Interface-resolved direct numerical simulations are performed to investigate bubble-induced transition from a laminar to elasto-inertial turbulent (EIT) state in a pressure-driven viscoelastic square channel flow. The Giesekus model is used to account for the viscoelasticity of the continuous phase, while the dispersed phase is Newtonian. Simulations are performed for both single- and two-phase flows for a wide range of Reynolds (${Re}$) and Weissenberg (${\textit{Wi}}$) numbers. In the absence of any discrete external perturbations, single-phase viscoelastic flow is transitioned to an EIT regime at a critical Weissenberg number ($Wi_{cr})$ that decreases with increasing ${Re}$. It is demonstrated that injection of bubbles into a laminar viscoelastic flow introduces streamline curvature that is sufficient to trigger an elastic instability leading to a transition to an EIT regime. The temporal turbulent kinetic energy spectrum shows a scaling of $-2$ for this multiphase EIT regime, and this scaling is found to be independent of size and number of bubbles injected into the flow. It is also observed that bubbles move towards the channel centreline and form a string-shaped alignment pattern in the core region at the lower values of ${Re}=10$ and ${\textit{Wi}}=1$. In this regime, there are disturbances in the core region in the vicinity of bubbles while flow remains essentially laminar. Unlike the solid particles, it is found that increasing shear-thinning effect breaks up the alignment of bubbles.

Turbulent convection under strong rotation can develop an inverse cascade of kinetic energy from smaller to larger scales. In the absence of an effective dissipation mechanism at the large scales, this leads to the pile up of kinetic energy at the largest available scale, yielding a system-wide large-scale vortex (LSV). Earlier works have shown that the transition into this state is abrupt and discontinuous. Here, we study the transition to the inverse cascade at Ekman number ${Ek}=10^{-4}$ and using stress-free boundary conditions, in the case where the inverse energy flux is dissipated before it reaches the system scale, suppressing the LSV formation. We demonstrate how this can be achieved in direct numerical simulations by using an adapted form of hypoviscosity on the horizontal manifold. We find that, in the absence of the LSV, the transition to the inverse cascade becomes continuous. This shows that it is the interaction between the LSV and the background turbulence that is responsible for the earlier observed discontinuity. We furthermore show that the inverse cascade in absence of the LSV has a more local signature compared with the case with LSV.

Particle motions under nonlinear gravity waves at the free surface of a two-dimensional incompressible and inviscid fluid are considered. The Euler equations are solved numerically using a high-order spectral method based on a Hamiltonian formulation of the water-wave problem. Extending this approach, a numerical procedure is devised to estimate the fluid velocity at any point in the fluid domain given surface data. The reconstructed velocity field is integrated to obtain particle trajectories for which an analysis is provided, focusing on two questions. The first question is the influence of a wave setup or setdown as is typical in coastal conditions. It is shown that such local changes in the mean water level can lead to qualitatively different pictures of the internal flow dynamics. These changes are also associated with rather strong background currents which dominate the particle transport and, in particular, can be an order of magnitude larger than the well-known Stokes drift. The second question is whether these particle dynamics can be described with a simplified wave model. The Korteweg–de Vries equation is found to provide a good approximation for small- to moderate-amplitude waves on shallow and intermediate water depth. Despite discrepancies in severe cases, it is able to reproduce characteristic features of particle paths for a wave setup or setdown.

In this study we focus on the collision rate and contact time of finite-sized droplets in homogeneous, isotropic turbulence. Additionally, we concentrate on sub-Hinze–Kolmogorov droplet sizes to prevent fragmentation events. After reviewing previous studies, we theoretically establish the equivalence of spherical and cylindrical formulations of the collision rate. We also obtained a closed-form expression for the collision rate of inertial droplets under the assumption of inviscid interactions. We then perform droplet-resolved simulations using the Basilisk solver with a multi-field volume-of-fluid method to prevent numerical droplet coalescence, ensuring a constant number of droplets of the same size within the domain, thereby allowing for the accumulation of collision statistics. The collision statistics are studied from numerical simulations, varying parameters such as droplet volume fraction, droplet size relative to the dissipative scale, density ratio and viscosity ratio. Our results show that the contact time is finite, leading to non-binary droplet interactions at high volume fractions. Additionally, the contact duration is well predicted by the eddy turnover time. We also find that the radial distribution at contact is significantly smaller than that predicted by the hard-sphere model due to droplet deformation in close proximity. Furthermore, we show that for neutrally buoyant droplets, the mean relative velocity is similar to the mean relative velocity of the continuous phase, except when the droplets are close. Finally, we demonstrate that the collision rate obeys the appropriate theoretical law, although a numerical prefactor weakly varies as a function of the dimensionless parameters, which differs from the constant prefactor from theory.

We develop an asymptotic theory of a compressible turbulent boundary layer on a flat plate, in which the mean velocity and temperature profiles can be obtained as exact asymptotic solutions of the boundary-layer equations, which are closed using functional relations of a general form connecting the turbulent shear stress and turbulent enthalpy flux to the mean velocity and enthalpy gradients. The outer region of the boundary layer is considered at moderate supersonic free-stream Mach numbers, when the relative temperature difference across the layer is of order one. A special change of variables allows us to construct the solution in the outer region in the form of asymptotic expansions at large values of the logarithm of the Reynolds number based on the boundary-layer thickness. As a result of asymptotic matching of the solutions for the outer region and logarithmic sublayer, the velocity and temperature defect laws are obtained, which allow us to describe the profiles of these quantities in the outer and logarithmic regions by universal curves known for the boundary layer of an incompressible fluid. Similarity rules for the Reynolds-tensor components and root-mean-square enthalpy fluctuation are given. The recovery and Reynolds-analogy factors are calculated. A friction law is established that is valid under arbitrary wall-heat-transfer conditions.

Hierarchical parcel swapping (HiPS) is a multiscale stochastic model of turbulent mixing based on a binary tree. Length scales decrease geometrically with increasing tree level, and corresponding time scales follow inertial range scaling. Turbulent eddies are represented by swapping subtrees. Lowest-level swaps change fluid parcel pairings, with new pairings instantly mixed. This formulation suitable for unity Schmidt number $Sc$ is extended to non-unity $Sc$. For high $Sc$, the tree is extended to the Batchelor level, assigning the same time scale (governing the rate of swap occurrences) to the added levels as the time scale at the base of the $Sc=3$ tree. For low $Sc$, a swap at the Obukhov–Corrsin level mixes all parcels within corresponding subtrees. Well-defined model analogues of turbulent diffusivity, and mean scalar-variance production and dissipation rates are identified. Simulations idealising stationary homogeneous turbulence with an imposed scalar gradient reproduce various statistical properties of viscous-range and inertial-range pair dispersion, and of the scalar power spectrum in the inertial-advective, inertial-diffusive and viscous-advective regimes. The viscous-range probability density functions of pair separation and scalar dissipation agree with applicable theory, including the stretched-exponential tail shape associated with viscous-range scalar intermittency. Previous observation of that tail shape for $Sc=1$, heretofore not modelled or explained, is reproduced. Comparisons to direct numerical simulation allow evaluation of empirical coefficients, facilitating quantitative applications. Parcel-pair mixing is a common mixing treatment, e.g. in subgrid closures for coarse-grained flow simulation, so HiPS can improve model physics simply by smarter (yet nearly cost-free) selection of pairs to be mixed.