ABSTRACT
An attempt has been made to develop & numerical method for the c a lc u la tio n of low-speed wing theory problems.
The method uses d is t r ib u t io n of s i n g u l a r i t i e s in t e rio r to the body surfa ce and solves the d is t r ib u t io n by s a t i s f y in g the condition o f zero normal fl o w on the body s urface. The
method is a p p lied to the fo llo w in g incom pressible fl o w problems:
( i ) a rb itr a ry n o n - liftin g three dimensional wings in steady in v is c id i r r o t a t i o n a l f l o w
( i i ) multi-element a e r o f o il s in steady inviscid.
i r r o t a t io n a l flo w
( i i i ) a e r o fo ils undergoing a rb itr a ry time-dependent motion in i n v i s c i d i r r o t a t i o n a l flow
( iv) a e r o f o il s (w ith or without control surfaces) in steady viscous flo w
(v ) o s c il l a t o r y a e r o f o i l s and o s c i l l a t o r y control surfaces in viscous flow
Computer programs are developed in FORTRAIJ IV fo r a l l these cases. To check the accuracy of the present method computer programs based on the ^ . . 1 . 0 . Smith technique
are developed in each case. Comparison, between the two methods shows that the present method i s about 3 to 4 times f a s t e r than the method based on the A .m . O . ^mith technique
f o r the same (sometimes improved) n um erical accuracy.
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points of its surface, whereas in other cases the boundary may 'transpire
1
i . e . have a finite flux of fluid through it . It is important to note that any model satisfying the boundary conditions of the mathematical problem can in principle be used subject to the restriction that Laplace's equation must be satisfied everywhere throughout the flow domain. In a small number of cases, exact analytical solutions can be obtained
0
More complex problems do not generally possess exact analytical models and demand numerical solution, but the same general
principle carries over to such techniques. A variety of numerical techniques have been developed for the solution of LaplaceTs
equation and without doubt the most powerful of these is at present the surface singularity or 'panel' technique.
In surface singularity or panel technique, the choice of surfaces on which singularities are located may consist of wetted surface only, or of fictitious internal surface only, or both of these. The satisfaction of given boundary conditions on a given boundary does not necessarily require singularities to be located on that boundary; approximate representation of the
geometry for the purpose of locating singularities can not by it s e lf be regarded as a fundamental cause of numerical error.
Numerous versions of panel techniques are available within the aircraft industry for a number of years. These techniques employ a variety of mathematical formulations and different forms of discretisations. The discretisation process is applied at three different levels : satisfaction of boundary conditions,
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approximation of geometric surface, and approximation of singu
larity distributions. These discretisations lead to certain sources of error of which the most significant error is the
‘ leakage* between collocation points arising from the discreti
sation in the satisfaction of boundary conditions. This error will in general reduce as the panelling is refined and the number of surface collocation points increased. However, this will also lead to increased number of unknowns which will consequently lead to longer computing time. It is therefore desirable to improve the basic accuracy of the model in such a way that leakage is less even with a smaller number of panels.
The attempt to reduce leakage without increasing number of panels has led to the development of the so-called higher order methods which use higher order forms for the spatial distribution of these singularities and retain the curvature effects for the panels on which the singularities are distributed. While this is true that such methods reduce leakage and hence can give sufficiently accurate results with smaller number of panels,
nevertheless such methods are in general highly complex and time- consuming.
However, the method developed by B a s u ^^ (a first-order method) for two-dimensional aerofoils indicates a reduction in
leakage with a small number of panels. His model essentially consists of distributing singularities on the mean camber line of the aerofoil; the boundary condition of zero normal flow is satisfied on the surface of the aerofoil. Comparison with the
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A.i'i.O. Smith method shows this method to be considerably faster for the same numerical accuracy.
In the present thesis, a model has been developed for
calculating the potential flow about three-dimensional non-lifting wings of arbitrary shape using the internal singularity concept in an attempt to reduce the leakage with a small number of panels but without resorting to higher order formulation. The method is first order in that source singularities of constant strengths are placed on the plane panels dividing the chordal plane. The strength of the source singularity distribution is then obtained by satisfying the boundary condition of zero normal flow on the wing surface* The above singularity model has also been adapted for a variety of two dimensional problems.
Methods developed by using the standard A.M .O. Smith technique formed a basis of comparison for all the cases studied in this thesis using internal singularity approach. A short description of the A.M.C. Smith technique has been included in each chapter before developing the method using internal singu
larity distribution.
Chapter 1 of this thesis deals with a survey of contem
porary methods used for solving low speed flow problems. The fundamental mathematical background of idealised flow problem using panel methods is given briefly, followed by a description of discretisation errors inherent in panel methods. A brief
survey of methods used for unsteady flow problems and for problems involving boundary layer effects in steady and oscillatory two
(2)
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dimensional flows are presented. In chapter 2 the three-dimen
sional non-lifting problem has been described. Special features of the present numerical model have been discussed in proper
detail to illustrate some of the advantages of the present numeri
cal model in solving aerofoil and wing problems. Problems of potential flow past multi-element aerofoil is described in
chapter 3. In this chapter, numerical results have been compared with the exact analytical solution of ri.K. Williams (3 ) . In
chapter 4, numerical solutions for general unsteady motion of an aerofoil in inviscid incompressible fluid are presented. Methods are applied to the case of sudden change in aerofoil incidence, to the case of an aerofoil oscillating in pitch and to the case of an aerofoil entering a 'sharp-edged g u s t '. In chapter 5,
numerical solutions for a steady two-dimensional aerofoil with or without control surface in incompressible viscous flow are given.
The estimation of boundary layer characteristics is obtained by the T h w a i t e s ^ (laminar) and the H o r t o n ^ (turbulent) methods.
The simulation of the boundary layer effects is done by both the techniques of 'surface displacement' and 'tra n s p ira tio n '. In
chapter
6
, numerical solutions of oscillatory aerofoils and oscillatory control surfaces in viscous flow are presented.The boundary layer is assumed to be turbulent over the entire surface of the aerofoil. The boundary layer characteristics
in oscillatory aerofoil problems are estimated by the approximate method developed by B a s u ^ . Concluding remarks are given in
chapter 7.
