On transonic viscous-inviscid interaction

The paper is concerned with the interaction between the boundary layer on a smooth body surface and the outer inviscid compressible flow in the vicinity of a sonic point. First, a family of local self-similar solutions of the Karman-Guderley equation describing the inviscid flow behaviour immediately outside the interaction region is analysed; one of them was found to be suitable for describing the boundary-layer separation. In this solution the pressure has a singularity at the sonic point with the pressure gradient on the body surface being inversely proportional to the cubic root dp_w/dx~(−x)^{-1/3} of the distance (−x) from the sonic point. This pressure gradient causes the boundary layer to interact with the inviscid part of the flow. It is interesting that the skin friction in the boundary layer upstream of the interaction region shows a characteristic logarithmic decay which determines an unusual behaviour of the flow inside the interaction region. This region has a conventional triple-deck structure. To study the interactive flow one has to solve simultaneously the Prandtl boundary-layer equations in the lower deck which occupies a thin viscous sublayer near the body surface and the Karman-Guderley equations for the upper deck situated in the inviscid flow outside the boundary layer. In this paper a numerical solution of the interaction problem is constructed for the case when the separation region is entirely contained within the viscous sublayer and the inviscid part of the flow remains marginally supersonic. The solution proves to be non-unique, revealing a hysteresis character of the flow in the interaction region

The generation of Tollmien–Schlichting waves by free-stream turbulence in transonic flow

This paper studies the generation of Tollmien–Schlichting waves by free-stream turbulence in transonic flow over a half-infinite flat plate with a roughness element using an asymptotic approach. It is assumed that the Reynolds number (denoted Re) is large, and that the free-stream turbulence is uniform so it can be modelled as vorticity waves. Close to the plate, a Blasius boundary layer forms at a thickness of O(Re−1/2), and a vorticity deformation layer is also present with thickness O(Re−1/4). The report shows that there is no mechanism by which the vorticity waves can penetrate from the vorticity deformation layer into the classical boundary layer; therefore, a transitional layer is introduced between them in order to prevent a discontinuity in vorticity. The flow in the interaction region in the vicinity of the roughness element is then analysed using the triple-deck model for transonic flow. A novel asymptotic expansion is used to analyse the upper deck, which enables a viscous–inviscid interaction problem to be derived. In order to make analytical progress, the height of the roughness element is assumed to be small, and from this, we find an explicit formula for the receptivity coefficient of the Tollmien–Schlichting wave far downstream of the roughness

Direct numerical simulation of a compressible boundary-layer flow past an isolated three-dimensional hump in a high-speed subsonic regime

This is the final version of the article. Available from American Physical Society via the DOI in this record.In this paper we study the boundary-layer separation produced in a high-speed subsonic boundary layer by a small wall roughness. Specifically, we present a direct numerical simulation (DNS) of a two-dimensional boundary-layer flow over a flat plate encountering a three-dimensional Gaussian-shaped hump. This work was motivated by the lack of DNS data of boundary-layer flows past roughness elements in a similar regime which is typical of civil aviation. The Mach and Reynolds numbers are chosen to be relevant for aeronautical applications when considering small imperfections at the leading edge of wings. We analyze different heights of the hump: The smaller heights result in a weakly nonlinear regime, while the larger result in a fully nonlinear regime with an increasing laminar separation bubble arising downstream of the roughness element and the formation of a pair of streamwise counterrotating vortices which appear to support themselves.This work was supported by the Laminar Flow Control Centre funded by Airbus/EADS and EPSRC under Grant No. EP/I037946 and computational resources were also provided through the UK Turbulence resource under EPSRC Grant No. EP/L000261/1. The authors would like to acknowledge the use of Imperial College High Performance Computing facility. S.J.S. would also like to acknowledge support under the RAEng Fellowship through Grant No. 10145/86

Discontinuous solutions of the unsteady boundary-layer equations for a rotating disk of finite radius

We consider the motion at large Reynolds number of an incompressible fluid around a thin circular disk of finite radius rotating in its plane. The disk is placed in a large tank filled with an initially stagnant fluid. Then it is brought into rotation about its centre with constant angular velocity. Due to viscosity, a layer of fluid adjacent to the disk gets involved in the circumferential motion. This activates centrifugal forces; the fluid particles start to deviate in the radial direction. When they cross the edge of the disk, a thin jet is formed. Firstly, we solve the classical boundary-layer equations for the flow in the boundary layer in the direct neighbourhood of the disk surface and in the jet. The solution is found to develop a discontinuity at the ‘head’ of the jet where the radial and circumferential velocity components experience a jump. This type of discontinuity, called a pseudo-shock, was previously observed by Ruban & Vonatsos (J. Fluid Mech., vol. 614, 2008, pp. 407–424). Then, we investigate the internal structure of the pseudo-shock. We find that the fluid motion is described by the Euler equations in the leading-order approximation. Their solution shows that, as the jet penetrates the stagnant fluid, it ejects the fluid from the boundary layer into the surrounding area. Analysis of the inviscid region outside the boundary layer reveals that the ejected fluid returns back to the boundary layer through the ‘entrainment process’. Finally, we conclude this paper with the study of the wake in the vicinity of the disk rim

Direct numerical simulation of a compressible boundary-layer flow past an isolated three-dimensional hump in a high-speed subsonic regime

In this paper we study the boundary-layer separation produced in a high-speed subsonic boundary layer by a small wall roughness. Specifically, we present a direct numerical simulation (DNS) of a two-dimensional boundary-layer flow over a flat plate encountering a three-dimensional Gaussian-shaped hump. This work was motivated by the lack of DNS data of boundary-layer flows past roughness elements in a similar regime which is typical of civil aviation. The Mach and Reynolds numbers are chosen to be relevant for aeronautical applications when considering small imperfections at the leading edge of wings. We analyze different heights of the hump: The smaller heights result in a weakly nonlinear regime, while the larger result in a fully nonlinear regime with an increasing laminar separation bubble arising downstream of the roughness element and the formation of a pair of streamwise counterrotating vortices which appear to support themselves

On receptivity of marginally separated flows

In this paper we study the receptivity of the boundary layer to suction/blowing in marginally separated flows, like the one on the leading edge of a thin aerofoil. We assume that the unperturbed laminar flow is two-dimensional, and investigate the response of the boundary layer to two-dimensional as well as to three-dimensional perturbations. In both cases, the perturbations are assumed to be weak and periodic in time. Unlike conventional boundary layers, the marginally separated boundary layers cannot be treated using the quasi-parallel approximation. This precludes the normal-mode representation of the perturbations. Instead, we had to solve the linearised integro-differential equation of the marginal separation theory, which was done numerically. For two-dimensional perturbations, the results of the calculations show that the perturbations first grow in the inside of the separation region, but then start to decay downstream. For three-dimensional perturbations, instead of dealing with the integro-differential equation of marginal separation, we found it convenient to work with the Fourier transforms of the fluid-dynamic functions. The equations for the Fourier transforms are also solved numerically. Our calculations show that a three-dimensional wave packet forms downstream of the source of perturbations in the boundary layer