Coherent structures of a three-dimensional (3D) separation due to an adverse pressure gradient are investigated experimentally. The flow set-up consists of a flat plate to develop a turbulent boundary layer upstream of an asymmetric two-dimensional diffuser with one diverging surface. The diffuser surface has an initial mild curvature followed by a flat section where flow separation occurs. The top and the two sidewalls of the diffuser are not equipped with any flow control mechanism to form a 3D separation. Planar particle image velocimetry (PIV) using four side-by-side cameras is applied to characterize the flow with high spatial resolution over a large streamwise-wall-normal field of view (FOV). Tomographic PIV (tomo-PIV) is also applied for volumetric measurement in a domain flush with the flat surface of the diffuser. The mean flow obtained from averaging instantaneous velocity fields of this intermittent unsteady flow appears as a vortex with an elliptical cross-section. The major axis of the ellipse is tilted with respect to the streamwise direction. As a result, the average velocity in the mid-span of the diffuser has an upstream forward flow and a downstream backward flow, separated by a point of zero wall shear stress. Sweep motions mainly carry out transport of turbulent kinetic energy upstream of this point, while ejections dominate at the downstream region. In the instantaneous flow fields, forward and backward flows have equivalent strength, and the separation front is extended in the spanwise direction. The conditional average of the separation instants forms a saddle-point structure with streamlines converging in the spanwise direction. Proper orthogonal decomposition (POD) of the tomo-PIV data demonstrates that about 42 % of the turbulent kinetic energy is present in the first pair of modes, with a strong spanwise component. The spatial modes of POD also show focus, node and saddle-point structures. The average of the coefficients of the dominant POD modes during the separation events is used to develop a reduced-order model (ROM). Based on the ROM, the instantaneous 3D separation over the diffuser is a saddle-point structure interacting with focus-type structures.

When the Galileo number is below the first bifurcation, the instability and transition of a vertical ascension or the fall of a free sphere affected by a vertical magnetic field are investigated numerically. A compact model is used to explain that the magnetic field can destabilize the fluid–solid system. When the interaction parameter exceeds a critical value, the sphere trajectory is transitioned from a steady vertical trajectory to a steady oblique one. Furthermore, the trajectory will remain vertical at a sufficiently large magnetic field because of a double effect of the magnetic field on the fluid–solid system. Under the influence of an external vertical magnetic field, four wake patterns at the rear of the sphere are found and the physical behaviour of the free sphere is independent of the density ratio. The wake or trajectory of the free sphere is only determined by the Galileo number $G$ and the interaction parameter $N$. A close relationship between the streamwise vorticity and the sphere motion is found. An interesting ‘agglomeration phenomenon’ is also found, which shows that the vertical velocities are agglomerated into a point for a certain magnetic field regardless of the Galileo number and satisfy a scaling law $V_{z}\sim N^{-1/4}$, when $N>1$. The principal results of the present work are summarized in a map of regimes in the $\{G,N\}$ plane.

Two expressions for the nonlinear dispersion relation for gravity waves on water of constant depth are derived, one for wave fields with discrete amplitude spectra, the other for wave fields with continuous wavenumber energy spectra. Numerical examples for wave quartets and for two-dimensional Pierson–Moskowitz spectra are given, and an important possible application is discussed.

The leading-edge vortex (LEV) is a powerful unsteady flow structure that can result in significant unsteady loads on lifting blades and wings. Using force, surface pressure and flow field measurements, this work represents an experimental campaign to characterize LEV behaviour in sinusoidally surging flows with widely varying amplitudes and frequencies. Additional tests were conducted in reverse flow surge, with kinematics similar to the tangential velocity profile seen by a blade element in recent high-advance-ratio rotor experiments. General results demonstrate the variability of LEV convection properties with reduced frequency, which greatly affected the average lift-to-drag ratio in a cycle. Analysis of surface pressure measurements suggests that LEV convection speed is a function only of the local instantaneous flow velocity. In the rotor-comparison tests, LEVs formed in reverse flow surge were found to convect more quickly than the corresponding reverse flow LEVs that form on a high-advance-ratio rotor, demonstrating that rotary motion has a stabilizing effect on LEVs. The reverse flow surging LEVs were also found to be of comparable strength to those observed on the high-advance-ratio rotor, leading to the conclusion that a surging-wing simplification might provide a suitable basis for low-order models of much more complex three-dimensional flows.

The objective of this work is to investigate linear modal and algebraic instability in Poiseuille flows with fluids close to their vapour–liquid critical point. Close to this critical point, the ideal gas assumption does not hold and large non-ideal fluid behaviours occur. As a representative non-ideal fluid, we consider supercritical carbon dioxide ($\text{CO}_{2}$) at a pressure of 80 bar, which is above its critical pressure of 73.9 bar. The Poiseuille flow is characterized by the Reynolds number ($Re=\unicode[STIX]{x1D70C}_{w}^{\ast }u_{r}^{\ast }h^{\ast }/\unicode[STIX]{x1D707}_{w}^{\ast }$), the product of the Prandtl ($Pr=\unicode[STIX]{x1D707}_{w}^{\ast }C_{pw}^{\ast }/\unicode[STIX]{x1D705}_{w}^{\ast }$) and Eckert numbers ($Ec=u_{r}^{\ast 2}/C_{pw}^{\ast }T_{w}^{\ast }$) and the wall temperature that in addition to pressure determine the thermodynamic reference condition. For low Eckert numbers, the flow is essentially isothermal and no difference with the well-known stability behaviour of incompressible flows is observed. However, if the Eckert number increases, the viscous heating causes gradients of thermodynamic and transport properties, and non-ideal gas effects become significant. Three regimes of the laminar base flow can be considered: the subcritical (temperature in the channel is entirely below its pseudo-critical value), transcritical and supercritical temperature regimes. If compared to the linear stability of an ideal gas Poiseuille flow, we show that the base flow is modally more unstable in the subcritical regime, inviscid unstable in the transcritical regime and significantly more stable in the supercritical regime. Following the principle of corresponding states, we expect that qualitatively similar results will be obtained for other fluids at equivalent thermodynamic states.

When a deep body of fluid with a stable salinity gradient is heated from below at a horizontal boundary a destabilizing temperature gradient develops and can lead to instabilities. We will focus on two variants of this problem: the sudden increase in the boundary temperature at the initial time and the sudden turning on of a constant heat flux. These generate time-dependent temperature profiles. We look at the growing phase of the linear instabilities as an initial value problem where the initial time for the instabilities is a parameter to be determined. We determine numerically the optimal initial conditions and the optimal starting time for the instabilities to ensure that the maximum growth occurs at some given later time. The method that is used is an extension of the method developed by Kerr & Gumm (J. Fluid Mech., vol. 825, 2017, pp. 1002–1034) in their investigation of the stability of developing temperature boundary layers at horizontal and vertical boundaries. This requires the use of an appropriate measure of the amplitude of the disturbances which is identified. The effectiveness of this approach is verified by looking at the classic problem of double-diffusive convection in a horizontal layer, where we look at both the salt-finger regime and the diffusive regime. We show that this approach is an effective way of investigating instabilities where the background gradients time dependent. For the problem of heating a salinity gradient from below, as the heat diffuses into the fluid the effective thermal Rayleigh number based on the instantaneous diffusion length scale grows. For the case of a sudden increase in the temperature by a fixed amount the effective thermal Rayleigh number is proportional to $t^{3/2}$, and for a constant heat flux it is proportional to $t^{2}$, where $t$ is the time since the onset of heating. However, the effective salt Rayleigh number also grows as $t^{2}$. We will show that for the constant temperature case the thermal Rayleigh number initially dominates and the instabilities undergo a phase where the convection is essentially thermal, and the onset is essentially instantaneous. As the salt Rayleigh number becomes more significant the instability undergoes a transition to oscillatory double-diffusive convection. For the constant heat flux the ratio of the thermal and salt Rayleigh numbers is constant, and the instabilities are always double diffusive in their nature. These instabilities initially decay. Hence, to achieve the largest growth at some given fixed time, there is an optimal time after the onset of heating for the instabilities to be initiated. These instabilities are essentially double diffusive throughout their growth.

We theoretically prove the existence in granular fluids of a thermal convection that is inherent in the sense that it is always present and has no thermal gradient threshold (convection occurs for all finite values of the Rayleigh number). More specifically, we study a gas of inelastic smooth hard disks enclosed in a rectangular region under a constant gravity field. The vertical walls act as energy sinks (i.e. inelastic walls that are parallel to gravity), whereas the other two walls are perpendicular to gravity and act as energy sources. We show that this convection is due to the combined action of dissipative lateral walls and a volume force (in this case, gravitation). Hence, we call it dissipative lateral walls convection (DLWC). Our theory, which also describes the limit case of elastic collisions, shows that inelastic particle collisions enhance the DLWC. We perform our study via numerical solutions (volume-element method) of the corresponding hydrodynamic equations in an extended Boussinesq approximation. We show that our theory describes the essentials of the results for similar (but more complex) laboratory experiments.

Particle-laden flows of sedimenting solid particles or droplets in a carrier gas have strong inter-phase coupling. Even at low particle volume fractions, the two-way coupling can be significant due to the large particle to gas density ratio. In this semi-dilute regime, the slip velocity between phases leads to sustained clustering that strongly modulates the overall flow. The analysis of perturbations in homogeneous shear reveals the process by which clusters form: (i) the preferential concentration of inertial particles in the stretching regions of the flow leads to the formation of highly concentrated particle sheets, (ii) the thickness of the latter is controlled by particle-trajectory crossing, which causes a local dispersion of particles, (iii) a transverse Rayleigh–Taylor instability, aided by the shear-induced rotation of the particle sheets towards the gravity normal direction, breaks the planar structure into smaller clusters. Simulations in the Euler–Lagrange formalism are compared to Euler–Euler simulations with the two-fluid and anisotropic-Gaussian methods. It is found that the two-fluid method is unable to capture the particle dispersion due to particle-trajectory crossing and leads instead to the formation of discontinuities. These are removed with the anisotropic-Gaussian method which derives from a kinetic approach with particle-trajectory crossing in mind.

Super-large-scale particle image velocimetry (SLPIV) and the associated flow visualization technique using natural snowfall have been shown to be effective tools to probe the turbulent velocity field and coherent structures around utility-scale wind turbines (Hong et al.Nat. Commun., vol. 5, 2014, article 4216). Here, we present a follow-up study using the data collected during multiple deployments from 2014 to 2016 around the 2.5 MW turbine at the EOLOS field station. These data include SLPIV measurements in the near wake of the turbine in a field of view of 115 m (vertical) $\times$ 66 m (streamwise), and the visualization of tip vortex behaviour near the elevation corresponding to the bottom blade tip over a broad range of turbine operational conditions. The SLPIV measurements provide velocity deficit and turbulent kinetic energy assessments over the entire rotor span. The instantaneous velocity fields from SLPIV indicate the presence of intermittent wake contraction states which are in clear contrast with the expansion states typically associated with wind turbine wakes. These contraction states feature a pronounced upsurge of velocity in the central portion of the wake. The wake velocity ratio $R_{w}$, defined as the ratio of the spatially averaged velocity of the inner wake to that of the outer wake, is introduced to categorize the instantaneous near wake into expansion ($R_{w}<1$) and contraction states ($R_{w}>1$). Based on the $R_{w}$ criterion, the wake contraction occurs 25 % of the time during a 30 min time duration of SLPIV measurements. The contraction states are found to be correlated with the rate of change of blade pitch by examining the distribution and samples of time sequences of wake states with different turbine operation parameters. Moreover, blade pitch change is shown to be strongly correlated to the tower and blade strains measured on the turbine, and the result suggests that the flexing of the turbine tower and the blades could indeed lead to the interaction of the rotor with the turbine wake, causing wake contraction. The visualization of tip vortex behaviour demonstrates the presence of a state of consistent vortex formation as well as various types of disturbed vortex states. The histograms corresponding to the consistent and disturbed states are examined over a number of turbine operation/response parameters, including turbine power and tower strain as well as the fluctuation of these quantities, with different conditional sampling restrictions. This analysis establishes a clear statistical correspondence between these turbine parameters and tip vortex behaviours under different turbine operation conditions, which is further substantiated by examining samples of time series of these turbine parameters and tip vortex patterns. This study not only offers benchmark datasets for comparison with the-state-of-the-art numerical simulation, laboratory and field measurements, but also sheds light on understanding wake characteristics and the downstream development of the wake, turbine performance and regulation, as well as developing novel turbine or wind farm control strategies.

This article solves numerically the equations of the leaky-dielectric model applied to cone jets. The solution is a function of the properties of the fluid and its flow rate, universal in that it does not depend on the geometry and potential of the electrodes. This is made possible by the use of the potential field generated by a semi-infinite Taylor cone as a far-field boundary condition. The numerical solution yields the current emitted by the electrospray, which compares well with experimental data, and detailed information about the velocity, surface charge, electric field and the position of the free surface. These characteristics are generally inaccessible through experiments, and are needed to understand the relative importance of competing processes and the dominant physics. The simulations investigate the liquids tributyl phosphate and propylene carbonate (dielectric constants of 8.91 and 64.9 respectively), in a wide range of electrical conductivities and flow rates. The simulations show that the position of the surface, expressed in units of the characteristic length $r_{c}$, is largely invariant regardless of the physical properties and flow rates of the liquids. The surface charge falls below its equilibrium value along the transition from cone to jet, with a deficit that increases with the ratio between the electrical relaxation and flow residence times. Several characteristics of the cone jet are functions of the dielectric constant, which is consistent with the importance of charge relaxation effects (i.e. with the absence of surface charge equilibrium). The electric energy transferred to the transition region is largely transformed into viscous and ohmic dissipation, and conversion into kinetic energy only dominates once most of the current is fixed on the surface.

The two-phase mixing layer formed between parallel gas and liquid streams is an important fundamental problem in turbulent multiphase flows. The problem is relevant to many industrial applications and natural phenomena, such as air-blast atomizers in fuel injection systems and breaking waves in the ocean. The velocity difference between the gas and liquid streams triggers an interfacial instability which can be convective or absolute depending on the stream properties and injection parameters. In the present study, a direct numerical simulation of a two-phase gas–liquid mixing layer that lie in the absolute instability regime is conducted. A dominant frequency is observed in the simulation and the numerical result agrees well with the prediction from viscous stability theory. As the interfacial wave plays a critical role in turbulence transition and development, the temporal evolution of turbulent fluctuations (such as the enstrophy) also exhibits a similar frequency. To investigate the statistical response of the multiphase turbulence flow, the simulation has been run for a long physical time so that time-averaging can be performed to yield the statistically converged results for Reynolds stresses and the turbulent kinetic energy (TKE) budget. An extensive mesh refinement study using from 8 million to about 4 billions cells has been performed. The turbulent dissipation is shown to be highly demanding on mesh resolution compared with other terms in TKE budget. The results obtained with the finest mesh are shown to be close to converged results of turbulent dissipation which allow us to obtain estimations of the Kolmogorov and Hinze scales. The estimated Kolmogorov scale is found to be similar to the cell size of the finest mesh used here. The computed Hinze scale is significantly larger than the size of droplets observed and does not seem to be a relevant length scale to describe the smallest size of droplets formed in atomization.

Liquid films can be entrained when the dewetting velocity attains a threshold, and this dynamical wetting transition has been well studied in the situation of plane substrates. We investigate the forced dewetting in a capillary tube using diffuse-interface simulations and lubrication analysis, focusing on the onset of wetting transition and subsequent interface evolution. Results show that the meniscus remains stable when the displacing rate is below a threshold, beyond which film entrainment occurs and eventually leads to the formation of Taylor bubbles separated by liquid slugs, as has also been observed in the recent experiments of Zhao et al. (Phys. Rev. Lett., vol. 120, 2018, 084501). We derive an analytical solution of the critical capillary number, and demonstrate that the wetting transition is accompanied by a vanishing apparent contact angle and an abrupt drop of the contact-line velocity. Both the bubble and slug lengths are found to depend on the capillary number and the wettability of the wall. A theoretical formula for the bubble length is also proposed and compares favourably with numerical and experimental results.

Slow droplet motion on chemically heterogeneous substrates is considered analytically and numerically. We adopt the long-wave approximation which yields a single partial differential equation for the droplet height in time and space. A matched asymptotic analysis in the limit of nearly circular contact lines and vanishingly small slip lengths yields a reduced model consisting of a set of ordinary differential equations for the evolution of the Fourier harmonics of the contact line. The analytical predictions are found, within the domain of their validity, to be in good agreement with the solutions to the governing partial differential equation. The limitations of the reduced model when the contact line undergoes stronger deformations are partially lifted by proposing a hybrid scheme which couples the results of the asymptotic analysis with the boundary integral method. This approach improves the agreement with the governing partial differential equation, but at a computational cost which is significantly lower compared to that required for the full problem.

Gravitational instability is a key process that may lead to fragmentation of gaseous structures (sheets, filaments, haloes) in astrophysics and cosmology. We introduce here a method to derive analytic expressions for the growth rate of gravitational instability in a plane stratified medium. First, the main strength of our approach is to reduce this intrinsically fourth-order eigenvalue problem to a sequence of second-order problems. Second, an interesting by-product is that the unstable part of the spectrum is computed by making use of its stable part. Third, as an example, we consider a pressure-confined, static, self-gravitating slab of a fluid with an arbitrary polytropic exponent, with either free or rigid boundary conditions. The method can naturally be generalised to analyse the stability of richer, more complex systems. Finally, our analytical results are in excellent agreement with numerical solutions. Their second-order expansions provide a valuable insight into how the rate and wavenumber of maximal instability behave as functions of the polytropic exponent and the external pressure (or, equivalently, the column density of the slab).

Wall modelling in large-eddy simulation (LES) is necessary to overcome the prohibitive near-wall resolution requirements in high-Reynolds-number turbulent flows. Most existing wall models rely on assumptions about the state of the boundary layer and require a priori prescription of tunable coefficients. They also impose the predicted wall stress by replacing the no-slip boundary condition at the wall with a Neumann boundary condition in the wall-parallel directions while maintaining the no-transpiration condition in the wall-normal direction. In the present study, we first motivate and analyse the Robin (slip) boundary condition with transpiration (non-zero wall-normal velocity) in the context of wall-modelled LES. The effect of the slip boundary condition on the one-point statistics of the flow is investigated in LES of turbulent channel flow and a flat-plate turbulent boundary layer. It is shown that the slip condition provides a framework to compensate for the deficit or excess of mean momentum at the wall. Moreover, the resulting non-zero stress at the wall alleviates the well-known problem of the wall-stress under-estimation by current subgrid-scale (SGS) models (Jiménez & Moser, AIAA J., vol. 38 (4), 2000, pp. 605–612). Second, we discuss the requirements for the slip condition to be used in conjunction with wall models and derive the equation that connects the slip boundary condition with the stress at the wall. Finally, a dynamic procedure for the slip coefficients is formulated, providing a dynamic slip wall model free of a priori specified coefficients. The performance of the proposed dynamic wall model is tested in a series of LES of turbulent channel flow at varying Reynolds numbers, non-equilibrium three-dimensional transient channel flow and a zero-pressure-gradient flat-plate turbulent boundary layer. The results show that the dynamic wall model is able to accurately predict one-point turbulence statistics for various flow configurations, Reynolds numbers and grid resolutions.

Microfluidic sorting of deformable particles finds many applications, for example, medical devices for cells. Deterministic lateral displacement (DLD) is one of them. Particle sorting via DLD relies only on hydrodynamic forces. For rigid spherical particles, this separation is to a great extent understood and can be attributed to size differences: large particles displace in the lateral direction with respect to the flow while small particles travel in the flow direction with negligible lateral displacement. However, the separation of non-spherical deformable particles such as red blood cells (RBCs) is more complicated than that of rigid particles. For example, is it possible to separate deformable particles that have the same size but different mechanical properties? We study deformability-based sorting of same-size RBCs via DLD using an in-house integral equation solver for vesicle flows in two dimensions. Our goal is to quantitatively characterize the physical mechanisms that enable the cell separation. To this end, we systematically investigate the effects of the interior fluid viscosity and membrane elasticity of a cell on its behaviour. In particular, we consider deep devices in which a cell can show rich dynamics such as taking a particular angular orientation depending on its mechanical properties. We have found out that cells moving with a sufficiently high positive inclination angle with respect to the flow direction displace laterally while those with smaller angles travel with the flow streamlines. Thereby, deformability-based cell sorting is possible. The underlying mechanism here is cell migration due to the cell’s positive inclination and the shear gradient. The higher the inclination is, the farther the cell can travel laterally. We also assess the efficiency of the technique for dense suspensions. It turns out that most of the cells in dense suspensions do not displace in the lateral direction no matter what their deformability is. As a result, separating cells using a DLD device becomes harder.

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.

A linear numerical study is conducted to quantify the effect of swirl on the response behaviour of premixed lean flames to general harmonic excitation in the inlet, upstream of combustion. This study considers axisymmetric M-flames and is based on the linearised compressible Navier–Stokes equations augmented by a simple one-step irreversible chemical reaction. Optimal frequency response gains for both axisymmetric and non-axisymmetric perturbations are computed via a direct–adjoint methodology and singular value decompositions. The high-dimensional parameter space, containing perturbation and base-flow parameters, is explored by taking advantage of generic sensitivity information gained from the adjoint solutions. This information is then tailored to specific parametric sensitivities by first-order perturbation expansions of the singular triplets about the respective parameters. Valuable flow information, at a negligible computational cost, is gained by simple weighted scalar products between direct and adjoint solutions. We find that for non-swirling flows, a mode with azimuthal wavenumber $m=2$ is the most efficiently driven structure. The structural mechanism underlying the optimal gains is shown to be the Orr mechanism for $m=0$ and a blend of Orr and other mechanisms, such as lift-up, for other azimuthal wavenumbers. Further to this, velocity and pressure perturbations are shown to make up the optimal input and output showing that the thermoacoustic mechanism is crucial in large energy amplifications. For $m=0$ these velocity perturbations are mainly longitudinal, but for higher wavenumbers azimuthal velocity fluctuations become prominent, especially in the non-swirling case. Sensitivity analyses are carried out with respect to the Mach number, Reynolds number and swirl number, and the accuracy of parametric gradients of the frequency response curve is assessed. The sensitivity analysis reveals that increases in Reynolds and Mach numbers yield higher gains, through a decrease in temperature diffusion. A rise in mean-flow swirl is shown to diminish the gain, with increased damping for higher azimuthal wavenumbers. This leads to a reordering of the most effectively amplified mode, with the axisymmetric ($m=0$) mode becoming the dominant structure at moderate swirl numbers.

We investigate the behaviour of large-scale coherent structures in a spanwise-heterogeneous turbulent boundary layer, using particle image velocimetry on multiple orthogonal planes. The statistical three-dimensionality is imposed by a herringbone riblet surface, although the key results presented here will be common to many cases of wall turbulence with embedded secondary flows in the form of mean streamwise vortices. Instantaneous velocity fields in the logarithmic layer reveal elongated low-momentum streaks located over the upwash-flow region, where their spanwise spacing is forced by the $2\unicode[STIX]{x1D6FF}$ periodicity of the herringbone pattern. These streaks largely resemble the turbulence structures that occur naturally (and randomly located) in spanwise-homogeneous smooth-/rough-wall boundary layers, although here they are directly formed by the roughness pattern. In the far outer region, the large spanwise spacing permits the streaks to aggressively meander. The mean secondary flows are the time-averaged artefact of the unsteady and spanwise asymmetric large-scale roll modes that accompany these meandering streaks. Interestingly, this meandering, or instability, gives rise to a pronounced streamwise periodicity (i.e. an alternating coherent pattern) in the spatial statistics, at wavelengths of approximately 4.5$\unicode[STIX]{x1D6FF}$. Overall, the observed behaviours largely resemble the streak-instability model that has been proposed for the buffer region, only here at a much larger scale and at a forced spanwise spacing. This observation further confirms recent observations that such features may occur at an entire hierarchy of scales throughout the turbulent boundary layer.

Water droplet freezing affects many aspects of our daily lives, although there is no comprehensive model which retrieves all of the experimentally observed features when a liquid water droplet deposited on a cold substrate turns to ice. In this paper, we present general governing equations to describe water droplet freezing on a solid substrate by accounting for the physical properties of each phase present, namely the liquid and ice, in addition to the solid substrate. The approach, which takes advantage of the full mean curvature expression of both the droplet–air and liquid–ice interfaces, disjoining pressure, the Gibbs–Thomson effect, natural convection and the substrate thermal and physico-chemical properties, enables us to model a more realistic frozen droplet shape, without a prior assumption for the freezing growth angle. In addition to correctly predicting the freezing time, we capture both qualitatively and quantitatively the key experimentally observed features during water droplet freezing such as volume expansion, concave ice front and the cusp singularity. Furthermore, the proposed equation for the tip angle seems to explain its experimentally observed variability.