This paper presents an approach to simulate the free molecular flow in vacuum systems by using a Monte Carlo method for solving the Boltzmann particle transport equation with no intermolecular collisions. Sometimes referred to as a point-to-source Monte Carlo path tracing in image rendering, in this paper the name is borrowed from the thermal radiation and heat transfer field, reverse Monte Carlo path tracing. It is shown that this method provides better accuracy and stability when computing the Clausing function when compared to the standard test particle Monte Carlo method used for free molecular flow. The Clausing function leads to the distribution function of positions and velocities from which the particle density map, pressure gradient, energy flux, and other local quantities can be computed. Using reverse path tracing, the particle concentration in a conical segment is computed, and the maximal flow input is determined by calculating the mean free path at the maximum density position.

Coherent jets containing most of the kinetic energy of the flow are a common feature in observations of atmospheric turbulence at the planetary scale. In the gaseous planets these jets are embedded in a field of incoherent turbulence on scales small relative to the jet scale. Large-scale coherent waves are sometimes observed to coexist with the coherent jets and the incoherent turbulence with a prominent example of this phenomenon being the distortion of Saturn's north polar jet (NPJ) into a distinct hexagonal form. Observations of this large-scale jet-wave-turbulence coexistence regime raise the question of identifying the mechanism responsible for forming and maintaining this turbulent state. The coherent planetary scale jet component of the turbulence arises and is maintained by interaction with the incoherent small-scale turbulence component. It follows that theoretical understanding of the dynamics of the jet-wave-turbulence coexistence regime can be facilitated by employing a statistical state dynamics (SSD) model in which the interaction between coherent and incoherent components is explicitly represented. In this work a two-layer $\beta$-plane SSD model closed at second order is used to develop a theory that accounts for the structure and dynamics of the NPJ. An asymptotic analysis is performed of the SSD equilibrium in the weak jet damping limit that predicts a universal jet structure in agreement with observations of the NPJ. This asymptotic theory also predicts the wave number of the prominent jet perturbation. Analysis of the jet-wave-turbulence regime dynamics using this SSD model reveals that jet formation is controlled by the effective value of $\beta$ and the required value of this parameter for correspondence with observation is obtained. As this is a robust prediction it is taken as an indirect observation of a deep poleward sloping stable layer beneath the NPJ. The slope required is obtained from observations of the magnitude of the zonal wind component of the NPJ. The amplitude of the wave-6 perturbation then allows identification of the effective turbulence excitation maintaining this combined structure. The observed jet structure is then predicted by the theory as is the wave-6 disturbance. The wave-6 perturbation, which is identified as the least stable mode of the equilibrated jet, is shown to be primarily responsible for equilibrating the jet with the observed structure and amplitude.

Parametric subharmonic instability (PSI) of internal wave beams in a uniformly stratified fluid is discussed, for the case where the beam frequency is nearly twice the inertial frequency due to background rotation. Compared with generic PSI, beams of finite width are expected on physical grounds to be more vulnerable to subharmonic perturbations of near-inertial frequency, as these disturbances have small group velocity and stay in contact with the underlying beam longer, thus extracting more energy. A weakly nonlinear theory for such near-inertial PSI is developed in the “distinguished limit” where the effects of triad nonlinear interactions, dispersion, and viscous dissipation are equally important. This model is used to examine the linear stability of a uniform beam to infinitesimal perturbations under a “pump-wave” approximation, as well as the nonlinear development of PSI that takes into account the effect of the growing perturbations on the beam evolution. Near-inertial PSI is possible for beams of general locally confined profile, in sharp contrast to generic PSI which can arise only for quasimonochromatic beams whose profile comprises a sinusoidal carrier modulated by a locally confined envelope. The theoretical predictions are consistent with earlier numerical simulations of semidiurnal internal tide beams generated over the continental shelf break at latitudes above and below the critical value $28.8^{\circ }\mathrm{N}$, at which the subharmonic semidiurnal frequency matches the local inertial frequency.

The steady radial stretching of a disk beneath a rigidly rotating flow with constant angular velocity is considered. The steady base flow is determined numerically for both a stretching and a shrinking disk. The convective instability properties of the flow are examined using temporal stability analysis of the governing Rayleigh equation, and typically for small to moderate radial wave numbers, the range of azimuthal wave numbers $\beta$, over which the flow is unstable increases for both a stretched and a shrinking disk, compared to the unstretched case. The inviscid absolute instability properties of the resulting base flows are also examined using spatiotemporal stability analysis. For suitably large stretching rates, the flow is absolutely unstable in only a small range of positive $\beta$. For small stretching rates there exists a second region of absolute instability for a range of negative $\beta$ values. In this region the “effective” two-dimensional base flow, comprised of a linear combination of the radial and azimuthal velocity profiles that enter the Rayleigh equation calculation, has a critical point (unlike for $\beta >0$) that can dominate the absolute instability growth rate contribution compared to the shear layer component. A similar behavior is found to occur for a radially shrinking disk, except these profiles have a strong shear layer structure and hence are more unstable than the stretching disk profiles. We thus find for a suitably large shrinking rate the absolute instability contribution from the critical point becomes subdominant to the shear layer contribution.

We study the coupled drainage mechanisms of a propagating viscous gravity current that leaks fluid through a permeable substrate and a fixed edge. Using both theoretical analyses and numerical simulations, we investigate the time evolution of the profile shape and the amount of fluid loss through each of the drainage mechanisms. For the case of a finite-volume release, asymptotic solutions are provided to describe the dynamics of the profile shapes. Specifically, for the case of buoyancy-driven drainage with finite-volume release, an early-time self-similar solution is obtained to describe the profile evolution and a late-time self-similar solution is approached in the limit of pure edge drainage. For the case of constant fluid injection, numerical and analytical solutions are given to describe the time evolution and the steady-state profile shapes, as well as the partition of the fluid loss through each mechanism. We also briefly discuss the practical implications of the theoretical predictions to the ${\mathrm{CO}}_2$ sequestration and leakage problems.

In a wide range of applications, microfluidic channels are implemented in soft substrates. In such configurations, where fluidic inertia and compressibility are negligible, the propagation of fluids in channels is governed by a balance between fluid viscosity and elasticity of the surrounding solid. The viscous-elastic interactions between elastic substrates and non-Newtonian fluids are particularly of interest due to the dependence of viscosity on the state of the system. In this work, we study the fluid-structure interaction dynamics between an incompressible non-Newtonian fluid and a slender linearly elastic cylinder under the creeping flow regime. Considering power-law fluids and applying the thin shell approximation for the elastic cylinder, we obtain a nonhomogeneous p-Laplacian equation governing the viscous-elastic dynamics. We present exact solutions for the pressure and deformation fields for various initial and boundary conditions for both shear-thinning and shear-thickening fluids. We show that in contrast to Stokes' problem where a compactly supported front is obtained for shear-thickening fluids, here the role of viscosity is inversed and such fronts are obtained for shear-thinning fluids. Furthermore, we demonstrate that for the case of a step in inlet pressure, the propagation rate of the front has a $t^{\frac{n}{n+1}}$ dependence on time $\left(t\right)$, suggesting the ability to indirectly measure the power-law index $\left(n\right)$ of shear-thinning liquids through measurements of elastic deformation.

The onset of time dependence in the cubic lid-driven cavity is surprisingly complicated, given the simplicity of the geometry and the modest value of the Reynolds number at which it occurs. The onset is characterized by finite-amplitude oscillations that appear to be stable for long times, but are subjected to intermittent bursts at irregular times during which the reflection symmetry about the spanwise midplane is broken. The complex dynamics are shown to be intimately related to the subcritical nature of the instability of the steady basic state. We use a spectral collocation numerical technique, solving both in the full three-dimensional space and in the symmetric subspace, and use selective frequency damping and Arnoldi iterations about the unstable basic state to determine its bifurcations. Edge tracking is also used to investigate a number of time-dependent saddle states. Putting all this together, we show that the complex dynamics are organized by two successive Hopf bifurcations, the first of which is shown to be subcritical. All local states are unstable in the full space at higher Reynolds numbers, leading to the intermittent bursting behavior.

Localized turbulent structures such as puffs or oblique stripes are building blocks of the intermittency regimes in subcritical wall-bounded shear flows. These turbulent structures are investigated in incompressible pressure-driven annular pipe flow using direct numerical simulations in long domains. For low enough radius ratio $\eta$, these coherent structures have a dynamics comparable to that of puffs in cylindrical pipe flow. For $\eta$ larger than 0.5, they take the shape of helical stripes inclined with respect to the axial direction. The transition from puffs to stripes is analyzed statistically by focusing on the axisymmetry properties of the associated large-scale flows. It is shown that the transition is gradual: as the azimuthal confinement relaxes, allowing for an azimuthal large-scale component, oblique stripes emerge as predicted in the planar limit. The generality of this transition mechanism is discussed in the context of subcritical shear flows.

While it is a common wisdom that initial conditions influence the evolution of the Richtmyer-Meshkov instability (RMI), the research in this area is focused primarily on the effects of the wavelength and amplitude of the interface perturbation. The information has hitherto largely ignored the influences on RMI dynamics of the relative phase of waves constituting a multiwave initial perturbation and the interference of the perturbation waves. In this work we systematically study the influence of the relative phase and the interference of waves constituting a multiwave initial perturbation on a strong-shock-driven Richtmyer-Meshkov unstable interface separating ideal fluids with contrast densities. We apply group theory analysis and smoothed particle hydrodynamics numerical simulations. For verification and validation of the simulations, qualitative and quantitative comparisons are performed with rigorous zeroth-order, linear, and nonlinear theories as well as with gas dynamics experiments achieving good agreement. For a sample case of a two-wave (two-mode) initial perturbation we select the first-wave amplitude enabling the maximum initial growth rate of the RMI and we vary the second-wave amplitude from 1% to 100% of the first-wave amplitude. We also vary the relative phase of the first and second waves and consider the in-phase, the antiphase and the random-phase cases. We find that the relative phase and the interference of waves are important factors of RMI dynamics influencing qualitatively and quantitatively the symmetry, morphology, and growth rate of the Richtmyer-Meshkov unstable interface, as well as the order and disorder in strong-shock-driven RMI.

A pair of bubbles starting from rest and rising side-by-side in a liquid have been shown earlier to display spherical and ellipsoidal shapes. In contrast to earlier computational studies on the two-dimensional dynamics of a pair of bubbles, we study the fully three-dimensional motion of the bubbles in the inertial regime. We reveal the destabilizing nature of the interaction between the wakes of the bubbles, which causes them to rise in an oscillatory path. Such three-dimensionality sets in earlier in time than for a single bubble and also at a lower inertia. The interaction leads to a mirror symmetry in the trajectories of the two bubbles, which persists for some time even in the high-inertia regime where each path is chaotic. The effect of the inertia and initial separation on the mirror symmetry of the path, the vortex shedding pattern, and the attraction and repulsion between the bubbles are examined. The bubble rise has been interestingly observed to be symmetrical about the plane perpendicular to the separation vector for all separation distances considered in the present study.

Outcomes from an isothermal drop impact onto a surface (rebound, deposition, or splash) are determined by the impact parameters, liquid material properties, substrate morphology, and wettability. Drop impact onto a hot surface can be accompanied by Marangoni effects, nucleate or film boiling, and evaporation. The main focus of this experimental study is the characterization of the hydrodynamic outcome of a single-drop impact onto a hot surface. Drop impact phenomena have been observed and characterized using a high-speed video system. The impact parameters of the drop and the surface temperature have been varied. The observations lead to a classification of impact outcomes into several regimes: evaporation, nucleate boiling, foaming, transitional boiling, and film boiling. The contact time of the impinging droplet on the surface has been measured, and a model for this time and for the heat flux from the substrate to the drop has been proposed for the nucleate boiling regime. Comparison of model predictions to existing literature data is good.

Oblique large-scale laminar-turbulent patterns are found near the onset of turbulence in subcritical planar shear flows. Their robustness to the introduction of wall roughness is investigated numerically in plane Couette flow as a function of the Reynolds number and the roughness height. The effect of roughness is considered on either one or two walls and is modeled numerically using a parametric model suggested recently [Busse and Sandham, J. Fluid Mech. 712, 169 (2012)]. In the case of two rough walls, the patterns are robust for a mean roughness height up to 10% of the wall gap, but the flow shows larger turbulent fractions with increasing roughness height. In the case of one rough wall only, the trend is similar, but the onset Reynolds number decreases faster with increasing roughness height. Roughness height levels above 15% of the wall gap give rise to new coherent structures, including turbulent bands with nonoblique interfaces. The energetic efficiency of the various regimes is investigated by monitoring the friction factor versus the friction Reynolds number. The mechanisms allowing for streamwise localization of the stripe patterns are discussed, with or without roughness, in the light of various low-order models.

In this work we create new understanding of wind turbine wakes recovery process as a function of wind farm density using large-eddy simulations of an atmospheric boundary layer diurnal cycle. Simulations are forced with a constant geostrophic wind and a time varying surface temperature extracted from a selected period of the Cooperative Atmospheric Surface Exchange Study field experiment. Wind turbines are represented using the actuator disk model with rotation and yaw alignment. A control volume analysis around each turbine has been used to evaluate wind turbine wake recovery and corresponding harvested power. Results confirm the existence of two dominant recovery mechanisms, advection and flux of mean kinetic energy, which are modulated by the background thermal stratification. For the low-density arrangements advection dominates, while for the highly loaded wind farms the mean kinetic energy recovers through fluxes of mean kinetic energy. For those cases in between, a smooth balance of both mechanisms exists. From the results, a low-order model for the wind farms' harvested power as a function of thermal stratification and wind farm density has been developed, which has the potential to be used as an order-of-magnitude assessment tool.

The mechanism of hydrodynamics-induced pairing of soft particles, namely closed bilayer membranes (vesicles, a model system for red blood cells) and drops, is studied numerically with a special attention paid to the role of the confinement (the particles are within two rigid walls). This study unveils the complexity of the pairing mechanism due to hydrodynamic interactions. We find both for vesicles and for drops that two particles attract each other and form a stable pair at weak confinement if their initial separation is below a certain value. If the initial separation is beyond that distance, the particles repel each other and adopt a longer stable interdistance. This means that for the same confinement we have (at least) two stable branches. To which branch a pair of particles relaxes with time depends only on the initial configuration. An unstable branch is found between these two stable branches. At a critical confinement the stable branch corresponding to the shortest interdistance merges with the unstable branch in the form of a saddle-node bifurcation. At this critical confinement we have a finite jump from a solution corresponding to the continuation of the unbounded case to a solution which is induced by the presence of walls. The results are summarized in a phase diagram, which proves to be of a complex nature. The fact that both vesicles and drops have the same qualitative phase diagram points to the existence of a universal behavior, highlighting the fact that with regard to pairing the details of mechanical properties of the deformable particles are unimportant. This offers an interesting perspective for simple analytical modeling.

Imposing a magnetic field on a turbulent flow of electrically conducting fluid incurs the Joule effect. A current paradigm is that the corresponding dissipation increases with the intensity of the magnetic field and as a result turbulent fluctuations are all the more damped as the magnetic field is strong. While this idea finds apparent support in the phenomenology of decaying turbulence, measurements of turbulence in duct flows and other, more complex configurations have produced seemingly contradicting results. The root of the controversy is that magnetic fields promote sufficient scale-dependent anisotropy to profoundly reorganize the structure of turbulence, so their net effect cannot be understood in terms of the additional dissipation only. Here we show that when turbulence is forced in a magnetic field that acts on turbulence itself rather than on the mechanisms that generate it, the field promotes large, nearly two-dimensional structures capturing sufficient energy to offset the loss due to Joule dissipation, with the net effect of increasing the intensity of turbulent fluctuations. This change of paradigm potentially carries important consequences for systems as diverse as the liquid cores of planets, accretion disks, and a wide range of metallurgical and nuclear engineering applications.

We study experimentally three-wave interactions between hydroelastic waves propagating on the surface of a fluid covered by an elastic sheet (where both tension and bending are important). We observe the generation of a resonant daughter wave by nonlinear interaction among two mother waves of almost perpendicular directions. By using local and spatiotemporal wave-height measurements, the frequency and wave vector of the daughter wave are found to satisfy the resonance conditions within the measurement accuracy. Its amplitude is also found to be reasonably well described by the resonant wave interaction theory. Finally, a phase-locking among interacting waves is also observed, as expected theoretically.

Separation of one type of chiral particle from mirror-image particles in a racemic mixture by shear flow and rotating electric field is studied theoretically. In shear flow the chiral particles and their mirror-image particles migrate in opposite directions along the vorticity direction. In a rotating electric field, they migrate in opposite directions along the axis of rotation. In each case, the migration velocity of the particles is calculated for a model chiral particle: a propeller-like particle which consists of two disks, at an angle to each other, rigidly connected to a thin rod. Effects of shear rate, field strength, and particle structure on the migration velocity are discussed. It is shown that the chirality of the particle is characterized by different parameters depending on the method of separation.

Hot-wire measurements are carried out in turbulent boundary layers over smooth and rough walls in order the assess the behavior of the skewness ($S$) and flatness ($F$) factors of the longitudinal velocity derivative as $y$, the distance from the wall, increases. The measurements are complemented by direct numerical simulations of a smooth wall turbulent channel flow. It is observed that, as the distance to the wall increases, $S$ and $F$ vary significantly before approaching a constant in the outer layer of the boundary layer. Further, $S$ and $F$ exhibit a nontrivial dependence on the Taylor microscale Reynolds number (${\text{Re}}_{\lambda }$). For example, in the region below about $0.2\delta$ ($\delta$ is the boundary layer thickness) where ${\text{Re}}_{\lambda }$ varies significantly, $S$ and $F$ strongly vary with ${\text{Re}}_{\lambda }$ and can be multivalued at a given ${\text{Re}}_{\lambda }$. In the outer region, between $0.3\delta$ and $0.6\delta$, $S$, $F$, and ${\text{Re}}_{\lambda }$ remain approximately constant. The channel flow direct numerical simulation data for $S$ and $F$ exhibit a similar behavior. These results point to the ambiguity that can arise when assessing the ${\text{Re}}_{\lambda }$ dependence of $S$ and $F$ in wall shear flows. In particular, the multivaluedness of $S$ and $F$ can lead to erroneous conclusions if $y/\delta$ is known only poorly, as is the case for the atmospheric shear layer (ASL). If the laboratory turbulent boundary layer is considered an adequate surrogate to the neutral ASL, then the behavior of $S$ and $F$ in the ASL is expected to be similar to that reported here.

We extend the vorticity-based modeling concept for stratified flows introduced by Borden and Meiburg [Z. Borden and E. Meiburg, J. Fluid Mech. 726, R1 (2013)] to unsteady flow fields that cannot be rendered quasisteady by a change of reference frames. Towards this end, we formulate a differential control volume balance for the conservation of mass and vorticity in the fully unsteady parts of the flow, which we refer to as the differential vorticity model. We furthermore show that with the additional assumptions of locally uniform parallel flow within each layer, the unsteady vorticity modeling approach reproduces the familiar two-layer shallow-water equations. To evaluate its accuracy, we then apply the vorticity model approach to partial-depth lock-release flows. Consistent with the shallow water analysis of Rottman and Simpson [J. W. Rottman and J. E. Simpson, J. Fluid Mech. 135, 95 (1983)], the vorticity model demonstrates the formation of a quasisteady gravity current front, a fully unsteady expansion wave, and a propagating bore that is present only if the lock depth exceeds half the channel height. When this bore forms, it travels with a velocity that does not depend on the lock height and the interface behind it is always at half the channel depth. We demonstrate that such a bore is energy conserving. The differential vorticity model gives predictions for the height and velocity of the gravity current and the bore, as well as for the propagation velocities of the edges of the expansion fan, as a function of the lock height. All of these predictions are seen to be in good agreement with the direct numerical simulation data and, where available, with experimental results. An energy analysis shows lock-release flows to be energy conserving only for the case of a full lock, whereas they are always dissipative for partial-depth locks.