Page 146 - Introduction to Computational Fluid Dynamics

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5.3 METHOD OF SOLUTION
Since none of these alternatives can be relied upon, it is advisable to ensure 12:28 125
that the overall mass balance for the domain is maintained throughout the iterative
process. This means that the exit-mass flow rate must equal the known inflow
rate. Thus, after effecting the boundary condition (marked by superscript *, say)
according to any of the alternatives just described, it is important to correct the
boundary velocities as
u 1b = u ∗ F, u 2b = u ∗ F, # ˙ m ∗ , (5.78)
1b 2b F = ˙ m in exit
where ˙ m ∗ is evaluated from the starred velocity boundary condition.
exit

Periodic Boundary
Figure 5.5(b) shows flow between parallel plates with attached fins. In this case,
after an initial development length, the flow between two fins will repeat exactly.
Such a flow is called periodically fully developed flow and the periodic boundary
condition will imply
1, j = IN,JN+1− j = 0.5( 2, j + IN−1,JN+1− j ),

u 2(1, j) =−u 2(IN, j) = 0.5(u 2(2, j) − u 2(IN−1,JN+1− j) ), (5.79)
where IN and JN are the total number of nodes in the i and j directions, respec-
tively. Note that in this boundary condition specification, the u 2 velocity has anti-
periodicity whereas all other shave even periodicity.

5.3.5 Boundary Condition for p
m
The boundary condition for p is given by Equation 5.33. The reason for this can

m
be understood from step 3 of the calculation procedure. When this step is executed,
l
the u fields along with their boundary values u l are already known. Now, when
i i,b
the p equation is solved, it is assumed that these boundary values are correct and,

therefore, require no further corrections.
l
If we now consider Equation 5.12 for = u l+1 and = u and subtract the
1 1
l

latter equation from the former, with u = u l+1 − u ,wehave
1
1
1

∂p
AP u = A k u − V m ,
1,P 1,k

∂x 1 P

where ks represent neighbours of P. Also, A k u = 0 through our assumption
1,k
introduced in Section 5.2.3. This explains the form of velocity correction introduced
in Equation 5.63 for an interior node. The same arguments apply to the u 2 velocity
corrections given in Equation 5.64.
Now, if the preceding equation is written for the boundary nodes (P = b),
clearly u = 0 because no corrections are to be applied to the boundary velocities.
1,b

Therefore, ∂p /∂x 1 | b = 0. This is boundary condition (5.33). In discretised form,
m