A. I. Ruban

Asymptotic Theory of Separated Flows

Preface 1. The theory of separation from a smooth surface 2. Flow separation from corners of a body contour 3. Flow in the vicinity of the trailing edge of a thin airfoil 4. Separation at the leading edge of a thin airfoil 5. The theory of unsteady separation 6. The asymptotic theory of flow past blunt bodies 7. Numerical methods for solving the equations of interaction theory References.

Once again on the supersonic flow separation near a corner

Laminar boundary-layer separation in the supersonic flow past a corner point on a rigid body contour, also termed the compression ramp, is considered based on the viscous–inviscid interaction concept. The ‘triple-deck model’ is used to describe the interaction process. The governing equations of the interaction may be formally derived from the Navier–Stokes equations if the ramp angle [theta] is represented as [theta] = [theta]0Re[minus sign]1/4, where [theta]0 is an order-one quantity and Re is the Reynolds number, assumed large. To solve the interaction problem two numerical methods have been used. The first method employs a finite-difference approximation of the governing equations with respect to both the streamwise and wall-normal coordinates. The resulting algebraic equations are linearized using a Newton–Raphson strategy and then solved with the Thomas-matrix technique. The second method uses finite differences in the streamwise direction in combination with Chebychev collocation in the normal direction and Newton–Raphson linearization. Our main concern is with the flow behaviour at large values of [theta]0. The calculations show that as the ramp angle [theta]0 increases, additional eddies form near the corner point inside the separation region. The behaviour of the solution does not give any indication that there exists a critical value [theta]*0 of the ramp angle [theta]0, as suggested by Smith & Khorrami (1991) who claimed that as [theta]0 approaches [theta]*0, a singularity develops near the reattachment point, preventing the continuation of the solution beyond [theta]*0. Instead we find that the numerical solution agrees with Neilands (1970) theory of reattachment, which does not involve any restriction upon the ramp angle.

An instability in supersonic boundary-layer flow over a compression ramp

Separation of a supersonic boundary layer (or equivalently a hypersonic boundary layer in a region of weak global interaction) near a compression ramp is considered for moderate wall temperatures. For small ramp angles, the flow in the vicinity of the ramp is described by the classical supersonic triple-deck structure governing a local viscous-inviscid interaction. The boundary layer is known to exhibit recirculating flow near the corner once the ramp angle exceeds a certain critical value. Here it is shown that above a second and larger critical ramp angle, the boundary-layer flow develops an instability. The instability appears to be associated with the occurrence of inflection points in the streamwise velocity profiles within the recirculation region and develops as a wave packet which remains stationary near the corner and grows in amplitude with time.

An effective numerical method for solving viscous–inviscid interaction problems

This paper presents a new numerical method to solve the equations of the asymptotic theory of separated flows. A number of measures was taken to ensure fast convergence of the iteration procedure, which is employed to treat the nonlinear terms in the governing equations. Firstly, we selected carefully the set of variables for which the nonlinear finite difference equations were formulated. Secondly, a Newton–Raphson strategy was applied to these equations. Thirdly, the calculations were facilitated by utilizing linear approximation of the boundary-layer equations when calculating the corresponding Jacobi matrix. The performance of the method is illustrated, using as an example, the problem of laminar two-dimensional boundary-layer separation in the flow of an incompressible fluid near a corner point of a rigid body contour. The solution of this problem is non-unique in a certain parameter range where two solution branches are possible.

On laminar separation at a corner point in transonic flow

The separation of the laminar boundary layer from a convex corner on a rigid body contour in transonic flow is studied based on the asymptotic analysis of the Navier-Stokes equations at large values of the Reynolds number. It is shown that the flow in a small vicinity of the separation point is governed, as usual, by strong interaction between the boundary layer and the inviscid part of the flow. Outside the interaction region the Karman-Guderley equation describing transonic inviscid flow admits a self-similar solution with the pressure on the body surface being proportional to the cubic root of the distance from the separation point. Analysis of the boundary layer driven by this pressure shows that as the interaction region is approached the boundary layer splits into two parts: the near-wall viscous sublayer and the main body of the boundary layer where the flow is locally inviscid. It is interesting that contrary to what happens in subsonic and supersonic flows, the displacement effect of the boundary layer is primarily due to the inviscid part. The contribution of the viscous sublayer proves to be negligible to the leading order. Consequently, the flow in the interaction region is governed by the inviscid-inviscid interaction

Hypersonic boundary-layer separation on a cold wall

An asymptotic theory of laminar hypersonic boundary-layer separation for large Reynolds number is described for situations when the surface temperature is small compared with the stagnation temperature of the inviscid external gas flow. The interactive boundary-layer structure near separation is described by well-known triple-deck concepts but, in contrast to the usual situation, the displacement thickness associated with the viscous sublayer is too small to influence the external pressure distribution (to leading order) for sufficiently small wall temperature. The present interaction takes place between the main part of the boundary layer and the external flow and may be described as inviscid-inviscid

Discontinuous solutions of the boundary-layer equations

(Received 5 December 2007 and in revised form 7 July 2008) Since 1904, when Prandtl formulated the boundary-layer equations, it has been presumed that due to the viscous nature of the boundary layers the solution of the Prandtl equations should be sought in the class of continuous functions. However, there are clear mathematical reasons for discontinuous solutions to exist. Moreover, under certain conditions they represent the only possible solutions of the boundarylayer equations. In this paper we consider, as an example, an unsteady analogue of the laminar jet problem first studied by Schlichting in 1933. In Schlichting’s formulation the jet emerges from a narrow slit in a flat barrier and penetrates into a semi-infinite region filled with fluid which would remain at rest if the slit were closed. Assuming the flow steady, Schlichting was able to demonstrate that the corresponding solution to the Prandtl equations may be written in an explicit analytic form. Here our concern will be with unsteady flow that is initiated when the slit is opened and the jet starts penetrating into the stagnant fluid. To study this process we begin with the numerical solution of the unsteady boundary-layer equations. Since discontinuities were expected, the equations were written in conservative form before finite differencing. The solution shows that the jet has a well-established front representing a discontinuity in the velocity field, similar to the shock waves that form in supersonic gas flows. Then, in order to reveal the ‘internal structure’ of the shock we turn to the analysis of the flow in a small region surrounding the discontinuity. With Re denoting the Reynolds number, the size of the inner region is estimated as an order Re −1/2 quantity in both longitudinal and lateral directions. We found that the fluid motion in this region is predominantly inviscid and may be treated as quasi-steady if considered in the coordinate frame moving with the jet front. These simplifications allow a simple formula for the front speed to be deduced, which proved to be in close agreement with experimental observation of Turner (J. Fluid Mech. vol. 13 (1962), p. 356).

Receptivity of the boundary layer to vibrations of the wing surface

In this paper we study the generation of Tollmien‐Schlichting waves in the boundary layer due to elastic vibrations of the wing surface. The subsonic flow regime is considered with the Mach number outside the boundary layer MD O.1/. The flow is investigated based on the asymptotic analysis of the Navier‐Stokes equations at large values of the Reynolds number, ReD 1V1L= 1. Here L denotes the wing section chord; and V1, 1 and 1 are the free stream velocity, air density and dynamic viscosity, respectively. We assume that in the spectrum of the wing vibrations there is a harmonic that comes in to resonance with the Tollmien‐Schlichting wave on the lower branch of the stability curve; this happens when the frequency of the harmonic is a quantity of the order of .V1=L/Re 1=4 . The wavelength, ‘, of the elastic vibrations of the wing is assumed to be ‘ LRe 1=8 , which has been found to represent a ‘distinguished limit’ in the theory. Still, the results of the analysis are applicable for ‘ LRe 1=8 and ‘ LRe 1=8 ; the former includes an important case when ‘D O.L/. We found that the vibrations of the wing surface produce pressure perturbations in the flow outside the boundary layer, which can be calculated with the help of the ‘piston theory’, which remains valid provided that the Mach number, M, is large as compared to Re 1=4 . As the pressure perturbations penetrate into the boundary layer, a Stokes layer forms on the wing surface; its thickness is estimated as a quantity of the order of Re 5=8 . When ‘D O.Re 1=8 / or ‘ Re 1=8 , the solution in the Stokes layer appears to be influenced significantly by the compressibility of the flow. The Stokes layer on its own is incapable of producing the Tollmien‐Schlichting waves. The reason is that the characteristic wavelength of the perturbation field in the Stokes layer is much larger than that of the Tollmien‐Schlichting wave. However, the situation changes when the Stokes layer encounters a wall roughness, which are plentiful in real aerodynamic flows. If the longitudinal extent of the roughness is a quantity of the order of Re 3=8 , then efficient generation of the Tollmien‐Schlichting waves becomes possible. In this paper we restrict our attention to the case when the Stokes layer interacts with an isolated roughness. The flow near the roughness is described by the triple-deck theory. The solution of the triple-deck problem can be found in an analytic form. Our main concern is with the flow behaviour downstream of the roughness and, in particular, with the amplitude of the generated Tollmien‐Schlichting waves.

Viscous–inviscid interaction in transonic Prandtl–Meyer flow

This paper presents a theoretical analysis of perfect gas flow over a convex corner of a rigid-body contour. It is assumed that the flow is subsonic before the corner. It accelerates around the corner to become supersonic, and then undergoes an additional acceleration in the expansion Prandtl–Meyer fan that forms in the supersonic part of the flow behind the corner. The entire process is described by a self-similar solution of the Karman–Guderley equation. The latter shows that the boundary layer approaching the apex of the corner is exposed to a singular pressure gradient,

Instabilities in supersonic compression ramp flow

{\rm d} p / {\rm d} x \sim (-x)^{-3/5}