Plane Shear Layer Research Articles

Natural instability of free shear layers

Under controlled small-amplitude excitation, an initially laminar free shear layer experiences maximum growth rate at a Strouhal number St(theta) of 0.017 (consistent with theory) and maximum growth at St(theta) = 0.011, while the natural instability frequency St(theta-n) (of an unexcited shear layer) is found to have an intermediate value. Investigations in both axisymmetric and plane shear layers in a number of independent facilities reveal that the St(theta-n) value falls in the range 0.0125-0.0155, depending on the exit boundary-layer fluctuation level and the spanwise radius of curvature. The St(theta-n) value decreases with increasing jet diameter or exit boundary-layer fluctuation level, but is not a direct function of the exit momentum thickness Reynolds number. For a given facility, the instability details are found to be independent of whether the entrainment at the lip is parallel to the stream or orthogonal (due to the addition of an end plate). The steamwise evolutions of the amplitudes at the fundamental frequency and its harmonics and subharmonics are unique functions of the downstream distance nondimensionalized by the exit momentum thickness, but their details remain functions of the flow geometry (i.e., axisymmetric or plane).

Three-dimensional tertiary motions in a plane shear layer

The nonlinear evolution of instabilities of a plane-parallel shear flow with an inflection-point profile is studied. The particular example of the cubic-profile flow generated in an inclined layer heated from above and cooled from below is chosen because it exhibits supercritical bifurcations for secondary and tertiary flows. Since the limit of small Prandtl number is assumed, buoyancy effects caused by temperature perturbations are negligible. The analysis describes first the transition to transverse roll-like vortices which become unstable at slightly supercritical Grashof numbers to a vortex-pairing instability with alternating pairing in the spanwise direction. Three-dimensional finite-amplitude solutions for this tertiary mode of motion are computed and discussed. Finally the question of the stability of the tertiary flow is addressed.

A spanwise structure in the plane shear layer

We study experimentally the spanwise structure of the mean flow field of a plane shear layer. The field is dominated by a lateral undulation that persists downstream to form long longitudinal structures. Both the amplitude and spacing of these structures vary in such a way as to suggest that they are due to a secondary instability of the flow field. Some models for this instability are discussed and compared with the experimental results.

Acoustic resonances and sound scattering by a shear layer

The energy reflection coefficient is evaluated numerically for plane waves incident on a plane shear layer having a linear velocity profile. The shear layer is found to exhibit no resonances and no Brewster angles. The behavior of the reflection coefficient depends crucially on the parameter tau, a nondimensional measure of the disturbance Strouhal number with respect to the disturbance Mach number in the mean flow direction. For moderate values of tau, the amplified reflection regime degenerates into the total reflection one, whereas in the ordinary reflection regime the variation of the reflection coefficient with tau depends on whether or not the corresponding vortex sheet has a Brewster angle. The results indicate that caution should be exercised in uncritically modeling a finite thickness shear layer by a corresponding vortex sheet.

Response of plane shear layers and wakes to strong three-dimensional disturbances

Shear layers and wakes formed downstream of a splitter plate whose trailing edge has spanwise variations were briefly explored using the reacting flow visualization technique. A fundamental difference in behavior was observed between the two types of flows. The shear layer (which has only one sign of mean vorticity) rapidly forms its characteristic two-dimensional vortex structures, whereas, in contrast, the wake (which has both signs of mean vorticity) forms closed vortex loops. These loops do not grow appreciably in the spanwise direction over the test section length, even though the transverse growth is comparable to the two-dimensional wake behavior. The global structure of the shear layer quickly forgets the initial perturbations, while the wake remembers them. The results strongly suggest that the presence of only one or both signs of mean vorticity in these plane turbulent flows has a fundamental bearing on the spanwise structure and stability of the vorticity field.

The calculation of free turbulent shear flows with strong density fluctuations

The turbulent mixing of two gases with highly different molecular masses in a plane shear layer is considered. A calculation model incorporating modelled transport equations for the partial density fluctuations is developed and the numerical results are compared with measurements showing good agreement.

Sound production at the edge of a steady flow

The theory initiated by Lighthill (1952) to describe the sound radiated by turbulence embedded in an uniform fluid at rest is here extended to the case where the turbulence exists on the edge of a uniformly moving stream. An exact analogy is developed between the distant real sound field and that which would be radiated by a particular quadrupole distribution adjacent to a vortex sheet positioned in the linearly disturbed flow. The equivalent sources in this analogy are quadrupoles identical in strength with those in Lighthill's model, but the quadrupoles are now shown to convect with the fluid-particle velocity. There is no amplifying effect of shear. The particular case of a plane shear layer is worked out in detail for sound waves of scale large in comparison with the shear-layer thickness.A downstream zone of silence is predicted as is the formation of highly directional beams associated with the interference of sound radiated directly and sound reflected from the fluid interface. A distinct structure results in which the variation of sound with flow velocity, density and angle is not easily accounted for by simple power-law scaling. Finally a comparison is made with some features of jet noise; the modelling of the high frequency jet noise problem by a single shear layer yields some features consistent with experiment.

Numerical Study Of A Mixing Layer In SpatialAnd Temporal Development Of A Stratified Flow

The two-dimensional plane shear layers for high Reynolds numbers are simulated using the transient explicit Spectral Collocation method. The fast and slow air streams are submitted to different densities. A temporally growing mixing layer, developing from a hyperbolic tangent velocity profile to which is superposed an infinitesimal white-noise perturbation over the initial condition, is studied. In the same way, a spatially growing mixing layer is also study. In this case, the flow in the inlet of the domain is perturbed by a time dependent white noise of small amplitude. The structure of the vorticity and the density are visualized for high Reynolds number for both cases studied. Broad-band kinetic energy and temperature spectra develop after the first pairing are observed. The results showed that similar conclusions are obtained for the temporal and spatial mixing layers. The spreading rate of the layer in particular is in good agreement with the works presented in the literature.

Three-Sound-Pressures Theorem, and Its Application, in Aerodynamically Generated Sound

A theorem is presented which is a direct consequence of Lighthill's theory of aerodynamic noise generation when expressed in the equivalent point-source approximation. It states that the sum of the sound pressures at points located at equal distances on each of three mutually perpendicular axes, with the origin in the flow, is zero. This is based on the incompressible-flow approximation and applies to a free flow which is inviscid and unconvected. The next approximation is given also, the sum varying as the Mach number raised to the fourth power. There is also an analogous form of the theorem for two-dimensional systems concerning the sum of the sound pressure on any two mutually perpendicular axes. The value of the theorem is demonstrated by showing how, for example, there can be, in general, no net longitudinal quadrupole strength for an inviscid noise-generating flow. This is illustrated by application to the sound radiators of a plane shear layer undergoing a two-dimensional disturbance.

Theorem Concerning the Aerodynamic Sound of a Free Inviscid Flow

A theorem is presented which is a direct consequence of the Lighthill theory of aerodynamic noise generation when expressed in the equivalent point-source approximation; the same theorem also follows from the vortex theory of sound generation. It states that the sum of the sound pressures at points located at equal distances on each of three mutually perpendicular axes, with the origin in the flow, is zero. This is based on the incompressible-flow approximation and applies to a flow which is inviscid and unconvected. There is also an analogous form of the theorem for two-dimensional systems, concerning the sum of the sound on any two mutually perpendicular axes. The value of the theorem is demonstrated by showing how there can be no net longitudinal quadrupole strength for an inviscid noise-generating flow. This is illustrated by considering the sound radiators of a plane shear layer undergoing a small two-dimensional disturbance. (This work has received partial support of the Office of Naval Research, under contract.)