Page 318 - Computational Fluid Dynamics for Engineers

308 10. Inviscid Compressible Flow

for a fixed value of K is shown in Fig. 10.10. Since the flow is mainly subsonic,
high values of the relaxation parameters result in better convergence rates.

10.6 Solution of Full-Potential Equation

The solution of the full-potential equation is the next step in obtaining more
accurate flow solutions over an airfoil surface for the following reasons:
1) the small perturbation assumption is removed, allowing flow solutions on
blunt leading edge airfoils (including the stagnation point).
2) the conservative shock relations are no longer a function of the incoming
freestream Mach number, which deteriorated the flow solutions for the TSD
equations as the Mach number changed from unity.
3) It remains valid for irrotational flow only, but the scalar equation is still
much simpler than the next available mathematical model, namely the Euler
equations.
The non-conservative full-potential equation, derived from the conservative form
Eq. (10.2.3) is
2
2
(a 2 - u )(f xx - 2uv(p xy + (a 2 - v )(p yy = 0 (10.6.1)
with u and v as defined as ip x and (p y, respectively and a is the local speed of
sound

2
a = /<& - ^ V + v - Ul) (10.6.2)
\
The presence of large disturbances, especially at the leading edge of an airfoil,
presents additional challenges that makes the solution of the full potential equa-
tion a much more complex problem than solving the TSD equation. Body fitted
grids are the preferred method for discretizing the computational flow domain
in order to capture the blunt leading edge of airfoils and this translates into the
use of a coordinate transformation to solve the equations in the computational
space rather than in the physical space.
Another and more complex problem occurs when we apply upwinding in
the hyperbolic flow region. For the TSD case, upwinding is achieved in the x-
direction. However, upwinding in the full potential equation could occur in the
x and/or y direction due to the fact that the equation type is now determined
2
by the sign of (u 2 -\- v 2 — a ). The solution to this problem, provided by Prof.
A. Jameson, is called the rotated difference scheme [3], aligning the differences
along the streamline by rewritting Eq. (10.6.1) as

2
2
2 + + V (fy Xy 2 - 2UV(f Xy + U (Py Xy\ _
2
( a - . U (p XX 2uV(f xy + a z V if XX
(10.6.3)