ALBERT

Description: We solve the problem of magnetic field generation by a laminar flow of conducting fluid with helical (screw-like) streamlines for large magnetic Reynolds numbers, Rm. Asymptotic solutions are obtained with help of the singular perturbation theory. The generated field concentrates within cylindrical layers whose position, the magnetic field configuration and the growth rate are determined by the distribution of the angular, Ω and longitudinal, Vz, velocities along the radius. The growth rate is proportional to Rm2. When Ω and Vz are identically distributed along the radius, the asymptotic forms are of the WKB type for different distributions, singular—layer asymptotics of the Prandtl type arise. The solutions are qualitatively different from those obtained for solid—body screw motion. The generation threshold strongly depends on the velocity profiles. © 1988, Cambridge University Press. All rights reserved.

Description: The stability of Couette flow in Hell is considered by an analysis of the HVBK equations. These equations are based on the Landau two-fluid model of Hell and include mutual friction between the normal and superfluid components, and the vortex tension due to the presence of superfluid vortices. We find that the vortex tension strongly affects the nature of the Taylor instability at temperatures below ≈ 2.05 K. The effect of the vortex tension is to make non-axisymmetric modes the most unstable, and to make the critical axial wavelength very long. We compare our results with experiments. © 1988, Cambridge University Press. All rights reserved.

Description: A numerical study has been conducted to determine the various modes of Taylor—Couette flow that exist between concentric vertical cylinders, as the aspect ratio F (height to gap width, H/d) and the Reynolds number Re (based on the inner cylinder speed) are varied. Furthermore, the effects of the introduction of buoyancy on the development of the flow are examined. This is accomplished by considering both cylinders to be isothermal, with the rotating inner cylinder at a higher temperature than the stationary outer cylinder. Results are presented for a wide range of the Grashof number Gr (based on the temperature difference ΔT across the annular gap). The structure of the Taylor vortices is observed to be distorted considerably with the buoyant flows, and the nature of the onset and subsequent development of the vortices is altered. The hysteresis between the different modes of cellular flow, characteristic of the bifurcation phenomena, is also substantially modified. © 1988, Cambridge University Press. All rights reserved.

Description: Substructures within a turbulent spot which develops in a slightly heated laminar boundary layer have been identified using arrays of cold wires aligned in either a streamwise direction or in a direction normal to the wall. At any given streamwise distance from the spot origin, histograms of the number of detected substructures exhibit a peak, defining the most probable spot or the spot with the most likely number of substructures. The number of substructures in the most probable spot increases with streamwise distance but all substructures are convected at approximately the same velocity for any given distance from the wall. This velocity is approximately equal to that of the leading edge of the spot and increases slightly with distance from the wall. The increase in the number of substructures accounts for the streamwise growth of the spot. A simple relation is derived for determining the number of substructures at a particular streamwise station and a geometrical construction is proposed for identifying the origin of a new substructure. There is sufficient evidence for suggesting that the new substructures are formed near the trailing edge of the spot. The convection velocity, inclination and lengthscales of the substructures compare favourably with the corresponding characteristics of hairpin vortices. © 1988, Cambridge University Press. All rights reserved.

Description: The instabilities of barotropic and baroclinic, quasi-geostrophic, f-plane, circular vortices are found using a linearized contour dynamics model. We model the vortex using a circular region of horizontally uniform potential vorticity surrounded by an annulus of uniform, but different, potential vorticity. We concentrate mostly upon isolated vortices with no circulation in the basic state outside the outer radius b. In addition to linear analyses, we also consider weakly nonlinear waves. The amplitude equation has a cubic nonlinearity and, depending upon the sign of the coefficient of the cubic term, may give nonlinear stabilization or nonlinear enhancement of the growth. Barotropic isolated eddies are unstable when the outer annulus is narrow enough; on the other hand, if the scale of the whole vortex is sufficiently small compared to the radius of deformation of a baroclinic mode, the break up may be preferentially to a depth-varying disturbance corresponding to a twisting and tilting of the vortex. As the vortex becomes more baroclinic, we find that large-scale vortices show an elliptical mode baroclinic instability as well which is relatively insensitive to the scale of the outer annulus. When the baroclinic currents in the basic state dominate, the twisting mode disappears, and we see only the instabilities associated with either strong enough shear in the annular region or sufficiently large vortices compared with the deformation radius. The finite amplitude results show that the baroclinic instability mode for large enough vortices is nonlinearly stabilized while in most cases. the other two kinds of instability are nonlinearly destabilized. © 1988, Cambridge University Press. All rights reserved.

Description: The finite-amplitude evolution of circular two-layer quasi-geostrophic vortices with piecewise uniform potential vorticity in each layer (also termed ‘heton’ clouds by Hogg & Stommel 1985a and Pedlosky 1985) is studied using the contour dynamics method. The numerical investigations are preceded by a linear stability analysis which shows the stabilizing influence of deepening the lower layer. Net barotropic flow may be either stabilizing or destabilizing. The contour dynamics calculations for baroclinic vortices show that supercritical (i.e. linearly unstable) conditions may lead to explosive break up of the vortex via the generation of continuous hetons at the cloud boundary. The number of vortex pairs is equal to the azimuthal mode number of the initial disturbance. An additional weakly supercritical regime in which amplitude vacillation occurs, but not explosive growth, is identified. Vortices with net barotropic circulation behave similarly except that the layer with vorticity opposite to the barotropic circulation will break up first. Strong barotropic circulation can inhibit the development of hetons. The stronger layer may eject thin filaments, but remain mostly intact. Calculations for initial conditions composed of several unstable modes show that the linearly most unstable mode dominates at finite amplitude. © 1988, Cambridge University Press. All rights reserved.

Description: The laminar pulsatile flow over a semi-infinite flat plate, on which is located a small (steady) surface distortion is investigated; triple-deck theory provides the basis for the study. The problem is of direct relevance to the externally imposed acoustic excitation of boundary layers. The investigation is primarily numerical and involves the solution of the nonlinear, unsteady boundary-layer equations which arise from the lower deck. The numerical method involves the use of finite differencing in the transverse direction, Crank-Nicolson marching in time, and Fourier transforms in the streamwise direction, and as such is an extension of the spectral method of Burggraf & Duck (1982). Supersonic and incompressible flows are studied. A number of the computations presented suggest that the small surface distortion can excite a large-wavenumber, rapidly growing instability, leading to a breakdown of the solution, with the wall shear at a point seeming to increase without bound as a finite time is approached. Rayleigh modes for the basic (undisturbed) velocity profile are computed and there is some correlation between the existence and magnitude of the growth rate of these unstable modes, and the occurrence of the apparent singularity. Streamline plots indicate that this phenomenon is linked to the formation of closed (or ‘cats-eye’) eddies in the main body of the boundary layer, away from the wall. Tollmien-Schlichting instabilities are clearly seen in the case of incompressible flows. © 1988, Cambridge University Press. All rights reserved.

Description: A general expression is derived for the fluid force on a body of simple shape moving with a velocity v through inviscid fluid in which there is an unsteady non-uniform rotational velocity field u0(x, t) in two or three dimensions. It is assumed that the radius is small compared with the scale over which the strain rate changes, though for the sphere it is also assumed that the changes in the ambient velocity field over the scale of the sphere are small compared with the velocity of the body relative to the flow. Given these approximations it is shown that the effects of the rate of change of the vorticity of the ambient flow is of second order and can be neglected. However the rate of change of the irrotational straining motion is included in the analysis. It is shown that the inertial forces derived by many authors for irrotational flow can be simply added to a generalization of the lift force derived by Auton (1987) in a companion paper. It is shown how this lift force is made up of a rotational and an inertial or added-mass component. For three-dimensional bluff bodies the latter is generally larger (by a factor of three for a sphere), and can be simply calculated from the added-mass coefficient. For illustration, the general expression is used to derive formulae for (i) the motion of a spherical bubble in a steady non-uniform flow to contrast with the motion in an unsteady flow, and (ii) the motion of rigid volumes of neutral density across an inviscid shear flow. These results show how added-mass (and lift) forces lead to different motions for a sphere and a cylinder. The general expression is useful in two-phase flow calculations, and for indicating the forces and motions of ‘lumps of fluid’ in turbulent flows. © 1988, Cambridge University Press. All rights reserved.

Description: The problem has been examined using a kinematic model for wall pliability, wherein a kinematic postulation of the wall boundary conditions is made. A form of the normalized wall-displacement and its phase are used as additional parameters in an extended eigenvalue problem. Using this technique the entire gamut of possibilities regarding stability of flow past (normally) pliable walls can be examined, yet without recourse to any specific material properties for the wall. Rather, the results based on the kinematic model can be used to back-calculate the material properties corresponding to any chosen model for the dynamics of the wall. A sample back calculation is discussed herein for the Benjamin—Landahl wall model, and based on this some predictions are made regarding both stabilization of the flow and physical realizability of modes. It is believed that the kinematic model will prove useful in further understanding of the problem, and in the design of stabilizing coatings. The results show that there are three important mode classes’ (distinct from ‘modes’), namely the Tollmien—Schlichting (TS), resonant (R) and Kelvin-Helmholtz (KH). Whereas the TS and R mode classes broadly agree with modes bearing similar names as found by earlier workers, the present KH mode class is difficult to classify based on earlier work. Moreover, there are also important transitional mode classes in the regions of bifurcations of the regular mode classes. Two important concepts evolve in connection with the TS and R mode classes, namely the existence of ‘stable pockets for the former and ‘ unstable pockets’ for the latter. It is also confirmed herein that there are conflicting requirements on the damping d to stabilize TS and R modes. Considering these points it has been suggested that TS and R modes be avoided by keeping soft surfaces as compliant coatings. However, this in turn leads to instabilities from one of the transitional mode classes. It is also seen that a soft surface that is also marginally active (i.e. having a small negative value of d) could render even better stabilization. © 1988, Cambridge University Press. All rights reserved.