P. G. Drazin

Shear layer instability of an inviscid compressible fluid. Part 2

The stability of parallel shear flow of an inviscid compressible fluid is investigated by a linear analysis. The extension of the Rayleigh stability criterion and Howards semi-circle theorem to compressible flows, obtained by Lees & Lin (1946) and Eckart (1963) respectively, are each rederived by a different approach. It is then shown that a subsonic neutral solution of the stability equation may be found when the basic flow is represented by the hyperbolic-tangent velocity profile. With the aid of this solution, the unstable eigenvalues, eigenfunctions and Reynolds stress are determined by numerical methods. A brief discussion of the results follows.

BIFURCATIONS OF TWO-DIMENSIONAL CHANNEL FLOWS

In this paper we study some instabilities and bifurcations of two-dimensional channel flows. We use analytical, numerical and experimental methods. We start by recapitulating some basic results in linear and nonlinear stability and drawing a connection with bifurcation theory. We then examine Jeffery–Hamel flows and discover new results about the stability of such flows. Next we consider two-dimensional indented channels and their symmetric and asymmetric flows. We demonstrate that the unique symmetric flow which exists at small Reynolds number is not stable at larger Reynolds number, there being a pitchfork bifurcation so that two stable asymmetric steady flows occur. At larger Reynolds number we find as many as eight asymmetric stable steady solutions, and infer the existence of another seven unstable solutions. When the Reynolds number is sufficiently large we find time-periodic solutions and deduce the existence of a Hopf bifurcation. These results show a rich and unexpected structure to solutions of the Navier–Stokes equations at Reynolds numbers of less than a few hundred.

Kelvin–Helmholtz instability of finite amplitude

Non-linear Kelvin–Helmholtz instability, of two parallel horizontal streams of inviscid incompressible fluids under the action of gravity, is studied theoretically. The lower stream is denser and there is surface tension between the streams. Some progressing waves of finite amplitude are found as the development of a slightly unstable wave of infinitesimal amplitude. In particular, the non-linear elevation of the interface between the fluids is calculated. The finite amplitude of the waves does not equilibrate to a constant after a long time, but varies periodically with time. In practice, slight dissipation should lead to equilibration at an amplitude close to a value given by the present theory.

The stability of Poiseuille flow in a pipe

Numerical calculations show that the flow of viscous incompressible fluid in a circular pipe is stable to small axisymmetric disturbances at all Reynolds numbers. These calculations are linked with known asymptotic results.

On transition to chaos in two-dimensional channel flow symmetrically driven by accelerating walls

A theoretical study of a viscous incompressible fluid in a parallel-walled channel, the flow driven by uniform steady suction through the porous and accelerating walls of the channel. Previous authors have discussed special cases of such flows, confining attention to flows which are symmetric, steady and 2-dimensional. A similarity form of solution is assumed, as used by Berman and originally due to Hiementz to reduce the Navier-Stokes equations to a nonlinear ordinary differential equation. The authors generalise their work by considering asymmetric flows, unsteady flows and three-dimensional perturbations. In particular it is shown in this paper that most of the previously found steady solutions are unstable to antisymmetric two-dimensional disturbances. this leads to pitchfork bifurcation, stable asymmetric steady solutions, a Hopf bifurcation, stable time-periodic solutions, stable quasi-periodic solutions and chaos in succession as the Reynolds number increases from zero.

On perturbations of Jeffery-Hamel flow

We examine various perturbations of Jeffery-Hamel flows, the exact solutions of the Navier-Stokes equations governing the steady two-dimensional motions of an incompressible viscous fluid from a line source at the intersection of two rigid plane walls. First a pitchfork bifurcation of the Jeffery-Hamel flows themselves is described by perturbation theory. This description is then used as a basis to investigate the spatial development of arbitrary small steady two-dimensional perturbations of a Jeffery-Hamel flow; both linear and weakly nonlinear perturbations are treated for plane and nearly plane walls. It is found that there is strong interaction of the disturbances up- and downstream if the angle between the planes exceeds a critical value 2α 2 , which depends on the value of the Reynolds number. Finally, the problem of linear temporal stability of Jeffery-Hamel flows is broached and again the importance of specifying conditions up- and downstream is revealed. All these results are used to interpret the development of flow along a channel with walls of small curvature. Fraenkels (1962) approximation of channel flow locally by Jeffery-Hamel flows is supported with the added proviso that the angle between the two walls at each station is less than 2α 2 .

Steady two-dimensional flow of fluid of variable density over an obstacle

A model of airflow over a mountain is treated mathematically in this paper. The fluid is inviscid, incompressible and of variable density. The flow is in a long channel, bounded above by a rigid horizontal lid and below by an obstacle. The variation with height of the horizontal velocity and of the density is specified far upstream. The details of flow are examined for particular conditions upstream which lead to a linear vorticity equation, although the non-linear inertial terms in the Euler equations of motion are exactly represented. In this case the flow is described by the superposition of solutions of some diffraction problems. Classical techniques of diffraction theory are then used to demonstrate the existence and some general properties of solutions for steady flow. Thus a steady solution is always possible if no restriction is placed on the amount of energy available to drive the flow, that is to say there is no critical internal Froude number (measuring the dynamical effect of buoyancy) for the existence of a steady flow. Finally the flows past a dipole and a vertical wall are computed.

Dust devil formation

Abstract A review of the literature on observations of dust devils suggests that these whirls form like the rolling-up of a vertical vortex sheet when the stratification is slightly unstable. The linearized equations governing the stability of such flows are formulated. These are solved when the basic flow is a vertical vortex sheet in a fluid with uniform Brunt-Vaisala frequency. Also, the effects of a shear layer of finite width are considered. It is found how unstable stratification strongly intensifies the shear instability. Increase of stable stratification may decrease stability of some special disturbances.

Discontinuous velocity profiles for the Orr-Sommerfeld equation

Simple ideas of dimensional analysis and of limiting cases are used to elucidate the stability characteristics of a steady basic parallel flow of a viscous incompressible fluid. The principal result is that the stability characteristics of a smoothly varying velocity profile for wave disturbances of small wave-number can be found by use of a discontinuous velocity profile. The boundary conditions for a disturbance at a discontinuity of the basic flow are derived, and are used to find the stability characteristics of broken-line representations of the hal-jet and jet. These findings are in agreement with previous ones.

Flow along a diverging channel

This paper treats the two-dimensional steady flow of a viscous incompressible fluid driven through a channel bounded by two walls which are the radii of a sector and two arcs (the ‘inlet’ and ‘outlet’), with the same centre as the sector, at which inflow and outflow conditions are imposed. The computed flows are related to both a laboratory experiment and recent calculations of the linearized ‘spatial’ modes of Jeffery–Hamel flows. The computations, at a few values of the angle between the walls of the sector and several values of the Reynolds number, show how the first bifurcation of the flow in a channel is related to spatial instability. They also show how the end effects due to conditions at the inlet and outlet of the channel are related to the spatial modes: in particular, Saint-Venants principle breaks down when the flow is spatially unstable, there being a temporally stable steady flow for which small changes at the inlet or outlet create substantial effects all along the channel. The choice of a sector as the shape of the channel is to permit the exploitation of knowledge of the spatial modes of Jeffery–Hamel flows, although we regard the sector as an example of channels with walls of moderate curvature.