One way that large-scale oceanic internal waves transfer their energy to small-scale mixing is through parametric subharmonic instability (PSI). But there is a disconnect between theory, which assumes the waves are periodic in space and time, and reality, in which waves are transient and localized. The innovative laboratory experiments and analysis techniques of Bourget et al. (J. Fluid Mech., vol. 723, 2013, pp. 1–20) show that theory can be applied to interpret the generation of subharmonic disturbances from a quasi-monochromatic wave beam. Their methodology and results open up new avenues of investigation into PSI through experiments, simulations and observations.

A laboratory investigation has been undertaken to quantify water wave attenuation rates as a function of rainfall rate. Vertical artificial rainfall is shown to generate weak near-surface velocity fluctuations that decline systematically away from the free surface and are independent of rainfall rate across the range of rainfall rates investigated (40–$170~\mathrm{mm} ~{\mathrm{h} }^{- 1} $). In the absence of rain, the observed attenuation of gravity waves is at levels consistent with classical viscous theory, but with a systematic finite-amplitude effect observed above a mean steepness of 0.10. Wave attenuation rates were found to be independent of the mean wave steepness and identical when artificial rainfall rates of 108 and $141~\mathrm{mm} ~{\mathrm{h} }^{- 1} $ were applied. Reassessment of complementary theoretical and experimental studies of individual droplets impacting on undisturbed water surfaces indicates that above a weak threshold rainfall rate of $30~\mathrm{mm} ~{\mathrm{h} }^{- 1} $, the surface irradiation becomes so frequent that droplet-generated violent surface motions directly interact with the incoming droplets. Present evidence is that a matching of time scales develops between the incoming surface irradiation and surface water motions generated by antecedent droplets as the rainfall rate increases. Consequently, at high rainfall rates, a highly dissipative surface regime is created that transmits little of the incident rainfall kinetic energy to the aqueous layers below. Rainfall-induced wave attenuation rates are compared with measurements of other wave attenuation processes to obtain a hierarchy of strength in both the laboratory and the field. Comparison is also made with wave attenuation theories that incorporate momentum and energy flux considerations. Rain-induced wave attenuation rates are weak or very strong depending on whether they are expressed in terms of energy scaling obtained from above or below the surface respectively, due to the high dissipation rate that occurs in the vicinity of the interface.

Experiments were performed in dump-stabilized axisymmetric flames to assess turbulent flame speeds (${S}_{T} $) and mean flamelets speeds (stretched laminar flame speeds, ${S}_{L, k} $). Fuels with significantly different thermodiffusive properties have been investigated, ranging from pure methane to syngas (${\mathrm{H} }_{2} \text{{\ndash}} \mathrm{CO} $ blends) and pure hydrogen, while the pressure was varied from 0.1 to 1.25 MPa. Flame front corrugation was measured with planar laser-induced fluorescence (PLIF) of the OH radical, while turbulence quantities were determined with particle image velocimetry (PIV). Two different analyses based on mass balance were performed on the acquired flame images. The first method assessed absolute values of turbulent flame speeds and the second method, by means of an improved fractal methodology, provided normalized turbulent flame speeds (${S}_{T} / {S}_{L, k} $). Deduced average Markstein numbers exhibited a strong dependence on pressure and hydrogen content of the reactive mixture. It was shown that preferential-diffusive-thermal (PDT) effects acted primarily on enhancing the stretched laminar flame speeds rather than on increasing the flame front corrugations. Interaction between flame front and turbulent eddies measured by the fractal dimension was shown to correlate with the eddy temporal activity.

We treat the dielectric decrement induced by excess ion polarization as a source of ion specificity and explore its impact on electrokinetics. We employ a modified Poisson–Nernst–Planck (PNP) model accounting for the dielectric decrement. The dielectric decrement is determined by the excess-ion-polarization parameter $\alpha $ and when $\alpha = 0$ the standard PNP model is recovered. Our model shows that ions saturate at large zeta potentials $(\zeta )$. Because of ion saturation, a condensed counterion layer forms adjacent to the charged surface, introducing a new length scale, the thickness of the condensed layer $({l}_{c} )$. For the electro-osmotic mobility, the dielectric decrement weakens the electro-osmotic flow owing to the decrease of the dielectric permittivity. At large $\zeta $, when $\alpha \not = 0$, the electro-osmotic mobility is found to be proportional to $\zeta / 2$, in contrast to $\zeta $ as predicted by the standard PNP model. This is attributed to ion saturation at large $\zeta $. In terms of the electrophoretic mobility ${M}_{e} $, we carry out both an asymptotic analysis in the thin-double-layer limit and solve the full modified PNP model to compute ${M}_{e} $. Our analysis reveals that the impact of the dielectric decrement is intriguing. At small and moderate $\zeta ~({\lt }6kT/ e)$, the dielectric decrement decreases ${M}_{e} $ with increasing $\alpha $. At large $\zeta $, it is known that the surface conduction becomes significant and plays an important role in determining ${M}_{e} $. It is observed that the dielectric decrement effectively reduces the surface conduction. Hence in stark contrast, ${M}_{e} $ increases as $\alpha $ increases. Our predictions of the contrast dependence of the mobility on $\alpha $ at different zeta potentials qualitatively agree with experimental results on the dependence of the mobility among ions and provide a possible explanation for such ion specificity. Finally, the comparisons between the thin-double-layer asymptotic analysis and the full simulations of the modified PNP model suggest that at large $\zeta $ the validity of the thin-double-layer approximation is determined by ${l}_{c} $ rather than the traditional Debye length.

This paper addresses the dam-break problem for particle suspensions, that is, the flow of a finite volume of suspension released suddenly down an inclined flume. We were concerned with concentrated suspensions made up of neutrally buoyant non-colloidal particles within a Newtonian fluid. Experiments were conducted over wide ranges of slope, concentration and mass. The major contributions of our experimental study are the simultaneous measurement of local flow properties far from the sidewalls (velocity profile and, with lower accuracy, particle concentration) and macroscopic features (front position, flow depth profile). To that end, the refractive index of the fluid was adapted to closely match that of the particles, enabling data acquisition up to particle volume fractions of 60 %. Particle migration resulted in the blunting of the velocity profile, in contrast to the parabolic profile observed in homogeneous Newtonian fluids. The experimental results were compared with predictions from lubrication theory and particle migration theory. For solids fractions as large as 45 %, the flow behaviour did not differ much from that of a homogeneous Newtonian fluid. More specifically, we observed that the velocity profiles were closely approximated by a parabolic form and there was little evidence of particle migration throughout the depth. For particle concentrations in the 52–56 % range, the flow depth and front position were fairly well predicted by lubrication theory, but taking a closer look at the velocity profiles revealed that particle migration had noticeable effects on the shape of the velocity profile (blunting), but had little impact on its strength, which explained why lubrication theory performed well. Particle migration theories (such as the shear-induced diffusion model) successfully captured the slow evolution of the velocity profiles. For particle concentrations in excess of 56 %, the macroscopic flow features were grossly predicted by lubrication theory (to within 20 % for the flow depth, 50 % for the front position). The flows seemed to reach a steady state, i.e. the shape of the velocity profile showed little time dependence.

The two-dimensional problem of an elastic-plate impact onto an undisturbed surface of water of infinite depth is analysed. The plate is forced to move with a constant horizontal velocity component which is much larger than the vertical velocity component of penetration. The small angle of attack of the plate and its vertical velocity vary in time, and are determined as part of the solution, together with the elastic deflection of the plate and the hydrodynamic loads within the potential flow theory. The boundary conditions on the free surface and on the wetted part of the plate are linearized and imposed on the initial equilibrium position of the liquid surface. The wetted part of the plate depends on the plate motion and its elastic deflection. To determine the length of the wetted part we assume that the spray jet in front of the advancing plate is negligible. A smooth separation of the free-surface flow from the trailing edge is imposed. The wake behind the moving body is included in the model. The plate deflection is governed by Euler’s beam equation, subject to free–free boundary conditions. Four different regimes of plate motion are distinguished depending on the impact conditions: (a) the plate becomes fully wetted; (b) the leading edge of the plate touches the water surface and traps an air cavity; (c) the free surface at the forward contact point starts to separate from the plate; (d) the plate exits the water. We could not detect any impact conditions which lead to steady planing of the free plate after the impact. It is shown that a large part of the total energy in the fluid–plate interaction leaves the main bulk of the liquid with the spray jet. It is demonstrated that the flexibility of the plate may increase the hydrodynamic loads acting on it. The impact loads can cause large bending stresses, which may exceed the yield stress of the plate material. The elastic vibrations of the plate are shown to have a significant effect on the fluid flow in the wake.

The diffusion of red blood cells (RBCs) in blood is important to the physiology and pathology of the cardiovascular system. In this study, we investigate flow-induced diffusion of RBCs in a semi-dilute system by calculating the pairwise interactions between RBCs in simple shear flow. A capsule with a hyperelastic membrane was used to model an RBC. Its deformation was resolved using the finite element method, whereas fluid motion inside and outside the RBC was solved using the boundary element method. The results show that shear-induced RBC diffusion is significantly anisotropic, i.e. the velocity gradient direction component is larger than the vorticity direction. We also found that the motion of RBCs during the interaction is strongly dependent on the viscosity ratio of the internal to external fluid, and the diffusivity decreases monotonically as the viscosity ratio increases. The scaling argument also suggests that the diffusivity is proportional to the shear rate and haematocrit, if the suspension is in a semi-dilute environment and the capillary number is invariant. These fundamental findings are useful to understand transport phenomena in blood flow.

Rayleigh–Bénard convection in glycerol (Prandtl number $\mathit{Pr}= 2547. 9$) in a cylindrical cell with an aspect ratio of $\Gamma = 1$ was studied by means of three-dimensional direct numerical simulations (DNS). For that purpose, we implemented temperature-dependent material properties into our DNS code, by prescribing polynomial functions up to seventh order for the viscosity, the heat conductivity and the density. We performed simulations with the common Oberbeck–Boussinesq (OB) approximation and with non-Oberbeck–Boussinesq (NOB) effects within a range of Rayleigh numbers of $1{0}^{5} \leq \mathit{Ra}\leq 1{0}^{9} $. For the highest temperature differences, $\Delta = 80~\mathrm{K} $, the viscosity at the top is ${\sim }360\hspace{0.167em} \% $ times higher than at the bottom, while the differences of the other material properties are less than $15\hspace{0.167em} \% $. We analysed the temperature and velocity profiles and the thermal and viscous boundary-layer thicknesses. NOB effects generally lead to a breakdown of the top–bottom symmetry, typical for OB Rayleigh–Bénard convection. Under NOB conditions, the temperature in the centre of the cell ${T}_{c} $ increases with increasing $\Delta $ and can be up to $15~\mathrm{K} $ higher than under OB conditions. The comparison of our findings with several theoretical and empirical models showed that two-dimensional boundary-layer models overestimate the actual ${T}_{c} $, while models based on the temperature or velocity scales predict ${T}_{c} $ very well with a standard deviation of $0. 4~\mathrm{K} $. Furthermore, the obtained temperature profiles bend closer towards the cold top plate and further away from the hot bottom plate. The situation for the velocity profiles is reversed: they bend farther away from the top plate and closer towards to the bottom plate. The top boundary layers are always thicker than the bottom ones. Their ratio is up to 2.5 for the thermal and up to 4.5 for the viscous boundary layers. In addition, the Reynolds number $\mathit{Re}$ and the Nusselt number $\mathit{Nu}$ were investigated: $\mathit{Re}$ is higher and $\mathit{Nu}$ is lower under NOB conditions. The Nusselt number $\mathit{Nu}$ is influenced in a nonlinear way by NOB effects, stronger than was suggested by the two-dimensional simulations. The actual scaling of $\mathit{Nu}$ with $\mathit{Ra}$ in the NOB case is $\mathit{Nu}\propto {\mathit{Ra}}^{0. 298} $ and is in excellent agreement with the experimental data.

In this paper we develop reduced-order models for the unsteady lift on a pitching and plunging aerofoil over a range of angles of attack. In particular, we analyse the pitching and plunging dynamics for two cases: a two-dimensional flat plate at $\mathit{Re}= 100$ using high-fidelity direct numerical simulations and a three-dimensional NACA 0006 aerofoil at $\mathit{Re}= 65\hspace{0.167em} 000$ using wind-tunnel measurements. Models are obtained at various angles of attack and they are verified against measurements using frequency response plots and large-amplitude manoeuvres. These models provide a low-dimensional balanced representation of the relevant unsteady fluid dynamics. In simulations, flow structures are visualized using finite-time Lyapunov exponents. A number of phenomenological trends are observed, both in the data and in the models. As the base angle of attack increases, the boundary layer begins to separate, resulting in a decreased quasi-steady lift coefficient slope and a delayed relaxation to steady state at low frequencies. This extends the low-frequency range of motions that excite unsteady effects, meaning that the quasi-steady approximation is not valid until lower frequencies than are predicted by Theodorsen’s classical inviscid model. In addition, at small angles of attack, the lift coefficient rises to the steady-state value after a step in angle, while at larger angles of attack, the lift coefficient relaxes down to the steady-state after an initially high lift state. Flow visualization indicates that this coincides with the formation and convection of vortices at the leading edge and trailing edge. As the angle of attack approaches the critical angle for vortex shedding, the poles and zeros of the model approach the imaginary axis in the complex plane, and some zeros cross into the right half plane. This has significant implications for active flow control, which are discussed. These trends are observed in both simulations and wind-tunnel data.

We use ultra-high-speed video imaging to look at the initial contact of a drop impacting on a liquid layer. We observe experimentally the vortex street and the bubble-ring entrapments predicted numerically, for high impact velocities, by Thoraval et al. (Phys. Rev. Lett., vol. 108, 2012, article 264506). These dynamics mainly occur within $50~\mathrm{\mu} \mathrm{s} $ after the first contact, requiring imaging at 1 million f.p.s. For a water drop impacting on a thin layer of water, the entrapment of isolated bubbles starts through azimuthal instability, which forms at low impact velocities, in the neck connecting the drop and pool. For Reynolds number $Re$ above ${\sim }12\hspace{0.167em} 000$, up to 10 partial bubble rings have been observed at the base of the ejecta, starting when the contact is ${\sim }20\hspace{0.167em} \% $ of the drop size. More regular bubble rings are observed for a pool of ethanol or methanol. The video imaging shows rotation around some of these air cylinders, which can temporarily delay their breakup into micro-bubbles. The different refractive index in the pool liquid reveals the destabilization of the vortices and the formation of streamwise vortices and intricate vortex tangles. Fine-scale axisymmetry is thereby destroyed. We show also that the shape of the drop has a strong influence on these dynamics.

A fully instrumented microscale shock tube, believed to be the smallest to date, has been fabricated and tested. This facility is used to study the transmission of a shock wave, produced in a large (37 mm) shock tube, into a 34 $\mathrm{\mu} \mathrm{m} $ hydraulic diameter and 2 mm long microchannel. Pressure microsensors of a novel design, with gigahertz bandwidth, are used to obtain pressure–time histories of the microchannel shock wave at five axial stations. In all cases the transmitted shock wave is found to be weaker than the incident shock wave, and is observed to decay both in pressure and velocity as it propagates down the microchannel. These results are compared with various analytical and numerical models, and the best agreement is obtained with a Navier–Stokes computational fluid dynamics computation, which assumes a no-slip isothermal wall boundary condition; good agreement is also obtained with a simple shock tube laminar boundary layer model. It is also found that the flow developing within the microchannel is highly dependent on conditions at the microchannel entrance, which control the mass flux entering into the device. Regardless of the micrometre dimensions of the present facility, shock wave propagation in a microchannel of that scale exhibits a behaviour similar to that observed in large-scale facilities operated at low pressures, and the shock attenuation can be explained in terms of accepted laminar boundary models.

The turbulent flow over a forward-facing step is studied using two-dimensional time-resolved particle image velocimetry. The structure and behaviour of the separation region in front of the step is investigated using conditional averages based on the area of reverse flow present. The relation between the position of the upstream separation and the two-dimensional shape of the separation region is presented. It is shown that when of ‘closed’ form, the separation region can become unstable resulting in the ejection of fluid over the corner of the step. The separation region is shown to grow simultaneously in both the wall-normal and streamwise directions, to a point where the maximum extent of the upstream position of separation is limited by the accompanying transfer of mass over the step corner. The conditional averages are traced backwards in time to identify the average behaviour of the boundary-layer displacement thickness leading up to such events. It is shown that these ejections are preceded by the convection of low-velocity regions from upstream, resulting in a three-dimensional interaction within the separation region. The size of the low-velocity regions, and the time scale at which the separation region fluctuates, is shown to be consistent with the large boundary layer structures observed in the literature. Instances of a highly suppressed separation region are accompanied by a steady increase in velocity in the upstream boundary layer.

The instability characteristics and resonance of a natural convection boundary layer adjacent to an isothermally heated vertical surface are investigated using direct stability analyses. The detailed streamwise evolution of the boundary-layer frequencies is visualized via the power spectra of the temperature time series in the thermal boundary layer. It is found that the entire thermal boundary layer may be divided into three distinct regions according to the frequency profile, which include an upstream low-frequency region, a transitional region (with both low- and high-frequency bands) and a downstream high-frequency region. The high-frequency band in the downstream region determines the resonance characteristics of the thermal boundary layer, which can be triggered by a single-mode perturbation at frequencies within the high-frequency band. The single-mode perturbation experiments further reveal that the maximum resonance of the thermal boundary layer is triggered by a perturbation at the characteristic frequency of the boundary layer. For the boundary-layer flow at $\mathit{Ra}= 3. 6\times 1{0}^{10} $ and $\mathit{Pr}= 7$, a net heat transfer enhancement of up to 44 % is achieved by triggering resonance of the boundary layer. This significant enhancement of heat transfer is due to the resonance-induced advancement of the laminar–turbulent transition, which is found to be dependent on the perturbation frequency and amplitude. Evidence from different perspectives revealing the same position of the transition are provided and discussed. The outcomes of this investigation demonstrate the prospect of a resonance-based approach for enhancing heat transfer.

We present a detailed experimental investigation of conditional statistics obtained from dissipation elements based on the passive scalar field $\theta $ and its instantaneous scalar dissipation rate $\chi $. Using high-frequency planar Rayleigh scattering measurements of propane discharging as a round turbulent jet into coflowing carbon dioxide, we acquire with Taylor’s hypothesis a highly resolved three-dimensional field of the propane mass fraction $\theta $. The Reynolds number (based on nozzle diameter and jet exit velocity) varies between 3000 and 8600. The experimental results for the joint probability density of the scalar difference $ \mathrm{\Delta} \theta $ and the length $l$ of dissipation elements resembles those previously obtained from direct numerical simulations of Wang & Peters (J. Fluid Mech., vol. 554, 2006, pp. 457–475). In addition, the normalized marginal probability density function $\tilde {P} (\tilde {l} )$ of the length of dissipation elements follows closely the theoretical model derived by Wang & Peters (J. Fluid Mech., vol. 608, 2008, pp. 113–138). We also find that the mean linear distance ${l}_{m} $ between two extreme points of an element is of the order of the scalar Taylor microscale ${\lambda }_{u} $. Furthermore, the conditional mean $\langle \mathrm{\Delta} \theta \vert l\rangle $ scales with Kolmogorov’s $1/ 3$ power law. The investigation of the orientation of long dissipation elements in the jet flow reveals a preferential alignment, perpendicular to the streamwise direction for long elements, while the orientation of short elements is close to isotropic. Following an approach proposed by Kholmyansky & Tsinober (Phys. Lett. A, vol. 373, 2009, pp. 2364–2367), we finally investigate the probability density function of the scalar increment $\delta \theta $ in the streamwise direction, when strong dissipative events are either retained in or excluded from the measurement volume. In the present study, however, these events are related to maximum points of the scalar dissipation rate $\chi $ together with their local extent. When these regions are excluded from the scalar field, we observe a tendency of the probability density function $P(\delta \theta (r))$ towards a Gaussian bell-shaped curve.

When traditional linearized theory is used to study gravity–capillary waves produced by flow past an obstruction, the geometry of the object is assumed to be small in one or several of its dimensions. In order to preserve the nonlinear nature of the obstruction, asymptotic expansions in the low-Froude-number or low-Bond-number limits can be derived, but here, the solutions invariably predict a waveless surface at every order. This is because the waves are in fact, exponentially small, and thus beyond-all-orders of regular asymptotics; their formation is a consequence of the divergence of the asymptotic series and the associated Stokes Phenomenon. By applying techniques in exponential asymptotics to this problem, we have discovered the existence of new classes of gravity–capillary waves, from which the usual linear solutions form but a special case. In this paper, we present the initial theory for deriving these waves through a study of gravity–capillary flow over a linearized step. This will be done using two approaches: in the first, we derive the surface waves using the standard method of Fourier transforms; in the second, we derive the same result using exponential asymptotics. Ultimately, these two methods give the same result, but conceptually, they offer different insights into the study of the low-Froude-number, low-Bond-number problem.

When traditional linearized theory is used to study gravity–capillary waves produced by flow past an obstruction, the geometry of the object is assumed to be small in one or several of its dimensions. In order to preserve the nonlinear nature of the obstruction, asymptotic expansions in the low-Froude-number or low-Bond-number limits can be derived, but here, the solutions are waveless to every order. This is because the waves are in fact, exponentially small, and thus beyond-all-orders of regular asymptotics; their formation is a consequence of the divergence of the asymptotic series and the associated Stokes Phenomenon. In Part 1 (Trinh & Chapman, J. Fluid Mech., vol. 724, 2013b, pp. 367–391), we showed how exponential asymptotics could be used to study the problem when the size of the obstruction is first linearized. In this paper, we extend the analysis to the nonlinear problem, thus allowing the full geometry to be considered at leading order. When applied to the classic problem of flow over a step, our analysis reveals the existence of six classes of gravity–capillary waves, two of which share a connection with the usual linearized solutions first discovered by Rayleigh. The new solutions arise due to the availability of multiple singularities in the geometry, coupled with the interplay of gravitational and cohesive effects.

This paper presents an extension of existing works dealing with the dynamics of a passive scalar in freely decaying isotropic turbulence, by accounting for a production mechanism of the passive scalar itself. The physically relevant case of the temperature dynamics in the presence of Joule heating via the dissipation of the turbulent kinetic energy is selected and analysed by theoretical and numerical means. In particular, the sensitivity of the temperature decay to the non-dimensional parameters Prandtl number ($\mathit{Pr}$) and Eckert number ($\mathit{Ec}$), the latter measuring the intensity of the internal energy production mechanism, is investigated. The time behaviour of the global quantities such as the temperature variance $ \overline{{\theta }^{2} } (t)$ and its destruction rate ${\varepsilon }_{\theta } (t)$ is analysed, and a detailed analysis of the temperature variance spectrum ${E}_{\theta } (k)$ is provided. In the case of a very strong heating mechanism, some important modifications of the temperature dynamics are observed. The time-decay-law exponents of the global physical quantities assume new values, which are governed only by features of the kinetic energy spectrum, while they depend on the shape of ${E}_{\theta } (k)$ in the classical free-decay case. The temperature variance spectrum ${E}_{\theta } (k)$ exhibits two new spectral ranges. One is a convective–production range such that ${E}_{\theta } (k)\propto {k}^{1/ 3} $ is observed for a finite time at all values of $\mathit{Pr}$. In the case of very diffusive fluids with $\mathit{Pr}\ll 1$, a convective–diffusive–production range with ${E}_{\theta } (k)\propto {k}^{- 7/ 3} $ is also detected.

We describe new experiments to examine the buoyancy-induced turbulent mixing which results from the injection of a small constant volume flux of dense fluid at the top of a long narrow vertical tank with square cross-section, in which a steady laminar upward flow of less dense fluid is present. To conserve volume of fluid in the tank, fluid leaves the tank through two small openings near the top of the tank. Dense source fluid vigorously mixes with the less dense fluid of the upward flow, such that a dense mixing region of turbulent fluid propagates downwards during the transient mixing phase of the experiment. Eventually, the transport of dense fluid associated with the buoyancy-induced turbulent flow is balanced by the transport of less-dense fluid associated with the steady upward flow, such that the mixing region evolves into a layer of finite extent which stays approximately constant in height during a statistically steady mixing phase of the experiment. With an ideal source of downward constant buoyancy flux ${B}_{s} $ at the top of the tank, tank width $d$, and speed of the upward flow ${u}_{u} $, we perform experiments with Froude numbers $\mathit{Fr}= {u}_{u} {d}^{1/ 3} / { B}_{s}^{1/ 3} $ ranging between $O(0. 01)$ and $O(1)$. The steady-state height of the mixing region and the maximum reduced gravity as found near the source of buoyancy flux at the top of the tank increase with decreasing Froude number. For the experiments with intermediate values of the Froude number, we find that the steady-state mixing region is small enough to be contained in the experimental tank, but large enough not to be dominated by developing turbulence near the source of buoyancy flux. For these experiments, we show that the key buoyancy-induced turbulent mixing properties are not significantly affected by the upward flow. We use a dye-attenuation technique to obtain vertical profiles of the time- and horizontally averaged reduced gravity to show a good agreement between the experimental profiles and the solution of a nonlinear turbulent advection–diffusion equation during the steady mixing phase. Furthermore, we discuss the characteristic time scale of the transient mixing phase. We compare our experimental results with the numerical solution of a time-dependent nonlinear turbulent advection–diffusion equation during the transient mixing phase. We also describe three reduced models for the evolution of the reduced gravity distribution in the mixing region, and we demonstrate these models’ usefulness by comparison with our experimental results and the numerical solution of the time-dependent nonlinear turbulent advection–diffusion equation.

The onset and development of turbulence from controlled disturbances in compressible ($\mathit{Ma}= 0. 2$), flat-plate boundary layers is studied by direct numerical simulation. We have validated the initial disturbance development, confirmed that H- and K-regime transitions were reproduced and, from these starting points, we carried these simulations beyond breakdown, past the skin-friction maximum and to higher Reynolds numbers than investigated before to evaluate how these two flow regimes converge towards turbulence and what transitional flow structures embody the statistics and mean dynamics of developed turbulence. We show that H- and K-type breakdowns both relax toward the same statistical structure typical of developed turbulence at high Reynolds number immediately after the skin-friction maximum. This threshold marks the onset of self-sustaining mechanisms of near-wall turbulence. At this point, computed power spectra exhibit a decade of Kolmogorov inertial subrange; this is further evidence of convergence to equilibrium turbulence at the late stage of transition. Here, visualization of the instantaneous flow structure shows numerous, tightly packed hairpin vortices (Adrian, Phys. Fluids, vol. 19, 2007, 041301). Strongly organized coherent hairpin structures are less perceptible farther downstream (at higher Reynolds numbers), but the flow statistics and near-wall dynamics are the same. These structurally simple hairpin-packet solutions found in the very late stages of H- and K-type transitions obey the statistical measurements of higher-Reynolds-number turbulence. Comparison with the bypass transition of Wu & Moin (Phys. Fluids, vol. 22, 2010, pp. 85–105) extends these observations to a wider class of transitional flows. In contrast to bypass transition, the (time- and spanwise-averaged) skin-friction maximum in both H- and K-type transitions overshoots the turbulent correlation. Downstream of these friction maxima, all three skin-friction profiles collapse when plotted versus the momentum-thickness Reynolds number, ${\mathit{Re}}_{\theta } $. Mean velocities, turbulence intensities and integral parameters collapse generally beyond ${\mathit{Re}}_{\theta } = 900$ in each transition scenario. Skin-friction maxima, organized hairpin vortices and the onset of self-sustaining turbulence found in controlled H- and K-type transitions are, in many dynamically important respects, similar to development of turbulent spots seen by Park et al. (Phys. Fluids, vol. 24, 2012, 035105). A detailed statistical comparison demonstrates that each of these different transition scenarios evolve into a unique force balance characteristic of higher-Reynolds-number turbulence (Klewicki, Ebner & Wu, J. Fluid Mech., vol. 682, 2011, pp. 617–651). We postulate that these dynamics of late-stage transition as manifested by hairpin packets can serve as a reduced-order model of high-Reynolds-number turbulent boundary layers.

A numerical study is conducted to investigate the influence of a uniform axial magnetic field on the global linear stability of the rotating-disc boundary layer. Simulation results obtained using a radially homogenized base flow were found to be in excellent agreement with an earlier linear stability analysis, which indicated that an axial magnetic field can locally suppress both convective and absolute instabilities. However, the numerical results obtained for the genuine, radially inhomogeneous, flow indicate that a global form of instability develops for sufficiently large magnetic fields. The qualitative nature of the global instability is similar to that which was observed in a previous study, where mass suction was applied at the rotating disc surface. It is shown that, just as for the case with mass suction, it is possible to explain the promotion of global instability by considering a model that includes detuning effects, which are associated with the radial variation of locally defined absolute temporal frequencies. The recurrence of the same type of instability behaviour when two distinct flow control strategies are implemented, one using suction and the other an axial magnetic field, indicates that the phenomena described by the model may be considered generic.