$$ c_n = \frac{1}{c} \left[ \int_0^c (C_{p,l} - C_{p,u}) \, dx + \int_0^c \left(c_{f, u} \frac{dy_u}{dx} + c_{f,l} \frac{dy_l}{dx} \right) dx \right] $$ $$ c_a = \frac{1}{c} \left[ \int_0^c \left(C_{p, u} \frac{dy_u}{dx} - C_{p,l} \frac{dy_l}{dx} \right) dx + \int_0^c (c_{f,u} + c_{f,l}) \, dx \right] $$

I am trying to calculate Cl vs alpha curve for NACA2412 airfoil. I have calculated the non-viscous Cp (pressure coefficient) using panel method and Cf (friction coefficient) using Thwaites, Michael's and Head's boundary layer integral equations. I now have both Cp and Cf for all the panels. In order to calculate Cl (lift coefficient), I need to calculate these force coefficients first. I am stuck here, I am having difficulties solving these two equations. Please somebody help me to solve these two integral equations so that I can insert my Cp and Cf values to get Cn (normal force coefficient) and Ca (axial force coefficient). The author left these equations for the readers as an exercise but seems like I am having huge confusions solving it. I can have dy/dx data from the panel method (it is just the tan(phi) for each panel).

Equation source - Fundamentals of Aerodynamics 6th edition(page number - 26)

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    $\begingroup$ We also have a Mathematics site. $\endgroup$ Dec 10, 2021 at 18:05
  • $\begingroup$ This is an excellent question! Many people are looking for a conceptual understanding of exactly this problem, and understanding the insights of others, both in the question, and given answer, are helpful. $\endgroup$ Dec 11, 2021 at 18:53

1 Answer 1


enter image description hereImage source

In order to find the resulting $C_P$ value over the length of the chord, we need to determine the enclosed area of the pressure distribution plot. Which is what solving an integral equation does.

The equation can be solved analytically if the resulting plots can be captured in a mathematical function of chord length, which would not be my preferred way of proceeding. Or more practically, the values found with the panel method can be linearly interpolated, which is a numerical way of solving.

enter image description here

Each blue rectangle is:

  • In vertical direction, the average of the two upper points minus the average of the two lower points.
  • In horizontal direction = 0.1 * chord $c$

Add all 10 rectangles to get the total area; divide by $c$ to obtain dimensionless unity.

Mutatis mutandis for the remaining bits of the equations.


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