BOUNDARY AND INFINITY CONDITIONS

Laplace’s equation for the velocity potential is the governing partial differential equation for the velocity for an inviscid, incompressible, and irrotational flow. It is an elliptic differential equation that results in a boundary-value problem. For aerodynamic problems the boundary conditions need to be specified on all solid surfaces and at infinity. One form of the boundary condition on a solid-fluid interface is given in Eq. (2.22). Another statement of this boundary condition, which will prove useful in applications, is obtained in the following way.

Let the solid surface be given by

F(x, y, z, t) = 0 (2.23)

in cartesian coordinates. Particles on the surface move with velocity qB such that F remains zero. Therefore the derivative of F following the surface particles must be zero:

/ D dF

Ы/-37+«*-vf = ° <2-24>

Equation (2.22) can be rewritten as

q • VF = qs • VF (2.25)

since the normal to the surface n is proportional to the gradient of F.

If Eq. (2.25) is now substituted into Eq. (2.24) the boundary condition

dF „ DF

—- l-q – VF = —— = 0

at 4 Dt

At infinity, the disturbance q due to the body moving through a fluid that was initially at rest decays to zero. In a space-fixed frame of reference the velocity of such fluid (at rest) is therefore zero at infinity (far from the solid boundaries of the body):

lim q = 0 (2.28)