Inverse Panel Method in 2D:

Specify a Cp distribution, and find the airfoil shape that produces the Cp distribution by solving inversely and iteratively the linear system, resulting from a panel method, w.r.t. the coordinates of the control points(not w.r.t. the singularity strengths).

QUESTION: How can I determine the strengths of the singularities for a given Cp distribution? (for example, vortex strength for each panel for constant vortex panel method.)

In the literature, no one explains how to do this, but they seem to have done this.

Thank you

Nibo

there have been several papers written on this subject. some techniques are general and can be applied to any aero solver (NS, euler etc). i wrote a paper for a class on this so i can give you refs. dulikravich g.s. "shape inverse design and optimization for three-dimensional aerodynamics" aiaa paper 95-0695 same author "aerodynamic shape design and and optimization: status and trends" read also the vki lecture series books titles "inverse design and optimization methods" also there are the proceedings of the "International conference on inverse design and optimization in engineering sciences, ICIDES" (I, II, III) there's also a book on inverse design by Elizarov,Il'insky, and Potashev titled "Mathematical methods of airfoil design: inverse boundary problems of aerohydrodynamics" published by Akademie Verlag if you'd like email me directly and i can send you a copy of the paper i wrote. it's on three dimensional Navier stokes inverse design techniques for turbomachinery

On the surface of the airfoil you have:

dg/dt = -(1/rho)(dP/ds)

where g is the surface gamma or the vortex sheet strength, t is time, dP/ds is the pressure gradient in the s direction along the contour of the airfoil. (rho is density)

Since C_p is non-dimensional Del_P, you can easily see from the above how the prescription of C_P is equivalent to prescribing the surface vorticity. Check, e.g., the book by R.I. Lewis, "Vortex Element Methods for Fluid Dynamic Analyis of Engineering Systems"

Another way of looking at it is:

C_P = 1 - (V/U_inf)^2

where U_inf is the freestream velocity and V is the local velocity. Now a boundary element (panel) formulation can be written for the potential velocity V on the surface to link C_P with the surface sources and doublets. For more details you can check, e.g., D. Mateescu, "A hybrid panel method for aerofoil aerodynamics," Boundary Elements XII, Vol. 2 Applications in Fluid Mechanics and Field Problems, Ed. Tanaka, et al, 1990

Adrin Gharakhani

THIS IS ALSO MY QUESTION, PLEASE SEND ME IF YOU FOIND THE ANSWERS AND RESULTS.

THANKS

HASSAN