An analysis of thermoacoustic instability is performed for a horizontal Rijke tube with an electrical resistance heater as the heat source. The governing equations for this fluid flow become stiff and are difficult to solve by the computational fluid dynamics (CFD) technique, as the Mach number of the steady flow and the thickness of the heat source (compared to the acoustic wavelength) are small. Therefore, an asymptotic analysis is performed in the limit of small Mach number and compact heat source to eliminate the above stiffness problem. The unknown variables are expanded in powers of Mach number. Two systems of governing equations are obtained: one for the acoustic field and the other for the unsteady flow field in the hydrodynamic zone around the heater. In this analysis, the coupling between the acoustic field and the unsteady heat release rate from the heater appears from the asymptotic analysis. Furthermore, a non-trivial additional term, referred to as the global-acceleration term, appears in the momentum equation of the hydrodynamic zone, which has serious consequences for the stability of the system. This term can be interpreted as a pressure gradient applied from the acoustic onto the hydrodynamic zone. The asymptotic stability of the system with the variation of system parameters is presented using the bifurcation diagram. Numerical simulations are performed using the Galerkin technique for the acoustic zone and CFD techniques for the hydrodynamic zone. The results confirm the importance of the global-acceleration term. Bifurcation diagrams obtained from the simulations with and without the above term are different. Acoustic streaming is shown to occur during the limit cycle and its effect on the unsteady heat release rate is discussed.

We investigate the organization of the momentum-carrying eddies in turbulent Couette–Poiseuille flows. The study relies on a direct numerical simulation (DNS) database covering a wide range of flow configurations from pure Couette to pure Poiseuille flows, at Reτ ≈ 250 (based on the flow properties at the stationary wall). The study highlights the occurrence of streaky patterns of alternating high and low momentum throughout the channel for all flow configurations, except near zeros of the mean shear, where streaks are suppressed. The mean streak spacing shows a relatively universal distribution in the core of the channel, where it ranges from 50 to 100 local viscous units. The validity of the local viscous scaling in collapsing flow features at different wall distances is confirmed by the analysis of the spanwise velocity spectra, which also highlights (in the case of Couette-like flows) the onset of a secondary low-wavenumber flow mode, superposed on the high-wavenumber flow mode that is responsible for the inner-layer dynamics. The effect of the former mode on the latter is studied by means of the two-point amplitude modulation coefficient, which brings to light a nonlinear modulation phenomenon. Physical mechanisms to explain the modulation effect are proposed, based on the interpretation of the conditional average events. Note that, although similar mechanisms have been previously observed in high-Reynolds-number turbulent boundary layers and channels, the modulation effect is here rather associated with the intrinsic large-scale dynamics of Couette-like flows, and takes place at DNS-accessible Reynolds numbers. We thus believe that the study of Couette-like flows may give an alternative avenue for probing inner/outer interaction effects than canonical channel flows.

The classical plume model due to Morton, Taylor & Turner (Proc. R. Soc. Lond. A, vol. 234, 1956, pp. 1–23) is re-cast in terms of the non-dimensional plume radius, the plume ‘laziness’ defined as the squared ratio of the source radius and the jet length, and the buoyancy flux. It is shown that many of the key results of this classical model can then be read straight from the equations without recourse to solving them. Based on this observation, derivative models that consider plumes propagating through stratified environments or undergoing chemical reactions are similarly re-cast. We show again that key results can be read straight from the governing equations and results that have previously only been demonstrated numerically can be found analytically. In particular, we unify two previously distinct models that consider plumes propagating through stable and unstable stratified environments whose stratification has a power-law dependence on height. We present analytical solutions for the range of stratification power-law decay rates for which straight-sided plumes are possible. This result unifies the sets of solutions by Batchelor (Q. J. R. Meteorol. Soc., vol. 80, 1954, pp. 339–358) and Caulfield & Woods (J. Fluid Mech., vol. 360, 1998, pp. 229–248). We are able to explain the unstable behaviour previously found when the power lies in the range (−4, −8/3). Finally we show that this method also has limited advantages when applied to plumes with unsteady source conditions.

In this study, we propose a non-linear continuous stochastic velocity process for simulations of monatomic gas flows. The model equation is derived from a Fokker–Planck approximation of the Boltzmann equation. By introducing a cubic non-linear drift term, the model leads to the correct Prandtl number of 2/3 for monatomic gas, which is crucial to study heat transport phenomena. Moreover, a highly accurate scheme to evolve the computational particles in velocity- and physical space is devised. An important property of this integration scheme is that it ensures energy conservation and honours the tortuosity of particle trajectories. Especially in situations with small to moderate Knudsen numbers, this allows to proceed with much larger time steps than with direct simulation Monte Carlo (DSMC), i.e. the mean collision time not necessarily has to be resolved, and thus leads to more efficient simulations. Another computational advantage is that no direct collisions have to be calculated in the proposed algorithm. For validation, different micro-channel flow test cases in the near continuum and transitional regimes were considered. Detailed comparisons with DSMC for Knudsen numbers between 0.07 and 2 reveal that the new solution algorithm based on the Fokker–Planck approximation for the collision operator can accurately predict molecular stresses and heat flux and thus also gas velocity and temperature profiles. Moreover, for the Knudsen Paradox, it is shown that good agreement with DSMC is achieved up to a Knudsen number of about 5.

We compare the results of two-dimensional, biased random walk models of individual swimming micro-organisms with advection–diffusion models for the whole population. In particular, we consider the influence of the local flow environment (gyrotaxis) on the resulting motion. In unidirectional flows, the results of the individual and population models are generally in good agreement, even in flows in which the cells can experience a range of shear environments, and both models successfully predict the phenomena of gravitactic focusing. Numerical results are also compared with asymptotic expressions for weak and strong shear. Discrepancies between the models arise in two cases: (i) when reflective boundary conditions change the orientation distribution in the random walk model from that predicted by the long-term asymptotics used to derive the advection–diffusion model; (ii) when the spatial and temporal scales are not large enough for the advection–diffusion model to apply. We also use a simple two-dimensional flow containing a variety of flow regimes to explore what happens when there are localized regions in which the generalized Taylor dispersion theory used in the derivation of the population model does not apply. For spherical cells, we find good agreement between the models outside the ‘break-down’ regions, but comparison of the results within these regions is complicated by the presence of nearby boundaries and their influence on the random walk model. In contrast, for rod-shaped cells which are reorientated by both vorticity and strain, we see qualitatively different spatial patterns between individual and advection–diffusion models even in the absence of gyrotaxis, because cells are advected between regions of differing rates of strain.

The flow at the leading edge and in the interior of a rectangular porous obstruction is described through experiments and scaling. The porous obstruction consists of an emergent, rectangular array of cylinders in shallow flow, a configuration that mimics aquatic vegetation. The main features of the flow depend upon the non-dimensional canopy flow-blockage, which is a function of the obstruction width and porosity. For the ranges of canopy flow-blockage tested in this paper, the fluid decelerates upstream of the obstruction over a length scale proportional to the array width. For high flow-blockage, the interior adjustment length within the porous obstruction is set by the array width. For low flow-blockage, the array's frontal area per unit volume sets the interior adjustment length. Downstream of the adjustment regions, the interior velocity is governed by a balance between the lateral divergence of the turbulent stress and canopy drag, or by a balance between the pressure gradient and canopy drag, depending on the lateral penetration into the array of Kelvin–Helmholtz (KH) vortices, which is set by the non-dimensional canopy flow-blockage. For a porous obstruction with two stream-parallel edges, the KH vortex streets along the two edges are in communication across the width of the array: a phenomenon that results in cross-array vortex organization, which significantly enhances the vortex strength and creates significant lateral transport within the porous obstruction.

We report observations of air bubble entrapment when a solid sphere impacts a quiescent liquid surface. Using high-speed imaging, we show that a small amount of air is entrapped at the bottom tip of the impacting sphere. This phenomenon is examined across a broad range of impact Reynolds numbers, 0.2 ≤ Re = (DU0/νl) ≤ 1.2 × 105. Initially, a thin air pocket is formed due to the lubrication pressure in the air layer between the sphere and the liquid surface. As the liquid surface deforms, the liquid contacts the sphere at a finite radius, producing a thin sheet of air which usually contracts to a nearly hemispherical bubble at the bottom tip of the sphere depending on the impact parameters and liquid properties. When a bubble is formed, the final bubble size increases slightly with the sphere diameter, decreases with impact speed but appears independent of liquid viscosity. In contrast, for the largest viscosities tested herein, the entrapped air remains in the form of a sheet, which subsequently deforms upon close approach to the base of the tank. The initial contact diameter is found to conform to scalings based on the gas Reynolds number whilst the initial thickness of the air pocket or ‘dimple’ scales with a Stokes' number incorporating the influence of the air viscosity, sphere diameter and impact speed and liquid density.