Publisher Summary The chapter reviews subdivisions of the boundary layers that become necessary under the impact of sudden stream wise changes. If the boundary layer is supersonic, a new phenomenon occurs that appears to have no counterpart in subsonic flow. It leads to a greater ease of study of the flow properties and helps to overcome the barrier of separation that appears to hinder progress in the incompressible studies. The phenomenon is the free-interaction boundary layer, first observed by Ackeret and independently by Liepmann in their study of the interaction between a shock wave and a boundary layer and more extensively studied subsequently by Liepmann, Chapman, Hakkinen, and many others. These studies show that when a shock, sufficiently strong to provoke separation, strikes a laminar boundary layer, the boundary layer actually separates ahead of the foot of the shock and the flow features of the separation region are independent of the characteristics of the shock and depend only on the local properties of the flow. The chapter provides example for incompressible boundary layers when the fluid is compressible and explains the modifications necessary to allow this effect.

Multistructured Boundary Layers on Flat Plates and Related Bodies

Advances in CFD prediction of shock wave turbulent boundary layer interactions

Asymptotic theory of separated flows

A method of calculation is constructed for the Navier-Stokes equations using asymptotic expansions for high Reynolds numbers and matching techniques to join solutions (for example, [1]). The equations and boundary conditions for the first approximation, the method, and the results of numerical integration for the region lying above the separation point, including the separation point as well, are presented. A comparison is made with experimental data, and corrections corresponding to the second approximation are estimated. On the basis of these results, the limits of applicability of the approximate theories that utilize the boundary layer equations are discussed.

Theory of laminar boundary layer separation in supersonic flow

Separation is a fluid dynamic phenomenon that influences the behaviour of a wide variety of liquid and gas flows. The difference between an attached flow and its separated counterpart is demonstrated in Figure 1 where the theoretical streamline pattern, given by the classical solution of the inviscid flow theory1

Asymptotic Theory of Separated Flows

A method is presented for calculating the distribution of the thermal fluxes, friction stresses, and pressure near the corner point of a body contour in whose vicinity the outer supersonic flow passes through an expansion wave. The method is based on a study of the asymptotic solutions of the Navier-Stokes equations as the Reynolds number R approaches infinity for the flow region in which the longitudinal gradients of the flow functions are large, invalidating conventional boundary layer theory. This problem was examined in part in [1], in which the distribution of the friction and pressure in a region with length on the order of a few thicknesses of the approaching boundary layer was obtained in the first approximation. The leading term of the expansion for the thermal flux to the surface of the body vanishes for a value of the Prandtl number equal to unity and for other values of the Prandtl number does not match directly with its value in the undisturbed boundary layer. The thermal-flux distribution is obtained for values of the Prandtl number σ approaching unity. For this purpose it was necessary to consider a more general double passage to the limit as σ→1 and e→0 for a finite value of the parameter B=[(σ−1)/σ] [−ln e1/4/e]1/4 characterizing the ratio of the effects of thermal conduction, viscous dissipation, and convection. The solution obtained previously [1] corresponds to the particular case ∥B∥ → ∞ and therefore for actual values of R=104–106, σ~ 0.7 overestimates considerably the effect of the dissipative term on heat transfer, although even in first approximation it describes the pressure distribution well and the friction distribution satisfactorily. For smooth matching of the solutions with the corresponding flow functions in the undisturbed boundary layer it was necessary to introduce a flow region with “free interaction” for the expansion flow. Equations and boundary conditions which describe the flow as a whole are presented. Examples are given of numerical calculations and comparison with experiment.

Asymptotic theory for calculating heat fluxes near the corner of a body

It is shown that, for hypersonic flows with moderate and strong degrees of interaction, perturbations brought about, for example, by a bottom opening or by any other sort of obstacle are propagated up to the leading edge of a solid body. Local regions with very large pressure gradients cannot arise in the flow. This is connected with the possibility of the development of breakaway zones with a length on the order of magnitude of the size of the solid body, described in the first approximation by the equations of the boundary layer. From a mathematical point of view the problem comes down to establishing the nonsingular nature of the solution near the leading edge, and to finding eigensolutions which make it possible to satisfy the boundary conditions at the trailing edge of the solid body. It is shown that, with a weak interaction between the hypersonic flow and the boundary layer, there may arise short flow regions with free interaction and locally nonviscous flows with large pressure gradients, within the limits of which the perturbations may move upstream.

Propagation of perturbations upstream with interaction between a hypersonic flow and a boundary layer

The flow structure behind the separation point of a laminar boundary layer in a supersonic stream has been investigated. Analytic and numerical solutions are obtained for simple semiinfinite separation zones starting from the leading edge or a point on the smooth surface. The question of the pressure plateau in a separation zone of finite length is discussed and its value is calculated on the basis of asymptotic theory. The asymptotic theory of flow [1, 2] in the neighborhood of the separation point of the laminar boundary layer in a supersonic gas stream (region of free interaction) is employed. The local solution obtained is subsequently used to construct the flow pattern in the separation zone [3]. An analysis is made of the behavior of the solution for the free-interaction region on transition to the region of reverse flows. The results make it possible actually to compute (in the first approximation) the pressure in the “plateau” region after establishing the mathematical significance of this concept, previously introduced on the basis of the experimental results. At the same time relatively simple solutions are obtained for semiinfinite separation zones.

V. Ya. Neiland

Papers

Theory of laminar boundary layer separation in supersonic flow

Asymptotic theory for calculating heat fluxes near the corner of a body

Propagation of perturbations upstream with interaction between a hypersonic flow and a boundary layer

Flow behind the boundary layer separation point in a supersonic stream

The asymptotic theory of the interaction of a supersonic flow with a boundary layer