Browsing by Author "Bragg, Andrew D"
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Item Open Access An Extended Rouse Model of Inertial Particles Settling in Turbulent Boundary Layers(2022) Zhang, YanThe settling of inertial particles in turbulence boundary layers plays an essential role in many meteorological, industrial and environmental processes, and is governed by multifarious mechanisms. First, turbulence alters the settling velocity of inertial particles through different effects, like preferential sweeping mechanism, loitering effect and vortex trapping. Second, the existence of a wall introduces extra effects that can influence particle settling, such as turbophoresis. The Rouse model was the most famous model in predicting particle settling in vertical wall-bounded settling. Nevertheless, it is only valid for inertia-less particles in the logarithmic region. A theory by Bragg et al., based on phase-space probability density theory, incorporates particle inertia into the Rouse model, and quantifies the contributions from the aforementioned mechanisms to the particle vertical velocity. The theory is valid for all particle Stokes numbers, yet it still lacks a closed form.In this work, one way to close the equations presented by Bragg et al. (the extended Rouse model) was examined. Using a central differencing scheme combined with an iterative method, the nonlinear second-order differential equation of the variance of vertical particle velocity was solved. The predictions of the variance of vertical particle velocity S and the particle concentration PDF ρ by the model were studied and compared to DNS. The comparison indicates that the extended Rouse model is able to predict many features of S and ρ, like the accumulation of particles close to the wall and turbophoretic drift. However, the quantitative agreement between the predictions by the model and DNS is poor. There are two probable reasons for the discrepancies between the predictions and DNS. First, the closure of the term in the equation may be a source of errors. Second, the lower boundary condition, whose validity is suspicious for particles with weak inertia, can be a reason for the discrepancies. In order to investigate the cause for the disagreement, three different boundary conditions (zero-gradient condition, asymptotic matching, iterative condition) were examined. The results indicate that the boundary conditions have a very limited influence on the predictions. As a result, the closure of the terms is more likely to be responsible for the discrepancies.
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 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 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.