M. Gaster

A note on the relation between temporally-increasing and spatially-increasing disturbances in hydrodynamic stability

The frequency and amplification rates for a disturbance growing with respect to time are compared with those of a spatially-growing wave having the same wave-number. For small rates of amplification it is shown that the frequencies are equal to a high order of approximation, and that the spatial growth is related to the time growth by the group velocity.

Large-scale structures in a forced turbulent mixing layer

The large-scale structures that occur in a forced turbulent mixing layer at moderately high Reynolds numbers have been modelled by linear inviscid stability theory incorporating first-order corrections for slow spatial variations of the mean flow. The perturbation stream function for a spatially growing time-periodic travelling wave has been numerically evaluated for the measured linearly diverging mean flow. In an accompanying experiment periodic oscillations were imposed on the turbulent mixing layer by the motion of a small flap at the trailing edge of the splitter plate that separated the two uniform streams of different velocity. The results of the numerical computations are compared with experimental measurements. When the comparison between experimental data and the computational model was made on a purely local basis, agreement in both the amplitude and phase distribution across the mixing layer was excellent. Comparisons on a global scale revealed, not unexpectedly, less good accuracy in predicting the overall amplification.

An experimental investigation of the formation and development of a wave packet in a laminar boundary layer

The formation and development of a wave packet in the laminar boundary layer of a flat plate has been studied experimentally. The packet was artificially generated by a short duration acoustic pulse, which was injected into the boundary layer flow through a small hole in the plate. The flow oscillations created by the passage of the packet were detected by a hotwire anemometer which was positioned just outside the boundary layer. Repeated excitation of the disturbance enabled the noise content of the data to be significantly reduced by signal averaging which was carried out online by a small computer. From records generated at a number of probe locations over the plate it has been possible to show how the packet developed as it propagated downstream. Contours of the signal amplitude from each of the downstream stations showed that the patch of waves was initially roughly elliptic, but this evolved into a distinctly bowed shape as the patch spread out as it progressed downstream. The smoothly contoured wave packet, with peak amplitudes close to the centre, gradually distorted and far downstream two amplitude maxima formed on either side of the centre line. The wave-like character of the fluctuations within the boundaries of the packet was revealed graphically through perspective projections of the signal records on the z, t plane (z is across the span). These displays showed the gradual development from a small smooth packet close to the source to the final warped and distorted pattern. Spectral decomposition of the fluctuations at each streamwise measuring station provided information relating to the growth of narrow bands of waves. These showed that a particular band of oblique waves, which grew very rapidly at some stage, were linked to the observed distortions.

Vortex shedding from slender cones at low Reynolds numbers

Water-tunnel experiments on two cones of taper ratio 36:1 and 18:1 have shown that the frequency of vortex shedding is controlled by the local diameter and has a value which is slightly lower than that on a parallel cylinder of the same size. The shedding process is modulated by a low-frequency oscillation which has a frequency dependent only on U 2 / v and is independent of any physical dimension of the model. A mathematical analogue of the wake, which helps to explain these features of the flow, has been constructed. The pattern of vortices in the wake of a cone suggested by this model is in reasonable agreement with the photographic evidence.

A theoretical model of a wave packet in the boundary layer on a flat plate

A linear model of a wave packet in a laminar boundary layer is proposed. The model wave packet was chosen to conform to a particular experimental situation where a packet was artificially excited by a localized pulsed perturbation at the wall boundary. Appropriate quantities were computed from this model and these have been compared directly with experimental measurements. The model disturbance was built up from a linear combination of spatially growing modes summed numerically over all wavenumbers and frequencies. The input spectrum was assumed to be flat (all modes were equally excited), and the downstream development of the frequency—wavenumber spectrum was calculated on the basis of linear stability theory. The development of the model wave packet was compared with that of the experiment. It was found that the overall shapes of the disturbed region and the way it spread out as the packet travelled downstream were well predicted by the model behaviour. Detailed comparisons of the wave motion within the wave packet were also made and although encouraging correlation was achieved at stations close to the source this was not maintained far downstream where the experimental packet showed various irregularities. The wave packet generated by the summation of modes developed smoothly downstream without forming any of the types of distortions which had been observed in the experiment.

Vortex shedding from circular cylinders at low Reynolds numbers

Experiments on slightly tapered models of circular cross-section have shown that the vortex wake structure exists in a number of discrete cells having different shedding frequencies. Within each cell shedding is regular and periodic, the frequency being somewhat lower than that from a parallel cylinder of the same diameter. A similar type of wake behaviour has also been observed on a parallel model in a non-uniform mean flow. These results suggest that the discontinuities in the shedding law observed by Tritton could arise through non-uniformities in the flow.

Growth of Disturbances in Both Space and Time

Betchov and Criminale found certain singularities in the eigenvalue relations for the combined stability problem in space and time. It is shown why these singularities must occur, and how they influence the perturbation generated by a pulse.

The development of three-dimensional wave packets in a boundary layer

The formation and growth of three-dimensional wave packets in a laminar boundary layer is treated as a linear problem. The asymptotic form of the disturbed region developing from a point source is obtained in terms of parameters describing two-dimensional instabilities of the flow. It is shown that a wave caustic forms and limits the lateral spread of growing disturbances whenever the Reynolds number is √2 times the critical value. The analysis is applied to the boundary layer on a flat plate and shapes of the wave-envelope are calculated for various Reynolds numbers. These show that all growing disturbances are contained within a wedge-shaped region of approximately 10° semi-angle.

THE VELOCITY FIELD CREATED BY A SHALLOW BUMP IN A BOUNDARY LAYER

The results of measurements of the disturbance velocity field generated in a boundary layer by a shallow three‐dimensional bump oscillating at a very low frequency on the surface of a flat plate are reported. The disturbance was entirely confined to the boundary layer. Profiles of the mean velocity, the disturbance velocity at the fundamental frequency, and at the first harmonic are presented. These profiles were measured both upstream and downstream of the oscillating bump. About 100 boundary layer thicknesses downstream of the bump the intensity had a maximum near η=2, with a magnitude of about 0.2%. Measurements of the disturbance velocity were also made at various spanwise and downstream locations at a fixed distance from the boundary of one displacement thickness. The spanwise profiles of the disturbance field changed dramatically from the near‐field region of the bump to the far downstream region. Finally, the spanwise spectrum of the disturbances at three locations downstream of the bump are presen...

Active control of boundary layer instabilities

Active control of spatially evolving three-dimensional instability waves in the boundary layer of a flat plate with zero pressure gradient has been investigated both numerically and experimentally. The boundary layer was artificially excited by various computer-generated perturbations in a low-turbulence wind tunnel so as to create a three-dimensional field of instability waves. Sensors were used to detect the oncoming disturbances and appropriate control signals sent to the downstream actuators to generate counteracting disturbances. The key element of this type of active control is the determination of the transfer function of the control system connecting the sensors with the actuators. Modelling by linear theory showed that the amplitudes of the disturbances downstream were significantly suppressed when the causal transfer function was applied. To simplify the control problem different strategies using approximations to the transfer function were used. It was shown that with a very simple transfer function almost the same result could be achieved as with the full causal transfer function. The modelling was validated experimentally, first by an offline control using different control modes, i.e. different transfer functions, and then by a real-time hardware implementation using one of the modes. Good agreement between the experimental measurements and numerical predictions indicated that a simple control strategy could be developed to inhibit the growth of the three-dimensional instability waves over extensive regions of the flow.