James P. Denier

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.

On the boundary-layer equations for power-law fluids

We reconsider the problem of the boundary–layer flow of a non-Newtonian fluid whose constitutive law is given by the classical Ostwald–de Waele power–law model. The boundary–layer equations are solved in similarity form. The resulting similarity solutions for shear–thickening fluids are shown to have a finite–width crisis resulting in the prediction of a finite–width boundary layer. A secondary viscous adjustment layer is required in order to smooth out the solution and to ensure correct matching with the far–field boundary conditions. In the case of shear–thinning fluids, the similarity forms have solutions whose decay into the far field is strongly algebraic. Smooth matching between these inner algebraically decaying solutions and an outer uniform flow is achieved via the introduction of a viscous diffusion layer.

Asymptotic matching constraints for a boundary-layer flow of a power-law fluid

We reconsider the three-dimensional boundary-layer flow of a power-law (Ostwald–de Waele) rheology fluid, driven by the rotation of an infinite rotating plane in an otherwise stationary system. Here we address the problem for both shear-thinning and shear-thickening fluids and show that there are some fundamental issues regarding the application of power-law models in a boundary-layer context that have not been mentioned in previous discussions. For shear-thickening fluids, the leading-order boundary-layer equations are shown to have no suitable decaying behaviour in the far field, and the only solutions that exist are necessarily non-differentiable at a critical location and of ‘finite thickness’. Higher-order effects are shown to regularize the singularity at the critical location. In the shear-thinning case, the boundary-layer solutions are shown to possess algebraic decay to a free-stream flow. This case is known from the existing literature; however here we shall emphasize the complexity of applying such solutions to a global flow, describing why they are in general inappropriate in a traditional boundary-layer context. Furthermore, previously noted difficulties for fluids that are highly shear thinning are also shown to be associated with the imposition of incorrect assumptions regarding the nature of the far-field flow. Based on Newtonian results, we anticipate the presence of non-uniqueness and through accurate numerical solution of the leading-order boundary-layer equations we locate several such solutions.

Unsteady fronts in the spin-down of a fluid-filled torus

We report the results of an experimental investigation into fluid motion induced by the deceleration to rest of a rigidly rotating fluid-filled torus. Transition to a transient turbulent state is found where the onset of the complicated motion is triggered by a small-scale wavelike instability. The wave forms on a front that propagates from the inner wall of the toroidal container after it is stopped. We reveal the origins of the front through a combination of careful experimental measurements, boundary-layer analysis, and computation of the axisymmetric Navier–Stokes equations.

Weakly nonlinear wave motions in a thermally stratified boundary layer

We consider weakly nonlinear wave motions in a thermally stratified boundary layer. Attention is focused on the upper branch of the neutral stability curve, corresponding to small wavelengths and large Reynolds number. In this limit the motion is governed by a first harmonic/mean flow interaction theory in which the wave-induced mean flow is of the same order of magnitude as the wave component of the flow. We show that the flow is governed by a system of three coupled partial differential equations which admit finite-amplitude periodic solutions bifurcating from the linear, neutral points.

On the growth (and suppression) of very short-scale disturbances in mixed forced–free convection boundary layers

The two-dimensional boundary-layer flow over a cooled/heated flat plate is investigated. A cooled plate (with a free-stream flow and wall temperature distribution which admit similarity solutions) is shown to support non-modal disturbances, which grow algebraically with distance downstream from the leading edge of the plate. In a number of flow regimes, these modes have diminishingly small wavelength, which may be studied in detail using asymptotic analysis. Corresponding non-self-similar solutions are also investigated. It is found that there are important regimes in which if the temperature of the plate varies (in such a way as to break self-similarity), then standard numerical schemes exhibit a breakdown at a finite distance downstream. This breakdown is analysed, and shown to be related to very short-scale disturbance modes, which manifest themselves in the spontaneous formation of an essential singularity at a finite downstream location. We show how these difficulties can be overcome by treating the problem in a quasi-elliptic manner, in particular by prescribing suitable downstream (in addition to upstream) boundary conditions.

The stability of boundary-layer flows under conditions of intense interfacial mass transfer: the effect of interfacial coupling

We consider the linear stability of a boundary-layer flow over a permeable surface under conditions of intense interfacial mass transfer. The stability of the flow is governed by an eigenvalue problem of Orr–Sommerfeld type coupled to a second-order differential equation for the concentration disturbance field through a flux boundary condition at the permeable surface. Previous studies on this problem have ignored the effect on the stability of the flow of this coupling. Curves of neutral stability and the critical Reynolds number for the flow are obtained. These show that the fully coupled system produce critical Reynolds numbers and wave-numbers that, in some cases, differ significantly from those obtained when the disturbance coupling is ignored.

On the nonlinear development of the most unstable Görtler vortex mode

Abstract : The nonlinear development of the most unstable Goertler vortex mode in boundary layer flows over curved walls is investigated. The most unstable Goertlermode is confined to a viscous wall layer of thickness O(G to the 1/5th power) and has spanwise wavelength O(G to the 1/5th power); it is, of course, most relevant to flow situations where the Gortler number G >> 1. The nonlinear equations governing the evolution of this mode over an O(G to the 3/5th power) streamwise lengthscale are derived and are found to be a fully nonparallel nature. The solution of these equations is achieved by making use of the numerical scheme used by Hall (1988) for the numerical solution of the nonlinear Goertler equations valid for O(1) Goertler numbers. Thus, the spanwise dependence of the flow is described by a Fourier expansion whereas the streamwise and normal variations of the flow are dealt with by employing a suitable finite difference discretization of the governing equations. Our calculations demonstrate that, given a suitable initial disturbance, after a brief interval of decay, the energy in all the higher harmonics grows until a singularity is encountered at some downstream position. The structure of the flow field as this singularity is approached suggests that the singularity is responsible for the vortices, which are initially confined to the thin viscous wall layer, moving away from the wall and into the core of the boundary layer.

The effect of wall compliance on the Goertler vortex instability

The stability of the flow of a viscous incompressible fluid over a curved compliant wall to longitudinal Goertler vortices is investigated. The compliant wall is modeled by a particularly simple equation relating the induced wall displacement to the pressure in the overlying fluid. Attention is restricted to the large Goertler number regime; this regime being appropriate to the most unstable Goertler mode. The effect of wall compliance on this most unstable mode is investigated.

The dominant wave mode within a trailing line vortex

We identify the dominant, or most unstable, wave mode for the flow in a trailing line vortex. This dominant mode is found to reside in a wavenumber regime between that of inviscid wave modes and the viscous upper branch neutral wave modes. A reevaluation of the growth rate in the vicinity of the upper branch of the curve of neutral stability allows us to predict the neutral value of the azimuthal and axial wavenumber as a function of the imposed swirl within the trailing line vortex.