Page 225 - Computational Fluid Dynamics for Engineers

214 7. Boundary-Layer Equations

the case of internal flows, the problem is somewhat easier because as discussed
in [4], the conservation of mass in integral form can be used to relate pressure
p(x) to velocity u(x, y) in terms of mass balance in the duct.
For two-dimensional external flows, two procedures have been developed to
couple the solutions of the inviscid and viscous equations for airfoil flows. In the
first procedure, developed by Le Balleur [7] and Carter and Wornom [8], the
solution of the boundary-layer equations is obtained by the standard method,
and a displacement-thickness, <5*°(x), distribution is determined. If this initial
calculation encounters separation, 6*°(x) is extrapolated to the trailing edge
of the airfoil. For the given 6*°(x) distribution, the boundary-layer equations
are then solved in the inverse mode to obtain an external velocity u ev(x). An
updated inviscid velocity distribution, u ei(x), is then obtained from the inviscid
flow method with the added displacement thickness. A relaxation formula is
introduced to define an updated displacement-thickness distribution,

6*(x) = «*°(x)|] (7.2.4)
=
6*"(x)il+u;
Uei(x)
where UJ is a relaxation parameter, and the procedure is repeated with this
updated mass flux.
In the second approach, developed by Veldman [9], the external velocity
u e(x) and the displacement thickness 6*(x) are treated as unknown quantities,
and the equations are solved in the inverse mode simultaneously in successive
sweeps over the airfoil surface. For each sweep, the external boundary condition
for the boundary-layer equation dimensionless form, with u e(x) normalized with
i^oo, is written as
u e(x) = u®(x) + 6u e(x) (7.2.5a)
Here u®(x) denotes the inviscid velocity and 6u e the perturbation due to the
displacement thickness, which is calculated from the Hilbert integral
1 C Xb d da
= ~ / - H T O O - ^ - (7.2.5b)
6u f 2
7T J Xa da x - a
The term ^(u e6*) in the above equation denotes the blowing velocity used to
simulate the boundary-layer in the region (x a, x^). This approach is more general
and can be used for two- and three-dimensional flows as discussed in [4].
Predicting the flowfield by solutions based on inviscid-flow theory is usually
adequate as long as the viscous effects are negligible. A boundary layer that
forms on the surface causes the irrotational flow outside it to be on a surface
displaced into the fluid by a distance equal to the displacement thickness <5*,
which represents the deficiency of mass within the boundary layer. Thus, a new
boundary for the inviscid flow, taking the boundary-layer effects into considera-
tion, can be formed by adding 6* to the body surface. The new surface is called
the displacement surface and, if its deviation from the original surface is not