On the solution of variational problems of supersonic gas dynamics

Abstract

In solving variational problems of supersonic gas dynamics by the method of Guderly and Armitage [1], it is essential to have linear dependence of the coefficients of the flow parameters being varied on the closing characteristic. This permitted the solution of a number of variatlonal problems [1 to 4] without having to consider the relations between the indicated variations, which arise from the equations of the characteristics. Some problems, however cannot be solved in this manner. In general, therefore, it is also necessary to include the relations on the closing characteristics in the auxilary functional. This is illustrated by the example below, in constructing the rear part of a minimum drag body with restriction on the length, when the contour may contain an end wall. Two cases are studied. In the first, the pressure on the end wall does not depend on the shapes of the desired contour. Here the Lagrange multipliers, introducing the relations on the characteristics, turn out to be zero, and the solution agrees with that obtained earlier [3]' In the second case, the pressure on the end wall is determined by a condition of the type of Korst's condition [5] and consequently, it depends on the shapes of the desired contour. Here it is necessary to introduce the relations on the closing characteristic. This example is also interesting in that for its solution, questions are considered which are connected with varying the position of the junction of the end wall and the segment of the two-sided extremum. The latter are important for the solution of other problems, e.g. in the construction of the nose part of minimum drag bodies.