Page 315 - Computational Fluid Dynamics for Engineers

10.5 Model Problem for the Transonic Small Disturbance Equation 305

coefficients introduces "if" statements in the computer logic that slows down the
calculations while also making the source code more difficult to read. Another
method of applying the boundary conditions consists of using halo cells lying
outside the computational domain (hence their name). Figure 10.5 shows the
computational space of Fig. 10.4 in the presence of the halo cells. The values of
the perturbation potential <p in the halo cells are determined by the boundary
conditions, and the algorithm Eq. (10.5.11) remains unaltered throughout the
computational domain. As an example, for a computational space extending
from j = 1 to j = jmax, the symmetry and airfoil surface boundary conditions
Eqs. (10.5.1) and (10.5.2) are used to update the halo cells located on the j = 0
line. Their values are obtained here with a first-order difference formula:

<Pi,o = <Pi,i (10.5.13)
for the symmetry boundary condition and

¥>i,o = <Pi,i ~ ^ % (10.5.14)
for the airfoil surface boundary condition.
The overall solution procedure is summarized below:
(a) generate the grid (subroutine generate grid)
(b) Initialize the perturbation potential (p (i.e. (p = 0 everywhere, subroutine
i n i t i a l conditions)
(c) apply the boundary conditions (subroutine boundary conditions)
(d) compute the coefficients a, 6, c and the RHS in Eq. (10.5.11). In this step, the
n
coefficient of the (p term in Eq. (10.5.1) must be computed at each point in
order to select appropriate central or upwind differencing, (subroutine slor)
(e) solve Eq. (10.5.11) using the Thomas algorithm (subroutine tridiagonal).
(f) update the values of tp using under/over relaxation [Eq. (10.5.12)]
(g) repeat steps (c) through (f) until convergence of the iterative process

One of the original test case used by Murman and Cole [2], namely the transonic
flow over a non-lifting circular arc airfoil has been programmed. The airfoil
coordinates are given by
2
= ( l - x ) for - 1 < x < 1 (10.5.15)
y r
where the parameter r controls the thickness-to-chord ratio. Analysis of the
TSD equation shows that a transonic similarity parameter relates r and the
freestream Mach number:
2 3
K=(l- M^)/r / (10.5.16)
and that the pressure distribution can be scaled as
3
C P = ^ ( T / M O O ) / 2 (10.5.17)