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SUBSONIC FLOW OVER THIN AIRFOILS
Recall the governing equations:
β^2 ϕxx + ϕyy = 0 where,
β = 1 − M∞^2 (1) the boundary condition requiring the flow be tangential to the airfoil surface y = Y(x):
ϕy = V (^) ∞
dY dx (2) imposed at the chord line y=0,
and the result we are interested in, namely, the surface pressure coefficient Cp:
C (^) p = − 2
ϕx V (^) ∞ (3) Note that the angle of attack information is built into the airfoil shape Y(x).
Transformation of Compressible Flow Problem into an Incompressible Flow problem:
Our first step is to transform this compressible flow problem into an incompressible flow problem. There are two reasons for this.
(1) Incompressible flows may be inexpensively modeled using panel methods on personal computers. A number of panel codes written in BASIC, Pascal, C or MATLAB are available in our school for this. Please contact Sankar (894-3014) if you are interested in getting a copy of these codes.
(2) In olden days, before computers, airfoils had to be tested in wind tunnels. It is always easier and less expensive to study or test an airfoil under low speed incompressible flow conditions than under compressible flow conditions.
We know that our governing equation and boundary condition are linear. Therefore, we seek simple linear transformations that will transform the flow from a compressible flow coordinate system (x , y) to an incompressible flow coordinate system (ξ , η).
ξ = B x η = C y (4)
The disturbance velocity potential Φ in the incompressible flow regime is different from ϕ in the compressible flow regime. We assume that these two are linearly related:
Φ = Aϕ (5) The airfoil shapes in the compressible flow Y(x) and incompressible flow problem Y 1 (ξ) will be different. We assume that they are affinely related. That is, their slopes differ from each other only by a constant, D:
dY dx
= D
dY 1 dξ (6) The freestream velocity may also be different in these two problems. We assume that the freestream velocity V∞ 1 in the incompressible flow regime is different from the freestream speed V∞ in the compressible flow regime. That is,
V∞ 1 = E V∞ (7) In the above relations, the constants A, B, C, D and E are at this time unknown.
The incompressible flow is governed by Laplace’s equation:
Φ (^) ξξ + Φ (^) ηη = 0 (8) The boundary condition applied at the airfoil chordline η=0 is
Φ (^) η = V∞ 1
dY 1 dξ (9) The surface pressure coefficient Cp1 in the incompressible flow is
C (^) p1 = − 2
Φξ V (^) ∞ 1 (10)
Now we have defined all the transformation relations. We now begin to transform the compressible flow equation, Boundary condition and definition of Cp given by equations (1), (2) and (3) to forms that resemble their incompressible flow counterparts, equations (8), (9) and (10).
E = 1
Equation (13) then becomes:
C = β (17) and equation (15) becomes
A=βD (18)
This still leaves us with two equations (17) and (18) and three constants, C, A and D. One of these constants may be chosen to be anything we want. There is no unique way this one constant should be chosen.
Historically, the following two choices became the most popular.
Prandtl-Glauert Rule:
In this transformation, the airfoil shape is the same in the compressible flow problem and the incompressible flow problem. This means their slopes are identical, and from equation (6),
D = 1 (18) This yields
C = β A = β (19) The surface pressure distributions in the compressible and incompressible flows may now be related. Consider equation (3). Replace the x- derivative of the disturbance potential ϕ in that equation with ∂Φ/∂ξ, using the transformations (11). Then,
C (^) p = − 2 ϕx V (^) ∞
E
V∞ 1
B
A
∂ξ Since E = 1 , A = β and B = 1
C (^) p =
β
V∞ 1
∂ξ
Comparing this with equation (10), we arrive at the Prandtl-Glauert Rule:
C (^) p,compressible =
C (^) p,Incompressble β if the airfoil shapes are the same in the compressible and incompressible flow problems. (21)
Thus, to apply Prandtl-Glauert rule, simply test (or analyze using Panel method) the same airfoil under incompressible flow conditions. For any compressible flow condition, divide the incompressible flow Cp by β to get the surface pressure coefficient under compressible conditions.
We can integrate the Cp distribution to get lift and drag coefficients. Then it is easy to show that.
C (^) l,compressible =
β
C (^) l,incompressible
C (^) d,compressible =
β
Cd, incompressible = 0
and,
C (^) m,compressible =
β
Cm, incompressible
if the airfoil shapes are the same. (22)
When computing and comparing the pitching moment, they should be about the same hinge point (quarter cord, leading edge etc.). Note that in 2-D incompressible potential flow, the drag is zero.
Gothert’s Rule:
In Gothert’s approach, A=1. From equation (5) the disturbance potentials ϕ and Φ are identical. Then, from equation (18) D equals 1/β. Since D links the slope of the airfoil in the compressible plane with the airfoil in the incompressible flow problem, the airfoils in these two problems will be different. The slope of airfoil in the incompressible flow problem will be higher, by a factor 1/β compared to the slope of the airfoil in the compressible flow. We can show that the Cp distributions of these two airfoils are linked using a procedure similar to Prandtl-Glauert rule. The final result is
