Page 215 - Introduction to Computational Fluid Dynamics

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2D CONVECTION – COMPLEX DOMAINS
i May 25, 2005 11:10
1
3.Knowing coordinates of nodes and vertices, calculate β , l x i , d x i for i = 1, 2 and
f m,c and A c for each face of every node. This is a once-and-for-all calculation
and all these quantities are stored in two-dimensional arrays (N , K). In addition,
V is calculated for each cell.
Solution Begins
l
4.At a given time step, guess the pressure field p .
l
l
5.Solve Equation 6.103 for = u and u . The solution is preceded by evaluation
1 2
of AE (N , K) and AP (N), SP (N), and the entire source term SU (N) in Equation
6.103. It is assumed that SU and SP are appropriately modified to account for
boundary conditions.
6.Perform a maximum of ten iterations on Equation 6.134 for p . Here AE (N , K)

are evaluated from Equation 6.133 and the source term containing mass fluxes
is evaluated from Equation 6.65.
l
l
l

7.Recover the p distribution from p = p − 0.5(p − p ), where p is evalu-

m m
ated from Equations 6.137 to 6.140.
8.Apply pressure and velocity corrections at each node. Thus
l
p l+1 = p + β p , 0 <β < 1, (6.141)
P P m,P

α V ∂p
u l+1 = u l − m , (6.142)
i,P i,P
AP u i ∂x i
P
where the pressure gradient is evaluated using Equations 6.105 and 6.106. The
mass-source residual R m is evaluated from Equation 5.73, where AP and A k
coefficients are the same as in Equation 6.134.
9.Solve Equation 6.103 for all other relevant scalar s.
10.Check convergence by evaluating residual R via the imbalance in Equation
6.103 for each as explained in Chapter 5. Special care is again needed in
evaluation of the mass residual R m . This is evaluated from the imbalance in
Equation 6.134 in which p is replaced by p .

m
l
l
11.If the convergence criterion is not satisfied, treat p l+1 = p and l+1 = and
return to step 5.
o
12.To execute the next time step, set all = l+1 and return to step 4.
6.4 Applications
Flow over Banks of Tubes
In shell-and-tube heat exchangers, the flow on the shell side takes place over a bank
of tubes several rows deep. The flow is aligned at various angles to the axis of
the tubes. However, for preliminary design work, the flow may be assumed to be
transverse to the axis (i.e., a cross flow). This configuration has been extensively re-
searched and experimentally determined data are available [91] for different values