Browsing by Subject "Turbulence"

A new method of stabilizing low-order, proper orthogonal decomposition based reduced-order models of the Navier Stokes equations is proposed. Unlike traditional approaches, this method does not rely on empirical turbulence modeling or modification of the Navier-Stokes equations. It provides spatial basis functions different from the usual proper orthogonal decomposition basis function in that, in addition to optimally representing the solution, the new proposed basis functions also provide stable reduced-order models. The proposed approach is illustrated with two test cases: two-dimensional flow inside a square lid-driven cavity and a two-dimensional mixing layer.

We 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.

Large-scale coherent structures are systematically investigated in terms of their geometric attributes, importance toward describing turbulent exchange of energy, momentum and mass as well as their relationship to landscape features in the context of land-atmosphere interaction. In the first chapter, we present the motivation of this work as well as a background review of large-scale coherent structures in land-atmosphere interaction. In the second chapter, the methodology of large-eddy simulation (LES) and the proper orthogonal decomposition (POD) is introduced. LES was used to serve as a virtual laboratory to simulate typical scenarios in land-atmosphere interaction and the POD was used as the major technique to educe the coherent structures from turbulent flows in land-atmosphere interaction. In the third chapter, we justify the use of the LES to simulate the realistic coherent structures in the atmospheric boundary layer (ABL) by comparing results obtained from LES of the ABL and direct numerical simulation (DNS) of channel flow. In the fourth chapter, we investigate the effects of a wide range of vegetation density on the coherent structures within the air space within and just above the canopy (the so-called canopy sublayer, CSL). The fifth chapter presents an analysis of the coherent structures across a periodic forest-clearing-forest transition in the steamwise direction. The sixth chapter focuses on the role of coherent structures in explaining scalar dissimilarity in the CSL. The seventh chapter summarizes this dissertation and provides suggestions for future study.

This dissertation describes two computational sensors that were used to demonstrate applications of generalized sampling of the optical field. The first sensor was an incoherent imaging system designed for compressive measurement of the power spectral density in the scene (spectral imaging). The other sensor was an interferometer used to compressively measure the mutual intensity of the optical field (coherence imaging) for imaging through turbulence. Each sensor made anisomorphic measurements of the optical signal of interest and digital post-processing of these measurements was required to recover the signal. The optical hardware and post-processing software were co-designed to permit acquisition of the signal of interest with sub-Nyquist rate sampling, given the prior information that the signal is sparse or compressible in some basis.

Compressive spectral imaging was achieved by a coded aperture snapshot spectral imager (CASSI), which used a coded aperture and a dispersive element to modulate the optical field and capture a 2D projection of the 3D spectral image of the scene in a snapshot. Prior information of the scene, such as piecewise smoothness of objects in the scene, could be enforced by numerical estimation algorithms to recover an estimate of the spectral image from the snapshot measurement.

Hypothesizing that turbulence between the scene and CASSI would introduce spectral diversity of the point spread function, CASSI's snapshot spectral imaging capability could be used to image objects in the scene through the turbulence. However, no turbulence-induced spectral diversity of the point spread function was observed experimentally. Thus, coherence functions, which are multi-dimensional functions that completely determine optical fields observed by intensity detectors, were considered. These functions have previously been used to image through turbulence after extensive and time-consuming sampling of such functions. Thus, compressive coherence imaging was attempted as an alternative means of imaging through turbulence.

Compressive coherence imaging was demonstrated by using a rotational shear interferometer to measure just a 2D subset of the 4D mutual intensity, a coherence function that captures the optical field correlation between all the pairs of points in the aperture. By imposing a sparsity constraint on the possible distribution of objects in the scene, both the object distribution and the isoplanatic phase distortion induced by the turbulence could be estimated with the small number of measurements made by the interferometer.

The 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.

The timing of the spring phytoplankton bloom in the subpolar North Atlantic Ocean has important consequences for the marine carbon cycle and ecosystems. There are currently several proposed mechanisms to explain the timing of this bloom. The conventional theory holds that the bloom begins when the ocean warms and the seasonal mixed layer shoals in the spring, decreasing the depth to which phytoplankton are mixed and increasing the light available to the population. Recent work has attributed the beginning of the bloom to decreases in turbulence within the upper ocean, driven by the onset of positive heat fluxes or decreases in the strength of local winds. Other studies have focused on the increase in the seasonal mixed layer in the winter as a driver of changes in ecosystem interactions and a control on the spring bloom. Finally, submesoscale eddies, occurring as a result of lateral density gradients, have been proposed as a stratification mechanism that can create phytoplankton blooms prior to the onset of ocean surface warming.

This dissertation critically examines and compares the proposed theories for the initiation of the spring bloom and draws on these theories to propose a new framework: that blooms begin when the active mixing depth shoals, a process generally driven by a weakening of surface heat fluxes and consequent shift from convective mixing to wind-driven mixing. Using surface forcing data, we develop a parameterization for the active mixing depth from estimates of the largest energy-containing eddies in the upper ocean.

Using in situ records of turbulent mixing and biomass, we find that the spring phytoplankton bloom occurs after mixing shifts from being driven by convection to being driven by wind, and that biomass increases as the active mixing depth shoals. Using remote sensing data, we examine patterns of bloom initiation in the North Atlantic at the basin scale, compare current theories of bloom initiation, and find that the shoaling of the active mixing depth better predicts the onset of the bloom across the North Atlantic subpolar basin and over multiple years than do other current theories. Additionally, using a process study model, we evaluate the importance of submesoscale eddy-driven stratification as a control on the initiation of the spring bloom, determining that this mechanism has a relatively minor effect on alleviation of phytoplankton light limitation. Finally, we describe potential techniques and tools to examine whether interannual variability in the active mixing depth acts as a control on variability in the timing of the spring bloom.

Turbulence 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.

Computational 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.

Problems in the area of land/biosphere-atmosphere interaction, hydrology, climate modeling etc. can be systematically organized as a study of turbulent flow in presence of boundary conditions in an increasing order of complexity. The present work is an attempt to study a few subsets of this general problem of turbulence in natural environments- in the context of neutral and thermally stratified atmospheric surface layer, the presence of a heterogeneous vegetation canopy and the interaction between air flow and a static water body in presence of flexible protruding vegetation. The main issue addressed in the context of turbulence in the atmospheric surface layer is whether it is possible to describe the macro-states of turbulence such as mean velocity and turbulent velocity variance in terms of the micro-states of the turbulent flow, i.e., a distribution of turbulent kinetic energy across a multitude of scales. This has been achieved by a `spectral budget approach' which is extended for thermal stratification scenarios as well, in the process unifying the seemingly different and unrelated theories of turbulence such as Kolmogorov's hypothesis, Heisenberg's eddy viscosity, Monin Obukhov Similarity Theory (MOST) etc. under a common framework. In the case of a more complex scenario such as presence of a vegetation canopy with edges and gaps, the question that is addressed is in what detail the turbulence is needed to be resolved in order to capture the bulk flow features such as recirculation patterns. This issue is addressed by a simple numerical framework and it has been found out that an explicit prescription of turbulence is not necessary in presence of heterogeneities such as edges and gaps where the interplay between advection, pressure gradients and drag forces are sufficient to capture the first order dynamics. This result can be very important for eddy-covariance flux calibration strategies in non-ideal environments and the developed numerical model can be used in related dispersion studies and coupled land atmosphere interaction models. For other more complex biosphere atmosphere interactions such as greenhouse gas emissions from wetlands, the interplay between air and water, often in presence of flexible aquatic vegetation, controls turbulence in water, which in turn affect the gas transfer processes. This process of wind shear induced wave-turbulent-vegetation interaction is studied for the first time in the laboratory and the state of turbulence as well as the bulk flow is found to be highly sensitive to environmental controls such as water height, wind speed, vegetation density and flexibility. This dissertation describes and gradually develops these concepts in an increasing order of complexity of boundary conditions. The first three chapters address the neutral and thermally stratified boundary layers and the last two chapters address the canopy edge problem and the air-water-vegetation experiments respectively.