AE/ME 339 Computational Fluid Dynamics (CFD)
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1 AE/ME 339 Computatioal Fluid Dyamics (CFD 0//004 Topic0_PresCorr_ Computatioal Fluid Dyamics (AE/ME 339 Pressure Correctio Method The pressure correctio formula (6.8.4 Calculatio of p. Coservatio form of the mometum equatios are as follows: Note that these equatios ca be obtaied from the o-coservatio form by usig the cotiuity equatio. See Patakar for a slightly differet formulatio. ( ρ ( ρu ( ρ u uv p u u = µ...(6.88 t x y x x y ( ρ ( ρv ( ρ v uv p v v = µ...(6.89 t y x y x y 0//004 Topic0_PresCorr_
2 Computatioal Fluid Dyamics (AE/ME 339 Pressure Correctio Method 0//004 Topic0_PresCorr_ 3 Computatioal Fluid Dyamics (AE/ME 339 Pressure Correctio Method Figure 6.6 shows a staggered grid where the pressures are calculated at the solid grid poits ad the velocities are evaluated at the ope grid poits. Write the differece equatio for Eq aroud the poit (/, show i the figure. We eed average values of v at poits a ad b. It is obtaied by iterpolatio. at Poit a: at Poit b: _ v j / ( vi, j v, j...(6.90 a vj / ( vi, j v, j...(6.90 b 0//004 Topic0_PresCorr_ 4
3 Computatioal Fluid Dyamics (AE/ME 339 Pressure Correctio Method Differece represetatio for Eq cetered aroud poit (/, ( ρu ( ρu ( ρu ( ρu /, j /, j 3/, j i /, j = _ ρuv ( ρ uv /, j /, j p, j j y u u u u u u µ ( ( y 3/, j /, j i /, j /, j /, j /, j...(6.9 0//004 Topic0_PresCorr_ 5 Computatioal Fluid Dyamics (AE/ME 339 Pressure Correctio Method or...(6.9 /, /, ( ρu = ( ρu A ( p, j j j where _ ρuv ( ( ( ρuv ρu ρ u /, j 3/, j i /, j /, j A = y u u u u u u µ ( ( y 3/, j /, j i /, j /, j /, j /, j 0//004 Topic0_PresCorr_ 6 3
4 Computatioal Fluid Dyamics (AE/ME 339 Pressure Correctio Method 0//004 Topic0_PresCorr_ 7 Computatioal Fluid Dyamics (AE/ME 339 Pressure Correctio Method At Poit c: At Poit d: u = u u ( i /, j i /, j _ u = u u ( /, j /, j Use forward differece i time ad cetral differece i space i Eq gives...(6.93, / i, j / ( ρv = ( ρv B ( pi, j i j 0//004 Topic0_PresCorr_ 8 4
5 Computatioal Fluid Dyamics (AE/ME 339 Pressure Correctio Method where _ ρvu B = y ( ρuv i, j / ( ρv ( ρv, j / i, j 3/ i, j / v v v v v v µ ( (, j / i, j / i, j / i, j 3/ i, j / i, j / 0//004 Topic0_PresCorr_ 9 Computatioal Fluid Dyamics (AE/ME 339 Pressure Correctio Method The staggered grid approach used ca be see from Figures 6.6 ad 6.7. The shaded areas i these two figures represet fiite volumes that would be used i the formulatio by Patakar ad Spaldig. Note that the two areas do t completely overlap. At the begiig of each iteratio p = p*. Therefore, for this step Eqs. 6.9 ad 6.93 become ( ρu = ( ρu A ( p, j /, j /, j ( ρv = ( ρv B ( pi, j i, j / i, j / y...( (6.95 0//004 Topic0_PresCorr_ 0 5
6 Computatioal Fluid Dyamics (AE/ME 339 Pressure Correctio Method Subtract Eq from Eq. 6.9 ad Eq from Eq to get the followig: ( ( ρu = ρu A ( p, j where /, j /, j ( ρ u = ( u ρ ( ρ u /, j /, j /, j ( ρ u = ( ρ u ( ρ u /, j /, j /, j...(6.96 A = A A, p p p, j =, j, j, p = p p i, j i, j i, j 0//004 Topic0_PresCorr_ Computatioal Fluid Dyamics (AE/ME 339 Pressure Correctio Method ( ( ρv = ρv B ( p i, j where i, j / i, j / ( ρv = ( ρv ( ρv y i, j / i, j / i, j / ( ρv = ( ρv ( ρv i, j / i, j / i, j /...(6.97 B = B B, p = p p p = p p, i, j i, j i, j i, j i, j i, j 0//004 Topic0_PresCorr_ 6
7 Computatioal Fluid Dyamics (AE/ME 339 Pressure Correctio Method Eqs ad 6.97 are the x ad y mometum equatios writte i terms of the velocity ad pressure correctio terms. The ext importat step is to get a expressio for pressure correctio by usig the coditio that the velocity field must satisfy the coservatio of mass equatio. The p formula that will be used is ot a exact represetatio. It is devised such that whe covergece is achieved: p 0 ad the formula for p teds to the physically correct cotiuity equatio. 0//004 Topic0_PresCorr_ 3 Computatioal Fluid Dyamics (AE/ME 339 Pressure Correctio Method Patakar sets A, B,( ρu,( ρv to zero i Eqs ad 6.97 which would yield the followig equatios. Recall that ( ρu = ( p, j pi, /, j ( ρv = ( p i, j pi, i, j / y ( ρ u = ( ρ u ( ρ u...( (6.99 /, j /, j /, j 0//004 Topic0_PresCorr_ 4 7
8 Computatioal Fluid Dyamics (AE/ME 339 Pressure Correctio Method Eq ca be writte as ( ρu = ( ρu ( p, j /, j /, j Similarly, Eq becomes ( ρv = ( ρv ( p i, j i, j / i, j / y...( (6.0 The cotiuity equatio cetered aroud the poit (i, usig cetral differecig (CD becomes ( ρu ( ρu ( ρv ( ρv = 0...(6.0 y /, j i /, j i, j / i, j / 0//004 Topic0_PresCorr_ 5 Computatioal Fluid Dyamics (AE/ME 339 Pressure Correctio Method Substitute Eqs. (6.00 ad ad (6.0 i Eq. (6.0 ad droppig the superscript gives ( ρu ( p, j ( ρu ( pi, j pi, /, j i /, j ( v t y ( p p i, j i, ( v ρ t y ( p p ρ i, j i, j i, j / i, j / y Eq. (6.03 ca be rearraged to give (see ext slide for expressios for a, b, c ad d = 0...(6.03 api, j bp, j bpi, j cpi, j cpi, j d = 0...(6.04 0//004 Topic0_PresCorr_ 6 8
9 Computatioal Fluid Dyamics (AE/ME 339 Pressure Correctio Method Where a, b, c ad d are give by the followig expressios a = c = b = ( ( y ( x ( y d = ( ρu ( ρu ( ρv ( ρv /, j i /, j y i, j / i, j / Eq. (6.04 gives the pressure correctio. 0//004 Topic0_PresCorr_ 7 Computatioal Fluid Dyamics (AE/ME 339 Pressure Correctio Method The SIMPLE algorithm: The pressure correctio formula (Eq is approximate because we set A, B,( ρu,( ρv equal to zero. Hece the term Semi-implicit i the ame. This makes the pressure effect to be localized. 0//004 Topic0_PresCorr_ 8 9
10 Computatioal Fluid Dyamics (AE/ME 339 Pressure Correctio Method for the staggered grid give i Figure 6.5. Guess ( p * at all pressure odes ad set ( ρu*,( ρv* arbitrarily at the appropriate velocity odes.. Solve for ( ρu*,( ρv* usig Eqs. (6,94 ad (6.95 respectively. 3. Substitute these values of ( ρu*,( ρv* i Eq. (04 ad solve for p at the iterior odes (boudary odes will be treated separately. Relaxatio procedure would work. 4. The values of (pobtaied i the previous step are used for 5. Calculate p = ( p* p at all odes. solvig the mometum equatios. 6. Repeat steps -5 util covergece criteria are satisfied. 0//004 Topic0_PresCorr_ 9 Computatioal Fluid Dyamics (AE/ME 339 Pressure Correctio Method The superscript ( ad ( used i the above equatios are pseudo-time i the sese that solutio obtaied from this procedure will ot be time-accurate. Therefore, the method essetially is a time-depedet method for steady state problems. ( ad ( therefore, ca be thought of as represetig sequetial iteratio steps. The above procedure ca cause the solutio to diverge. Extesive use of uder-relaxatio factors is employed as a remedy. The followig equatio is a example of how uder-relaxatio factor ca be used. p = p α p p...(6.06 where a is the uder relaxatio factor. 0 α p 0//004 Topic0_PresCorr_ 0 0
11 Computatioal Fluid Dyamics (AE/ME 339 Pressure Correctio Method Boudary coditios for pressure correctio method (6.8.6 Isert Figure 6.8 0//004 Topic0_PresCorr_ Computatioal Fluid Dyamics (AE/ME 339 Pressure Correctio Method At the iflow boudary: p ad v are specified, u is allowed to float. Therefore, p = 0 at the iflow boudary. Outflow boudary: p is specified ad u ad v are allowed to float. At the walls: No slip coditio gives all velocities to be zero. The y-mometum (Eq equatio at the wall ca be writte as: p v v = µ...(6.07 y x y w w Sice v = 0 at the wall, the first term o the RHS will be zero. Also we approximate the secod term to be zero because it is usually small. 0//004 Topic0_PresCorr_
12 Computatioal Fluid Dyamics (AE/ME 339 Pressure Correctio Method Therefore, we have the followig approximate coditio at the wall p y w = 0...(6.08 0//004 Topic0_PresCorr_ 3 Program Completed Uiversity of Missouri-Rolla Copyright 00 Curators of Uiversity of Missouri 0//004 Topic0_PresCorr_ 4
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