An isolated Leidenfrost droplet levitating over its own vapour above a superheated flat substrate is considered theoretically, the superheating for water being up to several hundred degrees above the boiling temperature. The focus is on the limit of small, practically spherical droplets of several tens of micrometres or less. This may occur when the liquid is sprayed over a hot substrate, or just be a late life stage of an initially large Leidenfrost droplet. A rigorous numerically assisted analysis is carried out within verifiable assumptions such as quasi-stationarities and small Reynolds/Péclet numbers. It is considered that the droplet is surrounded by its pure vapour. Simple formulae approximating our numerical data for the forces and evaporation rates are preliminarily obtained, all respecting the asymptotic behaviours (also investigated) in the limits of small and large levitation heights. They are subsequently used within a system of ordinary differential equations to study the droplet dynamics and take-off (drastic height increase as the droplet vapourises). A previously known quasi-stationary inverse-square-root law for the droplet height as a function of its radius (at the root of the take-off) is recovered, although we point out different prefactors in the two limits. Deviations of a dynamic nature therefrom are uncovered as the droplet radius further decreases due to evaporation, improving the agreement with experiment. Furthermore, we reveal that, if initially large enough, the droplets vanish at a universal finite height (just dependent on the superheat and fluid properties). Scalings in various distinguished cases are obtained along the way.

An exact solution is developed for bubble-induced acoustic microstreaming in the case of a gas bubble undergoing asymmetric oscillations. The modelling is based on the decomposition of the solenoidal, first- and second-order, vorticity fields into poloidal and toroidal components. The result is valid for small-amplitude bubble oscillations without restriction on the size of the viscous boundary layer $(2\nu /\omega )^{1/2}$ in comparison to the bubble radius. The non-spherical distortions of the bubble interface are decomposed over the set of orthonormal spherical harmonics $Y_{n}^{m}(\theta , \phi )$ of degree $n$ and order $m$. The present theory describes the steady flow produced by the non-spherical oscillations $(n,\pm m)$ that occur at a frequency different from that of the spherical oscillation, as in the case of a parametrically excited surface oscillation. The three-dimensional aspect of the streaming pattern is revealed as well as the particular flow signatures associated with different asymmetric oscillations.

Oscillatory flows induced by a monochromatic forcing frequency $\omega$ close to a planar surface are present in many applications involving fluid–matter interaction such as ultrasound, vibrational spectra by microscopic pulsating cantilevers, nanoparticle oscillatory magnetometry, quartz crystal microbalance and more. Numerical solution of these flows using standard time-stepping solvers in finite domains present important drawbacks. First, hydrodynamic finite-size effects scale as $1/L_{\parallel }^2$ close to the surface and extend several times the penetration length $\delta \sim \omega ^{-1/2}$ in the normal $z$ direction and second, they demand rather long transient times $O(L_z^2)$ to allow vorticity to diffuse over the computational domain. We present a new frequency-based scheme for doubly periodic (DP) domains in free or confined spaces which uses spectral-accurate solvers based on fast Fourier transform in the periodic $(xy)$ plane and Chebyshev polynomials in the aperiodic $z$ direction. Following the ideas developed for the steady Stokes solver (Hashemi et al. J. Chem. Phys. vol. 158, 2023, p. 154101), the computational system is decomposed into an ‘inner’ domain (where forces are imposed) and an outer domain (where the flow is solved analytically using plane-wave expansions). Matching conditions leads to a solvable boundary value problem. Solving the equations in the frequency domain using complex phasor fields avoids time-stepping and permits a strong reduction in computational time. The spectral scheme is validated against analytical results for mutual and self-mobility tensors, including the in-plane Fourier transform of the Green function. Hydrodynamic couplings are investigated as a function of the periodic lattice length. Applications are finally discussed.

A stochastic wavevector approach is formulated to accurately represent compressible turbulence subject to rapid deformations. This approach is inspired by the incompressible particle representation model of Kassinos & Reynolds (1994), and preserves the exact nature of compressible rapid distortion theory (RDT). The adoption of a stochastic – rather than Fourier – perspective simplifies the transformation of statistics to physical space and serves as a starting point for the development of practical turbulence models. We assume small density fluctuations and isentropic flow to obtain a transport equation for the pressure fluctuation. This results in four fewer transport equations compared with the compressible RDT model of Yu & Girimaji (Phys. Fluids, vol. 19, 2007, 041702). The final formulation is closed in spectral space and only requires numerical approximation for the transformation integrals. The use of Monte Carlo for unit wavevector integration motivates the representation of the moments as stochastic variables. Consistency between the Fourier and stochastic representation is demonstrated by showing equivalency between the evolution equations for the velocity spectrum tensor in both representations. Sample clustering with respect to orientation allows for different techniques to be used for the wavevector magnitude integration. The performance of the stochastic model is evaluated for axially compressed turbulence, serving as a simplified model for shock–turbulence interaction, and is compared with linear interaction approximations and direct numerical simulation (DNS). Pure and compressed sheared turbulence at different distortion Mach numbers are also computed and compared with RDT/DNS data. Finally, two additional deformations are applied and compared with solenoidal and pressure-released limits to demonstrate the modelling capability for generic rapid deformations.

Travelling wave charges lying on the insulating walls of an electrolyte-filled capillary give rise to oscillatory modes which vanish when averaged over the period of oscillation. They also give rise to a zero mode (a unidirectional, time-independent velocity component) which does not vanish. The latter is a nonlinear effect caused by continuous symmetry breaking due to the quadratic nonlinearity associated with the electric body force in the time-dependent Stokes equations. In this paper, we provide a unified view of the effects arising in boundary-driven electrokinetic flows (travelling wave electroosmosis) and establish the universal behaviour exhibited by the observables. We show that the incipient velocity profiles are self-similar implying that those obtained with a single experimental configuration can be employed again to attain further insights without the need of repeating the experiment. Certain results from the literature are recovered as special cases of our formulation and we resolve certain paradoxes having appeared in the past. We present simple theoretical expressions, depending on a single-fit parameter, that reproduce these profiles, which could thus provide a rapid test of consistency between our theory and future experiment. The effect becomes more pronounced when reducing the transverse dimension of the system, relative to the velocity direction, and increasing the excitation wavelength, and can therefore be employed for unidirectional transport of electrolytes in thin and long capillaries. General relations, expressing the zero mode velocity in terms of the electric potential and the geometry of the system only, can thus be easily adapted to alternative experimental settings.

Large-eddy simulation (LES) is performed to study the tip vortex flow in a ducted propulsor geometry replicating the experiments of Chesnakas & Jessup (2003, pp. 257–267), Oweis et al. (2006a J. Fluids Engng 128, 751–764) and Oweis et al. (2006b J. Fluids Engng 128, 751–764). Inception of cavitation in these marine propulsion systems is closely tied to the unsteady interactions between multiple vortices in the tip region. Here LES is used to shed insight into the structure of the tip vortex flow. Simulation results are able to predict experimental propeller loads and show agreement with laser Doppler velocimetry measurements in the blade wake at design advance ratio, $J=0.98$. Results show the pressure differential across the blade produces a leakage vortex which separates off the suction side blade tip upstream of the trailing edge. The separation sheet aft of the primary vortex separation point is shown to take the form of a skewed shear layer which produces a complex arrangement of unsteady vortices corotating and counter-rotating with the primary vortex. Blade tip boundary layer vortices are reoriented to align with the leakage flow and produce instantaneous low-pressure regions wrapping helically around the primary vortex core. Such low-pressure regions are seen both upstream and downstream of the propeller blade trailing edge. The trailing edge wake is found to only rarely have a low-pressure vortex core. Statistics of instantaneous low pressures below the minimum mean pressure are found to be concentrated downstream of the blade’s trailing edge wake crossing over the primary vortex core and continue in excess of 40 % chord length behind the trailing edge. The rollup of the leakage flow duct boundary layer behind the trailing edge is also seen to produce counter-rotating vortices which interact with the primary leakage vortex and contribute to strong stretching events.