The early phase of high-speed liquid droplet impact on a rigid wall is characterized by compressibility effects through the creation of a shock wave attached to the contact area periphery. Initially, the area of compressed liquid is assumed to be bounded by the shock envelope, which propagates both laterally and upwardly into the bulk of the liquid. In this paper, an analytical model accounting for the lateral liquid motion in the compressed area is developed and compared to the axisymmetric numerical solution of the inviscid (Euler) flow equations. It is shown that the often employed assumption that the compressed area is separated from the liquid bulk by a single shock wave attached to the contact line breaks down and results in an anomaly. This anomaly emerges prior to the time when the shock wave departs from the contact line, initiating lateral liquid jetting. In order to remove this anomaly, the analytical model presented in this paper proposes the transition from a single to a multiple wave structure in the contact line region, prior to jetting eruption. The occurrence of this more complex multiple wave structure is also supported by the numerical results.

Using slender-body hydrodynamics in the inertialess limit, we examine the motion of Purcell's swimmer, a planar, fore–aft-symmetric three-link flagellum or propulsive mechanism that translates by alternately moving its front and rear segments. Purcell (1976) concluded via symmetry arguments that the net displacement of such a swimmer must follow a straight line, but the direction and other details of the motion have never been investigated. Numerical results indicate that the direction of net translation and the speed of Purcell's swimmer depend on the angular amplitude of the swimming strokes as well as on the relative length of the links. Analytical results are presented for small rotations about the straightened configuration, and physical arguments are given to qualitatively explain the propulsive dynamics. The optimal swimmer configurations under the conditions of constant forcing and of minimum mechanical work are determined. We use a definition of efficiency based on the straightened configuration as a reference state to compare Purcell's swimmer with the previously treated swimming motions of an undulating rod and a rotating helix. Finally, we demonstrate the importance of the anisotropy in the local hydrodynamic slender-body drag to swimming motions at low Reynolds number by showing that, in general, any inextensible swimmer in an otherwise quiescent fluid cannot alter its average position under conditions of locally isotropic drag.

Spanwise structure and growth mechanisms in a turbulent boundary layer are investigated experimentally. PIV measurements are obtained in the streamwise–spanwise ($x$–$z$)-plane from the buffer layer to the top of the logarithmic region at Re$_\theta = 1015$ and 7705. The dominant motions of the flow are shown to be large-scale regions of momentum deficit elongated in the streamwise direction. Throughout the logarithmic layer, the regions are consistently bordered by vortices organized in the streamwise direction, offering strong support for a vortex packet model. Additionally, evidence is presented for the existence and organization of hairpin vortices in the region $y^+ < 60$. Statistical evidence is also presented for two important aspects of the vortex packet paradigm: vortex organization in the streamwise direction, and the clear association of the hairpin signature with local minima in streamwise velocity. Several spanwise lengthscales are shown to vary linearly with distance from the wall, revealing self-similar growth of spanwise structure in an average sense. Inspection of the data, however, suggests that individual structures do not grow strictly self-similarly in time. It is proposed that additional scale growth occurs by the merging of vortex packets on an eddy-by-eddy basis via a vortex re-connection mechanism similar to that suggested by Wark & Nagib (1989). The proposed mechanism provides a link between vortex-pairing concepts and the observed coalescence of streaky low-speed regions in the inner layer.

We have experimentally studied the effect of microjets on the flow field of a Mach 0.9 round jet. Planar and three-dimensional velocity field measurements using particle image velocimetry show a significant reduction in the near-field turbulent intensities with the activation of microjets. The axial and normal turbulence intensities are reduced by about 15% and 20%, respectively, and an even larger effect is found on the peak values of the turbulent shear stress with a reduction of up to 40%. The required mass flow rate of the microjets was about 1% of the primary jet mass flux. It appears that the microjets influence the mean velocity profiles such that the peak normalized vorticity in the shear layer is significantly reduced, thus inducing an overall stabilizing effect. Therefore, we seem to have exploited the fact that an alteration in the instability characteristics of the initial shear-layer can influence the whole jet exhaust including its noise field. We have found a reduction of about 2 dB in the near-field overall sound pressure level in the lateral direction with the use of microjets. This observation is qualitatively consistent with the measured reduced turbulence intensities.

We calculate the average swimming velocity and dispersion rate characterizing the transport of swimming gyrotactic micro-organisms suspended in homogeneous (simple) shear. These are requisite effective phenomenological coefficients for the macroscale continuum modelling of bioconvection and related collective-dynamics phenomena. The swimming cells are modelled as rigid axisymmetric dipolar particles subject to stochastic Brownian rotations. Calculations are effected via application of the generalized Taylor dispersion scheme. Attention is focused on finite (as opposed to weak) shear. Results indicate that the largest transverse average swimming velocities (essential to gyrotactic focusing) appear shortly after transition from the ‘tumbling’ mode of motion to cells swimming in the equilibrium direction. At sufficiently large shear rates, dispersivity is not monotonically decreasing with external-field intensity. Exceptional dispersion rates which are unique to non-spherical cells appear in the ‘intermediate domain’ of external fields. These are rationalized in terms of the corresponding deterministic problem (i.e. in the absence of diffusion) when cell rotary motion is governed by the simultaneous coexistence of multiple stable attractors.

The three-dimensional structure of the flow behind a heaving and pitching finite-span wing is investigated using dye flow visualization at a Reynolds number of 164. Phase-locked image sequences, which are obtained from two orthogonal views, are combined to create a set of composite images that give an overall sense of the three-dimensional structure of the flow. A model of the vortex system behind the wing is constructed from the image sequences. Variations of the Strouhal number, pitch amplitude and heave/pitch phase angle are qualitatively shown to affect the structure of the wake.

In an under-expanded supersonic jet, the interaction between shock-cell structure and vortices in a mixing layer generates intense tonal noise, called jet screech. This noise generation can be explained as a shock-leakage process through an unsteady vortex-laden mixing layer. This paper studies the shock-leakage mechanism based on a geometrical theory and direct numerical simulation (DNS) in two dimensions. In the limit of weak shocks, the analysis becomes analogous to geometrical acoustics: the eikonal equation demonstrates that shock waves tend to leak near the saddle points between vortices. Analysing the wavenumber vector, it is shown that the local vorticity behaves as a barrier against shocks. Using the unsteady DNS data, trajectories of the shock fronts are computed with the time dependent eikonal equation. Furthermore, the interaction between unsteady vortices and a compression wave is solved using DNS. The geometrical theory shows good agreement with DNS for shock-front evolution, but the amplitude of the leaked waves agrees only qualitatively. This study also investigates the effects of a temperature difference across the mixing layer. The analysis based on total internal reflection and the numerical results of both geometrical acoustics and DNS indicate that the direction of the radiated shock noise tends to rotate downstream as the jet temperature increases.

The stability of a separating boundary-layer flow at the rear of a two-dimensional bump mounted on a flat plate is numerically investigated. Above a critical Reynolds number, the flow field is shown to undergo self-sustained two-dimensional low-frequency fluctuations in the upstream region of the separation bubble, evolving into aperiodic vortex shedding further downstream. The computed steady flow states below the critical Reynolds number are shown to be convectively unstable. On extrapolating the flow field to Reynolds numbers above criticality, some evidence is found that the onset of the oscillatory behaviour coincides with topological flow changes near the reattachment point leading to the rupture of the (elongated) recirculation bubble. The structural changes near reattachment are shown to trigger an abrupt local transition from convective to absolute instability, at low frequencies. On preventing the separation bubble from bursting by reaccelerating the flow by means of a second bump further downstream, the separated flow remains steady for increasing Reynolds numbers, until a local region of absolute instability in the upper part of the geometrically controlled recirculation bubble is produced. The resulting global instability consists of self-sustained nonlinear saturated perturbations oscillating at a well-defined frequency, very distinct from the the low-frequency motion of the elongated recirculation bubble in the single-bump geometry. A frequency selection criterion based on local absolute frequencies, which has been successfully applied to wake flows, is shown to accurately predict the global frequency.

The baroclinic instability of a zonal current on the beta-plane is studied in the context of the two-layer model when the shear of the basic current is a periodic function of time. The basic shear is contained in a zonal channel and is independent of the meridional direction. The instability properties are studied in the neighbourhood of the classical steady-shear threshold for marginal stability. It is shown that the linear problem shares common features with the behaviour of the well-known Mathieu equation. That is, the oscillatory nature of the shear tends to stabilize an otherwise unstable current while, on the contrary, the oscillation is able to destabilize a current whose time-averaged shear is stable. Indeed, this parametric instability can destabilize a flow that at every instant possesses a shear that is subcritical with respect to the standard stability threshold. This is a new source of growing disturbances. The nonlinear problem is studied in the same near neighbourhood of the marginal curve. When the time-averaged flow is unstable, the presence of the oscillation in the shear produces both periodic finite-amplitude motions and aperiodic behaviour. Generally speaking, the aperiodic behaviour appears when the amplitude of the oscillating shear exceeds a critical value depending on frequency and dissipation. When the time-averaged flow is stable, i.e. subcritical, finite-amplitude aperiodic motion occurs when the amplitude of the oscillating part of the shear is large enough to lift the flow into the unstable domain for at least part of the cycle of oscillation. A particularly interesting phenomenon occurs when the time-averaged flow is stable and the oscillating part is too small to ever render the flow unstable according to the standard criteria. Nevertheless, in this regime parametric instability occurs for ranges of frequency that expand as the amplitude of the oscillating shear increases. The amplitude of the resulting unstable wave is a function of frequency and the magnitude of the oscillating shear. For some ranges of shear amplitude and oscillation frequency there exist multiple solutions. It is suggested that the nature of the response of the finite-amplitude behaviour of the baroclinic waves in the presence of the oscillating mean flow may be indicative of the role of seasonal variability in shaping eddy activity in both the atmosphere and the ocean.

Steep forced water waves generated by moving a rectangular tank are investigated both experimentally and numerically. Our main focus is on energetic events generated by two different types of external forcing. Horizontal motions are arranged to give wave impact on the sidewall. Steep standing waves forced by vertical acceleration can result in spectacular breaking modes similar to, and more energetic than, those reported by Jiang, Perlin & Schultz (1998, hereinafter J98). Among them we find thin sheets derived from sharp-crested waves, (‘mode A’ of J98) and the ‘flat-topped’ crest or ‘table-top’ breaker (‘mode B’ of J98). We report here on experimental observations of ‘table-top’ breakers showing remarkably long periods of free fall motion. Previously such breakers have only been observed in numerical computations. Both types of breakers often thin as they fall to give thin vertical sheets of water whose downward motion ends in either a small depression and a continuing smooth surface, or air entrainment to appreciable depths. Experimental results are compared graphically with numerical results of two theoretical models. One is an extended set of Boussinesq equations following Wei et al. (1995), which are successful up to wave slopes of O(1). The other numerical comparison is with a fully nonlinear irrotational flow solver (Dold 1992) which can follow the waves to breaking.

It is well known that the drag in a turbulent flow of a polymer solution is significantly reduced compared to Newtonian flow. Here we consider this phenomenon by means of a direct numerical simulation of a turbulent channel flow. The polymers are modelled as elastic dumbbells using the FENE-P model. In the computations the polymer model is solved simultaneously with the flow equations, i.e. the polymers are deformed by the flow and in their turn influence the flow structures by exerting a polymer stress. We have studied the results of varying the polymer parameters, such as the maximum extension, the elasticity and the concentration. For the case of highly extensible polymers the results of our simulations are very close to the maximum drag reduction or Virk (1975) asymptote. Our simulation results show that at approximately maximum drag reduction the slope of the mean velocity profile is increased compared to the standard logarithmic profile in turbulent wall flows. For the r.m.s. of the streamwise velocity fluctuations we find initially an increase in magnitude which near maximum drag reduction changes to a decrease. For the velocity fluctuations in the spanwise and wall-normal directions we find a continuous decrease as a function of drag reduction. The Reynolds shear stress is strongly reduced, especially near the wall, and this is compensated by a polymer stress, which at maximum drag reduction amounts to about 40% of the total stress. These results have been compared with LDV experiments of Ptasinski et al. (2001) and the agreement, both qualitatively and quantitatively, is in most cases very good. In addition we have performed an analysis of the turbulent kinetic energy budgets. The main result is a reduction of energy transfer from the streamwise direction, where the production of turbulent kinetic energy takes place, to the other directions. A substantial part of the energy production by the mean flow is transferred directly into elastic energy of the polymers. The turbulent velocity fluctuations also contribute energy to the polymers. The elastic energy of the polymers is subsequently dissipated by polymer relaxation. We have also computed the various contributions to the pressure fluctuations and identified how these change as a function of drag reduction. Finally, we discuss some cross-correlations and various length scales. These simulation results are explained here by two mechanisms. First, as suggested by Lumley (1969) the polymers damp the cross-stream or wall-normal velocity fluctuations and suppress the bursting in the buffer layer. Secondly, the ‘shear sheltering’ mechanism acts to amplify the streamwise fluctuations in the thickened buffer layer, while reducing and decoupling the motions within and above this layer. The expression for the substantial reduction in the wall drag derived by considering the long time scales of the nonlinear fluctuations of this damped shear layer, is shown to be consistent with the experimental data of Virk et al. (1967) and Virk (1975).

We present experimental measurements of the normal stresses in sheared Stokesian suspensions. Though the suspending fluid is Newtonian, dispersing rigid non-Brownian particles in it yields a suspension that is non-Newtonian, as it exhibits normal stress differences and an excess isotropic pressure in viscometric flows. At small to moderate concentrations, the normal stresses are very small in magnitude, and hence difficult to measure. This difficulty is compounded by the presence of noise due to unavoidable experimental artifacts. Owing to these limitations, most measurements reported earlier were carried out at relatively high particle concentrations, and some at shear rates large enough that the effects of particle and fluid inertia may have been significant. In our study, we have used a novel technique to measure the small stress levels. This was achieved by applying a sinusoidally varying shear rate with a fixed (low) frequency superimposed on a constant shear rate, and using a lock-in amplifier to measure the Fourier component of the same frequency in the stress signal. We have measured normal stresses in cylindrical-Couette and parallel-plate geometries, and combined these measurements to determine the two normal stress differences for particle volume fractions in the range 0.3–0.45. While the normal stresses are very small at low concentrations, they rise rapidly with increasing concentration. The normal stresses vary linearly with the magnitude of the shear rate, and are independent of its sign. In contrast to polymeric solutions, both normal stress differences are negative, and the first normal stress difference is significantly smaller in magnitude. We compare our data with the results of earlier studies, and observe good agreement.

Existing results on the linearized water-wave problem for a homogeneous fluid are extended to the case of a two-layer fluid. In particular, the appropriate form of Maz'ya's identity is presented and used to obtain results on the uniqueness of the solutions to forcing problems for a structure in a two-layer fluid. Further, examples of geometries are constructed for which trapped modes occur; such modes are finite-energy solutions of the unforced problem and provide examples of non-uniqueness.

Chandrasekhar (1961) extensively investigated the linear dynamics of Rayleigh–Bénard convection in an electrically conducting fluid exposed to a uniform vertical magnetic field and enclosed by rigid, stress-free, upper and lower boundaries. He determined the marginal stability boundary and critical horizontal wavenumbers for the onset of convection as a function of the Chandrasekhar number $Q$ or Hartmann number squared. No closed-form formulae appeared to exist and the results were tabulated numerically. We have discovered simple expressions that concisely describe the stability properties of the system. When the Prandtl number $Pr$ is greater than or equal to the magnetic Prandtl number $Pm$ the marginal stability boundary is described by the curve $Q = \pi^{-2}[R -R_c^{1/3}R^{2/3}]$ where $R$ is the Rayleigh number and $R_c = (27/4)\pi^4$ is Rayleigh's famous critical value for the onset of stationary convection in the absence of a magnetic field ($Q = 0$). When $Pm > Pr$ the marginal stability boundary is determined by this curve until intersected by the curve[Q = \frac{1}{\pi^2}\left[\frac{\Pm^2(1+\Pr)}{\Pr^2(1+\Pm)}R \right. -\left. \left(\frac{(1+\Pr)(\Pr+\Pm)}{\Pr^2}\right)^{1/3}\left( \frac{\Pm^2(1+\Pr)}{\Pr^2(1+\Pm)} \right)^{2/3} R_c^{1/3}R^{2/3}\right].]An expression for the intersection point is derived and also for the critical horizontal wavenumbers for which instability sets in along the marginal stability boundary either as stationary convection or in an oscillatory fashion. A simple formula is derived for the frequency of the oscillations. Also we show that in the limit of vanishing magnetic diffusivity, or infinite electrical conductivity, the system is unstable for sufficiently large $R$. Instability in this limit always sets in via overstability.