Philip Hall

Taylor—Gortler vortices in fully developed or boundary-layer flows: linear theory

The stability characteristics of some fluid flows at high Taylor or Gortler numbers are determined using perturbation methods. In particular, the stability characteristics of some fully developed flows between concentric cylinders driven either by a pressure gradient or the motion of the inner cylinder are investigated. The asymptotic structure of short-wavelength disturbances to these flows is obtained and used as a basis for a formal perturbation solution to the corresponding stability problem appropriate to a developing boundary layer. The non-parallel effect of the basic flow on the condition for neutral stability is discussed. The results obtained suggest that the disturbances are concentrated in internal viscous or critical layers well away from the wall and the free stream. The stability of a boundary layer on a concave wall to Gortler vortices that propagate downstream is also considered. These modes are found to be more stable than the usual time-independent modes and they propagate downstream with the speed of the basic flow in the critical layer. Some comparison with previous experimental and theoretical work is given.

On the stability of an infinite swept attachment line boundary layer

A number of numerical schemes were employed in order to gain insight in the stability problem of the infinite swept attachment line boundary layer. The basic flow was taken to be the classical Hiemenz flow. A number of assumptions for the perturbation flow quantities were considered. In all cases a pseudo- spectral approach was used; the chordwise and spanwise directions were treated spectrally, while an implicit Crank-Nicolson scheme was used temporally. Extensive use of the FFT algorithm has been made.

The linear inviscid secondary instability of longitudinal vortex structures in boundary layers

The inviscid instability of a longitudinal vortex structure within a steady boundary layer is investigated. The instability has wavelength comparable with the boundary-layer thickness so that a quasi-parallel approach to the instability problem can be justified. The generalisation of the Rayleigh equation to such a flow is obtained and solved for the case when the vortex structure is induced by curvature. Two distinct modes of instability are found; these modes correspond with experimental observations on the breakdown process for Gortler vortices.

On strongly nonlinear vortex/wave interactions in boundary-layer transition

The interactions between longitudinal vortices and accompanying waves considered here are strongly nonlinear, in the sense that the mean-flow profile throughout the boundary layer is completely altered from its original undisturbed state. Nonlinear interactions between vortex flow and Tollmien-Schlichting waves are addressed first, and some analytical and computational properties are described. These include the possibility in the spatial-development case of a finite-distance break-up, including a singularity in the displacement thickness. Second, vortex/Rayleigh-wave nonlinear interactions are considered for the compressible boundary layer, along with certain special cases of interest and some possible solution properties. Both types, vortex/Tollmien-Schlichting and vortex/Rayleigh, are short-scale/long-scale interactions and they have potential applications to many flows at high Reynolds numbers. Their strongly nonlinear nature is believed to make then very relevant to fully-fledged transition to turbulence.

Görtler vortices in growing boundary layers: the leading edge receptivity problem, linear growth and the nonlinear breakdown stage

Goertler vortices are thought to be the cause of transition in many fluid flows of practical importance. A review of the different stages of vortex growth is given. In the linear regime, nonparallel effects completely govern this growth, and parallel flow theories do not capture the essential features of the development of the vortices. A detailed comparison between the parallel and nonparallel theories is given and it is shown that at small vortex wavelengths, the parallel flow theories have some validity; otherwise nonparallel effects are dominant. New results for the receptivity problem for Goertler vortices are given; in particular vortices induced by free stream perturbations impinging on the leading edge of the walls are considered. It is found that the most dangerous mode of this type can be isolated and its neutral curve is determined. This curve agrees very closely with the available experimental data. A discussion of the different regimes of growth of nonlinear vortices is also given. Again it is shown that, unless the vortex wavelength is small, nonparallel effects are dominant. Some new results for nonlinear vortices of 0(1) wavelengths are given and compared to experimental observations.

An asymptotic investigation of the stationary modes of instability of the boundary layer on a rotating disc

The paper investigates high-Reynolds-number stationary instabilities in the boundary layer on a rotating disc. The investigation demonstrates that, in addition to the inviscid mode found by Gregory, Stuart & Walker (Phil. Trans. R. Soc. Lond. A 248, 155 (1955)) at high Reynolds numbers, there is a stationary short-wavelength mode. This mode has its structure fixed by a balance between viscous and Coriolis forces and cannot be described by an inviscid theory. The asymptotic structure of the wave-number and orientation of this mode is obtained, and a similar analysis is given for the inviscid mode. The expansion procedure provides the capacity of taking non-parallel effects into account in a self-consistent manner. The inviscid solution of Gregory et al. is modified to take account of viscous effects. The expansion procedure used is again capable of taking non-parallel effects into account. The results obtained suggest why the inviscid approach of Gregory et al. should give a good approximation to the experimentally measured orientation of the vortices. The results also explain partly why the inviscid analysis should not give such a good approximation to the wavenumber of the vortices. The asymptotic analysis of both modes provides a starting point for the corresponding nonlinear problems.

On the receptivity problem for Goertler vortices: Vortex motions induced by wall roughness

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.

Unsteady viscous flow in a pipe of slowly varying cross-section

The steady streaming generated in a pipe of slowly varying cross-section when a purely oscillatory pressure difference is maintained between its ends is considered. It is assumed that the perturbation of the pipe wall in the r , θ plane is small compared with the characteristic thickness of the Stokes layer associated with the oscillatory motion of the fluid. The first-order steady streaming is evaluated for the cases when this characteristic thickness is large and small compared with a typical radius of the pipe. In both these limits it is found that the geometry of the pipe is crucial in determining the nature of the induced steady streaming. If the ends of the pipe have the same mean radius it is found that the steady streaming consists of regions of recirculation between the nodes of the pipe. Otherwise the steady streaming is of a larger order of magnitude and has a component which represents a net flow towards the wider end of the pipe.

The Linear Stability of Flat Stokes Layers

The linear stability of a flat Stokes layer is investigated. The results obtained show that, in the parameter range investigated, the flow is stable. It is shown that the Orr-Sommerfield equation for this flow has a continuous spectrum of damped eigenvalues at all values of the Reynolds number. In addition, a set of discrete eigenvalues exists for certain values of the Reynolds number. The eigenfunctions associated with this set are confined to the Stokes layer while those corresponding to the continuous spectrum persist outside the layer. The effect of introducing a second boundary a long way from the Stokes layer is also considered. It is shown that the least stable disturbance of this flow does not correspond to the least stable discrete eigenvalue of the infinite Stokes layer when this boundary tends to infinity.

On the instability of a three-dimensional attachment-line boundary layer: weakly nonlinear theory and a numerical approach

The instability of a three dimensional attachment line boundary layer is considered in the nonlinear regime. Using weakly nonlinear theory, it is found that, apart from a small interval near the (linear) critical Reynolds number, finite amplitude solutions bifurcate subcritically from the upper branch of the neutral curve. The time dependent Navier-Stokes equations for the attachment line flow have been solved using a Fourier-Chebyshev spectral method and the subcritical instability is found at wavenumbers that correspond to the upper branch. Both the theory and the numerical calculations show the existence of supercritical finite amplitude (equilibrium) states near the lower branch which explains why the observed flow exhibits a preference for the lower branch modes. The effect of blowing and suction on nonlinear stability of the attachment line boundary layer is also investigated.