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.

Explicit Algebraic Stress Model for Three-Dimensional Turbulent Buoyant Flows Derived Using Tensor Representation

An explicit algebraic stress model (EASM) has been formulated for two-dimensional turbulent buoyant flows using a five-term tensor representation in a prior study. The derivation was based on partitioning the buoyant flux tensor into a two-dimensional and a three-dimensional component. The five-term basis was formed with the two-dimensional component of the buoyant flux tensor. As such, the derived EASM is limited to two-dimensional flows only. In this paper, a more general approach using a seven-term representation without partitioning the buoyant flux tensor is used to derive an EASM valid for two- and three-dimensional turbulent buoyant flows. Consequently, the basis tensors are formed with the fully three-dimensional buoyant flux tensor. The derived EASM has the two-dimensional flow as a special case. The matrices and the representation coefficients are further simplified using a four-term representation. When this four-term representation model is applied to calculate two-dimensional homogeneous buoyant flows, the results are essentially identical with those obtained previously using the two-dimensional component of the buoyant flux tensor. Therefore, the present approach leads to a more general EASM formulation that is equally valid for two- and three-dimensional turbulent buoyant flows.

Derivation of a second-order model for Reynolds stress using renormalization group analysis and the two-scale expansion technique

With the two-scale expansion technique proposed by Yoshizawa,the turbulent fluctuating field is expanded around the isotropic field.At a low-order two-scale expansion,applying the mode coupling approximation in the Yakhot-Orszag renormalization group method to analyze the fluctuating field,the Reynolds-average terms in the Reynolds stress transport equation,such as the convective term,the pressure-gradient-velocity correlation term and the dissipation term,are modeled.Two numerical examples：turbulent flow past a backward-facing step and the fully developed flow in a rotating channel,are presented for testing the efficiency of the proposed second-order model.For these two numerical examples,the proposed model performs as well as the Gibson-Launder （GL） model,giving better prediction than the standard k-ε model,especially in the abilities to calculate the secondary flow in the backward-facing step flow and to capture the asymmetric turbulent structure caused by frame rotation.

Algebraic-Stress Turbulent Model of Planar 2-D Pollutant Transportation in Curvilinear Coordinates

The shallow-water equations and pollutant convective-diffusive equation are transformed into curvilinear coordinate system. The anisotropic algebraic-stress turbulent model is introduced to simulate the turbulence items, and algebraic-stress turbulent model of planar 2-D pollutant convective-diffusive in curvilinear coordinates is build. The meandering channel with measured data of concentration in lab is adopted to validate the model, and the distribution figure of pollutant concentration field calculated through this model and that of the k-ε model, which show the model is superior to k-ε turbulent model in dealing with anisotropy character of flow.

COMPUTATION OF RECIRCULATING SWIRLING FLOW WITH THE GLM REYNOLDS STRESS CLOSURE

A Reynolds stress closure based on the generalized Langevin model (GLM), developed by Haworth and Pope, is applied to the flow calculation with swirl-induced recirculation. The purpose of the work is to assess the performance of this model under the complex flow conditions caused by the presence of strong swirl which gives rise to both unconventional recirculation in the vicinity of the symmetry axis and strong anisotropy in the turbulence field. Comparison of the computational results are made both with the experimental data of Roback and Johnson and the computational results obtained with the typical isotropization of production model (IPM) and the k-∈ type Boussinesq viscosity model.

MODELLING OF THE PRESSURE-VELOCITY CORRELATION IN TURBULENCE DIFFUSION

Most current computations of trubulent flows with second-moment closure adopt the diffusion mo- dels which neglect the effect of pressure-velocity correlation.ln the present paper the importance of this correlation effect is elucidated the neglect of this effect accounts for some major defects in the wide application of the se- cond-moment closures.Through the relation between and ,established by Lumley,we propose here a new turbulence diffusion model which takes into consideration the pressure effect.Applications of this new model in the computation of shearless turbulence mixing layer and plane-and round-jet flows show that the spreading rate of these flows can be satisfactorily captured.

Extended intrinsic mean spin tensor for turbulence modelling in non-inertial frame of reference

We investigate the role of extended intrinsic mean spin tensor introduced in this work for turbulence modelling in a non-inertial frame of reference. It is described by the Euclidean group of transformations and, in particular, its significance and importance in the approach of the algebraic Reynolds stress modelling, such as in a nonlinear K-ε model. To this end and for illustration of the effect of extended intrinsic spin tensor on turbulence modelling, we examine several recently developed nonlinear K-ε models and compare their performance in predicting the homogeneous turbulent shear flow in a rotating frame of reference with LES data. Our results and analysis indicate that, only if the deficiencies of these models and the like be well understood and properly corrected, may in the near future, more sophisticated nonlinear K-ε models be developed to better predict complex turbulent flows in a non-inertial frame of reference.

CURVAURE SENSITIVE NONLINEAR TURBULENCE MODEL

By modifying the Rodi assumption to take account of the influence of flow curvature, a new curvature modified algebraic stress model(CMASM) is de- veloped from the second moment closure in the generalized curvilinear coordinate system. And the explicit form of this ASM, a new curvature modified nonlinear k-ε model (CMNKE), is derived in the orthogonal curvilinear coordinate system. This new nonlinear k-ε model is further validated by a numerical simulation of a two- dimensional U-type turnaround duct flow. The results show that the CMNKE can effectively capture the main characteristic of this curvature flow and simulate the damping effect of the shear stress by a convex curvature and the enhancing effect by a concave curvature. So, this model is a rational and effective simplification to the second moment closure.

STUDY ON ANISOTROPIC BUOYANT TURBULENCE MODEL

By analyzing the components of Reynolds stresses of implicit algebraic stress model(IASM) in this paper, that Reynolds stresses in buoyant turbulent flows were produced by both strain and buoyancy is considered. Consequently, a nonlinear anisotropy buoyant turbulence model was developed by applying linearity of equilibrium hypothesis to Reynolds stress transports. The model avoids numerical singularity and its reliability is verified by the comparisons between predictions and experimental data.

New explicit algebraic stress and flux model for active scalar and simulation of shear stratified cylinder wake flow

On the numerical simulation of active scalar,a new explicit algebraic expression on active scalar flux was derived based on Wikstrm,Wallin and Johansson model (aWWJ model). Reynolds stress algebraic expressions were added by a term to account for the buoyancy effect. The new explicit Reynolds stress and active scalar flux model was then established. Governing equations of this model were solved by finite volume method with unstructured grids. The thermal shear stratified cylinder wake flow was computed by this new model. The computational results are in good agreement with laboratorial measurements. This work is the development on modeling of explicit algebraic Reynolds stress and scalar flux,and is also a further modification of the aWWJ model for complex situations such as a shear stratified flow.

Modeling and Numerical Simulation of Wings Effect on Turbulent Flow between two contra-rotating cylinders

Many industries in the world take part in the pollution of the environment. This pollution often comes from the reactions of combustion. To optimize these reactions and to minimize pollution, turbulence is a funda- mental tool. Several factors are at the origin of turbulence in the complex flows, among these factors, we can quote the effect of wings in the rotating flows. The interest of this work is to model and to simulate numeri- cally the effect of wings on the level of turbulence in the flow between two contra-rotating cylinders. We have fixed on these two cylinders eight wings uniformly distributed and we have varied the height of the wings to have six values from 2 mm to 20 mm by maintaining the same Reynolds number of rotation. The numerical tool is based on a statistical model in a point using the closing of the second order of the transport equations of the Reynolds stresses (Reynolds Stress Model: RSM). We have modelled wings effect on the flow by a source term added to the equation tangential speed. The results of the numerical simulation showed that all the average and fluctuating variables are affected the value of the kinetic energy of turbulence as those of Reynolds stresses increase with the height of the wings.

A CONSTITUTIVE THEORY FOR THE REYNOLDS－STRESS CLOSURES IN TURBULENCE MODELLING

For the Reynolds-stress closures in turbulence modelling, a constitutive theory is developed based on the principle of frame indifference and the theory of invariants. The present study sheds light on existing closure models and is instructive to developing new closure relations.

Modeling and Numerical Simulation of Wings Effect on Turbulent Flow between Two Contra-Rotating Discs

Turbulence is a fundamentally interesting physical phenomenon which is of fundamental interest. Indeed, it is at the origin of several industrial applications, the control of energy in these industrial applications pass by the comprehension and the modelling of turbulent flows. Several factors are at the origin of turbulence in the complex flows, among these factors, we can quote the effect of wings in the rotating flows. The interest of this work is to model and to simulate numerically the effect of wings on the level of turbulence in the flow between two contra-rotating discs. We have fixed on these two discs eight wings uniformly distributed and we have varied the height of the wings to have eleven values from 0 to 18 mm by maintaining the same Reynolds number of rotation. The numerical tool is based on a statistical model in a point using the closing of the second order of the transport equations of the Reynolds stresses （Reynolds Stress Model： RSM）. We have modelled wings effect on the flow by a source term added to the equation tangential speed. The results of the numerical simulation showed that all the average and fluctuating variables are affected the value of the kinetic energy of turbulence as those of Reynolds stresses increase with the height of the wings.

AN EXTENDED k-ε MODEL FOR NUMERICAL SIMULATION OF WIND FLOW AROUND BUILDINGS

It is assumed in this paper that for a high Reynolds number nearly homogeneouswind flow, the Reynolds stresses are uniquely related to the mean velocity gradientsand the two independent turbulent scaling parameters k and E. By applying dimensionalanalysis and owing to the Cayley-Hamilton theorem for tensors, a new turbulenceenclosure model so-called the axtended k-ε model has been developed. The coefficientsof the model expression were detemined by the wind tunnel experimental data ofhomogeneous shear turbulent flow. The model was compared with the standard k-εmodel in in composition and the prediction of the Reynold's normal Stresses. Using thenew model the numerical simulation of wind flow around a square cross-section tallbuilding was performed. The results show that the extended k-ε model improves theprediction of wind velocities around the building the building and wind pressures on the buildingenvelope.

Second-order modeling of non-premixed turbulent methane-air combustion

The main purpose of this research is the second-order modeling of flow and turbulent heat flux in nonpremixed methane-air combustion.A turbulent stream of non-premixed combustion in a stoichiometric condition,is numerically analyzed through the Reynolds averaged Navier-Stokes(RANS) equations.For modeling radiation and combustion,the discrete ordinates(DO) and eddy dissipation concept model have been applied.The Reynolds stress transport model(RSM) also was used for turbulence modeling.For THF in the energy equation,the GGDH model and high order algebraic model of HOGGDH with simple eddy diffusivity model have been applied.Comparing the numerical results of the SED model(with the turbulent Prandtl 0.85) and the second-order heat flux models with available experimental data follows that applying the second-order models significantly led to the modification of predicting temperature distribution and species mass fraction distribution in the combustion chamber.Calculation of turbulent Prandtl number in the combustion chamber shows that the assumption of Pr_(t) of 0.85 is far from reality and Pr_(t) in different areas varies from 0.4 to 1.2.

Simulation of Average Turbulent Pipe Flow: A Three-Equation Model

The aim of this study is to evaluate a three-equation turbulence model applied to pipe flow. Uncertainty is approximated by comparing with published direct numerical simulation results for fully-developed average pipe flow. The model is based on the Reynolds averaged Navier-Stokes equations. Boussinesq hypothesis is invoked for determining the Reynolds stresses. Three local length scales are solved, based on which the eddy viscosity is calculated. There are two parameters in the model;one accounts for surface roughness and the other is possibly attributed to the fluid. Error in the mean axial velocity and Reynolds stress is found to be negligible.

Application of deep learning method to Reynolds stress models of channel flow based on reduced-order modeling of DNS data

Recently,the methodology of deep learning is used to improve the calculation accuracy of the Reynolds-averaged Navier-Stokes (RANS) model.In this paper,a neural network is designed to predict the Reynolds stress of a channel flow of different Reynolds numbers.The rationality and the high efficiency of the neural network is validated by comparing with the results of the direct numerical simulation (DNS),the large eddy simulation (LES),and the deep neural network (DNN) of other studies.To further enhance the prediction accuracy,three methods are developed by using several algorithms and simplified models in the neural network.In the method 1.the regularization is introduced and it is found that the oscillation and the overfitting of the results are eflectively prevented.In the method 2,y^+ is embedded in the input variable while the combination of the invariants is simplified in the method 3.From the predicted results,it can be seen that by using the first two methods,the errors are reduced.Moreover,the method 3 shows considerable advantages in the DNS trend and the smoothness of a curve.Consequently,it is concluded that the DNNs can predict effectively the anisotropic Reynolds stress and is a promising technique of the computational fluid dynamics.

A theoretical model for Reynolds-stress and dissipation-rate budgets in near-wall region

A 3-D wave model for the turbulent coherent structures in near-wall region is proposed. The transport nature of the Reynolds stresses and dissipation rate of the turbulence kinetic energy are shown via computation based on the theoretical model. The mean velocity profile is also computed by using the same theoretical model. The theoretical results are in good agreement with those found from DNS, indicating that the theoretical model proposed can correctly describe the physical mechanism of turbulence in near wall region and it thus possibly opens a new way for turbulence modeling in this region.

Experimental investigation of Reynolds stress complex eddy viscosity model for coherent structure dynamics

Time sequence signals of streamwise and normal velocity components,as well as velocity strain rate,at different vertical locations in the turbulent boundary layer over a smooth flat plate in a wind tunnel have been finely examined by the use of double-sensor hot-wire anemometry.The local module maximum for wavelet coefficient of longitudinal velocity component,as a detecting index,is employed to educe the ejection and sweep process of the coherent structure burst in the turbulent boundary layer from the random fluctuating background.The coherent waveforms of Reynolds stress residual contribution term for random fluctuations to coherent structure,as well as the velocity strain rate of coherent structure,are extracted by the conditional phase average technique.Based on the theoretical analysis of eddy viscosity coefficient in complex eddy viscosity model for coherent structure,the macro-relaxation effect between Reynolds stress residual contribution term of random fluctuations to coherent structure and the velocity strain rate of coherent structure is studied and the variations of the phase difference between them across the turbulent boundary layer are investigated experimentally.The rationality of complex eddy viscosity model for coherent structure is confirmed through the investigation.

A NEW DEPTH-AVERAGED ALGEBRAIC STRESS MODEL AND ITS VERIFICATION TEST

Based on the 3D algebraic stress model and the Newton-Leibnitz equation,a new depth-average algebraic stress model (DAASM) is derived in this paper. The explicit Reynolds stress equations (ERSES), similar to those proposed by Boussinesq, arederived at the same time by an algebraic transformation from the depth-averaged Reynoldsstress equations. The ERSES can describe the anisotropic property of the Reynolds stress,so the DAASM can numerically simulate the anisotropic water transport. The new modelis tested by the hydraulic and thermal test data of drainage and intake in a shallow watercoolin pond. The results show the model can correctly predict the hydrulic and thermalproperties of large-volume water. In comparison with the depth-averaged k-ε model, it isfound that the new model is better.