Calculation of the equations of a laminar boundary space layer by the method of integral relations

Abstract

THE method of integral relations, put forward by A. A. Dorodnitsyn [1] for the solution of non-linear problems in aerodynamics, is widely used for numerical integration of the equation of a laminar boundary layer. Integration of the equation for a plane and axially-symmetric laminar boundary layer by this method [2–6] has shown that when the number of strips into which the field of flow is divided is increased the difference between two successive approximations is reduced, and the solution obtained agrees well with the results of calculations by other methods. This indicates that as the number of strips increases the accuracy of the claculation increases, and the solution itself is the solution of the original differential equation. In [3–5] the accuracy of the calculation of the laminary boundary layer equations by the method of integral relations is evaluated over the values of the local friction stress and the local heat flux. In particular, it was shown in [3] that in the neighbourhood of the plane critical point the local friction stress and local heat flux are calculated with an error of the order of 0.2 % in the fourth approximation. The solution of the laminar boundary layer on a blunt cylindrical slate [4] showed that if x ̄ ⩽ π 2 the fourth and third approximations differ by less than 3%; if x ̄ > π 2 the convergence deteriorates and the difference is about 6%. The fourth and fifth approximations on the cylindrical blunted part differ by less than 1.5% and outside it the difference is about 3% [5]. A comparison of the results of calculation of the characteristics of a laminar axially-symmetric boundary layer by the method of integral relations with the data of numerical integration [6] shows that if the field of flow is divided into five strips the method of integral relations in a region with zero or negative pressure gradient ensures an accuracy of calculation of the order of 1.5% in the case of strong heat transfer and of the order of 1.5–20% with no heat transfer. In a region with positive pressure gradient the results of calculations by these two methods differ significantly from one another. In the present paper the method of integral relations is used for the numerical integration of the equations for a laminar space boundary layer with finite external flow. In three particular examples we evaluate the accuracy of calculation of local coefficients of frictional resistance and local heat flow, and also of the profile of frictional stress, the rate of secondary flow and the total enthalpy. The author earlier used the method of integral relations to integrate the equations of a laminar boundary layer on infinite moving elliptic cylinders [7], but there the accuracy of the calculations was not evaluated.