In this work, model closures of the multiphase Reynolds-Average Navier-Stokes (RANS) equations are developed for homogeneous, fully-developed gas--particle flows. To date, the majority of RANS closures are based on extensions of single-phase turbulence models, which fail to capture complex two-phase flow dynamics across dilute and dense regimes, especially when two-way coupling between the phases is important. In the present study, particles settle under gravity in an unbounded viscous fluid. At sufficient mass loadings, interphase momentum exchange between the phases results in the spontaneous generation of particle clusters that sustain velocity fluctuations in the fluid. Data generated from Eulerian--Lagrangian simulations are used in a sparse regression method for model closure that ensures form invariance. Particular attention is paid to modelling the unclosed terms unique to the multiphase RANS equations (drag production, drag exchange, pressure strain and viscous dissipation). A minimal set of tensors is presented that serve as the basis for modelling. It is found that sparse regression identifies compact, algebraic models that are accurate across flow conditions and robust to sparse training data.
In recent years, there has been an explosion of machine learning techniques for turbulence closure modeling, though many rely on augmenting existing models. While this has proven successful in single-phase flows, it breaks down for multiphase flows, particularly when the system dynamics are controlled by two-way coupling between the phases. In this work, we propose an approach that blends sparse regression and gene expression programming (GEP) to generate closed-form algebraic models from simulation data. Sparse regression is used to determine a minimum set of functional groups required to cap…
Sub-Kolmogorov Intermittency and Multifractal Dissipation in Multiphase Turbulence
University of Modena and Reggio Emilia（摩德纳和雷焦艾米利亚大学）；Sorbonne Université（索邦大学）；Ecole Polytechnique Fédérale de Lausanne（洛桑联邦理工学院）；Université Paris-Saclay（巴黎萨克雷大学）；Inria Saclay - Tau team（法国国家信息与自动化研究所萨克莱-Tau团队）
We investigate the turbulence statistics in a {multiphase plume made of heavy particles (particle Reynolds number at terminal velocity is 450)}. Using refractive-index-matched stereoscopic particle image velocimetry, we measure the locations of particles {whose buoyancy drives the formation of a multiphase plume,} {together with the local velocity of the induced flow in the ambient salt-water}. {Measurements in the plume centerplane exhibit self-similarity in mean flow characteristics consistent with classic integral plume theories.} The turbulence characteristics resemble those measured in a …
This numerical study presents a simple but extremely effective way to considerably enhance heat transport in turbulent multiphase flows, namely by using oleophilic walls. As a model system, we pick the Rayleigh-Bénard setup, filled with an oil-water mixture. For oleophilic walls, e.g. using only $10\%$ volume fraction of oil in water, we observe a remarkable heat transport enhancement of more than $100\%$ as compared to the pure water case. In contrast, for oleophobic walls, the enhancement is then only about $20\%$ as compared to pure water. The physical explanation of the highly-efficient he…
The two-fluid model based Multiphase Computational Fluid Dynamics (MCFD) has been considered as one of the most promising tools to investigate two-phase flow and boiling system for engineering purposes. The MCFD solver requires closure relations to make the conservation equations solvable. The wall boiling closure relations, for example, provide predictions on wall superheat and heat partitioning. The accuracy of these closure relations significantly influences the predictive capability of the solver. In this paper, a study of validation and uncertainty quantification (VUQ) for the wall boilin…
The two-phase mixing layer formed between parallel gas and liquid streams is an important fundamental problem in turbulent multiphase flows. The problem is relevant to many industrial applications and natural phenomena, such as air-blast atomizers in fuel injection systems and breaking waves in the ocean. The velocity difference between the gas and liquid streams triggers an interfacial instability which can be convective or absolute depending on the stream properties and injection parameters. In the present study, a direct numerical simulation of a two-phase gas-liquid mixing layer that lie i…
The volume of fluid (VOF) method is a popular approach for multiphase flow modeling. The open-source computational fluid dynamics (CFD) software, OpenFOAM, implements a variety of VOF-based solvers and provides users a wide range of turbulence models. Since isothermal multiphase flows under the VOF framework belong to the variable-density incompressible flow category, the isothermal VOF-based solvers in OpenFOAM fail to use the correct turbulence models. varRhoTurbVOF is designed to solve this issue and with the hope to replace all the corresponding existing solvers in the future. With the obj…
Self-consistent closure theory for passive-scalar turbulence has been developed on the basis of the Hessian of the scalar field. As a primitive indicator of spatial structure of the scalar, we employ the Hessian into the core of the theory to properly characterize the time scale intrinsic to the scalar field itself. The resultant closure model is now endowed with several realistic features, i.e. the scale-locality of the interscale interaction, the detailed conservation, and the memory-fading effect. Applying the current theory to the inertial-convective range eventually leads to self-consiste…
The aim of the present paper is to introduce and to discuss inconsistencies errors that may arise when Eulerian and Lagrangian models are coupled for the simulations of turbulent poly-dispersed two-phase flows. In these hydrid models, two turbulence models are in fact implicitely used at the same time and it is essential to check that they are consistent, in spite of their apparent different formulations. This issue appears in particular in the case of very-small particles, or tracer-limit particles, and it is shown that coupling inconsistent turbulence models (Eulerian and Lagrangian) can res…
On dissipation time scales of the basic second-order moments: the effect on the Energy and Flux-Budget (EFB) turbulence closure for stably stratified turbulence
In this two--part study, we present the development and analysis of a stochastic theory for characterizing the relative positions of monodisperse, low-inertia particle pairs that are settling rapidly in homogeneous isotropic turbulence. In the limits of small Stokes number and Froude number such that $Fr \ll St_η\ll 1$, closures are developed for the drift and diffusion fluxes in the probability density function (PDF) equation for the pair relative positions. The theory focuses on the relative motion of particle pairs in the dissipation regime of turbulence, i.e., for pair separations smaller …
In this work, we perform an aposteriori error analysis on implicit and explicit large eddy simulation closure models for solving the Burgers turbulence problem. Our closure modeling efforts include both functional and structural models equipped with various low-pass filters. We introduce discrete binomial smoothing filters and an enhanced version of the Van Cittert algorithm to accelerate the convergence of approximate deconvolution processes including regularization and relaxation filtering approaches. Our implicit modeling efforts consist of various high-order schemes including compact and n…
This paper is the second part of a two-part series, which introduces and demonstrates a Validation and Uncertainty Quantification (VUQ) framework that serves two major purposes: i). quantify the uncertainties of the closure relation parameters and predictions of the Multiphase Computational Fluid Dynamics (MCFD) solver; ii). evaluate the agreement between the solver predictions and the experimental measurements. The framework, with the corresponding theory and method, are outlined in the first part paper. In this paper, the workflow of the framework is implemented and demonstrated for two rele…
A data-driven framework for formulation of closures of the Reynolds-Average Navier--Stokes (RANS) equations is presented. In recent years, the scientific community has turned to machine learning techniques to distill a wealth of highly resolved data into improved RANS closures. While the body of work in this area has primarily leveraged Neural Networks (NNs), we alternately leverage a sparse regression framework. This methodology has two important properties: (1) The resultant model is in a closed, algebraic form, allowing for direct physical inferences to be drawn and naive integration into e…
S3T (Stochastic Structural Stability Theory) employs a closure at second order to obtain the dynamics of the statistical mean turbulent state. When S3T is implemented as a coupled set of equations for the streamwise mean and perturbation states, nonlinearity in the dynamics is restricted to interaction between the mean and perturbations. The S3T statistical mean state dynamics can be approximately implemented by similarly restricting the dynamics used in a direct numerical simulation (DNS) of the full Navier-Stokes equations (referred to as the NS system). Although this restricted nonlinear sy…
In this work, we present a localized form of the dynamic eddy viscosity model for computationally efficient and accurate simulation of the turbulent flows governed by Euler equations. In our framework, we determine the dynamic model coefficient locally using the information from neighboring grid points through a test filtering process. We then develop an optimized Gaussian filtering kernel, using a consistent definition with respect to the test filtering ratio, which gives full attenuation at the grid cut-off wave number. A systematic a-posteriori analysis of our model is performed by solving …
This study presents a numerical procedure, which we call the macroscopic forcing method (MFM), which reveals the differential operators acting on the mean fields of quantities transported by underlying fluctuating flows. Specifically, MFM can precisely determine the eddy diffusivity operator, or more broadly said, it can reveal differential operators associated with turbulence closure for scalar and momentum transport. We present this methodology by considering canonical problems with increasing complexity. Starting from the well-known problem of dispersion of passive scalars by parallel flows…
The statistics of the velocity gradient tensor in turbulent flows are of both theoretical and practical importance. The Lagrangian view provides a privileged perspective for studying the dynamics of turbulence in general, and of the velocity gradient tensor in particular. Stochastic models for the Lagrangian evolution of velocity gradients in isotropic turbulence, with closure models for the pressure Hesssian and viscous Laplacian, have been shown to reproduce important features such as non-Gaussian probability distributions, skewness and vorticity strain-rate alignments. The Recent Fluid Defo…
We study the energy transfer properties of three dimensional homogeneous and isotropic turbulence where the non-linear transfer is altered in a way that helicity is made sign-definite, say positive. In this framework, known as homochiral turbulence, an adapted eddy-damped quasi-normal Markovian (EDQNM) closure is derived to analyze the dynamics at very large Reynolds numbers, of order $10^5$ based on the Taylor scale. In agreement with previous findings, an inverse cascade of energy with a kinetic energy spectrum like $\propto k^{-5/3}$ is found for scales larger than the forcing one. Conjoint…
We survey recent results in the mathematical literature on the equations of incompressible fluid dynamics, highlighting common themes and how they might contribute to the understanding of some phenomena in the theory of fully developed turbulence.
In this paper we investigate whether the features of the non-equilibrium cascade, which have been identified in recent studies using high-fidelity tools, can be captured in the case of the classical dissipation scaling by turbulence closures based on the statistical description of freely decaying isotropic turbulence. Numerical results obtained using the EDQNM model over a very large range of Reynolds numbers (from $Re_λ=50$ up to $Re_λ=10^6$) are analyzed to perform an extensive investigation of the scaling region identified as inertial range in Kolmogorov's theory. It is observed that EDQNM …
In this article, we utilize machine learning to dynamically determine if a point on the computational grid requires implicit numerical dissipation for large eddy simulation (LES). The decision making process is learnt through \emph{a priori} training on quantities derived from direct numerical simulation (DNS) data. In particular, we compute eddy-viscosities obtained through the coarse graining of DNS quantities and utilize their distribution to categorize areas that require dissipation. If our learning determines that closure is necessary, an upwinded scheme is utilized for computing the non-…
Manifestly Markovian closures for the interaction of two-dimensional inhomogeneous turbulent flows with Rossby waves and topography are formulated and compared with large ensembles of direct numerical simulations (DNS) on a generalized beta plane. Three versions of the Markovian inhomogeneous closure (MIC) are established from the quasi-diagonal direct interaction approximation (QDIA) theory by modifying the response function to a Markovian form and employing respectively the current-time (quasi-stationary) fluctuation dissipation theorem (FDT), the prior-time (non-stationary) FDT and the corr…
In a previous communication (W.J.T. Bos and J.-P. Bertoglio 2006, Phys. Fluids, 18, 031706), a self-consistent Markovian triadic closure was presented. The detailed derivation of this closure is given here, relating it to the Direct Interaction Approximation and Quasi-Normal types of closure. The time-scale needed to obtain a self-consistent closure for both the energy spectrum and the scalar variance spectrum is determined by evaluating the correlation between the velocity and an advected displacement vector-field. The relation between this latter correlation and the velocity-scalar correlati…
A discussion on Li et al. (2019) [Numerical computations of resonant sloshing using the modified isoAdvector method and the buoyancy-modified turbulence closure model. Appl. Ocean Res. 93, article no. 101829, DOI:10.1016.j.apor.2019.05.014] is provided. Some mis-characterizations regarding the work of Larsen and Fuhrman (2018) [On the over-production of turbulence beneath surface waves using Reynolds-averaged Navier-Stokes models. J. Fluid Mech. 853, 419-460, DOI:10.1017/jfm.2018.577], on stabilizing two-equation turbulence closures beneath surface waves, are clarified.
It has been demonstrated that the Euler equations of inviscid fluid are incomplete: according to the principle of release of constraints, absence of shear stresses must be compensated by additional degrees of freedom, and leads to Reynolds-type multivalued velocity field. however unlike the Reynolds equations, the enlarged Euler's (EE) model provides additional equations for fluctuations, and that eliminates the closure problem. Therefore the (EE) equations are applicable to fully developed turbulent motions where the physical viscosity is vanishingly small compare to the turbulent viscosity, …
Supernova explosions, active galactic nuclei jets, galaxy--galaxy interactions and cluster mergers can drive turbulence in the circumgalactic medium (CGM) and in the intracluster medium (ICM). However, the exact nature of turbulence forced by these sources and its impact on the different statistical properties of the CGM/ICM and their global thermodynamics is still unclear. To investigate the effects of different types of forcing, we conduct high resolution ($1008^3$ resolution elements) idealised hydrodynamic simulations with purely solenoidal (divergence-free) forcing, purely compressive (cu…
The central regions of cool-core galaxy clusters harbour multiphase gas, with gas temperatures ranging from $10$ $\mathrm{K}$--$10^7$$\mathrm{K}$. Feedback from active galactic nuclei (AGNs) jets prevents the gas from undergoing a catastrophic cooling flow. However, the exact mechanism of this feedback energy input is unknown, mainly due to the lack of velocity measurements of the hot phase gas. However, recent observations have measured the velocity structure functions ($\mathrm{VSF}$s) of the cooler molecular ($\sim10$$\mathrm{K}$) and H$α$ filaments ($\sim10^4$$\mathrm{K}$) and used them to…
Despite well-known limitations of Reynolds-averaged Navier-Stokes (RANS) simulations, this methodology remains the most widely used tool for predicting many turbulent flows, due to computational efficiency. Machine learning is a promising approach to improve the accuracy of RANS simulations. One major area of improvement is using machine learning models to represent the complex relationship between the mean flow field gradients and the Reynolds stress tensor. In the present work, modifications to improve the stability of previous optimal eddy viscosity approaches for RANS simulations are prese…
Mean-field-based Lagrangian framework is developed for the fluid turbulence theory. The space- time vector flow is naturally introduced from the mean velocity, which provides the Lagrangian picture based on the mean field in totally equivalent manner to that of the instantaneous field. The proposed framework is applied, on one hand, to one-point closure model, yielding an objective expression of the turbulence visco-elastic effect. Application to two-point closure, on the other hand, is also discussed, where natural extension of known Lagrangian correlation is discovered on the basis of extend…
This work introduces a novel data-driven framework to formulate explicit algebraic Reynolds-averaged Navier-Stokes (RANS) turbulence closures. Recent years have witnessed a blossom in applying machine learning (ML) methods to revolutionize the paradigm of turbulence modeling. However, due to the black-box essence of most ML methods, it is currently hard to extract interpretable information and knowledge from data-driven models. To address this critical limitation, this work leverages deep learning with symbolic regression methods to discover hidden governing equations of Reynolds stress models…
Carnegie Mellon University（卡内基梅隆大学）；Argonne National Laboratory（阿贡国家实验室）
Data-driven turbulence modeling is experiencing a surge in interest following algorithmic and hardware developments in the data sciences. We discuss an approach using the differentiable physics paradigm that combines known physics with machine learning to develop closure models for Burgers' turbulence. We consider the 1D Burgers system as a prototypical test problem for modeling the unresolved terms in advection-dominated turbulence problems. We train a series of models that incorporate varying degrees of physical assumptions on an a posteriori loss function to test the efficacy of models acro…
