We propose an approach to predict the modulation of wave packets in shock-containing jets. With a modeled ideally expanded mean flow as input, an approximation of the shock-cell structure is obtained from the parabolized stability equations (PSE) at zero frequency. This solution is then used to define a new shock-containing mean flow, which is a function of the shock-cell wave number at each streamwise station. Linearization of the Navier-Stokes equations around this quasiperiodic mean flow allows us to postulate a solution based on the Floquet ansatz, and further manipulation of the equations leads to a system called the parabolized Floquet equations [PFE; Ran et al., Phys. Rev. Fluids 4, 023901 (2019)] that bears several similarities to PSE. The modulation wave numbers are marched spatially together with the central Kelvin-Helmholtz wave number, leading to a modulated wave packet as the final solution. The limitations of PFE are highlighted, and the method is applied to two sample cases: a canonical slowly diverging jet at low supersonic Mach number and a heated overexpanded jet, for which large-eddy simulation (LES) data are available. Good agreement is observed between the wave packets predicted by PFE and the leading spectral proper orthogonal decomposition (SPOD) modes from the LES, suggesting that the method is able to capture the underlying physical mechanism associated with wave-packet modulation: the extraction of energy from the mean flow by the Kelvin-Helmholtz mode and a redistribution of energy to modulation wave numbers due to the interaction of this mode with the shock-cell structure.

Granular packing is often characterized by a linear relationship between packing fraction $\varphi$ and dimensionless shear rate. We examine this relationship in simple and complex heterogeneous flow geometries. At high inertial number $\left(I\right)$, we observe that the packing fraction depends approximately linearly on $I$ and is seemingly independent of the geometry. However, at low $I$ the packing fraction varies nonlinearly with $I$ and this variation is dependent on the specific geometry and conditions. This response is analogous to the behavior of the stress ratio $\left(\mu \right)$, where, at low $I$, $\mu$ depends on the geometry of the system, an effect that is often attributed to nonlocal effects of the granular rheology. We demonstrate that, in steady isochoric flow geometries, the $\varphi$ and $\mu$ responses are equivalent and $\varphi$ may be determined locally from $\mu$, even at low $I$. However, in a more complex, nonisochoric geometry, such as a pseudo-two-dimensional hopper, this $\varphi \left(\mu \right)$ relationship is not recovered in dense regions. We show that during transient startup in a shear cell $\varphi$ and $\mu$ follow a similar response to that seen in these nonisochoric flows. Hence we conclude that the observed discrepancy arises because the hopper is transient in a Lagrangian sense. The simple $\varphi \left(\mu \right)$ relationship seen in isochoric flows is insufficient in these more complex systems because of differences in the temporal evolution of the stress and packing fraction.

A turbulent round jet is experimentally manipulated using a pulsed periodic radial minijet placed near the nozzle exit. The actuation is characterized by the frequency $f_a$, duty cycle α, and mass-flow ratio $C_m$. The flow is measured at a Reynolds number of 8000, based on jet exit velocity and diameter, using the techniques of hot wire, flow visualization. and particle image velocimetry. Following Zhou et al. [J. Fluid Mech. 897, A27 (2020)], the jet decay rate of the centerline mean velocity $K$ is employed as a surrogate for jet mixing. The optimum $f_a$ corresponding to the maximum $K$ is observed to be one-half of the preferred mode frequency $f_0$ of main jet, i.e., $f_a/f_0=0.5$. Measurements indicate that $K$ depends on $C_m$ and α given $f_a/f_0=0.5$. It is found that once $K$, ${\mathrm{C}}_m$, and α are corrected and replaced by their “true values” $K^c$, $C_m^c$, and ${\alpha }^c$, $K=g_1\left(C_m,\alpha \right)$ may be reduced to $K^c=g_2\left({\xi }^c\right)$, where $g_1$ and $g_2$ are different functions and the scaling factor ${\xi }^c=C_m^c/{\alpha }^c$, which represents physically the effective penetration depth of the minijet and, unlike previous report, determines uniquely $K^c$. The manipulated jet is further classified into five typical flow structures, i.e., monomodal I, deflection I, bifurcation, monomodal II, and deflection II, which are uniquely determined by ${\xi }^c$. These representative flow structures are connected to the initial interactions between the minijet and the main jet.

We studied the anisotropization of homogeneous magnetohydrodynamic turbulence at low magnetic Reynolds numbers. Flows of this type are not only important for different engineering applications, but also provide an appealing framework for studies of quasi–two-dimensional turbulence with strongly modified transport properties. The results of large-scale forced, direct numerical simulations are presented and compared with those obtained with the quasi–normal scale elimination theory. For a weak magnetic field, the simulations validated the theoretical predictions, including the generation of the $k^{-7/3}$ range of the energy spectra and its propagation toward higher wave numbers with increasing magnetic field strength. In a strong magnetic field, the turbulence attains a quasi–two-dimensional state with an enstrophy cascade inertial range of the normal flow components in the normal plane and a passive scalar inertial-convective range of the parallel component. The corresponding energy spectra are in a good agreement with logarithmically corrected $k^{-3}$ and $k^{-1}$ theoretical predictions. With increasing Reynolds number at constant magnetic field the enstrophy cascade becomes unstable and is replaced by helicity cascade with $k^{-7/3}$ energy spectrum. The enstrophy cascade is restored with an increasing magnetic field. An investigation of the mechanism of energy injection into the parallel component in a strong magnetic field revealed that the energy is supplied directly by an external force. The spectrum of the parallel component depends on the isotropy of external forcing and is, thus, not universal.

Bubbles bursting at the surface of the sea water produce drops and is the main source of sea spray aerosol. The mechanisms underlying the drops production from a single bubble bursting event have been intensively studied and the influence of the bubble size and liquid parameters (density, viscosity, and surface tension) has been unified. However, despite the diversity of the surfactant molecules present in the oceans, their influence has been overlooked. In this paper we experimentally explore the influence of the surfactant concentration [sodium dodecyl surfate (SDS)] in a water solution on a single bubble collapse and subsequent drop production. We show that these surfactant molecules have an astonishing effect. In particular, we quantitatively show that they modify the bubble collapse, they induce less, smaller, and faster drops, and they can even completely prevent the drop production for a particular concentration. These results allow us to affirm that these important effects are mainly a consequence of the local surface tension gradients (Marangoni stresses), more than just the surface tension lowering, and to the related nonmonotonic variation of the surface elasticity with the surfactant concentration. Finally, this study shows that the role of water contamination by surface-active agents is important and needs to be understood to improve the prediction of the sea spray aerosol.

Studies of particle motion in vortical flows have mainly focused on pointlike particles. This approximation assumes that the velocity field that surrounds the particle is linear. We consider an inertialess rigid dumbbell in a two-dimensional steady vortex. While the system remains analytically tractable, the particle experiences the nonlinearity of the surrounding velocity field. By exploiting the rotational symmetry of the flow, we reduce the problem to that of a two-dimensional dynamical system, whose fixed points and periodic orbits can be used to explain the motion of the dumbbell. For all vortices in which the fluid angular velocity decreases with radial distance, the center of mass of the dumbbell follows a spirographic-like trajectory around the vortex center. This results from a periodic oscillation in the radial direction combined with revolution around the center. The shape of the trajectory depends strongly on the initial position and orientation of the dumbbell, but the dynamics is qualitatively the same irrespective of the form of the vortex. If the fluid angular velocity is not monotonic, the spirographic-like motion is altered by the existence of transport barriers, whose shape is now sensitive to the details of the vortex. This analysis may be extended to study the dynamics of inertial and/or self-propelled dumbbells in a two-dimensional vortex.

For the steady state of Burgers turbulence, Saffman [in Topics in Nonlinear Physics (Springer, Berlin, 1968), pp. 485–614] derived analytical expressions for the second-order structure function and energy spectrum in the inertial-dissipation range. By nondimensionalizing these quantities, we derive universal functions for the structure function and the energy spectrum and flux. We simulate Burgers turbulence with large-scale forcing for various injection rates and viscosity. For all these runs, the nondimensionalized structure functions and the energy spectra and fluxes collapse to the respective universal functions.

This paper introduces a local dynamic model for large-eddy simulation (LES) without averaging in the homogeneous directions. It is demonstrated that the widely used dynamic Smagorinsky model (DSM) has a singular dynamic model constant if it is used without averaging. The singularity can cause exceedingly large local values of the dynamic model constant. If these large values are not mitigated by the application of averaging, they can amplify discretization errors and impair the stability of simulations. To improve the local applicability of the DSM, the singularity is removed by replacing the resolved rate-of-strain tensors in the Smagorinsky model with the resolved velocity gradient tensor. This replacement results in the dynamic gradient Smagorinsky model (DGSM). Results of simulations of three canonical turbulent flows (decaying homogeneous isotropic turbulence, a temporal mixing layer, and turbulent channel flow) are presented to demonstrate the potential of this model. The DGSM provides improved stability compared to the local DSM and does not require averaging for stability at time step sizes that are typically used for a locally consistent static LES model. Results obtained with the DGSM are generally as accurate as results obtained with the DSM, while the DGSM has lower computational complexity. Moreover, the DGSM is easy to implement and does not require any homogeneous direction in space or time. It is therefore concluded that the DGSM is a promising local dynamic model for LES.

The strength and trajectory of a leading edge vortex (LEV) formed by a pitching-heaving hydrofoil (chord $c$) is studied. The LEV is identified using the $Q$-criterion method, which is calculated from the 2D velocity field obtained from particle image velocimetry measurements. The relative angle of attack at midstroke, ${\alpha }_{T/4}$, proves to be an effective method of combining heave amplitude ($h_0/c$), pitch amplitude (${\theta }_0$), and reduced frequency ($f^{*}$) into a single variable that predicts the maximum value of $Q$ over a wide range of operating conditions. Once the LEV separates from the foil, it travels downstream and rapidly weakens and diffuses. The downstream trajectory of the LEV has two characteristic shapes. At low values of ${\alpha }_{T/4}$, it travels straight downstream after separating from the foil, while at higher values of ${\alpha }_{T/4}$, an accompanying trailing edge vortex (TEV) forms and the induced velocity generates a cross-stream component to the vortex trajectories. This behavior is accurately predicted using a potential flow model for the LEV and TEV. Supervised machine learning algorithms, namely support vector regression and Gaussian process regression, are used to create regression models that predicts the vortex strength, shape, and trajectory during growth and after separation. The regression model successfully captures the features of two vortex regimes observed at different values of ${\alpha }_{T/4}$. However, the predicted LEV trajectories are somewhat smoother than observed in the experiments. The strengths of the vortex is often under-predicted. Both of these shortcomings may be attributed to the relatively small size of the training data set.

We investigate gunwale bobbing, a phenomenon in which a person jumping on the gunwales of a canoe achieves horizontal propulsion by forcing it with vertical oscillations. The canoe moves forward by surfing the resulting wave field. After an initial transient, the canoe achieves a cruising velocity which satisfies a balance between the thrust generated from pushing downwards into the surface gradients of the wave field and the resistance due to a combination of profile drag and wave drag. By superposing the linear wave theories of Havelock [Proc. R. Soc. A 95, 354 (1919)] for steady cruising and Helmholtz for an oscillating source, we demonstrate that such a balance can be sustained. We calculate the optimal parameter values to achieve maximum canoe velocity. We compare our theoretical result to accelerometer data taken from an enthusiastic gunwale bobber and to estimates from videos of other canoeing aficionados. Finally, we discuss the similarities and differences with other examples of macroscopic wave-driven bodies and comment on possible applications to competitive sports.

A series of direct numerical simulations of the Euler equation are conducted using a high-order spectral (HOS) method to investigate the nonlinear statistics and freak wave occurrences in crossing sea states. Several crossing sea states with varying frequency spectra, directional spreading, and crossing angles between two wave components are chosen for the computations. The dynamical statistics of surface waves are reported, including the wave spectra, the exceedance probability of wave crest amplitude, the probability density distribution of surface elevation, the kurtosis and skewness, the freak wave occurrence probability, and the freak wave shape. A Benjamin-Feir index, named as CBFI, is derived to measure the third-order nonlinearity effects for crossing seas. This parameter allows us to forecast the probability of freak waves, and it is validated by a series of HOS simulations. Furthermore, the freak wave shape is more notably influenced by changes to the crossing angle rather than each component's frequency or directional spectral bandwidth. Increasing the crossing angle reduces the vertical and horizontal asymmetries in the mean propagation direction.

Experiments were performed to quantify the interaction of a streamwise vortex with a series of oblique shock waves using stereoscopic particle image velocimetry. For these experiments, the streamwise vortex was created using a diamond-shaped, finite-span wing, pitched to one of two angles of attack $(\alpha =5^{\circ },{10}^{\circ })$. Downstream of the wing, the streamwise vortex encountered a two-dimensional oblique shock, generated by a wedge $(\theta ={15}^{\circ },{20}^{\circ },{25}^{\circ })$, resulting in a series of weak and moderate oblique shock-vortex interactions. While only moderate interactions produced a conical shock distortion, all interactions resulted in appreciable changes to the flow field downstream of the shock. These changes were reflected in increased velocity deficits and induced rotational motion; in the case of moderate interactions, increased turbulence kinetic energy was also apparent.

Dispersion of substances with reactive boundaries is relevant to a wide range of chemical, biological, and geophysical processes. A supplied crossflow, or equivalently sedimentation of the substance, is also expected to affect the dispersion process. We consider a setting with two infinite parallel plates, where the diffusive substance is adsorbed at the lower plate, simultaneously advected longitudinally by a main flow and vertically by a crossflow. Although the same configuration has been studied previously with the generalized Taylor dispersion (GTD) theory [M. Shapiro and H. Brenner, AIChE J. 33, 1155 (1987)] and a dual-perturbation method [T. Y. Lin and E. S. G. Shaqfeh, Phys. Rev. Fluids 4, 034501 (2019)], both of them focused on the long-time asymptotic dispersion regime, exclusive of the important transient dispersion process. As an extension of these works to the transient dispersion process, we utilize the classical method of moments along with the eigenfunction expansion to calculate the moments up to fourth order, and thus the effects of non-Gaussian properties can be reflected. Compared with the result of Brownian dynamics simulations, the present work is shown sufficient to cover the preasymptotic dispersion regime out of reach of the GTD and dual-perturbation method. Strong non-Gaussian properties are found in the preasymptotic regime, as reflected by the nonzero skewness and kurtosis as well as asymmetric longitudinal concentration distribution. Additionally, it is found that the duration of the preasymptotic regime is extended in the presence of both the crossflow and wall adsorption. Considering that most of the substance may have been adsorbed during the preasymptotic regime, it is necessary to use higher-order dispersion models such as the one presented herein.

A method to compute the thermal conductivity of a dilute gas from stochastic particle methods is described. The proposed approach is inspired by the Müller-Plathe method, in that the heat flux is imposed on the system and is precisely known, whereas the temperature gradient is sampled during the simulation. In this manner, the issue of large statistical scatter associated with sampling the heat flux is circumvented. The modified Müller-Plathe (MMP) method is first verified with a standard variable hard sphere direct simulation Monte Carlo calculation. Then, it is used in direct molecular simulation (DMS) equipped with an ab initio $\mathrm{N}_2+\mathrm{N}_2$ potential energy surface. The most significant sources of uncertainty are clearly identified and discussed. The DMS results for molecular nitrogen thermal conductivity agree remarkably well with the experimental evidence up to 2 000 K. The proposed MMP technique is applicable to arbitrary interatomic potentials and gas compositions, and it provides a direct way to evaluate thermal conductivity without relying on empirical extrapolations to high temperatures or approximations based on viscosity.

We investigate by direct numerical simulation (DNS) the dynamics of a liquid metal in a cylindrical cell under the conjugate effects of a temperature gradient and an ac magnetic field. These numerical simulations are performed for a range of Hartmann and shielding parameters, $30\le \mathrm{Ha}\le 301$ and $1\le S_{\omega }\le 68$, which correspond to intermediate regimes between low and high frequency of the applied magnetic field. The scales of the physical parameters are discussed in order to analyze the DNS results. In our simulations, the Lorentz force overbalances the buoyancy force and is responsible for the flow patterns made of two tori. Each torus is characterized by a different average temperature which leads to different temperature gradients at the bottom and the top walls, and consequently different heat transfer to the outside. Thermal plumes advected between the two tori appear intermittently. The Lorentz force enhances the thermal exchanges even at a low Ha. We establish that a single master curve represents the evolution of the heat transfer characteristic of the different cases by plotting the $\mathrm{Nu}/\mathrm{Pe}$ vs $Q_J/Q_c$, where Nu and Pe are the Nusselt and Péclet numbers, and $Q_J$ and $Q_c$ are the total power deposited by the Joule effect and the total power transferred without motion, respectively.

The breakup of surfactant-covered liquid jets or threads is central to inkjet printing, microarraying, and crop spraying. During the thinning of the thread, convection and diffusion—the relative importance of which is characterized by a Peclet number Pe—compete to determine the distribution of surfactant along the interface. As fluid evacuates the thinning neck, surfactant is convected away from that locale. However, the resulting concentration gradient gives rise to diffusion which tries to replenish the neck with surfactant. When surface rheological effects are negligible, regardless of Pe, the dynamics in the vicinity of the finite-time singularity is self-similar and, asymptotically, there is always a transition from a diffusion-dominated scaling regime to a convection-dominated one as the space-time singularity is approached. Theory and simulations are used to show that a highly viscous thread undergoing breakup under conditions in which surface viscous stresses are present gives rise to an exceptional type of dynamics in the vicinity of the neck. In contrast to previous studies of breakup in which there is always a dynamical transition between different scaling regimes as pinch-off nears, the presence of surface viscous stresses results in dynamics that cuts off this universal type of response. It is demonstrated that when $\text{Pe}<{\text{Pe}}_c$, the dynamics is always diffusion-dominated and the rate of thread thinning is exponential in time. If, however, $\text{Pe}>{\text{Pe}}_c$, the dynamics is self-similar and exhibits a power-law dependence on time until pinch-off. The fact that a transition between the two regimes is not possible is also demonstrated.

The erosion and transport of particles by an impinging turbulent jet in air is observed in various situations, such as the cleaning of a surface or during the landing of a spacecraft. The presence of interparticle cohesive forces modifies the erosion threshold beyond which grains are transported. The cohesion also influences the resulting formation and shape of the crater. In this paper we characterize the role of the cohesive forces on the erosion of a flat granular bed by an impinging normal turbulent jet in air. We perform experiments using a cohesion-controlled granular material to finely tune the cohesion between particles while keeping the other properties constant. We investigate the effects of the cohesion on the erosion threshold and show that the results can be rationalized by a cohesive Shields number that accounts for the interparticle cohesion force. Despite the complex nature of a turbulent jet, we can provide a scaling law to correlate the jet erosion threshold, based on the outlet velocity at the nozzle, to a local cohesive Shields number. The presence of cohesion between the grains also modifies the shape of the resulting crater, the transport of grains, and the local erosion process.

Many aquatic microorganisms are able to swim. In natural environments they typically do so in the presence of flows. In recent years it has been shown that the interplay of swimming and flows can give rise to interesting and biologically relevant phenomena, such as accumulation of microorganisms in specific flow regions and local alignment with the flow properties. Here, we consider a mechanical model for elongated microswimmers in a Kolmogorov flow, a prototypic shear flow, both in steady and in turbulent conditions. By means of direct numerical simulations, supported by analytical calculation in a simplified stochastic setting, we find that the alignment of the swimming direction with the local velocity is a general phenomenon. We also explore how the accumulation of microorganisms, typically observed in steady flows, is modified by the presence of unsteady fluctuations.

Spherical approximation is often adopted in modeling planetary and stellar fluid dynamics, which implicitly assumes the rotational flattening effect is small. This simplification, to the leading order of the basic reference state, makes gravity, pressure, chemical, and thermal variables only depend on the radial coordinate. However, a rotating and self-gravitating fluid body is necessarily nonspherical. In order to represent fundamental mechanical equilibrium and thermal state of rapidly rotating planets or stars, where rotational flattening is not to be neglected, we construct a nonspherical, stably stratified Boussinesq fluid model of uniform rotation. Closed-form gravity and temperature formulations of the basic reference state are derived in terms of oblate spheroidal coordinates. The key emphasis of this paper is that the uniformly heated model is mathematically confirmed to be motionless in the corotating frame of reference. Based on this hydrostatic model of rotating stable stratification, we show that although treating the centrifugal terms within a spherical geometry context is convenient, it can lead to incorrect flows. The neglected terms produced by the oblateness are of the same order as the baroclinic terms included, and can indeed cancel them out in some circumstances. This paper proposes a foundation for the analysis of thermally driven flows in nonspherical geometries, which will be carried on in a series of future papers.

In this paper we extend the harmonic resolvent analysis to study the dynamics of subharmonic perturbations about a periodically time-varying base flow. In doing so, we recover an input-output operator that is closely related to the harmonic transfer function introduced in Wereley [Ph.D. thesis, MIT (1991)], and we also elucidate the nature of the cross-frequency interactions between subharmonic flow structures in the proximity of the base flow. We first demonstrate the use of this method on the Rössler system, under conditions for which the dynamics are sensitive to period-doubling perturbations. We then apply it to a forced incompressible axisymmetric jet, and we study how the jet's sensitivity to subharmonic perturbations varies as a function of the Reynolds number. This analysis suggests that as the Reynolds number is increased, the cross-frequency interactions between subharmonic structures with period $2T$ and the $T$-periodic base flow become increasingly important. Remarkably, we also demonstrate that the well-known nonlinear vortex pairing phenomenon is driven by the spatiotemporal structures contained within the first right singular vector (or input mode) of the harmonic resolvent evaluated at the $1/2$-subharmonic of the fundamental frequency. In particular, we show that if the nonlinear flow is forced with an input that is orthogonal to the first right singular vector of the harmonic resolvent, then no pairing will occur.