Page 312 - Computational Fluid Dynamics for Engineers

302 10. Inviscid Compressible Flow

10.5 Model Problem for the Transonic Small Disturbance
Equation: Flow Over a Non-Lifting Airfoil

We consider a non-lifting airfoil placed in a uniform flow (Fig. 10.4).
Since the flow is symmetrical on the upper and lower surfaces of the airfoil,
only the upper surface will be considered with a symmetrical boundary condition
along the dividing streamline to reproduce the physics of the flow field:

= 0 (10.5.1)
(p y
The surface boundary condition on the perturbation velocity potential is

10 5 2
Vv = % ( - - )
where y = f(x) describes the airfoil coordinates. The upstream, downstream
and upper boundaries are placed far enough to ensure uniform flow:

if = 0 (10.5.3)

An alternative boundary condition on the upstream and downstream boundary
can be used:
<Px = 0 (10.5.4)

whereas an alternate boundary condition on the upper boundary can be

<p y = 0 (10.5.5)
The incoming flow field is assumed to be subsonic, but subsonic or supersonic
(transonic) flow conditions can be attained above the airfoil surface. Most of the
flowfield will then be subsonic, governed by the theory of elliptic equations, while
only a small portion of the computational domain will be governed by hyperbolic
equations. Therefore, the block iteration method described in subsection 4.5.2
will be used to solve the discretized flow equations, with the hyperbolic terms
properly discretized.

9=0

( p = 0 _ df <p = 0 Fig. 10.4. Non-lifting airfoil in a uni-
y y
^y " dx form flow.