The asymmetrical Mach reflection configuration is studied analytically in this paper, using an asymmetrical model extended from a recent symmetrical model and accounting for the new features related to asymmetry of the two wedges. It is found that the two sliplines do not turn parallel to the incoming flow at the same horizontal location and the sonic throat locates at the position where the difference of slopes of the two sliplines vanishes. This allows us to define a new sonic throat compatibility condition essential to determine the size of the Mach stem. The present model gives the height of the Mach stem, declined angle of the Mach stem from vertical axis, sonic throat location and shape of all shock waves and sliplines. The accuracy of the model is checked by computational fluid dynamics (CFD) simulation. It is found that the Mach stem height is strongly dependent on asymmetry of the wedge angles and almost linearly dependent on the asymmetry of the wedge lower surface lengths. The Mach stem height is shown to be insensitive to the asymmetry of the horizontal positions of the two wedges. The mechanisms for these observations are explained. For instance, it is demonstrated that the Mach reflection configuration remains closely similar when there is horizontal shift of either wedge.

The central theme of this work is that a stable levitation of a denser non-magnetizable liquid droplet, against gravity, inside a relatively lighter ferrofluid – a system barely considered in ferrohydrodynamics – is possible, and exhibits unique interfacial features; the stability of the levitation trajectory, however, is subject to an appropriate magnetic field modulation. We explore the shapes and the temporal dynamics of a plane non-magnetizable droplet levitating inside a ferrofluid against gravity due to a spatially complex, but systematically generated, magnetic field in two dimensions. The coupled set of Maxwell’s magnetostatic equations and the flow dynamic equations is integrated computationally, utilizing a conservative finite-volume-based second-order pressure projection algorithm combined with the front-tracking algorithm for the advection of the interface of the droplet. The dynamics of the droplet is studied under both the constant ferrofluid magnetic permeability assumption as well as for more realistic field-dependent permeability described by Langevin’s nonlinear magnetization model. Due to the non-homogeneous nature of the magnetic field, unique shapes of the droplet during its levitation, and at its steady state, are realized. The complete spatio-temporal response of the droplet is a function of the Laplace number $La$, the magnetic Laplace number $La_{m}$ and the Galilei number $Ga$; through detailed simulations we separate out the individual roles played by these non-dimensional parameters. The effect of the viscosity ratio, the stability of the levitation path and the possibility of existence of multiple stable equilibrium states is investigated. We find, for certain conditions on the viscosity ratio, that there can be developments of cusps and singularities at the droplet surface; we also observe this phenomenon experimentally and compare with the simulations. Our simulations closely replicate the singular projection on the surface of the levitating droplet. Finally, we present a dynamical model for the vertical trajectory of the droplet. This model reveals a condition for the onset of levitation and the relation for the equilibrium levitation height. The linearization of the model around the steady state captures that the nature of the equilibrium point goes under a transition from being a spiral to a node depending upon the control parameters, which essentially means that the temporal route to the equilibrium can be either monotonic or undulating. The analytical model for the droplet trajectory is in close agreement with the detailed simulations.

The motion of a free-moving plate atop turbulent thermal convection exhibits diverse dynamics that displays characteristics of both deterministic and chaotic motions. Early experiments performed by Zhong & Zhang (Phys. Rev. E, vol. 75 (5), 2007, 055301) found an oscillatory and a trapped state existing for a plate floating on convective fluid in a rectangular tank. They proposed a piecewise smooth physical model (ZZ model) that successfully captures this transition of states. However, their model was deterministic and therefore could not describe the stochastic behaviours. In this study, we combine the ZZ model with a novel approach that models the stochastic aspects through a variational inequality structure. With the powerful mathematical tools for stochastic variational inequalities, the properties of the Markov process and corresponding Kolmogorov equations could be studied both numerically and analytically. Moreover, this framework also allows one to compute the transition probabilities. Our present work captures the stochastic aspects of the two aforementioned boundary–fluid coupling states, predicts the stochastic behaviours and shows excellent qualitative and quantitative agreements with the experimental data.

Diffusive convection in an infinite two-dimensional fluid with linear vertical gradients of temperature and salinity is studied numerically and analytically. When the density gradient ratio exceeds a critical value above which diffusive convection grows according to the linear stability analysis, spontaneous layer formation is found to occur. At the first stage nearly steady oscillating motions, the horizontal scale of which is of the order of the buoyancy boundary layer scale δ, arise. After several tens of the oscillation cycle, a transition to the second stage occurs in which overturning convective motions develop and well-mixed regions are formed. The convective motions resemble Rayleigh–Bénard convection at a high Rayleigh number. The well-mixed regions are gradually organized into horizontal layers, a typical thickness of which is of the order of δ. A detailed analysis of the nonlinear process during the layer formation reveals that four modes are responsible for the layer formation: The first mode is the linear fastest-growing mode with wavenumber vector (k0, 0). The second mode with (k0, m0) is weakly growing. The third mode with (0, m0) is dissipating, and the fourth mode is its higher harmonic having (0, 2m0). It is shown that a truncated spectral model with the four modes well reproduces the results of the full numerical simulation.

A weakly nonlinear stability analysis of an axisymmetric Newtonian liquid jet is presented. The calculation is based on a small-amplitude perturbation method and performed to second order in the perturbation parameter. The obtained solution includes terms derived from a polynomial approximation of a viscous contribution containing products of Bessel functions with different arguments. The use of such an approximation is not needed in the inviscid case and the planar case, since the equations of those problems can be solved in an exact form. The developed model depends on three dimensionless parameters: the initial perturbation amplitude, the perturbation wavenumber and the liquid Ohnesorge number, the latter being the dimensionless liquid viscosity. The influence of the approximate terms was shown to be relatively small for a large range of Ohnesorge numbers so that they can be ignored. This simplification provides a jet model as simple to use as the previous ones, but taking into account the liquid viscosity and the cylindrical geometry. The jet model is used to reveal the effect of both the wavenumber and the Ohnesorge number on the formation of satellite drops, which is known as a nonlinear effect. Results are found in good agreement with direct numerical simulations and forced liquid jet experiments for wavenumbers lower than a threshold value. Satellite drop formation is retarded with increasing Ohnesorge number and wavenumber, as expected by the damping and size effects of viscosity. The threshold number corresponds to the maximum wavenumber for which satellite drop formation is predicted before jet breakup, and for which volume conservation is satisfied within a certain amount. The volume conservation criterion is imposed to ensure that the conclusions inferred by our model are safe.

The relaminarization phenomenon in the laminar–turbulent transition process of spherical Couette flow with the inner sphere rotating and the outer sphere fixed has been experimentally investigated for $0.06\,{ \leqslant}\,\beta\,{\leqslant}\,0.206$, where $\beta$ is the ratio of the clearance to the inner-sphere radius. The relaminarization occurs for $0.13\,{<}\,\beta\,{<}\,0.17$, and is observed as a reverse Hopf bifurcation from the limit cycle to the fixed point. The kinetic energy in the high-frequency region of the fluctuating azimuthal velocity component continues to increase with Reynolds number Re in the case without relaminarization, but in the case with relaminarization it first increases and then decreases with increasing Re and finally vanishes with the onset of the relaminarization. For $\beta\,{=}\,0.14$ with relaminarization, a small artificial disturbance introduced externally into the spherical Couette flow has no influence on the lowest critical Reynolds number of the first instability that can be defined universally and uniquely. On the other hand, the external disturbance decreases the onset Reynolds numbers of both the second instability (occurrence of spiral Taylor–Görtler (TG) vortices) and the relaminarization, although it has no influence on the fundamental frequencies of velocity fluctuation caused by spiral TG vortices and travelling waves on TG vortices.

Turbulent two-phase flows are characterized by the presence of multiple time and length scales. Of particular interest in flows with non-negligible interphase momentum coupling are the time scales associated with interphase turbulent kinetic energy transfer (TKE) and inertial particle dispersion. Point-particle direct numerical simulations (DNS) of homogeneous turbulent flows laden with sub-Kolmogorov size particles report that the time scale associated with the interphase TKE transfer behaves differently with Stokes number than the time scale associated with particle dispersion. Here, the Stokes number is defined as the ratio of the particle momentum response time scale to the Kolmogorov time scale of turbulence. In this study, we propose a two-way coupled stochastic model (CSM), which is a system of two coupled Langevin equations for the fluctuating velocities in each phase. The basis for the model is the Eulerian–Eulerian probability density function formalism for two-phase flows that was established in Pai & Subramaniam (J. Fluid Mech., vol. 628, 2009, pp. 181–228). This new model possesses the unique capability of simultaneously capturing the disparate dependence of the time scales associated with interphase TKE transfer and particle dispersion on Stokes number. This is ascertained by comparing predicted trends of statistics of turbulent kinetic energy and particle dispersion in both phases from CSM, for varying Stokes number and mass loading, with point-particle DNS datasets of homogeneous particle-laden flows.

The boundary layer that forms on the walls of a shock tube, after the diaphragm which initially separates two gases at different pressures is burst, is investigated. Both the driver and driven gases are assumed to have the same thermal properties and the shock tube wall is maintained at constant temperature. Crocco variables are used and a method is presented for solving the compressible boundary-layer equations within the tube in similarity variables. Three cases, corresponding to different initial pressure ratios of the driver and driven gases, are calculated which are representative of weak and medium-strength shock and expansion waves.

Linear stability of Couette flow modified by transpiration applied at the lower wall is considered. It is shown that transpiration can induce flow instability resulting in the appearance of streamwise-vortex-like structures. It is argued that the instability is driven by centrifugal forces associated with streamline curvature. The conditions leading to the onset of the instability depend on the amplitude and wavelength of the transpiration and can be expressed in terms of the critical Reynolds number. The global critical conditions describing the minimum critical Reynolds number required for the onset of the instability for the specified amplitude of the transpiration regardless of its wavelength are also given. The threshold amplitude required for the onset varies approximately as ${\sim}\hbox{\it Re}^{-1.15}$ for large $Re$, where the Reynolds number used is based on the velocity difference between the walls and the channel half-width. The existence of a global threshold, below which the instability cannot occur regardless of the amplitude of the transpiration, has been demonstrated. This threshold corresponds approximately to $\hbox{\it Re}=84$.

We propose a model for the acceleration of micro-bubbles (smaller than the dissipative scale of the flow) subjected to the drag and fluid inertia forces in a homogeneous and isotropic turbulent flow. This model, that depends on the Stokes number, Reynolds number and the density ratio, reproduces the evolution of the acceleration variance as well as the relative importance and alignment of the two forces as observed from direct numerical simulations (DNS). We also report that the bubble acceleration statistics conditioned on the local kinetic energy dissipation rate are invariant with the Stokes number and the dissipation rate. Based on this observation, we propose a stochastic model for the instantaneous bubble acceleration vector accounting for the small-scale intermittency of the turbulent flows. The norm of the bubble acceleration is obtained by modelling the dissipation rate along the bubble trajectory from a log-normal stochastic process, whereas its orientation is given by two coupled random walks on a unit sphere in order to model the evolution of the joint orientation of the drag and inertia forces acting on the bubble. Furthermore, the proposed stochastic model for the bubble acceleration is used in the context of large eddy simulations (LES) of turbulent flows laden with small bubbles. To account for the turbulent motion at scales smaller than the mesh resolution, we decompose the instantaneous bubble acceleration in its resolved and residual parts. The first part is given by the drag and fluid inertia forces computed from the resolved velocity field, and the second term refers to the random contribution of small unresolved turbulent scales and is estimated with the stochastic model proposed in the paper. Comparisons with DNS and standard LES, show that the proposed model improves significantly the statistics of the bubbly phase.

On the basis of the Saint-Venant equations for flowing granular matter, we study the various travelling waveforms that are encountered in chute flow for growing Froude number. Generally, for $Fr<2/3$ one finds either a uniform flow of constant thickness or a monoclinal flood wave, i.e. a shock structure monotonically connecting a thick region upstream to a shallower region downstream. For $Fr>2/3$ both the uniform flow and the monoclinal wave cease to be stable; the flow now organizes itself in the form of a train of roll waves. From the governing Saint-Venant equations we derive a dynamical system that elucidates the transition from monoclinal waves to roll waves. It is found that this transition involves several intermediate stages, including an undular bore that had hitherto not been reported for granular flows.

Three-dimensional transitional flow of a rarefied monatomic gas over a spinning sphere is studied numerically by the direct simulation Monte Carlo method. Gas molecules interact with each other as hard spheres. The Maxwell model of specular–diffuse scattering is used to describe the interaction between gas molecules and the sphere surface. The effect of all dimensionless governing parameters of the problem on the flow structure, distributions of stresses and heat flux density on the sphere surface, aerodynamic force and torque exerted on the sphere, and heat flux on the sphere surface is analysed. Simulations are conducted at Mach numbers raging from to and Knudsen numbers raging from to . Two qualitatively different streamline patterns are observed around the sphere depending on its dimensionless rotational velocity. The direction and magnitude of the transverse Magnus force in transitional flow depend on the Knudsen and Mach numbers. The torque and heat flux coefficients are found to be functions of the Mach number and dimensionless rotational velocity. The effect of rotation on the sphere aerodynamics weakens with an increase in the sphere temperature with respect to the gas temperature in the free stream. A complete set of equations is developed to fit the computed values of the aerodynamic and heat flux coefficients in a case when the sphere temperature is equal to the temperature of the free stream.

The system composed of a circular cylinder, either fixed or elastically mounted, and immersed in a current linearly sheared in the cross-flow direction, is investigated via numerical simulations. The impact of the shear and associated symmetry breaking are explored over wide ranges of values of the shear parameter (non-dimensional inflow velocity gradient, $\unicode[STIX]{x1D6FD}\in [0,0.4]$) and reduced velocity (inverse of the non-dimensional natural frequency of the oscillator, $U^{\ast }\in [2,14]$), at Reynolds number $Re=100$; $\unicode[STIX]{x1D6FD}$, $U^{\ast }$ and $Re$ are based on the inflow velocity at the centre of the body and on its diameter. In the absence of large-amplitude vibrations and in the fixed body case, three successive regimes are identified. Two unsteady flow regimes develop for $\unicode[STIX]{x1D6FD}\in [0,0.2]$ (regime L) and $\unicode[STIX]{x1D6FD}\in [0.2,0.3]$ (regime H). They differ by the relative influence of the shear, which is found to be limited in regime L. In contrast, the shear leads to a major reconfiguration of the wake (e.g. asymmetric pattern, lower vortex shedding frequency, synchronized oscillation of the saddle point) and a substantial alteration of the fluid forcing in regime H. A steady flow regime (S), characterized by a triangular wake pattern, is uncovered for $\unicode[STIX]{x1D6FD}>0.3$. Free vibrations of large amplitudes arise in a region of the parameter space that encompasses the entire range of $\unicode[STIX]{x1D6FD}$ and a range of $U^{\ast }$ that widens as $\unicode[STIX]{x1D6FD}$ increases; therefore vibrations appear beyond the limit of steady flow in the fixed body case ($\unicode[STIX]{x1D6FD}=0.3$). Three distinct regimes of the flow–structure system are encountered in this region. In all regimes, body motion and flow unsteadiness are synchronized (lock-in condition). For $\unicode[STIX]{x1D6FD}\in [0,0.2]$, in regime VL, the system behaviour remains close to that observed in uniform current. The main impact of the shear concerns the amplification of the in-line response and the transition from figure-eight to ellipsoidal orbits. For $\unicode[STIX]{x1D6FD}\in [0.2,0.4]$, the system exhibits two well-defined regimes: VH1 and VH2 in the lower and higher ranges of $U^{\ast }$, respectively. Even if the wake patterns, close to the asymmetric pattern observed in regime H, are comparable in both regimes, the properties of the vibrations and fluid forces clearly depart. The responses differ by their spectral contents, i.e. sinusoidal versus multi-harmonic, and their amplitudes are much larger in regime VH1, where the in-line responses reach $2$ diameters ($0.03$ diameters in uniform flow) and the cross-flow responses $1.3$ diameters. Aperiodic, intermittent oscillations are found to occur in the transition region between regimes VH1 and VH2; it appears that wake–body synchronization persists in this case.