ALBERT

Description: We analyse the steady streaming generated in an infinite elliptical tube containing a viscous, incompressible fluid when the boundary oscillates in such a way that the area and ellipticity of the cross-section vary with time but remain independent of the longitudinal coordinate. The parameters a−1 = (v/Ωa%): ande = U0/a0Ω, where v is the kinematic viscosity, Ω is the oscillation frequency, a0is the undisturbed semi-major axis and UQis a typical wall velocity, are taken to be small, so that the Stokes layer is thin and the interaction which leads to the steady streaming can be analysed as a small perturbation. Coupled axial and transverse velocities, both oscillatory and steady, are generated. A complication is the need to specify the tangential as well as the normal velocity component on the tube wall, which requires an assumption concerning its elastic properties. We have considered two cases: (i) constant major axis, in which all boundary points move parallel to the minor axis, and (ii) an inextensible wall. The three-dimensional steady streaming in the core of the tube is computed only in the limit of small steady-streaming Reynolds number, Rs= 2α2. © 1987, Cambridge University Press. All rights reserved.

Description: We describe flow-visualization experiments and theory on the two-dimensional unsteady flow of an incompressible fluid in a channel with a time-dependent indentation in one wall. There is steady Poiseuille flow far upstream, and the indentationmoves in and out sinusoidally, its retracted position being flush with the wall. The governing parameters are Reynolds number Re, Strouhal number (frequency parameter) St and amplitude parameter e (themaximum fraction of the channel width occupied by the indentation);most of the experiments were performed with e « 0.4. For St ≤ 0.005 the flow is quasi-steady throughout the observed range of Re (360 〈 Re 〈 1260). For St 〉 0.005 a propagating train of waves appears, during every cycle, in the core flow downstream of the indentation, and closed eddies form in the separated flow regions on the walls beneath their crests and above their troughs. Later in the cycle, a second, corotating eddy develops upstream of the first in the same separated-flow region ('eddy doubling'), and, later still, three-dimensional disturbances appear, before being swept away downstream to leave undisturbed parallel flow at the end of the cycle. The longitudinal positions of the wave crests and troughs and of the vortex cores aremeasured as functions of time formany values of the parameters; they vary with St but not with Re. Our inviscid, long-wavelength, small-amplitude theory predicts the formation of a wavetrain during each cycle, in which the displacement of a core-flow streamline satisfies the linearized Kortewegde Vries equation downstream of the indentation. The waves owe their existence to the non-zero vorticity gradient in the oncoming flow. Eddy formation and doubling are not described by the theory. The predicted positions of the wave crests and troughs agree well with experiment for the larger values of St used (up to 0.077), but less well for small values. Analysis of the viscous boundary layers indicates that the inviscid theory is self-consistent for sufficiently small time, the time of validity increasing as St increases (for fixed e). © 1985, Cambridge University Press. All rights reserved.

Description: A new model is presented to describe flow in segments of collapsible tube mounted between two rigid tubes and surrounded by a pressurized container. The new features of the model are the inclusion of (a) longitudinal wall tension and (b) energy loss in the separated flow downstream of the time-dependent constriction in a collapsing tube, in a manner which is consistent with the one-dimensional equations of motion. As well as accurately simulating steady-state collapse, the model predicts self-excited oscillations whose amplitude is large enough to be observable only if the flow in the collapsible tube becomes supercritical somewhere (fluid speed exceeding long-wave propagation speed). The dynamics of the oscillations is dominated by longitudinal movement of the point of flow separation, in response to the adverse pressure gradient associated with waves propagating backwards and forwards between the (moving) narrowest point of the constriction and the tube outlet. © 1985, Cambridge University Press. All rights reserved.

Description: A laser-Doppler anemometer (LDA) was used to measure turbulent velocities in drag-reducing fibre suspensions. Measurements of streamwise velocities (and, in one case, the circumferential velocity as well) were made in flow through a straight pipe at x/d = 190, and at Reynolds numbers in the range 1.4 x 104-5.3 x 104. The fibres used were chrysotile asbestos of high aspect ratio (~ 105), at a concentration of 300 w.p.p.m. They were dispersed in an aqueous solution of a surfactant (0.5% by weight Aerosol OT). In some experiments, the fibre suspensions were supplemented by a drag-reducing polymer (Separan AP30) at a concentration of 150 w.p.p.m. A complete experiment involved passing a quantity of fibre suspension through the apparatus a number of times (at a given Reynolds number) and measuring the velocity distribution across the pipe during each pass. As the amount of drag reduction generally declined with the number of passes (i.e. due to fibre degradation), this provided a convenient way of varying the percentage drag reduction as an experimental parameter. Results were obtained for mean velocity and intensity profiles, autocorrelations, and one-dimensional energy spectra. The mean period of turbulent bursts was determined by measuring autocorrelations with short sampling times. At the lowest Reynolds number (Re = 1.4 x 104), drag reductions of about 70% were obtained during the first two passes. This was accompanied by a reduction in the streamwise intensity below the level obtained in the surfactant solution alone. (Note: The opposite behaviour is found in drag-reducing polymer solutions, where intensity levels are larger than those in the solvent alone.) A measurement of the r.m.s. circumferential velocity showed an increased level (relative to surfactant alone) during this part of the experiment. During further passes, there was a transition to ‘polymer-like’ behaviour, with increased streamwise intensity, which subsequently declined with pass number (and hence drag reduction) towards the result for surfactant alone. This effect had previously been found in preliminary experiments at Re = 9xl03(McComb & Chan 1979). Repetition of the experiment at Re = 1.4 x 104, with the addition of Separan AP30, confirmed the existence of this transition from ‘fibre-like’ to ‘polymer-like’ drag reduction. In this case, the drag reduction was smaller (at about 60%), but the mixed suspension was much more resistant to degradation, with transition occurring at the ninth pass. However, such behaviour was not found at higher Reynolds numbers (Re = 3.2 x 104and 5.3 x 104), in fibre suspensions where increased streamwise intensities occurred, even at high levels of drag reduction (about 70%). Anomalous streamwise autocorrelations were found during ‘fibre-like’ drag reduction but in the ‘ polymer-like ’ regime they were very similar to those measured in polymer solution, and showed characteristically increased lengthscales. On the other hand, energy spectra were found to be anomalous in all cases and showed an energy deficit at lengthscales of the same order as the fibre length. Finally, mean bursting periods were found to be much increased, with the increases being about the same as those in polymer solutions at the same Reynolds number and percentage drag reduction. © 1985, Cambridge University Press. All rights reserved.

Description: ‘Bioconvection’ is the name given to pattern-forming convective motions set up in suspensions of swimming micro-organisms. ‘Gyrotaxis describes the way the swimming is guided through a balance between the physical torques generated by viscous drag and by gravity operating on an asymmetric distribution of mass within the organism. When the organisms are heavier towards the rear, gyrotaxis turns them so that they swim towards regions of most rapid downflow. The presence of gyrotaxis means that bioconvective instability can develop from an initially uniform suspension, without an unstable density stratification. In this paper a continuum model for suspensions of gyrotactic micro-organisms is proposed and discussed; in particular, account is taken of the fact that the organisms of interest are nonspherical, so that their orientation is influenced by the strain rate in the ambient flow as well as the vorticity. This model is used to analyse the linear instability of a uniform suspension. It is shown that the suspension is unstable if the disturbance wavenumber is less than a critical value which, together with the wavenumber of the most rapidly growing disturbance, is calculated explicitly. The subsequent convection pattern is predicted to be three-dimensional (i.e. with variation in the vertical as well as the horizontal direction) if the cells are sufficiently elongated. Numerical results are given for suspensions of a particular algal species (Chlamy-domonas nivalis); the predicted wavelength of the most rapidly growing disturbance is 5—6 times larger than the wavelength of steady-state patterns observed in experiments. The main reasons for the difference are probably that the analysis describes the onset of convection, not the final, nonlinear steady state, and that the experimental fluid layer has finite depth. © 1988, Cambridge University Press. All rights reserved.

Description: Experiments are described in which the populations of metastable levels in ionizing argon are measured through spatially resolved hook interferometry. The results are compared with the present model for shock-induced ionization and a recently proposed mechanism to explain observed flow instabilities. It is found that the experimental measurements support the presently accepted model, which states that electron–atom collisions play the dominant role in the excitation process, but contradicts recent proposals which predict a rapid build-up of anomalously high metastable populations through atom–atom collisions. © 1986, Cambridge University Press. All rights reserved.