# Browsing by Subject "Fluid mechanics"

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Item Embargo Acoustic-based automated manipulation of particles for biological applications(2023) Zhu, HaodongAcoustic-based techniques have emerged as a promising avenue for the precise manipulation of particles, combining the disciplines of acoustics, physics, and biotechnology. Utilizing sound waves, this method allows for the gentle, non-invasive movement and positioning of particles, from minute biological entities to larger synthetic materials. Such automated manipulation harnesses the intricacies of acoustic radiation forces and streaming, offering advantages in terms of scalability, precision, and integration into various systems. As biotechnological demands grow, the potential of acoustic-based platforms to influence fields like drug delivery, diagnostics, and cellular research becomes increasingly evident. This defense delves into the development of two platforms utilizing automated acoustic technologies for particle manipulation aimed at advancing biological applications. The first part showcases a digital piezoelectric-based platform, adept at dynamic particle manipulation through the modulation of acoustic streaming, enhanced with surrounding barrier structures. We built a programmable droplet-handling platform to demonstrate the basic functions of planar-omnidirectional droplet transport, merging droplets, and in situ mixing via a sequential cascade of biochemical reactions. The ensuing part unveils a novel platform tailored for the meticulous long-term observation of single cell physical attributes, founded on 2D acoustic patterning of single cell array and automatic phase modulation. By adaptively segmenting and fitting the movement, we are able to monitor the density, compressibility and size fluctuation of the sample at the same time. These innovations have the potential to revolutionize biological endeavors, notably in large-scale drug screening and the proactive surveillance of cellular responses to distinct environmental stimulations over extended periods.

Item Open Access An Investigation into the Multiscale Nature of Turbulence and its Effect on Particle Transport(2022) Tom, JosinWe study the effect of the multiscale properties of turbulence on particle transport, specifically looking at the physical mechanisms by which different turbulent flow scales impact the settling speeds of particles in turbulent flows. The average settling speed of small heavy particles in turbulent flows is important for many environmental problems such as water droplets in clouds and atmospheric aerosols. The traditional explanation for enhanced particle settling speeds in turbulence for a one-way coupled (1WC) system is the preferential sweeping mechanism proposed by Maxey (1987, J. Fluid Mech.), which depends on the preferential sampling of the fluid velocity gradient field by the inertial particles. However, Maxey's analysis does not shed light on role of different turbulent flow scales contributing to the enhanced settling, partly since the theoretical analysis was restricted to particles with weak inertia.

In the first part of the work, we develop a new theoretical result, valid for particles of arbitrary inertia, that reveals the multiscale nature of the preferential sweeping mechanism. In particular, the analysis shows how the range of scales at which the preferential sweeping mechanism operates depends on particle inertia. This analysis is complemented by results from Direct Numerical Simulations (DNS) where we examine the role of different flow scales on the particle settling speeds by coarse-graining (filtering) the underlying flow. The results explain the dependence of the particle settling speeds on Reynolds number and show how the saturation of this dependence at sufficiently large Reynolds number depends upon particle inertia. We also explore how particles preferentially sample the fluid velocity gradients at various scales and show that while rapidly settling particles do not preferentially sample the fluid velocity gradients, they do preferentially sample the fluid velocity gradients coarse-grained at scales outside of the dissipation range.

Inspired by our finding that the effectiveness of the preferential sweeping mechanism depends on how particles interact with the strain and vorticity fields at different scales, we next shed light on the multiscale dynamics of turbulence by exploring the properties of the turbulent velocity gradients at different scales. We do this by analyzing the evolution equations for the filtered velocity gradient tensor (FVGT) in the strain-rate eigenframe. However, the pressure Hessian and viscous stress are unclosed in this frame of reference, requiring in-depth modelling. Using data from DNS of the forced Navier-Stokes equation, we consider the relative importance of local and non-local terms in the FVGT eigenframe equations across the scales using statistical analysis. We show that the anisotropic pressure Hessian (which is one of the unclosed terms) exhibits highly non-linear behavior at low values of normalized local gradients, with important modeling implications. We derive a generalization of the classical Lumley triangle that allows us to show that the pressure Hessian has a preference for two-component axisymmetric configurations at small scales, with a transition to a more isotropic state at larger scales. We also show that the current models fail to capture a number of subtle features observed in our results and provide useful guidelines for improving Lagrangian models of the FVGT.

In the final part of the work, we look at how two-way coupling (2WC) modifies the multiscale preferential sweeping mechanism. We comment on the the applicability of the theoretical analysis developed in the first part of the work for 2WC flows. Monchaux & Dejoan (2017, Phys. Rev. Fluids) showed using DNS that while for low particle loading the effect of 2WC on the global flow statistics is weak, 2WC enables the particles to drag the fluid in their vicinity down with them, significantly enhancing their settling, and they argued that two-way coupling suppresses the preferential sweeping mechanism. We explore this further by considering the impact of 2WC on the contribution made by eddies of different sizes on the particle settling. In agreement with Monchaux & Dejoan, we show that even for low loading, 2WC strongly enhances particle settling. However, contrary to their study, we show that preferential sweeping remains important in 2WC flows. In particular, for both 1WC and 2WC flows, the settling enhancement due to turbulence is dominated by contributions from particles in straining regions of the flow, but for the 2WC case, the particles also drag the fluid down with them, leading to an enhancement of their settling compared to the 1WC case. Overall, the novel results presented here not only augments the current understanding of the different physical mechanisms in producing enhanced settling speeds from a fundamental physics perspective, but can also be used to improve predictive capabilities in large-scale atmospheric modeling.

Item Open Access Analyzing Hydrodynamic Properties of the North Atlantic Right Whales with Computer Solutions(2020) Wu, Chen-YiAnimals experience hydrodynamic forces (lift, drag, and side) and moments (pitching, yawing, and rolling) as a result of motion in an aqueous medium. Under selective pressure, most cetaceans, including porpoises, dolphins, and whales, developed a streamlined body shape and modified limbs, which delay the separation of flow, create lower drag when they swim, and therefore decrease their locomotor cost. In order to calculate the locomotor cost and propulsive efficiency of cetaceans, accurate estimates of drag on marine animals are required. However, extra momentum imparted into the fluid from lift and side forces as well as pitching, rolling, and yawing moments (here, the parasitic loads) results in extra drag force on the animal. Therefore, in addition to streaming and delaying flow separation, animals must also minimize excess fluid momentum resulting from parasitic loads. Given the endangered status of the North Atlantic right whale (Eubalaena glacialis; hereafter NARW), analyzing the hydrodynamic characteristics of the NARWs was the focus of this work. Additionally, previous studies showed that body shape of NARWs changes with life stages, reproduction status, nutritive conditions or prey abundance, and the effects of entanglement in fishing gear. Therefore, in this study, computational fluid dynamics (CFD) analysis was performed on multiple 10 m three-dimensional NARW models with different body shapes (e.g., normal condition, emaciated, and pregnant) to measure baseline measurements of flow regimes and hydrodynamic loads on the animal. Swimming speeds covering known right whale speed range (0.125 m/s to 8 m/s) were simulated in most scenarios. In addition to the hydrodynamic effects of different body shapes, drag was also considered a function of parasitic loads. The NARW models were embedded with bone segments that allowed one to manipulate the body pose of the model via adjusting the flippers or the spine of the animal before measuring hydrodynamic drag. By doing so, momentum from parasitic loads was expected to be eliminated. CFD simulations revealed that drag on NARWs is dictated by its irregular outline and that the drag coefficient (0.0071-0.0059; or dimensionless drag) of on NARWs is approximately twice that of many previous estimates for large cetaceans. It was also found that pregnant NARW model encounters the lowest drag coefficient due to delayed flow separation resulting from enlarged abdomen, whereas the emaciated NARW model experiences the highest drag coefficient possibly due to the concavity at the post-nuchal region. These results suggested that drag on NARWs and their thrust power requirements were indeed affected by its body shape but the differences between the three NARW models tested were small. Lastly, minimum drag, which corresponds to the elimination of the parasitic loads, can be obtained by adjusting the pose of the animal. Thus, minimum drag occurs at the neutral trim pose. For the static, normo-nourished NARW model, simulations revealed that by changing the angle of attack of the flippers by 4.03° (relative to the free-stream flow) and pitching the spine downward by 5° while maintaining fluke angle, the drag was lowered by approximately 11% across the flow speeds tested. This drag reduction was relative to the drag study conducted on the same animal model but without body pose adjustments. Together the studies included in the present work explored and highlighted the capability of numerical methods in investigating the hydrodynamics and energetics of cetaceans. Future studies should address how computer solutions can be used to solve problems from a wider aspect. For instance, extra parasitic loads caused by attached gear as well as possible injuries due to the encounter with fishing gear should also be considered while evaluating the energy budget of the North Atlantic right whales.

Item Open Access Capillary Water Behavior During Evaporation of Granular Media(2018) Yang, ShuIntense studies on evaporation in fluid systems have been conducted over the past decades, mainly on sessile droplets which are the simplest bodies to simulate. In this thesis, the evaporation process of a water bridge between two spherical grains is studied. A multi-physics approach is adopted focusing on the evaporation dynamics, the flow motion in liquid, the pressure distribution inside the water bridge, and the vapor behavior combined with the movement of free interface.

Item Embargo Digital Hydraulics Simulation in Mathematica on Sudden Expansion Flows(2023) Frechette, AugustIn this work, we offer readers the ability to numerically simulate flow through a sudden expansion themselves. We choose to study the sudden expansion due to its prevalence in engineered and natural water distribution networks (i.e., pipes and rivers, respectively). The simulation is written in the Wolfram Language, also known as Mathematica. The symbolic nature of this programming language enables readers to implement physical theory directly, resulting in a highly readable numerical flow solver; a stark contrast with commonplace commercial flow solvers, which operate like “black box” technologies, and low-level programming languages, which require an advanced level of syntax knowledge and programming proficiency. Upon completion of this laboratory exercise, users should be able to: (i) describe the main principles underpinning the numerical simulation of non-linear models, (ii) apply numerical models to investigate the accuracy of simplified analytical models, (iii) demonstrate a beginner-level understanding of Mathematica and, more broadly, symbolic coding environments, (ii) and most generally, (iv) understand the proper context for physical and numerical experimentation. The novelty of this work is attributed to the fact that no such simulation tool is detailed and provided in the literature for readers to utilize and alter at their discretion.

This work was developed and undertaken in collaboration with my co-authors, Dr. Anil Ganti (A.G.), and Dr. Zbigniew Kabala (Z.J.K), my master’s advisor. Author contributions are as follows: conceptualization, Z.J.K.; methodology, A.H.F, A.G. and Z.J.K.; software, A.H.F and A.G.; validation, A.H.F, A.G. and Z.J.K.; formal analysis, A.H.F; investigation, A.H.F, A.G. and Z.J.K.; resources, Z.J.K; data curation, A.H.F, A.G. and Z.J.K.; writing—original draft preparation, A.H.F and Z.J.K.; writing—review and editing, A.H.F, A.G. and Z.J.K.; visualization, A.H.F.; supervision, Z.J.K.; project administration, A.H.F and Z.J.K.

Partial funding for this project has been received from Duke University Undergraduate Program Enhancement Fund (UPEF) grant 399-000226.

Item Open Access Digital Microfluidics for the Detection of Inorganic Ions in Aerosols(2018) Huang, ShuquanThe quantitative measurement of inorganic ions in the atmosphere is an important aspect in environmental science. The three most important inorganic ions are sulfate, nitrate and ammonium, which are the most abundant components of atmospheric pollutants and have a significant impact on rainfall, atmospheric visibility and human health. To accurately and quickly measure the distribution of inorganic ions in the vertical and horizontal directions of the atmosphere, a compact and automatic real-time detection system is in need.

The research performed in this study is aimed at developing the science and technology for an aerosol detection system that combines digital microfluidics technology, aerosol impaction and chemical detection on the same chip. The system will be smaller and faster with respect to current aerosol analyzing instruments. The chip in this study performs the integrated functions of aerosol collection, extraction, and quantitative detection in real-time, unlike current benchtop methods that require operator handling and laboratory equipment. All functions are realized in dedicated sections on a digital microfluidic platform.

This thesis will present the design and test of individual components of the aforementioned functions. The digital microfluidics chip design includes transparent top and bottom plates for light absorbance measurement. The droplets are dispensed, transported and mixed on chip with other droplets by activating electrodes individually with a 50V AC sine voltage.

In Chapters 3 and 4, the issues involving droplet transportation are addressed, including droplet movement between the air and silicone oil media and droplet transport across the aerosol impaction area. Next, an aerosol impactor and a chip-to-world chamber are demonstrated and tested with lab generated sulfate aerosol. The collected aerosol showed a clear pattern on the impaction plate, and the collection efficiency inside the chip was 96%.

In Chapter 5, the development of colorimetric methods are described as well as experimental testing for inorganic ion detection. Three well-known tests for detecting sulfate, nitrate and ammonium were first adjusted to adapt to on-chip measurement conditions, the adjustments including the choices of solvent, concentration ranges and mixing ratios. The particle measurement results using a conventional spectrometer were compared with on-chip measurements in terms of absorbance range, limit of detection, sensitivity (based on the coefficient of determination and the slope of the linear regression) and signal-to- noise ratio (presented with standard deviation/average of absorbance measurements).

The thin oil film between the droplet and the top/bottom plate, which is naturally formed, plays an important role in lubrication and reduces contact angle hysteresis. However, these oil films are not always uniform in thickness. During the absorbance measurement tests, varied sizes of oil lenses were observed at the oil/top plate interface, and the size and position of the oil lenses randomly changed when a droplet moved between electrodes. The absorbance measured in the normal direction to the chip’s surface was affected by these oil lenses and, thus, not stable for multiple measurements of the same droplet or for different droplets. To solve this problem, optical fibers were introduced horizontally inside the chip, and measurements taken in this direction proved to produce stable results.

Prototypes of the chip have been fabricated, and the impaction and on-chip colorimetric tests for sulfate and ammonium were successful. Although this study was designed to build the fundamentals of a novel detection system of inorganic ions in aerosol, the potential use of the designed system is not limited to atmospheric studies. Applications can extend to testing the quality of drinking water, detection of nitroaromatic explosives or other experiments based on colorimetry.

Item Open Access Flexural Wave Based Acoustofluidic Devices(2020) Bachman, HunterMicrofluidic technologies, and the subset of devices that integrate acoustics into their designs (known as acoustofluidic devices), present great potential for solving the challenges of the future. One specific subset of these technologies, termed sharp-edge based acoustofluidics, has shown promise in a variety applications; specifically, previous work has explored the use of this technology in applications such as fluid pumping and mixing, cell stimulation, and bio-sample preparation. However, even though there are a vast number of applications that sharp-edge based acoustofluidics have been applied to, there are several shortcomings that need to be addressed.

First, and perhaps most critically, very little is known about the fundamental mechanism of this platform’s operation. The search for novel applications has left a gap in the knowledge base for understanding how these devices work on a fundamental level; gaining a better understanding of how the technology works may open the door to finding new and previously unimagined applications. Second, although not a problem that is specifically limited to sharp-edge based acoustofluidic devices, the technology suffers from serious limitations in real world applicability. That is, even though these devices have advantages over traditional techniques, including speed, cost, and ease of use, they are unable to be taken advantage of. For this reason, there is a critical need to demonstrate a viable pathway to real-world usage.

In an attempt to tackle these shortcomings, we begin our research by investigating the vibrational profile generated within a sharp-edge mixer. Throughout this exploration we uncover that the mechanism behind the technology’s success is relatively low frequency flexural waves which have wavelengths commensurate with the overall dimensions of the technology. This is in contrast to the previous belief that waves with lengths many times larger than the device itself were dominating; as a result, we developed and explored a novel platform for particle manipulation based on wave interference (not unlike high frequency based acoustofluidic platforms). This technology offers a new technique for interacting with micro particles and cells in an open fluid chamber. In order to improve the technology’s adoptability, we also developed and characterized two unique and portable control platforms towards eventual point-of-care (POC) use.

Altogether, this work serves to further the knowledge and relevance of sharp-edge based technology. It is our hope that this work can serve as a starting point for future explorations into novel platforms which make use of the small wavelength vibrations achievable with this low cost setup. Additionally, we hope that this work may motivate the broader field to transition their technology into equally accessible platforms, such that the microfluidics community as a whole can bring their useful technology to practical applications.

Item Open Access Models To Derive the Resonant Frequency of a Liquid in a Rectangular Tank With a Curved Bottom(2021) Garcia, Alejandro DanielThis thesis investigates the resonant frequency of a partially-filled rectangular tank of water with a curved bottom that is subject to a horizontal harmonic excitation. The primary goal was to find a model that can accurately find the resonant frequency to study the change in the natural frequency when the parameters of the curved base and system were changed. The EOM model, the h ̅ model, and the ω ̅_n model were derived all from the same linear assumptions and approximation for the velocity potential. Frequency sweeps were done for several curved base systems and compared to each of the models’ predictions. It was found that the h ̅ and ω ̅_n models both agreed well with the data generally, while the EOM model did not. An additional investigation was done on this system to understand the presence of nonlinearities and damping and their significance to the problem. It was found that while several nonlinearities exist like additional harmonic frequency content and surface tension, they are not significant in determining the resonant frequency. Furthermore, the accuracy in the h ̅ and ω ̅_n models show that the linear assumptions and simplifications made for the velocity potential equation were feasible to a degree. Despite this, it is clear that this approximation of the velocity potential needs further work as the EOM model utilizes it fully and is inaccurate.

Item Open Access On Improving the Predictable Accuracy of Reduced-order Models for Fluid Flows(2020) Lee, Michael WilliamThe proper orthogonal decomposition (POD) is a classic method to construct empirical, linear modal bases which are optimal in a mean L2 sense. A subset of these modes can form the basis of a dynamical reduced-order model (ROM) of a physical system, including nonlinear, chaotic systems like fluid flows. While these POD-based ROMs can accurately simulate complex fluid dynamics, a priori model accuracy and stability estimates are unreliable. The work presented in this dissertation focuses on improving the predictability and accuracy of POD-based fluid ROMs. This is accomplished by ensuring several kinematically significant flow characteristics -- both at large scales and small -- are satisfied within the truncated bases. Several new methods of constructing and employing modal bases within this context are developed and tested. Reduced-order models of periodic flows are shown to be predictably accurate with high confidence; the predictable accuracy of quasi-periodic and chaotic fluid flow ROMs is increased significantly relative to existing approaches.

Item Embargo Pore-Scale Flow Mechanisms and the Hydrodynamic Porosity of Porous Media in Surface Water Treatment and Groundwater Remediation(2023) Frechette, AugustAs climate change and growing demand exacerbate water scarcity, it will become more imperative than ever to remediate our natural resources and treat our waste streams. This is especially true if we are to successfully provide clean water for all and ensure the future of endangered species and habitats. Thus, we look to surface water treatment technologies (e.g., granular media and filtration membranes) and groundwater remediation strategies (e.g., the vertical circulation well, rapidly pulsed pump and treat, and bioremediation) to add to our freshwater stores and reduce environmental pollution.

Complicating the matter is the fact that both surface water treatment and groundwater remediation are reliant, to varying degrees, on flow through porous media. Even without the added complexities of multiphase flows, immiscible fluids, and the time-dependent processes associated with chemical reactions and biofouling, characterizing flow through porous media, properly, is a cumbersome and arduous task. Heterogeneities in the morphology of the medium range from the pore scale, to, in the case of groundwater flows, meters. Resulting is a random distribution of the shape, size, and connectivity of the pore space. To quantify flow through porous media, researchers are forced to either make a set of simplifying assumptions, some more appropriate than others, or more recently, use black-box machine learning models that have little basis in the physicality of the flow. In this work, we choose to focus on one of the standard assumptions researchers make when calculating the pore-scale velocity (i.e., the supposed “static” nature of flow porosity). In relaxing this assumption, we provide a paradigm shift in the modeling of flows through porous media. We apply our theory to flow through and along the walls of microporous membranes, granular media, and aquifer substrates.

We choose to study pore-scale flow velocity because it is an essential parameter in determining transport through porous media, but it is often miscalculated. Researchers use a static porosity value to relate volumetric or superficial velocities to pore-scale flow velocities. We know this modeling assumption to be an oversimplification. The porosity conducive to flow, what we define as hydrodynamic porosity, exhibits a quantifiable dependence on Reynolds number (i.e., pore-scale flow velocity) in the laminar flow regime. This fact remains largely unacknowledged in the literature. In this work, we quantify the dependence of hydrodynamic porosity on Reynolds number via numerical flow simulation at the pore scale. We demonstrate that, for the tested flow geometries, hydrodynamic porosity decreases by as much as 42% over the laminar flow regime. Moreover, hydrodynamic porosity exhibits an exponential dependence on Reynolds number. The fit quality is effectively perfect, with a coefficient of determination of approximately 1 for each set of simulation data. We then demonstrate the applicability of this model by validating a high fit quality for a range of rectangular and non-rectangular cavity geometries. Finally, we show that this exponential dependence can be easily solved for pore-scale flow velocity using only a few Picard iterations, even with an initial guess that is over 10 orders of magnitude off. Not only is this relationship a more accurate definition of pore-scale flow velocity, but it is also a necessary modeling improvement that can be easily implemented.

In the chapters that follow our introduction of hydrodynamic porosity, we apply the concept to subsurface flow modeling for groundwater remediation via the vertical circulation well and flows over patterned membrane surfaces for surface water treatment – supposing that a hydrodynamic porosity parameter could be defined for the surface pattern of a membrane and then correlated to the rate of particle deposition (and therefore fouling) at the membrane surface.

In the future, we aim to explore the applicability of the hydrodynamic porosity model to microporous membrane wall flows. Although the characteristic length scale of the membrane wall is admittedly much smaller than the characteristic length scale of granular media, microporous membranes, like granular media, have dead-end pores. Thus, it remains necessary to determine the effect of these dead-end pore volumes on membrane wall flows. Preliminary experimental data previously collected from a hollow-fiber ultrafiltration membrane will be used to verify our numerical results.

Following our study of steady flows, we pivot to the analysis of rapidly pulsed flows and the mixing mechanisms these flows induce at the pore scale (i.e., the deep sweep and vortex ejection) in cavities and other effectively immobile zones. These mechanisms have been shown to significantly reduce contaminant recovery time in media with significant immobile zone volume. This finding suggests substantial cost-savings for treatment and remediation methods that utilize rapidly pulsed flows.

Regarding groundwater remediation, we estimate that the cost savings from utilizing rapidly pulsed flows could be on the order of magnitude of 100 billion USD. But this calculation assumes that we can remediate the entirety of a contaminated groundwater matrix with the mixing mechanisms induced by rapidly pulsed pump-and-treat. In application, induced oscillations will only reach a small volume of the flow field before dissipating to a negligible amplitude. Equally important, these oscillations will only induce a deep sweep or vortex ejection if the mean pore-scale flow velocity is above a Reynolds number of 0.1. Following, we use our model of hydrodynamic porosity to determine the magnitude of the volume we expect to benefit from rapidly pulsed pumping in a vertical circulation well.

Finally, given the similarity in characteristic length scale, we liken flow in the dead-end pore space of groundwater matrices, to flow past the channels in patterned membrane surfaces. We find that for the studied surface pattern, the vortex ejection and deep sweep are still present in highly laminar flows (i.e., a Reynolds number of 1600 for pipe flows). We hypothesize that these mechanisms can prevent particle deposition at the membrane surface, and when used as a cleaning mechanism, can remove loose deposits that would otherwise adhere to the membrane surface. It is also likely that these mechanisms would speed up the regeneration of fouled granular media used to remove suspended solids, microorganisms, and organics (i.e., sand and granulated activated carbon) from wastewater.

Item Open Access Random Splitting of Fluid Models: Ergodicity, Convergence, and Chaos(2022) Melikechi, Omar EmlenIn this thesis we study random splitting and apply our results to random splittings of fluid models. Random splitting is loosely defined as follows. Consider the differential equation $\dot{x}=V(x)$ where $\dot{x}$ is a time derivative and the vector field $V$ on $\mathbb{R}^D$ splits as the sum $V=\sum_{j=1}^n V_j$. In traditional operator splitting one approximates solutions of $\dot{x}=V(x)$ by composing solutions of $\dot{x}=V_j(x)$ over (typically small) deterministic time steps. Here we take these times to be independent and identically distributed random variables. This turns the aforementioned compositions into Markov chains, which we call \textit{random splittings of $V$} or simply \textit{random splittings}. We prove under relatively mild conditions that these random splittings possess a unique invariant measure (ergodicity), that their trajectories converge on average and almost surely to trajectories of the original system $\dot{x}=V(x)$ (convergence), and that, in certain cases, their top Lyapunov exponent is positive (chaos). After proving these general results, we construct random splittings of four fluid models: the conservative Lorenz-96 and Lorenz-96 equations, and Galerkin approximations of the 2d Euler and 2d Navier-Stokes equations on the torus. We prove all these random splittings are ergodic and converge to their deterministic counterparts in a certain sense, and, for conservative Lorenz-96 and 2d Euler, that their top Lyapunov exponent is positive.

Item Open Access Statistical Learning of Particle Dispersion in Turbulence and Modeling Turbulence via Deep Learning Techniques(2021) Momenifar, RezaTurbulence is a complex dynamical system that is strongly high-dimensional, non-linear, non-local and chaotic with a broad range of interacting scales that vary over space and time. It is a common characteristic of fluid flows and appears in a wide range of applications, both in nature and industry. Moreover, many of these flows contain suspended particles. Motivated by this, the research presented here aims at (i) studying particle motion in turbulence and (ii) modeling turbulent flows using modern machine learning techniques.

In the first research objective, we conduct a parametric study using numerical experiments (direct numerical simulations) to examine accelerations, velocities and clustering of small inertial settling particles in statistically stationary isotropic turbulent flow under different values of the system control parameters (Taylor Reynolds number $Re_\lambda$, particle Stokes number $St$ and settling velocity $Sv$). To accomplish our research goals, we leveraged a wide variety of tools from applied mathematics, statistical physics and computer science such as constructing the probability density function (PDF) of quantities of interest, radial distributionfunction (RDF), and three-dimensional Vorono\text{\"i} analysis. Findings of this study have already been published in two journal papers (PhysRevFluids.4.054301 and PhysRevFluids.5.034306), both of which received editors' suggestion awards. In the following paragraphs, some of the important results are highlighted.

The results for the probability density function (PDF) of the particle relative velocities show that even when the particles are settling very fast, turbulence continues to play a key role in their vertical relative velocities, and increasingly so as $Re_\lambda$ is increased. Thisoccurs because although the settling velocity may be much larger than typical velocities of the turbulence, due to intermittency, there are significant regions of the flow where the contribution to the particle motion from turbulence is of the same order as that from gravitational settling.

In agreement with previous results using global measures of particle clustering, such as the RDF, we find that for small Vorono\text{\"i} volumes (corresponding to the most clustered particles), the behavior is strongly dependent upon $St$ and $Sv$, but only weakly dependent upon $Re_\lambda$, unless $St>1$. However, larger Vorono\text{\"i} volumes (void regions) exhibit a much stronger dependence on $Re_\lambda$, even when $St\leq 1$, and we show that this, rather than the behavior at small volumes, is the cause of the sensitivity of the standard deviation of the Vorono\text{\"i} volumes that has been previously reported. We also show that the largest contribution to the particle settling velocities is associated with increasingly larger Vorono\text{\"i} volumes as $Sv$ is increased.

Our local analysis of the acceleration statistics of settling inertial particles shows that clustered particles experience a net acceleration in the direction of gravity, while particles in void regions experience the opposite. The particle acceleration variance, however, is a convex function of the Vorono\text{\"i} volumes, with or without gravity, which seems to indicate a non-trivial relationship between the Vorono\text{\"i} volumes and the sizes of the turbulent flow scales. Results for the variance of the fluid acceleration at the inertial particle positions are of the order of the square of the Kolmogorov acceleration and depend only weakly on Vorono\text{\"i} volumes. These results call into question the ``sweep-stick'' mechanism for particle clustering in turbulence which would lead one to expect that clustered particles reside in regions where the fluid acceleration is zero (or at least very small).

In the second research objective, we propose two cutting-edge, data-driven, deep learning simulation frameworks, with the capability of embedding physical constraints corresponding to properties of three-dimensional turbulence. The first framework aims to reduce the dimensionality of data resulting from large-scale turbulent flow simulations (static mapping), while the second framework is designed to emulate the spatio-temporal dynamics of a three-dimensional turbulent flow (dynamic mapping).

In the static framework, we apply a physics-informed Deep Learning technique based on vector quantization to generate a discrete, low-dimensional representation of data from simulations of three-dimensional turbulent flows. The deep learning framework is composed of convolutional layers and incorporates physical constraints on the flow, such as preserving incompressibility and global statistical characteristics of the velocity gradients.A detailed analysis of the performance of this lossy data compression scheme, with evaluations based on multiple sets of data having different characteristics to that of the training data, show that this framework can faithfully reproduce the statistics of the flow, except at the very smallest scales, while offering 85 times compression. %Compared to the recent study of Glaws. et. al. (Physical Review Fluids, 5(11):114602, 2020), which was based on a conventional autoencoder (where compression is performed in a continuous space), our model improves the CR by more than $30$ percent, and reduces the MSE by an order of magnitude. Our compression model is an attractive solution for situations where fast, high quality and low-overhead encoding and decoding of large data are required. Our proposed framework for dynamic mapping consists of two deep learning models, one for dimension reduction and the other for sequence learning. In the model, we first generate a low-dimensional representation of the velocity data and then pass it to a sequence prediction network that learns the spatio-temporal correlations of the underlying data. For the sequence forecasting, the idea of Transformer architecture is used and its performance compared against a standard Recurrent Network, Convolutional LSTM. These architectures are designed to perform a sequence to sequence multi-class classification task, which is attractive for modeling turbulence. The diagnostic tests show that our Transformer based framework can perform short-term predictions that retain important characteristics of large and inertial scales of flow across all the predicted snapshots.

Item Open Access Surface Energy Powered Processes upon Drop Coalescence(2015) Liu, FangjieSurface energy-powered motion is useful for a variety of autonomous functions such as passive cooling and self-cleaning, where independence from external forces is highly desirable. Drop coalescence offers a convenient process to release surface energy, which can be harvested to power self-propelled fluid motion.

On superhydrophobic surfaces, out-of-plane jumping motion spontaneously results from drop coalescence. However, less than 4\% of the released surface energy is converted to useful kinetic energy giving rise to the jumping motion. Using three-dimensional interfacial flow simulations that are experimentally validated, we elucidate the mechanism of low energy conversion efficiency. The non-wetting substrate interferes with the expanding liquid bridge between the coalescing drops at a relatively late stage, forcing a small fraction of the merged drop to "bounce" back from the non-wetting substrate. The substrate breaks the symmetry of surface energy release, leading to self-propelled jumping that is perpendicular to the solid substrate. The intercepting substrate imparts a relatively small translational momentum on the overall merged drop, giving rise to a small energy conversion efficiency.

This mechanistic understanding has provided guidance on how to increase the energy conversion efficiency by changing the geometry of the intercepting solid surface, e.g. to a pillared substrate which has additional intercepting planes, or to a cylindrical fiber which interferes with the coalescence process at a much earlier stage. These topographical changes have already led to a 10-fold increase in energy conversion efficiency. The directional control of surface energy-powered motion is achieved by breaking the symmetry of oscillations induced by drop coalescence, such as by adding additional intercepting planes on pillared substrates. The work has applications ranging from self-sustained dropwise condensers, drop coalescers to ballistospore discharge in some fungi species in nature.

The ballistospore discharge process is powered by surface energy released from the coalescence between a spherical Buller's drop and an adaxial drop on the spore. The disturbance to the adaxial drop from coalescing Buller's drop results in the capillary-inertial oscillations of the liquid system. The oscillations redirect the mass and momentum transfer and yields a tensile force along the adaxial direction with negligible momentums in other directions, ensuring the preferable launching along the adaxial direction. The findings offer insights for applications of biomimicry involving self-propelled jumping with payloads which takes advantage of the high power density of the process.

Item Open Access The Lid-Driven Cavity's Many Bifurcations - A Study of How and Where They Occur(2017) Lee, MichaelComputational simulations of a two-dimensional incompressible regularized lid-driven cavity were performed and analyzed to identify the dynamic behavior of the flow through multiple bifurcations which ultimately result in chaotic flow. Pseudo-spectral numerical simulations were performed at Reynolds numbers from 1,000 to 25,000. Traditional as well as novel methods were implemented to characterize the system's behavior. The first critical Reynolds number, near 10,250, is found in agreement with existing literature. An additional bifurcation is observed near a Reynolds number of 15,500. The largest Lyapunov exponent was studied as a potential perspective on chaos characterization but its accurate computation was found to be prohibitive. Phase space and power spectrum analyses yielded comparable conclusions about the flow's progression to chaos. The flow's transition from quasi-periodicity to chaos between Reynolds numbers of 18,000 and 23,000 was observed to be gradual and of the form of a toroidal bifurcation. The concepts of frequency shredding and power capacity are introduced which, paired with an existing understanding of frequency entrainment, can help explain the system's progression through quasi-periodicity to chaos.

Item Open Access Understanding and predicting the dynamics of scalar turbulence using multiscale analysis, computational simulations, and stochastic models(2023) Zhang, XiaolongWe investigate the dynamics of turbulent flows and scalar fields based on multiscale analysis, numerical simulations, and modeling. Specifically, we study the fundamental mechanisms of multiscale energy transfers in stratified turbulence where both the turbulent fluid flow and scalar field are present and exchanging energies (i.e., kinetic and potential energies). We also have developed a Lagrangian model which shows great capabilities for predicting the important dynamics of passive scalars in isotropic turbulence. Further evaluations and analysis of the scalar gradient diffusion term (which is approximated by the Lagrangian closure model) are also performed based on direct numerical simulation (DNS) data at higher Reynolds numbers $Re$, to potentially improve the model capability for higher $Re$.

In the first part of the work, we analyze the budgets of turbulent kinetic energy (TKE) and turbulent potential energy (TPE) at different scales $\ell$ in sheared, stably stratified turbulence using a filtering approach. We consider the competing effects in the flow along with the physical mechanisms governing the energy fluxes between scales. The theoretical work of our energy budget analysis is used to analyze data from direct numerical simulation (DNS) at buoyancy Reynolds number $Re_b=O(100)$. Various quantities in the energy budget equations are evaluated based on DNS data of SSST, with detailed discussions on both the mean-field behavior of the flow, as well as fluctuations about this mean-field state. Importantly, it is shown that the TKE and TPE fluxes between scales are both downscale on average and their instantaneous values are positively correlated, but not strongly. The relative weak correlation occurs mainly due to the different physical mechanisms that govern the TKE and TPE fluxes. Moreover, the contribution to these fluxes arising from the sub-grid fields (i.e., small scales) are shown to be significant, in addition to the filtered scale contributions associated with the processes of strain-self amplification, vortex stretching, and density gradient amplification.

Motivated by our findings that the average downscale flux of TKE and TPE are due to different mechanisms and that the contributions to the energy fluxes from small scale (i.e., sub-grid) dynamics are significant, in the second part we develop a Lagrangian model for studying the small-scale scalar dynamics in isotropic turbulence. It is known that the equation for the fluid velocity gradient along a Lagrangian trajectory immediately follows from the Navier-Stokes equation, and such an equation involves two terms that cannot be determined from the velocity gradient along the chosen Lagrangian path: the pressure Hessian and the viscous Laplacian; similarly, the equation for passive scalar gradients also involves an unclosed term in the Lagrangian frame, namely the scalar gradient diffusion term which needs to be closed. For the fluid velocity gradient, a recent model handles the unclosed terms using a multi-level version of the recent deformation of Gaussian fields (RDGF) closure (Johnson \& Meneveau, Phys.~Rev.~Fluids, 2017). The model is in remarkable agreement with DNS data and works for arbitrary Taylor Reynolds numbers $Re_\lambda$. Inspired by this, our Lagrangian model for the passive scalar gradients is developed using the RDGF approach. However, comparisons of the statistics obtained from this model with direct numerical simulation (DNS) data reveal substantial errors due to erroneously large fluctuations generated by the model. We address this defect by incorporating into the closure approximation information regarding the scalar gradient production along the local trajectory history of the particle. This modified model makes predictions for the scalar gradients, their production rates, and alignments with the strain-rate eigenvectors that are in very good agreement with DNS data. However, while the model yields valid predictions up to around $Re_\lambda\approx 500$, beyond this, the model breaks down.

In consideration of the model failure beyond $Re_\lambda\approx 500$, the final part of work conducts further investigations via theoretical analysis and computations of more DNS data at various Reynolds numbers $Re_\lambda$. We theoretically analyzed the governing equations and identified two key mechanisms preventing the divergence of the scalar gradient magnitude. The conditional average of the scalar gradient diffusion term is also analyzed via its reduced forms which are used to test the model closure against DNS results. The model closure shows considerable errors in terms of its linear predictions of the conditional averages, in contrast to the strongly nonlinear dependencies on the condition quantities shown in DNS data. Such revealed errors potentially could be the reason why the model collapses beyond $Re_\lambda\approx 500$. Also discussed are the local relations of the scalar gradient diffusion term and various relevant quantities. It has been found that the diffusion term acts strictly to dissipate fluctuations of the scalar gradients in all regions where the scalar gradients are being either amplified or suppressed. Scalar gradients are dissipated most strongly in the regions where the straining motions are strong and the TKE are most strongly dissipated. Overall, the presented work here gains novel insights into the dynamics of scalar turbulence, reveals important implications/defects of the existing model closures, and in the meantime provides useful guidance for further improvements of existing model closures or for developing new models so that the complex scalar dynamics can be better captured in a more accurate manner.