We present an experimental study of Rayleigh–Bénard convection using liquid metal alloy gallium-indium-tin as the working fluid with a Prandtl number of $Pr=0.029$. The flow state and the heat transport were measured in a Rayleigh number range of $1.2\times 10^{4} \le Ra \le 1.3\times 10^{7}$. The temperature fluctuation at the cell centre is used as a proxy for the flow state. It is found that, as $Ra$ increases from the lower end of the parameter range, the flow evolves from a convection state to an oscillation state, a chaotic state and finally a turbulent state for $Ra>10^5$. The study suggests that the large-scale circulation in the turbulent state is a residual of the cell structure near the onset of convection, which is in contrast with the case of $Pr\sim 1$, where the cell structure is transiently replaced by high order flow modes before the emergence of the large-scale circulation in the turbulent state. The evolution of the flow state is also reflected by the heat transport characterised by the Nusselt number $Nu$ and the probability density function (p.d.f.) of the temperature fluctuation at the cell centre. It is found that the effective local heat transport scaling exponent $\gamma$, i.e. $Nu\sim Ra^{\gamma }$, changes continuously from $\gamma =0.49$ at $Ra\sim 10^4$ to $\gamma =0.25$ for $Ra>10^6$. Meanwhile, the p.d.f. at the cell centre gradually evolves from a Gaussian-like shape before the transition to turbulence to an exponential-like shape in the turbulent state. For $Ra>10^6$, the flow shows self-similar behaviour, which is revealed by the universal shape of the p.d.f. of the temperature fluctuation at the cell centre and a $Nu=0.19Ra^{0.25}$ scaling for the heat transport.

An isolated charge-neutral droplet in a uniform electric field experiences no net force. However, a droplet pair can move in response to field-induced dipolar and hydrodynamic interactions. If the droplets are identical, the centre of mass of the pair remains fixed. Here, we show that if the droplets have different properties, the pair experiences a net motion due to non-reciprocal electrohydrodynamic interactions. We analyse the three-dimensional droplet trajectories using asymptotic theory, assuming spherical droplets and large separations, and numerical simulations based on a boundary integral method. The dynamics can be quite intricate depending on the initial orientation of the droplets line-of-centres relative to the applied field direction. Drops tend to migrate towards a configuration with line-of-centres either parallel or perpendicular to the applied field direction, while either coming into contact or indefinitely separating. We elucidate the conditions under which these different interaction scenarios take place. Intriguingly, we find that in some cases droplets can form a pair (tandem) that translates either parallel or perpendicular to the applied field direction.

We consider nonlinear dynamical systems driven by stochastic forcing. It has been largely evidenced in the literature that the linear response of non-normal systems (e.g. fluid flows) may exhibit a large variance amplification, even in a linearly stable regime. This linear response, however, is relevant only in the limit of vanishing forcing intensity. As the intensity increases, the frequency distribution and the variance of the response may be strongly modified due to nonlinear effects. To go beyond this limitation, we propose a theoretical approach to derive an amplitude equation governing the weakly nonlinear evolution of these systems. This approach does not rely on the presence of an eigenvalue close to the neutral axis, applying instead to any sufficiently non-normal operator, and the Fourier components of the response are allowed to be arbitrarily different from any eigenmode. The methodology is outlined for a generic nonlinear dynamical system, and the application case highlights a common non-normal mechanism in hydrodynamics: convective non-normal amplification in the flow past a backward-facing step.

Whereas streamwise effective slope ($ES_{x}$) is accepted as a key topographical parameter in the context of rough-wall turbulent flows, the significance of its spanwise counterpart ($ES_{y}$) remains largely unexplored. Here, the response of turbulent channel flow over irregular, three-dimensional rough walls with systematically varied values of $ES_{y}$ is studied using direct numerical simulation. All simulations were performed at a fixed friction Reynolds number 395, corresponding to a viscous-scaled roughness height $k^{+}\approx 65.8$ (where $k$ is the mean peak-to-valley height). A surface generation algorithm is used to synthesise a set of ten irregular surfaces with specified $ES_{y}$ for three different values of $ES_{x}$. All surfaces share a common mean peak-to-valley height and are near-Gaussian, which allows this study to focus on the impact of varying $ES_{y}$, since roughness amplitude, skewness and $ES_{x}$ can be eliminated simultaneously as parameters. Based on an analysis of first- and second-order velocity statistics, as well as turbulence co-spectra and the fractional contribution of pressure and viscous drag, the study shows that $ES_{y}$ can strongly affect the roughness drag penalty – particularly for low-$ES_{x}$ surfaces. A secondary observation is that particular low-$ES_{y}$ surfaces in this study can lead to diminished levels of outer-layer similarity in both mean flow and turbulence statistics, which is attributed to insufficient scale separation between the outer length scale and the in-plane spanwise roughness wavelength.

Turbulence amplification and the large-scale coherent structures in shock wave/turbulent boundary layer interaction flows have been studied at length in previous research, while the direct association between these two flow features is still lacking. In the present study, the transport equation of turbulent kinetic energy spectra is derived and utilized to analyse the scale-by-scale energy budget across the interaction zone, enabling us to reveal the association between the genesis of the large-scale motions and the turbulence amplification. For the presently considered flow with incipient shock-induced separation, we identified in turbulent kinetic energy spectra distribution that the most energetic motions are converted from the near-wall small-scale motions to large-scale motions consisting of velocity streaks and cross-stream circulations as they go through the interaction zone. The amplification of streamwise velocity fluctuation is triggered first, resulting in the emergence of large-scale velocity streaks, which is attributed to the adverse pressure gradient, as indicated by the spectra of the production term. The energy carried by large-scale velocity streaks is transferred to other velocity components by the pressure-strain term, producing large-scale cross-stream circulations. When large-scale motions are convected downstream, their energy is transferred via turbulent cascade to smaller scales and dissipated by viscosity. The spanwise uniform fluctuations, reminiscent of the unsteadiness of the separation bubble, are contributed primarily by the inter-scale energy transfer from the finite spanwise scale motions.

Using branch continuation in the FENE-P model, we show that finite-amplitude travelling waves borne out of the recently discovered linear instability of viscoelastic channel flow (Khalid et al., J. Fluid Mech., vol. 915, 2021, A43) are substantially subcritical reaching much lower Weissenberg ($Wi$) numbers than on the neutral curve at a given Reynolds ($Re$) number over $Re \in [0,3000]$. The travelling waves on the lower branch are surprisingly weak indicating that viscoelastic channel flow is susceptible to (nonlinear) instability triggered by small finite-amplitude disturbances for $Wi$ and $Re$ well below the neutral curve. The critical $Wi$ for these waves to appear in a saddle node bifurcation decreases monotonically from, for example, $\approx 37$ at $Re=3000$ down to $\approx 7.5$ at $Re=0$ at the solvent-to-total-viscosity ratio $\beta =0.9$. In this latter creeping flow limit, we also show that these waves exist at $Wi \lesssim 50$ for higher polymer concentrations, $\beta \in [0.5,0.97)$, where there is no known linear instability. Our results therefore indicate that these travelling waves, found in simulations and named ‘arrowheads’ by Dubief et al. (Phys. Rev. Fluids, vol. 7, 2022, 073301), exist much more generally in $(Wi,Re, \beta )$ parameter space than their spawning neutral curve and, hence, can either directly, or indirectly through their instabilities, influence the dynamics seen far away from where the flow is linearly unstable. Possible connections to elastic and elasto-inertial turbulence are discussed.

The present research will provide an analytical explanation to experiments destabilising turbulence in pipe flow reported in Kuehnen et al. (Nat. Phys., vol. 14, 2018, 386–390). Those experiments show four methods by which turbulence vanishes from steady-state pipe flow, without decreasing its bulk velocity, until it becomes completely laminar. The explanation is based on our theory of underlying laminar flow (TULF), which has already been successfully applied to account for other uncommon experiments reported in the literature. The TULF is founded on the Reynolds-averaged Navier–Stokes equations and thus is a theory of ensemble-averaged flows. The zero theorem for steady-state flow is introduced as a universal result that will help explain the laminarisation process described in experiments. After presenting the most comprehensive solution for the mean pipe flow governing equation that, to our knowledge, has ever been reported, we uncover a general sequence for laminarisation, called the laminarisation pattern, and we introduce a mathematical model for it. We show that a drastic decrease in a flow's mean-pressure gradient, while maintaining constant its Reynolds number, is necessary and sufficient to erase turbulence. Equations derived from our model are used to calculate the minimum pressure gradient necessary to cause complete laminarisation in each experiment. Results are then contrasted with reported experimental data, with noticeable agreement. We also propose a figure of merit to assess the efficiency of each laminarisation method. Having disclosed the intrinsic mechanism leading to complete laminarisation, we expect researchers will propose other ingenious methods to achieve it.

We consider axisymmetric Boussinesq convection in a shallow cylinder of radius $L$ and depth $H (\ll L)$, which rotates with angular velocity $\varOmega$ about its axis of symmetry aligned to the vertical. Constant heat flux boundary conditions, top and bottom, are adopted, for which the onset of instability occurs on a long horizontal length scale provided that $\varOmega$ is sufficiently small. We investigate the nonlinear development by well-established two-scale asymptotic expansion methods. Comparisons of the results with the direct numerical simulations (DNS) of the primitive governing equations are good at sufficiently large Prandtl number $\sigma$. As $\sigma$ is reduced, the finite amplitude range of applicability of the asymptotics reduces in concert. Though the large meridional convective cell, predicted by the DNS, is approximated adequately by the asymptotics, the azimuthal flow fails almost catastrophically, because of significant angular momentum transport at small $\sigma$, exacerbated by the cylindrical geometry. To appraise the situation, we propose hybrid methods that build on the meridional streamfunction $\psi$ derived from the asymptotics. With $\psi$ given, we solve the now linear azimuthal equation of motion for the azimuthal velocity $v$ by DNS. Our ‘hybrid’ methods enable us to explain features of the flow at large Rayleigh number, found previously by Oruba et al. (J. Fluid Mech., vol. 812, 2017, pp. 890–904).

One of the most dramatic effects of vibrational and chemical non-equilibrium in hypersonic flows occurs in the bow-shock detachment process in flow over a wedge. This was shown theoretically and in reflected shock tunnel experiments by Hornung & Smith (J. Fluid Mech., vol. 93, 1979, pp. 225–239). In the present work, the effect is first demonstrated by computation of two-dimensional non-equilibrium flows. The effect of the finite transverse extent of the wedge is then studied by three-dimensional computations of non-relaxing flows. An analytical formula is obtained that gives the shock detachment distance of a finite wedge for ideal-gas and equilibrium flows. In the experiment, the finite transverse extent of the wedge competes with the non-equilibrium effects, as each introduces a new length scale. The carbon dioxide and nitrogen flows of the experiment are therefore computed in three dimensions and with two-temperature chemistry accounting for vibrational and chemical non-equilibrium. In the case of nitrogen flow, the agreement between experiment and computation is not good, the experimental detachment distance being larger. A number of possible reasons are quantitatively examined. A conclusive resolution of the discrepancy is considered to require a repeat of the experiment with more accurately characterized conditions. In the case of the carbon dioxide experiments, the computed results agree remarkably well with experiment. This is partially due to the fact that the condition is very close to equilibrium, where the sensitivity of the detachment process to relaxation effects is small. The analytical expression for the dimensionless detachment distance agrees very well with all the three-dimensional computations of non-relaxing flows.

Rapidly rotating convection-driven dynamos are investigated under different kinematic and magnetic boundary conditions using direct numerical simulations. At a fixed rotation rate, represented by the Ekman number $Ek=5\times 10^{-7}$, the thermal forcing is varied from $2$ to $20$ times its value at the onset of convection ($\mathcal {R}=Ra/Ra_c=2\unicode{x2013}20$, where $Ra$ is the Rayleigh number), keeping the fluid properties constant ($Pr=Pr_m=1$, where $Pr$ and $Pr_m$ are the thermal and magnetic Prandtl numbers). The statistical behaviour of the dynamos, including the force balance, energetics, and heat transport, depends on the boundary conditions that dictate both the boundary layer and the interior dynamics. At a fixed thermal forcing ($\mathcal {R}=3$), the structure and strength of the magnetic field produced by the dynamos, especially near the walls, depend on both velocity and magnetic boundary conditions. Though the leading-order force balance in the bulk remains geostrophic, the Lorentz force becomes comparable to the Coriolis force inside the thermal boundary layer with no-slip, perfectly conducting conditions. In this case, a term signifying the work done by the Lorentz force in the turbulent kinetic energy (TKE) equation is found to have some components that extract energy from the velocity field to produce the magnetic field, while some other components extract energy from the magnetic field to produce TKE. However, with no-slip, pseudo-vacuum conditions, all the components of the work done by the Lorentz force perform unidirectional energy transfer to produce magnetic energy from the kinetic energy of the fluid to sustain dynamo action. We find heat transfer enhancement in the rotating dynamo convection, as compared with non-magnetic rotating convection, with the peak enhancement lying in the range $\mathcal {R}=3\unicode{x2013}4$. For free-slip conditions, in the absence of an Ekman layer, the dynamo action may alter the heat transport significantly by suppressing the formation of large-scale vortices. However, the highest heat transfer enhancement is found at $\mathcal {R}=3$ with no-slip, perfectly conducting walls, which can be attributed to a local magnetorelaxation of the rotational constraint due to enhanced Lorentz force inside the thermal boundary layer.

The aim of this paper is twofold: the first aim is to formulate and validate a multi-scale discrete Boltzmann method (DBM) based on density functional kinetic theory for thermal multiphase flow systems, ranging from continuum to transition flow regime; the second aim is to present some new insights into the thermo-hydrodynamic non-equilibrium (THNE) effects in the phase separation process. Methodologically, for bulk flow, DBM includes three main pillars: (i) the determination of the fewest kinetic moment relations, which are required by the description of significant THNE effects beyond the realm of continuum fluid mechanics; (ii) the construction of an appropriate discrete equilibrium distribution function recovering all the desired kinetic moments; (iii) the detection, description, presentation and analysis of THNE based on the moments of the non-equilibrium distribution ( $f-f^{(eq)}$). The incorporation of appropriate additional higher-order thermodynamic kinetic moments considerably extends the DBM's capability of handling larger values of the liquid–vapour density ratio, curbing spurious currents, and ensuring mass/momentum/energy conservation. Compared with the DBM with only first-order THNE (Gan et al., Soft Matt., vol. 11 (26), 2015, pp. 5336–5345), the model retrieves kinetic moments beyond the third-order super-Burnett level, and is accurate for weak, moderate and strong THNE cases even when the local Knudsen number exceeds $1/3$. Physically, the ending point of the linear relation between THNE and the concerned physical parameter provides a distinct criterion to identify whether the system is near or far from equilibrium. Besides, the surface tension suppresses the local THNE around the interface, but expands the THNE range and strengthens the THNE intensity away from the interface through interface smoothing and widening.

The present work investigates the linear instability of three-dimensional boundary layers in thermodynamically non-ideal regimes. As a representative fluid, we consider carbon dioxide at supercritical pressure (80 bar). The flow set-up is matched to the redesigned DLR (German Aerospace Center) experiment on cross-flow instability, with identical pressure-coefficient distribution (accelerating the flow), sweep angle and Reynolds number, at a low Mach number. The flow temperature relative to the Widom line – also known as the pseudocritical line – thus characterises the non-ideality of the flow. We consider supercritical (gas-like), subcritical (liquid-like) and transcritical (pseudoboiling) regimes, where the flow temperature remains above, below or crosses the Widom line. The stability analyses of the parabolised Navier–Stokes baseflows indicate that wall heating destabilises the flow in the supercritical regime while wall cooling stabilises both effects similar to the ideal-fluid situation but being stronger. On the contrary, wall heating/cooling exhibits reversed effects in the subcritical regime, like for an ideal liquid. In the transcritical regime, with its sharp gradients of the thermodynamic and transport properties, wall heating stabilises the flow. Most substantially, however, wall cooling provokes a changeover of the leading instability mechanism: the accelerated streamwise flow attains inflectional wall-normal profiles, and the invoked inviscid Tollmien–Schlichting instability prevails with growth rates up to one order of magnitude larger than those of the cross-flow mode. We establish a two-fold mathematical relation from the momentum equation that explains the consequence of non-ideality and wall heating/cooling. The streamwise perturbation patterns of the flows in their linear instability regime are shown by mimicking wave trains emanating from virtual point-disturbance sources. From the viewpoint of keeping laminar flows, the transcritical thermodynamic state with a cooling wall must be avoided.

We report a systematic experimental study of the mean temperature profile $\theta (\delta z)$ and temperature variance profile $\eta (\delta z)$ across a stable and immiscible liquid–liquid (water–FC770) interface formed in two-layer turbulent Rayleigh–Bénard convection. The measured $\theta (\delta z)$ and $\eta (\delta z)$ as a function of distance $\delta z$ away from the interface for different Rayleigh numbers are found to have the scaling forms $\theta (\delta z/\lambda )$ and $\eta (\delta z/\lambda )$, respectively, with varying thermal boundary layer (BL) thickness $\lambda$. By a careful comparison with the simultaneously measured BL profiles near a solid conducting surface, we find that the measured $\theta (\delta z)$ and $\eta (\delta z)$ near the liquid interface can be well described by the BL equations for a solid wall, so long as a thermal slip length $\ell _T$ is introduced to account for the convective heat flux passing through the liquid interface. Direct numerical simulation results further confirm that the turbulent thermal diffusivity $\kappa _t$ near a stable liquid interface has a complete cubic form, $\kappa _t(\xi )/\kappa \sim (\xi +\xi _0)^3$, where $\kappa$ is the molecular thermal diffusivity of the convecting fluid, $\xi =\delta z/\lambda$ is the normalized distance away from the liquid interface and $\xi _0$ is the normalized slip length associated with $\ell _T$.

The Richtmyer–Meshkov instability (RMI) results from the impulsive acceleration of a density interface where either it or the acceleration is perturbed. Density interfaces may arise due to a change in gas species, isotope, temperature or a combination of these. We computationally investigate the effect of interface type on the plasma RMI, which is relevant for a range of applications, including inertial confinement fusion. We simulate the evolution of single-mode perturbed thermal, species and isotope interfaces in an ideal ion–electron plasma using the multi-fluid plasma (MFP) model. We find that, in the MFP model, the evolution of different types of interface differs significantly, in contrast to single-fluid models where they behave similarly if the Atwood number is matched. The thermal and species interfaces produce the most severe response to shock acceleration, experiencing the secondary instabilities and enhanced primary mode growth. The isotope interface evolution is restrained in comparison with the former cases, resembling the response predicted by single-fluid models. The determining factor in the severity of the MFP RMI is the density ratio across the initial interface in the electron fluid, which is unity for an isotope interface. We observe that, as the density ratio across the electron interface decreases, so do the magnitudes of the self-generated fields and consequently the severity of the growth amplification. Generally, the evolution of the RMI with different types of interface becomes more similar as the level of coupling between the ion and electron fluids is increased, characterised by reducing the plasma non-dimensional skin depth.

A core unresolved challenge in rectified diffusion is the lack of predictive models, which allow accurate modelling of the growth of bubbles over millions of cycles of oscillation. The spherically symmetric problem involves two asymptotic regions in space for the convection and diffusion of the dissolved gas in the liquid, an inner region adjacent to the bubble and an outer region. It also involves three time scales: the shortest time scale associated with the bubble oscillation, the intermediate and the longest time scales for the mass transport across the bubble interface and the outer region, respectively. The problem is analysed systematically using matched asymptotic expansions in space and multi-scales in time, the large parameter being the Péclet number for mass transfer. A tractable mathematical model is formulated for rectified diffusion. It comprises the Rayleigh–Plesset equation for the spherical oscillations of the bubble on the shortest time scale, an ordinary differential equation for the mass of gas in the bubble on the intermediate time scale and a diffusion equation on the longest time scale and the long length scale. Predictions of this simplified model have excellent agreement with experimental results for a single spherical bubble in the bulk over millions of cycles of oscillation. A parametric study is undertaken with several new phenomena/features observed for acoustic frequencies in the range 12 to 62 kHz, acoustic field $0.12$ to $0.22$ bar and the initial gas concentration in the liquid being $0.95$ to $1.0$ of the saturation concentration.

We report that vertical vibration with small amplitude and high frequency can tame convective heat transport in Rayleigh–Bénard convection in a turbulent regime. When vertical vibration is applied, a dynamically averaged ‘anti-gravity’ results that stabilizes the thermal boundary layer and inhibits the eruption of thermal plumes. This eventually leads to the attenuation of the intensity of large-scale mean flow and a significant suppression of turbulent heat transport. Accounting for both the thermally led buoyancy and the vibration-induced anti-gravitational effects, we propose an effective Rayleigh number that helps to extend the Grossmann–Lohse theory to thermal vibrational turbulence. The prediction of the reduction on both the Nusselt and Reynolds numbers obtained by the extended model is found to agree well with the numerical data. In addition, vibrational influences on the mean flow structure and the temporal evolution of Nusselt and Reynolds numbers are investigated. The non-uniform characteristic of vibration-induced ‘anti-gravity’ is discussed. The present findings provide a powerful basis for studying thermal vibrational turbulence and put forward a novel strategy for actively controlling thermal turbulence.

Viscous effects on an ideal gas flow in a supersonic convergent–divergent nozzle are a well-studied subject in classical gas dynamics. However, the ideal assumption fails on fluids that exhibit complex behaviours such as near-critical-region and non-ideal dense vapours. Under those conditions, a realistic equation of state (EOS) plays a vital role for a precise and realistic computation. This work examines the problem for solving the quasi-one-dimensional viscous compressible flow using a realistic EOS. The governing equations are discretised and solved using the fourth-order Runge–Kutta method coupled with a state-of-the-art EOS to calculate the thermodynamic properties. The role of the Grüneisen parameter along with viscous and real gas effects and their influence on the sonic point formation are discussed. The study shows that the flow may not achieve the supersonic regime for any pressure ratio depending on the combination of that parameter and the normalised friction factor. In addition, the analysis yields the discharge coefficient and the isentropic nozzle efficiency, which may achieve maximum values as a function of the stagnation conditions. Finally, the study also evaluates the formation and intensity of normal shock waves by using the Rankine–Hugoniot relations, which now depend on the real gas and viscous effects in opposition to the inviscid solution. Moreover, the methodology used captures the sonic point and shock wave position by a space marching algorithm using the Brent method for scalar minimisation. Experimental data available in the open literature corroborate the approach.

The average wave drag in unsteady motion is studied experimentally with force measurements. Towing hulls of size $L$ at sinusoidal speed, the mean drag is measured for different amplitudes and frequencies of the fluctuating velocity, as well as different Froude numbers $\mathcal {F}_0$ associated with the mean velocity $V_0$ ($\mathcal {F}_0 = V_0/\sqrt {gL}$). The wave drag is reported to be either increased or decreased by velocity fluctuations depending on $\mathcal {F}_0$. For small fluctuation amplitudes, this drag change is proportional to the square of the amplitude. The effect is maximized for a resonance frequency identified as the Wehausen frequency, which scales as $\sqrt {g/L}$ times the inverse of the Froude number. All these results are rationalized by developing an extension to Havelock's theory.

Small drag-reducing riblets and larger drag-increasing ridges are longitudinally invariant and laterally periodic surface structures that differ only in the details of their lateral periodicity and their size in viscous units. Due to their different drag behaviour, typically riblets and ridges have been analysed separately. By studying experimentally trapezoidal-grooved surfaces of different sizes, we address systematically the transition from riblet-like to ridge-like behaviour in a unified framework. The structure height and lateral wavelength are varied both physically, by considering eight different surfaces, and in their viscous-scaled form, by spanning a wide range of bulk Reynolds number $Re_b$. The effective skin-friction coefficient $C_f$ is determined via pressure-drop measurement in a turbulent channel flow facility designed for accurate drag measurements. An unexpectedly rich drag behaviour is unveiled, in which different drag regimes are distinguished depending on the value of $l_g^+$, the viscous-scaled square root of the groove area. The well-known drag-reducing regime of riblets that spans up to $l_g^+=17$ is followed by a regime in which the roughness function ${\rm \Delta} U^+$ increases logarithmically with $l_g^+$, indicating an apparent fully rough behaviour up to $l_g^+\approx 40$. Further increase of $l_g^+$ leads to a clear departure from the fully rough regime, and an unexpected non-monotonic behaviour of the roughness function ${\rm \Delta} U^+$ for $50< l_g^+<200$ is reported for the first time. For sufficiently large $Re_b$ and $l_g$, it is shown that a single parameter, similar to the classical hydraulic diameter, is sufficient to describe the drag behaviour of ridges. We find that an appropriate definition of the effective channel height is crucial for interpreting the drag behaviour. When the longitudinal protrusion height of the structured surface is accounted for in the channel height definition, a laminar flow exhibits the same $C_f(Re_b)$ relation known for flat surfaces. This approach thus allows us to discern the modification of $C_f$ induced by turbulence. We provide predictive correlations for the fully rough regime and the high Reynolds number range of trapezoidal-grooved surfaces that become possible thanks to the chosen channel height definition.

Shallow mixing layers (SMLs) behind a splitter plate were studied in a tilted rectangular open-channel flume for a range of flow depths and the initial shear parameter ${\lambda = (U_{2}-U_{1})/(U_{2}+U_{1})}$, where $U_1$ and $U_2$ are streamwise velocities of the slow and fast streams, respectively. The main focus of the study is on (i) key parameters controlling the time-averaged SMLs; and (ii) the emergence and spatial development of Kelvin–Helmholtz coherent structures (KHCSs) and large- and very-large-scale motions (LSMs and VLSMs) and associated turbulence statistics. The time-averaged flow features of the SMLs are mostly controlled by bed-friction length scale $h/c_f$ and shear parameter $\lambda$ as well as by time-averaged spanwise velocities $V$ and momentum fluxes $UV$, where $h$ and $c_f$ are flow depth and bed-friction coefficient, respectively. For all studied cases, the effect of shear layer turbulence on the SML growth is comparatively weak, as the fluxes $UV$ dominate over the spanwise turbulent fluxes. Initially, the emergence of KHCSs and their length scales largely depend on $\lambda$. The KHCSs cannot form if ${\lambda \lessapprox 0.3}$ and the turbulence behind the splitter plate resembles that of free mixing layers. Further downstream, shear layer turbulence becomes dependent on the bed-friction number $S = c_f \delta _v /(4 h \lambda )$, where $\delta _v$ is vorticity thickness. When $S \gtrapprox 0.01$, the KHCSs are longitudinally stretched and the scaled transverse turbulent fluxes decrease with increasing $S$. The presence and streamwise development of LSMs/VLSMs away from the splitter plate depends on the $\lambda$-value, particularly when $\lambda > 0.3$, resembling LSMs/VLSMs in conventional open-channel flows when $\lambda$ is small.