Numerical methods - Computational River Dynamics - page 165

and the pressure correction has the following relation:

p n + 1 p ∗ + p = (4.191)

The momentum interpolation technique proposed by Rhie and Chow (1983)
calculates the values of u i at face w as

u i , w = α u [ ( G 1 ∗

i , PW

f x , P G 1 ∗
i , P

a PW + a P ] 1 − f x , P ) + ]+ α u [ ( 1 − f x , P )/ f x , P /

1
i

p ∗ W − p P ) + ( u i , W + f x , P u i , P ] × ( J α η) ( 1 − α ) [ ( 1 − f x , P ) (4.192) w u

where G 1 ∗
i , P

H i , P + D i ( p s − p n ) , and G 1 ∗

i , PW and a PW are the values of G 1 ∗ i , P and a P

=

for the neighboring control volume centered at point W .
Similarly, the values of u i at face s are calculated by

u i , s = α G 2 ∗

i , PS

f y , P G 2 ∗ a PS + a P ]

[ ( 1 − f y , P ) + ]+ α [ ( 1 − f y , P )/ f y , P /

u u

i , P

i

p S − p P ) + ( u i , S + f y , P u i , P ] × ( J α ξ) ( 1 − α ) [ ( 1 − f y , P ) (4.193)

s u

where G 2 ∗
i , P

H i , P + D i ( p w − p e ) , and G 2 ∗

i , PS and a PS are the values of G 2 ∗ i , P and a P for

=

the neighboring control volume centered at point S .
Subtracting Eqs. (4.192) and (4.193) from their counterparts for u n + 1
i , w

and u n + 1
i , s

under the pressure field p n + 1 and neglecting the terms G 1 ∗
i , P

G i , P , G 2 ∗ G i , P , etc.,

− −

i , P

leads to

u n + 1
i , w

u i , w + α u Q i , w ( p W − p P ) = (4.194)

u n + 1
i , s

u i , s + α

u Q i , s (

p S − p P )

= (4.195)

where Q i , w =[ ( a PW + a P ] (

i

η) w , and Q i , s =[ ( a PS + a P ] 1 − f x , P )/ f x , P / J α 1 − f y , P )/ f y , P /

2
i

( J α

s .
Using the definition (4.132) of the fluxes at cell faces yields

ξ)

a p
W

F w + p W − p P ) F w = ( (4.196)

a p
S

F s p S − p P )

F s = + ( (4.197)

where a p W = α

w Q i , w , a S s Q i , s , and F w and F s are

the fluxes determined using Eq. (4.132) in terms of the approximate velocities u i , w
abd u i , s .
Integrating the continuity equation (4.185) over the control volume shown in
Fig. 4.21 and discretizing the time-derivative term with the backward difference

n + 1 1

i

n + 1 2

i

ρ ( J α η) = α ρ ( J α ξ)

u u

w s