Page 192 - Computational Fluid Dynamics for Engineers

180 6. Inviscid Flow Equations for Incompressible Flows

d 2 d 2
72 2
V = 3 3 + 1 o-5 (6-2-3)
=
dx 2 dy 2
The use of the stream function ^,
«-£•—£ <«•<>

allows the equation resulting from the irrotationality condition,

du dv
(6.2.5)
dy dx
to be written as Laplace's equation in ^,
2
V ^ = 0 (6.2.6)
The assumption of an irrotational flow is a useful one in that it removes the
nonlinearity in the momentum equations and allows them to be replaced by
the Bernoulli equation (2.4.16), which provides an algebraic relation between
velocity and pressure. For a two-dimensional flow, it is given by

2
P + ^Q(u 2 + v ) = const. (6.2.7)

Equations (6.2.2) and (6.2.6) apply to any incompressible irrotational flow
and can be used to compute the velocity field about a given body. What dis-
tinguishes one flow from another are the boundary conditions. For example, to
predict the flowfield about a body at rest in an onset flow, V^, moving in the
increasing x-direction (an onset flow is the flow that would exist if the body
is not present), it is necessary to impose the condition that the surface of the
body is a streamline of the flow, that is,

%b — constant or —— = 0 (6.2.8)
on
at the surface on which n is the direction of the normal, and that far away from
the body. The velocity components are

» = ! = ! = " - < « • ' • * >

«=£=-£=• <"•«»

since the onset flow is irrotational.
The requirement of an irrotational flow and the conditions imposed by Eq.
(6.2.9) are independent of the body shape. The relation given by Eq. (6.2.8),
which does not allow any flow through the body surface, brings in the body
shape about which a given circulation exists; in other words, the uniqueness of