Development of a fully coupled control‐volume finite element method for the incompressible Navier–Stokes equations

Abstract

SUMMARYThis paper proposes and investigates fully coupled control-volume ÿnite element method (CVFEM) for solving the two-dimensional incompressible Navier-Stokes equations. The proposed method borrows many of its features from the segregated CVFEM described by Baliga et al. Thus ÿnite-volume discretization is employed on a colocated grid using either the MAW or the FLO schemes and an element-by-element assembling procedure is applied for the construction of the discretizations equations. In this paper, and unlike the case for most fully coupled formulations available in the literature, the Poisson pressure equation has been retained from the segregated approach. The use of a pressure equation leads to an unfavourable size increase of the fully coupled linear system, but signiÿcantly improves the system's conditioning. The fully coupled system obtained is solved using an ILUT preconditioned GMRES algorithm. The other important element in this paper is the proposal of a Newton linearization of the convection terms in lieu of the common Picard iteration procedure. A systematic comparison between two segregated and four fully coupled fomulations has been presented which has allowed for an evaluation of the individual beneÿts and strengths of the coupling and linearization procedure by studying lid-driven cavity problems and ows past a circular cylinder. All coupled formulations have proven to be signiÿcantly superior both in robustness and e ciency, as compared with the segregated formulation. In some circumstances, the coupled methods yield a converged solution of the system of discretized equations constructed using the FLO scheme, while the segregated formulations diverge. Compared to Picard's linearization, Newton's linearization is more e cient at reducing the number of iterations needed to converge, but requires more computational e ort per iteration from the linear equation solver. Furthermore, the Jacobian matrix should include contributions from the nonlinearity appearing at both the governing-equation level and the interpolation-scheme level to ensure Newton's method convergence. The key element in guaranteeing successful, fully coupled solutions lies in the use of an e cient linear equation solver and preconditioner.

A coupled finite volume solver for the solution of incompressible flows on unstructured grids

“…Because the cost per iteration is higher for the coupled solver, it is more meaningful to compare the CPU time consumed by both solvers. Results in Table 3 indicate that as the grid size increases from 10 4 to 3 Â 10 5 quadrilateral (triangular) control volumes, the corresponding ratio of the CPU time needed by the segregated solver to the CPU time required by the coupled algorithm increases from 13 to 115 (13-104), 18 to 71 (8-58), 4 to 31 (3-33), 8 to 56 (6-54), 8 to 22 (6)(7)(8)(9)(10)(11)(12)(13)(14)(15)(16)(17)(18)(19)(20)(21)(22)(23)(24)(25), and 5 to 38 for the above problems. This represents a tremendous savings as the total time required by the coupled approach to solve all problems on the coarsest and densest quadrilateral (triangular) grids used are 209.6 and 11660.1 (339.7 and 12849.3) seconds while the times required by the segregated method are 1844.89 and 525911.74 (2113.11 and 564206) seconds with the average S/C ratio varying from 8.8 to 45.1 (6.22-43.9).…”

“…In the second approach a pressure equation is derived either through the addition of pseudo-velocities [23] as in the segregated SIMPLER algorithm [24] or without the addition of new variables [25] as in the segregated SIMPLE algorithm [9]. Following either method, a set of diagonally dominant equations is obtained.…”

Preconditioning Based on a Partially Implicit Implementation of Momentum Interpolation for Coupled Solvers

“…The successive substitution method has been traditionally used with segregated solvers for the Navier-Stokes equations. To the authors' knowledge, only Deng et al [3,11] and Ammara and Masson [4] have used such a method for coupled solvers. The application of the Picard procedure to a coupled system implies the solution of larger linear systems when compared with segregated ones (and, therefore, a greater computational cost), but additional cycles due to the coupling of the transport equations are avoided.…”

“…Their proposal is to add a new set of unknowns (the so-called pseudo-velocities) for implementing the discretization of the pressure equation without increasing the size of the computational molecule. Ammara and Masson [4] carried out a comparison with segregated formulations and found that their proposal was more efficient for all the problems studied (and the gain became greater for larger meshes). Density-based algorithms are extended to low-Mach flows by using the pseudo-compressibility technique [5], which is based on including an artificial transient term for pressure in the continuity equation, viz., ð1=b c Þ qp=qt.…”

“…Different preconditioning strategies are used for pressure-based and density-based methods. Deng et al [3], and more recently, Ammara and Masson [4], derived an equation for pressure from the continuity equation which avoids the presence of zeros in the main diagonal of the coefficient matrix. Their proposal is to add a new set of unknowns (the so-called pseudo-velocities) for implementing the discretization of the pressure equation without increasing the size of the computational molecule.…”