Page 172 - Aerodynamics for Engineering Students

Potential flow 155

What are the advantages of the panel method compared to other numerical
methods such as finite differences and finite elements? Both of the latter are
field methods that require that the whole of the flow field be discretized. The
panel method, on the other hand, only requires the discretization of the body
surface - the boundary of the flow field. The dimensions of the solution are
thereby reduced by one compared to the field method. Thus for the aerofoil calcula-
tion presented above the panel method required N node points along the aerofoil
contour, whereas a field method would require N x N points throughout the flow
field. However, this advantage is more apparent than real, since for the panel method
the N x N influence coefficients need to be calculated. The real advantages of panel
methods lie elsewhere. First, like finite-element methods, but unlike finite difference
methods, the panel method can readily accommodate complex geometries. In fact, an
alternative and perhaps more appropriate term to panel method is boundary-element
method. This name makes the connection with finite elements more clear. A second
advantage compared to any field method is the ease with which panel methods can deal
with an infinite flow field; note that the aerofoil in Fig. 3.39 is placed in an airflow of
infinite extent, as is usual. Thirdly, as can readily be seen from the example in Fig. 3.39,
accurate results can be obtained by means of a relatively coarse discretization, i.e. using
a small number of panels. Lastly, and arguably the most important advantage from the
viewpoint of aerodynamic design, is the ease with which modifications of the design can
be incorporated with a panel method. For example, suppose the effects of under-wing
stores, such as additional fuel tanks or missiles, were being investigated. If an additional
store were to be added it would not be necessary to repeat the entire calculation with a
panel method. It would be necessary only to calculate the additional influence coeffi-
cients involving the new under-wing store. This facility of ,panel methods allows the
effects of modifications to be investigated rapidly during aerodynamic design.

Exercises

1 Define vorticity in a fluid and obtain an expression for vorticity at a point with
polar coordinates (r, e), the motion being assumed two-dimensional. From the
definition of a line vortex as irrotational flow in concentric circles determine the
variation of velocity with radius, hence obtain the stream function ($), and the velocity
potential (+), for a line vortex. (U of L)
2 A sink of strength 120 m2sP1 is situated 2 m downstream from a source of equal
strength in an irrotational uniform stream of 30 m s-l. Find the fineness ratio of the
oval formed by the streamline $ = 0. (Answer: 1.51)(CU)
3 A sink of strength 20 m2 s-' is situated 3 m upstream of a source of 40 m2 s-' , in a
uniform irrotational stream. It is found that at the point 2.5 m equidistant from both
source and sink, the local velocity is normal to the line joining the source and sink.
Find the velocity at this point and the velocity of the undisturbed stream.
(Answer: 1.02 m s-l , 2.29 m s-')(CU)

4 A line source of strength m and a sink of strength 2m are separated a distance c.
Show that the field of flow consists in part of closed curves. Locate any stagnation
points and sketch the field of flow. (U of L)
5 Derive the expression giving the stream function for irrotational flow of an
incompressible fluid past a circular cylinder of infinite span. Hence determine the
position of generators on the cylinder at which the pressure is equal to that of the
undisturbed stream. (Answer: f. 30°, f. 150°)(U of L)