Inverse problem for two–dimensional flow around a profile

1. Abstract The main purpose of this paper is to describe a numerical method of an inverse problem for two– dimensional flow of fluid around a profile. The principle of the method is based on construction of a contractive operator. The practical use is getting the possibility of constructing or modifying the shape of a profile by the velocity distribution on the profile surface. We formulate an operator equation for an inverse problem. Our task is to find a shape ψ of a profile for a given chord presented by segment 〈0, l〉 and defined velocity function f = (fu, fl) in this segment for the upper and lower sides of the profile. It is possible to construct a wrong linear inverse explicit operator ψ = Lf using linear theory of singularits. Further, it is known how to construct a correct direct implicit operator f = P(ψ) for a given shape ψ to get a velocity function f around the profile. We can compose these two operators and solve a new operator equation P(Lu) = f for an unknown function u. This problem can be transformed to a problem of ”fix point”. The equation is u = u + α(f − PLu), where α is a suitable real number from interval (0, 1). Number α is chosen so that the sequence {ui}i=0, ui+1 = ui + α(f −PLui), u0 = f has a limit u. The operator P can be the system of the Euler equations for non-viscous compressible flow or the Fredholm integral equation of the second kind for potential incompressible flow. Using this trick we will find the fix point u of the contractive operator. ψ = Lu is the found curve for the shape of the profile. The right solution ψ is checked in the step where the direct operator P is used.