Variable-density mixing in shock bubble interaction, a canonical flow of Richtermyer-Meshkov instability, is studied with high-resolution simulation. While the dissipation mainly controls the passive scalar mixing rate, an objective definition of variable-density mixing rate characterizing the macroscopic mixing formation is still lacking, and the fundamental behavior of mixing rate evolution is not yet well understood. Here we first show that the variable-density mixing of shock bubble interaction is distinctly different from the previous observations of the passive scalar mixing. The widely accepted hyperbolic conservation of the first moment of concentration in the scalar mixing, i.e., the conservation of the mean concentration, is violated in variable-density flows. We further combine the compositional transport equation and the divergence relation for the miscible flows to provide evidence that the existence of a density gradient accelerated mixing rate, decomposed by the accelerated dissipation term and redistributed diffusion term, contributes to the anomalous decrease or increase of the mean concentration depending on Atwood number. Further analyzing a number of simulations for the cylindrical or spherical bubbles under a broad range of shock Mach numbers, Reynolds numbers, and Péclet numbers, the density gradient accelerated mixing rate exhibits weak dependence on Péclet numbers, and we identify an Atwood number range with high mixing rate, which can be theoretically predicted based on the mode of hyperbolic conservation violation behavior.

The orientation vector of a spheroid subjected to a shear flow undergoes periodic Jeffrey orbits on a unit sphere. If the spheroid is polarizable and is subjected to a magnetic field, there is a transition between a rotating state at low magnetic field strength and a steady orientation along the field direction at high magnetic field strength. For the special case where the magnetic field is parallel to the flow plane, it has been shown [V. Kumaran, Phys. Rev. Fluids 6, 043702 (2021)] that the spheroid exhibits complex dynamics, including a transition from rotating to static states and the possibility of multiple steady states. Here the more general case is analyzed where the magnetic field is not parallel to the flow plane, for Langevin, linear, and signum magnetization models. At low field strength, there are two possible states, a steady orientation almost perpendicular to the flow plane and a rotating state with orientation close to the flow plane. The magnetic field orientation for transition between these two states is derived for a general magnetic model. For high magnetic field strength, the particle aligns close to the magnetic field direction. The special cases of a spherical particle and a thin rod are analyzed for arbitrary magnetic field strength. Analytical results are obtained for the critical magnetic field strength for transition between rotating and steady states, and the scaling of the frequency of rotation with magnetic field strength in the rotating state for the linear and signum models. These are compared with numerical results for the Langevin model. It is found that a thin rod has rotating states only when the cross-stream component of the magnetic field is antiparallel to the velocity gradient, and the orientation is always steady when the cross-stream component of the magnetic field is parallel to the velocity gradient. For steady orientation, the torque in the streamwise direction is zero, and the ratio of the torques in the cross-stream and spanwise directions is only a function of the magnetic field orientation.

An analytical solution is derived for the flow generated by a self-propelling two-dimensional Marangoni boat driven by reactive insoluble surfactant on a deep layer of fluid of viscosity $\mu$ at zero Reynolds number, capillary number, and surface Péclet number. In the model, surfactant emitted from the edges of the boat causes a surface tension disparity across the boat. Once emitted, the surfactant diffuses along the interface and sublimates to the upper gas phase. A linear equation of state relates the surface tension to the surfactant concentration. The propulsion speed of the boat is shown to be $U_0=\mathrm{\Delta }\sigma {\left(2\pi \mu \right)}^{-1}{e}^{\sqrt{\text{Da}}}{K}_0\left(\sqrt{\text{Da}}\right)$ where Da is a Damköhler number measuring the reaction rate of the surfactant to its surface diffusion, $\mathrm{\Delta }\sigma$ is the surface tension disparity between the front and rear of the boat, and $K_0$ is the order-zero modified Bessel function. Explicit expressions for the streamfunction associated with the Stokes flow beneath the boat are found facilitating ready examination of the Marangoni-induced streamlines. An integral formula, derived using the reciprocal theorem, is also given for the propulsion speed of the boat in response to a more general Marangoni stress distribution.

Objects released into a granular packing close to incipient fluidization may float or sink depending on their density. Contrary to intuition, Oshitani et al. [Phys. Rev. Lett. 116, 068001 (2016)] reported that under certain conditions, a lighter sphere can sink further and slower than a heavier one. While this phenomenon has been attributed to a local fluidization around the sinking sphere, its physical mechanisms have not yet been understood. Here, we studied this intriguing phenomenon using both magnetic resonance imaging and discrete particle simulation. Our findings suggest that local fluidization around the sinking sphere and the formation and detachment of gas bubbles play a critical role in driving this anomaly. An analysis of forces acting on the intruder revealed that the upward-directed fluid force acting on a sphere is almost fully counterbalanced by the sum of the net contact forces and the gravitational force acting downward, when the sphere density is close to the bulk density of the granular packing (${\rho }_{\mathrm{sphere}}/{\rho }_{\mathrm{bulk}}\approx 1$). At the time when bubbles detach from the sphere, the gas pressure gradient experienced by the sphere is slightly attenuated and the sphere is pushed downward by the particle cap located on top of the sphere. Because the deviations from the force equilibrium are small, the sphere sinks slowly. Even after the sphere has reached its final stable depth, local fluidization in combination with bubble formation remains in the proximity of the sphere.

We discuss the dynamics of unidirectional random wave fields that propagate against an opposing current through laboratory experiments and direct numerical simulations of the Euler equations solved with a high-order spectral method. Both approaches demonstrate that the presence of a negative horizontal velocity gradient increases the probability of the occurrence of extreme and rogue waves in the course of their propagation with the emergence of a rapid transition from weakly to strongly non-Gaussian properties. Numerical simulations capture quantitatively well the statistical properties of laboratory observations and substantiate that underlying physics are associated to quasiresonant nonlinear interactions triggered by the background current.

We report a resonance-driven convective heat transfer enhancement downstream of a natural convection boundary layer that is perturbed by a moderate impinging jet. Flow resonance is experimentally and numerically confirmed between the boundary layer and a periodic flow driven by the unbalance of jet momentum and buoyancy. When the oscillation frequency in the impinging region is a submultiple of the characteristic frequency of natural convection, downstream heat transfer enhancement $E$ from the baseline of natural convection exceeds 40%. In contrast, further increasing the jet momentum fourfold yields $E<20\%$. The results support the development of efficient heat transfer enhancement methods.

Most widely used Reynolds-averaged Navier-Stokes (RANS) models employ the Boussinesq approximation, which assumes a linear relationship between the turbulent Reynolds stresses and the mean-velocity gradient tensor. This assumption, which can be very stringent, is more suited for simple shear-flows and is regarded as an important shortcoming for the improvement of the representation of turbulence in complex geometries. Correction of the local turbulence length scales, as achieved, for example, by the introduction of a correction term in the eddy-viscosity equation of a Spalart-Allmaras model, then indeed only allows relatively small corrections to the base-line model and may not be sufficiently flexible to adapt to common situations where the Reynolds stresses are not aligned with the velocity gradient tensor. For the variational data-assimilation step, we consider a vectorial source correction term in the momentum equations (output quantity) together with the Spalart-Allmaras model, which allows full flexibility to adapt to any mean-flow topology. We show how the machine-learning framework should be adapted, in particular with respect to the vectorial nature of the correction term and given invariance properties. As for the input quantities, we discuss the impact of considering either local quantities for the nondimensionalization or global ones that characterize the configuration. We showcase the procedure on the periodic hill configuration, for which a rich direct numerical simulation database is available in the literature: in particular, the availability of the full mean-flow solution for a range of Reynolds numbers and geometries ensures identification of “physical” correction terms and allows learning of turbulence models that accurately extrapolate to new Reynolds numbers and geometries.

The dip-coating process consists of withdrawing immersed objects from a liquid reservoir. After withdrawal, a significant layer of liquid remains on the object. Various industrial processes (food and beverage industry, automotive industry) use this technique to coat or treat surfaces. Recent studies have shown that the thickness of deposit is determined by the flow inside the reservoir for yield-stress fluids. This is different from the behavior of simple liquids for which the coating thickness is solely determined by the flow inside the meniscus. In this work, we reexamine this question and propose a complete phase diagram linking the Newtonian case and the yield-stress fluid case. We provide asymptotic scaling laws for extreme cases. A good agreement with experiments is obtained.

Experiments on a spherical shock wave propagating across an unheated- or a heated-cylinder wake are performed in a wind tunnel to investigate the effects of velocity and temperature fluctuations of turbulence on the shock wave. The temperature of the heated cylinder is low enough for the buoyancy effect to be negligible in the wake development, and comparisons between the heated- and unheated-cylinder experiments highlight the effects of temperature fluctuations on the shock wave. Peak overpressure of the spherical shock wave is measured on a wall after the shock wave has passed the wake. Along with the overpressure measurement, temperature and velocity are measured in the heated and unheated wakes, respectively. Larger peak-overpressure fluctuations are obtained when the shock wave interacts with the heated-cylinder wake than with the unheated-cylinder wake. Correlation coefficients are calculated between the velocity/temperature fluctuations of the unheated/heated-cylinder wakes and peak-overpressure fluctuations. The temperature fluctuations and overpressure fluctuations are found to be negatively correlated, which is explained by the shock deformation caused by speed-of-sound fluctuations in front of the shock wave. By comparing the correlation coefficients between velocity and overpressure fluctuations with those between temperature and overpressure fluctuations, it is also discovered that the temperature fluctuations of the heated-cylinder wake have a stronger correlation with the overpressure fluctuations than the velocity fluctuations of the unheated-cylinder wake.

The effect of freestream turbulence on the dynamics of an incompressible flow past a cylindrical roughness element in subcritical conditions (i.e., for Reynolds numbers below the onset of linear instability) has been investigated by the joint application of direct numerical simulations, linear modal and nonmodal stability analyses, and dynamic mode decomposition. At first, the influence of the Reynolds number and the ratio of the boundary layer's thickness to roughness height on the three-dimensional spatiotemporal (global) stability of the flow has been investigated. Depending on the operating conditions, the leading instability can either be varicose (symmetric) or sinuous (antisymmetric). In both cases, when the flow is excited by broadband frequency forcing, dynamic mode decomposition extracts only varicose coherent structures even though optimal response analysis predicts a strong amplification of sinuous disturbances having frequency close to that of the marginally stable sinuous eigenmode. This apparent discrepancy is attributed to the fact that the sinuous instability is sensitive to a very limited range of frequencies barely excited by freestream turbulence while varicose disturbances are associated with high amplification in a much wider frequency range. Hence, in this case the flow behaves as an amplifier of varicose perturbations rather than a resonator. Consequences on the subsequent transition to turbulence have been studied, highlighting that varicose perturbations extract energy from the near-wake region, get continuously amplified due to the excitation provided by freestream turbulence, and eventually give rise to a shedding of hairpin vortices.

We study the Rayleigh-Taylor instability of gravity-driven viscous liquid films flowing under a uniformly heated or cooled inclined substrate. The long-wave assumption is adopted to derive the evolution equation of the film, which is characterized by five dimensionless parameters including Marangoni number $\mathrm{Ma}$, Biot number $\mathrm{Bi}$, Reynolds number $\mathrm{Re}$, Weber number $\mathrm{We}$, and the inclination angle $\alpha$ of the substrate. Based on the long-wave equation, we systematically examine the temporal and spatiotemporal stability of the system. Temporal stability analysis shows that the thermocapillary stress reinforces the Rayleigh-Taylor instability of a heated film but counteracts the instability of a cooled film, as verified by the numerical solutions of linearized Navier-Stokes equation. In particular, this instability can be completely inhibited if a composite Marangoni number ${\mathrm{Ma}}^{*}$ is below a critical value ${\mathrm{Ma}}_{\text{1}}^{*}$. We further perform a spatiotemporal stability analysis to delineate the absolute and convective nature of the temporally unstable system. We find that the thermocapillary stress in the heated film enhances the absolute instability and suppresses the convective instability. The trend reverses for a cooled film that is featured by suppressed absolute instability and enhanced convective instability. More importantly, the transition between the absolute and convective instability can be characterized by another critical value, ${\mathrm{Ma}}_2^{*}$, beyond which the flow stability will be triggered from the convectively into the absolutely unstable. The predictions from linear stability analysis are confirmed by numerical solutions of the full long-wave evolution equation.

The objective in the present paper is to study binary fluids with phase field modeling coupled with Navier-Stokes equations. An extended free energy is proposed to account for the continuous path from immiscible to miscible states. We consider fluid pairs that are immiscible for temperatures below the critical one (consolute temperature) and miscible above it. Our extended phase field equation permits us to move from the immiscible state (governed by the Cahn-Hilliard equation) to the miscible state (defined by the species diffusion equation). The scaling of interface tension and interface width with the distance to the critical point is highlighted. The whole system is mechanically excited showing Faraday instability of a flat interface. A linear stability analysis is performed for the stable case (interface waves) as well as for the unstable Faraday one. For the latter, a Floquet analysis shows the well-known Arnold's tongues as a function of the consolute temperature and depth layer. Moreover, two-dimensional finite difference simulations have been performed allowing us to model nonlinear flow patterns both in miscible and immiscible phases. Linear theory and nonlinear simulations show interesting results such as the diminishing of the wavelength of Faraday waves or a shift of the critical vibration amplitude when the consolute temperature is approached.

The computational cost of fluid simulations increases rapidly with grid resolution. This has given a hard limit on the ability of simulations to accurately resolve small-scale features of complex flows. Here we use a machine learning approach to learn a numerical discretization that retains high accuracy even when the solution is under-resolved with classical methods. We apply this approach to passive scalar advection in a two-dimensional turbulent flow. The method maintains the same accuracy as traditional high-order flux-limited advection solvers, while using $4\times$ lower grid resolution in each dimension. The machine learning component is tightly integrated with traditional finite-volume schemes and can be trained via an end-to-end differentiable programming framework. The solver can achieve near-peak hardware utilization on CPUs and accelerators via convolutional filters.

We focus on the wake function $F\left(Y\right)$, where $Y=y/\delta$: the excess velocity above the logarithmic profile in wall-bounded turbulence. It is first measured using direct numerical simulation data of turbulent channel flow at ${\text{Re}}_{\tau }\simeq 5200$, which has a distinguishable overlap layer. The function $Q\left(Y\right)=Y{F}^{\prime }\left(Y\right)$ is seen to be significant only beyond $Y_c\simeq 0.16$, and also linear (with a slope $\alpha \simeq 1.15$) up to $Y\simeq 0.5$. Our simplest $Q\left(Y\right)$ model is then a linear ramp function that starts at the measured intercept. Our $Q\left(Y\right)$ model is furthermore extended up to the channel center, by subtracting a quadratic term which satisfies the boundary condition at $Y=1$. The analytical integration of $Q\left(Y\right)$ provides the model for the wake function with offset $Y_c$, and which covers the full range; it is seen to fit very well the DNS data. Our $Q\left(Y\right)$ model is also compared to the “extended law of the wall” model without offset. Furthermore, and to comply with some recent literature, a version of our model is also developed with some small slope ${\alpha }_0\ll \alpha$ contribution to $Q\left(Y\right)$ within the overlap layer. The DNS of a zero pressure gradient boundary layer at ${\text{Re}}_{\tau }\simeq 2300$ is considered next: the amplitude of the wake function is much larger than in channel flow $(\alpha \simeq 7.3)$ and the intercept is smaller $(Y_c\simeq 0.11)$. The obtained $F\left(Y\right)$ model also compares very well with the DNS data over the full range. Lastly, we also revisit the Coles wake function model and we show that adding the offset improves it significantly.

Experiments are performed examining particle settling from constant-flux surface gravity currents and from a near-stationary layer of particle-bearing fluid overlying a saline ambient fluid. The particles consisted either of clay or of near-spherical glass ballotini with four different mean diameters. In the case of ballotini with relatively large diameters ($\gtrsim 46\mu \mathrm{m}$), the particles formed a settling front within the current the settling speed of which was consistent with the Stokes settling velocity for a single particle. A single particle plume descended below the current over the extent of the settling front. In the case of clay and of $6\mu \text{m}$ diameter ballotini, no settling within or below the current was observed before the current nose reached the end of the tank and the source was turned off. After a time on the order of minutes, these particles were found to descend from the near-stationary surface layer through convective instability, though this may have been preconditioned by a fingering instability in the stratified interface between the upper and lower layers. Even though the particle concentration in the current and upper layer was less than 0.5% in all experiments, the front of the particle plumes descended at significantly greater speeds than the settling velocity of an individual particle. The relative descent speed was larger for smaller particles, with clay, in particular, descending at a speed on the order of 1000 times that of an individual clay particle.

One of the crucial aspects in the case of describing flow through a fluid-porous double layer configuration is the quantification of the flow behavior near the fluid-porous interface. Both theoretical as well as experimental studies indicate that in the case of real-life fluid-porous geometries, the shift from the fluid layer to the porous layer is not sharp and immediate. Instead, there exists a transition layer in between the fluid and the porous layers. In order to get a realistic picture of flow transition characteristics for such systems, it is imperative that this transition layer be taken into consideration in the hydrodynamic stability analysis. The same has been carried out in the present study, and the effect of the transition layer on the stability characteristics has been assessed. The current study has investigated the effect of the transition layer on the flow of a power-law fluid overlying a porous layer in a channel flow configuration, as the earlier studies were restricted to Newtonian fluids only. A rigorous three-dimensional perturbation framework has been adopted, as Squire's theorem is not applicable for the power-law rheology unlike Newtonian rheology. Both modal and nonmodal analyses are attempted in order to reveal the long-time response as well as short-time amplifications of the system to external disturbances. The intricate interaction between the non-Newtonian rheology of the fluid and the thin transition layer is found to affect the transient amplifications of the flow system significantly.

Multiscale statistical analyses of inertial particle distributions are presented to investigate the statistical signature of clustering and void regions in particle-laden incompressible isotropic turbulence. Three-dimensional direct numerical simulations of homogeneous isotropic turbulence at high Reynolds number (${\text{Re}}_{\lambda }\gtrsim 200$) are performed. Lagrangian motion of inertial particles are tracked by the one-way coupled point-particle method assuming the Stokes drag. Up to ${10}^9$ inertial particles for each Stokes number ranging from 0.01 to 5.0 are computed. Orthogonal wavelet analysis is then applied to the computed particle-number density fields. Scale-dependent skewness and flatness values of the particle-number density distributions are calculated and the influence of Reynolds number ${\text{Re}}_{\lambda }$ and Stokes number St is assessed. For $\text{St}\sim 1$, both the scale-dependent skewness and flatness values become larger as the scale decreases, suggesting intermittent clustering at small scales. For $\text{St}\le 0.2$, the flatness at intermediate scales, i.e., scales larger than the Kolmogorov scale and smaller than the integral scale of the flow, increases as St increases, and the skewness exhibits negative values at the intermediate scales. The negative values of the skewness are a signature of void regions. These results indicate that void regions at the intermediate scales are pronounced and intermittently distributed for such small Stokes numbers. As ${\text{Re}}_{\lambda }$ increases, the flatness increases weakly. For ${\text{Re}}_{\lambda }\ge 328$, the skewness shows negative values at large scales, suggesting that even for $\text{St}\sim 1$, void regions are pronounced at large scales, while clusters are pronounced at small scales.

The dynamics of small, yet heavy, identical particles in turbulence exhibits singularities, called caustics, that lead to large fluctuations in the spatial particle-number density, and in collision velocities. For large particle inertia, the fluid velocity at the particle position is essentially a white-noise signal and caustic formation is analogous to Kramers escape. Here we show that caustic formation at small particle inertia is different. Caustics tend to form in the vicinity of particle trajectories that experience a specific history of fluid-velocity gradients, characterized by low vorticity and a violent strain exceeding a large threshold. We develop a theory that explains our findings in terms of an optimal path to caustic formation that is approached in the small inertia limit.

Ventilated cavitating flow of a circular cylinder is investigated by numerical simulation. The coupled level set and volume of fluid method is used to capture the interface between the cavity and surrounding water. The simulation results indicate that the bubble size distribution in the wake is closely related to the turbulence intensity. The bubble number reaches its peak value in the closure region owing to the high turbulence intensity there. When the gas entrainment coefficient $Q_v$ increases, the length of the cavity increases and the turbulent kinetic energy decreases, leading to a decrease in the bubble number and an increase in the Sauter mean diameter. It is also found that the presence of the ventilated cavitation delays the formation of vortices and affects the vortex shedding. In the single phase flow, the enstrophy is concentrated in the shear layer and closure region. In the ventilated cavitating flow, on the other hand, the enstrophy is mainly distributed at the cavity interface and the re-entrant region, which indicates that the instability of the cavity interface and the existence of the re-entrant jet play important roles in the formation of vortices.

In this paper we present a novel hydrodynamic experiment using liquid ${}^4\mathrm{He}$. Lagrangian trajectories are obtained using two-dimensional particle tracking on hollow glass microspheres in a cryogenic liquid helium turbulent flow. The flow is forced inertially by a canonical oscillating grid, both below and above the superfluid transition. This allows for a direct comparison of the Lagrangian statistics in the normal (He I) and superfluid (He II) phases. The high temporal resolution allows us to resolve the velocity fluctuations at integral and inertial scales and, most importantly, assess the noise contribution. The careful analysis of velocity fluctuations, acceleration fluctuations, and pair dispersion, allows us to extract estimates of the energy-injection rate at large scale, the energy flux cascading through inertial scales, and the dissipation rate at small scale, and therefore build the energy budget in both the normal and superfluid phases. We find that, within experimental uncertainty, the statistical features of turbulence and the energy budget in superfluid helium is indistinguishable from those of normal helium, highlighting the importance of conducting experiments in both He I and He II to draw meaningful conclusions, because deviations from the theoretical predictions may arise from noise contributions or deviation from the homogeneous and isotropic approximations.