Airfoils(2-D flow)
 

Objective:Find the pressure distribution around an airfoil in subsonic compressibleflow.

Method: Transformthe linearized potential equation into an equation describing a "related"incompressible flow. This will be the Laplace equation. Its solution willgive the pressure distribution around the "related" airfoil in incompressibleflow. Alternately, the pressure distribution in incompressible flow canbe obtained from the extensive wind tunnel data available in the literature(incompressible flow wind tunnels are far cheaper to run than high-Machnumber tunnels). Transform this solution back to the compressible-flowcase.

Linearized potential equation for 2-D subsonicflow

(x and y subscripts denote derivatives)

or where , 

The Laplace equation describing incompressibleflow is

i.e. 

Equation (1) can then be transformed to(2) by setting

and 

and where'm' is some constant.

Note: The text by Anderson refersto as and .

Thus, equation (1) becomes,

or, 

Note that 'm' is still undetermined. Wecan decide later what we want it to be.
 

BoundaryConditions

1. At ¥ , 
2. Surface tangency

Subsonic compressible flow where y = f (x) is the surface

Incompressible flow where describesthe surface

Note that the same isused in both flows. If we used different ones, everything cancels out eventually:it makes no difference.

Now, 

or 

However, the boundary condition in theincompressible flow is

i.e. 

or Relation between surface slopes of the original airfoil and the "related"incompressible flow airfoil.

Note that the slope is a function of chordwiselocation on the airfoil. Different choices of will give different relations between the slopes.
 

PressureCoefficients

Incompressible flow: 

Depending on the problem to be solved,we can make various choices of m
 

Considerthe choice m=1

This implies that ;i.e. the chordwise pressure distributions in the incompressible and compressibleflows are the same. However, we see also that it implies that .(Note 0<b <1). The compressible flow airfoil must have a lower surfaceslope than the incompressible-flow airfoil, in order for the pressure distributionsto be the same.

The other usual question is: "How doesthe pressure distribution over a given airfoil change as the Mach # isincreased?"

To answer this, make a choice of 'm' thatwill give .This is m = b .

c_p increases with Mach #.

We have so far  seen the general idea of transforming the linearizedpotential equation and the boundary conditions for our problem, to an equivalentLaplace equation with appropriate boundary conditions in another coordinatesystem. We saw that by leaving an "undetermined constant" in the transformation,we have the flexibility to pick useful transformations. Below, we extendthis basic concept to the problems of a thin wing, and then to the caseof an axisymmetric body (a body of "revolution"). This is in fact a 2-Dproblem, because things look the same everywhere around a given section.