Direct numerical simulations (DNS) of turbulent fronts spreading into an irrotational flow region are used to analyse the turbulent entrainment mechanism for viscoelastic fluids. The simulations use the FENE-P fluid model and are initiated from DNS of isotropic turbulence with Weissenberg and turbulence Reynolds numbers varying in the ranges $1.30 \le Wi \le 3.46$ and $206 \le Re_{\lambda }^{0} \le 404$, respectively. The enstrophy dynamics near the turbulent/non-turbulent interface (TNTI) layer, that separates regions of turbulent and irrotational flow, includes a new mechanism – the viscoelastic production – caused by the interaction between the vorticity field and the polymer stresses. This term can be a sink or a source of enstrophy in the turbulent core region of the flow, depending on the Weissenberg number, and contributes to the initial growth of the enstrophy in the viscous superlayer, together with the viscous diffusion, which is the only mechanism present for Newtonian fluids. For low and moderate Weissenberg numbers the scaling of the TNTI layer is similar to the scaling of TNTI layers for Newtonian fluids, but this is no longer the case at high Weissenberg numbers where the enstrophy tends to be concentrated into thin vortex sheets instead of vortex tubes. Finally, it is shown that the substantial decrease of the entrainment rates observed in turbulent flows of viscoelastic fluids, compared with Newtonian fluids, is caused by a reduction of the surface area and fractal dimension of the irrotational boundary, originated by the depletion of ‘active’ scales of motion in the fluid solvent caused by the viscoelasticity.

Clustering of inertial particles is important for many types of astrophysical and geophysical turbulence, but it has been studied predominately for incompressible flows. Here, we study compressible flows and compare clustering in both compressively (irrotationally) and vortically (solenoidally) forced turbulence. Vortically and compressively forced flows are driven stochastically either by solenoidal waves or by circular expansion waves, respectively. For compressively forced flows, the power spectrum of the density of inertial particles is a useful tool for displaying particle clustering relative to the fluid density enhancement. Power spectra are shown to be particularly sensitive for studying large-scale particle clustering, while conventional tools such as radial distribution functions are more suitable for studying small-scale clustering. Our primary finding is that particle clustering through shock interaction is particularly prominent in turbulence driven by spherical expansion waves. It manifests itself through a double-peaked distribution of spectral power as a function of Stokes number. The two peaks are associated with two distinct clustering mechanisms; shock interaction for smaller Stokes numbers and the centrifugal sling effect for larger values. The clustering of inertial particles is associated with the formation of caustics. Such caustics can only be captured in the Lagrangian description, which allows us to assess the relative importance of caustics in vortically and compressively forced turbulence. We show that the statistical noise resulting from the limited number of particles in the Lagrangian description can be removed from the particle power spectra, allowing us a more detailed comparison of the residual spectra. We focus on the Epstein drag law relevant for rarefied gases, but show that our findings apply also to the usual Stokes drag.

This paper studies numerically the convection of water vapour in snowpacks using an Eulerian–Eulerian two-phase approach. The convective water vapour transport in snow and its effects on snow density are often invoked to explain observed density profiles, e.g. of thin Arctic snow covers, but this process has never been numerically simulated and analysed in a systematic manner. Here, the impact of convection on the thermal and phase change regimes as a function of different snowpack depths, thermal boundary conditions and Rayleigh numbers is analysed. We find considerable impact of natural convection on the snow density distribution with a layer of significantly lower density at the bottom of the snowpack and a layer of higher density located higher in the snowpack or at the surface. We find that emergent heterogeneity in the snow porosity results in a feedback effect on the spatial organization of convection cells causing their horizontal displacement. Regions where the snowpack is most impacted by phase changes are found to be horizontally extended and vertically thin, ‘pancake’-like layers at the top and bottom of the snowpack. We further demonstrate that among the parameters important for natural convection, the snowpack depth has the strongest influence on the heat and mass transfer. Despite our simplifying assumptions, our study represents a significant improvement over the state of the art and a first step to accurately simulate snowpack dynamics in diverse regions of the cryosphere.

Vortex-induced vibrations (VIV) of a cylinder in a Newtonian fluid is a model problem in fluid–structure interactions and has been studied extensively. In this work, we study the influence of shear-thinning and shear-thickening fluids on the VIV response of a one-degree-of-freedom flexibly-mounted cylinder. We consider a system with a mass ratio of $m^*=2$ and zero structural damping in shear-thinning and shear-thickening power-law fluids at $Re_0 = 15$ and $Re_0 = 200$, respectively, defined based on the zero-shear-rate viscosity of the fluids. We investigate how the VIV amplitude and frequency, flow forces, and the vorticity contours change as the reduced velocity, $U^*$, and fluid's time constant, $\lambda$, change. When the results are compared based on $Re_0$, shear-thinning fluids enhance the oscillations while shear-thickening fluids suppress them. If, however, we define a characteristic Reynolds number, $Re_{char}$, based on a viscosity evaluated at the characteristic shear rate, $\dot {\gamma } = U/D$, then at a constant $Re_{char}$, the amplitude of response stays very similar for the shear-thinning, shear-thickening and Newtonian fluids. Despite this similarity, the observed far wake is different: shear thinning amplifies the generation of vorticity and reduces the extent of the wake, whereas shear thickening limits the generation of vorticity and extends the wake. Our findings show that the local apparent viscosity observed close to the cylinder placed in shear-thinning or shear-thickening fluids governs the VIV response of the cylinder.

Following the findings by Wangsawijaya et al. (J. Fluid Mech., vol. 894, 2020, A7), we re-examine the turbulent boundary layers developing over surfaces with spanwise heterogeneous roughness of various roughness half-wavelengths $0.32 \leq S/\bar {\delta } \leq 3.63$, where $S$ is the width of the roughness strips and $\bar {\delta }$ is the spanwise-averaged boundary-layer thickness. The heterogeneous cases induce counter-rotating secondary flows, and these are compared with the large-scale turbulent structures that occur naturally over the smooth wall. Both appear as meandering elongated high- and low-momentum streaks in the instantaneous flow field. Results based on the triple decomposed velocity fluctuations suggest that the secondary flows are spanwise-locked turbulent structures, with $S/\bar {\delta }$ governing the strength of the turbulent structures and the efficacy of the surface in locking the structures in place (most effective when $S/\bar {\delta } \approx 1$). In terms of unsteadiness, we find additional evidence from conditional averages of the fluctuating velocity fields showing that the secondary flows exhibit maximum unsteadiness (or meandering) when $S/\bar {\delta } \approx 1$. The conditional averages of both spanwise heterogeneous and smooth-wall cases result in structures that are reminiscent of those proposed for the streak-vortex instability model for the inner cycle of wall-bounded turbulence. However, in this case these structures are larger and do not necessarily share the same formation mechanism with the inner cycle. Secondary flows and large-scale structures coexist in the limits where either $S/\bar {\delta } \gg 1$ or $S/\bar {\delta } \ll 1$, where the secondary flows scale on $\delta$ or $S$, respectively. When $S/\bar {\delta } \gg 1$, the secondary flows are locked about the roughness transition, while relatively unaltered large-scale structures occur further from the transition. In the case where $S/\bar {\delta } \ll 1$, $S$-scaled secondary flows are confined close to the surface, coexisting with unaltered larger-scale turbulent structures that penetrate much deeper into the layer.

The dynamics of wind-generated water waves in the principal stage of the Phillips theory (Phillips, J. Fluid Mech., vol. 2, 1957, pp. 417–445) is investigated by a combined numerical and analytical approach. We perform a number of high-resolution direct numerical simulation (DNS) of turbulent wind over initially calm water to capture the multistage generation of water waves. Detailed analyses are conducted to evaluate the Phillips theory in both physical space and wavenumber space. Numerical evidence is obtained for the existence of a principal stage when the surface elevation variance grows linearly with time. We further propose a random sweeping turbulence pressure–wave interaction model by introducing the random sweeping hypothesis of air pressure fluctuations to the Phillips theory, and obtain an asymptotic solution of the mean square of surface wave elevations over time. This asymptotic analysis captures the temporal oscillations of surface elevation variance in the principal stage, which is also confirmed by our DNS results. The wavenumber spectrum of surface wave elevations is analysed using a time-dependent norm to elucidate the role of the resonance mechanism on wave generation. In physical space, we use the random sweeping turbulence pressure–wave interaction model to obtain a quantitative prediction of the growth rate of surface elevation variance in the principal stage, which is found to agree with the DNS results better than the original Phillips model.

Doubly diffusive convection is considered in a vertical slot where horizontal temperature and solutal variations provide competing effects to the fluid density while allowing the existence of a conduction state. In this configuration, the linear stability of the conductive state is known, but the convection patterns arising from the primary instability have only been studied for specific parameter values. We have extended this by determining the nature of the primary bifurcation for all values of the Lewis and Prandtl numbers using a weakly nonlinear analysis. The resulting convection branches are extended using numerical continuation and we find large-amplitude steady convection states can coexist with the stable conduction state for sub- and supercritical primary bifurcations. The stability of the convection states is investigated and attracting travelling waves and periodic orbits are identified using time stepping when these steady states are unstable.

The temporal stability of a two-layer plane Couette flow with an interfacial heat source/sink is studied, in the presence of a viscosity stratification, predicting unstable streamwise and spanwise perturbations. Based on the wavenumber at onset, the perturbations are further classified as long-wave and finite-wave, where the long-wave mode is investigated using asymptotic and numerical analyses. The finite-wave instability is expected to be relevant for highly viscous fluids. Generally, a heat source (sink) at the interface is found to have a stabilising (destabilising) effect on the flow. However, if the rate of heat release exceeds a certain threshold, a new instability emerges, referred to here as an ‘explosive’ instability, characterised by an unbounded growth rate of the perturbations. The interaction between the viscosity stratification, thermocapillarity and inertia leads to the emergence of another new mode of streamwise instability, termed the ‘interaction’ instability. The spanwise perturbations possess higher growth rates than the streamwise perturbations for the heat sink along the interface when inertia has a stabilising influence, and vice versa. The heat sink leads to the emergence of thermocapillary instability by generating a negative temperature gradient across the fluids. In contrast, a heat source increases the temperature of the interface, resulting in a positive temperature gradient across the fluids, leading to the suppression of the instabilities by the thermocapillary effect. If the heat source provides energy at a faster rate than the energy lost due to Marangoni convection and heat diffusion, then it leads to an accumulation of energy, which presumably leads to the explosive instability.

A large eddy simulation wall model is developed based on a formal interpretation of quasi-equilibrium that governs the momentum balance integrated in the wall-normal direction. The model substitutes the law-of-the-wall velocity profile for a smooth surface into the wall-normal integrated momentum balance, leading to a Lagrangian relaxation towards equilibrium (LaRTE) transport equation for the friction–velocity vector ${\boldsymbol u}_\tau (x,z,t)$. This partial differential equation includes a relaxation time scale governing the rate at which the wall stress can respond to imposed fluctuations due to the inertia of the fluid layer from the wall to the wall-model height. A priori tests based on channel flow direct numerical simulation (DNS) data show that the identified relaxation time scale ensures self-consistency with assumed quasi-equilibrium conditions. The new approach enables us to formally distinguish quasi-equilibrium from additional, non-equilibrium contributions to the wall stress. A particular model for non-equilibrium contributions is derived, motivated by laminar Stokes layer dynamics in the viscous sublayer when applying fast-varying pressure gradients. The new wall modelling approach is first tested in standard equilibrium channel flow in order to document various properties of the approach. The model is then applied in large eddy simulation of channel flow with a suddenly applied spanwise pressure gradient (SSPG). The resulting mean wall-stress evolution is compared with DNS with good agreement. At the onset of the SSPG, the laminar Stokes layer develops rapidly while the LaRTE portion of the stress has a delayed response due to its inherent relaxation dynamics. Results also highlight open challenges such as modelling the response of near-wall turbulence occurring above the viscous sublayer and at time scales faster than quasi-equilibrium conditions.

Enhanced fluctuations, steep gradients, and intensified heat transfer are characteristics of wall-bounded turbulence at transcritical conditions. Although such conditions are prevalent in numerous technical applications, the structure of the thermal boundary layer under realistic density gradients and heating conditions remains poorly understood. Specifically, statistical descriptions of the temperature field in such flows are provided inconsistently using existing models. To address this issue, direct numerical simulations are performed by considering fully developed transcritical turbulent channel flow at pressure and temperature conditions that cause density changes of a factor of up to $O(20)$ between the hot and cold walls. As a consequence of the proximity of the Widom line to the hot wall, significant asymmetries are observed when comparing regions near the cold wall and near the hot wall. Previous transformations that attempt to collapse the near-wall mean temperature profiles among different cases to a single curve are examined. By addressing model deficiencies of these transformations, a formulation for an improved mean temperature transformation is proposed, with appropriate considerations for real fluid effects that involve strong variations in thermodynamic quantities. Our proposed transformation is shown to perform well in collapsing the slope of the logarithmic region to a single universal value with reduced uncertainty. Coupled with a predictive framework to estimate the non-universal shift parameter of the logarithmic region using a priori information, our transformation provides an analytic profile to model the near-wall mean temperature. These results thus provide a framework to guide the development of models for wall-bounded transcritical turbulence.

Turbulent secondary flows are defined as Prandtl's secondary flow of the first or second kind, the former produced by stretching and/or tilting of vorticity, the latter produced via spatial heterogeneity of Reynolds stresses. Both mechanisms are instantaneously active within inertia-dominated wall turbulence; Reynolds stress spatial heterogeneity is required for Reynolds-averaged secondary flows. Spanwise-variable surface roughness can induce turbulent stress spatial heterogeneity in the spanwise–wall-normal plane and provide sustenance for streamwise-aligned mean secondary flows. Herein, we demonstrate that turbulent secondary flows can also be sustained by spanwise variability in the surface heat flux in unstably stratified turbulent channels, defined hereafter as Prandtl's secondary flow of the third kind. Support for this mechanism is established with scaling arguments, while large-eddy simulation is used to model inertia-dominated channel turbulence responding to a lower boundary with uniform aerodynamic/hydrodynamic roughness but spanwise-variable surface heat flux. Transport equations for streamwise vorticity and turbulent kinetic energy, $k$, outline the conditions needed for third-kind production: shear and buoyancy production over the elevated heat flux regions necessitates lateral entrainment of low-$k$ fluid, inducing mean counter-rotating secondary cells aligned such that upwelling and downwelling occur over the high and low heat flux regions, respectively. Buoyancy-driven production of $k$ alters aggregate flow response and thus is a distinctly different mechanism responsible for sustenance of secondary flows.

In this work we study experimentally and numerically the stability of hydrodynamic focusing to produce microemulsions. The jetting regime was produced experimentally for the lowest outer viscosity, while microdripping was obtained for the highest outer viscosity. The liquid ejection for small capillary numbers stabilizes as the distance between the feeding capillary and the focusing orifice decreases. The comparison between confined selective withdrawal and stretched flow focusing for the largest outer viscosity shows that confined selective withdrawal significantly enhances the stability of the microdripping mode, which entails a significant reduction of the minimum value of the droplet diameter. When surfactants are dissolved in the inner liquid at sufficiently large concentrations, we observe a significant decrease in both the minimum flow rate leading to tip streaming and the droplet diameter. The surfactant monolayer stabilizes the meniscus and promotes the transition from microdripping to jetting. The global stability analysis shows that jetting via tip streaming becomes unstable due to the growth of an oscillatory perturbation. The unstable perturbation affects the tapering meniscus for the lowest outer viscosity, which explains why the meniscus oscillates in the experiments. The critical flow rate ratio is accurately predicted by the global stability analysis in most of the experiments with the lowest outer viscosity. For the highest outer viscosity, the global stability analysis correctly predicts the microdripping mode and allows one to calculate the droplet emission frequency in that mode.