8/10/2019 CFD Notes 06
1/13
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
8/10/2019 CFD Notes 06
2/13
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
8/10/2019 CFD Notes 06
3/13
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
8/10/2019 CFD Notes 06
4/13
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
8/10/2019 CFD Notes 06
5/13
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
8/10/2019 CFD Notes 06
6/13
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
8/10/2019 CFD Notes 06
7/13
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 ****
8/10/2019 CFD Notes 06
8/13
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:
8/10/2019 CFD Notes 06
9/13
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
8/10/2019 CFD Notes 06
10/13
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'
8/10/2019 CFD Notes 06
11/13
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
8/10/2019 CFD Notes 06
12/13
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 **
8/10/2019 CFD Notes 06
13/13
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