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Momentum and Continuity are the basic equations for fluid flow modeling. The momentum equations in their final form are known as Navier-Stokes equations and can be solved using different numerical methods. There are several approaches such as SIMPLE, PISO, and Fractional Step for solving these equations. In solution procedure, it needs to decide where to store the velocity components. Staggered and Collocated grids can be used to evaluate this problem. On Staggered grids, the velocity components are stored at the cell face, and the scalar variables such as pressure are stored at the central nodes. However, on Collocated grids, all parameters are defined at the same location at the central nodes. The Staggered grids method gives more accurate pressure gradient estimation. However, Collocated grids method is simpler for solving the equations. In this paper, for solving Navier-Stokes equations, Collocated and Staggered grids are employed. Comparison of horizontal and vertical velocities and stream lines at various Reynolds numbers was performed. The results were validated using standard tests such as lid-driven cavity, channel and backward facing step. Discussion is made on accuracy of these methods to estimate horizontal and vertical velocity profiles.

Thefundamental equations of motion known as Navier-Stokes (N-S) equations are basically valid for laminar flow. To use them for turbulent flow, time, and spatial averaged form of these equations are used [

There are several methods such as SIMPLE, PISO, and N-S equations. The SIMPLE algorithm was originally developed by Patankar and Spalding (1972) on the Staggered grids arrangement and involves one predictor step and one corrector step. While PISO method can be regarded as an extension of SIMPLE method with further correction steps [

In this paper, a turbulent model was developed using Staggered and Collocated grids. To validate the model some standard tests such as lid-driven cavity, Channel flow and Backward-facing step were used.

Several researchers studied these tests. Prasad and Koseff [

The governing equations of incompressible fluids which are Continuity and N-S equations can be expressed as

In this research, LES model is used to simulate turbulent flow. It relies on the definition of large and small scales in which the former are solved and the later are modeled. Figure

Schematic view of the simplest scale separation operator in LES model.

To separate the large scales from the small ones, filtering process is used. A filtered variable, denoted by an over bar, is defined as

Finally, the total kinematic viscosity is derived as

The solution algorithm for the governing equations is consisting of discretization of flow domain and storing the velocity vectors. Two methods denoted as Collocated Grids (C.G) and Staggered Grids (S.G) are used in this procedure. On C.G, all velocity components and scalar variables such as pressure are stored in the same locations (see Figure

Control volume in C.G.

Although application of C.G is simple, it cannot distinguish between a non-uniform pressures field and a uniform one. In S.G method, velocities are stored at the faces of control volume (see Figure

Control volume in a S.G.

One of the advantages of the S.G arrangement is that velocities are generated at locations where needed for the convection computations. While on C.G, stored velocities at the center of control volumes cannot be used directly. In this situation, for calculation of convection term, Rie-Chow interpolation is used.

The solution algorithm employed in this study is a finite volume method based on the PISO predictor corrector algorithm coupled with a pressure correction to discretize the governing equations on structure grids.

In this method, the filtered N-S equations, are discretised using the finite volume method where the domain

In this research, to estimate the coefficients power law scheme is used as

The discritized N-S equations must be solved for all control volumes to estimate the velocity components. But in this step, the unknown pressures in the control volumes should be predicted.

In next step, Momentum equation is solved using predicted pressure field and Three Diagonal Matrix Algorithm (TDMA) as

In next step, the pressure correction equation is derived as

In this research, the N-S equations are discretized using a finite volume method of second-order accuracy in space based on Eulerian description at each time step. The N-S solver is developed based on S.G and C.G with a LES model as turbulent.

To validate the model, some standard tests such as lid-driven cavity, Channel flow and Backward-facing step are used.

The fluid flow is in a rectangular container which moves tangentially to itself and parallel to one of the side walls. Due to the simplicity of the cavity geometry, applying a numerical method on this flow problem in terms of coding is quite easy and straight forward. Despite its simple geometry, the driven cavity flow retains a rich fluid flow physics manifested by multiple counter rotating recirculation regions on the corners of the cavity depending on the Reynolds number.

The geometry of the problem consist of boundary condition of velocity and the successive of eddies which is shown in Figure

Lid-driven cavity test, (a) boundary condition (b) arrangement of eddies.

A sensitivity analysis is performed on mesh size (see Figures

Mesh independency on S.G for (a)

Mesh independency on C.G for (a)

The model is then performed on S.G and C.G for various Reynolds numbers at a range of 100 to 10000 and results were compared with those of Ghia et al. [

Comparison of horizontal velocity at various Reynolds numbers (a) C.G and (b) S.G.

Comparison of vertical velocity at various Reynolds numbers (a) C.G and (b) S.G.

Comparison of horizontal velocity on C.G and S.G and Ghia et al. results.

Comparison of vertical velocity on C.G and S.G and Ghia et al. results.

The streamlines of fluid flow in cavity on C.G and S.G are shown on Figure

Comparison of stream lines for various Reynolds numbers (a) S.G and (b) C.G.

For

For

Figures

However increasing the Reynolds number increases the inconsistency between results so that the results of S.G are more close to Ghia results. These differences in the upstream and downstream of the domain are greater. While in the middle point, the results of C.G and S.G at various Reynolds numbers are predicted accurately.

Figure

For low Reynolds number the stream lines on C.G and S.G are similar. However as the Reynolds number increases, the shape and the number of eddies change. For example at

In this section, the solution of the channel flow for a long channel with a fully developed laminar flow is presented. The length and height of the channel are selected as

Channel test, comparison of fully developed profile on C.G and S.G and Analytical solution.

The extension of boundary layer on C.G and S.G at

Channel test, development of Boundary layer on C.G and S.G.

To assess the accuracy of numerical method, the flow over a backward-facing step in a channel is a good test case. A dissipative scheme cannot predict the correct reattachment length of the recirculation zone downstream of the step [

Geometry of step test.

The regime of fluid flow on Backward facing step, for

In this section, the governing equations are solved and reattachment length at various Reynolds numbers on C.G and S.G is calculated. Finally the numerical and Armaly et al. results [

Comparison of Reattachment point on C.G and S.G and Armaly result.

It is evident from this figure that the results of S.G are better than C.G. But this result is not accurate for all the Reynolds numbers. For example for

The reattachment length of the recirculation zone downstream of the step is decreased for Reynolds numbers more than 1000. This is because of the existence of second eddy at the top of step (see Figure

Step test, existence of second eddy at the top of step for

In this paper, three test cases as lid-driven cavity, channel flow and backward-facing step are considered to compare the accuracy of S.G and C.G methods. In the light of different results for N-S solver, following conclusion can be drawn:

In lid-driven cavity test, for low Reynolds numbers the results of S.G and C.G methods are nearly the same. Nevertheless for high Reynolds numbers, the results of S.G demonstrate better agreement. In lid-driven cavity test, for low Reynolds number the stream lines on C.G and S.G are similar. However as the Reynolds number increases, the shape and the number of eddies change. In the channel test, two grids present very good results for fully development and the extension of boundary layer. In backward-facing step, the reattachment length of the recirculation zone downstream of the step is decreased for Reynolds number more than 1000. This is because of the existence of second eddy at the top of step. In backward-facing step, the results of S.G are better than C.G.

It can be concluded from these test cases that, for problems such as lid-driven cavity and backward-facing step associated with eddy on the fluid flow, application of S.G leads to more accurate results. However for other problems such as channel flow, both methods nearly have the same accuracy.

A constant in the range

Static pressure

Strain rate tensor

Source term in the

Time

Filtered velocity in

Filtered velocity in

Velocity components in

Velocity components in

Velocity vector.

Length of filter

Mesh size in

Mesh size in

Mesh size in

Turbulence eddy viscosity

Dynamic fluid viscosity

Fluid density.