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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 suitability of a coupled scheme based on parabolic/elliptic Navier‐Stokes equations for calculating film cooling flows and heat transfer downstream of flush, angled injection slots is explored. The coupled algorithm that combined the coarse mesh ‘outer’ Navier‐Stokes and fine grid ‘inner’ parabolic Navier‐Stokes codes makes retention of the current high resolution model desirable because an acceptable accuracy and economy of computation time are attainable using only mini‐computer resources. The ‘inner‐code’ includes the FLARE approximation to permit small reverse flow. The inner and outer codes are coupled by adopting an approach analogous to classical multigrid methods. It is found that for high blowing mass flow rate of 1.0 with the case of greater than 40° injection angle, the fully parabolic procedure is unable to cope with an extensive separation region immediately downstream of the slot; the present coupling methodology is crucial. The study involves the calculation of heat transfer rates on the surface downstream of the angled slot. Predicted film cooling effectiveness distribution together with the effects of governing parameters are described and show close agreement with the experimental data.

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.

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.

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.

A comparison of the accuracy of the central discretization scheme with artificial dissipation and the upwind flux‐difference TVD scheme has been made for the compressible Navier‐Stokes equations for high Reynolds number flows. First, a comparison is made on two one‐dimensional model problems. Then the schemes are compared on flat plate boundary layer flow. It is shown that a central scheme basically has poor accuracy due to the isotropic nature of the artificial dissipation. An upwind scheme decomposes the flow into different components and adapts the dissipation to the velocity of the components. The associated ansitropic dissipation results in a good accuracy. It is further discussed how a central discretization scheme with artificial dissipation can be improved at the expense of the same complexity of an upwind scheme.

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.

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.

The purpose of this paper is to propose a parallel partition of unity method (PPUM) to solve the nonstationary Navier-Stokes equations.

This paper opted for the nonstationary Navier-Stokes equations by using the finite element method and the partition of unity method.

This paper provides one efficient parallel algorithm which reaches the same accuracy as the standard Galerkin method but saves a lot of computational time.

In this paper, a PPUM is proposed for nonstationary Navier-Stokes. At each time step, the authors only need to solve a series of independent local sub-problems in parallel instead of one global problem.

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.