Motivated by previous work on kinetic energy cascades in the ocean, atmosphere, plasmas, and other fluids, we develop a spatiotemporal spectral transfer tool that can be used to study scales of variability in generalized dynamical systems. In particular, we use generalized time-frequency methods from signal analysis to broaden the applicability of frequency transfers from theoretical to practical fluids applications such as the study of observational data or simulation output. We also show that triad interactions in wavenumber used to study kinetic energy and enstrophy cascades can be generalized to study triad interactions in frequency or wavenumber frequency. We study the effects of sweeping on the locality of frequency transfers and frequency triad interactions to better understand the locality of spatiotemporal frequency transfers. As an illustrative example, we use the spatiotemporal spectral transfer tool to study the results of a simulation of two-dimensional homogeneous isotropic turbulence. This simulated fluid is forced at a well-defined wavenumber and frequency with dissipation occurring at both large and small scales, making this one of the first studies of “modulated turbulence” in two dimensions. Our results show that the spatiotemporal transfers we develop in this paper are robust to potential practical problems such as low sampling rates or nonstationarity in time series of interest. We anticipate that this method will be a useful tool in studying scales of spatiotemporal variability in a wide range of fluids applications as higher resolution observations and simulations of fluids become more widely available.

To investigate the effects of finite arithmetic precision on large-scale direct numerical simulations (DNS) of three-dimensional turbulence using a spectral method, we performed a comparative analysis between two sets of DNS: one using single arithmetic precision (AP) and the other using double AP. Each set simulated the turbulent flows of an incompressible fluid under periodic boundary conditions at Reynolds numbers ($R_{\lambda }$) of 170 and 268, with a resolution defined by $k_{\max }\eta =4$, where $k_{\max }$ denotes the maximum wave number in the DNS and $\eta$ represents the Kolmogorov length scale. Special focus was given to the temporal evolution of the maximum values of local enstrophy ($\mathrm{\Omega }$) and local energy dissipation rate ($\epsilon$), as well as the moments $\langle {\mathrm{\Omega }}^p\rangle$ and $\langle {\epsilon }^p\rangle$. The comparison shows that at $R_{\lambda }=268$, the differences in the time-dependent maximum values between double AP and single AP DNSs are negligible for short duration ($t<1.6T$), where $T$ denotes the eddy turnover time. However, these differences become significant for longer duration ($t>1.6T$). For moments with $p<4$, no notable differences were observed, but significant discrepancies emerged for $p>4$ and $t>1.6T$. In DNSs at $R_{\lambda }=170$, the differences in both maximum values and moments were insignificant throughout the entire simulated time range. The results in the present paper suggest that the difference becomes more significant in DNSs of turbulence at higher Reynolds numbers.

The velocity gradient tensor (VGT) has been extensively employed to describe many fundamental and intrinsic spatial characteristics of turbulence. However, the temporal evolution characteristics following fluid particles are crucial for understanding the dynamics of turbulence. Therefore, we utilize the acceleration gradient tensor (AGT) including both temporal evolution and spatial characteristics of flow as a tool to probe turbulence. Our focus is directed towards the flow characteristics described by three principal invariants of the AGT. It is confirmed that the AGT principal invariants include not only the spatial characteristics described by VGT principal invariants, but also the evolution features such as the temporal evolutions of dilatation, strain rate, and rotation rate. Moreover, the statistical analysis of AGT principal invariants is conducted in compressible channel flows to discern the flow characteristics affected by compressibility. The distributions of normalized AGT principal invariants with different bulk Mach numbers, ${\text{Ma}}_b$, collapse together far from the viscous sublayer but exhibit significant differences in the viscous sublayer. These differences are attributed to the intense temporal evolutions of dilatation, strain rate, and rotation rate at high ${\text{Ma}}_b$, induced by the generation of alternating positive and negative structures (APNSs) near the wall in high-speed flows with strong compressibility. By examining the principal invariants of filtered AGT, it is confirmed that the compressibility effect on flow characteristics near the wall becomes negligible as the flow scale increases. We further prove that AGT is determined by the spatial nonuniformities of the density and the force acting on the fluid particles. Hence, the compelling spatial nonuniformity of the force in the viscous sublayer has been identified as the primary dynamics driving the intense temporal evolutions and the generation of APNSs at high ${\text{Ma}}_b$.

Our experiments on a sphere falling from rest under gravity in Stokes flow show the development of unsteady flow structures and significant history effects. Unlike in steady Stokes flow, we observe a vortex ring forming around the sphere and drifting away, which is captured by our unsteady Stokes theory. Consistently, we observe an algebraic, not exponential, relaxation rate of the sphere to its terminal velocity, validating the Basset-Boussinesq-Oseen equation, which is based on unsteady Stokes theory. As the Reynolds number nears unity, the vortex ring lags behind the sphere, departing from Stokesian theory, though the sphere's algebraic response persists. These findings are critical for particle interactions in the Stokes limit.

Representing the glaciation of mixed-phase clouds in terms of the Wegener–Bergeron–Findeisen process is a challenge for many weather and climate models, which tend to overestimate this process because cloud dynamics and microphysics are not accurately represented. As turbulence is essential for the transport of water vapor from evaporating liquid droplets to ice crystals, we developed a statistical model using established closures to assess the role of small-scale turbulence. The model successfully captures results of direct numerical simulations and we use it to assess the role of small-scale turbulence. We find that small-scale turbulence broadens the droplet-size distribution somewhat, but it does not significantly affect the glaciation time on submeter scales. However, our analysis indicates that turbulence on larger spatial scales is likely to affect ice growth. While the model must be amended to describe larger scales, the present work facilitates a path forward to understanding the role of turbulence in the Wegener–Bergeron–Findeisen process.

Buoyant or dense plumes form in porous media during various environmental applications such as groundwater contamination and subsurface carbon storage. In this study, we investigate the spatiotemporal evolution of dense plumes in porous media as a function of permeability, injection rate, and density difference between the injected and ambient fluids. A series of laboratory experiments are conducted using light attenuation technique, focusing particularly on the evolution of plume length and volume, which in turn accounts for the mixing between the plume and the ambient. Based on the measurement of the aspect ratio of the plumes, we quantify transition time that separates a fully developed late plume from an early developing plume. Further, we propose semiempirical correlations for plume length and volume, which exhibit power-law behavior on time in both the early and late time regimes. The exponents defining the plume growth rate that we obtain here are less than the values presented by several analytical and computational models in the literature, e.g., for the convective dissolution that occurs during an underground carbon sequestration process.

Recent experiments of active particles encapsulated in various soft matter materialize their exciting potential for biomedical applications such as drug delivery. They also provide a means to validate theoretical models proposed in the last decade. The active particles in these experiments can be classified as field-driven or self-propelled. Given these options, it is natural to ponder which of these types is most appropriate for drug delivery. The answer will surely depend on design specification and existing clinical practices. However, one consideration, and the focus of this study, is the energy expenditure of these potential delivery systems. This paper uses Lighthill's swimming efficiency (the ratio of power dissipation generated by a towed rigid particle to the power dissipation generated by a self-propelled swimmer) to differentiate between active particles. As a first approximation, we consider the configuration of a spherical squirmer enclosed in a nondeforming spherical droplet. Letting the squirmer-droplet system move through homogeneous fluids and porous media, we derive exact analytical expressions for the power dissipation of the confined squirmer. To obtain the swimming efficiency of the confined squirmer, we also derive analytical formulas for the drag force on a rigid sphere that is towed at the same speed as the squirmer. Our findings reveal lower energy expenditure for field-driven active particles when the bulk fluid is homogeneous. However, in porous media, the swimming efficiency shows a strong dependence on the droplet to particle size ratio. We also consider a different squirming mechanism (torque driven), and find that it does not influence the squirmer's swimming efficiency. We conclude our analysis with a discussion of the limitations of our approach to differentiate between active particles, and expand on relevant future research directions.

We present a laboratory experiment to model convection in the atmosphere driven by latent heat of condensation (“moist convection”). The system consists of a tank filled with an aqueous solution containing a color indicator that renders the fluid yellow and transparent when acidic and blue and opaque when basic. A sodium lamp positioned above the tank serves as a heat source. Initially, the solution is entirely acidic and stably stratified in temperature. A basic layer is progressively generated at the tank's bottom through water electrolysis, forming an initially stable, blue region. This layer is internally heated by absorbing the heat from the sodium lamp, modeling the latent heat of condensation in the atmosphere. Over time, instabilities develop at the interface between the yellow and blue layers. We investigate the variation of the instability wavelength and complement our observations with a linear stability analysis to elucidate the underlying mechanisms and the process of wavelength selection. Direct numerical simulations are then employed to explore the first steps of the nonlinear regime in connection with experimental observations.

In the context of turbulent dispersal of aerosol pollutants from a source within a ventilated indoor space, the present work addresses the importance of going beyond accurate prediction of the mean ensemble-averaged exposure, by evaluating the expected level of variability in individual realizations. This uncertainty quantification requires a statistical description of the inherent stochastic turbulent dispersal process and the inhomogeneous nature of the indoor flow. We leverage large datasets from turbulence-resolving Euler-Lagrange simulations of aerosol dispersal in varying indoor geometries. The datasets provide time-resolved concentrations of pollutants emitted by a source located anywhere in a room and reaching a sink (susceptible individual) that may be located anywhere else in the same room. The statistical information is used to accurately predict the mean pollutant concentration as a function of source-sink separation distance and associated variability in two different experimental measurements to demonstrate the importance of estimating the uncertainty associated with the predicted average concentration. The average pollutant concentration results are compared with the predictions of three other commonly used models (well-mixed model, near-field/far-field model, and eddy diffusivity model).

We conduct direct numerical simulations of developed turbulence in a periodic cube to investigate the formation mechanism of the power spectral density of the Lagrangian velocity. We compare the power spectral density of the Lagrangian velocity of turbulent flows with different forcing methods and Reynolds numbers. This systematic comparison demonstrates that universal behavior is observed in a narrow high-frequency regime, whereas nonuniversality originating from the forcing method broadly appears in a low-frequency regime. To reveal the formation mechanism of the spectra in terms of the hierarchy of coherent structures in turbulence, we propose a scale-decomposition method for the Lagrangian velocity, which enables us to evaluate the contribution of vortices at different scales. This scale-decomposition analysis directly demonstrates that the largest-scale flows driven by the external force can contaminate the Kolmogorov scaling of the Lagrangian velocity spectra formed by small-scale vortices in the inertial range, thus leading to the narrow Lagrangian inertial range. Furthermore, we provide evidence that this remarkable effect by the largest-scale flows is specific to the Lagrange velocity by demonstrating that the power spectral density of the Eulerian velocity is less sensitive to the forcing method.

We study numerically the dynamical evolution of two different vorticity arrays with an initially uniform magnetic field in two-dimensional incompressible magnetohydrodynamics. We concentrate particularly on the role of the strength of the background magnetic field, which is always assumed weak in the sense that its energy is much less than the kinetic energy of the vortical flows. Within the context of a weak background field, we are able to identify four distinct regimes. When the field is so weak that the back-reaction of the Lorentz force can be ignored (the kinematic regime), classical flux expulsion occurs. As the field strength is increased, the first signs of the dynamical influence of the small-scale field generated is in the disruption of vortex filaments, with flux expulsion still occurring in the vortex cores. A further increase in field strength leads to the regime of vortex disruption, in which the magnetic field is expelled, but is then of sufficient strength to disrupt or destroy the vortices. For yet stronger fields, even the large-scale field can be sufficiently strong to be dynamically active; flux expulsion is then prevented, and the field is dynamically active throughout the evolution. Furthermore, in the case of a row of vortices, we show that the orientation of the background field significantly influences the evolution, especially at higher field strengths.

A bottom Ekman layer is defined by a dynamic balance between the Coriolis force, pressure gradient force, and turbulent transport. We investigate how this balance is modified by bottom topography by analyzing Large Eddy Simulations of separating flow over a two-dimensional ridge. The focus is on two cases that include rotation: one where the exterior flow is driven across the ridge (Cross-Hill with Rotation: “CHR”), and another where the flow is driven along the ridge axis (Along-Hill with Rotation: “AHR”). In both cases, the bottom portion of the boundary layer is dominated by topographic disturbances, including a speedup over the hill and a recirculation in the wake. While these near-bottom modifications are rooted in nonrotating flow dynamics (evidenced in nonrotating simulations presented here), they are impacted by the Coriolis force, leading to a wake jet in the along-ridge flow component of CHR and a wake deficit in the along-ridge component of AHR. These features drive magnifications in the ageostrophic transport that cannot be predicted by the bottom stress. Analysis of the transport budget suggests the wake dynamics are sensitive to the strength of the horizontal divergence of the Reynold's stresses, being stronger in CHR than in AHR. This leads to a large nonlocal influence in CHR, where averaging the ageostrophic transport over a distance that is 3 times the hill's total length leads to a more than $30\%$ increase in magnitude when compared to the flat case. Averaging over the same distance in AHR shows negligible differences to the bulk transport relative to the flat case, as the wake is more confined to the topography vicinity. These results show that the influence of rotation on the speedup and recirculation play an essential role in facilitating topographically induced changes to the bulk transport, with an evident dependence on the strength of flow separation.

The goal of this paper is to propose a theoretical framework to study homogeneous and isotropic turbulence in a viscoelastic fluid, regarded as a perturbation of a Newtonian incompressible fluid, where the fluid relaxation time, or else the Weissenberg number, plays the role of small parameter. We use a Martin-Siggia-Rose framework to obtain a formal expression for the velocity correlation function of the non-Newtonian flow, and we expand this formal expression to linear order in the relaxation time. The coefficients in this expansion are correlation functions of the base Newtonian flow. We do not derive these correlations, instead we replace them by their values according to K41 theory, which could be regarded as an extreme form of renormalization. While substantial work will be necessary to validate the model against numerical and experimental data, preliminary results are encouraging.

We study a two-dimensional isotropic rotating system and obtain both theoretically and numerically a $K^{-2}$ energy spectrum under the rapidly rotating condition ($\text{Ro}\ll 1$), which was initially obtained by Zeman [Phys. Fluids 6, 3221 (1994)] and Zhou [Phys. Fluids 7, 2092 (1995)]. In rotating turbulence, the $K^{-2}$ energy spectrum was proposed under the assumption of isotropy; however, the direction selectivity of rotation breaks isotropy, making this $K^{-2}$ spectrum not easily observable. To fill the gap between theoretical assumptions and realizability, we study the turbulence of inertial waves in an artificial two-dimensional isotropic rotating turbulence system. In the limit of a small Rossby number, we asymptotically derive a nonlinear amplitude equation for inertial waves, which gives the $K^{-2}$ spectrum using a strong turbulence argument. This scaling is justified by numerical simulations of both the amplitude equation and the original system.

This study presents a theoretical model for gravity wave attenuation in a multilayered fluid system comprising fluidized mud, water, an interfacial fluid layer (typically enriched with surface-active compounds including oil), and air, which closely mimics the atmosphere-ocean configuration. A rigorous mathematical analysis initiated from the boundary value problem yields the dispersion relation, which is then solved numerically to obtain the complex-valued wave number, whose imaginary part is the spatial attenuation rate. Parametric studies are conducted to investigate how the attenuation rate responds to changes in wave frequencies, fluid and interfacial properties, and layer depths. Results obtained are compared with published experimental data in select cases, showing good agreement. In addition, the model is applied to reduced systems by setting the densities or depths of select layers to zero. For a water-mud system, maximum attenuation rates occur when the mud layer's depth is comparable to the wavelength and exceeds the boundary layer thickness by 100%–200%. Similarly, in an oil-water-mud system, maximum attenuation rates are obtained when the mud layer's depth exceeds the boundary layer thickness by 10%. Further analysis highlights the significance of air dynamics through the ratio, $\alpha$, defined as the attenuation rate in an air-oil-water-mud system relative to that in a corresponding vacuum-oil-water-mud system. Results reveal that $\alpha \gg 1$ for low-frequency waves in deep water systems, with values reaching up to ${10}^3$ in specific cases. Such high values underscore the critical role of air dynamics in predicting attenuation rates accurately. Moreover, in an air-oil-water-mud system, the maximum value of $\alpha$ and the corresponding frequency at which it occurs exhibit power-law dependencies on water layer depth ($d_2$), scaling as $d_2^{2/3}$ and $d_2^{-1/2}$, respectively. It is inferred that maximum attenuation rates are linked to strong velocity gradients in the fluid layers; they may also be understood by analyzing variations in the group velocity and the penetration depth, which is related to the rotational part of the velocity field.

The microhydrodynamics of particle suspensions in polymeric fluids has a wide range of applications in industry and biology. To discern the dynamics of particles in such systems, it is important to analyze the stress response of the suspension to applied flow fields. While such investigations have been theoretically done for suspensions of rigid spheres in weakly viscoelastic fluids, the effect of nonsphericity of particles on the stress remains relatively unexplored. The interplay between the response of the polymeric fluid and the particle orientation yields rich physics. The viscoelastic torques make the particle inhabit a preferred orientation in a given flow, resulting in time-dependent stresses. In this paper, we determine the average extra stress in a dilute suspension of rigid, non-Brownian spheroids in a second-order fluid subject to shear and extensional flows. We perform this task by examining the flow around a single spheroid in the limit of small Weissenberg number ($\text{Wi}\ll 1$) and perform an ensemble average of the stress tensor over all particle configurations. There are two contributions to the extra stress: one from the force dipole on the particles (stresslet) and another from the fluctuations in the velocity in the bulk fluid (fluid-induced particle stress), the latter of which does not arise in a zero Reynolds number Newtonian fluid. We present results for the $O\left(\varphi \text{Wi}\right)$ corrections to the long-time effective shear viscosity, normal stress coefficients, and extensional viscosities in the suspension in shear, uniaxial extensional, and planar extensional flows, where $\varphi$ is the particle volume fraction. To elucidate the effect of particle shape on the effective viscosity, we repeat this analysis for different aspect ratios ($A_R$) for prolate (needlelike) and oblate (disklike) spheroids.

In this study, we numerically investigate the impact of droplets on elastic plates using a two-phase lattice Boltzmann method, with a particular focus on how vertical surface movements influence the spreading and retraction dynamics of the droplet. The results show that, during the spreading phase, the spreading diameter is smaller on elastic surfaces compared to rigid ones due to the vertical velocity of the surface. A universal linear evolution of the drop spreading is derived for the early stage across both rigid and elastic substrates, accounting for the surface motion by rescaling time, and this relationship is in good agreement with the numerical results. In the retraction phase, unlike the nearly constant retraction speed observed on rigid surfaces, the retraction speed $V_{\mathrm{ret}}$ oscillates with the vibrations of the elastic surface, with the oscillation period remaining relatively consistent. Further analysis reveals that the variation in $V_{\mathrm{ret}}$ is not influenced by the surface's velocity but rather by its acceleration, as additional inertia is introduced during surface acceleration. Based on this understanding, a predictive model for $V_{\mathrm{ret}}$ during droplet impacts on moving surfaces is proposed, which demonstrates strong agreement with the numerical findings.

Is ice always covered by a thin layer of water? This question has been discussed for over 150 years. Here we show that the apparent contact angle of a droplet of water on ice increases steeply with decreasing ice temperature, from around 12 degrees near the melting point, to close to 160 degrees at $-{100}^{\circ }\mathrm{C}$. This indicates that ice is never completely wetted. We quantitatively model the temperature dependence of the apparent contact angle by assuming the droplet's contact line gets pinned due to the crystallization of a thin layer of ice on the cold surface. However close to the melting temperature, where the formation of the ice layer is slowest, surface energy considerations need to be included to explain the nonzero contact angle observed in experiments.

Previous studies on peristalsis, the pumping of fluid along a channel by wavelike displacements of the channel walls, have shown that the elastic properties of the channel and the peristaltic wave shape can influence the flow rate. We model a prescribed wave motion of a single boundary which pumps fluid at small Reynolds number near an elastic half space. We investigate the relationship between flow rate and elastic deformation, considering transverse and longitudinal motion of the driving peristaltic wave. The phase of the periodic solid deformation is found to depend on peristalsis being driven transversely or longitudinally. It is concluded that softness of the elastic material always reduces the net flow rate for transversely driven flows but can increase flow rate for longitudinally driven flows when in narrow confinement. In an applied example, we use measured values of cerebrospinal fluid flow parameters with our model and find agreement with in vivo velocity data, suggesting that the mechanism of perivascular pumping is a plausible driver of cerebrospinal fluid flow in the brain.

Understanding the dynamics of the turbulent velocity gradient tensor (VGT) is essential to gain insights into the Navier-Stokes equations and improve small-scale turbulence modeling. However, characterizing the VGT dynamics conditional on all its relevant invariants in a continuous fashion is extremely difficult. Hence, in this paper, we represent the Lagrangian dynamics using a network where each node represents a unique flow state in the space of the invariants of the VGT. The sign and relative magnitude ranking of the invariants resulting from a Schur decomposition of the VGT determines the discrete flow state vector. Our analysis reveals intriguing features of the resulting network dynamics, such as a grouping of the influential nodes where the eigenvalues of the VGT are real, in the proximity of the right Vieillefosse tail. We relate our complex network approach to the well-established VGT discretization based on the sign of its principal invariants, $Q$ and $R$, and its discriminant, $\mathrm{\Delta }$ into six regions representing very different physical and topological states. We study the shortest paths on the network based on the six regions to which their starting and ending nodes belong. This analysis shows that the typically clockwise path on the $Q\text{−}R$ diagram arises in a varying manner: sometimes region transitions are based on multiple pathways of more equal probability, while others are focused on specific nodes that are pivotal in these interregion dynamics. Comparisons to an enhanced Gaussian closure model show that the model captures the out-degree distribution for the nodes well. However, it uses a greater number of nodes to transition from real to complex eigenvalues and is far less efficient at generating clockwise motions that are counter to the restricted Euler dynamics.