# 2-D Inviscid, Linearized, Thin Airfoil Theories

14.5.1.1 Incompressible Flow (Mo = 0)

Definition

In 2-D, define the following aerodynamic coefficients C;, Cm, o, and Cd in terms of the forces and moment per unit span (L’, M’o, D’), the density (p), incoming flow velocity (U) and chord (c). Give the expressions for Ci(a), Cm, o(a) and Cd for a symmetric profile.

Suction Force

Consider a symmetric profile (d(x) = 0). The lifting problem corresponds to the flow past a flat plate at incidence. Sketch the flat plate, the force due to pressure integration and the suction force.

Calculate the suction force coefficient Cs = F’s/( 1 pU2c), where F’s is the suction force per unit span, from the Ci expression.

In what sense does the thickness distribution contribute to recuperating the suction force? Explain.

Center of Pressure

Give the definition of the center of pressure.

Find the location of the center of pressure for a symmetric profile.

Calculate Cm, ac for a symmetric profile.

Verify your result for Cm, o(a) using the change of moment formula, from the aerodynamic center to the profile leading edge.

14.5.1.2 Supersonic Flow (M0 > 1, в = M^ — 1)

A thin double wedge airfoil equips the fins of a missile cruising at Mach number M0 > 1 in a uniform atmosphere. The chord of the airfoil is c. The profile camber and thickness are:

d (x) = 0 20x, 0 < x < c/2
29(c – x), c/2 < x < c

Fig. 14.2 Double wedge in г

supersonic flow at zero Mo  incidence *■

with z±(x) = d(x) ± e(x)/2 + a(c — x). See Fig. 14.2.

Pressure Distributions

Plot —C + and —C— versus x for this airfoil at a = 0.

Lift Coefficient

Give the lift coefficient C; (a) for this airfoil in supersonic flow.

Drag Coefficient

Calculate (Cd)a=0.

Give the value of the coefficient Cd for the general case a = 0.

Moment Coefficient

Calculate the zero incidence moment coefficient (Cm, o) 0.

Give the values of the coefficient Cm, o for the general case a = 0.

Maximum Finess

Form the expression of the inverse of the finess, 1/f = Cd /Ci and find the value of a that maximizes f (minimizes 1 /f).

Sketch the profile at the maximum finess incidence and indicate on your drawing the remarkable waves (shocks, expansions).