Convection in a closed porous domain is a temporally and spatially complex flow which evolves over long time scales as the driving buoyancy contrasts are eliminated by mixing. In a contribution that combines numerical, experimental and asymptotic approaches, Hewitt, Neufeld & Lister (J. Fluid Mech., vol. 719, 2013, pp. 551–586) demonstrate that the essential dynamics can be captured by simple ‘box’ models, both when the buoyancy supply is imposed at the upper boundary and when the domain contains a moving interface between different fluids. This work provides insights into the dynamics and viability of schemes for the geological sequestration of CO2.

The input of energy by wind to water waves is compared with the observed growth of the waves using a suite of microphysical measurement techniques in the laboratory. These include measured tangential stresses in the water and air immediately adjacent to the interface with corresponding form drag measurements above wind-forced freely propagating waves. The drag data sets are consistent but the comparison has highlighted important issues in relation to the measurement of fluctuating pressures above freely propagating waves. Derived normalized wind input values show good collapse as a function of mean wave steepness and are significantly in excess of the assembly of net wave growth measurements by Peirson & Garcia (J. Fluid Mech., vol. 608, 2008, pp. 243–274) at low steepness. Sheltering coefficients in the form of Jeffreys (Proc. R. Soc. Lond. Ser. A, vol. 107, 1925, pp. 189–206) are derived that are consistent with values previously obtained by Donelan & Pierson (J. Geophys. Res., vol. 92, 1987, pp. 4971–5029), Donelan (Wind-over-Wave Couplings: Perspectives and Prospects, Clarendon, 1999, pp. 183–194) and Donelan et al. (J. Phys. Oceanogr., vol. 36, 2006, pp. 1672–1689). The sheltering coefficients exhibit substantial scatter. By carefully measuring the associated growth of the surface wave fields, systematic energy budgets for the interaction between wind and waves are obtained. For non-breaking waves, there is a significant and systematic misclose in the radiative transfer equation if wave–turbulence interactions are not included. Significantly higher levels of turbulent wave attenuation are found in comparison with the theoretical estimates by Teixeira & Belcher (J. Fluid Mech., vol. 458, 2002, pp. 229–267) and Ardhuin & Jenkins (J. Phys. Oceanogr., vol. 36, 2006, pp. 551–557). Suitable normalizations of attenuation for wind-forced wave fields exhibit consistent behaviour in the presence and absence of wave breaking. Closure of the surface energy flux budget is obtained by comparing the normalized energy loss rates due to breaking with the values previously determined by Banner & Peirson (J. Fluid Mech., vol. 585, 2007, pp. 93–115) and Drazen et al.(J. Fluid Mech., vol. 611, 2008, pp. 307–332) when expressed as a function of mean wave steepness. Their normalized energy loss rates obtained for non-wind forced breaking wave groups are remarkably consistent with the levels found during this present study when breaking waves are subject to wind forcing.

The flow around the three-dimensional blunt geometry presented in the work of Ahmed, Ramm & Faitin (Tech. Rep., 1984) is investigated experimentally at $\mathit{Re}= {U}_{0} H/ \nu = 9. 2\times 1{0}^{4} $ (where ${U}_{0} $ is free-stream velocity, $H$ the height of the body and $\nu $ viscosity). The very large recirculation on the base responsible for the dominant part of the drag is characterized. The analyses of the coherent dynamics of the wake reveal the presence of two very distinctive time scales. At long time scales ${T}_{l} \sim 1{0}^{3} H/ {U}_{0} $, the recirculation region shifts between two preferred reflectional-symmetry-breaking positions leading to a statistically symmetric wake; the sequence of these asymmetric states is random. This bi-stable behaviour is independent of the Reynolds number but occurs only above a critical value of ground clearance. At short time scales ${T}_{s} \sim 5H/ {U}_{0} $, the wake presents weak coherent oscillations in the vertical and lateral directions. They are respectively associated with the interaction of the top/bottom and lateral shear layers; when normalized by the height and width of the body, the Strouhal numbers are close to 0.17. These results suggest an alternate shedding associated with the vertical oscillation and a one-sided vortex shedding in the lateral direction with an orientation linked to the current asymmetric position. Finally, the impact of these coherent wake motions on the base pressure is discussed to motivate further drag reduction strategies.

An experimental and numerical investigation of the unsteady phenomena induced in a hydrogen-fuelled scramjet combustor under high-equivalence-ratio conditions is carried out, focusing on the processes leading up to unstart. The configuration for the study is the fuelled flow path of the HyShot II flight experiment. Experiments are performed in the HEG reflected-shock wind tunnel, and results are compared with those obtained from unsteady numerical simulations. High-speed schlieren and OH∗ chemiluminescence visualization, together with time-resolved surface pressure measurements, allow links to be drawn between the experimentally observed flow and combustion features. The transient flow structures signalling the onset of unstart are observed to take the form of an upstream-propagating shock train. Both the speed of propagation and the downstream location at which the shock train originates depend strongly on the equivalence ratio. The physical nature of the incipient shock system, however, appears to be similar for different equivalence ratios. Both experiments and computations indicate that the primary mechanism responsible for the transient behaviour is thermal choking, though localized boundary-layer separation is observed to accompany the shock system as it moves upstream. In the numerical simulations, the global choking behaviour is dictated by the limited region of maximum heat release around the shear layer between the injected hydrogen and the incoming air flow. This leads to the idea of ‘local’ thermal choking and results in a lower choking limit than is predicted by a simple integral analysis. Such localized choking makes it possible for new quasi-steady flow topologies to arise, and these are observed in both experiments and simulation. Finally, a quasi-unsteady one-dimensional analytical model is proposed to explain elements of the shock-propagation behaviour.

Most rigid, torque-free, low-Reynolds-number, axisymmetric particles undergo a time-periodic tumbling motion in a simple shear flow, with their axes of symmetry following a set of closed Jeffery orbits. We have identified a class of rigid, ring-like particles whose axes of symmetry instead reach a permanent alignment near the velocity gradient direction with the plane of the particle aligning near the flow–vorticity plane. An asymptotic analysis for small particle aspect ratio (ratio of length parallel to the axis of symmetry to diameter perpendicular to the axis) shows that an appropriate asymmetry of the ring cross-section with a thinner outer edge and thicker inner edge leads to a tendency to rotate in a direction opposite to the vorticity; this tendency can balance the usual rotation rate associated with the finite thickness of the particle. Boundary integral computations for finite particle aspect ratios are used to determine the conditions of aspect ratio and degree of asymmetry that lead to the aligning behaviour and the final orientation of the axis of symmetry of the aligned particles. The aligning particle follows an equation of motion similar to the Leslie–Erickson equation for the director of a small-molecule nematic liquid crystal. However, whereas the alignment of the director arises from intermolecular interactions, the ring-like particle aligns solely due to its intrinsic rotational motion in a low-Reynolds-number flow.

This article describes the radial drift of a suspended sphere in a cylinder-bound Poiseuille flow where the Reynolds number is small but finite. Unlike past studies, it considers a circular narrow conduit whose cross-sectional diameter is only $1. 5$–$6$ times the particle diameter. Thus, the analysis quantifies the effect of fluid inertia on the radial motion of the particle in the channel when the flow field is significantly influenced by the presence of the suspended body. To this end, the hydrodynamic fields are expanded as a series in Reynolds number, and a set of hierarchical equations for different orders of the expansion is derived. Accordingly, the zeroth-order fields in Reynolds number satisfy the Stokes equation, which is accurately solved in the presence of the spherical particle and the cylindrical conduit. Then, recognizing that in narrow vessels Stokesian scattered fields from the sphere decrease exponentially in the axial direction, a simpler regular perturbation scheme is used to quantify the first-order inertial correction to hydrodynamic quantities. Consequently, it is possible to obtain two results. First, the sphere is assumed to follow the axial motion of a freely suspended sphere in a Stokesian condition, and the radial lift force on it due to the presence of fluid inertia is evaluated. Then, the approximate motion is determined for a freely suspended body on which net hydrodynamic force including first-order inertial lift is zero. The results agree well with the available experimental results. Thus, this study along with the measured data would precisely describe particle dynamics inside narrow tubes.

The role of streamline curvature-driven favourable pressure gradients in modifying the turbulence structure of a Mach 4.9, high-Reynolds-number (${\mathit{Re}}_{\theta } = 43\hspace{0.167em} 000$) boundary layer is examined. Three pressure gradient cases ($\beta = (\mathrm{d} p/ \mathrm{d} x)({\delta }^{\ast } / {\tau }_{w} )= 0. 07, - 0. 3$ and $- 1. 0$) are characterized via particle image velocimetry. The expected stabilizing trends in the Reynolds stresses are observed, with a sign reversal in the Reynolds shear stress in the outer part of the boundary layer for the strongest favourable pressure gradient considered. The increased transverse normal strain rate and reduced principal strain rate are the primary factors. Reynolds stress quadrant events are redistributed, such that the relative differences between the quadrant magnitudes decreases. Very little preferential quadrant mode selection is observed for the strongest pressure gradient considered. Two-point correlations suggest that the turbulent structures are reoriented to lean farther away from the wall, accompanied by a slight reduction in their characteristic size, consistent with previous flow visualization studies. This reorientation is more pronounced in the outer, dilatation-dominated region of the boundary layer, whereas the alteration in structure size is more pronounced nearer the wall, where the principal strain rates are larger. In addition, integration of a simplified form of a Reynolds stress transport closure model provided a framework to assess the role of the strain-rate field on the observed Reynolds shear stresses. Given the simple geometry, the present data provide a suitable test bed for Reynolds stress transport and large-eddy model development and validation.

In small vessels blood is usually treated as a Newtonian fluid down to diameters of ${\sim }200~\mathrm{\mu} \mathrm{m} $. We investigate the flow of red blood cell (RBC) suspensions driven through small tubes (diameters $10\text{{\ndash}} 150~\mathrm{\mu} \mathrm{m} $) in the range marking the transition from arterioles and venules to the largest capillary vessels. The results of the simulations combined with previous simulations of uniform shear flow and experimental data show that for diameters less than ${\sim }100~\mathrm{\mu} \mathrm{m} $ the suspension’s stress cannot be described as a continuum, even a heterogeneous one. We employ the dissipative particle dynamics (DPD) model, which has been successfully used to predict human blood bulk viscosity in homogeneous shear flow (Fedosov et al. Proc. Natl Acad. Sci. USA, vol. 108, 2011, pp. 11772–11777). In tube flow the cross-stream stress gradient induces an inhomogeneous distribution of RBCs featuring a centreline cell density peak, and a cell-free layer (CFL) next to the wall. For a neutrally buoyant suspension the imposed linear shear-stress distribution together with the differentiable velocity distribution allow the calculation of the local viscosity across the tube section. The viscosity across the section as a function of the strain rate is found to be essentially independent of tube size for the larger diameters and is determined by the local haematocrit ($H$) and shear rate. Other RBC properties such as asphericity, deformation, and cell-flow orientation exhibit similar dependence for the larger tube diameters. As the tube size decreases below ${\sim }100~\mathrm{\mu} \mathrm{m} $ in diameter, the viscosity in the central region departs from the large-tube similarity function of the shear rate, since $H$ increases significantly towards the centreline. The dependence of shear stress on tube size, in addition to the expected local shear rate and local haematocrit, implies that blood flow in small tubes cannot be described as a heterogeneous continuum. Based on the analysis of the DPD simulations and on available experimental results, we propose a simple velocity-slip model that can be used in conjunction with continuum-based simulations.

Binary fluid mixtures with a negative separation ratio heated from below exhibit steady spatially localized states called convectons for supercritical Rayleigh numbers. With no-slip, fixed-temperature, no-mass-flux boundary conditions at the top and bottom stationary odd- and even-parity convectons fall on a pair of intertwined branches connected by branches of travelling asymmetric states. In appropriate parameter regimes the stationary convectons may be stable. When the boundary condition on the top is changed to Newton’s law of cooling the odd-parity convectons start to drift and the branch of odd-parity convectons breaks up and reconnects with the branches of asymmetric states. We explore the dependence of these changes and of the resulting drift speed on the associated Biot number using numerical continuation, and compare and contrast the results with a related study of the Swift–Hohenberg equation by Houghton & Knobloch (Phys. Rev. E, vol. 84, 2011, art. 016204). We use the results to identify stable drifting convectons and employ direct numerical simulations to study collisions between them. The collisions are highly inelastic, and result in convectons whose length exceeds the sum of the lengths of the colliding convectons.

An active technique for friction drag reduction in a turbulent channel flow is studied by direct numerical simulations. The flow modification is induced by the steady rotation of rigid flush-mounted discs, located next to one another on the walls. The effect of the disc motion on the turbulent drag is investigated at a Reynolds number of ${R}_{\tau } = 180$ , based on the friction velocity of the stationary-wall case and the half channel height. For a fixed maximum disc tip velocity, drag reduction can be achieved when the disc diameter is larger than a threshold, while below this threshold the drag increases. A maximum drag reduction of 23% is computed. The net power saved, obtained by taking into account the power spent to enforce the rotational motion against the fluid viscous resistance, is found to be positive and reach 10%. The disc-flow parameters required for commercial aircraft flight conditions and flows over high-speed trains and ship hulls are estimated and future implementations based on existing micro-electromagnetic motor and micro-air turbine technologies are discussed.

The dynamics and stability of the flow past two cylinders sliding along a wall in a tandem configuration is studied numerically for Reynolds numbers ($\mathit{Re}$) between 20 and 200, and streamwise separation distances between 0.1 and 10 cylinder diameters. For cylinders at close separations, the onset of unsteady two-dimensional flow is delayed to higher $\mathit{Re}$ compared with the case of a single sliding cylinder, while at larger separations, this transition occurs earlier. For Reynolds numbers above the threshold, shedding from both cylinders is periodic and locked. At intermediate separation distances, the wake frequency shifts to the subharmonic of the leading-cylinder shedding frequency, which appears to be due to a feedback cycle, whereby shed leading-cylinder vortices interact strongly with the downstream cylinder to influence subsequent leading-cylinder shedding two cycles later. In addition to the shedding frequency, the drag coefficients for the two cylinders are determined for both the steady and unsteady regimes. The three-dimensional stability of the flow is also investigated. It is found that, when increasing the Reynolds number at intermediate separations, an initial three-dimensional instability develops, which disappears at higher $\mathit{Re}$. The new two-dimensional steady flow again becomes unstable, but with a different three-dimensional instability mode. At very close spacings, when the two cylinders are effectively seen by the flow as a single body, and at very large spacings, when the cylinders form independent wakes, the flow characteristics are similar to those of a single cylinder sliding along a wall.

Bubbly turbulent Taylor–Couette (TC) flow is globally and locally studied at Reynolds numbers of $\mathit{Re}= 5\times 1{0}^{5} $ to $2\times 1{0}^{6} $ with a stationary outer cylinder and a mean bubble diameter around 1 mm. We measure the drag reduction (DR) based on the global dimensional torque as a function of the global gas volume fraction ${\alpha }_{global} $ over the range 0–4 %. We observe a moderate DR of up to 7 % for $\mathit{Re}= 5. 1\times 1{0}^{5} $. Significantly stronger DR is achieved for $\mathit{Re}= 1. 0\times 1{0}^{6} $ and $2. 0\times 1{0}^{6} $ with, remarkably, more than $40\hspace{0.167em} \% $ of DR at $\mathit{Re}= 2. 0\times 1{0}^{6} $ and ${\alpha }_{global} = 4\hspace{0.167em} \% $. To shed light on the two apparently different regimes of moderate DR and strong DR, we investigate the local liquid flow velocity and the local bubble statistics, in particular the radial gas concentration profiles and the bubble size distribution, for the two different cases: $\mathit{Re}= 5. 1\times 1{0}^{5} $ in the moderate DR regime and $\mathit{Re}= 1. 0\times 1{0}^{6} $ in the strong DR regime, both at ${\alpha }_{global} = 3\pm 0. 5\hspace{0.167em} \% $. In both cases the bubbles mostly accumulate close to the inner cylinder (IC). Surprisingly, the maximum local gas concentration near the IC for $\mathit{Re}= 1. 0\times 1{0}^{6} $ is ${\approx }2. 3$ times lower than that for $\mathit{Re}= 5. 1\times 1{0}^{5} $, in spite of the stronger DR. Evidently, a higher local gas concentration near the inner wall does not guarantee a larger DR. By defining and measuring a local bubble Weber number ($\mathit{We}$) in the TC gap close to the IC wall, we observe that the cross-over from the moderate to the strong DR regime occurs roughly at the cross-over of $\mathit{We}\sim 1$. In the strong DR regime at $\mathit{Re}= 1. 0\times 1{0}^{6} $ we find $\mathit{We}\gt 1$, reaching a value of $9(+ 7, - 2)$ when approaching the inner wall, indicating that the bubbles increasingly deform as they draw near the inner wall. In the moderate DR regime at $\mathit{Re}= 5. 1\times 1{0}^{5} $ we find $\mathit{We}\approx 1$, indicating more rigid bubbles, even though the mean bubble diameter is larger, namely $1. 2(+ 0. 7, - 0. 1)~\mathrm{mm} $, as compared with the $\mathit{Re}= 1. 0\times 1{0}^{6} $ case, where it is $0. 9(+ 0. 6, - 0. 1)~\mathrm{mm} $. We conclude that bubble deformability is a relevant mechanism behind the observed strong DR. These local results match and extend the conclusions from the global flow experiments as found by van den Berg et al. (Phys. Rev. Lett., vol. 94, 2005, p. 044501) and from the numerical simulations by Lu, Fernandez & Tryggvason (Phys. Fluids, vol. 17, 2005, p. 95102).

A low-dimensional model capturing the fully coupled dynamics of a wavy liquid film in interaction with a strongly confined laminar gas flow is introduced. It is based on the weighted residual integral boundary layer approach of Ruyer-Quil & Manneville (Eur. Phys. J. B, vol. 15, 2000, pp. 357–369) and accounts for viscous diffusion up to second order in the film parameter. The model is applied to study two scenarios: a horizontal pressure-driven water film/air flow and a gravity-driven liquid film interacting with a co- or counter-current air flow. In the horizontal case, interfacial viscous drag is weak and interfacial waves are primarily driven by pressure variations induced by the velocity difference between the two layers. This produces an extremely thin interfacial shear layer which is pinched at the main and capillary wave humps, creating several elongated vortices in the wave-fixed reference frame. In the capillary wave region preceding a large wave hump, flow separation occurs in the liquid in the form of a vortex transcending the liquid–gas interface. For the gravity-driven film, a twin vortex (in the wave-fixed reference frame) linked to the occurrence of rolling waves has been identified. It consists of the known liquid-side vortex within the wave hump and a previously unknown counter-rotating gas-side vortex, which are connected by the same interfacial stagnation points. At large counter-current gas velocities, interfacial waves on the falling liquid film are amplified and cause flooding of the channel in a noise-driven scenario, while this can be delayed by forcing regular waves at the most amplified linear wave frequency. Our model is shown to exactly capture the long-wave linear stability threshold for the general case of two-phase channel flow. Further, for the two considered scenarios, it predicts growth rates and celerity of linear waves in convincing agreement with Orr–Sommerfeld calculations. Finally, model calculations of nonlinear interfacial waves are in good agreement with film thickness and velocity measurements as well as streamline patterns in both phases obtained from direct numerical simulations.

We consider electrokinetic flows about a freely suspended liquid drop, deriving a macroscale description in the thin-double-layer limit where the ratio $\delta $ between Debye width and drop size is asymptotically small. In this description, the electrokinetic transport occurring within the diffuse part of the double layer (the ‘Debye layer’) is represented by effective boundary conditions governing the pertinent fields in the electro-neutral bulk, wherein the generally non-uniform distribution of $\zeta $, the dimensionless zeta potential, is a priori unknown. We focus upon highly conducting drops. Since the tangential electric field vanishes at the drop surface, the viscous stress associated with Debye-scale shear, driven by Coulomb body forces, cannot be balanced locally by Maxwell stresses. The requirement of microscale stress continuity therefore brings about a unique velocity scaling, where the standard electrokinetic scale is amplified by a ${\delta }^{- 1} $ factor. This reflects a transition from slip-driven electro-osmotic flows to shear-induced motion. The macroscale boundary conditions display distinct features reflecting this unique scaling. The effective shear-continuity condition introduces a Lippmann-type stress jump, appearing as a product of the local charge density and electric field. This term, representing the excess Debye-layer shear, follows here from a systematic coarse-graining procedure starting from the exact microscale description, rather than from thermodynamic considerations. The Neumann condition governing the bulk electric field is inhomogeneous, representing asymptotic matching with transverse ionic fluxes emanating from the Debye layer; these fluxes, in turn, are associated with non-uniform tangential ‘surface’ currents within this layer. Their appearance at leading order is a manifestation of dominant advection associated with the large velocity scale. For weak fields, the linearized macroscale equations admit an analytic solution, yielding a closed-form expression for the electrophoretic velocity. When scaled by Smoluchowski’s speed, it reads

$${\delta }^{- 1} \frac{\sinh ( \overline{\zeta } / 2)/ \overline{\zeta } }{1+ { \textstyle\frac{3}{2} }\mu + 2\alpha {\mathop{\sinh }\nolimits }^{2} ( \overline{\zeta } / 2)} ,$$

$ \overline{\zeta } $

$\zeta $

$\mu $

$\alpha $

${\delta }^{- 1} $

$\alpha $

$\delta $

We have performed fully resolved three-dimensional numerical simulations of a statically unstable monochromatic inertia–gravity wave using the Boussinesq equations on an $f$-plane with constant stratification. The chosen parameters represent a gravity wave with almost vertical direction of propagation and a wavelength of 3 km breaking in the middle atmosphere. We initialized the simulation with a statically unstable gravity wave perturbed by its leading transverse normal mode and the leading instability modes of the time-dependent wave breaking in a two-dimensional space. The wave was simulated for approximately 16 h, which is twice the wave period. After the first breaking triggered by the imposed perturbation, two secondary breaking events are observed. Similarities and differences between the three-dimensional and previous two-dimensional solutions of the problem and effects of domain size and initial perturbations are discussed.

Streamwise variation in vessel geometry is a feature of many multiphase flows of practical interest, ranging from natural porous media flows to man-made lab-on-the-chip applications. The variable streamwise geometry typically exerts a dominant influence on bubble motion, and can lead to undesirable phenomena such as clogging of the vessel. Here, we study clogging in a fundamental configuration, where a tube of square cross-section is suddenly expanded over a short streamwise distance. The extent to which a bubble driven by constant flux flow broadens to partially fill the expansion depends on the balance between viscous and surface tension stresses, measured by the capillary number $\mathit{Ca}$. This broadening is accompanied by the slowing and momentary arrest of the bubble as $\mathit{Ca}$ is reduced towards its critical value for trapping. For $\mathit{Ca}\lt {\mathit{Ca}}_{c} $ the pressure drag forces on the quasi-arrested bubble are insufficient to force the bubble out of the expanded region so it remains trapped. We examine the conditions for trapping by varying bubble volume, flow rate of the carrier fluid, relative influence of gravity and length of expanded region. We find specifically that ${\mathit{Ca}}_{c} $ depends non-monotonically on the size of the bubble. We verify, with experiments and a capillary static model, that a bubble is released if the work of the pressure forces over the length of the trap exceeds the surface energy required for the trapped bubble to reenter the constricted square tube.

The instability due to resonant interactions of finite-amplitude water waves is examined in the long-wave limit. In contrast to the well-known case of a small-amplitude limit in which the resonance is considered for a flat surface, here we consider a periodic approximate of the finite-amplitude solitary wave which is the long-wave limit of the periodic wave. The resonance conditions for the corresponding perturbations yield a new family of resonance curves that are totally different from those of the small-amplitude limit obtained by Phillips and Mclean. Under these resonance conditions, we conduct a systematic asymptotic analysis for small wavenumbers to obtain the growth rates of the perturbations explicitly and clarify whether each resonance curve is associated with instability. These results are verified numerically by showing that the instability bands for finite-amplitude periodic waves in shallow water are located along these unstable resonance curves.

In this work the purely electro-osmotic flow of a viscoelastic liquid, which obeys the simplified Phan-Thien–Tanner (sPTT) constitutive equation, is solved numerically and asymptotically by using the lubrication approximation. The analysis includes Joule heating effects caused by an imposed electric field, where the viscosity function, relaxation time and electrical conductivity of the liquid are assumed to be temperature-dependent. Owing to Joule heating effects, temperature gradients in the liquid make the fluid properties change within the microchannel, altering the electric potential and flow fields. A consequence of the above is the appearance of an induced pressure gradient along the microchannel, which in turn modifies the normal plug-like electro-osmotic velocity profiles. In addition, it is pointed out that, depending on the fluid rheology and the used values of the dimensionless parameters, the velocity, temperature and pressure profiles in the fluid are substantially modified. Also, the finite thermal conductivity of the microchannel wall was considered in the analysis. The dimensionless temperature profiles in the fluid and the microchannel wall are obtained as function of the dimensionless parameters involved in the analysis, and the interactions between the coupled momentum, thermal energy and potential electric equations are examined in detail. A comparison between the numerical predictions and the asymptotic solutions was made, and reasonable agreement was found.

Two-dimensional direct numerical simulation (DNS) of the receptivity of a flat-plate boundary layer to temperature spottiness in the Mach 6 free stream is carried out. The influence of spottiness parameters on the receptivity process is studied. It is shown that the temperature spots propagating near the upper boundary-layer edge generate mode F inside the boundary layer. Further downstream mode F is synchronized with unstable mode S (Mack second mode) and excites the latter via the inter-modal exchange mechanism. Theoretical assessments of the mode F amplitude are made using the biorthogonal eigenfunction decomposition method. The DNS results agree with the theoretical predictions. If the temperature spots are initiated in the free stream and pass through the bow shock, the dominant receptivity mechanism is different. The spot–shock interaction leads to excitation of acoustic waves, which penetrate into the boundary layer and excite mode S. Numerical simulations show that this mechanism provides the instability amplitudes an order of magnitude higher than in the case of receptivity to the temperature spots themselves.

We consider long-time simulations of two-dimensional turbulence body forced by $\sin 4y\hat {\boldsymbol{x}} $ on the torus $(x, y)\in \mathop{[0, 2\mathrm{\pi} ] }\nolimits ^{2} $ with the purpose of extracting simple invariant sets or ‘exact recurrent flows’ embedded in this turbulence. Each recurrent flow represents a sustained closed cycle of dynamical processes which underpins the turbulence. These are used to reconstruct the turbulence statistics using periodic orbit theory. The approach is found to be reasonably successful at a low value of the forcing where the flow is close to but not fully in its asymptotic (strongly) turbulent regime. Here, a total of 50 recurrent flows are found with the majority buried in the part of phase space most populated by the turbulence giving rise to a good reproduction of the energy and dissipation p.d.f. However, at higher forcing amplitudes now in the asymptotic turbulent regime, the generated turbulence data set proves insufficiently long to yield enough recurrent flows to make viable predictions. Despite this, the general approach seems promising providing enough simulation data is available since it is open to extensive automation and naturally generates dynamically important exact solutions for the flow.