Previous proposed modifications in the respective constitutive forms of the Newtonian deviatoric stress tensor and the no-slip boundary condition imposed upon viscous fluids at solid surfaces, wherein the fluid's mass velocity is replaced by its volume velocity, furnishes a complete continuum-hydrodynamic description of thermal transpiration phenomena occurring in a closed capillary tube filled with a single-component gas or liquid, the former at negligibly small Knudsen numbers. The resulting expression for the steady-state thermomolecular pressure difference $\Delta p$ existing between the two ends of the capillary, the latter maintained at different temperatures, is free of empirical parameters, such as Maxwell's thermal-slip coefficient, upon which current non-continuum theories of the phenomenon are based. The predicted $\Delta p$ (with the pressure highest at the hotter end) is shown to agree well with experimental data for gases in the near-continuum limit of vanishingly small Knudsen number. Also discussed is the experimentally observed lack of dependence of $\Delta p$ upon the physicochemical properties of the capillary walls, an observation which accords with the predictions of our theory. Our proposed volume velocity-based rationalization of the phenomenon of thermal transpiration offers a strictly continuum no-slip alternative to Maxwell's widely-accepted thermal creep explanation thereof, involving slip of the fluid's mass velocity at a non-isothermal surface. The agreement of our theoretical predictions of the thermomolecular pressure difference with experimental data, which is essentially indistinguishable in accuracy from that provided by Maxwell's thermal creep theory, provides further support for the viability of the generic volume velocity-based framework underlying our theory, the latter having recently been used to also rationalize related thermophoretic and diffusiophoretic phenomena in gases, as well as thermal diffusion in liquids.

Magnetic field amplification by the motion of an electrically conducting fluid is studied, using a rotating plane-layer geometry. The fluid flow is driven by convection, and by a moving bottom boundary, which leads to an Ekman layer localized at the base of the system. The system thus has the structure of an interface dynamo, with convection lying over a thin layer of shear.

The combination of shear in the Ekman layer and convection above leads to amplification of seed magnetic fields. In kinematic regimes the magnetic field is mostly localized in sheets in the shear layer, but thin tongues are pulled out by the convection above and folded. The nonlinear saturation of these growing fields is studied at moderately high values of magnetic Reynolds number and Taylor number. It is found that the sheets of field tend to gain fine-scale structure when the dynamo saturates, breaking up into tubes, and the fluid flow shows complex time-dependence. Although the magnetic field lies predominantly within the highly sheared Ekman layer, this flow remains remarkably unchanged despite the action of Lorentz forces. Instead, the effect of the field is to suppress or modify the convection above. A simple alpha-omega dynamo model is set up, and gives some insights into the dynamo processes occurring in the full magnetohydrodynamic simulation.

Direct numerical simulations (DNS) of turbulent Rayleigh–Bénard convection in a wide cylindrical container (aspect ratio $\Gamma\,{=}\,10$) with a lateral wall have been performed for the first time for the Rayleigh numbers $10^5$, $10^6$ and $10^7$ and Prandtl number $Pr\,{=}\,0.7$. Evaluating the thermal dissipation rates from the generated DNS data, the formation and development of the thermal plumes and their interaction are highlighted. Two new functions $\sigma$ and $\tau$ are defined to determine quantitatively the role of the turbulent background. Evaluating these functions from the DNS data, it is shown that the turbulent background pushes the thermal plumes back and that its contribution to the volume-averaged thermal dissipation rate increases with the Rayleigh number. Further, it is proven analytically that the ratio of the area-averaged (over the top or the bottom plates) to the volume-averaged thermal dissipation rate is greater than or equal to the Nusselt number for all aspect ratios, and Prandtl and Rayleigh numbers.

The effect of vertical vibration on the long-wave instability of a Marangoni system is studied. The vibration augments the stabilizing effect of surface tension in bounded systems. In laterally unbounded systems nonlinear terms can stabilize non-flat states and prevent the appearance of dry spots. The effect of a slight inclination of the system is also considered.

This paper examines the existence of a localized mode of sound in a planar waveguide between two parallel rigid walls with a pair of identical Helmholtz resonators connected to the upper and lower walls. The localized mode means that linear free oscillations at a particular frequency are trapped only in the vicinity of the resonators and decay exponentially away from them. Assuming the waveguide extends infinitely, a two-dimensional problem is solved fully within lossless theory. It is revealed that, in a waveguide with the resonators connected exactly opposite to each other, the localized mode can exist at a frequency lower than the lowest cutoff frequency of the waveguide and the natural frequency of the resonator. Then the pressure field is antisymmetric spanwise with respect to the centreline of the waveguide, and symmetric axially with respect to the resonators. The localized mode is represented by the superposition of an infinite number of anti-symmetric evanescent modes and no plane-wave mode is involved. The absence of the latter mode makes it possible to localize sound without it being accompanied by radiation damping. This explains why no symmetric localized mode over the width exists. In a waveguide with the resonators offset, no localized mode is shown to exist generally except for a particular spacing. Generalization to cases with multiple pairs of resonators is also considered.

The dynamics of a single bubble rising in a viscous Newtonian fluid was investigated both experimentally by a particle image velocimetry (PIV) device and numerically using the free-energy-based lattice Boltzmann (LB) model. The rise velocity, bubble shape and flow field were considered for various bubble volumes in axisymmetric flow conditions. Experimentally, the flow measurements by the PIV device revealed the wake increasing with the bubble volume. Such an evolution is linked to the deformation of bubble shape from spherical for small bubbles to flattened at the bottom for large bubbles. The LB simulations compare satisfactorily with our experimental data for both the bubble shape and drag coefficient over the range of Reynolds number ($ 0.033 \leq Re \leq 1.8 $). With a more extended flow structure around the bubble compared to experiments, the two-dimensional approach shows some limitations in its quantitative description. Fully three-dimensional simulations are necessary, especially for bigger bubbles with $ Re > 1.8 $.

The adjoint of the perturbed and linearized compressible viscous flow equations is formulated in such a way that its solution can be used to optimize control actuation in order to reduce flow-generated sound. We apply it to a direct numerical simulation of a randomly excited two-dimensional mixing layer, with inflow vorticity-thickness Reynolds number 500 and free-stream Mach numbers 0.9 and 0.2. The control actuation is implemented as general source terms in the flow equations (body forces, mass sources, and internal energy sources) with compact support near the inflow boundary. The noise to be reduced is defined by a space-time integral of the mean-square pressure fluctuations on a line parallel to the mixing layer in the acoustic field of the low-speed stream. Both the adjoint and flow equations are solved numerically and without modelling approximations. The objective is to study the mechanics of the noise generation and its control. All controls reduce targeted noise with very little required input power, with the most effective (the internal energy control) reducing the noise intensity by 11 dB. Numerical tests confirm that the control is not by any simple acoustic cancellation mechanism but instead results from a genuine change of the flow as a source of sound. The comparison of otherwise identical flows with and without control applied shows little change of the flow's gross features: the evolution and pairings of the energetic structures, turbulence kinetic energy, spreading rate, and so on are superficially unchanged. However, decomposition of the flow into empirical eigenfunctions, as surrogates for Fourier modes in the non-periodic streamwise direction, shows that the turbulence structures advect downstream more uniformly. This change appears to be the key to reducing their acoustic efficiency, a perspective that is clarified by comparing the randomly excited mixing layer to a harmonically excited mixing layer, which is relatively quiet because it is highly ordered. Unfortunately, from the perspective of any practical implementation with actuators, the optimized control identified has a complex spatial and temporal structure, but it can be simplified. Two empirical eigenmodes were required to represent it sufficiently to reduce the targeted noise intensity by about 50%. Optimization of a simple single-degree-of-freedom control with an ad hoc spatial structure is less effective.

Two-dimensional incompressible and heterogeneous free jets are studied numerically. They are characterized by two linearly unstable modes: the sinuous mode and the varicose mode. When the density is constant, the sinuous mode dominates, in a sense that it has a maximum growth rate nearly three times greater than that of the varicose mode, both in the temporal and spatial stability frameworks. This suggests that jet evolution at the linear and nonlinear regimes, at least at early stage, would be sinuous with the frequency and wavenumber of the most unstable mode. This is confirmed in the present work. When the jet density is decreased, it is found that the varicose mode growth rate is enhanced and its maximum value can exceed that of the sinuous mode. Furthermore, in the spatial case, only the varicose mode is subject to a convective/absolute instability transition. It is shown, however, that the maximum varicose growth rate exceeds that of the sinuous mode before the transition occurs. Therefore, a natural question is: does the convective/absolute transition dictate its frequency to the nonlinear evolution of the jet, leading to a global oscillating varicose mode, or is it, simply, the frequency of the maximum varicose growth rate that does? The main part of this paper is devoted to answering this question by direct numerical simulation for a wide range of jet-exit to ambient density ratios.

We study the first bifurcation in the axisymmetric flow between two exactly counter-rotating disks with very large aspect ratio $ \Gamma\,{\equiv}\, R/H$, where $R$ is the disk radius and $2 H$ is the inter-disk spacing. The scaling law for the critical Reynolds number is found to be $\Rey_c \propto \Gamma^{-1/2}$, with $\Rey \,{\equiv}\, \Omega H^2/\nu$, $\Omega$ being the magnitude of the angular velocity and $\nu$ the kinematic viscosity. An asymptotic analysis for large $\Gamma$ is developed, in which curvature is neglected, but the centrifugal acceleration term is retained. The Navier–Stokes equations then reduce to leading order to those in a Cartesian frame, and the axisymmetric base flow to a parallel flow. This allows us locally to use a Fourier decomposition along the radial direction. In this framework, we explain the physical mechanism of the instability invoking the linear azimuthal velocity profile and the effect of centrifugal acceleration.

We consider the effect of surface contamination, modelled by Marangoni elasticity with insoluble surfactant and surface viscosity, in promoting drift instabilities in spatially uniform standing Faraday waves. It is seen that contamination enhances drift instabilities that lead to various steadily propagating and (both standing and propagating) oscillatory patterns. In particular, steadily propagating waves appear to be quite robust, as a seminal experiment by Douady, Fauve & Thual (1989).

Experiments in the HMX-based condensed explosive PBX-9501 were carried out to validate a reduced, asymptotically derived description of detonation shock dynamics (DSD) where it is assumed that the normal detonation shock speed is determined by the total shock curvature. The passover experiment has a lead disk embedded in a right circular cylindrical charge of PBX-9501 and is initiated from the bottom. The subsequent detonation shock experiences a range of dynamic states with both diverging (convex) and converging (concave) configurations as the detonation shock passes over the disk. The time of arrival of the detonation shock at the top surface of the charge is recorded and compared against DSD simulation and direct multi-material simulation. A new wide-ranging equation of state (EOS) and rate law that is constrained by basic explosive characterization experiments is introduced as a constitutive description of the explosive. This EOS and rate law is used to compute the theoretical normal shock velocity, curvature relation of the explosive for the reduced description, and is also used in the multi-material simulation. The time of arrival records are compared against the passover experiment and the dynamic motion of the shock front and states within the explosive are analysed. The experiment and simulation data are in excellent agreement. The level of agreement, both qualitative and quantitative, of theory and simulation with experiment is encouraging because it indicates that descriptions such as the wide-ranging EOS/rate law and the corresponding reduced DSD description can be used effectively to model real explosives and predict complex dynamic behaviors.

Thermal imprint and heat transfer on the target surface in multiple impinging jet arrays have been investigated in conjunction with flow pattern and large-scale eddy structure. Surface temperature was measured with liquid crystal thermography (LCT) in a range of jet configurations with hexagonal and in-line orifice arrangements for different combinations of distances between the orifices $s/D_m\,{=}\,2\hbox{--}6$ and between the orifice plate and the impingement surface $H/D_m\,{=}\,3\hbox{--}10$ (where $D_m$ is the orifice diameter). The hexagonal arrangement was considered with two different orifice shapes: sharp-edged and contoured. For selected configurations, the distribution of Nusselt number and its peculiarities were analysed in relation with the underlying eddy structure educed by proper orthogonal decomposition (POD) of the snapshots of fluid velocity measured with particle image velocimetry (PIV). Owing to the breakdown of the jets, heat transfer deteriorates with increasing orifice-to-plate distance. The jet interaction and breakdown become more severe as the jets are placed closer to each other. The large-scale eddies originating from the jet-edge shear layers grow as they are convected towards the impingement plate. Eddies of sizes between 0.2 and 0.3 orifice diameters are shown to break up the jets and cause mixing of fresh and spent fluid, lowering the beneficial temperature gradient. In some configurations, an asymmetric flow pattern is generated with embedded weak eddies on only one side of the diagonal symmetry line, which is reflected in an asymmetrical heat transfer distribution on the impingement plate. For $H/D_m\,{>}\,4$, the Nusselt number shows peak values in and around the jet impingement centres, but relatively uniform distribution of turbulence kinetic energy with local negative energy production close to the impingement surface.

We consider the motion of a finger of low-viscosity fluid as it propagates into a branching network of fluid-filled microchannels – a scenario that arises in many appli- cations, such as microfluidics, biofluid mechanics (e.g. pulmonary airway reopening) and the flow in porous media. We perform experiments to investigate the behaviour of the finger as it reaches a single bifurcation and determine under what conditions the finger branches symmetrically. We find that if the daughter tubes have open ends, the finger branches asymmetrically and will therefore tend to reopen a single path through the branching network. Conversely, if the daughter tubes terminate in elastic chambers, which provide a lumped representation of the airway wall elasticity in the airway reopening problem, the branching is found to be symmetric for sufficiently small propagation speeds. A mathematical model is developed to explain the experimentally observed behaviour.

This paper describes one-dimensional (parallel) laminar natural convection in a viscous stably stratified fluid owing to temporally periodic variations in the surface temperature of infinite vertical plates and circular cylinders. Analytical solutions of the one-dimensional (parallel) Boussinesq equations of motion and thermodynamic energy are obtained for the periodic regime for arbitrary values of ambient stratification, Prandtl number and forcing frequency. The solutions for plates and cylinders are qualitatively similar and show that (i) the flows are composed of two waves that decay exponentially with distance from the surface: a fast long wave and a slow short wave; (ii) for forcing frequencies greater than the natural frequency of the corresponding inviscid system, these two waves propagate away from the surface; and (iii) for forcing frequencies less than this natural frequency, the short wave propagates away from the surface while the long wave propagates toward the surface. This latter case provides an example of a flow for which the conventional radiation condition is not appropriate. The analytical results are complemented, for the plate problem, with three-dimensional numerical simulations of flows that start from rest and are suddenly subjected to a periodic thermal forcing at the plate. The numerical results depict the transient (start-up) stage of the flow and the approach to a periodic regime. These results confirm that the analytical solutions provide the appropriate description of the periodic regime.

We present numerical simulations of randomly forced internal gravity waves in a uniformly stratified Boussinesq fluid, and compare the resulting vertical wavenumber energy spectra with the saturation spectrum $E_z(k_z)\,{=}\,c\,N^2k_z^{-3}$ ($N$ is the Brunt–Väisälä frequency) observed in the atmosphere and ocean. Overall, we have been unsuccessful at reproducing the observed spectrum in our simulations. Our spectra are shallower than $k_z^{-3}$, although they steepen towards it with increasing stratification as long as wave breaking (in the form of static instability) is resolved. The spectral amplitude increases like $N^{1.1}$ rather than $N^2$. For a single stratification, our spectrum agrees well with the saturation spectrum with $c\,{=}\,0.1$, but only because it is spuriously steepened by insufficient resolution. We show that overturning occurs when the length scale $l_c\,{=}\,u_{rms}/N$ is larger than the dissipation scale, where $u_{rms}$ is the root mean square velocity. This scale must be at least three times larger than the dissipation scale for the energy spectrum to be independent of Reynolds number in our simulations. When this condition is not satisfied, the computed energy spectrum must be interpreted with caution. Finally, we show that for strong stratifications, the presence of vortical energy can have a dramatic effect on the spectrum of wave energy due to the efficiency of interactions between two waves and a vortical mode. Any explanation of the energy spectrum involving resonant interactions must take into account the effect of vortical motion.

The flow fields behind the Mach stem in steady-state Mach reflections are analysed theoretically and numerically. When the angle between the slipline and the reflecting plane is sufficiently small, the subsonic flow just behind the Mach stem can be described by the isentropic small-disturbance equation. Using this analytical model, a very simple algebraic expression for the shape of the Mach stem is obtained. It is found that the Mach stem is well-approximated by a circular arc with very small curvature. The prediction by the model agrees very well with numerical computations.

Experiments have shown that when a mixture of fuel and oxygen is passed through a zirconia tube whose inner surface is coated with a catalyst, and then ignited at the end of the tube, a reaction front, or flame, propagates back along the tube towards the fuel inlet. The reaction front is visible as a red hot region moving at a speed of a few millimetres per second. In this paper we study a model of the flow, which takes into account diffusion, advection and chemical reaction at the inner surface of the tube. By assuming that the flame propagates at a constant speed without change of form, we can formulate a steady problem in a frame of reference moving with the reaction front. This is solved using the method of matched asymptotic expansions, assuming that the Reynolds and Damköhler numbers are large. We present numerical and, where possible, analytical results, first when the change in fluid density is small (a simplistic but informative limit) and secondly in the variable-density case. The speed of the travelling wave decreases as the critical temperature of the surface reaction increases and as the mass flow rate of fuel increases. We also make a comparison between our results and some preliminary experiments.

Recent theoretical findings show that the curvature of streamsurfaces naturally warping in the crossflow direction of a three-dimensional boundary layer can maintain centrifugal forces, provoking unsteady short-scaled vortices. The wave/vortex eigenmode coupling takes place in the linear stage of disturbance evolution and results in streamwise absolute instability. The two-dimensional flow past a curved cylindrical surface under consideration here provides an example where centrifugal forces are associated with the fixed curvature of a solid wall bending in the direction of the main stream. The spiral-type Görtler vortices develop in proximity to the surface and their interaction with the Tollmien–Schlichting eigenmodes creates a mechanism driving disturbances both downstream and upstream of a perturbing agency. The vortex eigenmodes arising from centrifugal forces are balanced out by the normal-to-wall pressure gradient. Some higher-order terms need to be kept in its expansion to achieve the modal coupling. As a consequence, a side band appears in the spectra of eigen-frequencies and wavenumbers featuring the classical triple-deck scheme. The extended composite asymptotic model proves to be self-consistent, with the Cauchy problem well-posed in the limit of large Reynolds numbers. It follows from the extended model that the boundary layer on a concave surface, much like the one with crossflow, suffers absolute instability in the streamwise direction. This unusual property may lead to earlier transition or, conversely, be exploited to artificially excite the nonlinear vortex structures with delayed transition.

Flow visualization is used to interrogate the wake structure produced by a rigid flat panel of aspect ratio (span/chord) 0.54 pitching in a free stream at a Strouhal number of 0.23. At such a low aspect ratio, the streamwise vorticity generated by the plate tends to dominate the formation of the wake. Nevertheless, the wake has the appearance of a three-dimensional Kármán vortex street, as observed in a wide range of other experiments, and consists of horseshoe vortices of alternating sign shed twice per flapping cycle. The legs of each horseshoe interact with the two subsequent horseshoes in an opposite-sign, then like-sign interaction in which they become entrained. A detailed vortex skeleton model is proposed for the wake formation.