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x-momentumequation

or for i=1 ( x direction) ui=u 1=u :

general momentum equation :

i j

i

ji j

ji x

p xu

xuu

xu

t

x p

zu

yu

xu

uw z

uv y

uu x

ut

2

2

2

2

2

2

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x-momentumequation discretized

in the x direction:

e E P nbnbee A p pbuaua

PW Ew e

N

S

n

s

ee

ne

se

x

y

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y-momentumequation

y p

zv

yv

xv

vw z

vv y

uv x

vt

2

2

2

2

2

2

or for i=2 ( y direction) ui=u 2=v :

general momentum equation :

i j

i

ji j

ji x

p xu

xuu

xu

t

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y-momentumequation discretized

in the y direction:

n N P nbnbnn A p pbvava

x

y

S

PW Ew e

N

n

s

nn

nenw

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z- momentumequation

z p

zw

yw

xw

ww z

vw y

uw x

wt

2

2

2

2

2

2

or for i=3 ( z direction) ui=u 3=w :

general momentum equation :

i j

i

ji j

ji x

p xu

xuu

xu

t

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z-momentumequation discretized

in the z direction:

t T P nbnbt t A p pbwawa

Bx

z

PW Ew e

T

t

b

tt

tetw

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First guess

e E P nbnbee A p pbuaua ****

for a guessed (wrong) pressure field p* we get a wrong velocity field u*, v*, w* by solving the momentum equations:

n N P nbnbnn A p pbvava ****

t T P nbnbt t A p pbwawa ****

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Pressure andvelocity corrections

p p p *

uuu *

vvv *

www *

one needs to calculate:the pressure corrections p'

and the velocity corrections u', v', w'

thus we need to make corrections:

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Link of pressure andvelocity corrections

e E P nbnbee A p pbuaua

e E P nbnbee A p pbuaua ****

e E P nbnbee A p puaua

e E P ee A p pua

giving:0

E P ee p pd u

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Link of pressure andvelocity corrections

eee uuu *

E P eee p pd uu *

nnn vvv *

t t t www *

N P nnn p pd vv *

T P t t t p pd ww *

one still needs an equation for:the pressure corrections p' orthe velocity corrections u', v', w'

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Pressure correctionequation

make use of the continuity equation:

0w z

v y

u xt

discretized:

t

z y xold P P

z yuu we z xvv sn

0 y xww bt

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Pressure correctionequation

b pa pa

pa pa

pa pa pa

B BT T

S S N N

W W E E P P

BT S N W E P

bb Bt t T

s sS nn N

wwW ee E

aaaaaaa

y xd a y xd a z xd a z xd a

z yd a z yd a

t

z y xb old P P

z yuu we **

z xvv sn **

y xww bt **

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SIMPLE algorithm

Guess the pressure field p* Solve the momentum equations toobtain u*, v*, w*Solve the p' equation (continuity)Correct pressure field p = p* + p'Calculate the velocity corrections

u', v', w' Correct velocity field

u =u*+u',v =v*+v'w =w*+w'

Treat the corrected pressure p as anew guess p* and go to