The growth rate and turbulent structure of the compressible, plane shear layer are investigated experimentally in a novel facility. In this facility, it is possible to flow similar or dissimilar gases of different densities and to select different Mach numbers for each stream. Ten combinations of gases and Mach numbers are studied in which the free-stream Mach numbers range from 0.2 to 4. Schlieren photography of 20-ns exposure time reveals very low spreading rates and large-scale structures. The growth of the turbulent region is defined by means of Pitot-pressure profiles measured at several streamwise locations. A compressibility-effect parameter is defined that correlates and unifies the experimental results. It is the Mach number in a coordinate system convecting with the velocity of the dominant waves and structures of the shear layer, called here the convective Mach number. It happens to have nearly the same value for each stream. In the current experiments, it ranges from 0 to 1.9. The correlations of the growth rate with convective Mach number fall approximately onto one curve when the growth rate is normalized by its incompressible value at the same velocity and density ratios. The normalized growth rate, which is unity for incompressible flow, decreases rapidly with increasing convective Mach number, reaching an asymptotic vaue of about 0.2 for supersonic convective Mach numbers.

/pdf/the-compressible-turbulent-shear-layer-an-experimental-study-53ojxzr9zi.pdf

The compressible turbulent shear layer: an experimental study

The Orr-Sommerfeld equation is solved numerically using expansions in Chebyshev polynomials and the QR matrix eigenvalue algorithm. It is shown that results of great accuracy are obtained very economically. The method is applied to the stability of plane Poiseuille flow; it is found that the critical Reynolds number is 5772·22. It is explained why expansions in Chebyshev polynomials are better suited to the solution of hydrodynamic stability problems than expansions in other, seemingly more relevant, sets of orthogonal functions.

/pdf/accurate-solution-of-the-orr-sommerfeld-stability-equation-30ap5ab7oe.pdf

Accurate solution of the Orr–Sommerfeld stability equation

Advanced Mathematical Methods for Scientists and Engineers

Publisher Summary The chapter reviews the meteorological phenomena that are associated with topography. The study of airflow past mountains is complicated by the wide range of scales that must be considered. The ratios of the mountain width to each of the natural length scales are important in determining the physical regime of the flow. This idea is emphasized in the chapter by treating the effects of boundary layers and buoyancy. The theory of two-dimensional mountain waves with the help of its governing equations is presented and the observations of mountain waves are presented. The chapter also examines the influence of the boundary layer on mountain flows and slope winds and mountain and valley winds. It considers the perturbation to the wind flow caused by a mountain of intermediate scale where the rotation of the Earth cannot be neglected. For this the flow near mesoscale and synoptic-scale mountains, quasi-geostrophic flow over a mountain, the effect of inertia on the flow over mesoscale mountains, and theories of lee cyclogenesis are discussed. Finally the chapter describes planetary-scale mountain waves; a vertically integrated model of topographically forced planetary waves; the vertical structure of planetary waves; models of stationary planetary waves allowing meridional propagation and lateral; and variation in the background wind.

The Influence of Mountains on the Atmosphere

The Rayleigh stability equation of inviscid linearized stability theory was integrated numerically for amplified disturbances of the hyperbolic-tangent velocity profile. The evaluation of the eigenvalues and eigenfunctions is followed by a discussion of the streamline pattern of the disturbed flow. Here no qualitative distinction is found between an amplified and the neutral disturbance. But considering the vorticity distribution of the disturbed flow it is shown that in the case of amplified disturbances two concentrations of vorticity occur within a disturbance wavelength, while in the neutral case only one maximum of vorticity exists. The results are discussed with respect to the instability mechanism of free boundary-layer flow.

https://www.cambridge.org/core/services/aop-cambridge-core/content/view/S0022112064000908

On the inviscid instability of the hyperbolictangent velocity profile

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 Howard's 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.

https://www.cambridge.org/core/services/aop-cambridge-core/content/view/S0022112077000640

Shear layer instability of an inviscid compressible fluid. Part 2

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.

https://www.cambridge.org/core/services/aop-cambridge-core/content/view/S0022112086001441

Bifurcations of two-dimensional channel flows

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.

Kelvin–Helmholtz instability of finite amplitude

A model of steady airflow past a three-dimensional mountain is considered. The fluid is inviscid and incompressible. The flow is under the influence of gravity, but for most of the work the earth's rotation is neglected. Far upstream the flow is parallel and horizontal, the velocity U ( z ) and the density ?( z ) varying with height z . By use of conservation of density and head along streamlines, and thence of the dependence of density and head on one variable z 0 only, the equations of motion are simplified, z 0 (r) being the height far upstream of the streamline through the point with position vector r. The flow is considered first with small and then with large density variations far upstream. In the case when these variations are small, results of the known method of simple-shear secondary flows are rederived and extended. The chief consideration of this paper is the buoyancy effects of large variations of density in the oncoming stream. For simplicity, the shear far upstream is neglected. The primary and secondary velocities are found for large Richardson number (proportional to gd ?/? dz 0 ). The method is time-symmetric and analogous to taking the first two terms of the Janzen-Rayleigh expansion, the inverse of the Richardson number being the analogue of the square of the Mach number. The primary velocity, for infinite buoyancy forces, is horizontal, the flow in each horizontal plane being the same as the potential flow about the section of the obstacle in that plane. In general there is shear between horizontal planes, with horizontal primary vorticity. The secondary flow is found explicitly for a circular cylinder with vertical axis and for a hemisphere resting in a horizontal plane. The secondary approximation is not uniformly valid near the horizontal plane at the height of the top (if any) of the obstacle. Near that plane there is an inviscid shear layer, and the velocity gradient cannot be neglected, however large the Richardson number. DOI: 10.1111/j.2153-3490.1961.tb00081.x

On the Steady Flow of a Fluid of Variable Density Past an Obstacle

http://iopscience.iop.org/article/10.1016/0169-5983(88)90021-4/pdf

P. G. Drazin

Papers

Shear layer instability of an inviscid compressible fluid. Part 2

Bifurcations of two-dimensional channel flows

Kelvin–Helmholtz instability of finite amplitude

On the Steady Flow of a Fluid of Variable Density Past an Obstacle

On the flow of a viscous fluid driven along a channel by suction at porous walls