Example: Thin Airfoil Theory

As a practical application of the tools we developed in this chapter, let's use the simpsonsRule() method to solve for the lift and moment coefficients of an airfoil using thin airfoil theory. Once again, when reading through this section concentrate on the process. Even if you never need to compute the aerodynamic characteristics of a thin airfoil, you can apply the same solution process to your own integral problems.

Before we discuss what thin airfoil theory is, let's define some terminology about an airfoil. Consider the airfoil shown in Figure 21.2. A straight line drawn from the wing leading edge to trailing edge is called the chord line . The length of the chord line is the chord of the wing. The mean camber line is the line from the leading to trailing edge such that there is an equal airfoil thickness above and below the line. The difference between the mean camber and chord lines is the camber of the wing. A symmetrical airfoil would be one that has zero camber. The angle that an airfoil's chord line makes with the airfoil's direction of travel is called the angle of attack .

Figure 21.2. Generic airfoil configuration

Now let's move on to thin airfoil theory. The German aerodynamicist Ludwig Prandtl developed this theory in the early 1900s. It starts by modeling an airfoil as a thin plate that follows the mean camber line. Because you can't have air flow through the solid surface of the airfoil, the normal velocity at the airfoil surface is balanced by the induced normal velocity of a sheet of vortices distributed along the airfoil. This assumption results in the following integral equation.

Equation 21.24

In Eq. (21.24), U is the freestream velocity, c is the chord length, a is the angle of attack, and z = z ( x ) is the equation that describes the mean camber line. The quantity g is the unknown vorticity distribution function that we need to compute. It turns out that the vorticity distribution can be approximated using a Fourier series.

Equation 21.25

Eq. (21.25) uses a change of variables from x to q with the relation in Eq. (21.26).

Equation 21.26

The q quantity varies from 0 to p . After performing some trigonometric manipulations, we arrive at the expressions in Eq. (21.27) for A and A _n .

Equation 21.27

Once the A coefficients are known, the lift and leading edge moment coefficients can be computed from the relations shown in Eq. (21.28) and Eq. (21.29). The terms r and U are the freestream density and velocity, L is the lifting force on the airfoil, and M is the moment about the leading edge.

Equation 21.28

Equation 21.29

We will now write a program to compute the A coefficients for an arbitrarily cambered wing using Simpson's rule integration method we developed earlier in this chapter. Once the A coefficients have been found, the program will calculate the lift and leading edge moment coefficients for the airfoil.

We will apply our program to compute the aerodynamic coefficients for a NACA 2412 airfoil traveling at an angle of attack of 5 degrees. The NACA 2412 is an airfoil with a maximum thickness of 0.12*chord and has a 2 percent maximum camber at 40 percent chord. The equation for its camber line is defined in two pieces.

Equation 21.30

Taking the derivatives of Eq. (21.30) and converting x to q , we obtain expressions (Eq. 21.31) for z / x that we need to compute the A coefficients.

Equation 21.31

The first code to write is a Function subclass that will return the function z / x cos(n q ). This function is needed to evaluate the A integrals. The class is named CamberFunction and its source code is shown next . The power to be applied to the cosine term is passed to the CamberFunction constructor.

 package TechJava.MathLib; //  This class represents the camber //  function for a NACA 2412 airfoil public class CamberFunction extends Function {   private int n;   public CamberFunction(int n) {     this.n = n;   }   //  getValue() returns the height of the camber   //  line for a given theta where   //  cos(theta) = 1.0  2x/c   public double getValue(double theta) {     if (theta < 1.36944) {       return (0.125*Math.cos(theta) - 0.025)*               Math.cos (theta*n);     } else {       return (0.0555*Math.cos(theta) - 0.0111)*               Math.cos (theta*n);     }   } } 

Now we can write our thin airfoil analysis code, which we will call ThinAirfoil.java . In this case, the program computes the aerodynamic coefficients for a 5 degree angle of attack. With a little extra effort, you could modify the code to take the angle of attack as an input argument. After declaring some variables, the code calculates the A , A ₁ , and A ₂ coefficients. It does this by creating a CamberFunction object and then integrating the associated function using the simpsonsRule() method. Once the A coefficients are in hand, the lift and moment coefficients are computed and the results are printed to the console.

Here is the ThinAirFoil class source code ”

 import TechJava.MathLib.*; public class ThinAirfoil {   public static void main(String args[]) {     double alpha, Cl, Cm;     double A[] = new double[3];     CamberFunction naca2412;     //  Compute the conditions for a 5 degree     //  angle of attack.     alpha = 5.0*Math.PI/180.0;     //  Calculate the A0, A1, and A2 coefficients     double temp = 1.0/Math.PI;     naca2412 = new CamberFunction(0);     A[0] = alpha  temp*Integrator.simpsonsRule(naca2412,0.0,Math.PI,1.0e-4);     for(int i=1; i<3; ++i) {       naca2412 = new CamberFunction(i);       A[i] = 2.0*temp*Integrator.simpsonsRule(naca2412,0.0,Math.PI,1.0e-4);     }     //  Compute Cl and Cm(leading edge)     Cl = Math.PI*(2.0*A[0] + A[1]);     Cm = -0.5*Math.PI*(A[0] + A[1] - 0.5*A[2]);     //  Print out the results     for(int i=0; i<3; ++i) {       System.out.println("A["+i+"] = "+A[i]);     }     System.out.println("\nCl = "+Cl);     System.out.println("Cm = "+Cm);   } } 

Output ”

 A[0] = 0.08274980955387196 A[1] = 0.08146092605250739 A[2] = 0.01387204625538765 Cl = 0.7758494344019757 Cm = -0.02470465406592427 

The compute lift coefficient value is within 3.5 percent of the experimentally obtained value ¹ of 0.75.