Picano et al. (J. Fluid Mech., vol. 764, 2015, pp. 463–487) have conducted fully resolved numerical simulations of many thousands of spherical particles in a turbulent channel flow with $\mathit{Re}=5600$. Their results give a tantalizing demonstration of the vastness of the vistas that this line of research is about to open.

In very large wind farms, the vertical interaction with the atmospheric boundary layer plays an important role, i.e. the total energy extraction is governed by the vertical transport of kinetic energy from higher regions in the boundary layer towards the turbine level. In the current study, we investigate optimal control of wind-farm boundary layers, considering the individual wind turbines as flow actuators, whose energy extraction can be dynamically regulated in time so as to optimally influence the flow field and the vertical energy transport. To this end, we use large-eddy simulations of a fully developed pressure-driven wind-farm boundary layer in a receding-horizon optimal control framework. For the optimization of the wind-turbine controls, a conjugate-gradient optimization method is used in combination with adjoint large-eddy simulations for the determination of the gradients of the cost functional. In a first control study, wind-farm energy extraction is optimized in an aligned wind farm. Results are accumulated over one hour of operation. We find that the energy extraction is increased by 16 % compared to the uncontrolled reference. This is directly related to an increase of the vertical fluxes of energy towards the wind turbines, and vertical shear stresses increase considerably. A further analysis, decomposing the total stresses into dispersive and Reynolds stresses, shows that the dispersive stresses increase drastically, and that the Reynolds stresses decrease on average, but increase in the wake region, leading to better wake recovery. We further observe also that turbulent dissipation levels in the boundary layer increase, and overall the outer layer of the boundary layer enters into a transient decelerating regime, while the inner layer and the turbine region attain a new statistically steady equilibrium within approximately one wind-farm through-flow time. Two additional optimal control cases study penalization of turbulent dissipation. For the current wind-farm geometry, it is found that the ratio between wind-farm energy extraction and turbulent boundary-layer dissipation remains roughly around 70 %, but can be slightly increased by a few per cent by penalizing the dissipation in the optimization objective. For a pressure-driven boundary layer in equilibrium, we estimate that such a shift can lead to an increase in wind-farm energy extraction of 6 %.

Mixed convection in a channel with flow driven by a pressure gradient and subject to spatially periodic heating along one of the walls has been studied. The pattern of the heating is characterized by the wavenumber ${\it\alpha}$ and its intensity is expressed in terms of the Rayleigh number $\mathit{Ra}_{p}$. The primary convection has the form of counter-rotating rolls with the wavevector parallel to the wavevector of the heating. The resulting net heat flow between the walls increases proportionally to $\mathit{Ra}_{p}$ but the growth saturates when $\mathit{Ra}_{p}=O(10^{3})$. The most effective heating pattern corresponds to ${\it\alpha}\approx 1$, as this leads to the most intense transverse motion. The primary convection is subject to transition to secondary states with the onset conditions depending on ${\it\alpha}$. The conditions leading to transition between different forms of secondary motion have been determined using the linear stability theory. Three patterns of secondary motion may occur at small Reynolds numbers $\mathit{Re}$, i.e. longitudinal rolls, transverse rolls and oblique rolls, with the critical conditions varying significantly as a function of ${\it\alpha}$. An increase of ${\it\alpha}$ leads to the elimination of the longitudinal rolls and, eventually, to the elimination of the oblique rolls, with the transverse rolls assuming the dominant role. For large ${\it\alpha}$, the transition is driven by the Rayleigh–Bénard mechanism; while for ${\it\alpha}=O(1)$, the spatial parametric resonance dominates. The global flow characteristics are identical regardless of whether the heating is applied at the lower or the upper wall.

We consider two-dimensional solitary water waves on a shear flow with an arbitrary distribution of vorticity. Assuming that the horizontal velocity in the fluid never exceeds the wave speed and that the free surface lies everywhere above its asymptotic level, we give a very simple proof that a suitably defined Froude number $F$ must be strictly greater than the critical value $F=1$. We also prove a related upper bound on $F$, and hence on the amplitude, under more restrictive assumptions on the vorticity.

The viscous and inviscid linear stability of the incompressible flow past a square open cavity is studied numerically. The analysis shows that the flow first undergoes a steady three-dimensional bifurcation at a critical Reynolds number of 1370. The critical mode is localized inside the cavity and has a flat roll structure with a spanwise wavelength of about 0.47 cavity depths. The adjoint global mode reveals that the instability is most efficiently triggered in the thin region close to the upstream tip of the cavity. The structural sensitivity analysis identifies the wavemaker as the region located inside the cavity and spatially concentrated around a closed orbit. As the flow outside the cavity plays no role in the generation mechanisms leading to the bifurcation, we confirm that an appropriate parameter to describe the critical conditions in open cavity flows is the Reynolds number based on the average velocity between the two upper edges. Stabilization is achieved by a decrease of the total momentum inside the shear layer that drives the core vortex within the cavity. The mechanism of instability is then studied by means of a short-wavelength approximation considering pressureless inviscid modes. The closed streamline related to the maximum inviscid growth rate is found to be the same as that around which the global wavemaker is concentrated. The structural sensitivity field based on direct and adjoint eigenmodes, computed at a Reynolds number far higher than that of the base flow, can predict the critical orbit on which the main instabilities inside the cavity arise. Further, we show that the sub-leading unstable time-dependent modes emerging at supercritical conditions are characterized by a period that is a multiple of the revolution time of Lagrangian particles along the orbit of maximum growth rate. The eigenfrequencies of these modes, computed by global stability analysis, are in very good agreement with the asymptotic results.

Cavitation occurrence about a jet in crossflow is investigated experimentally in a variable-pressure water tunnel using still and high-speed photography. The 0.012 m diameter jet is injected on the centreplane of a 0.6 m square test section at jet to freestream velocity ratios ranging from 0.2 to 1.6, corresponding to jet-velocity-based Reynolds numbers of $25\times 10^{3}$ to $160\times 10^{3}$ respectively. Measurements were made at a fixed freestream-based Reynolds number, for which the ratio of the undisturbed boundary layer thickness to jet diameter is 1.18. The cavitation number was varied from inception (up to about 10) down to 0.1. Inception is investigated acoustically for bounding cases of high and low susceptibility to phase change. The influence of velocity ratio and cavitation number on cavity topology and geometry are quantified from the photography. High-speed photographic recordings made at 6 kHz provide insight into cavity dynamics, and derived time series of spatially averaged pixel intensities enable frequency analysis of coherent phenomena. Cavitation inception was found to occur in the high-shear regions either side of the exiting jet and to be of an intermittent nature, increasing in occurrence and duration from 0 to 100 % probability with decreasing cavitation number or increasing jet to freestream velocity ratio. The frequency and duration of individual events strongly depends on the cavitation nuclei supply within the approaching boundary layer. Macroscopic cavitation develops downstream of the jet with reduction of the cavitation number beyond inception, the length of which has a power-law dependence on the cavitation number and a linear dependence on the jet to freestream velocity ratio. The cavity closure develops a re-entrant jet with increase in length forming a standing wave within the cavity. For sufficiently low cavitation numbers the projection of the re-entrant jet fluid no longer reaches the cavity leading edge, analogous to supercavitation forming about solid cavitators. Hairpin-shaped vortices are coherently shed from the cavity closure via mechanisms of shear-layer roll-up similar to those shed from protuberances and jets in crossflow in single-phase flows. These vortices are shed at an apparently constant frequency, independent of the jet to freestream velocity ratio but decreasing in frequency with reducing cavitation number and cavity volume growth. Highly coherent cavitating vortices form along the leading part of the cavity due to instability of the jet upstream shear layer for jet to freestream velocity ratios greater than about 0.8. These vortices are cancelled and condense as they approach the trailing edge in the shear layer of opposing vorticity associated with the cavity closure and the hairpin vortex formation. For lower velocity ratios, where there is decreased jet penetration, the jet upstream shear velocity gradient reverses and vortices of the opposite sense form, randomly modulated by boundary layer turbulence.

Direct numerical simulations (DNS) were performed to investigate the laminar–turbulent transition in a boundary layer on a sharp cone with an isothermal wall at Mach 6 and at zero angle of attack. The motivation for this research is to make a contribution towards understanding the nonlinear stages of transition and the final breakdown to turbulence in hypersonic boundary layers. In particular, the role of second-mode fundamental resonance, or (K-type) breakdown, is investigated using high-resolution ‘controlled’ transition simulations. The simulations were carried out for the laboratory conditions of the hypersonic transition experiments conducted at Purdue University. First, several low-resolution simulations were carried out to explore the parameter space for fundamental resonance in order to identify the cases that result in strong nonlinear interactions. Subsequently, based on the results from this study, a set of highly resolved simulations that proceed deep into the turbulent breakdown region have been performed. The nonlinear interactions observed during the breakdown process are discussed in detail in this paper. A detailed description of the flow structures that arise due to these nonlinear interactions is provided and an analysis of the skin friction and heat transfer development during the breakdown is presented. The controlled transition simulations clearly demonstrate that fundamental breakdown may indeed be a viable path to complete breakdown to turbulence in hypersonic cone boundary layers at Mach 6.

Studying the nature of transient reflections of shock waves from surfaces is important in many engineering fields, e.g. blast protection, supersonic flights, shock focusing, medical and industrial applications and more. The recent advancements in this field reveal that the major obstacle in better understanding this phenomenon by means of experimental investigations is the limited temporal and spatial resolution. An alternative approach to commonly used high-speed photography is based on the use of a single-lens reflex (SLR) camera that captures only one image per experiment. Using this method to study a transient reflection process necessitates repeating each experiment many times while retaining extremely high repeatability. In the present study, we present a solution to this obstacle by means of a fully automated shock tube facility, which has been developed in the course of this study. A typical experiment can be executed a few hundred times with a repeatability of less than 0.01 in the incident-shock-wave Mach number at moderate shock strengths ($M=1.2{-}1.4$). The system offers a very high spatial and temporal resolution description of the transient reflection process of a shock wave over a coupled convex–concave surface. The study of this complex configuration using a fully automated shock tube enables one to observe, in greater detail than ever before, both the transient transition from regular reflection, RR, to Mach reflection, MR, and the reverse transient transition from MR to RR. The geometry studied can also be found in blunt leading-edge reflectors in which higher pressures were recorded, and the results presented also describe in detail the shock reflection process inside such a reflector. The results highlight and strengthen the recent understanding of the importance of high spatial and temporal resolution in determining the transition process from RR to MR over a coupled concave–convex surface. However, despite achieving very high statistical certainty in the experimental measurements, the question of the difference between the pseudo-steady transition criterion and the experimental results remains unresolved.

The effect of airfoil design parameters, such as airfoil thickness and camber, are well understood in steady-state aerodynamics. But this knowledge cannot be readily applied to the flapping flight in insects and birds: flow visualizations and computational analyses of flapping flight have identified that in many cases, a leading-edge vortex (LEV) contributes substantially to the generation of aerodynamic force. In flapping flight, very high angles of attack and partly separated flow are common features. Therefore, it is expected that airfoil design parameters affect flapping wing aerodynamics differently. Existing studies have focused on force measurements, which do not provide sufficient insight into the dominant flow features. To analyse the influence of wing morphology in slow-speed bird flight, the time-resolved three-dimensional flow field around different flapping wing models in translational motion at a Reynolds number of $22\,000<\mathit{Re}<26\,000$ was studied. The effect of several Strouhal numbers ($0.2<\mathit{St}<0.4$), camber and thickness on the flow morphology and on the circulation was analysed. A strong LEV was found on all wing types at high $\mathit{St}$. The vortex is stronger on thin wings and enhances the total circulation. Airfoil camber decreases the strength of the LEV, but increases the total bound circulation at the same time, due to an increase of the ‘conventional’ bound circulation at the inner half of the wing. The results provide new insights into the influence of airfoil shape on the LEV and force generation at low $\mathit{Re}$. They contribute to a better understanding of the geometry of vertebrate wings, which seem to be optimized to benefit from LEVs in slow-speed flight.

Experiments have been performed on a series of high-Reynolds-number flat-plate turbulent boundary layers formed over rough and smooth walls. The boundary layers were fully rough, yet the elements remained a very small fraction $({<}1.4\,\%)$ of the boundary-layer thickness, ensuring conditions free of transitional effects. The wall-pressure spectrum and its scaling were studied in detail. One of the major findings is that the rough-wall turbulent pressure spectrum at vehicle relevant conditions is comprised of three scaling regions. These include a newly discovered high-frequency region where the pressure spectrum has a viscous scaling controlled by the friction velocity, adjusted to exclude the pressure drag on the roughness elements.

The development and structure of flows in eccentric annular channels and their dependence on inlet conditions, inner-to-outer diameter ratio $d/D$, eccentricity $e=2{\rm\Delta}y/(D-d)$, where ${\rm\Delta}y$ is the distance between the axes of the inner and outer cylinders, and Reynolds number $\mathit{Re}$, based on the hydraulic diameter and the bulk velocity, were studied experimentally, with focus on the phenomena of gap instability and the resulting vortex street. Experimental conditions covered a Reynolds number range between 0 and 19 000, an eccentricity range from 0 to 0.9 and inner-to-outer diameter ratios equal to 0.25, 0.50 and 0.75. Much of the discussion is based on measurements in the middle of the narrow annular gap, where the phenomena of interest could be observed most vividly. In the range $\mathit{Re}<10\,000$, the Strouhal number, the normalized mid-gap axial flow velocity and the normalized axial and cross-flow fluctuations at mid-gap were found to increase with increasing $\mathit{Re}$ and to depend strongly on inlet conditions. At higher Reynolds numbers, however, these parameters reached asymptotic values that were less sensitive to inlet conditions. We constructed a map for the various stages of periodic motions versus eccentricity and Reynolds number and found that for $e<0.5$ or $\mathit{Re}<1100$ the flow was unconditionally stable, as far as gap instability is concerned. For $e\leqslant 0.5$, transition to turbulence occurred at $\mathit{Re}\approx 6000$, whereas, for $0.6\leqslant e\leqslant 0.9$, the critical Reynolds number for the formation of periodic motions was found to increase with eccentricity from 1100 for $e=0.6$ to 3800 for $e=0.9$.

We study the dynamics of turbulent boundary layer flow over a heterogeneous topography composed of roughness patches exhibiting relatively high and low correlation in the streamwise and spanwise directions, respectively (i.e. the roughness appears as streamwise-aligned ‘strips’). It has been reported that such roughness induces a spanwise-wall normal mean secondary flow in the form of mean streamwise vorticity associated with counter-rotating boundary-layer-scale circulations. Here, we demonstrate that this mean secondary flow is Prandtl’s secondary flow of the second kind, both driven and sustained by spatial gradients in the Reynolds-stress components, which cause a subsequent imbalance between production and dissipation of turbulent kinetic energy that necessitates secondary advective velocities. In reaching this conclusion, we study (i) secondary circulations due to spatial gradients of turbulent kinetic energy, and (ii) the production budgets of mean streamwise vorticity by gradients of the Reynolds stresses. We attribute the secondary flow phenomena to extreme peaks of surface stress on the relatively high-roughness regions and associated elevated turbulence production in the fluid immediately above. An optimized state is attained by entrainment of fluid exhibiting the lowest turbulent stresses – from above – and subsequent lateral ejection in order to preserve conservation of mass.

The use of spanwise wall oscillations to attenuate the growth of laminar streaks within the incompressible Blasius boundary layer is investigated. As in the case of the flow above a stationary flat plate, studied by Leib et al. (J. Fluid Mech., vol. 380, 1999, pp. 169–203), free-stream convected gusts interact with the boundary layer to drive the streak growth. Spanwise wall oscillations can either reduce or increase the total energy of the laminar streaks, depending upon the wall oscillation amplitude and frequency, as well as the free-stream gust properties. Reductions in streak energies of up to 90 % are obtained, indicating that spanwise wall oscillations are an effective technique for attenuating the laminar streak growth. Therefore they may suppress secondary boundary-layer instabilities and delay transition. The laminar boundary-layer base flow matches the Blasius profile in the streamwise and wall-normal directions, while in the spanwise direction a generalized version of the classical Stokes layer profile (generated by a wall oscillating beneath a quiescent fluid) occurs, which evolves downstream due to non-parallel flow effects. Via a Wentzel–Kramers–Brillouin–Jeffreys analysis, this generalized Stokes layer is shown to approach the classical Stokes layer in the limit of large downstream distances or high-frequency plate oscillations. The laminar streaks forced by the generalized and the classical Stokes flows differ significantly, which implies that the choice of the spanwise base flow may affect the secondary instability and transition in this flow. The analysis also proves that the use of the classical Stokes layer as spanwise base flow, as employed by Hack & Zaki (Phys. Fluids. vol. 24 (3), 2012, 034101), is inappropriate.

We study how confinement affects the viscous spatiotemporal instability of a two-phase mixing layer. We show that the absolute instability triggered by the inclusion of finite liquid and gas thicknesses leads to a good prediction of experimental data. In addition, this new mechanism provides a justification for the relevance of both simplified inviscid scaling laws and more sophisticated viscous approaches.

We present an internal-flow multiscale method (‘unsteady-IMM’) for compressible, time-varying/unsteady flow problems in nano-confined high-aspect-ratio geometries. The IMM is a hybrid molecular–continuum method that provides accurate flow predictions at macroscopic scales because local microscopic corrections to the continuum-fluid formulation are generated by spatially and temporally distributed molecular simulations. Exploiting separation in both time and length scales enables orders of magnitude computational savings, far greater than seen in other hybrid methods. We apply the unsteady-IMM to a converging–diverging channel flow problem with various time- and length-scale separations. Comparisons are made with a full molecular simulation wherever possible; the level of accuracy of the hybrid solution is excellent in most cases. We demonstrate that the sensitivity of the accuracy of a solution to the macro–micro time-stepping, as well as the computational speed-up over a full molecular simulation, is dependent on the degree of scale separation that exists in a problem. For the largest channel lengths considered in this paper, a speed-up of six orders of magnitude has been obtained, compared with a notional full molecular simulation.

This paper extends the resolvent formulation proposed by McKeon & Sharma (J. Fluid Mech., vol. 658, 2010, pp. 336–382) to consider turbulence–compliant wall interactions. Under this formulation, the turbulent velocity field is expressed as a linear superposition of propagating modes, identified via a gain-based decomposition of the Navier–Stokes equations. Compliant surfaces, modelled as a complex wall admittance linking pressure and velocity, affect the gain and structure of these modes. With minimal computation, this framework accurately predicts the emergence of the quasi-two-dimensional propagating waves observed in recent direct numerical simulations. Further, the analysis also enables the rational design of compliant surfaces, with properties optimized to suppress flow structures energetic in wall turbulence. It is shown that walls with unphysical negative damping are required to interact favourably with modes resembling the energetic near-wall cycle, which could explain why previous studies have met with limited success. Positive-damping walls are effective for modes resembling the so-called very-large-scale motions, indicating that compliant surfaces may be better suited for application at higher Reynolds number. Unfortunately, walls that suppress structures energetic in natural turbulence are also predicted to have detrimental effects elsewhere in spectral space. Consistent with previous experiments and simulations, slow-moving spanwise-constant structures are particularly susceptible to further amplification. Mitigating these adverse effects will be central to the development of compliant coatings that have a net positive influence on the flow.

High-speed images of driven sessile water drops recorded under frequency scans are analysed for resonance peaks, resonance bands and hysteresis of characteristic modes. Visual mode recognition using back-lit surface distortion enables modes to be associated with frequencies, aided by the identifications in Part 1 (Bostwick & Steen, J. Fluid Mech., vol. 760, 2014, pp. 5–38). Part 1 is a linear stability analysis that predicts how inviscid drop spectra depend on base state geometry. Theoretically, the base states are spherical caps characterized by their ‘flatness’ or fraction of the full sphere. Experimentally, quiescent shapes are controlled by pinning the drop at a circular contact line on the flat substrate and varying the drop volume. The response frequencies of the resonating drop are compared with Part 1 predictions. Agreement with theory is generally good but does deteriorate for flatter drops and higher modes. The measured frequency bands agree better with an extended model, introduced here, that accounts for forcing and weak viscous effects using viscous potential flow. As the flatness varies, regions are predicted where modal frequencies cross and where the spectra crowd. Frequency crossings and spectral crowding favour interaction of modes. Modal interactions of two kinds are documented, called ‘mixing’ and ‘competing’. Mixed modes are two pure modes superposed with little evidence of hysteresis. In contrast, modal competition involves hysteresis whereby one or the other mode disappears depending on the scan direction. Perhaps surprisingly, a linear inviscid irrotational theory provides a useful framework for understanding observations of forced sessile drop oscillations.

We derive an equation that relates the evolution in time of the maximum of the horizontal fluid velocity at the surface of an irrotational deep-water plunging or spilling breaker to the first component of the pressure gradient at the surface. The approach applies to overhanging wave profiles, up to breaking time.

The double diffusive convection between two parallel plates is numerically studied for a series of parameters. The flow is driven by the salinity difference and stabilised by the thermal field. Our simulations are directly compared with experiments by Hage & Tilgner (Phys. Fluids, vol. 22, 2010, 076603) for several sets of parameters and reasonable agreement is found. This, in particular, holds for the salinity flux and its dependence on the salinity Rayleigh number. Salt fingers are present in all simulations and extend through the entire height. The thermal Rayleigh number seems to have a minor influence on the salinity flux but affects the Reynolds number and the morphology of the flow. In addition to the numerical calculation, we apply the Grossmann–Lohse theory for Rayleigh–Bénard flow to the present problem without introducing any new coefficients. The theory successfully predicts the salinity flux both with respect to the scaling and even with respect to the absolute value for the numerical and experimental results.

Water drops falling on a deep pool can either coalesce to form a vortex ring or splash, depending on the impact conditions. The transition between coalescence and splashing proceeds via a number of intermediate steps, such as thick and thin jet formation and gas-bubble entrapment. We perform simulations to determine the conditions under which bubble entrapment and jet formation occur. A regime map is established for Weber numbers ranging from 50 to 300 and Froude numbers from 25 to 600. Vortex ring formation is seen for all of the regimes; it is greater for the coalescence regime and less in the case of the thin jet regime.