A new method for constructing the coefficients of pressure correction equation for colocated unstructured grids

INTRODUCTION

In the last two decades, solution of Navier Stokes equations using colocated arrangement has received great interests. On the other hand, unstructured grids are popular for solution of flow field in complex geometries. Colocated arrangement has some obvious advantages over staggered grids especially in case of non-orthogonal meshes. In colocated arrangement, all variables share the same location, hence only one set of volumes is considered. Also, by applying the colocated grids, the convection contribution to the coefficients in discretized equations, are the same for all variables. Finally, Cartesian velocity components can be used in conjunction with non-orthogonal grids for complex geometries which result simpler discretized equations.

After the original work by Rhie and Chow (1), Peric (2) generalized the same idea for three dimensional flows and calculated several two and three dimensional flow situations. Majumdar (3) studied the role of under relaxation parameter in momentum interpolation for calculation of flow with colocated grids. Lien (4) used the colocated arrangement for unstructured grids successfully. He applied the momentum interpolation as a concept for derivation of pressure correction equation. However, different authors have different ideas to construct the pressure correction equation. For example the approach used by Thomadakis et al. (5) for constructing the coefficients is quiet different from that used by Lien (4). Here the emphasis is on the fact that these coefficients can affect the numerical solution. Therefore, a new form of the coefficients is proposed and it's rate of convergence and performance is examined.

In this study, two examples are solved by structured and unstructured grids:

* Laminar flow in a lid-driven square cavity

* Two dimensional parallel flow

FLOW EQUATIONS

For incompressible Newtonian fluid flows, the conservation equation for mass and momentum are as:

[[vector].[nabla]]*([rho][[vector].V]) = 0 (1)

[[partial derivative]/[[partial derivative]t]]([rho]u) + [[vector].[nabla]].([rho][[vector].V]u) = [[vector].[nabla]]*([mu][[vector].[nabla]]u) - [[[partial derivative]p]/[[partial derivative]x]] (2)

[[partial derivative]/[[partial derivative]t]]([rho]v) + [[vector].[nabla]]*([rho][[vector].V]v) = [[vector].[nabla]]*([mu][[vector].[nabla]]v) - [[[partial derivative]p]/[[partial derivative]y]] (3)

In above equation, u and v are Cartesian components of velocity vector.

Transport equation for a general variable, such as [phi], can be written as:

[[partial derivative]/[[partial derivative]t]]([rho][phi]) + [[partial derivative]/[[partial derivative][x.sub.j]]]([rho][u.sub.j][phi]) = [[partial derivative]/[[partial derivative][x.sub.j]]]([GAMMA][[partial derivative][phi]]/[[partial derivative][x.sub.j]]) + [q.sub.[phi]] (4)

DATA STRUCTURE

As mentioned before, colocated arrangement for velocity components and pressure within an arbitrary finite volume is adopted here. The basic idea of this data structure is shown in Fig. 1. As shown, forming points are the vertices of control volume and the center of control volume is assumed for storage of all variables.

[FIGURE 1 OMITTED]

In Fig. 1, each cell has some adjacent cells with two or three shared nodes. For example, in triangular meshes, the adjacent cells have two shared nodes and in tetrahedral meshes, they have three shared nodes.

The forming points and adjacent of a cell are numbered in the counterclockwise direction suitable for divergence theorem.

FINITE VOLUME DISCRETIZATION

By integrating the Eq. 5, over a control volume, using divergence theorem, the general transport equation in integral form is obtained:

[[partial derivative]/[[partial derivative]t]][[integral].[OMEGA]][[rho][phi]d[OMEGA]] + [[integral].s][[rho][phi][[vector].V]*[[vector].dS] = [[integral].s][GAMMA][[vector].[nabla]][phi]*[[vector].dS] + [[integral].[OMEGA]][q.sub.[phi]]d[OMEGA] (5)

where, [[vector].S] is the surface vector with the Cartesian components, namely [S.sub.x], [S.sub.y]. This integral form consists of four parts; transient term, convection term, diffusion and source term.

On triangular meshes, the convective term can be approximated as:

[[integral].s][rho][phi][[vector].V].[[vector].dS] = [3.summation over (i = 1)][C.sub.si][[phi].sub.si] (6)

The parameter [C.sub.si] is the mass flux over the i'th surface of control volume. The mass fluxes are:

[C.sub.si] = [rho]([u.sub.si][S.sub.xi] + [V.sub.si][S.sub.yi]) (7)

Variables [u.sub.si] and [v.sub.si] are the components of velocities at the i th face of the control volume.

The value of [phi] on the faces (i.e.[[phi].sub.si]) can be approximated by the following second-order upwind scheme (1):

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (8)

The vectors of [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] and [[vector].sub.ci-si] are shown in Fig. 2.

[FIGURE 2 OMITTED]

In Fig. 2, [C.sub.0] is the Center of the control volume [OMEGA] and [C.sub.i] is the centre of its i'th adjacent cell