Thakkar et al. (J. Fluid Mech., vol. 837, 2018, R1) represents a significant advancement in the ability to computationally model rough wall flows. Direct numerical solution (DNS) of turbulent boundary layer flow over an industrial grit blasted surface at relevant roughness Reynolds numbers, from hydraulically smooth to fully rough regimes, is a path forward to parametrically study a wide range of surface roughness. The methodology described in this paper, coupled with validation experiments, ultimately should lead to improved frictional drag predictions.

The wake meandering characteristics of four different wind turbine designs with diameters ranging from a few centimetres (wind tunnel scale) to a hundred metres (utility scale) are investigated using large-eddy simulation with the turbine blades and nacelle parametrised using a new actuator surface model. Different velocity fields and meandering behaviours are observed at near-wake locations. At far-wake locations, on the other hand, the mean velocity deficit profiles begin to collapse when scaled by the centreline velocity deficit based on the incoming wind speed at turbine hub height, suggesting far-wake similarity across scales. The turbine-added turbulence kinetic energy profiles are shown to also nearly collapse with each other in the far wake when normalised using a velocity scale defined by the thrust on the turbine rotor. Moreover, we show that at far-wake locations, the simulated flow fields for all four turbine designs exhibit similar wake meandering characteristics in terms of (1) a Strouhal number independent of rotor designs of different sizes and (2) the distributions of wake meandering wavelengths and amplitudes when normalised by the rotor diameter and a length scale defined by the turbine thrust respectively.

We present a theoretical study of the statics and dynamics of a partially wetting liquid droplet, of equilibrium contact angle $\unicode[STIX]{x1D703}_{e}$, confined in a solid wedge geometry of opening angle $\unicode[STIX]{x1D6FD}$. We focus on a mostly non-wetting regime, given by the condition $\unicode[STIX]{x1D703}_{e}-\unicode[STIX]{x1D6FD}>90^{\circ }$, where the droplet forms a liquid barrel – a closed shape of positive mean curvature. Using a quasi-equilibrium assumption for the shape of the liquid–gas interface, we compute the changes to the surface energy and pressure distribution of the liquid upon a translation along the symmetry plane of the wedge. Our model is in good agreement with numerical calculations of the surface energy minimisation of static droplets deformed by gravity. Beyond the statics, we put forward a Lagrangian description of the droplet dynamics. We focus on the overdamped limit, where the driving capillary force is balanced by the frictional forces arising from the bulk hydrodynamics, the corner flow near the contact lines and the contact-line friction. Our results provide a theoretical framework to describe the motion of partially wetting liquids in confinement, and can be used to gain further understanding on the relative importance of dissipative processes that span from microscopic to macroscopic length scales.

The predictive control of the self-sustained single spiral vortex breakdown mode is addressed in the three-dimensional flow geometry of Ruith et al. (2003) for a constant swirl number $S=1.095$. Based on adjoint optimization algorithms, two different control strategies have been designed. First, a quadratic objective function minimizing the radial velocity intensity, taking advantage of the physical mechanism underpinning spiral vortex breakdown. The second strategy focuses on the hydrodynamic instability properties using as objective function the growth rate of the most unstable global eigenmode. These minimization algorithms seek for an optimal volume force in an axisymmetric domain avoiding therefore expensive three-dimensional computations. In addition to considering eigenvalues around the base flow, we also investigate the stability around the mean flow and we find that it correctly predicts the frequency of the self-sustained single spiral vortex breakdown mode for Reynolds numbers up to $Re=500$. Close to the instability threshold, at a Reynolds value of $Re=180$, all these control strategies successfully quench the spiral vortex breakdown. The related volume force is found identical for the base and mean flow eigenvalue control even if the uncontrolled growth rates differ significantly. The control of the least unstable eigenvalue of the mean flow is not only found optimal at $Re=180$, it also stabilizes the flow at a Reynolds value as large as $Re=300$, which opens promising extensions to industrial applications.

This experimental study is focused on the mechanisms of thermal atomisation of a drop impacting onto a hot substrate. This phenomenon is characterised by the wetting and dewetting of the substrate, caused not by the rim dynamics, but induced by thermal effects. These thermal effects lead to the lamella evaporation, levitation and disintegration, generation of a vertical spray of fine droplets and consequently, drop breakup. A typical contact time of the drop before complete detachment is theoretically estimated. This estimation agrees very well with the experiments. It is shown that the Weber number, often used for describing splashing drops, is not a relevant parameter for thermal atomisation. Finally, a regime map is plotted, using a combination of the dimensionless contact time and the dimensionless heat flux at the substrate.

Three-dimensional (3-D) wake transition for flow past a square cylinder aligned with sides perpendicular and parallel to the approaching flow is investigated using direct numerical simulation. The secondary wake instability, namely a Mode A instability, occurs at a Reynolds number ($Re$) of 165.7. A gradual wake transition from Mode A* (i.e. Mode A with vortex dislocations) to Mode B is observed over a range of $Re$ from 185 to 210, within which the probability of occurrence of vortex dislocations decreases monotonically with increasing $Re$. The characteristics of the Strouhal–Reynolds number relationship are analysed. At the onset of Mode A*, a sudden drop of the 3-D Strouhal number from its two-dimensional counterpart is observed, which is due to the subcritical nature of the Mode A* instability. A continuous 3-D Strouhal–Reynolds number curve is observed over the mode swapping regime, since Mode A* and Mode B have extremely close vortex shedding frequencies and therefore only a single merged peak is observed in the frequency spectrum. The existence of hysteresis for the Mode A and Mode B wake instabilities is examined. The unconfined Mode A and Mode B wake instabilities are hysteretic and non-hysteretic, respectively. However, a spanwise confined Mode A could be non-hysteretic. It is proposed that the existence of hysteresis at a wake instability can be identified by examining the sudden/gradual variation of the 3-D flow properties at the onset of the wake instability, with sudden and gradual variations corresponding to hysteretic (subcritical) and non-hysteretic (supercritical) flows, respectively.

We study the large-scale motions in turbulent plane Couette flows at moderate friction Reynolds number up to $Re_{\unicode[STIX]{x1D70F}}=500$. Direct numerical simulation (DNS) domains were as large as $100\unicode[STIX]{x03C0}\unicode[STIX]{x1D6FF}\times 2\unicode[STIX]{x1D6FF}\times 5\unicode[STIX]{x03C0}\unicode[STIX]{x1D6FF}$, where $\unicode[STIX]{x1D6FF}$ is half the distance between the walls. The results indicate that there are streamwise vortices filling the space between the walls that remain correlated over distances in the streamwise direction and that increase strongly with the Reynolds number, so that for the largest Reynolds number studied here, they are correlated across the entire $100\unicode[STIX]{x03C0}\unicode[STIX]{x1D6FF}$ length of the domain. The presence of these very long structures is apparent in the spectra of all three velocity components and the Reynolds stress. In DNS using a smaller domain, the large structures are constrained, eliminating the streamwise variations present in the larger domain. Near the centre of the domain, these large-scale structures contribute as much as half of the Reynolds shear stress.

A dynamical systems approach is used to devise a linear estimation tool for channel flow at a friction Reynolds number of $Re_{\unicode[STIX]{x1D70F}}=1000$. The estimator uses time-resolved velocity measurements at a single wall-normal location to estimate the velocity field at other wall-normal locations (the data coming from direct numerical simulations). The estimation tool builds on the work of McKeon & Sharma (J. Fluid Mech., vol. 658, 2010, pp. 336–382) by using a Navier–Stokes-based linear model and treating any nonlinear terms as unknown forcings to an otherwise linear system. In this way nonlinearities are not ignored, but instead treated as an unknown model input. It is shown that, while the linear estimator qualitatively reproduces large-scale flow features, it tends to overpredict the amplitude of velocity fluctuations – particularly for structures that are long in the streamwise direction and thin in the spanwise direction. An alternative linear model is therefore formed in which a simple eddy viscosity is used to model the influence of the small-scale turbulent fluctuations on the large scales of interest. This modification improves the estimator performance significantly. Importantly, as well as improving the performance of the estimator, the linear model with eddy viscosity is also able to predict with reasonable accuracy the range of wavenumber pairs and the range of wall-normal heights over which the estimator will perform well.

The low value of the speed of sound in dilute granular media permits the study of the properties of supersonic flows for a wide range of Mach numbers. In this paper, we report the experimental observation of a subsonic–supersonic transition in a vibrated granular gas. The shock fronts studied are obtained by simply pushing a rectangular obstacle into the granular gas for different obstacle velocities. The supersonic regime is characterized by the formation of normal shock waves whose width increases when the Mach number decreases to values close to 1. The bimodal model proposed by Mott-Smith in the 1950s provides a good description for the velocity distributions as well as the macroscopic quantities for shock waves in molecular gases but remains inadequate for dissipative media like granular gases and plasmas. Here by examining the shock front structure for a wide range of Mach numbers, we adapt the Mott-Smith bimodal description to a dissipative medium. By using balance equations from granular kinetic theory and taking into account different dissipation sources, the proposed model allows us to understand how this dissipation modifies temperature, mean velocity and volume fraction profiles through the shock front.

This paper revisits the problem of forces on obstacle arrays in combined waves, an in-line steady current and structural dynamic motions. The intended application is the design and re-assessment of dynamically responding offshore platforms. Planar grids of perforated plates are moved in forced motion on three scales through otherwise stationary water. A new analytical wave–current–structure blockage model is developed by building on the existing wave–current blockage model presented by Santo et al. (J. Fluid Mech., vol. 739, 2014b, pp. 143–178) using a similar set of experiments but with forced motion on two scales. The new model, which is an improved Morison relative-velocity formulation, is tested against the experimental data for a range of structural to wave oscillation frequency ratios, $f_{s}/f_{w}=2$, 2.5 and 3. For relatively small current speed ($u_{c}$) and oscillatory structural velocity ($u_{s}$) compared with the oscillatory wave velocity ($u_{w}$), the drag force time history on grids is well approximated by a summation of the wave drag and the current drag components independently, without a $u_{w}\times u_{c}$ cross-term, consistent with the previous model. The wave drag component contains an additional $u_{s}$ contribution, while the current drag component may or may not contain an additional $u_{s}$ contribution depending on $f_{s}/f_{w}$. The measured drag force is observed to be asymmetric in time due to biasing from the mean flow. This is supported by numerical simulation using a porous block as a numerical model of the grids, although the simulated force asymmetry is weaker. All these effects can be sufficiently accounted for in the analytical model. The new model is shown to fit the variation of the experimental forces and force harmonics in time well for a wide range of cases, requiring only calibration of the Morison type drag and inertia coefficients. In contrast, the industry-standard version of the Morison relative-velocity formulation cannot reproduce the variation of the measured force in time, so present practice should be regarded as inadequate for combined steady, low frequency and high frequency motion acting on obstacle arrays.

This work focuses on the dynamics of a train of unconfined bubbles flowing in a microchannel. We investigate the transverse position of the train of bubbles, its velocity and the associated pressure drop when flowing in a microchannel, depending on the internal forces due to viscosity, inertia and capillarity. Despite the small scales of the system, the inertial migration force plays a crucial role in determining the transverse equilibrium position of the bubbles. Besides inertia and viscosity, other effects may also affect the transverse migration of bubbles, such as the Marangoni surface stresses and the surface deformability. We look at the influence of surfactants in the limit of infinite Marangoni effect, which yields a rigid bubble interface. The resulting migration force may balance external body forces, if present, such as buoyancy, centrifugal or magnetic ones. This balance not only determines the transverse position of the bubbles but, consequently, the surrounding flow structure, which can be determinant for any mass/heat transfer process involved. Finally, we look at the influence of the bubble deformation on the equilibrium position and compare it with the inertial migration force at the centred position, explaining the stable or unstable character of this position accordingly. A systematic study of the influence of the parameters, such as the bubble size, uniform body force, Reynolds and capillary numbers, has been carried out using numerical simulations based on the finite element method, solving the full steady Navier–Stokes equations and their asymptotic counterparts for the limits of small Reynolds and/or capillary numbers.

The wake structure and transition process of an incompressible viscous fluid flow past a sphere affected by an imposed streamwise magnetic field are investigated numerically over flow regimes that include steady and unsteady laminar flows at Reynolds numbers up to 300. For cases without a magnetic field, a subregion with the existence of a limit cycle is found in the range $210<Re<270$. The point of division is between $Re=220$ and $Re=230$. For cases with a streamwise magnetic field, five wake patterns are the steady axisymmetric wake with an attached separation bubble, the steady plane symmetric wake with a small spiral dismissed, the steady plane symmetric wake with a limit cycle, the steady plane symmetric wake with a small spiral fed by the upstream fluid and the unsteady plane symmetric wake with a wave-like oscillation or vortex shedding. Under the influence of an imposed streamwise magnetic field, the wake will be transitioned to various patterns. An interesting ‘reversion phenomenon’, which describes the topological structure behind a sphere with a higher Reynolds number and a certain interaction parameter which corresponds to a lower Reynolds number case with a certain interaction parameter or a much lower Reynolds number case without a magnetic field, is also found. The principal results of the present work are summarized in a map of regimes in the $\{N,Re\}$ plane.

An analytic model of a stationary hypersonic magnetohydrodynamic (MHD) shock with an externally applied magnetic field is proposed. Basically, original jump conditions at a plane oblique shock, analogous to the Rankine–Hugoniot formulae, with a moderately resistive air plasma downstream are derived. Viscous, thermal and Hall effects are neglected, but the plasma dissociation behind the shock causing a jump of isentropic exponent is also a major input of the model. Then, a shock-fitting procedure with ambient atmospheric conditions is worked out by the coupling of these MHD jumps with thermodynamic correlations and an electric conductivity model. For an application to atmospheric entry problems, the flow behind an axisymmetric blunt-body shock is modelled with a stream function satisfying these MHD jump conditions as boundary conditions. An important feature put into evidence is a similarity rule involving the hypersonic parameter $M_{1}\cos \unicode[STIX]{x1D712}_{1}$, which shows an aerodynamic correspondence between the upstream Mach number $M_{1}$ and the velocity angle $\unicode[STIX]{x1D712}_{1}$. It also emerges that curvature effects become important past $30^{\circ }$ and the assumption of a spherical shock also becomes untenable past $50^{\circ }$; therefore, we limit the model of shock thickness used in the MHD fitting to $\unicode[STIX]{x1D712}_{1}<50^{\circ }$.

A new multi-layer irrotational Boussinesq-type model is proposed for both linear and nonlinear surface water waves over mildly sloping seabeds. The model is formulated in terms of computational horizontal and vertical velocity components within each layer and satisfies exact kinematic and dynamic free-surface conditions as well as kinematic seabed conditions. Using a Stokes-type expansion, a theoretical analysis of the new multi-layer model is carried out to examine both linear and nonlinear properties, including wave celerity, velocity profiles, shoaling amplitude, second- and third-order transfer functions and amplitude dispersion. The dispersive coefficients in the governing equations are determined by optimizing the linear celerity or linear velocity profiles. For example, the four-layer model shows extremely high accuracy and is applicable up to $kh=667$–800 (where $k$ is the wavenumber and $h$ is a typical water depth) with a 1 % error in wave phase celerity, and up to $kh=352$–423 with a 1 % error in the linear velocity components. The super- and subharmonic transfer functions are extremely accurate up to $kh=300$ (1 % error), the third-order harmonics and amplitude dispersion are accurate up to $kh=477$ (1 % error), and the shoaling property is optimized to cover the range of $0<kh<300$, which presents a 0.06 % tolerance error in shoaling amplitude. The high-accuracy nature of the model increases its suitability for simulating random wave propagation from extremely deep to shallow waters over mildly sloping topographies. The model is implemented numerically on a non-staggered grid via a composite fourth-order Adams–Bashforth–Moulton time integration. The numerical results show good agreement with both the analytical solutions and experimental data.

The kinematics of a fully developed passive scalar is modelled using the hierarchical random additive process (HRAP) formalism. Here, ‘a fully developed passive scalar’ refers to a scalar field whose instantaneous fluctuations are statistically stationary, and the ‘HRAP formalism’ is a recently proposed interpretation of the Townsend attached eddy hypothesis. The HRAP model was previously used to model the kinematics of velocity fluctuations in wall turbulence: $u=\sum _{i=1}^{N_{z}}a_{i}$, where the instantaneous streamwise velocity fluctuation at a generic wall-normal location $z$ is modelled as a sum of additive contributions from wall-attached eddies ($a_{i}$) and the number of addends is $N_{z}\sim \log (\unicode[STIX]{x1D6FF}/z)$. The HRAP model admits generalized logarithmic scalings including $\langle \unicode[STIX]{x1D719}^{2}\rangle \sim \log (\unicode[STIX]{x1D6FF}/z)$, $\langle \unicode[STIX]{x1D719}(x)\unicode[STIX]{x1D719}(x+r_{x})\rangle \sim \log (\unicode[STIX]{x1D6FF}/r_{x})$, $\langle (\unicode[STIX]{x1D719}(x)-\unicode[STIX]{x1D719}(x+r_{x}))^{2}\rangle \sim \log (r_{x}/z)$, where $\unicode[STIX]{x1D719}$ is the streamwise velocity fluctuation, $\unicode[STIX]{x1D6FF}$ is an outer length scale, $r_{x}$ is the two-point displacement in the streamwise direction and $\langle \cdot \rangle$ denotes ensemble averaging. If the statistical behaviours of the streamwise velocity fluctuation and the fluctuation of a passive scalar are similar, we can expect first that the above mentioned scalings also exist for passive scalars (i.e. for $\unicode[STIX]{x1D719}$ being fluctuations of scalar concentration) and second that the instantaneous fluctuations of a passive scalar can be modelled using the HRAP model as well. Such expectations are confirmed using large-eddy simulations. Hence the work here presents a framework for modelling scalar turbulence in high Reynolds number wall-bounded flows.

The effect of the variations of the permeability tensor on the close-to-the-wall behaviour of a turbulent channel flow bounded by porous walls is explored using a set of direct numerical simulations. It is found that the total drag can be either reduced or increased by more than 20 % by adjusting the permeability directional properties. Drag reduction is achieved for the case of materials with permeability in the vertical direction lower than the one in the wall-parallel planes. This configuration limits the wall-normal velocity at the interface while promoting an increase of the tangential slip velocity leading to an almost ‘one-component’ turbulence where the low- and high-speed streak coherence is strongly enhanced. On the other hand, strong drag increase is found when high wall-normal and low wall-parallel permeabilities are prescribed. In this condition, the enhancement of the wall-normal fluctuations due to the reduced wall-blocking effect triggers the onset of structures which are strongly correlated in the spanwise direction, a phenomenon observed by other authors in flows over isotropic porous layers or over ribletted walls with large protrusion heights. The use of anisotropic porous walls for drag reduction is particularly attractive since equal gains can be achieved at different Reynolds numbers by rescaling the magnitude of the permeability only.

This work introduces a mathematical approach to analysing the polymer dynamics in turbulent viscoelastic flows that uses a new geometric decomposition of the conformation tensor, along with associated scalar measures of the polymer fluctuations. The approach circumvents an inherent difficulty in traditional Reynolds decompositions of the conformation tensor: the fluctuating tensor fields are not positive definite and so do not retain the physical meaning of the tensor. The geometric decomposition of the conformation tensor yields both mean and fluctuating tensor fields that are positive definite. The fluctuating tensor in the present decomposition has a clear physical interpretation as a polymer deformation relative to the mean configuration. Scalar measures of this fluctuating conformation tensor are developed based on the non-Euclidean geometry of the set of positive definite tensors. Drag-reduced viscoelastic turbulent channel flow is then used an example case study. The conformation tensor field, obtained using direct numerical simulations, is analysed using the proposed framework.

A direct-numerical-simulation study of spatially evolving compressible zero-pressure-gradient turbulent boundary layers is presented for a fine-meshed range of Mach numbers from 0.3 to 2.5. The use of an identical set-up for all subsonic and supersonic cases warrants proper comparability and allows a highly reliable quantitative evaluation of compressible mean-flow scaling laws and the settlement on a commonly accepted compressible mean-flow velocity profile in the considered Mach and Reynolds number range. All data are compared to the literature data-base where significant data scattering can be observed. The skin-friction distribution was found in excellent agreement with the prediction by the van Driest-II transformation. Contrary to the prevailing appraisal, the wake region of the mean-velocity profile is observed to scale much better with the momentum-thickness Reynolds number calculated with the far-field-viscosity than with the wall-viscosity. The time-averaged velocity fluctuations, density-scaled according to Morkovin’s hypothesis, are found to be noticeably influenced by compressibility effects in the inner layer as well as in the wake region. Allowing wall-temperature fluctuations affects neither the density nor velocity fluctuations.

The flow response of a rapidly rotating fluid-filled cube to low-amplitude librational forcing is investigated numerically. Librational forcing is the harmonic modulation of the mean rotation rate. The rotating cube supports inertial waves which may be excited by libration frequencies less than twice the rotation frequency. The response is comprised of two main components: resonant excitation of the inviscid inertial eigenmodes of the cube, and internal shear layers whose orientation is governed by the inviscid dispersion relation. The internal shear layers are driven by the fluxes in the forced boundary layers on walls orthogonal to the rotation axis and originate at the edges where these walls meet the walls parallel to the rotation axis, and are hence called edge beams. The relative contributions to the response from these components is obscured if the mean rotation period is not small enough compared to the viscous dissipation time, i.e. if the Ekman number is too large. We conduct simulations of the Navier–Stokes equations with no-slip boundary conditions using parameter values corresponding to a recent set of laboratory experiments, and reproduce the experimental observations and measurements. Then, we reduce the Ekman number by one and a half orders of magnitude, allowing for a better identification and quantification of the contributions to the response from the eigenmodes and the edge beams.

Combined free-stream disturbance measurements and receptivity studies in hypersonic wind tunnels were conducted by means of a slender wedge probe and direct numerical simulation. The study comprises comparative tunnel noise measurements at Mach 3, 6 and 7.4 in two Ludwieg tube facilities and a shock tunnel. Surface pressure fluctuations were measured over a wide range of frequencies and test conditions including harsh test environments not accessible to measurement techniques such as Pitot probes and hot-wire anemometry. A good agreement was found between normalized Pitot pressure fluctuations converted into normalized static pressure fluctuations and the wedge probe readings. Quantitative results of the tunnel noise are provided in frequency ranges relevant for hypersonic boundary-layer transition. Complementary numerical simulations of the leading-edge receptivity to fast and slow acoustic waves were performed for the applied wedge probe at conditions corresponding to the experimental free-stream conditions. The receptivity to fast acoustic waves was found to be characterized by an early amplification of the induced fast mode. For slow acoustic waves an initial decay was found close to the leading edge. At all Mach numbers, and for all considered frequencies, the leading-edge receptivity to fast acoustic waves was found to be higher than the receptivity to slow acoustic waves. Further, the effect of inclination angles of the acoustic wave with respect to the flow direction was investigated. An inclination angle was found to increase the response on the wave-facing surface of the probe and decrease the response on the opposite surface for fast acoustic waves. A frequency-dependent response was found for slow acoustic waves. The combined numerical and experimental approach in the present study confirmed the previous suggestion that the slow acoustic wave is the dominant acoustic mode in noisy hypersonic wind tunnels.