P. J. Blennerhassett

Supercritical analysis of strongly nonlinear vortices in magnetized ferrofluids

The structure of two–dimensional vortices in a thin layer of magnetized ferrofluid heated from above is examined in the limit as the critical wavenumber, a, of the roll cells becomes large. In particular, we present a nonlinear asymptotic description of the vortex pattern that occurs directly above the critical point in parameter space where instability first sets in. Two cases are examined. First, an idealized case where the fluid layer has boundary conditions appropriate for a free surface and second, a more physical situation where the fluid is confined between rigid, horizontal magnetic pole-pieces. The idealized problem has a relatively simple solution structure, which is a leading–order approximation to the solution of the physical problem. As the critical wavenumber increases, boundary layers of thickness O(a-4/3) develop at the walls in the physical problem and the critical value of the instability parameter has an asymptotic expansion in inverse powers of a2. Weakly nonlinear solutions are extended into the unstable region of parameter space where the convection is described by fully nonlinear equations and the perturbations to the basic state have finite amplitude. This analysis is new since similar large–wavenumber investigations in other problems do not apply at critical conditions. As the instability parameter is increased above its critical value, a hierarchy of bifurcations in the equations governing the temperature perturbation occurs, necessitating the inclusion of progressively more harmonics in the solution. Numerical solutions for the first few of these temperature equations are presented and asymptotic expressions for the heat transfer across the fluid layer for both sets of boundary conditions are derived. Although the conduction state still dominates the heat transfer, the solution structure is fundamentally different from previous weakly nonlinear theories. Finally, some numerical solutions of the full steady–state governing equations are presented and compared with the asymptotic structures. These results verify that the solutions obtained with both sets of boundary conditions are asymptotic to one another in the large–wavenumber limit and that the asymptotic solution is a useful approximation in this limit.

The linear stability of oscillating pipe flow

An investigation is made of the three-dimensional linear stability of the Stokes layer generated within a fluid contained inside a long oscillating cylinder. Both longitudinal and torsional vibrations are examined and the system of disturbance equations derived using Floquet theory are solved using pseudospectral methods. Critical parameters for instability are obtained for an extensive range of pipe radii and longitudinal and azimuthal wavenumbers. For sufficiently small pipe diameters, three-dimensional perturbations are sometimes found to be more unstable than their two-dimensional counterparts. In contrast, at larger radii, the three-dimensional disturbance modes are less important and the two-dimensional versions are expected to be observed in practice. These results imply constraints on experiments that are designed to exhibit shear modes in oscillatory flow.

The Structure of Highly Nonlinear Vortices in Curved Channel Flow

The structure of highly nonlinear, small wavelength Gӧrtler vortices is analysed for fully developed flow in a curved channel. The Taylor number is taken to be T0ϵ-5, which is asymptotically larger than the neutral stability Taylor number for vortices of small wavelength 2πϵ. Over most of the flow domain the motion is determined by the interaction of the mean flow and terms proportional to exp {iz/ϵ}, whereas adjacent to the outer channel wall there is a boundary layer of thickness O(ϵ) which requires all the harmonics of the disturbance to be considered. Numerical solutions of the wall layer equations are presented for T0 up to 14000 with the main feature of these results being the development of a weak vortex on the boundary layer lengthscale.

Strongly nonlinear vortices in the Stokes layer on an oscillating cylinder

The Stokes layer on a torsionally oscillating circular cylinder is susceptible to axially periodic vortices which are driven by a centrifugal mechanism which has some similarities to that which is responsible for Görtler vortices in boundary layers on concave surfaces. For small wavelength vortices an analytic description of linear, weakly nonlinear and fully nonlinear perturbations is possible and, as the growth rates of these modes are asymptotically greater than the timescale of the underlying Stokes layer, a quasi-steady stability analysis is attempted in preference to a Floquet-type approach. It is shown that highly nonlinear vortices can be expected to form on the surface of the cylinder; it is noted that these vortices alter the basic mean flow at leading order and occupy a zone which has depth comparable to the Stokes thickness. Depending on the Taylor number of the flow, three distinct types of vortex behaviour are found. In the first of these, corresponding to Taylor numbers just above that at which the Stokes layer becomes unstable over a part of its cycle, the vortices are relatively weak and remain attached to the cylinder throughout their lifetime. At larger Taylor numbers, the vortices initially grow whilst remaining on the cylinder but at some stage they break away and concentrate in a region within the Stokes layer. However, their strength reduces with time and they soon decay away. In the third regime identified, before this vortex activity region dies, the residual flow next to the cylinder becomes unstable once more. In this case a second vortex zone begins to grow so that for a certain time interval two active zones exist. The governing equations for the strongly nonlinear modes take the form of a free boundary value problem subject to ordinary differential constraints. Numerical solutions are presented and, although periodic solutions are by no means guaranteed by use of the quasi-steady approach, the computations show that remarkably periodic vortex configurations tend to result.

Strongly nonlinear vortices in magnetized ferrofluids

Nonlinear convective roll cells that develop in thin layers of magnetized ferrofluids heated from above are examined in the limit as the wavenumber of the cells becomes large. Weakly nonlinear solutions of the governing equations are extended to solutions that are valid at larger distances above the curves of marginal stability. In this region, a vortex flow develops where the fundamental vortex terms and the correction to the mean are determined simultaneously rather than sequentially. The solution is further extended into the nonlinear region of parameter space where the flow has a core-boundary layer structure characterized by a simple solution in the core and a boundary layer containing all the harmonics of the vortex motion. Numerical solutions of the boundary layer equations are presented and it is shown that the heat transfer across the layer is significantly greater than in the conduction state.

A three-dimensional analogue of the Prandtl-Batchelor closed streamline theory

For steady laminar flow with closed streamlines Batchelor (1956) has shown how an integral condition arising from the effect of viscosity can be used with the inviscid flow equations to determine the vorticity distribution when the Reynolds number is large. Here a condition analogous to that used by Batchelor is derived for a class of flows with helical streamlines. An exact integral condition relating the constant axial pressure gradient and the viscous terms is obtained, which when combined with the inviscid flow equations leads to the result that the axial velocity is proportional to the stream function for the motion in the plane normal to the axial velocity.

LARGE WAVENUMBER VORTICES IN TIME-PERIODIC FLOWS

The linear stability properties of short wavelength vortices within time–periodic flows are examined using a WKB–based asymptotic analysis. Such vortices typically have an instantaneous growth rate considerably larger than the time–scale of the underlying flow and thus have often been studied using quasi–steady techniques. A major drawback of these quasi–steady approaches is that although they are able to determine the instantaneous flow structure they are of no use in describing the evolution of the modes. It is shown how rational asymptotic analysis based on WKB ideas overcomes this deficiency and enables the form of the neutral stability curve to be obtained for short wavelengths. The technique is applied to three problems. First, the key features of the analysis are exemplified by careful consideration of a model problem which is motivated by the physical situation of the sinusoidal heating of a horizontal flat plate bounding a semi–infinite layer of fluid. The governing linearized stability equations take the form of a system which is sixth order in the spatial variable and second order in time; however, the important features of this system are captured by our model which is only second order in the spatial coordinate and first order in time. The second physically important flow examined is the motion induced in a viscous fluid surrounding a long torsionally oscillating cylinder (a curved Stokes flow). Detailed asymptotic and numerical studies of both the model and the two physically motivated flows are undertaken and it is shown how the analytical results provide good qualitative agreement with the numerical findings at quite modest wavenumbers. The methodology adopted here may be used as a basis for the rational study of the stability properties of other time–periodic flows. A key result forthcoming from this study is the demonstration that the first few terms in the relevant high wavenumber form of the neutral curve for modes in an oscillatory flow may be derived relatively quickly with minimal computational effort.

The linear impulse response for disturbances in an oscillatory Stokes layer

Numerical simulation results are reported for the linear evolution of disturbances in a flat oscillatory Stokes layer. A spatially localised form of impulsive forcing is applied. The subsequent disturbance development displays an intriguing family tree-like structure, which involves the birth of successive generations of wavepackets. The complexity of this response is likely to lead to some difficulties in the interpretation of data obtained from physical experiments. It is also noted that important features of the wavepacket behaviour can be accounted for using linear stability results based upon Floquet theory.

Long wavelength vortices in time-periodic flows

The linear stability properties are examined of long wavelength vortex modes in two timeperiodic flows. These flows are the motion which is induced by a torsionally oscillating cylinder within a viscous fluid and, second, the flow which results from the sinusoidal heating of an infinite layer of fluid. Previous studies concerning these particular configurations have shown that they are susceptible to vortex motions and linear neutral curves have been computed for wavenumbers near their critical value. These computations become increasingly difficult for long wavelength motions and here we consider such modes using asymptotic methods. These yield simple results which are formally valid for small wavenumbers and we show that the agreement between these asymptotes and numerical solutions is good for surprisingly large wavenumbers. The two problems studied share a number of common features but also have important differences and, between them, our methods and results provide a basis which can be extended for use with other time-periodic flows.