Title:

Problems in transonic flow

The object of this research is to examine two particular problems of transonic flow. The first problem of axisymmetric nature is solved in the physical plane. The second problem is of two dimensional character and the solution is obtained by a transformation to the hodograph plane. Thus part I of this thesis deals with the case of transonic flow past a slender pointed parabolicarc body of revolution at zero angle of attack. Part II deals with the design of a straight walled wind tunnel with a finite porous section to give reduced blockage interference in high subsonic compressible flow. Continuous solutions for the problem in Part I have been obtained by Spreiter and Alksne and by Cole and Royce. These approximate solutions were determined from the second order linearised partial differential equations obtained by replacing one of the partial differential coefficients in the nonlinear term of the transonic small disturbance flow equation by a linear parameter. The method we use to obtain our solution is very similar to that used by Spreitar and Alksne. The difference in the complete solution to the problem is that they used the solutions of three different linearised equations to obtain a continuous solution while we use only the solutions of two linearised equations along with a shock surface to give a solution. As it is not possible to give a rigorous mathematical Justification for the approximate methods used, the only way whereby their validity may be established is to compare the values obtained for the coefficients of pressure on the surface of the body with experimental results. Over the forebody, Where our solution and that of Spreiter and Alksne are identical, the values obtained for the coefficient of pressure agree very well with those obtained in the theory of Cole and Royce and with the experimental results. Over the rear part of the body the values obtained by Spreiter and Alksne and by Cole and Royce, and lower than those given by the experimental results while our values are in a good agreement with them. In Part II it is assumed that a solution to the problem can be determined by a perturbation from the solution found by Helliwell for a tunnel with solid straight walls. This approximate solution was derived from Tricomi's equation which is the second order linear partial differential equation obtained by interchanging the dependent and independent variables in the transonic small disturbance flow equation. From the "perturbation" solution it is shown that it is possible to eliminate some of the blockage interference and that it should be possible to eliminate the blockage interference entirely by the use of materials with greater values of porosity than those for which the present theory is valid. It should be noted that the solution presented here may not be strictly justified for the flow of an ideal gas as the order of the approximations made in deriving the basic Tricomi solution are of the same order as those made in deriving the "perturbation" solution. One may however consider the identical problem for the flow of a "Tricomi" gas. In this case the exact governing equation for the flow is Tricomi's equation and a perturbation theory based upon this equation is then fully justified.
