The gas-particle flow with multiple dispersed solid phases is associated with a complicated multiphase flow dynamics. In this paper, a unified algorithm is proposed for the gas-particle multiphase flow. The gas-kinetic scheme (GKS) is used to simulate the gas phase and the multiscale unified gas-kinetic wave–particle (UGKWP) method is developed for the multiple dispersed solid particle phase. For each disperse solid particle phase, the decomposition of deterministic wave and statistic particle in UGKWP is based on the local cell's Knudsen number. The method for solid particle phase can become the Eulerian fluid approach at the small cell's Knudsen number and the Lagrangian particle approach at the large cell's Knudsen number. This becomes an optimized algorithm for simulating dispersed particle phases with a large variation of Knudsen numbers due to different physical properties of the individual particle phase, such as the particle diameter, material density, etc. The GKS-UGKWP method for gas-particle flow unifies the Eulerian–Eulerian and Eulerian–Lagrangian methods. The particle and wave decompositions for the solid particle phase and their coupled evolution in UGKWP come from the consideration to balance the physical accuracy and numerical efficiency. Two cases of a gas–solid fluidization system, i.e. one circulating fluidized bed and one turbulent fluidized bed, are simulated. The typical flow structures of the fluidized particles are captured, and the time-averaged variables of the flow field agree well with the experimental measurements. In addition, the shock particle–bed interaction is studied by the proposed method, which validates the algorithm for the polydisperse gas-particle system in the highly compressible case, where the dynamic evolution process of the particle cloud is investigated.

It is generally believed that the velocity and passive scalar fields share many similarities and differences in wall-bounded turbulence. In the present study, we conduct a series of direct numerical simulations of compressible channel flows with passive scalars and employ the two-dimensional spectral linear stochastic estimation and the correlation function as diagnostic tools to shed light on these aspects. Particular attention is paid to the relevant multiphysics couplings in the spectral domain, i.e. the velocity–temperature ($u-T$), scalar–temperature ($g-T$) and velocity–scalar ($u-g$) couplings. These couplings are found to be utterly different at a given wall-normal position in the logarithmic and outer regions. Specifically, in the logarithmic region, the $u-T$ and $u-g$ couplings are tight at the scales that correspond to the attached eddies and the very large-scale motions (VLSMs), whereas the $g-T$ coupling is robust in the whole spectral domain. In the outer region, $u-T$ and $u-g$ couplings are only active at the scales corresponding to the VLSMs, whereas the $g-T$ coupling is diminished but still strong at all scales. Further analysis indicates that although the temperature field in the vast majority of zones in a channel can be roughly treated as a passive scalar, its physical properties gradually deviate from those of a pure passive scalar as the wall-normal height increases due to the enhancement of the acoustic mode. Furthermore, the deep involvement of the pressure field in the self-sustaining process of energy-containing motions also drives the streamwise velocity fluctuation away from a passive scalar. The current work is an extension of our previous study (Cheng & Fu, J. Fluid Mech., vol. 964, 2023, A15), and further uncovers the details of the multiphysics couplings in compressible wall turbulence.

The viscoelasticity of a dilute bubble suspension is theoretically derived from the constitutive equation originally for a dilute emulsion proposed by Frankel & Acrivos (J. Fluid Mech., vol. 44, issue 1, 1970, pp. 65–78). Non-dimensionalization of the original tensor equation indicates that the viscoelasticity is systematized for a given void fraction by the capillary number $Ca$ and dynamic capillary number $Cd$, representing the bubble deformability and unsteadiness of bubble deformation. Comprehensive evaluation of the viscoelasticity according to the volume fraction, $Ca$ and $Cd$ reveals that whether the viscosity increases or decreases depends on whether $Ca$ or $Cd$ exceeds a common critical value. In addition, it is indicated that the bubble suspension has the most prominent viscoelasticity when the time scale of the shear deformation is the same as the relaxation time of the suspended bubble and when the bubbles keep a spherical shape, that is, $Ca \ll 1$ and $Cd = 1$. The applicability of this theory in flow prediction was examined in a Taylor–Couette system, and experimentally good agreement was confirmed.

The column collapse experiment is a simplified version of natural and industrial granular flows. In this set-up, a column built with grains collapses and spreads over a horizontal plane. Granular flows are often studied with a monodisperse distribution; however, this is not the case in natural granular flows where a variety of grain sizes, known as polydispersity, is a common feature. In this work, we study the effect of polydispersity, and of the inherent changes that polydispersity causes in the initial packing fraction, in dry and immersed columns. We show that dry columns are not significantly affected by polydispersity, reaching similar distances at similar times. In contrast, immersed columns are strongly affected by the polydispersity and packing fraction, and the collapse sequence is linked to changes of the basal pore fluid pressure $P$. At the collapse initiation, negative changes of $P$ beneath the column produce a temporary increase of the column strength. The negative change of $P$ lasts longer in polydisperse columns than in monodisperse columns, delaying the collapse sequence. Conversely, during the column spreading, positive changes of $P$ lead to a decrease of the shear strength. For polydisperse collapses, the excess of $P$ lasts longer, allowing the material to reach farther distances, compared with the collapses of monodisperse materials. Finally, we show that a mobility model that scales the final runout with the collapse kinetic energy remains true for different polydispersity levels in a three-dimensional configuration, capturing the scaling between the micro to macro controlling features.

The interaction between the flow in a channel with an obstruction on the bottom and an elastic sheet representing the ice covering the liquid is considered for the case of steady flow. The mathematical model based on the velocity potential theory and the theory of thin elastic shells fully accounts for the nonlinear boundary conditions at the elastic sheet/liquid interface and on the bottom of the channel. The integral hodograph method is employed to derive the complex velocity potential of the flow, which contains the velocity magnitude at the interface in explicit form. This allows one to formulate the coupled ice/liquid interaction problem and reduce it to a system of nonlinear equations in the unknown magnitude of the velocity at the interface. Case studies are carried out for a semi-circular obstruction on the bottom of the channel. Three flow regimes are studied: a subcritical regime, for which the interface deflection decays upstream and downstream; an ice supercritical and channel subcritical regime, for which two waves of different lengths may exist; and a channel supercritical regime, for which the elastic wave is found to extend downstream to infinity. All these regimes are in full agreement with the dispersion equation. The obtained results demonstrate a strongly nonlinear interaction between the elastic and the gravity wave near the first critical Froude number where their lengths approach each other. The interface shape, the bending moment and the pressure along the interface are presented for wide ranges of the Froude number and the obstruction height.

A Lagrangian approach to both hydrostatic non-dispersive in the short-wave range and non-hydrostatic dispersive rotating shallow-water magnetohydrodynamics is developed, and used to analyse weakly and fully nonlinear waves described by the model. Hyperbolic structure in the non-dispersive case is displayed and Riemann invariants are constructed. Characteristic equations are used to establish criteria of breaking and formation of shocks by magneto-gravity waves, and conditions of the appearance of contact discontinuities in Alfvén waves. As in the case of non-magnetic rotating shallow water, rotation cannot prevent breaking. The Lagrangian equations of the model are reduced to a single partial differential ‘master’ equation, which is used to analyse the propagation of weakly nonlinear waves of both families, with or without weak rotation, and with or without weak short-wave dispersion. Corresponding modulation equations are constructed and their main properties sketched. The same master equation is used to obtain fully nonlinear finite-amplitude wave solutions in particular cases of no short-wave dispersion or no rotation.

Direct numerical simulations (DNSs) of three-dimensional cylindrical release gravity currents in a linearly stratified ambient are presented. The simulations cover a range of stratification strengths $0< S\leq 0.8$ (where $S=(\rho _b^*-\rho _0^*)/(\rho _c^*-\rho _0^*), \rho _b^*, \rho _0^*$ and $\rho _c^*$ are the dimensional density at the bottom of the domain, top of the domain and the dense fluid, respectively) at two different Reynolds numbers. A comparison between the stratified and unstratified cases illustrates the influence of stratification strength on the dynamics of cylindrical gravity currents. Specifically, the front velocity in the slumping phase decreases with increasing stratification strength whereas the duration of the slumping phase increases with increments of $S$. The Froude number calculated in this phase shows a good agreement with models proposed by Ungarish & Huppert (J. Fluid Mech., vol. 458, 2002, pp. 283–301) and Ungarish (J. Fluid Mech., vol. 548, 2006, pp. 49–68), originally developed for planar gravity currents in a stratified ambient. In the inertial phase, the front velocity across cases with different stratification strengths adheres to a power-law scaling with an exponent of $-$1/2. Higher Reynolds numbers led to more frequent lobe splitting and merging, with lobe size diminishing as stratification strength increased. Strong interactions among inner vortex rings occurred during the slumping phase, leading to the early formation of hairpin vortices in weakly stratified cases, while strongly stratified cases exhibited delayed vortex formation and less turbulence.

This work presents a simple analytical model for the streamwise and radial variations of turbulent kinetic energy dissipation in an incompressible round turbulent jet. The key assumptions in the model are: similarity in the axial velocity profile with a Gaussian shape, axisymmetric flow and the dominance of radial derivatives of the mean velocity over axial direction derivatives (similar to boundary layer theory). Initially, a simplified eddy-viscosity relation for turbulent stresses is derived using the algebraic stress model by Gatski & Speziale (J. Fluid Mech., vol. 254, 1993, pp. 59–78). Subsequently, with this eddy-viscosity relation, the relation for variations of turbulent kinetic energy dissipation is formulated using the conservation of turbulent kinetic energy. To extract the necessary constants of the model, experimental velocity statistics for round jets are obtained through particle image velocimetry measurements. The experimental results of the mean entrainment coefficient for turbulent jets are also analysed. When comparing the radial variation of turbulent kinetic energy dissipation from the model with experimental results at Reynolds number $1.4\times 10^5$ and numerical results at Reynolds number $1200$ from the available literature, we observe a maximum error of $10\,\%$ and $15\,\%$, respectively. Finally, using the validated model, we analyse the impact of mean velocity evolution parameters on the behaviour of turbulent kinetic energy dissipation and discuss its potential significance in future studies.

The feedback between particle-size segregation and rheology in bidisperse granular flows is studied using the Stokes’ problem configuration. A method of lines scheme is implemented to solve the coupled momentum and segregation equations for a normally graded particle size distributed bulk at constant solids volume fraction. The velocity profiles develop quickly into a transient state, decoupled from segregation yet determined by the particle size. From this transient state, the velocity profile changes due to the particles’ relative movement, which redistributes the frictional response, hence its rheology. Additionally, the particles’ relative friction is modified via a frictional coefficient ratio, by analogy with the particles’ size ratio. While positive values of this coefficient exacerbate the nonlinearity of the velocity profiles induced by size differences, negative values dampen this behaviour. The numerical solutions reproduce well the analytical solutions for the velocity profile, which can be obtained from the steady-state conditions of the momentum and segregation equations for the transient and steady states, respectively. Segregation–momentum balances and four characteristic time scales can be established to propose two non-dimensional quantities, including specific Schmidt and Péclet numbers that describe broadly the segregation–rheology feedback. The proposed scheme, theoretical solutions and non-dimensional numbers offer a combined approach to understand segregation and flow dynamics within a granular bulk, extensible across many flow configurations.

An improved resolvent analysis is proposed in the regime of compressible turbulent boundary layers. To better model nonlinear processes in the input, the resolvent framework is augmented by adding eddy viscosity. To this end, we propose two eddy-viscosity models: a modified Cess eddy-viscosity model coupling the compressibility transformation and outer-layer correction, and a new eddy-viscosity model based on an empirical relationship and mixing-length theory. Both are incorporated into the resolvent operator to examine the performance of the eddy-viscosity-improved resolvent-based reduced-order modelling. Results of the augmented resolvent analysis are compared qualitatively and quantitatively with the first leading mode of spectral proper orthogonal decomposition, by checking the profiles and cross-spectral densities of velocities, density and temperature in two hypersonic turbulent boundary layers under different wall conditions. Higher accuracy of the turbulence prediction is achieved by adding the proposed eddy-viscosity models, particularly for the energetic cycle in the outer-layer region where strong nonlinear energy transfer exists.

The unresolved scalar variance in large-eddy simulations of turbulent flows is a fundamental physical and modelling parameter. Despite its importance, relatively few algebraic models have been developed for this important variable with the most prominent models to date being the classic scale-similarity and gradient models. In this work a new generalized modelling framework based on reconstruction has been developed, which in contrast to classic modelling approaches allows the construction of base static variance models of arbitrary accuracy. It is demonstrated that higher-order reconstructions naturally lead to base static variance models of increased accuracy, and that the classic scale-similarity and gradient models are subsets of more general and higher-order models. The classic scale-similarity assumption for developing dynamic models is also revisited, and it is demonstrated that this can essentially be reinterpreted as a two-level reconstruction approach. Based on this result, a new general methodology is proposed that allows the construction of dynamic models for any given base static model, and a corresponding general reconstruction operator, algebraic or iterative. Consequently, improved static and dynamic models for the scalar variance are developed. The newly developed models are then thoroughly tested a priori using two high-fidelity direct numerical simulation databases corresponding to two substantially different flame and flow configurations, and are shown to outperform classic algebraic models for the variance.