Numerical methods - Computational River Dynamics - page 150

Figure 4.21 Typical 2-D control volume.

The numerical schemes previously introduced for the 1-D case can be extended to
determine the convection and diffusion fluxes at faces w , e , s , and n . For example,
inserting the exponential scheme (4.116) into Eq. (4.130) leads to

F e

n + 1 n + 1 n + 1 n + 1

n
P

n
P

ρ φ − ρ φ

+ φ

− φ

+

P P n 1 P E

A P +

φ

P

t exp ( F e / D e ) − 1

F w F n

n + 1 n + 1 n + 1 n + 1

φ − φ

+ φ

− φ

n + 1 W P n + 1 P N

− φ + + φ

W P

exp ( F w / D w ) − 1 exp ( F n / D n ) − 1

F s

n + 1 n + 1

+ φ

− φ

n + 1 S P

− φ = S A P (4.131)

S

(

F s / D s ) − exp 1

where

P is the area of the control volume at point P ; F w , F e , F s , and
F n are the convection fluxes at cell faces w , e , s , and n , respectively, approximated by
the midpoint integral rule as follows:

A P = ( J ξη)

n + 1

u n + 1
ξ

n + 1

u n + 1
ξ

F w = ρ ( J η) ˆ

, w ,

F e = ρ ( J η) ˆ

, e ,

w e

w e

n + 1

u n + 1
η

n + 1

u n + 1
η

F s = ρ ( J ξ) ˆ , F n = ρ ( J ξ) ˆ

, n ;

(4.132)

s n

s , s n

and D w , D e , D s , and D n are the diffusion parameters:

1
j

1
j

1
j

1
j

D w = ( J α α η) w /ξ w , D e = ( J α α η) e /ξ e ,

2
j

2
j

2
j

2
j

D s = ( J α α ξ) /η

s , D n

= ( J α α ξ) /η

n .

(4.133)

s n