The Method of Complex Characteristics for Design of Transonic Compressors.

We calculate shockless transonic flows past two -dimensional cascades of airfoils characterized by a prescribed speed distribution. The approach is to find solutions of the partial differential equation (c('2)-u('2)) (PHI)(,xx) - 2uv (PHI)(,xy) + (c('2)-v('2)) (PHI)(,yy) = 0 by the method of complex characteristics. Here (PHI) is the velocity potential, so (DEL)(PHI) = (u,v), and c is the local speed of sound. Our method consists in noting that the coefficients of the equation are analytic, so that we can use analytic continuation, conformal mapping, and a spectral method in the hodograph plane to determine the flow. After complex extension we obtain canonical equations for (PHI) and for the stream function (psi) as well as an explicit map from the hodograph plane to complex characteristic coordinates. In the subsonic case, a new coordinate system is defined in which the flow region corresponds to the interior of an ellipse. We construct special solutions of the flow equations in these coordinates by solving characteristic initial value problems in the ellipse with initial data defined by the complete system of Chebyshev polynomials. The condition (psi) = 0 on the boundary of the ellipse is used to determine the series representation of (PHI) and (psi). The map from the ellipse to the complex flow coordinates is found from data specifying the speed q as a function of the arc length s. The transonic problem for shockless flow becomes well posed after appropriate modifications of this procedure. The nonlinearity of the problem is handled by an iterative method that determines the boundary value problem in the ellipse and the map function in sequence. We have implemented this method as a computer code to design two-dimensional cascades of shockless compressor airfoils with gap-to-chord ratios as low as .5 and supersonic zones on both the upper and lower surfaces. The method may be extended to solve more general boundary value problems for second order partial differential equations in two independent variables.