Page 193 - Computational Fluid Dynamics for Engineers

6.2 Laplace Equation and Its Fundamental Solutions 181

the irrotational flow about a given body is established if the boundary conditions
on the body and at infinity are satisfied and, in addition, if the magnitude of
the circulation around the body is specified. Thus, the problem of determining
the flowfield of an incompressible inviscid flow about a body is reduced to a
purely mathematical one of finding a suitable solution of Laplace's equation in
either 0 or ip.
The solution of the Laplace equation for the potential function, Eq. (6.2.2),
or for the stream function, Eq. (6.2.6), subject to the boundary conditions that
the resultant velocity is equal to the freestream value at points far from the
surface and that the component of the velocity normal to the surface is zero,
can be obtained by the finite-difference methods discussed in Section 4.5. In this
case, for a given transformation, the Laplace equation in the physical plane can
be written in the form shown in Problem 2.18. With the metrics determined
from the transformation, the transformed Laplace equation can then be solved
in the computational plane subject to its boundary conditions. In the following
section we shall describe this choice to compute the external flow over a circular
cylinder.
A more efficient choice for external flows, however, is to use a panel method
which takes account of the linearity of the Laplace equation and avoids the
need to generate the grid in both physical and computational planes. The panel
method is based on the superposition of flows: functions that individually satisfy
the Laplace equation may be added together to describe the desired flowfield.
A popular approach is to express the flowfield in terms of the velocity potential
based on two or more elemental flows in the presence of an onset flow. The
two elemental flows (sometimes called singularities) often used in this approach,
referred to as the surface-singularity method, include a source, sink and a vortex.
The first is the radially symmetric solution of the Laplace equation, and the
second is its complement, the solution independent of radius. A source flows
radially outward (see Fig. 6.1) such that the continuity equation is satisfied
everywhere but at the singularity that exists at the source's center. The potential
function for a two-dimensional source centered at the origin is

= ^ l n r (6.2.10)
0 s
where r is the radial coordinate from the center of the source and q is the source
strength.
A sink is a negative source, that is, fluid flows into a sink along radial
streamlines. The potential function for a sink centered at the origin is the same
as Eq. (6.2.10) except for the minus sign (—) on the right-hand side.
A potential vortex is defined as a singularity about which fluid flows along
concentric streamlines (see Fig. 6.2). The potential function for a vortex centered
at the origin is
<t>v = - ^ 0 (6.2.11)