. Thin Aerofoil Theory
We saw that the shock-expansion theory gives a simple method for computing lift and drag acting over a body kept in a supersonic stream. This theory is applicable as long as the shocks are attached. This theory may be further simplified by approximating it by using the approximate relations for the weak shocks and expansion, when the aerofoil is thin and is kept at a small angle of attack, that is, if the flow inclinations are small. This approximation will result in simple analytical expressions for lift and drag.
At this stage, we may have a doubt about the difference between shock-expansion theory and thin aerofoil theory. The answer to this doubt is the following:
“In shock-expansion theory, the shock is essentially a non-isentropic wave causing a finite increase of entropy. Thus, the total pressure of the flow decreases across the shock. But in thin aerofoil theory even the shock is regarded as an isentropic compression wave. Therefore, the Bow across this compression wave is assumed to be isentropic. Thus the pressure loss across the compression wave is assumed to be negligibly small."
From our studies on weak oblique shocks, we know that the basic approximate expression [Equation (9.174)] for calculating pressure change across a weak shock is:
— ^ / 1 Ah.
pi vm? -1
Because the wave is weak, the pressure p behind the shock will not be significantly different from p1, nor will the Mach number M behind the shock be appreciably different from the freestream Mach number M1. Therefore, we can express the above relation for pressure change across a weak shock, without any reference to the freestream state (that is, without subscript 1 to the pressure and Mach number) as:
— ^ / Ав.
p – JM2 – 1
Now, assuming all direction changes to the freestream direction to be zero and freestream pressure to be p1, we can write:
p-p ^YML(h – 0),
p1 vM2 – 1
where в is the local flow inclination relative to the freestream direction. The pressure coefficient Cp is defined as:
p – p1
where p is the local static pressure and p1 and q1 are the freestream static pressure and dynamic pressure, respectively. In terms of freestream Mach number M1, the pressure coefficient Cp can be expressed as:
Substituting the expression for (p — p1)/p1 in terms of в and M1, we get:
The above equation, which states that the pressure coefficient is proportional to the local Bow direction, is the basic relation for thin aerofoil theory.