ctesi1

8/14/2019 ctesi1

1/18

International Journal of Fluid Dynamics (2001) Vol 5 , Article 5, 59-7 5 Available online at: http://elecpress.monash.edu.au/ijfd

Removal of Temporal and Under-Relaxation Terms from the Pressure-

Correction Equation of the SIMPLE Algorithm

V.A.O. Anjorin and I.E. Barton

Department Mechanical Engineering, Brunel University, Uxbridge, Middlesex, UB8 3PH, UK. E-mail: [email protected]

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

The SIMPLEV (SIMPLEVincent) algorithm is an improved version of the standard SIMPLEalgorithm where the under-relaxation and temporal terms are removed from the pressure correction

equation of the SIMPLE algorithm.This paper focuses on the methodologies of the SIMPLEV algorithm and how it is used to solve thediscretized momentum equations that yield the pressure correction terms of the pressure correctionequation. The SIMPLE and SIMPLEV algorithms rely on under-relaxation to avoid divergence andthere 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 smallpressure under-relaxation factor must be employed. The SIMPLE algorithm is assessed with a versionthat has the terms for the pressure correction equation that increases the pressure correction as thevelocity under-relaxation factors or the time step terms decrease. It is shown that the SIMPLEValgorithm gives the reverse effect of this.

1. INTRODUCTION

A method for solving the momentum equations for laminar flow problems has been developed byPatankar and Spalding (1972). They proposed the SIMPLE (Semi-Implicit Method for Pressure-LinkedEquations) algorithm to iteratively solve the momentum equations in their discretized form. Themethodology 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 realdifficulty in calculating the velocity field lies in determining the unknown pressure field. There is noobvious equation to solve thus pressure is used to satisfy the condition for continuity. To solve this thepressure field is indirectly specified via the continuity equation. This indirect specification is achievedby obtaining a whole set of discretized equations from the momentum and continuity equations andsolving the discretized equations by a direct solution.

The advantage of the SIMPLE algorithm is that it uses an iterative approach. Iterative solutions arecommonly used to solve a whole set of discretized equations so that they may be applied to one singledependent 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 flowproblems. If the flow is compressible (Zaiyong et al, 1998), the pressure may be obtained from thedensity and temperature by using the equation of state where the density is regarded as a dependentvariable of the continuity equation as discussed by Versteeg and Malalasekera (1995). Since the flowwe 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 onthe solution of the flow field. If the correct pressure field is applied in the momentum equations, theresulting velocity field satisfies the continuity equation. A brief explanation of the methodology of theSIMPLE 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 ismade to enhance the convergence rate (Wanik and Schnell, 1989) of this algorithm. This is achieved bydeveloping a new algorithm known as the SIMPLEV algorithm. This is aimed at giving a faster rate of

8/14/2019 ctesi1

2/18

The SIMPLEV Algorithm

60

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 samemethodology as that of the SIMPLE algorithm in solving the velocity fields that satisfy the continuityequation except that the under-relaxation and temporal terms are removed from the pressure-correctionequation of the SIMPLE algorithm. When this is performed the pressure correction tends to zerotherefore 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 setof partial differential equations. The momentum and continuity equations in their primitive form areshown in equations (1-3) where the equation for conservation of mass is given by

( ) ( )0=

+

y

v

x

u . (1)

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

( ) ( ) ( ) uSxP

t+

+

=

++

y

u

y x

u

xvu

yuu

x

u

, (2)

as well as the v-momentum equation

( ) ( ) ( ) vSyP

t+

+

=

++

y

v

y x

v

xvv

yuv

x

v

. (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 ( ), thevelocity components for the x and y directions of the two-dimensional control volume ( u, v), the u andv momentum source terms ( Su, Sv), the pressure field ( P ), and the molecular viscosity ( ).

2.2. DISCRETIZATION

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

y

=

y v

x = xu

y

=

y v

Figure 1: Staggered grid showing locations of the flow variables: = u ; = v ; P .

8/14/2019 ctesi1

3/18


61

This yields the governing equations in their discretized form as shown in equations (4-6). Thestaggered grid evaluates the scalar variables, in this case only the pressure, which are stored at thescalar nodes marked ( ), and located at the intersection of two unbroken grid lines. Such points areindicated by the capital letters P, W, E, N and S. The u-velocity components are stored at the east andwest 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. Thehorizontal arrows ( ) shown in Fig. 1 indicate the locations of the ue and uw velocity components andthe vertical arrows ( ) indicate the locations of the vn and vs velocity components. Forward staggeredvelocity grids are used. The uniform grids are forward staggered since the u-velocity, ue, is at a distanceof 1/2 xu from the scalar node P p. Similarly, the location for the v-velocity, vn, is at a distance of 1/2 yv from the scalar node. After the process of discretization, the discretized continuity equationbecomes:

( ) ( )[ ] ( ) ( )[ ] 0snwe =+ xvv yuu , (4)

and the discretized u-momentum equation becomes

( )( )

tbaa

1e

uPEnbnbenewe

++=k u

yPPuu , (5)

and finally the discretized v-momentum equation can be written

( )( )

tbaa

1n

vPNnbnbnnewn

++=k v

xPPvv . (6)

The time derivative terms are incorporated into the coefficients newea andnewna of equations (5) and (6)

and the source terms Su and Sv of equations (2) and (3). The coefficientsnewea and

newna are expressed as

follows

taa e

newe

+= y x t

aa nnewn

+= y x .

The values of the coefficients a e and a n at the east and north cell faces of the control volume areobtained 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 a nb account for thecombined 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. Theterm ( P E P P) y represents the pressure force acting on the u control volume. The pressure differenceacts on the control volume of width y. Similarly ( P N P P) x represents the pressure force acting onthe v control volume where the pressure difference acts on the control volume of width x. The firstterms on the right hand side of equations (5) and (6) ( anbunb and anbvnb) represents the summation of the product of the neighbour coefficients and the velocity fields at the neighbouring nodes. When thesource terms of equations (2) and (3) are integrated over the control volume they give V S u and

V S v which are approximated by the final volume method (Fletcher, 1988; Ranade, 1997) as

( ) tb 1e +=k

uu uV S , (7)

( ) tb 1n +=k

vv vV S . (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 andMalalasekera (1995) where they show how to employ differencing schemes to successfully interpolatebetween the node points.

8/14/2019 ctesi1

4/18


62

The velocity and pressure fields of the SIMPLE and SIMPLEV algorithms are all solved on astaggered 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 discussionof this is presented by Simoneau and Pollard (1994). In the past, a majority of applications use thestaggered grid arrangement to solve all the flow variables for numerical modelling purposes. As anexample 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. Thedifferential equations for the computation of turbulent wind flow conditions around buildings werediscretized 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 Chowinterpolation method (Oliveira et al, 1998; Rhie and Chow, 1983) which is based on a non-staggeredgrid. This approach cures the problem of odd-even coupling between the pressure and velocity fieldsby 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 eu , nv .

( )( )

tbaa

1e

uPEnbnbenewe ++=

k u yPPuu (9)

( )

t

1n

vbPNnbnbannewna

++

=

k v xPPvv . (10)

Next solve the corrected-velocity fields eu ,

nv 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).( )

t

1e

ubPEnbnbaenewea

++

=

k u yPPuu (11)

( ) ( )t

baa1

nvPNnbnbt

newn

++=

k v

xPPvv . (12)

Then the pressure correction equation (24) is derived by substituting the terms eu and

nv into thecontinuity equation (4) using the correction formulae equations (13) and (14).

( )'P'Enewea

p p y

uu ee = (13)

( )'P'Nnewna

p p x

vv nn = . (14)

Equations (13) and (14) are obtained by subtracting those from the second step away from those fromthe first step. After the pressure correction equation (24) is solved, the solution of the pressure field issubstituted back into equations (9) and (10) to obtain the terms eu , nv (Barton, 1998). To under-relax

the discretized u and v momentum equations (11) and (12) of the SIMPLE algorithm, we begin with therelationship between the corrected pressure and its under-relaxation factor P as well as the relationshipbetween the velocity fields and their under-relaxation factors u and v. These are derived to determinethe effect that their under-relaxation has on the convergence of the algorithm. The pressure field pnew isunder-relaxed as follows:

'p

new p p p += (15)

where p is the pressure under-relaxation factor. The relaxation factor is taken between 0 and 1 so thatthe 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 neweu and newnv areunder-relaxed in the following manner:

8/14/2019 ctesi1

5/18


63

( ) ( )1u1**eunewe += nuuu (16)

( ) ( )1vnvnewn 1 += n** vvv . (17)

The terms **n**

e , vu are the corrected velocity fields without under-relaxation and ( ) ( )11 , nn vu

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 intothe general discretized equation (11). To recall, this is:

( )( )

t

1e

ub**

P****

nbnba**

enewe

a

++=

k u y p

E puu , (18)

which can be re-written as

( )( )

newe

1e

u**

PEnbnb**

ea

tba

++=

k ** u y p pu

u , (19)

so that under-relaxation can be introduced.

Next we assume that **eu can be replaced by neweu and introduce the previous obtained values of theunder-relaxed velocity fields to give

( )( )

( )

( )

++

+=

1newe

1e

u**

P**

E**

nbnb

1newe

at

ba

n

k

n u

u y p pu

uu. (20)

Here ( )1nu is taken as the value of **

eu from the previous iteration. The variables in the parenthesis

represent the change in neweu produced by the current iteration. So this change can be modified by theintroduction of the velocity under-relaxation factor u so that

( )( )

( )

( )

++

+=

1newe

1e

u**

P**

E**

nbnb

u1newe

at

ban

k

n u

u y p pu

uu . (21)

Substituting the under-relaxed velocity field of equation (16) into equation (21) and re-arranging givesthe discretized u-momentum equation with under-relaxation:

( ) ( ) ( ) ( )1u

newe

1e

u**

P**

E**

nbnbnewe

newe a1

tba

a

+++= nu

k

uu

u y p puu

. (22)

Similarly the v-momentum equation with under-relaxation becomes:

( )( )

( ) ( )1v

newn

v

1n

v**

P**

N**

nbnbnewn

v

newn a1

tba

a

+++= n

k v

v x p pvv

. (23)

To derive the under-relaxed pressure-correction equation of the SIMPLE algorithm, the velocity fieldsneweu and

newnv of equations (16) and (17) are substituted into equations (22) and (23) to give the

following corrected velocity fields

y p p

uu

=

newe

'N

'E

uee a & x

p pvv

=

newn

'P

'N

vnn a .

8/14/2019 ctesi1

6/18


64

Substituting the corrected velocity fields into the discretized continuity equation (4) yields

( ) ( )[ ].

aaaa

snwe

news

'S

'P

vnewn

'P

'N

vneww

'W

'P

unewe

'P

'E

u

xvv yuu

x x p p

x p p

y y p p

y p p

+=

+

(24)

4. SOLUTION PROCEDURE OF THE SIMPLEV ALGORITHM

The pressure correction equation of the SIMPLEV algorithm is derived by first determining if thereis a connection between the under-relaxation factor ( u and v) and the time-step t. Using the under-relaxed discretized u-momentum equation (22), the temporal term ( ) t1-e k u is disregarded so that acomparison can be made of the effect of under-relaxation and the original temporal term. Taking theleft-hand side of equation (22) and removing the temporal term gives

u

eu

newee

at

a

=

+ u y x . (25)

Also taking the right-hand side of equation (22) and removing the temporal term leads to

( )e

u

u1-ee

ua

1t

a1

=

+

k u u y x

. (26)

Using equations (25) and (26), an investigation is made as to whether there is a connection between u and t . For equation (25) we first of all make the newea coefficient equal to a e / u as

u

ee

newe

at

aa

=

+= y x . (27)

Also for equation (26) we make

eu

u a1

t

=

y x. (28)

Using equation (27) to compare t and u gives

( )u y x

== 1

aa

at u

eeue . (29)

Rearranging equation (29) for t gives

( )ueu

1at

= y x . (30)

From equation (30) it can be seen that there is a connection between t and u. Taking the inverse of equation (30) results in equation (28). Alternatively equation (27) can be rearranged for u as

+

=

ta

a

e

eu y x

. (31)

8/14/2019 ctesi1

7/18


65

From equation (31) there is another connection between the under-relaxation factor ( u) and the timestep t . To examine the effect of what happens to the pressure-correction equation of the SIMPLEalgorithm as the under-relaxation terms tends to zero, equation (28) is substituted into equation (24)which gives

.'S'P

v

v1sasa

vs'P

'N

v

v1nana

vn

'W

'P

u

u1wawa

uw'P

'E

u

u1eaea

ue

snwe

x p p x

p p x

y p p y

p p y

xvv yuu

+

+

+

+

+

=

+

(32)Equation (32) can be simplified as follows by factorizing the a e, aw, an, a s coefficients for the right-handside terms of equation (32) to give

( ) ( )( ) ( ) ( )( )[ ]

( ) ( ) ( ) ( )

( )( )

( )( ) .

11

a11

a

11

a11

a

'P

'N

vvn

v'S

'P

vvs

v

'P

'E

uue

u'W

'P

uuw

u

snwe

x p p x

p p x

y p p y

p p y

xuu yuu

ns

ew

+

++

+

+=

+

(33)

An important point that can be seen from equation (33) is that, as the velocity under-relaxationfactors u and v tend to zero, the pressure correction increases. Logically, we actually require thereverse result of this such that as u and v 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) issubstituted into equation (22). Next the inverse of the newea coefficient term of equation (27) is takenand substituted into equation (22) to give the final implicit discretized u and v momentum equationswhich are presented as:

( )( ) ( )( )

( ) ( )1e1-

euPEnbnbu1euee a1t

ba1aa

+++=+ nuk

n uu

yPPuuu , (34)

( )( ) ( )( )

( ) ( )1nv1

nvPNnbnbv1nvnn a1t

ba1aa

+++=+ n

k

n vv

xPPuvv . (35)

This set of equations is equivalent to

( )( ) ( )( )

( ) ( )1e1

euPEnbnbu1euee a1t

ba1aa

+

++=+ nu

k

n uu

yPPuuu , (36)

( )( ) ( )( )

( ) ( )1nv1

nvPNnbnbv1nvnn a1t

ba1aa

+++=+ n

k

n vv

xPPuvv . (37)

Subtracting equations (36) and (37) from (34) and (35) gives

) ) )yPPPPuuuu = PPEE'nbnbeee aa , (38)

8/14/2019 ctesi1

8/18


66

) ) )xPPPPvvv = PPNN'nbnbnnvn aa . (39)Rearranging and omitting terms 'nbnba u and

'nbnba v of equations (38) and (39) gives

( )'P'Eue

eea

PP y

uu =

, (40)

( )'P'Nvn

nn aPP

xvv =

. (41)

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

( ) ( )( ) ( ) ( )( )[ ] ( ) ( )

( ) ( ) .aa

aa

'P

'N

vn

n'S

'P

vs

s

'P

'E

uee

'W

'P

uwwsnwe

x p p x

p p x

y p p y

p p y

xvv yuu

+

=+

(42)

So, as u and v 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. Fromthe pressure-correction equation (24) of the SIMPLE algorithm, it is noticed that as the time step t of the a new coefficient tends to zero, the pressure correction p becomes large. The reverse result of this ast and p tends to zero is obtained by first removing the under-relaxation terms from equation (24) asunder-relaxation is not required for this case. A parameter is introduced into equation (24) by takingthe inverse value of the a new term of equation (27). This gives

1

enewe

taa1

+== y x . (43)

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

( ) ( )( ) ( ) ( )( )[ ] ( )( ) ( )( )[ ]( )( ) ( )( )[ ] .'P'Nn'S'Ps

'P

'Ee

'W

'Pwsnwe

xPP xPP x

yPP yPP y y pv pv y pu pu

+=+

(44)

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

( )

t

1

eubPEnbnbaetea

++ =

+

k u yPPuu

y x , (45)

( )

t

1n

vbPNnbnbantna

++

=

+

k v xPPvv

y x . (46)

The a e and a n coefficients are upgraded to thenewea and

newna in which the parameter of equation

(43) is introduced into equations (45) and (46) to give

( )

t

1e

ubPEnbnbaet

++

=

+

k u yPPuu

y x , (47)

( )1n

vbPNnbnban

++

=

+

k v

xPPvv

y x . (48)

8/14/2019 ctesi1

9/18


67

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

( ) ( )( ) ( ) ( )( )[ ] ( ) ( )

( ) ( ) .aa

aa

'S

'P

vss

'P

'N

vnn

'W

'P

uww

'P

'E

ueesnwe

xPP x

PP x

yPP y

PP y

y pv pv y pu pu

+

=+

(49)

As the under-relaxation and time-step terms of equation (49) tends to zero, the velocity fields willprogressively satisfy the continuity equation so that the converged solution of equation (49) can beachieved. A flow chart of the SIMPLEV algorithm is shown in Fig. 2.

Figure 2: Flow chart for the SIMPLEV algorithm.

The next section presents two simple laminar flow problems of the steady state solutions of laminarflow 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 theconvergence rate of the SIMPLE and SIMPLEV algorithms.

p

8/14/2019 ctesi1

10/18


68

5. RESULTS AND DISCUSSION

Comparisons were performed between the SIMPLEV and SIMPLE algorithms. Our investigationfocuses 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 bychoosing 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 isobtained 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 SIMPLEValgorithms.

The results of the converged solution of the pressure correction and the number of iterations to reachconvergence for the SIMPLE and SIMPLEV algorithms were obtained by the use of a BrunelUniversity CFD code known as AFLOW (Barton, 1995). Computations are performed on a Cartesiangrid system (Guo et al, 1998) with 30x30 and 60x60 grid points respectively for the two laminar flowapplications. 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 isan easy problem in which the SIMPLEV algorithm can be applied to investigate how the pressure fieldon a control volume can be improved in order for the velocity fields to satisfy continuity and hencehow the converged solution of the pressure-correction equation can be determined. An inlet Reynoldsnumber of 800 was used. The Reynolds number for this case is defined using the diameter of the squarecylinder, the freestream velocity and the viscosity as mentioned by Barton (1998). A schematicshowing the geometry and boundary conditions is shown in Fig. 3. The grid used was compressedtowards the upper and lower boundaries, as well as the solid boundaries of the square cylinder to betterresolve the higher gradients in those regions. The upper and lower boundaries are situated 4 D awayfrom 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.

Figure 3: Representation of flow around a square cylinder.

5.1.2. Flow Predictions around Square Cylinder

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

regions.

8/14/2019 ctesi1

11/18


69

Fig. 5 shows the flow around a square cylinder where the flow approaches, separates and is forcedaway from the square cylinder forming two recirculation regions behind the cylinder. The upstreamflow 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).

Figure 4: Velocity profiles are various positions for the flow past a square cylinder.

Figure 5: Streamlines showing the flow past the square cylinder.

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 designand geometry, and is used in a variety of engineering applications. The sudden changes in pressure inwhich our SIMPLEV algorithm numerically evaluates can be studied using backward-facing stepconfigurations. The configuration of the flow over a backward facing step is shown in Fig. 6 which hasan inlet channel of height ( h ) and a parabolic inlet flow profile.

Figure 6: Schematic of laminar flow over a backward-facing step showing main reattachmentregion, which reattaches at x 1, and an upper recirculation region which separates at x 2 andreattaches at x 3.

The channel expansion number was set to 2 and this is defined as the ratio of the total height of themain channel to the height of the inlet channel ( h ). The inlet Reynolds number was set at Re = 800,

8/14/2019 ctesi1

12/18


70

where the Reynolds number is based on the total channel height (2 h), the average inlet velocity and thedynamic viscosity .

5.2.1. Flow Predictions for the Backward-Facing Step Problem

Shown in Fig. 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). Ithas a main recirculation point where the flow reattaches. As the flow recovers downstream, arecirculation region is formed that separates and later recovers as the pressure recovers. Initially theflow has four recirculation regions. The two recirculation regions furthest downstream eventuallydisappear with increasing time because the vorticity of the flow decreases finally enabling the flow tohave two recirculation regions.

Figure 7: Streamline representation of flow for the backward-facing step problem.

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

Table 1: Predictions of the growth of the main reattachment and separationpositions with non-dimensionalised time (T) for backward-facing step problem.

SIMPLE SIMPLEV

T X1 X2 X3 X1 X2 X310 12.244 2.3216 14.262 12.969 3.5539 14.36420 12.270 2.4300 14.279 12.969 3.6707 14.36630 12.302 2.6079 14.292 12.972 3.7237 14.36740 12.973 3.0919 14.370 12.977 4.4357 14.36850 12.973 3.1079 14.370 12.980 4.5573 14.36960 12.972 3.1238 14.370 12.982 4.6060 14.37070 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 searchfor the minimum number of iterations and the optimum under-relaxation factors that enable thesolution to converge. During the computational process, the pressure-correction equations of theSIMPLE 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 thepressure correction equations of the SIMPLE and SIMPLEV algorithms. The convergence criterionthat was used for the TDMA solver was 5x10 -5 for the backward-facing step problem and 7x10 -4 for theproblem of laminar flow around a square cylinder. For both algorithms, the convergence criteriaremained the same so that there can be a comparison between the two schemes. This happens when thevelocity and pressure residuals reach their desired convergence criteria.

The effectiveness of the SIMPLEV algorithm depends on the chosen set of values of the velocity andpressure under-relaxation factors. Shown in Fig. 8 are the results of the influence of the under-

relaxation factors ( u, v and p) (Johansson and Davidson, 1997; Jun Zhang, 1996) on the number of iterations to yield convergence. Optimal values ( u, v and p) are marked as well as the allowed

8/14/2019 ctesi1

13/18


71

intervals of changes of these factors. However if coefficients are taken from the outside of theseintervals, divergence of the computational process occurs.

In this computational test, the velocity under-relaxation factors ( u, v) were varied simultaneouslywhile keeping the pressure under-relaxation factor ( p) constant. Also the pressure under-relaxationfactor was varied while keeping the velocity under-relaxation factors constant. The set of optimalvalues 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 Fig. 8 that thesaving 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 greaterthan the SIMPLE method. The values shown in Fig. 8 are recommended.

Figure 8: Comparisons of the convergence rates for the SIMPLE and SIMPLEV methods.Iterations required for various under-relaxation parameters. Results are for a 30x30 mesh. Top:Convergence rates for the backward-facing step flow. Bottom: Convergence rates for the flow pasta square cylinder.

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 thesolutions are therefore obtained with more CPU time than in the case of the SIMPLE method. Analternative way to consider the algorithms is by introducing a time-step instead of under-relaxation.The results of Fig. 9 presents the convergence rates of the SIMPLE and SIMPLEV schemes expressedin non-dimensional time intervals. For both schemes it can be shown that the converged solution iseasily obtained as the time step decreases. However using very small number of time steps leads to anincreasing number of iterations that are required to enable the solution of the pressure correctionequation to converge. For the case of laminar flow around a square cylinder of Fig. 9 an initial timestep of 0.01 units was applied for both schemes in which convergence was obtained after 782 iterationsfor 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, theSIMPLE and SIMPLEV algorithms require more iterations than the 30x30 grid system to obtainconvergence. 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 bothmethods 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

0.9

0.6

1

0.4

0

1000

2000

3000

4000

5000

6000

0 0.2 0.4 0.6 0.8 1 1.2

Alphau,Alphav

Alphap

Result using SIMPLE algorithm

0.8

0.7

0.9

0.3

0

500

1000

1500

2000

2500

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

Alphau,AlphavAlphap

Result using SIMPLEV algorithm

1

0.8

1.1

0.8

0

500

1000

1500

2000

2500

3000

3500

4000

4500

0 0.2 0.4 0.6 0.8 1 1.2

Alphau,AlphavAlphap

Result using SIMPLE algorithm

0.8

1

0.2

1.1

0

500

1000

1500

2000

2500

0 0.2 0.4 0.6 0.8 1 1.2

Alphau,AlphavAlphap

Result using SIMPLEV algorithm

8/14/2019 ctesi1

14/18


72

time, therefore causing slow convergence of both schemes. With more number of node points in thecomputational domain, the SIMPLEV algorithm is therefore not able to improve the efficiency of computations. The computational tests for the two grid systems of Fig. 9 were performed in order toinvestigate the influence of the number of nodes on the SIMPLE and SIMPLEV algorithms. Above allthe SIMPLEV scheme yields better convergence as fewer iterations are required for the convergesolution of the pressure correction equation.

Fig.10 shows the results of the number of iterations of the iterative versions of the SIMPLE andSIMPLEV algorithms as well as the changes of the residual values of the velocities and pressuresduring the iterative computation of the AFLOW code. It can be shown that the SIMPLEV algorithmattains the required value of the convergence criterion in less time than the SIMPLE algorithm. Itshould be noted that the results shown in Fig.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 everyiteration and the convergence criterion in all cases is attained fastest than the SIMPLE method.However it can be observed from Fig.10 that the SIMPLEV algorithm is not able to improve theefficiency of computations if a grid system with large number of nodes is employed. This makes itquite difficult for the converging solutions to be obtained. The set of optimal under-relaxationcoefficients for which there is higher computational efficiency is shown in Table 2. This set wasdetermined 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 thecomputational process to reach convergence.

Figure 9: Comparisons of the convergence rates for the SIMPLE and SIMPLEV methods.Number of iterations required against non-dimensionalised time to convergence. Results are for a

30x30 and 60x60 mesh. Top: Convergence rates for the flow past a square cylinder. Bottom:Convergence rates for the backward-facing step flow.

80

60 80

70

0

200

400

600

800

1000

1200

1400

1600

1800

0 10 20 30 40 50 60 70 80 90

SIMPLE

SIMPLEV

30x30 grid system

70

60

80

50

0

1000

2000

3000

4000

5000

6000

0 10 20 30 40 50 60 70 80 90

SIMPLE

SIMPLEV

60X60 grid system

50

60

40

50

0

200

400

600

800

1000

1200

1400

1600

1800

2000

0 10 20 30 40 50 60 70

SIMPLEV

SIMPLE

60x60 grid system

80

70

90

70

0

100

200

300

400

500

600

700

800

900

1000

0 10 20 30 40 50 60 70 80 90 100

SIMPLE

SIMPLEV30x30 grid system

8/14/2019 ctesi1

15/18


73

Figure 10: Convergence comparison between SIMPLE and SIMPLEV for different mesh sizes.The vertical axis corresponds to the logarithm of the residual for the momentum and continuityequations. The horizontal axis shows the number of iterations. Top to bottom: (a) convergencehistory for the backward-facing step problem, 30x30 mesh; (b) convergence history for thebackward-facing step problem, 60x60 mesh; (c) and (d) as for (a) and (b) but for the flow past asquare cylinder.

8/14/2019 ctesi1

16/18


74

6. CONCLUSIONS

The work presented in this paper compares the performance of the SIMPLEV and SIMPLEalgorithms 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 appliedto steady state laminar flow over a backward-facing step and steady state laminar flow around a squarecylinder. 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 arerequired. However this gives an increase in the number of iterations to achieve convergence for theSIMPLE and SIMPLEV algorithms.

3) The optimal under-relaxation factors and time-step values indicate higher efficiency of theSIMPLEV algorithm.

4) The converged solution of the pressure correction equation in all cases is obtained fastest when theSIMPLEV algorithm is employed. However, for grid systems with large number of nodes, theefficiency 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.

Table 2. The optimal sets of under-relaxation factors for 30x30 grid systemshowing the quickest convergence

Problem Backward-facing step Square cylinderParameter u v p u v pMethod

SIMPLE 0.6 0.6 0.4 0.8 0.8 0.8No of iterations 422 422 892 92 92 392Method

SIMPLEV 0.7 0.7 0.3 0.8 0.8 0.2No of iterations 272 272 882 72 72 252

Table 3. Optimal time step values for SIMPLE and SIMPLEV methodsProblem Laminar flow around square cylinder Backward-facing stepGrid 30x30 60x60 30x30 60x60Method SIMPLENumber ofiterations

152 522 542 3172

Time 70 40 60 50Method SIMPLEVNumber of

iterations

132 462 102 2362

Time 70 50 70 60

8/14/2019 ctesi1

17/18

8/14/2019 ctesi1

18/18


76

24. Robert R. and Hwang., (2000), Ground Effect on Vortical Flows behind a Square Cylinder,Proceedings of the International Offshore and Polar Engineering Conference , 3 , 467-470.

25. Simoneau, J. and Pollard, A., (1994), Finite-Volume Methods for Laminar and Turbulent Flowusing a Penalty Function Approach, International Journal for Nunerical Methods in Fluids ., 18 ,733-746.

26. Spalding, D. B., (1972), A Novel Finite-difference Formulation for Differential Expressions

Involving Both First and Second Derivatives, Int . J . Numer . Methods . Eng ., 4 , 551.27. Stathopoulos,T. Baskaran, B. A.,(1996), Computer Simulation of Wind Environmental Conditionsaround Buildings, Engineering Structures , 18 , Issue 11, 876-885.

28. Versteeg, H. K and Malalasekera, W., (1995), An Introduction to Computational Fluid Dynamics:The Finite Volume Method, McGraw-Hill, Loughborough University.

29. Wanik, A. & Schnell, U., (1989), Some Remarks on the PISO and SIMPLE Algorithms for SteadyTurbulent Flow Problems, Computers & Fluids ., 17 , No.4, 555-570.

30. Wengle, H. Huppertz, A. Barwolff, G. Janke, G., (2001), Manipulated Transitional Backward-Facing Step Flow: An Experimental and Direct Numerical Simulation Investigation, European

Journal of Mechanics, B/Fluids , 20 , Issue 1, 25-46.31. Zaiyong, S. Hongqing, H. and Timin, C., (1998), Solving Viscous Flow in Nozzles using

Improved SIMPLE Algorithm, Tuijin Jishu/Journal of Propulsion Technology, 19 , Issue 3, 48-51.

Documents

ctesi1