A sub-grid scale model for Burgers turbulence based on the artificial neural network method

The present study proposes a sub-grid scale model for the one-dimensional Burgers turbulence based on the neuralnetwork and deep learning method.The filtered data of the direct numerical simulation is used to establish thetraining data set,the validation data set,and the test data set.The artificial neural network(ANN)methodand Back Propagation method are employed to train parameters in the ANN.The developed ANN is applied toconstruct the sub-grid scale model for the large eddy simulation of the Burgers turbulence in the one-dimensionalspace.The proposed model well predicts the time correlation and the space correlation of the Burgers turbulence.

Applying resolved-scale linearly forced isotropic turbulence in rational subgrid-scale modeling

In previous attempts of rational subgrid-scale (SGS) modeling by employing the Kolmogorov equation of filtered (KEF) quantities, it was necessary to assume that the resolved-scale second-order structure function is stationary. Forced isotropic turbulence is often used as a framework for establishing and validating such SGS models based on stationary restrictions, for it generates statistical stationary samples. However, traditional forcing method at low wavenumbers cannot provide an analytic form of forcing term for a complete KEF in physical space, which has been illustrated to be essential in the modeling of such SGS models. Thus, an alternative forcing method giving an analytic forcing term in physical space is needed for rational SGS modeling. Giving an analytic linear driving term in physical space, linearly forced isotropic turbulence should be considered an ideal theoretical framework for rational SGS modeling. In this paper, we demonstrate the feasibility of establishing a rational SGS model with stationary restriction based on linearly forced isotropic turbulence. The performance of this rational SGS model is validated. We, therefore, propose the use of linearly forced isotropic turbulence as a complement to free-decaying isotropic turbulence and low-wavenumber forced isotropic turbulence for SGS model validations.

Stochastic modeling of subgrid-scale effects on particle motion in forced isotropic turbulence

The subgrid-scale effects on particle motion were investigated in forced isotropic turbulence by DNS and priorLES methods.In the DNS field,the importance of Kolmogorov scaling to preferential accumulation was validated by comparing the radial distribution functions under various particle Stokes numbers.The prior-LES fields were generated by filtering the DNS data.The subgrid-scale Stokes number(StSGS)is a useful tool for determining the effects of subgrid-scale eddies on particle motion.The subgrid-scale eddies tend to accumulate particles with StSGSb 1 and disperse particles with 1 b StSGSb 10.For particles with StSGS?1,the effects of subgrid-scale eddies on particle motion can be neglected.In order to restore the subgrid-scale effects,the Langevin-type stochastic model with optimized parameters was adopted in this study.This model is effective for the particles with StSGS N 1 while has an adverse impact on the particles with StSGSb 1.The results show that the Langevin-type stochastic model tends to smooth the particle distribution in the isotropic turbulence.

A Closure for Isotropic Turbulence Based on Extended Scale Similarity Theory in Physical Space

The closure of a turbulence field is a longstanding fundamental problem, while most closure models are introduced in spectral space. Inspired by Chou＇s quasi-normal closure method in spectral space, we propose an analytical closure model for isotropic turbulence based on the extended scale similarity theory of the velocity structure function in physical space. The assumptions and certain approximations are justified with direct numerical simulation. The asymptotic scaling properties are reproduced by this new closure method, in comparison to the classical Batchelor model.

Numerical simulation of gas–liquid flow in the bubble column using Wray–Agarwal turbulence model coupled with population balance model

In this paper,an improved computational fluid dynamic(CFD)model for gas-liquid flow in bubble column was developed using the one-equation Wary-Agarwal(WA)turbulence model coupled with the population balance model(PBM).Through 18 orthogonal test cases,the optimal combination of interfacial force models,including drag force,lift force,turbulent dispersion force.The modified wall lubrication force model was proposed to improve the predictive ability for hydrodynamic behavior near the wall of the bubble column.The values simulated by optimized CFD model were in agreement with experimental data,and the errors were within±20%.In addition,the axial velocity,turbulent kinetic energy,bubble size distribution,and the dynamic characteristic of bubble plume were analyzed at different superficial gas velocities.This research work could provide a theoretical basis for the extension of the CFD-PBM coupled model to other multiphase reactors..

New perspective in statistical modeling of wall-bounded turbulence

Despite dedicated effort for many decades,statistical description of highly technologically important wall turbulence remains a great challenge.Current models are unfortunately incomplete,or empirical,or qualitative.After a review of the existing theories of wall turbulence,we present a new framework,called the structure ensemble dynamics （SED）,which aims at integrating the turbulence dynamics into a quantitative description of the mean flow.The SED theory naturally evolves from a statistical physics understanding of non-equilibrium open systems,such as fluid turbulence, for which mean quantities are intimately coupled with the fluctuation dynamics.Starting from the ensemble-averaged Navier-Stokes（EANS） equations,the theory postulates the existence of a finite number of statistical states yielding a multi-layer picture for wall turbulence.Then,it uses order functions（ratios of terms in the mean momentum as well as energy equations） to characterize the states and transitions between states.Application of the SED analysis to an incompressible channel flow and a compressible turbulent boundary layer shows that the order functions successfully reveal the multi-layer structure for wall-bounded turbulence, which arises as a quantitative extension of the traditional view in terms of sub-layer,buffer layer,log layer and wake. Furthermore,an idea of using a set of hyperbolic functions for modeling transitions between layers is proposed for a quantitative model of order functions across the entire flow domain.We conclude that the SED provides a theoretical framework for expressing the yet-unknown effects of fluctuation structures on the mean quantities,and offers new methods to analyze experimental and simulation data.Combined with asymptotic analysis,it also offers a way to evaluate convergence of simulations.The SED approach successfully describes the dynamics at both momentum and energy levels, in contrast with all prevalent approaches describing the mean velocity profile only.Moreover,the SED theoretical framework is general,independent of the flow system to study, while the actual functional form of the order functions may vary from flow to flow.We assert that as the knowledge of order functions is accumulated and as more flows are analyzed, new principles（such as hierarchy,symmetry,group invariance,etc.） governing the role of turbulent structures in the mean flow properties will be clarified and a viable theory of turbulence might emerge.

Influence of polymer additives on turbulent energy cascading in forced homogeneous isotropic turbulence studied by direct numerical simulations

Direct numerical simulations （DNS） were performed for the forced homogeneous isotropic turbulence （FHIT） with/without polymer additives in order to elaborate the characteristics of the turbulent energy cascading influenced by drag-reducing effects. The finite elastic non-linear extensibility-Peterlin model （FENE-P） was used as the conformation tensor equation for the viscoelastic polymer solution. Detailed analyses of DNS data were carried out in this paper for the turbulence scaling law and the topological dynamics of FHIT as well as the important turbulent parameters, including turbulent kinetic energy spectra, enstrophy and strain, velocity structure function, small-scale intermittency, etc. A natural and straightforward definition for the drag reduction rate was also proposed for the drag-reducing FHIT based on the decrease degree of the turbulent kinetic energy. It was found that the turbulent energy cascading in the FHIT was greatly modified by the drag-reducing polymer additives. The enstrophy and the strain fields in the FH1T of the polymer solution were remarkably weakened as compared with their Newtonian counterparts. The small-scale vortices and the small-scale intermittency were all inhibited by the viscoelastic effects in the FHIT of the polymer solution. However, the scaling law in a fashion of extended self-similarity for the FHIT of the polymer solution, within the presently simulated range of Weissenberg numbers, had no distinct differences compared with that of the Newtonian fluid case.

Modeling of Fluid Turbulence Modification Using Two-time-scale Dissipation Models and Accounting for the Particle Wake Effect

Presently developed two-phase turbulence models under-predict the gas turbulent fluctuation, because their turbulence modification models cannot fully reflect the effect of particles. In this paper, a two-time-scale dis- sipation model of turbulence modification, developed for the two-phase velocity correlation and for the dissipation rate of gas turbulent kinetic energy, is proposed and used to simulate sudden-expansion and swirling gas-particle flows. The proposed two-time scale model gives better results than the single-time scale model. Besides, a gas tur- bulence augmentation model accounting for the finite-size particle wake effect in the gas Reynolds stress equation is proposed. The proposed turbulence modification models are used to simulate two-phase pipe flows. It can prop- erly predict both turbulence reduction and turbulence enhancement for a certain size of particles observed in ex- periments.

Modeling of Fluid Turbulence Modification Using Two-time-scale Dissipation Models and Accounting for the Particle Wake Effect

Presently developed two-phase turbulence models under-predict the gas turbulent fluctuation, because their turbulence modification models cannot fully reflect the effect of particles. In this paper, a two-time-scale dissipation model of turbulence modification, developed for the two-phase velocity correlation and for the dissipation rate of gas turbulent kinetic energy, is proposed and used to simulate sudden-expansion and swirling gas-particle flows. The proposed two-time scale model gives better results than the single-time scale model. Besides, a gas turbulence augmentation model accounting for the f＇mite-size particle wake effect in the gas Reynolds stress equation is proposed. The proposed turbulence modification models are used to simulate two-phase pipe flows. It can properly predict both turbulence reduction and turbulence enhancement for a certain size of particles observed in experiments.

Subgrid-scale contributions to Lagrangian time correlations in isotropic turbulence

The application of large-eddy simulation （LES） to particle-laden turbulence raises such a fundamental question as whether the LES with a subgrid scale （SGS） model can correctly predict Lagrangian time correlations （LTCs）. Most of the currently existing SGS models are constructed based on the energy budget equations. Therefore, they are able to correctly predict energy spectra, but they may not ensure the correct prediction on the LTCs. Previous researches investigated the effect of the SGS modeling on the Eulerian time correlations. This paper is devoted to study the LTCs in LES. A direct numerical simulation （DNS） and the LES with a spectral eddy viscosity model are performed for isotropic turbulence and the LTCs are calculated using the passive vector method. Both a priori and a posteriori tests are carried out. It is observed that the subgrid;scale contributions to the LTCs cannot be simply ignored and the LES overpredicts the LTCs than the DNS. It is concluded from the straining hypothesis that an accurate prediction of enstrophy spectra is most critical to the prediction of the LTCs.

Influence of moderate-to-strong anisotropic non-Kolmogorov turbulence on intensity fluctuations of a Gaussian–Schell model beam in marine atmosphere

The scintillation index（SI） of a Gaussian–Schell model（GSM） beam in a moderate-to-strong anisotropic nonKolmogorov turbulent atmosphere is developed based on the extended Rytov theory. The on-axis SI in a marine atmosphere is higher than that in a terrestrial atmosphere, but the off-axis SI exhibits the opposite trend. The on-axis SI first increases and then begins to decrease and saturate as the turbulence strength increases. Turbulence inner and outer scales have different effects on the on-axis SI in different turbulent fluctuation regions. The anisotropy characteristic of atmospheric turbulence leads to the decline in the on-axis SI, and the rise in the off-axis SI. The on-axis SI can be lowered by increasing the anisotropy of turbulence, wavelength, and source partial coherence before entering the saturation attenuation region. The developed model may be useful for evaluating ship-to-ship/shore free-space optical communication system performance.

Corrections to the scaling of the second-order structure function in isotropic turbulence

The approach of Obukhov assuming a constant skewness was used to obtain analytical corrections to the scaling of the second order structure function, starting from Kolmogorov＇s 4/5 law. These corrections can be used in model applications in which explicit expressions, rather than numerical solutions are needed. The comparison with an interpolation formula proposed by Batchelor, showed that the latter gives surprisingly precise results. The modification of the same method to obtain analytical corrections to the scaling law, taking into account the possible corrections induced by intermittency, is also proposed.

Assessment of the modulated gradient model in decaying isotropic turbulence

A recently introduced nonlinear model undergoes evaluations based on two isotropic turbulent cases:a University of Wiscosion-Madison case at a moderate Reynolds number and a Johns Hopkins University case at a high Reynolds number.The model uses an estimation of the subgrid-scale(SGS) kinetic energy to model the magnitude of the SGS stress tensor,and uses the normalized velocity gradient tensor to model the structure of the SGS stress tensor.Testing is performed for the first case through a comparison between direct numerical simulation(DNS) results and large eddy simulation(LES) results regarding resolved kinetic energy and energy spectrum.In the second case,we examine the resolved kinetic energy,the energy spectrum,as well as other key statistics including the probability density functions of velocities and velocity gradients,the skewness factors,and the flatness factors.Simulations using the model are numerically stable,and results are satisfactorily compared with DNS results and consistent with statistical theories of turbulence.

The influence of sub-grid scale motions on particle collision in homogeneous isotropic turbulence

The absence of sub-grid scale(SGS) motions leads to severe errors in particle pair dynamics, which represents a great challenge to the large eddy simulation of particle-laden turbulent flow. In order to address this issue,data from direct numerical simulation(DNS) of homogenous isotropic turbulence coupled with Lagrangian particle tracking are used as a benchmark to evaluate the corresponding results of filtered DNS(FDNS). It is found that the filtering process in FDNS will lead to a non-monotonic variation of the particle collision statistics, including radial distribution function, radial relative velocity, and the collision kernel. The peak of radial distribution function shifts to the large-inertia region due to the lack of SGS motions, and the analysis of the local flowstructure characteristic variable at particle position indicates that the most effective interaction scale between particles and fluid eddies is increased in FDNS. Moreover,this scale shifting has an obvious effect on the odd-order moments of the probability density function of radial relative velocity, i.e. the skewness, which exhibits a strong correlation to the variance of radial distribution function in FDNS.As a whole, the radial distribution function, together with radial relative velocity, can compensate the SGS effects for the collision kernel in FDNS when the Stokes number based on the Kolmogorov time scale is greater than 3.0. However,it still leaves considerable errors for St< 3.0.

Mathematical concepts and their physical foundation in the nonstandard analysis theory of turbulence

Main mathematical concepts and their physical foundation in the nonstandard analysis theory of turbulence are presented and discussed. The underlying fact is that there does not exist the absolute zero fluid-volume. Therefore, the physical object corresponding to the absolute point is just the uniform fluid-particle. The fluid-particle, in general, corresponds to the monad. The uniform fluid-particle corresponds to the uniform monad, while the nonuniform fluid-particle to the nonuniform monad. There are two kinds of the differentiations, one is based on the absolute point, and the other based on the monad. The former is adopted in the Navier-Stokes equations, and the latter in the fundamental equations presented in this paper for the nonstandard analysis theory of turbulence. The continuity of fluid is elucidated by virtue of the concepts of the fluid-particle and fluid-particle at a lower level. Furthermore, the characters of the continuity in two cases, i.e. in the standard and nonstandard analyses, are presented in this paper. And the difference in discretization between the Navier-Stokes equations and the fundamental equations given herein is also pointed out.

Direct numerical simulation study of the interaction between the polymer effect and velocity gradient tensor in decaying homogeneous isotropic turbulence

Direct numerical simulation of decaying homogeneous isotropic turbulence （DHIT） of a polymer solution is performed. In order to understand the polymer effect on turbulence or additive-turbulence interaction, we directly investigate the influence of polymers on velocity gradient tensor including vorticity and strain. By visualizing vortex tubes and sheets, we observe a remarkable inhibition of vortex structures in an intermediate-scale field and a small-scale field but not for a large scale field in DHIT with polymers. The geometric study indicates a strong relevance among the vorticity vector, rate-of-strain tensor, and polymer conformation tensor. Joint probability density functions show that the polymer effect can increase ＂strain generation resistance＂ and ＂vorticity generation resistance＂, i.e., inhibit the generation of vortex sheets and tubes, ultimately leading to turbulence inhibition effects.

An iterative data-driven turbulence modeling framework based on Reynolds stress representation

Data-driven turbulence modeling studies have reached such a stage that the basic framework is settled,but several essential issues remain that strongly affect the performance.Two problems are studied in the current research:(1)the processing of the Reynolds stress tensor and(2)the coupling method between the machine learning model and flow solver.For the Reynolds stress processing issue,we perform the theoretical derivation to extend the relevant tensor arguments of Reynolds stress.Then,the tensor representation theorem is employed to give the complete irreducible invariants and integrity basis.An adaptive regularization term is employed to enhance the representation performance.For the coupling issue,an iterative coupling framework with consistent convergence is proposed and then applied to a canonical separated flow.The results have high consistency with the direct numerical simulation true values,which proves the validity of the current approach.

Statistical Theory of Turbulence by the Late Lamented Dr. Shunichi TsugéCase Study on Flow through a Grid in Wind Tunnel

This paper is concerned with statistical theory of turbulence by the late lamented Dr. Shunichi Tsugé.? The theory has been applied to the primary flow through a grid fixed vertically with respect to the horizontal axis of the wind tunnel.?The first analytical solution has been obtained and explained the well-known “the inverse-linear decay law” of the turbulent intensity.? It is believed that the present result is the first exact solution in the theory of turbulence.

Modeling of Air Temperature for Heat Exchange due to Vertical Turbulence and Horizontal Air Flow

In order to calculate the air temperature of the near surface layer in urban environment,the surface layer air was divided into several layers in the vertical direction,and some energy balance equations were developed for each air layer,in which the heat exchange due to vertical turbulence and horizontal air flow was taken into account.Then,the vertical temperature distribution of the surface layer air was obtained through the coupled calculation using the energy balance equations of underlying surfaces and building walls.Moreover,the measured air temperatures in a small area(with a horizontal scale of less than 500 m)and a large area(with a horizontal scale of more than 1 000 m)in Guangzhou in summer were used to validate the proposed model.The calculated results agree well with the measured ones,with a maximum relative error of 4.18%.It is thus concluded that the proposed model is a high-accuracy method to theoretically analyze the urban heat island and the thermal environment.

THE DOUBLE AND TRIPLE VELOCITY CORRELATIONS OF SMALL VORTEXES IN THREE-VORTEX MODEL THEORY AND THE DECAY OF GRID-PRODUCED TURBULENCE

In the three-vortex model theory of turbulence[1,2] and double and the triple velocity correlation functions of small vortexes were employed. In this paper, the double and the triple velocity correlation functions of small vortexes are further discussed, and the expressions of some coefficients in the expansions in terms of relative displacement of two points are given. Finally, by using these coefficients, the decay of grid-produced trubulence is calculated. The result of calculations gives good agreement with the experimental data of G.K. Batchelor and A.A.Townsend[3]