The problem of gas motion inside a resonance tube, closed at one end by a plug and fitted at the other with an oscillating piston is treated analytically and numerically. An analytical model is derived for arbitrary piston motion and gas oscillations about the first resonance frequency, where the gas flow is characterized by a shock wave travelling periodically back and forth in the tube. The model is obtained by a perturbation analysis in terms of a small-amplitude parameter $\varepsilon$. All the hydrodynamic properties of the gas are predicted with accuracy up to the second-order terms of $\varepsilon$. Isentropic and adiabatic problem formulations are addressed. Expressions for spatial distributions of the time-averaged hydrodynamic gas properties are derived for any frequency within the resonance band. It is shown that they are determined by the gas adiabatic exponent $\gamma$ and the law that governs the motion of the piston. The analytical model is verified by comparison with a numerical solution, showing good agreement.

Rarefied gas flow through a thin orifice is studied on the basis of the direct simulation Monte Carlo method. The mass flow rate and the flow field are calculated over the whole range of the Knudsen number for various values of the pressure ratio. It is found that at all values of the pressure ratio a significant variation of the flow rate occurs in the transition regime between the free-molecular and hydrodynamic regimes. In the hydrodynamic regime the flow rate tends to a constant value. In the case of finite pressure ratio the flow field qualitatively differs from that for outflow into vacuum, namely vortices appear in the downflow container on approaching the hydrodynamic regime. Then, in the hydrodynamic regime the gas flow forms a strong jet. A comparison of the numerical results with experimental data available in the open literature has been performed.

In this paper, the sedimentation of a sphere and its radial migration in a Poiseuille flow in a vertical tube filled with a Newtonian fluid are simulated with a finite-difference-based distributed Lagrange multiplier (DLM) method. The flow features, the settling velocities, the trajectories and the angular velocities of the spheres sedimenting in a tube at different Reynolds numbers are presented. The results show that at relatively low Reynolds numbers, the sphere approaches the tube axis monotonically, whereas in a high-Reynolds-number regime where shedding of vortices takes place, the sphere takes up a spiral trajectory that is closer to the tube wall than the tube axis. The rotation motion and the lateral motion of the sphere are highly correlated through the Magnus effect, which is verified to be an important (but not the only) driving force for the lateral migration of the sphere at relatively high Reynolds numbers. The standard vortex structures in the wake of a sphere, for Reynolds number higher than 400, are composed of a loop mainly located in a plane perpendicular to the streamwise direction and two streamwise vortex pairs. When moving downstream, the legs of the hairpin vortex retract and at the same time a streamwise vortex pair with rotation opposite to that of the legs forms between the loops. For Reynolds number around 400, the wake structures shed during the impact of the sphere on the wall typically form into streamwise vortex structures or else into hairpin vortices when the sphere spirals down. The radial, angular and axial velocities of both neutrally buoyant and non-neutrally buoyant spheres in a circular Poiseuille flow are reported. The results are in remarkably good agreement with the available experimental data. It is shown that suppresion of the sphere rotation produces significant large additional lift forces pointing towards the tube axis on the spheres in the neutrally buoyant and more-dense-downflow cases, whereas it has a negligible effect on the migration of the more dense sphere in upflow.

The interaction of an isolated spherical particle with an isotropic turbulent flow is considered using direct numerical simulations (DNS). The particle Reynolds number is varied from about 50 to 600 and the particle diameter is varied from about 1.5 to 10 times the Kolmogorov scale. The Reynolds number based on the Taylor microscale of the free-stream turbulent field considered here is 164. The DNS technique employed here is the first of its kind to address particle–turbulence interaction and it resolves the smallest scales in the free-stream turbulent flow and the complex vortical structures in the particle wake. The primary objective of this paper is to present new results on the effect of the free-stream turbulence on the particle wake and vortex shedding, and the modulation of free-stream turbulence in the particle wake. The parameters of the present simulations are comparable to those of the experimental study by Wu & Faeth (1994$a$, $b$), and agreement between the present computational results and the experimental measurement is demonstrated.

The effect of free-stream turbulence on the mean and instantaneous wake structure is studied. The time-averaged mean wake in a turbulent ambient flow shows a lower velocity deficit and a flatter profile. However, in agreement with the experimental results of Wu & Faeth the mean wake in a turbulent flow behaves like a self-preserving laminar wake. At Reynolds numbers below about 210 the effect of free-stream turbulence is to introduce wake oscillations. For Reynolds numbers in the range 210 to 280, free-stream turbulence is observed to promote early onset of vortex shedding. The nature of the shed vortices is somewhat different from that in a uniform flow. Increasing the free-stream turbulence intensity suppresses the process of vortex shedding, and only marginally increases the wake oscillation. The modulation of free-stream turbulence in the wake is studied in terms of the distribution of kinetic energy and RMS velocity fluctuation. The free-stream energy lost in the wake is recovered faster in a turbulent ambient flow than in a uniform ambient flow. The energy of the velocity fluctuation is enhanced in the wake at low free-stream intensities, and is damped or marginally increased at higher intensities. The fluctuation energy is not equi-partitioned among the streamwise and cross-stream components. The RMS streamwise fluctuation is always enhanced, whereas the RMS cross-stream fluctuation is enhanced only at low free-stream intensities, and damped at higher intensities.

An analysis is made of steady-state flow of a compressible fluid in an infinite rapidly rotating pipe. Flow is induced by imposing a small azimuthally varying thermal forcing at the pipe wall. The Ekman number is small. Analyses are conducted to reveal both the axisymmetric-type and non-axisymmetric-type solutions. The axisymmetric solution is based on the azimuthally averaged wall boundary condition. The non-axisymmetric solution stems from the azimuthally fluctuating part of the wall boundary condition. It is shown that the two-dimensional (uniform in the axial direction) non-axisymmetric solution exists for $\sigma (\gamma - 1)M^2 \,{\gg}\, O(E^{1 / 3})$. However, an axially dependent solution is found if $\sigma (\gamma - 1)M^2 \,{\lesssim}\,O(E^{1 / 3})$, in which $E$ denotes the Ekman number, $M$ the Mach number, $\gamma $ the specific heat ratio and $\sigma $ the Prandtl number. The axisymmetric solution prevails over the whole flow region; the two-dimensional non-axisymmetric solution is confined to the near-wall thermal layer of thickness $O(E^{1 / 3})$. As a canonical example, a detailed description is given for the case of a highly conducting wall with differential heating.

Weis & Hutter (J. Fluid Mech. vol. 476, 2003, p. 63) obtained an implicit algebraic Reynolds stress model from a differential Reynolds stress transport equation valid in an arbitrarily rotating time-dependent coordinate frame (relative to an inertial system). Although the form of this implicit algebraic equation differed from previous implicit forms, its correctness was argued based on the objective tensor form of the implicit algebraic equation. It is shown here that such conclusions based on simple coordinate frame transformations are incomplete, and that additional considerations taking into account flow rotation and curvature, for example, are necessary. By properly accounting for both the arbitrary motions of the observer coordinate frame as well as the motion of the flow itself, it is shown that previous formulations and application of the weak-equilibrium condition are correct in contrast to the results of Weis & Hutter.

The trajectories of 20 and 40 $\umu$m radius water droplets colliding with a gas–water interface are observed to determine the conditions under which drops will bounce or coalesce on impact and the apparent coefficient of restitution of the drops that bounce. The experiments were performed in a pressure chamber so that the pressure and composition of the gas could be varied to explore the effects of the viscosity and mean-free path of the gas. The impact velocity is varied by producing drops with a velocity larger than their terminal velocity using a piezoelectric drop generator and adjusting the distance between the generator and the liquid interface. The geometry of the collisions is axisymmetric and the Weber number, $\hbox{\it We}\,{=}\,\rho_{l}U^{2}a/\sigma$, is O (1) or smaller, so as to facilitate comparison with a theory for weakly deformable gas–liquid interfaces. Here, $\rho_{l}$ is the liquid density, $U$ the impact velocity, $a$ the drop radius and $\sigma$ the surface tension. After a low-Weber-number drop coalesces, a smaller, daughter drop is emitted from the interface with a velocity higher than the incident velocity of the mother drop. The daughter drop radius is about 0.55$a$ and the daughter velocity is 0.38($\sigma /(\rho_{l}a))^{1/2}$ for We$\,{<}\,0.01$.

The experimental results are compared with a theory in which the small deformations of the drop and surface are expanded in Legendre polynomials and Fourier modes, respectively, the non-continuum lubrication stresses are computed in the thin gas film between the drop and interface, and the liquid flow is approximated as an inviscid potential flow. The coefficient of restitution decreases with increasing Weber number and becomes insensitive to the viscosity of the gas at Weber numbers larger than about 1. At smaller Weber numbers, drops in a less viscous gas lose less energy during the collision. Drops are observed to undergo a transition from coalescence to bouncing as the drop velocity (Weber number) is increased. However, the marginal condition for drop bouncing is much more sensitive to gas mean-free path (Knudsen number) and gas viscosity (Ohnesorge number) than to Weber number. The Knudsen and Ohnesorge numbers are defined as $\hbox{\it Kn}\,{=}\,{\lambda/a}$ and $\hbox{\it Oh}\,{=}\,\mu_{g}/(\rho_{l}a\sigma )^{1/2}$ where $\lambda $ is the mean-free path and $\mu_{g}$ is the gas viscosity. Theory and experiment show similar trends of increasing critical Weber number with decreasing Ohnesorge number and increasing Knudsen number. Theoretical results are also derived for the coalescence–bounce transition and coefficient of restitution for head-on collisions of equal sized drops.

This paper is concerned with flow-rate limitations in open capillary channels under low-gravity conditions. The channels consist of two parallel plates bounded by free liquid surfaces along the open sides. In the case of steady flow the capillary pressure of the free surface balances the differential pressure between the liquid and the surrounding constant-pressure gas phase. A maximum flow rate is achieved when the adjusted volumetric flow rate exceeds a certain limit leading to a collapse of the free surfaces.

In this study the steady one-dimensional momentum equation is solved numerically for perfectly wetting incompressible liquids to determine important characteristics of the flow, such as the free-surface shape and limiting volumetric flow rate. Using the ratio of the mean liquid velocity and the longitudinal small-amplitude wave speed a local capillary speed index $S_{ca}$ is introduced. A reformulation of the momentum equation in terms of this speed index illustrates that the volumetric flow rate is limited. The maximum flow rate is reached if $S_{ca}\,{=}\,1$ locally, a phenomenon called choking in compressible flows. Experiments with perfectly wetting liquids in the low-gravity environment of a drop tower and aboard a sounding rocket are presented where the flow rate is increased in a quasi-steady manner up to the maximum value. The experimental results are in very good agreement with the numerical predictions. Furthermore, the influence of the $S_{ca}$ on the flow-rate limit is confirmed.

Numerical experiments are performed to investigate the oscillatory flow over a two-dimensional wavy wall characterized by a large amplitude, such as to induce flow separation. Even though the Reynolds number is moderate, a three-dimensional turbulent flow is observed. The turbulence dynamics is characterized by the presence of coherent ribs superimposed on the main spanwise vortices generated by the roll-up of the free vortex sheets shed at the crests of the wall waviness. The ribs are formed by the stretching of vorticity patches which are generated by the instability of the two-dimensional flow at two different locations. The first are the saddle points of the flow field created, far from the wall, by the vortex pairs generated every half-cycle. The second are the saddle points created, close to the upstream side of the wavy wall, by the combined action of the free-stream flow and of the vortex structures shed by the ripple crests. Later, the ribs wrap around the main spanwise vortices and cause the distortion of these vortices and the alignement of the vortex lines with the free-stream flow, thus inducing large contributions to the coherent helicity. Simultaneously, regions of high dissipation appear which tend to be separated from those characterized by large values of helicity.

The goal of this study is to provide a window into the physics of extinction and reignition via three-dimensional simulations of non-premixed combustion in isotropic decaying turbulence using one-step global reaction and neglecting density variations. Initially non-premixed fields of fuel and oxidant are developing in a turbulent field. Due to straining, the scalar dissipation rate is initially increasing and its fluctuations create extinguished regions on the stoichiometric surface. Later in the process, the stoichiometric surface again becomes uniformly hot. Besides using Eulerian data, this research applies flame element tracking and investigates the time history of individual points (‘flame elements’) along the stoichiometric surface. The main focus of the study is the discussion of the different scenarios of reignition. This paper identifies three major scenarios: independent flamelet scenario, reignition via edge (triple) flame propagation, and reignition through engulfment by a hot neighbourhood. The results give insight into the role different scenarios play in the reignition process, reveal the physical processes associated with each scenario, and provide the relative frequency of reignition for each scenario.

We reconsider the three-dimensional boundary-layer flow of a power-law (Ostwald–de Waele) rheology fluid, driven by the rotation of an infinite rotating plane in an otherwise stationary system. Here we address the problem for both shear-thinning and shear-thickening fluids and show that there are some fundamental issues regarding the application of power-law models in a boundary-layer context that have not been mentioned in previous discussions. For shear-thickening fluids, the leading-order boundary-layer equations are shown to have no suitable decaying behaviour in the far field, and the only solutions that exist are necessarily non-differentiable at a critical location and of ‘finite thickness’. Higher-order effects are shown to regularize the singularity at the critical location. In the shear-thinning case, the boundary-layer solutions are shown to possess algebraic decay to a free-stream flow. This case is known from the existing literature; however here we shall emphasize the complexity of applying such solutions to a global flow, describing why they are in general inappropriate in a traditional boundary-layer context. Furthermore, previously noted difficulties for fluids that are highly shear thinning are also shown to be associated with the imposition of incorrect assumptions regarding the nature of the far-field flow. Based on Newtonian results, we anticipate the presence of non-uniqueness and through accurate numerical solution of the leading-order boundary-layer equations we locate several such solutions.

We present a study of the drag reduction induced by rigid fibres in a turbulent channel flow using direct numerical simulation. The extra stresses due to the fibres are calculated with the well-known constitutive equation involving the moments of the orientation vector. Drag reductions of up to 26% are calculated, with the largest drag reductions observed using non-Brownian fibres and semi-dilute concentrations. These findings suggest that elasticity is not necessary to achieve turbulent drag reduction. Flow statistics show trends similar to those observed in simulation of polymeric drag reduction: Reynolds stresses are reduced, velocity fluctuations in the wall-normal and spanwise directions are reduced while streamwise fluctuations are increased, and streamwise vorticity is reduced. We observe strong correlations between the fibre stresses and inter-vortex extensional flow regions. Based on these correlations and instantaneous visualizations of the flow field, we propose a mechanism for turbulent drag reduction by rigid fibre additives.

Thermal convection in rapidly rotating, self-gravitating Boussinesq fluid spheres is characterized by three parameters: the Rayleigh number $R$, the Prandtl number Pr and the Ekman number $E$. Two different asymptotic limits were considered in the previous studies of the linear problem. In the double limit $E \,{\ll}\, 1$ and $\hbox{\it Pr}/E \,{\gg}\, 1$, the local asymptotic theory showed that the convective motion is strongly non-axisymmetric, columnar, highly localized and described by the asymptotic scalings, $({1}/{s}) {\zpt}/{\zpt \phi} \,{=} {O}\big(E^{-1/3}\big),\; {\zpt}/{\zpt z} \,{=}\, {O}(1), \; R_c\,{=}\,{O}\big(E^{-1/3}\big)$, where $R_c$ denotes the critical Rayleigh number and $(s,\phi,z)$ are cylindrical polar coordinates with the axis of rotation at $s\,{=}\,0$. A global asymptotic theory with novel features for the limit $E \,{\ll}\, 1$ and $\hbox{\it Pr}/E \,{\gg}\, 1$, indicating the radial asymptotic scaling ${\zpt}/{\zpt s} \,{=}\, {O}\big(E^{-1/3}\big)$, was recently developed by Jones et al. (J. Fluid Mech. vol. 405, 2000, p. 157). In the different double limit $E \,{\ll}\, 1$ and $ \hbox{\it Pr}/E \,{\ll}\, 1 $, an asymptotic theory for the onset of convection building upon the theory of inertial waves was developed by Zhang (J. Fluid Mech. vol. 268, 1994 p. 211). It was shown that the convective motion at the leading-order approximation is represented by a single inertial-wave mode with a quadratic polynomial of $s$ and $z$, obeying the asymptotic dependence ${\zpt}/{\zpt s} \,{\sim}\, ({1}/{ s}) {\zpt}/{\zpt \phi} \,{=}\, {O}(1), {\zpt}/{\zpt z} \,{=}\, {O}(1)$ and $ R_c\,{=}\,{O}(E)$ for stress-free spheres.

There exist no simple asymptotic scalings for $ E \,{\ll}\, 1 $ appropriate to all values of $\hbox{\it Pr}/E$. For an arbitrary small but non-zero $E$, the highly localized convection spreads out spatially with decreasing $\hbox{\it Pr}$, suggesting that the scaling laws such as ${\zpt}/{\zpt s} \,{=}\, {O}\big(E^{-1/3}\big)$ are no longer valid when $\hbox{\it Pr}/E$ is not sufficiently large. This paper represents an attempt to develop a new asymptotic method for the analysis of convection in rapidly rotating spheres valid for asymptotically small $E$ and for $0 \,{\le}\,\hbox{\it Pr}/E \,{<}\, \infty$. The new method is based on the following three hypotheses. The first is that the leading-order velocity of convection for $0 \,{\le}\,\hbox{\it Pr}/E \,{<}\, \infty$ at $E \,{\ll}\, 1$ is represented by either a single quasi-geostrophic-inertial-wave mode or by a combination of several quasi-geostrophic-inertial-wave modes convectively excited and sustained. Secondly, we assume that the convective motion for $0 \,{\le}\,\hbox{\it Pr}/E \,{<}\, \infty$ at $E \,{\ll}\, 1$ always has columnar structure, i.e. ${\zpt}/{\zpt z} \,{\sim}\, {O}(1)$, but without the general asymptotic scalings in the radial and azimuthal direction. Thirdly, we assume that there always exists a boundary flow that is non-zero only in the Ekman boundary layer on the bounding spherical surface and plays an important role even in the case of stress-free boundaries. Comparison between the result of the new method and the corresponding fully numerical simulation demonstrates a satisfactory quantitative agreement for all values of $0 \,{\le}\,\hbox{\it Pr}/E \,{\le}\,{O}(10^6)$ when $ {O} (10^{-5}) \le E \,{\le}\,{O}(10^{-6})$. The new method is asymptotic in the sense that it is valid only for an asymptotically small $E \,{\ll}\, 1$.

In addition to the linear problem of thermal convection in rapidly rotating spheres, the corresponding weakly nonlinear problem is also solved to obtain an analytical expression for the convection-driven differential rotation generated by the nonlinear interaction of quasi-geostrophic-inertial-wave modes through the Reynolds stresses. The new method not only reveals the underlying nature of thermal convection in rapidly rotating spheres but also unites the two previously disjointed subjects in rotating fluids: the inertial-wave problem and the convective instability problem.

Observations and measurements are reported on the patterns and rates of growth in time of the double-diffusive cells that form adjacent to a cooled sidewall in a saltstratified environment. Fluid near the wall is cooled and sinks a distance $h$ where its density, increased by cooling, matches that of the salt-stratified ambient. The fluid separates from the wall, moving outwards as a cool, fresher layer beneath a warmer, more saline region. This leads to growing double-diffusive cells that advance outward at a rate, found by dimensional reasoning, to initially be proportional to $N_{0}h,$ where $N_{0}$ is the initial buoyancy frequency in the ambient and $h$ is the intrusion's vertical thickness. Near the wall at the top of each cell, the sinking colder fluid is continually replaced by selective withdrawal from the ambient ‘far field’. The fluid being withdrawn from the ambient is always the least dense in the cell, and as the experiment proceeds, the straining of the fluid in the ambient region reduces the stratification. The vertical density gradient inside the cell relaxes by continuous hydrostatic adjustment (CHA) to match the ambient and the speed of advance reduces. Measurements of the rate of advance of the cell nose were made in tanks of different lengths $L$ with a range of initial salinity gradients and temperature differences. A simple two-dimensional model is developed to describe the rate of extension of the cells and the internal density gradient as functions of time in which the tank length appears as an important variable. This effect does not seem to have been recognized previously. The rates of evolution in each run involve the time scale $\tau \,{=}\, L /( {C_H hN_0 })$, where $C_H \,{\approx}\, 10^{ - 2}$ is a heat transfer coefficient. The mean length of the cells $\skew2\bar {l}(t)$and the internal buoyancy frequency as functions of time are given by \[ \skew2\bar {l}(t) / L = t/\tau - ( t/2\tau)^2,\quad N = N_0 (1 - t / 2\tau ). \] Inversion of the first of these expressions results in $t/\tau \,{=}\, 2\,{-}\, 2\{ {1 - (\skew2\bar {l}(t) / L)}\}^{1 / 2}$ from which a time scale $\tau ^{ - 1}$ can be estimated. The measurements from individual runs when plotted in this way generally produce accurate straight lines as the model predicts, from which $C_H $ is found. This should be approximately the same for each run; the mean over all runs was found to be $9.3\,{\times}\, 10^{ - 3}$ with standard deviation 2.4$\,{\times}\, 10^{ - 3}$. The velocity scale of the intrusions at the beginning of an experiment is of order 10$^{ - 2}$ cm s$^{-1}$, for typical parameters of water at temperature 20 $^\circ$C, cooled wall temperature of 0 $^\circ$C and mean salinity of 5%.

This paper investigates the spatio-temporal instability of the natural-convection boundary-layer flow adjacent to a vertical heated flat plate immersed in a thermally stratified ambient medium. The temperature on the plate surface is distributed linearly. By introducing a temperature gradient radio $a$ between the wall and the medium, we obtain a similarity solution which can describe in a smooth way the evolution between the states with isothermal and uniform-heat-flux boundary conditions. It is shown that the flow reversal in the basic flow vanishes when $a$ is larger than a critical value. A new absolute–convective instability transition of this flow is identified in the context of the coupled Orr–Sommerfeld and energy equations. Increasing $a$ decreases the domain of absolute instability, and when $a$ is large enough the absolute instability disappears. In particular, when $a\,{=}\,0$ (isothermal surface), the interval of absolute instability becomes narrower for fluids of larger Prandtl numbers, and the absolute instability does not occur for Prandtl numbers greater than 70; when $a\,{=}\,1$ (uniform-heat-flux surface) the instability remains convective in a wide Prandtl number range. Analysis of the Rayleigh equations for this problem reveals that the basic flows supporting this new instability transition have inviscid origin of convective instability. Based on the steep global mode theory, the effects of $a$ and Prandtl number on the global frequency are discussed as well.