International Journal of Fluid Dynamics (2001), Vol. 5, Article 5
 
Removal of Temporal and Under-Relaxation Terms from the Pressure-Correction Equation of the SIMPLE Algorithm
V.A.O. Anjorin & I.E. Barton

Department of Mechanical Engineering, Brunel University, Uxbridge, Middlesex, UB8 3PH, UK.
E-mail:    Vincent.Anjorin@brunel.ac.uk
 

Keywords: SIMPLE algorithm, Flow solver, Pressure-correction equation, Navier-Stokes equations


Abstract

The SIMPLEV (SIMPLE-Vincent) algorithm is an improved version of the standard SIMPLE algorithm where the under-relaxation and temporal terms are removed from the pressure correction equation of the SIMPLE algrithm. This paper focuses on the methodologies of the SIMPLEV algorithm and how it is used to solve the discretized momentum equations that yield the pressure correction terms of the pressure correction equation. The SIMPLE and SIMPLEV algorithms rely on under-relaxation to avoid divergence and there is the problem that by using a very small velocity under-relaxation factor or a very small time step, the convergence rate of the SIMPLE and SIMPLEV algorithms becomes extremely slow since a small pressure under-relaxation factor must be employed. The SIMPLE algorithm is assessed with a version that has the terms for the pressure correction equation that increases the pressure correction as the velocity under-relaxation factors or the time step terms decrease. It is shown that the SIMPLEV algorithm gives the reverse effect of this.
 

1. Introduction

A method for solving the momentum equations for laminar flow problems has been developed by Patankar and Spalding (1972). They proposed the SIMPLE (Semi-Implicit Method for Pressure-Linked Equations) algorithm to iteratively solve the momentum equations in their discretized form. The methodology of the SIMPLE algorithm has been recently discussed by Versteeg and Malalasekera (1995), explaining how to calculate the velocity field for a two-dimensional control volume. The real difficulty in calculating the velocity field lies in determining the unknown pressure field. There is no obvious equation to solve thus pressure is used to satisfy the condition for continuity. To solve this the pressure field is indirectly specified via the continuity equation. This indirect specification is achieved by obtaining a whole set of discretized equations from the momentum and continuity equations and solving the discretized equations by a direct solution.

 
The advantage of the SIMPLE algorithm is that it uses an iterative approach. Iterative solutions are commonly used to solve a whole set of discretized equations so that they may be applied to one single dependent variable or even to a single point. Using a direct solution for solving the entire sets of velocity and pressure components is more difficult. Another advantage of the SIMPLE algorithm is that it can be applied to solve incompressible flow problems. If the flow is compressible (Zaiyong et al, 1998), the pressure may be obtained from the density and temperature by using the equation of state where the density is regarded as a dependent variable of the continuity equation as discussed by Versteeg and Malalasekera (1995). Since the flow we are dealing with is incompressible, the density is constant and is therefore not linked to the pressure. For this case, there exists a coupling between the pressure and velocity that introduces a constraint on the solution of the flow field. If the correct pressure field is applied in the momentum equations, the resulting velocity field satisfies the continuity equation. A brief explanation of the methodology of the SIMPLE algorithm (Barton and Kirby, 1998) is explained in section 3, and in more detail by Patankar (1980).
 
Since the SIMPLE algorithm possesses a rather slow convergence rate, a proposed improvement is made to enhance the convergence rate (Wanik and Schnell, 1989) of this algorithm. This is achieved by developing a new algorithm known as the SIMPLEV algorithm. This is aimed at giving a faster rate of convergence of the solution of the pressure-correction equation (Versteeg and Malalasekera, 1995; Barton and Kirby, 1998) than the SIMPLE algorithm. The SIMPLEV algorithm uses the same methodology as that of the SIMPLE algorithm in solving the velocity fields that satisfy the continuity equation except that the under-relaxation and temporal terms are removed from the pressure-correction equation of the SIMPLE algorithm. When this is performed the pressure correction tends to zero therefore satisfying the continuity equation to obtain better convergence.


2. Numerical Analysis

2.1. Governing Equations

Two-dimensional incompressible laminar constant-density flow (Melaaen, 1993) is governed by a set of partial differential equations. The momentum and continuity equations in their primitive form are shown in equations (1-3) where the equation for conservation of mass is given by

        (1)

The conservation of momentum in the x and y directions are governed by the u-momentum equation expressed as

          (2)
as well as the v-momentum equation
          (3)


The terms on the left-hand side of equations (2) and (3) include the time-derivative and convective terms, and the terms on the right hand side include the pressure gradient, viscous and source terms (Jang et al, 1986). The symbols appearing in equations (1-3) are classified as the density (r), the velocity components for the x and y directions of the two-dimensional control volume (u, v), the u and v momentum source terms (Su, Sv), the pressure field (P), and the molecular viscosity (m).


2.2. Discretisation


In order to numerically solve the velocity and pressure fields that obey the discretized momentum and continuity equations, the finite-volume method was applied. This method involves integrating the continuity and momentum equations over a two-dimensional control volume on a staggered differential grid (Patankar and Spalding, 1972; Harlow and Welch, 1965) shown in Figure 1.
 

  This yields the governing equations in their discretized form as shown in equations (4-6). The staggered grid evaluates the scalar variables, in this case only the pressure, which are stored at the scalar nodes marked (·), and located at the intersection of two unbroken grid lines. Such points are indicated by the capital letters P, W, E, N and S. The u-velocity components are stored at the east and west cell faces of the scalar control volume and are indicated by the lower case letters e and w. The v-velocity components are stored at the north and south cell faces of the scalar control volume which are indicated by the lower case letters n and s. These velocity components are located at the intersection of a dashed and unbroken line that construct the scalar cell faces and are indicated by arrows. The horizontal arrows shown in Figure 1 indicate the locations of the ue and uw velocity components and the vertical arrows  indicate the locations of the vn and vs velocity components. Forward staggered velocity grids are used. The uniform grids are forward staggered since the u-velocity, ue, is at a distance of 1/2 dxu from the scalar node Pp. Similarly, the location for the v-velocity, vn, is at a distance of 1/2 dyv from the scalar node. After the process of discretization, the discretized continuity equation becomes:

,          (4)

and the discretized u-momentum equation becomes
,          (5)
and finally the discretized v-momentum equation can be written
.          (6)
 
The time derivative terms are incorporated into the coefficients and of equations (5) and (6) and the source terms Su and Sv of equations (2) and (3). The coefficients and are expressed as follows
          & .
The values of the coefficients ae and an at the east and north cell faces of the control volume are obtained by the use of the differencing methods such as the upwind (Huang et al, 1985), hybrid (Spalding, 1972) and QUICK (Leonard, 1979) schemes. The neighbour coefficients anb account for the combined convection-diffusion influence at the control-volume faces. The velocity components, unb and vnb, in equations (5) and (6) are those at the neighbouring nodes outside the control volume. The term (PE - PP)Dy represents the pressure force acting on the u control volume. The pressure difference acts on the control volume of width Dy. Similarly (PN - PP)Dx represents the pressure force acting on the v control volume where the pressure difference acts on the control volume of width Dx. The first terms on the right hand side of equations (5) and (6) (Sanbunb and Sanbvnb) represents the summation of the product of the neighbour coefficients and the velocity fields at the neighbouring nodes. When the source terms of equations (2) and (3) are integrated over the control volume they give and which are approximated by the final volume method (Fletcher, 1988; Ranade, 1997) as
,          (7)

.          (8)

The second terms on the right-hand side of equations (7) and (8) are the temporal terms where (k-1) implies values of the velocity components from the previous time step. The first terms on the right-hand side of equations (7) and (8) stand for the constant part of the average value of the source terms. The method of discretizing the momentum and continuity equations is fully explained by Versteeg and Malalasekera (1995) where they show how to employ differencing schemes to successfully interpolate between the node points.

The velocity and pressure fields of the SIMPLE and SIMPLEV algorithms are all solved on a staggered differential grid arrangement. This arrangement is used to prevent the pressure-velocity (PV) coupling that links the mass conservation equation and the momentum equations. A further discussion of this is presented by Simoneau and Pollard (1994). In the past, a majority of applications use the staggered grid arrangement to solve all the flow variables for numerical modelling purposes. As an example of a typical problem, Stathopoulos and Baskaran (1996) carried out simulations of the mean wind environmental conditions around buildings for the assessment of the dispersion of pollutants. The differential equations for the computation of turbulent wind flow conditions around buildings were discretized and represented on a non-uniform staggered grid arrangement containing 235,000 nodes. However more recently, finite-volume flow calculations have often used the Rhie and Chow interpolation method (Oliveira et al, 1998; Rhie and Chow, 1983) which is based on a non-staggered grid. This approach cures the problem of odd-even coupling between the pressure and velocity fields by employing the technique of interpolating the cell-face velocities via momentum interpolation.
 

3. Solution Procedure of the SIMPLE Algorithm

The standard SIMPLE algorithm (Patankar, 1980) can be broken down into the following steps:

First solve the discretized u and v momentum equations (9) and (10) by using the current guessed pressure field p*to yield the intermediate velocity fields .

          (9)
       (10)
Next solve the corrected-velocity fields and the correct pressure field p** in order to satisfy the discretized u and v momentum equations (11) and (12) and the continuity equation (4).
       (11)
       (12)
Then the pressure correction equation (24) is derived by substituting the terms and into the continuity equation (4) using the correction formulae equations (13) and (14).
       (13)

       (14)

Equations (13) and (14) are obtained by subtracting those from the second step away from those from the first step. After the pressure correction equation (24) is solved, the solution of the pressure field is substituted back into equations (9) and (10) to obtain the terms (Barton, 1998). To under-relax the discretized u and v momentum equations (11) and (12) of the SIMPLE algorithm, we begin with the relationship between the corrected pressure and its under-relaxation factor aP as well as the relationship between the velocity fields and their under-relaxation factors au and av These are derived to determine the effect that their under-relaxation has on the convergence of the algorithm. The pressure field pnew is under-relaxed as follows:
       (15)
where ap is the pressure under-relaxation factor. The relaxation factor is taken between 0 and 1 so that the guessed pressure field has added to it a fraction of the correction pressure field p¢ in order for the iteration improvement process to be carried forward. Similarly the velocity fields and are under-relaxed in the following manner:
       (16)

.       (17)

The terms are the corrected velocity fields without under-relaxation and represent their previous values obtained from the previous iteration (Barton, 1998).The discretized u-momentum equation with under-relaxation is obtained by substituting the under-relaxation factors into the general discretized equation (11). To recall, this is:
,       (18)
which can be re-written as
,       (19)
so that under-relaxation can be introduced.
Next we assume that can be replaced by and introduce the previous obtained values of the under-relaxed velocity fields to give
.       (20)
Here is taken as the value of from the previous iteration. The variables in the parenthesis represent the change in produced by the current iteration. So this change can be modified by the introduction of the velocity under-relaxation factor au so that
.       (21)
Substituting the under-relaxed velocity field of equation (16) into equation (21) and re-arranging gives the discretized u-momentum equation with under-relaxation:
.       (22)
Similarly the v-momentum equation with under-relaxation becomes:
.       (23)
To derive the under-relaxed pressure-correction equation of the SIMPLE algorithm, the velocity fields and equations (16) and (17) are substituted into equations (22) and (23) to give the following corrected velocity fields
&.
Substituting the corrected velocity fields into the discretized continuity equation (4) yields
       (24)

4. Solution Procedure of the SIMPLEV Algorithm

The pressure correction equation of the SIMPLEV algorithm is derived by first determining if there is a connection between the under-relaxation factor (au and av) and the time-step dt.  Using the under-relaxed discretized u-momentum equation (22), the temporal term is disregarded so that a comparison can be made of the effect of under-relaxation and the original temporal term. Taking the left-hand side of equation (22) and removing the temporal term gives

.       (25)
Also taking the right-hand side of equation (22) and removing the temporal term leads to
.       (26)
Using equations (25) and (26), an investigation is made as to whether there is a connection between au and dt. For equation (25) we first of all make the coefficient equal to ae/au as
.       (27)
Also for equation (26) we make
.       (28)
Using equation (27) to compare dt and au gives
.       (29)
Rearranging equation (29) for dt gives
.       (30)
From equation (30) it can be seen that there is a connection between dt and au. Taking the inverse of equation (30) results in equation (28). Alternatively equation (27) can be rearranged for au as
.       (31)
 
From equation (31) there is another connection between the under-relaxation factor (au) and the time step dt. To examine the effect of what happens to the pressure-correction equation of the SIMPLE algorithm as the under-relaxation terms tends to zero, equation (28) is substituted into equation (24) which gives
       (32)
Equation (32) can be simplified as follows by factorizing the ae, aw, an, as coefficients for the right-hand side terms of equation (32) to give
 
       (33)
An important point that can be seen from equation (33) is that, as the velocity under-relaxation factors au and av tend to zero, the pressure correction increases. Logically, we actually require the reverse result of this such that as au andav tends to zero, the pressure correction p¢ also tends to zero. To obtain the reverse result of equation (33) the under-relaxed velocity field of equation (16) is substituted into equation (22). Next the inverse of the coefficient term of equation (27) is taken and substituted into equation (22) to give the final implicit discretized u and v momentum equations which are presented as:
,       (34)

.       (35)

This set of equations is equivalent to
,       (36)

.       (37)

Subtracting equations (36) and (37) from (34) and (35) gives
,       (38)
.       (39)
Rearranging and omitting terms and of equations (38) and (39) gives
,       (40)

.       (41)

Equations (40) and (41) are substituted into the continuity equation (4) to give

       (42)


So, as au and av tend to zero in equation (42), the pressure correction p¢ tends to zero. Equation (42) forms part of the development of the pressure-correction equation of the SIMPLEV algorithm. From the pressure-correction equation (24) of the SIMPLE algorithm, it is noticed that as the time step dt of the anew coefficient tends to zero, the pressure correction p¢ becomes large. The reverse result of this as dt and p¢ tends to zero is obtained by first removing the under-relaxation terms from equation (24) as under-relaxation is not required for this case. A parameter e is introduced into equation (24) by taking the inverse value of the anew term of equation (27). This gives
.       (43)


Substituting equation (43) into equation (24) gives the following pressure-correction equation

      (44)


From the pressure correction equation (44) it can be seen that as the time step dt tends to zero, the pressure correction p¢ tends to zero. Alternatively, the pressure-correction equation (44) can be obtained from the discretized u and v momentum equations (9) and (10). To recall this is
,       (45)

.       (46)


The ae and an coefficients are upgraded to the and in which the parameter eof equation (43) is introduced into equations (45) and (46) to give
,       (47)

.       (48)


Using the method outlined by equations (34-41) gives the pressure-correction equation (44). A final form of the pressure-correction equation of the SIMPLEV algorithm is obtained by combining equations (42) and (44) to give
       (49)


As the under-relaxation and time-step terms of equation (49) tends to zero, the velocity fields will progressively satisfy the continuity equation so that the converged solution of equation (49) can be achieved. A flow chart of the SIMPLEV algorithm is shown in Figure 2.
The next section presents two simple laminar flow problems of the steady state solutions of laminar flow around a square cylinder (Breuer et al, 2000; Robert and Hwang, 2000; Alvaro Valencia, 1995)and laminar flow over a backward-facing step (Wengle et al, 2001; Iwai et al, 2000; Keskar and Lyn, 1999) in which numerical results are presented. These two problems were used to study the convergence rate of the SIMPLE and SIMPLEV algorithms.

5. Results and Discussion
 

Comparisons were performed between the SIMPLEV and SIMPLE algorithms. Our investigation focuses on the following issues:
 
  1. Number of iterations of the SIMPLEV algorithm that are required to reach convergence.
  2. How quickly the converged solution of the pressure correction equation is obtained by choosing a set of under-relaxation factors and time step values.
  3. The optimal values of the under-relaxation factors and time step values where convergence is obtained quickest. This indicates higher effectivity of the SIMPLE and SIMPLEV algorithms.
  4. The influence of the number of nodes on the grid systems for the SIMPLE and SIMPLEV algorithms.
The results of the converged solution of the pressure correction and the number of iterations to reach convergence for the SIMPLE and SIMPLEV algorithms were obtained by the use of a Brunel University CFD code known as AFLOW (Barton, 1995). Computations are performed on a Cartesian grid system (Guo et al, 1998) with 30x30 and 60x60 grid points respectively for the two laminar flow applications. The two simple laminar flow problems are discussed below.
 
5.1. Flow around a Square Cylinder

The problem of the flow around a square cylinder was chosen because the geometry is simple. It is an easy problem in which the SIMPLEV algorithm can be applied to investigate how the pressure field on a control volume can be improved in order for the velocity fields to satisfy continuity and hence how the converged solution of the pressure-correction equation can be determined. An inlet Reynolds number of 800 was used. The Reynolds number for this case is defined using the diameter of the square cylinder, the freestream velocity and the viscosity as mentioned by Barton (1998). A schematic showing the geometry and boundary conditions is shown in Figure 3. The grid used was compressed towards the upper and lower boundaries, as well as the solid boundaries of the square cylinder to better resolve the higher gradients in those regions. The upper and lower boundaries are situated 4D away from the cylinder, where D is known as the diameter of the cylinder. The length scales are non-dimensionalised by D. The inlet and outlet boundaries are placed 6D from the cylinder.

5.1.2. Flow Predictions around Square Cylinder

The predicted velocity profiles using the SIMPLEV scheme are shown in Figure 4. The u-velocity profiles were predicted at the positions x = 0, 8, 12, 14, 20 & 25 downstream from the inlet boundary. It can be shown that the u-velocity profiles are symmetric about the centreline of the flow field just as the flow approaches the cylinder. Behind the cylinder the U-velocity profile changes due to the flow separation in front of the cylinder and the formation of recirculation regions behind the cylinder. As the flow moves towards the outlet boundary, the U-velocity increases further away from the recirculation regions.

Figure 5 shows the flow around a square cylinder where the flow approaches, separates and is forced away from the square cylinder forming two recirculation regions behind the cylinder. The upstream flow slowly recovers downstream returning to its original profile. Using non-dimensionalized time-steps at particular time intervals the recirculation regions increase with respect to time (Barton, 1998).
 

5.2. Flow over a Backward-Facing Step

The problem of the flow over a backward-facing step was chosen because it is fundamental in design and geometry, and is used in a variety of engineering applications. The sudden changes in pressure in which our SIMPLEV algorithm numerically evaluates can be studied using backward-facing step configurations. The configuration of the flow over a backward facing step is shown in Figure 6 which has an inlet channel of height (h) and a parabolic inlet flow profile.

The channel expansion number was set to 2 and this is defined as the ratio of the total height of the main channel to the height of the inlet channel (h). The inlet Reynolds number was set at Re = 800, where the Reynolds number is based on the total channel height (2h), the average inlet velocity and the dynamic viscosity n.

5.2.1. Flow Predictions for the Backward-Facing Step Problem

Shown in Figure 7 is the flow configuration of the backward-facing step problem where the flow moves downstream from the inlet channel forming a recirculation region close to the step (Barton, 1998). It has a main recirculation point where the flow reattaches. As the flow recovers downstream, a recirculation region is formed that separates and later recovers as the pressure recovers. Initially the flow has four recirculation regions. The two recirculation regions furthest downstream eventually disappear with increasing time because the vorticity of the flow decreases finally enabling the flow to have two recirculation regions.
 

Shown in Table 1 are the predictions of the growth of the main reattachment and separation positions with time. The table shows that the SIMPLEV algorithm gives the longest reattachment and separation positions for iterations expressed in non-dimensional time units.
 
 

 
Table 1: Predictions of the growth of the main reattachment and separation positions with non-dimensional time (T) for backward-facing step flow.
 
     
 
SIMPLE
SIMPLEV
 
           
        T
         X1
          X2
           X3
           X1
          X2
           X3
10
12.244
2.3216
14.262
12.969
3.5539
14.364
20
12.270
2.4300
14.279
12.969
3.6707
14.366
30
12.302
2.6079
14.292
12.972
3.7237
14.367
40
12.973
3.0919
14.370
12.977
4.4357
14.368
50
12.973
3.1079
14.370
12.980
4.5573
14.369
60
12.972
3.1238
14.370
12.982
4.6060
14.370
70
12.972
3.3184
14.370
12.983
4.9592
14.370

5.3. Performance of the SIMPLEV Algorithm

The performance of the SIMPLEV algorithm was compared with the SIMPLE algorithm to search for the minimum number of iterations and the optimum under-relaxation factors that enable the solution to converge. During the computational process, the pressure-correction equations of the SIMPLE and SIMPLEV algorithms were iteratively solved until the tri-diagonal matrix algorithm (TDMA) solver (Anderson et al, 1984) terminates operation. This algorithm solves the matrix of the pressure correction equations of the SIMPLE and SIMPLEV algorithms. The convergence criterion that was used for the TDMA solver was 5x10-5 for the backward-facing step problem and 7x10-4 for the problem of laminar flow around a square cylinder. For both algorithms, the convergence criteria remained the same so that there can be a comparison between the two schemes. This happens when the velocity and pressure residuals reach their desired convergence criteria.

 
The effectiveness of the SIMPLEV algorithm depends on the chosen set of values of the velocity and pressure under-relaxation factors. Shown in Figure 8 are the results of the influence of the under-relaxation factors (au, av and ap) (Johansson and Davidson, 1997; Jun Zhang, 1996) on the number of iterations to yield convergence. Optimal values (au, av and ap) are marked as well as the allowed intervals of changes of these factors. However if coefficients are taken from the outside of these intervals, divergence of the computational process occurs.
 
In this computational test, the velocity under-relaxation factors (au, av) were varied simultaneously while keeping the pressure under-relaxation factor (ap) constant. Also the pressure under-relaxation factor was varied while keeping the velocity under-relaxation factors constant. The set of optimal values of under-relaxation factors obtained for the 30x30 grid system was then employed in the SIMPLE and SIMPLEV algorithms for further iterative computations. It follows from Figure 8 that the saving in using the SIMPLEV algorithm strongly depends on the chosen set of under-relaxation factors. With an improperly chosen set, the number of iterations for the SIMPLEV scheme may be even greater than the SIMPLE method. The values shown in Fig. 8 are recommended.

In order to obtain convergence, smaller values of under-relaxation factors should be used. However this poses a problem that an increasing number of iterations are required to yield convergence and the solutions are therefore obtained with more CPU time than in the case of the SIMPLE method. An alternative way to consider the algorithms is by introducing a time-step instead of under-relaxation. The results of Figure 9 presents the convergence rates of the SIMPLE and SIMPLEV schemes expressed in non-dimensional time intervals. For both schemes it can be shown that the converged solution is easily obtained as the time step decreases. However using very small number of time steps leads to an increasing number of iterations that are required to enable the solution of the pressure correction equation to converge. For the case of laminar flow around a square cylinder of Figure 9 an initial time step of 0.01 units was applied for both schemes in which convergence was obtained after 782 iterations for the SIMPLEV scheme and 812 iterations for the SIMPLE scheme. This was marched in time of 0.01 units until a maximum time step of 0.085 units was employed. For the 60x60 grid system, the SIMPLE and SIMPLEV algorithms require more iterations than the 30x30 grid system to obtain convergence. This is because by having more grid points in the computational domain increases the number of algebraic equations that need to be solved for the pressure and velocity fields. Because both methods are iterative, more equations almost always means slower convergence measured in terms of the total number of iterations required. In addition, each individual TDMA step requires more CPU time, therefore causing slow convergence of both schemes. With more number of node points in the computational domain, the SIMPLEV algorithm is therefore not able to improve the efficiency of computations. The computational tests for the two grid systems of Figure 9 were performed in order to investigate the influence of the number of nodes on the SIMPLE and SIMPLEV algorithms. Above all the SIMPLEV scheme yields better convergence as fewer iterations are required for the converge solution of the pressure correction equation.

Figure 10 shows the results of the number of iterations of the iterative versions of the SIMPLE and SIMPLEV algorithms as well as the changes of the residual values of the velocities and pressures during the iterative computation of the AFLOW code. It can be shown that the SIMPLEV algorithm attains the required value of the convergence criterion in less time than the SIMPLE algorithm. It should be noted that the results shown in Figure 10 were obtained for the 30x30 and 60x60 grid systems. When the SIMPLEV algorithm is employed, there is a reduction in the residual values for every iteration and the convergence criterion in all cases is attained fastest than the SIMPLE method. However it can be observed from Figure 10 that the SIMPLEV algorithm is not able to improve the efficiency of computations if a grid system with large number of nodes is employed. This makes it quite difficult for the converging solutions to be obtained. The set of optimal under-relaxation coefficients for which there is higher computational efficiency is shown in Table 2. This set was determined for the 30x30 grid system. A comparison of the minimal number of iterations and the set of optimal time steps for the SIMPLE and SIMPLEV methods for which there is higher computational efficiency is shown in Table 3. They indicate the necessary number of time steps for the solution of the computational process to reach convergence.
 
 


 
 
 




6. Conclusions
 
The work presented in this paper compares the performance of the SIMPLEV and SIMPLE algorithms investigating whether the SIMPLEV method is able to solve for the converged solution of the pressure correction in less CPU time than the SIMPLE algorithm. Both methods have been applied to steady state laminar flow over a backward-facing step and steady state laminar flow around a square cylinder. On the basis of the result of the calculations, the following conclusions can be formulated:
  1. The higher efficiency of the SIMPLEV algorithm strongly depends on the chosen set of under-relaxation factors and the time-step values.
  2. To obtain convergence, smaller values of under-relaxation factors and time-step values are required. However this gives an increase in the number of iterations to achieve convergence for the SIMPLE and SIMPLEV algorithms.
  3. The optimal under-relaxation factors and time-step values indicate higher efficiency of the SIMPLEV algorithm.
  4. The converged solution of the pressure correction equation in all cases is obtained fastest when the SIMPLEV algorithm is employed. However, for grid systems with large number of nodes, the efficiency of the SIMPLEV algorithm reduces.
  5. From all the results obtained it is shown that the SIMPLEV algorithm will converge more rapidly than the SIMPLE algorithm.
7. Acknowledgements

A special thanks to Mr Callum Downie for his help in setting up the AFLOW code on the UNIX workstations.

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