Page 189 - Aerodynamics for Engineering Students

172 Aerodynamics for Engineering Students

u’ and v’ represent the departure of the local velocity from the undisturbed free
stream, and are commonly known as the disturbance or perturbation velocities. In
fact, thin-aerofoil theory is an example of a small perturbation theory.
The velocity component perpendicular to the aerofoil profile is zero. This
constitutes the boundary condition for the potential flow and can be expressed
mathematically as:
-usinp+vcosp=O at y=yu and y1
Dividing both sides by cos p, this boundary condition can be rewritten as

-(Ucosa+ul)-+Usina+v’=O at y=yu and y1 (4.11)
dY
dx
By making the thin-aerofoil assumptions mentioned above, Eqn (4.11) may be
simplified. Mathematically, these assumptions can be written in the form

dYu dyl
<<
yu and yl e c; a,- and - 1
dx dx
Note that the additional assumption is made that the slope of the aerofoil profile is
small. These thin-aerofoil assumptions imply that the disturbance velocities are small
compared to the undisturbed free-steam speed, i.e.

ut and VI<< U
Given the above assumptions Eqn (4.1 1) can be simplified by replacing cos a and
sina by 1 and a respectively. Furthermore, products of small quantities can be
neglected, thereby allowing the term u‘dyldx to be discarded so that Eqn (4.1 1)
becomes

(4.12)

One further simplification can be made by recognizing that if yu and y1 e c then to
a sufficiently good approximation the boundary conditions Eqn (4.12) can be applied
at y = 0 rather than at y = y, or y1.
Since potential flow with Eqn (4.12) as a boundary condition is a linear system, the
flow around a cambered aerofoil at incidence can be regarded as the superposition of
two separate flows, one circulatory and the other non-circulatory. This is illustrated
in Fig. 4.1 1. The circulatory flow is that around an infinitely thin cambered plate and
the non-circulatory flow is that around a symmetric aerofoil at zero incidence. This
superposition can be demonstrated formally as follows. Let
yu=yc+yt and H=yc-yt
--
y = yc(x) is the function describing the camber line and y = yt = (yu - y1)/2 is known
as the thickness function. Now Eqn (4.12) can be rewritten in the form

dYt
dYc
VI= u-- Ua f u- dx
dx
Circulatory Non-circulatory
where the plus sign applies for the upper surface and the minus sign for the lower
surface.