On the receptivity problem for Görtler vortices: vortex motions induced by wall roughness

Abstract

The receptivity problem for Görtler vortices induced by wall roughness is investigated. The roughness is modelled by small amplitude perturbations to the curved wall over which the flow takes place. The amplitude of these perturbations is taken to be sufficiently small for the induced Görtler vortices to be described by linear theory. The roughness is assumed to vary in the spanwise direction on the boundary-layer lengthscale, whilst in the flow direction the corresponding variation is on the lengthscale over which the wall curvature varies. In fact the latter condition can be relaxed to allow for a faster streamwise roughness variation so long as the variation does not become as fast as that in the spanwise direction. The function that describes the roughness is assumed to be such that its spanwise and streamwise dependences can be separated; this enables us to make progress by taking Fourier or Laplace transforms where appropriate. The cases of isolated and distributed roughness elements are investigated and the coupling coefficient which relates the amplitude of the forcing and the induced vortex amplitude is found asymptotically in the small wavelength limit. It is shown that this coefficient is exponentially small in the latter limit so that it is unlikely that this mode can be stimulated directly by wall roughness. The situation at O (1) wavelengths is quite different and this is investigated numerically for different forcing functions. It is found that an isolated roughness element induces a vortex field which grows within a wedge at a finite distance downstream of the element. However, immediately downstream of the obstacle the disturbed flow produced by the element decays in amplitude. The receptivity problem at larger Görtler numbers appropriate to relatively large wall curvature is discussed in detail. It is found that the fastest growing linear mode of the Görtler instability equations has wavenumber proportional to the one-fifth power of the Gortler number. The mode can be related to both inviscid disturbances and the disturbances appropriate to the right-hand branch of the neutral curve for Görtler vortices. The coupling coefficient between this, the fastest growing vortex, and the forcing function is found in closed form.

Stability of the flow around a cylinder: The spin-up problem

“…It may be shown that the growth rate is proportional to G~ following Denier, Hall & Seddougui (1991). Thus the disturbance in this regime takes the form (3.2a) where (3.2b) This form of the disturbance may be substituted into (3.1), and the two equations can be combined to produce, is plotted against k in figure 1.…”

“…In this paper the flow quantity chosen to be representative of the vortex size is We now go to investigate inviscid Gortler modes, where we have at' '" 1 and Gt' ~ 1 described in Denier, Hall, & Seddougui (1991). We also consider the right hand branch problem such that for at' ~ 1, we have Gt' rv a~, as investigated in .…”

“…Various forms of this hump are chosen and give rise to different regions of vortex activity, subsequently referred to as vortex wedges. In this paper we do not have the freedom of Denier, Hall & Seddougui (1991) to introduce the disturbance at a certain spatial position as we have said that the impulsive motion starts at time equals zero, and the hump must be present for all time.…”

“…The development of the vortices is caused by the inclusion of a small hump. This hump is considered to be the same size as the boundary layer, this technique was employed in Denier, Hall, & Seddougui (1991) in the spatially varying Gortler problem. Various forms of this hump are chosen and give rise to different regions of vortex activity, subsequently referred to as vortex wedges.…”

“…We also consider matching between the inviscid solutions and right hand branch solutions as in Denier, Hall & Seddougui (1991).…”