By incorporating the traditionally overlooked advective term in the wall-normal momentum equation, a new momentum integral equation is developed for two-dimensional incompressible turbulent boundary layers under arbitrary pressure gradients. The classical Kármán's integral arises as a special instance of the new momentum integral equation when the pressure gradient is weak. The new momentum integral equation's validity is substantiated by direct numerical simulation data. Unlike the classical Kármán's integral, which is limited to predicting wall shear stress within mild pressure gradients, the new momentum integral equation accurately computes wall shear stress across a broad range of pressure gradients, even in the presence of strong adverse pressure gradients that lead to flow separation. Moreover, a new pressure parameter $\beta _\kappa$ is introduced through examining terms in the new momentum integral equation. This parameter naturally quantifies the pressure gradient's influence on turbulent boundary layers and offers guidance for applying the classical Kármán's integral. Additionally, to facilitate experimental determination of wall shear stress under strong pressure gradients, an approximate integral equation is proposed that relies solely on easily measurable variables. Validation against direct numerical simulation data demonstrates that this simplified equation provides reasonably accurate estimates of wall shear stress in turbulent boundary layers experiencing strong pressure gradients.

Simultaneous time-resolved measurements of wall deformation and the 3-D velocity field in boundary layers over a compliant surface are performed by integrating Mach Zehnder interferometry with tomographic particle tracking velocimetry. The pressure is calculated by spatially integrating the material acceleration. Combining data obtained from several references, trends of the deformation r.m.s. scaled by the compliant wall thickness collapse when plotted vs pressure fluctuations scaled by the material shear modulus. For the present data, at all Reynolds numbers, the deformation waves travel at 53% of the free-stream velocity and have a preferred wavelength of three times the thickness. The latter is consistent with theoretical models. Adopting insight derived from atmospheric wind–wave interactions, the pressure–deformation correlations peak at or slightly above the ‘critical layer’, where the mean flow speed is equal to the surface wave speed. This layer is located within the log layer, and when expressed using inner variables, increases in elevation with increasing Reynolds number. For the entire region below the critical layer, wavenumber–frequency spectra of pressure and vertical velocity fluctuations indicate that the turbulence is phase locked and travels with the deformation, even for deformation amplitudes much smaller than a wall unit. In contrast, above the critical layer, the turbulence is advected at the local mean streamwise velocity, and its correlation with the deformation decays rapidly. These findings indicate that the height of the zone dominated by flow-deformation interactions is determined by the surface wave speed, and its variations are caused by deformation-induced modifications to the mean velocity profile.

We present a suite of large-eddy simulations (LES) of a wind farm operating in conventionally neutral boundary layers. A fixed 1.6 GW wind farm is considered for 40 different atmospheric stratification conditions to investigate effects on wind-farm efficiency and blockage, as well as related gravity-wave excitation. A tuned Rayleigh damping layer and a wave-free fringe-region method are used to avoid spurious excitation of gravity waves, and a domain-size study is included to evaluate and minimize effects of artificial domain blockage. A fully neutral reference case is also considered, to distinguish between a case with hydrodynamic blockage only, and cases that include hydrostatic blockage induced by the air column above the boundary layer and the excitation of gravity waves therein. We discuss in detail the dependence of gravity-wave excitation, flow fields and wind-farm blockage on capping-inversion height, strength and free-atmosphere lapse rate. In all cases, an unfavourable pressure gradient is present in front of the farm, and a favourable pressure gradient in the farm, with hydrostatic contributions arising from gravity waves at least an order of magnitude larger than hydrodynamic effects. Using respectively non-local and wake efficiencies $\eta _{nl}$ and $\eta _{w}$, we observe a strong negative correlation between the unfavourable upstream pressure rise and $\eta _{nl}$, and a strong positive correlation between the favourable pressure drop in the farm and $\eta _{w}$. Using a simplified linear gravity-wave model, we formulate a simple scaling for the ratio $(1-\eta _{nl})/\eta _{w}$, which matches reasonably well with the LES results.

Flux conditions are semi-analytical expressions that can be used to determine the flux at the grounding line of marine ice sheets. In the glaciology literature, such flux conditions have initially been established for the Weertman and Coulomb friction laws. However, the extension to more general and complex friction laws, such as the Budd friction law, for which basal friction depends on both the basal velocity and the effective pressure, is a topic of recent research. Several studies have also shown that hybrid friction laws, which consider a transition between a power-law friction far from the grounding line and a plastic behaviour close to it, were good candidates for improved modelling of marine ice sheets. In this article, we show that the flux conditions previously derived for the Weertman and Coulomb friction laws can be generalised to flux conditions for the Budd friction law with two different effective-pressure models. In doing so, we build a bridge between the results obtained for these two friction laws. We provide a justification for the existence and uniqueness of a solution to the boundary-layer problem based on asymptotic developments. We also generalise our results to hybrid friction laws, based on a parametrisation of the flux condition. Finally, we discuss the validity of the assumptions made during the derivation, and we provide additional explicit expressions for the flux that stay valid when the bedrock slopes are important or when the friction coefficients are relatively small.

A mathematical model is developed to investigate seabed heat transfer processes under long-crested ocean waves. The unsteady convection–diffusion equation for water temperature includes terms depending on the velocity field in the laminar boundary layer, analogous to mass transfer near the seabed. Here we consider regular progressive waves and standing waves reflected from a vertical structure, which complicate the convective term in the governing equation. Rectangular and Gaussian distributions of seabed temperature and heat flux are considered. Approximate analytical solutions are derived for uniform and trapezoidal currents, and compared against predictions from a numerical solver of the full equations. The effects of heat source profile, location and strength on heat transfer dynamics in the thermal boundary layer are explained, providing insights into seabed temperature forced convection mechanisms enhanced by free-surface waves.

The study concerns a slender, heavy body moving with streamwise velocity in a boundary layer. Modelling assumptions on body size reduce the governing equations for the body motion to a pair of nonlinear integro-differential equations (IDEs) which displays a wide range of distinguished behaviours, including eventual collision with the wall (‘crash’), escape to infinity (‘fly away’) and repeatedly travelling far from the wall and back again without ever colliding or escaping (‘bouncing’). The paper gives a survey of the variety of behaviour, as well as asymptotic analysis and insight into each category of fluid/body interaction and the conditions under which crash, fly away and bouncing occur.

Recently, the authors considered a thin steady developed viscous free-surface flow passing the sharp trailing edge of a horizontally aligned flat plate under surface tension and the weak action of gravity, acting vertically, in the asymptotic slender-layer limit (J. Fluid Mech., vol. 850, 2018, pp. 924–953). We revisit the capillarity-driven short-scale viscous–inviscid interaction, on account of the inherent upstream influence, immediately downstream of the edge and scrutinise flow detachment on all smaller scales. We adhere to the assumption of a Froude number so large that choking at the plate edge is insignificant but envisage the variation of the relevant Weber number of $O(1)$. The main focus, tackled essentially analytically, is the continuation of the structure of the flow towards scales much smaller than the interactive ones and where it no longer can be treated as slender. As a remarkable phenomenon, this analysis predicts harmonic capillary ripples of Rayleigh type, prevalent on the free surface upstream of the trailing edge. They exhibit an increase of both the wavelength and amplitude as the characteristic Weber number decreases. Finally, the theory clarifies the actual detachment process, within a rational description of flow separation. At this stage, the wetting properties of the fluid and the microscopically wedge-shaped edge, viewed as infinitely thin on the larger scales, come into play. As this geometry typically models the exit of a spout, the predicted wetting of the wedge is related to what in the literature is referred to as the teapot effect.

The present study focuses on acoustically induced thermal effects on Rayleigh streaming inside a resonator. Firstly, we consider the effect of the transverse (or wall-normal) mean temperature gradient on the acoustic streaming flow generated by a standing wave between two parallel plates. Analytical expressions for acoustic quantities are developed and used to express the sources of linear streaming. The influence of a transverse temperature variation on the streaming velocity is clearly identified through a term proportional to the temperature difference and to the square of the half-width of the guide. This term modifies the Rayleigh streaming pattern and may generate an additional vortex. On the other hand, the longitudinal (or wall-parallel) temperature difference is calculated as a cumulated effect of thermoacoustic heat transport in the fluid, heat conduction in the wall and heat convection of the air outside the resonator. At high acoustic levels, heat is significantly convected by the streaming flow and the resulting transverse temperature difference is proportional to the longitudinal temperature difference. Combining these expressions brings out a new criterion parameter for the nonlinear Reynolds number ($Re_{NL}$) characterizing the transition in streaming patterns at high acoustic levels. This result explains previous experimental and numerical observations of the streaming flow dynamics at high acoustic amplitudes, under different temperature boundary conditions, and can provide a powerful prediction tool for streaming pattern transitions.

Recent comparisons between classical Wagner theory for the impact of two liquid droplets and direct numerical simulations in Cimpeanu & Moore (J. Fluid Mech., vol. 856, 2018, pp. 764–796) show that, in some regimes, the inviscid theory over-predicts the thickness of the root of the splash jet that forms in the impact, while also struggling to predict the angle at which the jet is emitted. The effect of capillary and viscous perturbations to Helmholtz flows was investigated in a previous study, see Moore et al. (J. Fluid Mech., vol. 742, 2014, R1). However, the paper in question ignored a term in the second-order perturbation analysis, which needs to be included in order to predict the displacement of the inviscid free boundary to lowest order. In this paper, we derive a singular integro-differential equation for the free-surface perturbations caused by viscosity in Helmholtz flows and discuss its application both in the context of Wagner theory and more generally. In particular, viscosity can induce non-monotonic behaviour in the free boundary profiles near points of maximum curvature.

Acoustic streaming generated by a plane standing wave between two infinite plates or inside a cylindrical tube is considered, under the isentropic flow assumption. A two-dimensional analysis is performed in the linear case of slow streaming motion, based on analytical formal solutions of separate problems, each associated with a specific source term (Reynolds stress term). In order to obtain these analytical solutions, a necessary geometrical hypothesis is that $(R/L)^{2}\ll 1$, where $R$ and $L$ are the guide half-width (or radius) and length. The effect of the two source terms classically taken into account is quantified in order to derive the dependence of the maximum axial streaming velocity on the axis as a function of the ratio $R/\unicode[STIX]{x1D6FF}_{\unicode[STIX]{x1D708}}$, where $\unicode[STIX]{x1D6FF}_{\unicode[STIX]{x1D708}}$ is the acoustic boundary layer thickness. The effect of two other source terms that are usually neglected, is then analysed. It is found that one of these terms can generate a counter-rotating streaming flow. While negligible for very narrow guides, this term can become important for some values of the aspect ratio $L/R$.

For a family of elliptic operators with periodically oscillating coefficients, $-{\rm div}(A(\cdot /\varepsilon )\nabla )$ with tiny ε > 0, we comprehensively study the first-order expansions of eigenvalues and eigenfunctions (eigenspaces) for both the Dirichlet and Neumann problems in bounded, smooth and strictly convex domains (or more general domains of finite type). A new first-order correction term is introduced to derive the expansion of eigenfunctions in L2 or $H^1_{\rm loc}$. Our results rely on the recent progress on the homogenization of boundary layer problems.

The free-surface flow formed by a circular jet impinging on a stationary disk is analysed theoretically. We develop a simple and coherent model to predict the location and height of the jump for high-viscosity liquids. The study explores the effect of gravity in the supercritical flow. The formulation reduces to a problem, involving only one parameter: $\unicode[STIX]{x1D6FC}=Re^{1/3}Fr^{2}$, where $Re$ and $Fr$ are the Reynolds and Froude numbers based on the flow rate and the jet radius. We show that the jump location coincides with the singularity in the thin-film equation when gravity is included, suggesting that the jump location can be determined without the knowledge of downstream flow conditions such as the jump height, the radius of the disk, which corroborates earlier observations in the case of type I circular hydraulic jumps. Consequently, there is no need for a boundary condition downstream to determine the jump radius. Our results corroborate well existing measurements and numerical simulation. Our predictions also confirm the constancy of the Froude number $Fr_{J}$ based on the jump radius and height as suggested by the measurements of Duchesne et al. (Europhys. Lett., vol. 107, 2014, 54002). We establish theoretically the conditions for $Fr_{J}$ to remain independent of the flow rate. The subcritical flow and the height of the hydraulic jump are sought subject to the thickness at the edge of the disk, comprising contributions based on the capillary length and minimum flow energy. The thickness at the edge of the disk appears to be negligibly small for high-viscosity liquids.

The flow induced by a disk rotating at the bottom of a cylindrical tank is characterised using numerical techniques – computation of steady solutions or time-averaged two-dimensional and three-dimensional direct simulations – as well as laser-Doppler velocimetry measurements. Axisymmetric steady solutions reveal the structure of the toroidal flow located at the periphery of the central solid body rotation region. When viewed in a meridional plane, this flow cell is found to be bordered by four layers, two at the solid boundaries, one at the free surface and one located at the edge of the central region, which possesses a sinuous shape. The cell intensity and geometry are determined for several fluid-layer aspect ratios; the flow is shown to depend very weakly on Froude number (associated with surface deformation) or on Reynolds number if sufficiently large. The paper then focuses on the high Reynolds number regime for which the flow has become unsteady and three-dimensional while the surface is still almost flat. Direct numerical simulations show that the averaged flow shares many similarities with the above steady axisymmetric solutions. Experimental measurements corroborate most of the numerical results and also allow for the spatio-temporal characterisation of the fluctuations, in particular the azimuthal structure and frequency spectrum. Mean azimuthal velocity profiles obtained in this transitional regime are eventually compared to existing theoretical models.

Dynamic stall due to a ramp-type pitching motion is investigated on the NACA 0012 airfoil at chord Reynolds number of $Re_{c}=1.0\times 10^{6}$ through the use of wall-resolved large-eddy simulation. Emphasis is placed on the unsteady boundary-layer interactions that develop as the airfoil approaches stall. At this Reynolds number it is shown that turbulent separation moves upstream across much of the airfoil suction surface. When turbulent separation reaches the leading-edge separation bubble, a bursting event is initiated leading to a strong coherent leading-edge vortex structure. This vortex wraps up the turbulent shear layer to form a large dynamic stall vortex. The use of large-eddy simulation elucidates the roll of the laminar separation bubble in defining the onset of the dynamic stall process. Comparisons are made to identical simulations at lower Reynolds numbers of $Re_{c}=0.2\times 10^{6}$ and $0.5\times 10^{6}$. This comparison demonstrates trends in the boundary-layer mechanics that explain the sensitivity of the dynamic stall process to Reynolds number.

Direct numerical simulation data of bypass transition in flat-plate boundary layers are analysed to examine the characteristics of turbulence in the transitional regime. When intermittency is 50 % or less, the flow features a juxtaposition of turbulence spots surrounded by streaky laminar regions. Conditionally averaged turbulence statistics are evaluated within the spots, and are compared to standard time averaging in both the transition region and in fully turbulent boundary layers. The turbulent-conditioned root-mean-square levels of the streamwise velocity perturbations are notably elevated in the early transitional boundary layer, while the wall-normal and spanwise components are closer to the levels typical for fully turbulent flow. The analysis is also extended to include ensemble averaging of the spots. When the patches of turbulence are sufficiently large, they develop a core region with similar statistics to fully turbulent boundary layers. Within the tip and the wings of the spots, however, the Reynolds stresses and terms in the turbulence kinetic energy budget are elevated. The enhanced turbulence production in the transition zone, which exceeds the levels from fully turbulent boundary layers, contributes to the higher skin-friction coefficient in that region. Qualitatively, the same observations hold for different spot sizes and levels of free-stream turbulence, except for young spots which do not yet have a core region of developed turbulence.

We examine disturbances leading to optimal energy growth in a spatially developing, zero-pressure-gradient turbulent boundary layer. The slow development of the turbulent mean flow in the streamwise direction is modelled through a parabolized formulation to enable a spatial marching procedure. In the present framework, conventional spatial optimal disturbances arise naturally as the homogeneous solution to the linearized equations subject to a turbulent forcing at particular wavenumber combinations. A wave-like decomposition for the disturbance is considered to incorporate both conventional stationary modes as well as propagating modes formed by non-zero frequency/streamwise wavenumber and representative of convective structures naturally observed in wall turbulence. The optimal streamwise wavenumber, which varies with the spatial development of the turbulent mean flow, is computed locally via an auxiliary optimization constraint. The present approach can then be considered, in part, as an extension of the resolvent-based analyses for slowly developing flows. Optimization results reveal highly amplified disturbances for both stationary and propagating modes. Stationary modes identify peak amplification of structures residing near the centre of the logarithmic layer of the turbulent mean flow. Inner-scaled disturbances reminiscent of near wall streaks, and amplified over short streamwise distances, are identified in the computed streamwise energy spectra. In all cases, however, propagating modes surpass their stationary counterpart in both energy amplification and relative contribution to total fluctuation energy. We identify two classes of large-scale energetic modes associated with the logarithmic and wake layers of the turbulent mean flow. The outer-scaled wake modes agree well with the large-scale motions that populate the wake layer. For high Reynolds numbers, the log modes increasingly dominate the energy spectra with the predicted streamwise and wall-normal scales in agreement with superstructures observed in turbulent boundary layers.

The linear stability of high-speed boundary layers can be altered by distortions to the base velocity and temperature profiles. An analytic expression for the sensitivity is derived for parallel and spatially developing boundary layers, the latter using linear parabolized stability equations and their adjoint. Both the slow mode, S, and the fast mode, F, are investigated at Mach number 4.5. The mode S is more sensitive with respect to distortion in base velocity than in base temperature. The sensitivity is largest within the boundary layer away from the wall. Near the critical layer, where the phase speed of the mode equals the base streamwise velocity, the sensitivity to the base streamwise velocity is negative. For the mode F, there is a discontinuous jump in the sensitivity when the phase speed is below unity, and a critical layer is established. The sensitivity of the two modes increases with the Reynolds number, but there is a sudden drop and a jump in the sensitivities of the modes S and F, respectively, near the synchronization point where the phase speeds of the two modes are equal. Furthermore, the maximum uncertainty bounds are obtained for the distorted base state that maximizes the destabilization or stabilization of the modes by solving the Lagrangian optimization problem for the sensitivity. The sensitivity of the flow stability to surface heating is then studied, and changes in growth rate and the $N$-factor are evaluated. The formulation provides a clear physical interpretation of these changes, and establishes uncertainty bounds for stability predictions for a given level of uncertainty in wall temperature.