracy of the CICSAM and HRIC schemes, with their performance approaching that of the UPWIND scheme as the Courant number increases. On the other hand, predictions obtained with the new STACS scheme are by far more accurate and less diffusive, preserving interface sharpness and boundedness at all Courant number values considered.

The Finite Volume Method

Similar to other numerical methods developed for the simulation of fluid flow, the finite volume method transforms the set of partial differential equations into a system of linear algebraic equations. Nevertheless, the discretization procedure used in the finite volume method is distinctive and involves two basic steps. In the first step, the partial differential equations are integrated and transformed into balance equations over an element. This involves changing the surface and volume integrals into discrete algebraic relations over elements and their surfaces using an integration quadrature of a specified order of accuracy. The result is a set of semi-discretized equations. In the second step, interpolation profiles are chosen to approximate the variation of the variables within the element and relate the surface values of the variables to their cell values and thus transform the algebraic relations into algebraic equations. The current chapter details the first discretization step and presents a broad review of numerical issues pertaining to the finite volume method. This provides a solid foundation on which to expand in the coming chapters where the focus will be on the discretization of the various parts of the general conservation equation. In both steps, the selected approximations affect the accuracy and robustness of the resulting numerics. It is therefore important to define some guiding principles for informing the selection process.

A coupled incompressible flow solver on structured grids

This article deals with the formulation, implementation, and testing of a fully coupled velocity–pressure algorithm for the solution of laminar incompressible flow problems. The tight velocity–pressure coupling is developed within the context of a collocated structured grid, and the systems of equations involving velocity and pressure are solved simultaneously. The pressure and momentum equations are derived in a way similar to a segregated SIMPLE algorithm [1], yielding an extended set of diagonally dominant equations. An algebraic multigrid solver is used to accelerate the solution of the extended system of equations. The performance of the newly developed coupled algorithm is evaluated by solving three test problems showing the effects of grid size, mesh skewness, large pressure gradients, and large source terms on the convergence behavior. Results are presented in the form of convergence history plots and tabulated values of the maximum number of required iterations, the total CPU time, and the CPU time per control volume. This latter performance indicator is shown to be nearly independent of the grid size.

A Fully Coupled Navier-Stokes Solver for Fluid Flow at All Speeds

This article deals with the formulation and testing of a newly developed, fully coupled, pressure-based algorithm for the solution of fluid flow at all speeds. The new algorithm is an extension into compressible flows of a fully coupled algorithm developed by the authors for laminar incompressible flows. The implicit velocity–pressure–density coupling is resolved by deriving a pressure equation following a procedure similar to a segregated SIMPLE algorithm using the Rhie-Chow interpolation technique. The coefficients of the momentum and continuity equations are assembled into one matrix and solved simultaneously, with their convergence accelerated via an algebraic multigrid method. The performance of the coupled solver is assessed by solving a number of two-dimensional problems in the subsonic, transsonic, supersonic, and hypersonic regimes over several grid systems of increasing sizes. For a desired level of convergence, results for each problem are presented in the form of convergence history plots, tabulated values of the maximum number of required iterations, the total CPU time, and the CPU time per control volume.

New bounded skew central difference scheme, Part I: Formulation and testing

The skew central difference scheme is combined with the normalized variable formulation to yield a new bounded skew central difference scheme. The newly developed scheme is tested and compared with the upwind scheme, the bounded skew upwind scheme, and the high-resolution SMART scheme by solving four problems: (1) pure convection of a step profile in an oblique velocity field; (2) sudden expansion of an oblique flow field in a rectangular cavity; (3) driven flow in a skew cavity; and (4) gradual expansion in an axisymmetric, nonorthogonal channel. Results generated reveal the new scheme to be bounded and to be the most accurate among those investigated.

A COMPARATIVE ASSESSMENT WITHIN A MULTIGRID ENVIRONMENT OF SEGREGATED PRESSURE-BASED ALGORITHMS FOR FLUID FLOW AT ALL SPEEDS

This article deals with the evaluation of six segregated high-resolution pressure-based algorithms, which extend the SIMPLE, SIMPLEC, PISO, SIMPLEX, SIMPLEST, and PRIME algorithms, originally developed for incompressible flow, to compressible flow simulations. The algorithms are implemented within a single grid, a prolongation grid, and a full multigrid method and their performance assessed by solving problems in the subsonic, transonic, supersonic, and hypersonic regimes. This study clearly demonstrates that all algorithms are capable of predicting fluid flow at all speeds and qualify as efficient smoothers in multigrid calculations. In terms of CPU efficiency, there is no global and consistent superiority of any algorithm over the others, even though PRIME and SIMPLEST are generally the most expensive for inviscid flow problems. Moreover, these two algorithms are found to be very unstable in most of the cases tested, requiring considerable upwind bleeding (up to 50%) of the high-resolution scheme to promote convergence. The most stable algorithms are SIMPLEC and SIMPLEX. Moreover, the reduction in computational effort associated with the prolongation grid method reveals the importance of initial guess in segregated solvers. The most efficient method is found to be the full multigrid method, which resulted in a convergence acceleration ratio, in comparison with the single grid method, as high as 18.4.

Parallelization of an Additive Multigrid Solver

This article deals with the implementation and performance analysis of a parallel algebraic multigrid solver (pAMG) for a finite-volume, unstructured computational fluid dynamics (CFD) code. The parallelization of the solver is based on the domain decomposition approach using the single program, multiple data paradigm. The Message Passing Interface library (MPI) is used for communication of data. An ILU(0) iterative solver is used for smoothing the errors arising within each partition at the different grid levels, and a multi-level synchronization across the computational domain partitions is enforced in order to improve the performance of the parallelized multigrid solver. Two synchronization strategies are evaluated. In the first the synchronization is applied across the multigrid levels during the restriction step in addition to the base level, while in the second the synchronization is enforced during the restriction and prolongation steps. The effect of gathering the coefficients across partitions for the coarsest level is also investigated. Tests on grids up to 800,000 elements are conducted for a number of diffusion and advection problems on up to 20 processors. Results show that synchronization across partitions for multigrid levels plays an essential role in ensuring good scalability. Furthermore, for a large number of partitions, gathering coefficients across partitions is important to ensure a convergence history that is consistent with the sequential solver, thus yielding the same number of iterations for parallel and sequential runs, which is crucial for retaining high scalability. The shadow-to-core elements ratio is also shown to be a good indicator for scalability.

Fully Implicit Coupling for Non‐Matching Grids

The efficient solution of flow problems depends on quality meshing the computational domain. In problems with complex geometries or having a large spectrum of time or length scale, the meshing process greatly benefits from the subdivision of the original geometry (domain decomposition) into sub‐domains, that are meshed independently with suitable elements and mesh density. Procedures for solving multiblock meshes can be of two types explicit or implicit. In either case it is essential that the fluxes at the regions interfaces be conserved. In this paper an efficient fully implicit multi‐region coupling discretization procedure is presented. A test problem involving 1, 2, 4 and 8 blocks with a mesh size of about 100,000 elements, is solved to show that the coupling procedure yields the same number of iteration for multiple block as for a single block.

Fully implicit method for coupling multiblock meshes with nonmatching interface grids

ABSTRACT A newly developed efficient and fully implicit method for multiblock mesh coupling that preserves the convergence characteristics of single-block meshing is presented. The technique is developed in the context of an unstructured pressure-based collocated finite-volume method, is applicable to both segregated and coupled flow solvers, and is ideal for code parallelization. The discretization at interfaces is performed in a separate step to stitch the regions sub-matrices into a global matrix. By solving the global matrix, the solution achieved to the multiregion problem is exactly the one that would result from a single-mesh discretization. The method is tested by solving three laminar flow problems. Solutions are obtained by meshing the domain as one block or by subdividing it into a number of blocks with non-matching grids at interfaces. Results show the very tight coupling at interfaces with a convergence rate that is independent of the number of blocks used.