We examine the dynamics of a semi-infinite vortex sheet attached not to a semi-infinite plate but instead to a rigid right-angled wedge, with the sheet aligned along one of its edges. Our approach to this problem, which was suggested by David Crighton, accords well with the fundamental ethos of Crighton’s work, which was characterized by ‘the application of rigorous mathematical approximations to fluid mechanical idealizations of practically relevant problems’ (Ffowcs Williams, Annu. Rev. Fluid Mech., vol. 34, 2002, pp. 37–49). The resulting linearised unsteady potential flow is forced by an oscillatory dipole in the uniform stream passing along the top of the wedge, while there is stagnant fluid in the remaining quadrant. Spatial instability is considered according to well-established methods: causality is enforced by allowing the frequency to become temporarily complex. The essentially quadrant-type geometry replaces the usual Wiener–Hopf technique by the Mellin transform. The core difficulty is that a first-order difference equation of period 4 requires a solution of period unity. As a result, the complex fourth roots $(\pm 1\pm \text{i})$ of $-4$ appear in the complementary function. The Helmholtz instability wave is excited and requires careful handling to obtain explicit results for the amplitude of the instability wave.

Unsteady turbulent flame evolution in non-premixed combustion has been computationally investigated using large eddy simulations. A simple coaxial combustion chamber, subjected to highly unsteady, turbulent recirculating flow is considered, following the experimental study of Owen et al. (Proc. Combust. Inst., vol. 16, 1976, pp. 105–117). Large-scale flame fluctuations, reported in the above experimental study, such as pulsating flames in swirling and non-swirling conditions, were identified here in our computation. New criteria for flame three-dimensional inhomogeneity are suggested and implemented in the present study, providing the ability to quantify the flame unsteadiness. Using this technique, it is shown that local, large quenched regions develop in the flame’s mixing area and rotate continuously, even when swirl is not imposed on the inlet. However, this rotation appears to be disordered, abruptly changing its direction. On the other hand, our study shows that when swirl is imposed on the inlet, a larger quenched region is identified, rotating in steady ordered rotation in the direction of the imposed swirl. In addition, large-scale radial flame fluctuations are increased downstream with the increase of swirl number. Consequently, significant correlations between radial and circumferential flame fluctuation frequencies were retrieved. Proper orthogonal decomposition analysis reveals coherent flame structures of five dominant modes that contain most of the energy in the fluctuating flame. A simplified analytical stability model is derived and implemented here to assess the hydrodynamic contribution to the flame instability; it is shown that radial fluctuations are excited by circumferential perturbations in the mixing region, providing new insight into the mechanism responsible for the onset of radial fluctuations. The computed radial flame fluctuation spectrum is predicted well using the linear stability analysis. Thus, our findings may therefore be applicable to a large class of non-premixed turbulent combustion problems.

Numerical simulations of three-dimensional rapidly rotating Rayleigh–Bénard convection are performed by employing an asymptotic quasi-geostrophic model that incorporates the effects of no-slip boundaries through (i) parametrized Ekman pumping boundary conditions and (ii) a thermal wind boundary layer that regularizes the enhanced thermal fluctuations induced by pumping. The fidelity of the model, obtained by an asymptotic reduction of the Navier–Stokes equations that implicitly enforces a pointwise geostrophic balance, is explored for the first time by comparisons of simulations against the findings of direct numerical simulations (DNS) and laboratory experiments. Results from these methods have established Ekman pumping as the mechanism responsible for significantly enhancing the vertical heat transport. This asymptotic model demonstrates excellent agreement over a range of thermal forcing for Prandtl number $Pr\approx 1$ when compared with results from experiments and DNS at maximal values of their attainable rotation rates, as measured by the Ekman number ($E\approx 10^{-7}$); good qualitative agreement is achieved for $Pr>1$. Similar to studies with stress-free boundaries, four spatially distinct flow morphologies exists. Despite the presence of frictional drag at the upper and/or lower boundaries, a strong non-local inverse cascade of barotropic (i.e. depth-independent) kinetic energy persists in the final regime of geostrophic turbulence and is dominant at large scales. For mixed no-slip/stress-free and no-slip/no-slip boundaries, Ekman friction is found to attenuate the efficiency of the upscale energy transport and, unlike the case of stress-free boundaries, rapidly saturates the barotropic kinetic energy. For no-slip/no-slip boundaries, Ekman friction is strong enough to prevent the development of a coherent dipole vortex condensate. Instead, vortex pairs are found to be intermittent, varying in both time and strength. For all combinations of boundary conditions, a Nastrom–Gage type of spectrum of kinetic energy is found, where the power-law exponent changes from ${\approx}-3$ to ${\approx}-5/3$, i.e. from steep to shallow, as the spectral wavenumber increases.

Horizontal convection, with a focus on the essential physics it can disclose for the oceanic overturning circulation, is studied in an annular gap where differential heating is applied along the bottom. Based on the three-dimensional distribution of both velocity and temperature fields, we propose that in addition to differential buoyancy forcing and diapycnal mixing, potential vorticity (PV) can also be a limiting factor for the overturning circulation by imposing a constraint on the meridional flow. With rotation, flow has to be along PV contours excepting that effects of diffusion or friction exist to break the PV constraint. In the axisymmetric rotating experiment, the flow along isotherms is to zero-order azimuthal, due to the fact that PV varies only along the radial direction. Consequently, the radial flow appears at the next order with much smaller amplitude, leading to a much weaker overturning circulation compared with that of the non-rotating experiment. In the presence of a full radial barrier, the annular gap can be viewed as a four-sided domain with its northern boundary along the inner cylinder, and the meridional flow shows remarkable inhomogeneity in space. In the absence of any background PV gradient, the flow field equilibrates itself to a state where the PV contours above the bottom boundary layer are largely radial, particularly in a sub-region near the western boundary, allowing a relatively strong radial geostrophic flow and greatly strengthening the overturning circulation from the case without the barrier.

When the surface of a liquid has a non-uniform distribution of a surfactant that lowers surface tension, the resulting variation in surface tension drives a flow that spreads the surfactant towards a uniform distribution. We study the spreading dynamics of an insoluble and non-diffusing surfactant on an initially motionless liquid. We derive solutions for the evolution over time of sinusoidal variations in surfactant concentration with a small initial amplitude relative to the average concentration. In this limit, the coupled flow and surfactant transport equations are linear. In contrast to exponential decay when the inertia of the flow is negligible, the solution for unsteady Stokes flow exhibits oscillations when inertia is sufficient to spread the surfactant beyond a uniform distribution. This oscillatory behaviour exhibits two properties that distinguish it from that of a simple harmonic oscillator: the amplitude changes sign at most three times, and the decay at late times follows a power law with an exponent of $-3/2$. As the surface oscillates, the structure of the subsurface flow alternates between one and two rows of counter-rotating vortices, starting with one row and ending with two during the late-time monotonic decay. We also examine numerically the evolution of the surfactant distribution when the system is nonlinear due to a large initial amplitude.

Direct numerical simulations (DNS) are used to investigate the unsteady flow over a model turbine blade tip at engine-scale Reynolds and Mach numbers. The DNS are performed with an in-house multiblock structured compressible Navier–Stokes solver. The particular case of a transonic tip flow is studied since previous work has suggested that compressibility has an important effect on the turbulent nature of the separation bubble at the inlet to the tip–casing gap and subsequent flow reattachment. The flow is simulated over an idealized tip geometry where the tip gap is represented by a constant-area channel with a sharp inlet corner to represent the pressure side edge of the turbine blade. The effects of free-stream disturbances, cross-flow and the pressure side boundary layer on the tip flow aerodynamics and heat transfer are studied. For ‘clean’ inflow cases we find that even at engine-scale Reynolds numbers the tip flow is intermittent in nature, i.e. neither laminar nor fully turbulent. The breakdown to turbulence occurs through the development of spanwise streaks with wavelengths of approximately 15 %–20 % of the gap height. Multidimensional linear stability analysis confirms the two-dimensional base state to be most unstable with respect to spanwise wavelengths of 25 % of the gap height. The linear stability analysis also shows that the addition of cross-flows with 25 % of the streamwise gap exit velocity increases the stability of the tip flow. This is confirmed by the DNS, which also show that the turbulence production is significantly reduced in the separation bubble. For the case when free-stream disturbances are added to the inlet flow, viscous dissipation and the rapid acceleration of the flow at the inlet to the tip–casing gap cause significant distortion of the vorticity field and reductions of turbulence intensity as the flow enters the tip gap. The DNS results also suggest that the assumption of the Reynolds analogy and a constant recovery factor are not accurate, in particular in regions where the skin friction approaches zero while significant temperature gradients remain, such as in the vicinity of flow reattachment.

Under steady flow conditions, the topological complexity inherent to all random three-dimensional (3D) porous media imparts complicated flow and transport dynamics. It has been established that this complexity generates persistent chaotic advection via a 3D fluid mechanical analogue of the baker’s map which rapidly accelerates scalar mixing in the presence of molecular diffusion. Hence, pore-scale fluid mixing is governed by the interplay between chaotic advection, molecular diffusion and the broad (power-law) distribution of fluid particle travel times which arise from the non-slip condition at pore walls. To understand and quantify mixing in 3D porous media, we consider these processes in a model 3D open porous network and develop a novel stretching continuous time random walk (CTRW), which provides analytic estimates of pore-scale mixing which compare well with direct numerical simulations. We find that the chaotic advection inherent to 3D porous media imparts scalar mixing which scales exponentially with the longitudinal advection, whereas the topological constraints associated with two-dimensional porous media limit the mixing to scale algebraically. These results decipher the role of wide transit time distributions and complex topologies on porous media mixing dynamics, and provide the building blocks for macroscopic models of dilution and mixing which resolve these mechanisms.

The asymptotic suction boundary layer (ASBL) is a parallel shear flow that becomes turbulent in a bypass transition in parameter regions where the laminar profile is stable. We here add a temperature gradient perpendicular to the plate and explore the interaction between convection and shear in determining the transition. We find that the laminar state becomes unstable in a subcritical bifurcation and that the critical Rayleigh number and wavenumber depend strongly on the Prandtl number. We also track several secondary bifurcations and identify states that are localized in two directions, showing different symmetries. In the subcritical regime, transient turbulent states which are connected to exact coherent states and follow the same transition scenario as found in linearly stable shear flows are identified and analysed. The study extends the bypass transition scenario from shear flows to thermal boundary layers and highlights the intricate interactions between thermal and shear forces.

This study deals with the motion and deformation of a compound drop system, subject to arbitrary but Stokesian far-field flow conditions, in the presence of bulk-insoluble surfactants. We derive solutions for fluid velocities and the resulting surfactant concentrations, assuming the capillary number and surface Péclet number to be small, as compared with unity. We first focus on a concentric drop configuration and apply Lamb’s general solution, assuming the far-field flow to be arbitrary in nature. As representative case studies, we consider two cases: (i) flow dynamics in linear flows and (ii) flow dynamics in a Poiseuille flow, although for the latter case, the concentric configuration does not remain valid in general. We further look into the effective viscosity of a dilute suspension of compound drops, subject to linear ambient flow, and compare our predictions with previously reported experiments. Subsequently, the eccentric drop configuration is addressed by using a bipolar coordinate system where the far-field flow is assumed to be axisymmetric but otherwise arbitrary in nature. As a specific example for eccentric drop dynamics, we focus on Poiseuille flow and study the drop migration velocities. Our analysis shows that the presence of surfactant generally opposes the imposed flows, thereby acting like an effective augmented viscosity. Our analysis reveals that maximizing the effects of surfactant makes the drops behave like solid particles suspended in a medium. However, in uniaxial extensional flow, the presence of surfactants on the inner drop, in conjunction with the drop radius ratio, leads to a host of interesting and non-monotonic behaviours for the interface deformation. For eccentric drops, the effect of eccentricity only becomes noticeable after it surpasses a certain critical value, and becomes most prominent when the two interfaces approach each other. We further depict that surfactant and eccentricity generally tend to suppress each other’s effects on the droplet migration velocities.

The two-point theory of homogeneous isotropic turbulence is extended to source terms appearing in the equations for higher-order structure functions. For this, transport equations for these source terms are derived. We focus on the trace of the resulting equations, which is of particular interest because it is invariant and therefore independent of the coordinate system. In the trace of the even-order source term equation, we discover the higher-order moments of the dissipation distribution, and the individual even-order source term equations contain the higher-order moments of the longitudinal, transverse and mixed dissipation distribution functions. This shows for the first time that dissipation fluctuations, on which most of the phenomenological intermittency models are based, are contained in the Navier–Stokes equations. Noticeably, we also find the volume-averaged dissipation $\unicode[STIX]{x1D700}_{r}$ used by Kolmogorov (J. Fluid Mech., vol. 13, 1962, pp. 82–85) in the resulting system of equations, because it is related to dissipation correlations.

We examine the partitioning of the energy transferred to the water column by deep-water wave breaking; in this case between the turbulent and mean flow. It is found that more than 95 % of the energy lost by the wave field is dissipated in the first four wave periods after the breaking event. The remaining energy is in the coherent vortex generated by breaking. A scaling argument shows that the ratio between the energy in this breaking generated mean current and the total energy lost from the wave field to the water column due to breaking scales as $(hk)^{1/2}$, where $hk$ is the local slope at breaking. This model is examined using direct numerical simulations of breaking waves solving the full two-phase air–water Navier–Stokes equations, as well as the limited available laboratory data, and good agreement is found for strong breaking waves.

We examine the geometry, kinematics, and dynamics of weakly nonlinear narrow-banded deep-water wave packets governed by the modified nonlinear Schrödinger equation (Dysthe, Proc. R. Soc. Lond. A., vol. 369, 1979, pp. 105–114; MNLSE). A new derivation of the spatial MNLSE, by a direct application of Whitham’s method, elucidates its variational structure. Using this formalism, we derive a set of conserved quantities and moment evolution equations. Next, by examining the MNLSE in the limit of vanishing linear dispersion, analytic solutions can be found. These solutions then serve as trial functions, which when substituted into the moment evolution equations form a closed set of equations, allowing for a qualitative and quantitative examination of the MNLSE without resorting to numerically solving the full equation. To examine the theory we consider initially symmetric, chirped and unchirped wave packets, chosen to induce wave focusing and steepening. By employing the ansatz for the trial function discussed above, we predict, a priori, the evolution of the packet. It is found that the speed of wave packets governed by the MNLSE depends on their amplitude, and in particular wave groups speed up as they focus. Next, we characterize the asymmetric growth of the wave envelope, and explain the steepening of the forward face of the initially symmetric wave packet. As the packet focuses, its variance decreases, as does the chirp of the signal. These theoretical results are then compared with the numerical predictions of the MNLSE, and agreement for small values of fetch is found. Finally, we discuss the results in the context of existing theoretical, numerical and laboratory studies.

A droplet obliquely impacting a bath surface of the same fluid can traverse along the interface while slowing at an exponential rate. The droplet rests on a thin film of air, deforms the bath surface creating a dimple and travels along the surface similarly to a wave pulse. Viscous coupling of the droplet and bath surfaces through the air film leads to viscous drag on the bath and perturbs the wave motion of the otherwise free surface. Even though the Reynolds numbers are greater than unity ($\mathit{Re}\,O(10{-}100)$), we show that the droplet’s deceleration is only due to viscous coupling through the air gap. The rate of deceleration is found to increase linearly with droplet diameter.

Direct numerical simulations based on the incompressible nonlinear Navier–Stokes equations of the flow over the surface of a rotating disk have been conducted. An impulsive disturbance was introduced and its development as it travelled radially outwards and ultimately transitioned to turbulence has been analysed. Of particular interest was whether the nonlinear stability is related to the linear stability properties. Specifically three disk-edge conditions were considered; (i) a sponge region forcing the flow back to laminar flow, (ii) a disk edge, where the disk was assumed to be infinitely thin and (iii) a physically realistic disk edge of finite thickness. This work expands on the linear simulations presented by Appelquist et al. (J. Fluid. Mech., vol. 765, 2015, pp. 612–631), where, for case (i), this configuration was shown to be globally linearly unstable when the sponge region effectively models the influence of the turbulence on the flow field. In contrast, case (ii) was mentioned there to be linearly globally stable, and here, where nonlinearity is included, it is shown that both cases (ii) and (iii) are nonlinearly globally unstable. The simulations show that the flow can be globally linearly stable if the linear wavepacket has a positive front velocity. However, in the same flow field, a nonlinear global instability can emerge, which is shown to depend on the outer turbulent region generating a linear inward-travelling mode that sustains a transition front within the domain. The results show that the front position does not approach the critical Reynolds number for the local absolute instability, $R=507$. Instead, the front approaches $R=583$ and both the temporal frequency and spatial growth rate correspond to a global mode originating at this position.

The invariants of the velocity gradient tensor, $R$ and $Q$, and their enstrophy and strain components are studied in the logarithmic layer of an incompressible turbulent channel flow. The velocities are filtered in the three spatial directions and the results are analysed at different scales. We show that the $R$–$Q$ plane does not capture the changes undergone by the flow as the filter width increases, and that the enstrophy/enstrophy-production and strain/strain-production planes represent better choices. We also show that the conditional mean trajectories may differ significantly from the instantaneous behaviour of the flow since they are the result of an averaging process where the mean is 3–5 times smaller than the corresponding standard deviation. The orbital periods in the $R$–$Q$ plane are shown to be independent of the intensity of the events, and of the same order of magnitude as those in the enstrophy/enstrophy-production and strain/strain-production planes. Our final goal is to test whether the dynamics of the flow is self-similar in the inertial range, and the answer turns out to be that it is not. The mean shear is found to be responsible for the absence of self-similarity and progressively controls the dynamics of the eddies observed as the filter width increases. However, a self-similar behaviour emerges when the calculations are repeated for the fluctuating velocity gradient tensor. Finally, the turbulent cascade in terms of vortex stretching is considered by computing the alignment of the vorticity at a given scale with the strain at a different one. These results generally support a non-negligible role of the phenomenological energy-cascade model formulated in terms of vortex stretching.

The steady incompressible viscous flow in the wide gap between spheres rotating rapidly about a common axis at slightly different rates (small Rossby number) has a long and celebrated history. The problem is relevant to the dynamics of geophysical and planetary core flows, for which, in the case of electrically conducting fluids, the possible operation of a dynamo is of considerable interest. A comprehensive asymptotic study, in the small Ekman number limit $E\ll 1$, was undertaken by Stewartson (J. Fluid Mech., vol. 26, 1966, pp. 131–144). The mainstream flow, exterior to the $E^{1/2}$ Ekman layers on the inner/outer boundaries and the shear layer on the inner sphere tangent cylinder $\mathscr{C}$, is geostrophic. Stewartson identified a complicated nested layer structure on $\mathscr{C}$, which comprises relatively thick quasigeostrophic $E^{2/7}$- (inside $\mathscr{C}$) and $E^{1/4}$- (outside $\mathscr{C}$) layers. They embed a thinner ageostrophic $E^{1/3}$ shear layer (on $\mathscr{C}$), which merges with the inner sphere Ekman layer to form the $E^{2/5}$-equatorial Ekman layer of axial length $E^{1/5}$. Under appropriate scaling, this $E^{2/5}$-layer problem may be formulated, correct to leading order, independent of $E$. Then the Ekman boundary layer and ageostrophic shear layer become features of the far-field (as identified by the large value of the scaled axial coordinate $z$) solution. We present a numerical solution of the previously unsolved equatorial Ekman layer problem using a non-local integral boundary condition at finite $z$ to account for the far-field behaviour. Adopting $z^{-1}$ as a small parameter we extend Stewartson’s similarity solution for the ageostrophic shear layer to higher orders. This far-field solution agrees well with that obtained from our numerical model.

Supercritical ship internal wave wakes with $Fr=U/c_{0}\sim 4{-}12$ (where $U$ is the ship speed and $c_{0}$ is the linear internal long-wave speed) are calculated by a strongly nonlinear two-layer model in three dimensions, accounting for the complex ship geometry, and compared with field measurements. The degree of nonlinearity, defined by the ship draught ($d_{0}$) to average depth of pycnocline ($h_{0}$) ratio, is explored in the range $d_{0}/h_{0}\sim 0.1{-}1.2$, comparing nonlinear and linear calculations. For $d_{0}/h_{0}\sim 1$, the wave amplitude far downstream is overpredicted by up to 50 % by the linear method. The nonlinear trough amplitudes decay algebraically in the lateral coordinate with decay exponents in the range 0.16–0.61. The nonlinear leading trough is systematically somewhat ahead of the classical asymptotic pattern, and its amplitude and forerunning slope are appreciable, while the linear counterparts are very small. The calculated and asymptotic patterns are close for large $Fr$. Field measurements in a Canadian fjord of the internal wave wake of a 221 m long cargo ship of dead weight 43 000 tonnes moving at $Fr=6.6$ document a series of waves of 1–2 m isopycnal displacements at an off-track distance of 700 m. The nonlinear computations of a similar ship predict a wave train of height 1.5 m at a similar position, matching the field observation, whereas linear calculations give a wave train of height 2.3 m. Two- and three-layer theoretical models predict speeds that match the observed speeds of the second and third wave troughs. The observed leading signature of the diverging wave wake is moving at supercritical speed. This may be explained by the position and speed of the nonlinear wave slope moving ahead of the leading trough. Nonlinear computations of the surface velocity and strain rate compare well with measurements in the Loch Linnhe experiment of Watson et al. (J. Geophys. Res., vol. 97 (C6), 1992, pp. 9689–9703). The calculated nonlinear wave resistance of the observed cargo ship is comparable to the frictional force for $Fr\sim 4{-}6$, exceeds the surface wave resistance and increases the total drag by 40 %. A linear force prediction is useless when $d_{0}/h_{0}\sim 1$. The results show that nonlinearity of the dead water wake depends on the relative ship volume as well as the relative ship draught, with the pycnocline depth as the relevant length scale.

We analyse the influence of the specific features of time-dependent variable-density Kelvin–Helmholtz (VDKH) roll-ups on the development of three-dimensional secondary instabilities. Due to inertial (high Froude number) baroclinic sources of spanwise vorticity at high Atwood number (up to 0.5 here), temporally evolving mixing layers exhibit a layered structure associated with a strain field radically different from their homogeneous counterpart. We use a direct-adjoint non-modal linear approach to determine the fastest growing perturbations over a single period of the time-evolving two-dimensional base flow during a given time interval $[t_{0},T]$. When perturbations are seeded at the initial time of the primary KH mode growth, i.e. $t_{0}=0$, it is found that additional mechanisms of energy growth are onset around a bifurcation time $t_{b}$, a little before the saturation of the primary two-dimensional instability. The evolution of optimal perturbations is thus observed to develop in two distinct stages. Whatever the Atwood number, the first period $[t_{0},t_{b}]$ is characterised by a unique route for optimal energy growth resulting from a combination of the Orr and lift-up transient mechanisms. In the second period $[t_{b},T]$, the growing influence of mass inhomogeneities raises the energy gain over the whole range of spanwise wavenumbers. As the Atwood number increases, the short spanwise wavelength perturbations tend to benefit more from the onset of variable-density effects than large wavelength ones. The extra energy gain due to increasing Atwood numbers relies on contributions from spanwise baroclinic sources. The resulting vorticity field is structured into two elongated dipoles localised along the braid on either side of the saddle point. In return they yield two longitudinal velocity streaks of opposite sign which account for most of the energy growth. This transition towards three-dimensional motions is in marked contrast with the classic streamwise rib vortices, so far accepted as the paradigm for the transition of free shear flows, either homogeneous or not. It is argued that the emergence of these longitudinal velocity streaks is generic of the transition in variable-density shear flows. Among them, the light round jet is known to display striking side ejections as a result of the loss of axisymmetry. The present analysis helps to renew the question of the underlying flow structure behind side jets, otherwise based on radial induction between pairs of counter-rotating longitudinal vortices (Monkewitz & Pfizenmaier, Phys. Fluids A, vol. 3 (5), 1991, pp. 1356–1361). Instead, it is more likely that side ejections would result from the convergence of the longitudinal velocity streaks near the braid saddle point. When the injection time is delayed so as to suppress the initial stage of energy growth, a new class of perturbations arises at low wavenumber with energy gains far larger than those observed so far. They correspond to the two-dimensional Kelvin–Helmholtz secondary instability of the baroclinically enhanced vorticity braid discovered by Reinaud et al. (Phys. Fluids, vol. 12 (10), pp. 2489–2505), leading potentially to another route to turbulence through a two-dimensional fractal cascade.

We determine the asymptotic conditions under which the Boussinesq approximation is valid for oscillatory convection in a rapidly rotating fluid. In the astrophysically relevant parameter regime of small Prandtl number, we show that the Boussinesq prediction for the onset of convection is valid only under much more restrictive conditions than those that are usually assumed. In the case of an ideal gas, we recover the Boussinesq results only if the ratio of the domain height to a typical scale height is much smaller than the Prandtl number. This requires an extremely shallow domain in the astrophysical parameter regime. Other commonly used ‘sound-proof’ approximations generally perform no better than the Boussinesq approximation. The exception is a particular implementation of the pseudo-incompressible approximation, which predicts the correct instability threshold beyond the range of validity of the Boussinesq approximation.

At room temperature, when a drop impacts against a smooth solid surface at a velocity above the so-called critical velocity for splashing, the drop loses its integrity and fragments into tiny droplets violently ejected radially outwards. Below this critical velocity, the drop simply spreads over the substrate. Splashing is also reported to occur for solid substrate temperatures above the Leidenfrost temperature, $T_{L}$, for which a vapour layer prevents the drop from touching the solid. In this case, the splashing morphology differs from the one reported at room temperature because, thanks to the presence of the gas layer, the shear stresses acting on the liquid can be neglected. Our purpose here is to predict, for wall temperatures above $T_{L}$, the critical Weber number for splashing as well as the maximum spreading radius. First, making use of boundary integral simulations, we calculate both the time evolution of the liquid velocity as well as the height of the sheet which is ejected tangentially to the substrate. These results are then used as boundary conditions for the one-dimensional mass and momentum equations describing the dynamics of the rim limiting the expanding liquid sheet. Our predictions for both the maximum spreading radius and for the critical Weber number for splashing are in good agreement with experimental observations.