Search results

The purpose of this paper is to develop Triple Finite Volume Method (tFVM), the author discretizes incompressible Navier-Stokes equation by tFVM, which leads to a special linear system of saddle point problem, and most computational efforts for solving the linear system are invested on the linear solver GMRES.

In this paper, by recently developed preconditioner Hermitian/Skew-Hermitian Separation (HSS) and the parallel implementation of GMRES, the author develops a quick solver, HSS-pGMRES-tFVM, for fast solving incompressible Navier-Stokes equation.

Computational results show that, the quick solver HSS-pGMRES-tFVM significantly increases the solution speed for saddle point problem from incompressible Navier-Stokes equation than the conventional solvers.

Altogether, the contribution of this paper is that the author developed the quick solver, HSS-pGMRES-tFVM, for fast solving incompressible Navier-Stokes equation.

The indefinite nature of the mixed finite element formulation of the Navier‐Stokes equations is treated by segregation of the variables. The segregation algorithm assembles the coefficients which correspond to the velocity variables in the upper part of the equation matrix and the coefficients which corresponds to the pressure variables in the lower part of the equation matrix. During the incomplete; elimination of the velocity matrix, fill‐in will occur in the pressure matrix, hence, divisions with zero are avoided. The fill‐in rule applied here is related to the location of the node in the finite element mesh, rather than the magnitude of the fill‐in or the magnitude of the coefficient at the location of the fill‐in. Different orders of fill‐in are explored for ILU preconditioning of the mixed finite element formulation of the Navier‐Stokes equations in two dimensions.

A simple algebraic multigrid (AMG) solver for linear equations is presented, and its performance compared with a conjugate gradient scheme. This multigrid method is extended to solve the discrete Navier—Stokes equations, obtained by applying a finite volume approach to three‐dimensional incompressible flow on a finite element mesh. The resulting multigrid solver is incorporated into a general purpose flow code (ASTEC), where it proves faster than the original solution algorithm, based upon SIMPLE. The linear AMG solver is both efficient and robust, but the extension to include coupling in the Navier—Stokes equations does not converge on all problems.

We describe a novel mathematical approach to deriving and solving covolume models of the incompressible 2‐D Navier‐Stokes flow equations. The approach integrates three technical components into a single modelling algorithm: 1. Automatic Grid Generation. An algorithm is described and used to automatically discretize the flow domain into a Delaunay triangulation and a dual Voronoi polygonal tessellation. 2. Covolume Finite Difference Equation Generation. Three covolume discretizations of the Navier‐Stokes equations are presented. The first scheme conserves mass over triangular control volumes, the second scheme over polygonal control volumes and the third scheme conserves mass over both. Simple consistent finite difference equations are derived in terms of the primitive variables of velocity and pressure. 3. Dual Variable Reduction. A network theoretic technique is used to transform each of the finite difference systems into equivalent systems which are considerably smaller than the original primitive finite difference system.

We describe a new mathematical approach for deriving and solving covolume models of the three‐dimensional, incompressible Navier—Stokes flow equations. The approach integrates three technical components into a single modelling algorothm: automatic grid generation; covolume equation generation; dual variable reduction.

Providing a much easier (direct) approach to calculate the Lie point symmetries of (3 + 1) unsteady Navier‐Stokes equations for viscous, incompressible flow in cylindrical polar coordinates.

Lie group theory, is applied to the equations of motion. Symmetries obtained through a direct approach are then used to reduce (3 + 1) Navier‐Stokes system to a system of ordinary differential equations.

We observed that the approach applied here to calculate the symmetries of the group is entirely straightforward and involves less calculation as compared to the computer programs such as LIE, Symmgrp.max (MACSYMA) or other symbolic manipulation systems. Further, results obtained here will be practical and useful in comprehending the fluid flow behavior.

We only obtained the exact solution through basic transformations (translation and scaling). The similarity reduction through other subalgebras (finite and infinite dimensions) can be used to explore more facts about the Navier‐Stokes equations.

Direct approach provided in this paper can be utilized to achieve symmetries of other physically important PDEs.

This research aims to investigate numerically the influence of braking jets on separation process of a flying object.

The flying object is at supersonic regime and axial separation of its stages is accomplished with the aid of braking jets of separated stage. The simulation is three‐dimensional and relative motion of the stages is considered three degree of freedom. Full Navier‐Stokes equations in conjunction with κ−ε (RNG) turbulence model equations are considered as governing equations. These equations are solved using the finite volume technique. The separation process is analysed as an unsteady process and the problem is solved in a moving grid domain. The local remeshing method is adapted to regenerate computational cells around moving boundaries.

Time history of flow field around the vehicle components, time history of aerodynamic coefficients, and instantaneous relative position of the body components are the results of this research. Numerical modelling results are compared with the results of other references.

Most of the similar works in this area have used Euler or thin layer Navier‐Stokes (TLNS) equations as governing equations and the use of full Navier‐Stokes equations to analyse such a complicated problem (3D axial separation with braking jets) have not been reported in the literature. Since there are some recirculation zones inside the flow field and Euler or even TLNS equations cannot predict their behaviours, the use of full Navier‐Stokes equations may lead to more accurate prediction of these regions and aerodynamic forces.

To‐date, several segregated finite element algorithms have been proposed that solve the Navier—Stokes equations. These have considered only steady‐state cases. This paper describes the addition of the time‐dependent terms to one such segregated solution scheme. Several laminar flow examples have been computed and comparisons made to predictions obtained with both finite difference and finite volume solution schemes. The finite element results compare very well with the results from the other schemes, both in terms of accuracy and the qualitative behaviour of the iterative schemes.

Most studies of power‐law fluids are carried out using stress‐based system of Navier‐Stokes equations; and least‐squares finite element models for vorticity‐based equations of power‐law fluids have not been explored yet. Also, there has been no study of the weak‐form Galerkin formulation using the reduced integration penalty method (RIP) for power‐law fluids. Based on these observations, the purpose of this paper is to fulfill the two‐fold objective of formulating the least‐squares finite element model for power‐law fluids, and the weak‐form RIP Galerkin model of power‐law fluids, and compare it with the least‐squares finite element model.

For least‐squares finite element model, the original governing partial differential equations are transformed into an equivalent first‐order system by introducing additional independent variables, and then formulating the least‐squares model based on the lower‐order system. For RIP Galerkin model, the penalty function method is used to reformulate the original problem as a variational problem subjected to a constraint that is satisfied in a least‐squares (i.e. approximate) sense. The advantage of the constrained problem is that the pressure variable does not appear in the formulation.

The non‐Newtonian fluids require higher‐order polynomial approximation functions and higher‐order Gaussian quadrature compared to Newtonian fluids. There is some tangible effect of linearization before and after minimization on the accuracy of the solution, which is more pronounced for lower power‐law indices compared to higher power‐law indices. The case of linearization before minimization converges at a faster rate compared to the case of linearization after minimization. There is slight locking that causes the matrices to be ill‐conditioned especially for lower values of power‐law indices. Also, the results obtained with RIP penalty model are equally good at higher values of penalty parameters.

Vorticity‐based least‐squares finite element models are developed for power‐law fluids and effects of linearizations are explored. Also, the weak‐form RIP Galerkin model is developed.

The steady compressible Navier—Stokes equations coupled to the k—ε turbulence equations are discretized within a vertex‐centered finite volume formulation. The convective fluxes are obtained by the polynomial flux‐difference splitting upwind method. The first order accurate part results directly from the splitting. The second order part is obtained by the flux‐extrapolation technique using the minmod limiter. The diffusive fluxes are discretized in the central way and are split into a normal and a tangential contribution. The first order accurate part of the convective fluxes together with the normal contribution of the diffusive fluxes form a positive system which allows solution by classical relaxation methods. The source terms in the low‐Reynolds k‐ε equations are grouped into positive and negative terms. The linearized negative source terms are added to the positive system to increase the diagonal dominance. The resulting positive system forms the left hand side of the equations. The remaining terms are put in the right hand side. A multigrid method based on successive relaxation, full weighting, bilinear interpolation and W‐cycle is used. The multigrid method itself acts on the left hand side of the equations. The right hand side is updated in a defect correction cycle.