Stochastic Analysis and Modeling of Velocity Observations in Turbulent Flows

Highly turbulent water flows,often encountered near human constructions like bridge piers,spillways,and weirs,display intricate dynamics characterized by the formation of eddies and vortices.These formations,varying in sizes and lifespans,significantly influence the distribution of fluid velocities within the flow.Subsequently,the rapid velocity fluctuations in highly turbulent flows lead to elevated shear and normal stress levels.For this reason,to meticulously study these dynamics,more often than not,physical modeling is employed for studying the impact of turbulent flows on the stability and longevity of nearby structures.Despite the effectiveness of physical modeling,various monitoring challenges arise,including flow disruption,the necessity for concurrent gauging at multiple locations,and the duration of measurements.Addressing these challenges,image velocimetry emerges as an ideal method in fluid mechanics,particularly for studying turbulent flows.To account for measurement duration,a probabilistic approach utilizing a probability density function(PDF)is suggested to mitigate uncertainty in estimated average and maximum values.However,it becomes evident that deriving the PDF is not straightforward for all turbulence-induced stresses.In response,this study proposes a novel approach by combining image velocimetry with a stochastic model to provide a generic yet accurate description of flow dynamics in such applications.This integration enables an approach based on the probability of failure,facilitating a more comprehensive analysis of turbulent flows.Such an approach is essential for estimating both short-and long-term stresses on hydraulic constructions under assessment.

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..

Artificial neural network-based one-equation model for simulation of laminar-turbulent transitional flow

A mapping function between the Reynolds-averaged Navier-Stokes mean flow variables and transition intermittency factor is constructed by fully connected artificial neural network(ANN),which replaces the governing equation of the intermittency factor in transition-predictive Spalart-Allmaras(SA)-γmodel.By taking SA-γmodel as the benchmark,the present ANN model is trained at two airfoils with various angles of attack,Mach numbers and Reynolds numbers,and tested with unseen airfoils in different flow states.The a posteriori tests manifest that the mean pressure coefficient,skin friction coefficient,size of laminar separation bubble,mean streamwise velocity,Reynolds shear stress and lift/drag/moment coefficient from the present two-way coupling ANN model almost coincide with those from the benchmark SA-γmodel.Furthermore,the ANN model proves to exhibit a higher calculation efficiency and better convergence quality than traditional SA-γmodel.

Artificial neural network-based subgrid-scale models for LES of compressible turbulent channel flow

Fully connected neural networks(FCNNs)have been developed for the closure of subgrid-scale(SGS)stress and SGS heat flux in large-eddy simulations of compressible turbulent channel flow.The FCNNbased SGS model trained using data with Mach number Ma=3.0 and Reynolds number Re=3000 was applied to situations with different Mach numbers and Reynolds numbers.The input variables of the neural network model were the filtered velocity gradients and temperature gradients at a single spatial grid point.The a priori test showed that the FCNN model had a correlation coefficient larger than 0.91 and a relative error smaller than 0.43,with much better reconstructions of SGS unclosed terms than the dynamic Smagorinsky model(DSM).In a posteriori test,the behavior of the FCNN model was marginally better than that of the DSM in predicting the mean velocity profiles,mean temperature profiles,turbulent intensities,total Reynolds stress,total Reynolds heat flux,and mean SGS flux of kinetic energy,and outperformed the Smagorinsky model.

Effect of RANS Turbulence Model on Aerodynamic Behavior of Trains in Crosswind

The numerical simulation based on Reynolds time-averaged equation is one of the approved methods to evaluate the aerodynamic performance of trains in crosswind.However,there are several turbulence models,trains may present different aerodynamic performances in crosswind using different turbulence models.In order to select the most suitable turbulence model,the inter-city express 2(ICE2)model is chosen as a research object,6 different turbulence models are used to simulate the flow characteristics,surface pressure and aerodynamic forces of the train in crosswind,respectively.6 turbulence models are the standard k-ε,Renormalization Group(RNG)k-ε,Realizable k-ε,Shear Stress Transport(SST)k-ω,standard k-ωand Spalart-Allmaras(SPA),respectively.The numerical results and the wind tunnel experimental data are compared.The results show that the most accurate model for predicting the surface pressure of the train is SST k-ω,followed by Realizable k-ε.Compared with the experimental result,the error of the side force coefficient obtained by SST k-ωand Realizable k-εturbulence model is less than 1%.The most accurate prediction for the lift force coefficient is achieved by SST k-ω,followed by RNG k-ε.By comparing 6 different turbulence models,the SST k-ωmodel is most suitable for the numerical simulation of the aerodynamic behavior of trains in crosswind.

Effect of Time Step Size and Turbulence Model on the Open Water Hydrodynamic Performance Prediction of Contra-Rotating Propellers

A growing interest has been devoted to the contra-rotating propellers （CRPs） due to their high propulsive efficiency, torque balance, low fuel consumption, low cavitations, low noise performance and low hull vibration. Compared with the single-screw system, it is more difficult for the open water performance prediction because forward and aft propellers interact with each other and generate a more complicated flow field around the CRPs system. The current work focuses on the open water performance prediction of contra-rotating propellers by RANS and sliding mesh method considering the effect of computational time step size and turbulence model. The validation study has been performed on two sets of contra-rotating propellers developed by David W Taylor Naval Ship R ＆ D center. Compared with the experimental data, it shows that RANS with sliding mesh method and SST k-ω turbulence model has a good precision in the open water performance prediction of contra-rotating propellers, and small time step size can improve the level of accuracy for CRPs with the same blade number of forward and aft propellers, while a relatively large time step size is a better choice for CRPs with different blade numbers.

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.

A USM-Θ two-phase turbulence model for simulating dense gas-particle flows

A second-order moment two-phase turbulence model for simulating dense gas-particle flows （USM-Θ model）, combining the unified second-order moment twophase turbulence model for dilute gas-particle flows with the kinetic theory of particle collision, is proposed. The interaction between gas and particle turbulence is simulated using the transport equation of two-phase velocity correlation with a two-time-scale dissipation closure. The proposed model is applied to simulate dense gas-particle flows in a horizontal channel and a downer. Simulation results and their comparison with experimental results show that the model accounting for both anisotropic particle turbulence and particle-particle collision is obviously better than models accounting for only particle turbulence or only particle-particle collision. The USM-Θ model is also better than the k-ε-kp-Θ model and the k-ε-kp-εp-Θ model in that the first model can simulate the redistribution of anisotropic particle Reynolds stress components due to inter-particle collision, whereas the second and third models cannot.

Mechanism of local scour around submarine pipelines based on numerical simulation of turbulence model

The mechanism of local scour around submarine pipelines is studied numerically based on a renormalized group （RNG） turbulence model. To validate the numerical model, the equilibrium profiles of local scour for two cases are simulated and compared with the experimental data. It shows that the RNG turbulence model can give an appropriate prediction for the configuration of equilibrium scour hole, and it is applicable to this situation. The local scour mechanism around submarine pipelines including the flow structure, shear stress distribution and pressure field is then analyzed and compared with experiments. For further comparison and validation, especially for the flow structure, a numerical calculation employing the large eddy simulation （LES） is also conducted. The numerical results of RNG demonstrate that the critical factor governing the equilibrium profile is the seabed shear stress distribution in the case of bed load sediment transport, and the two-equation RNG turbulence model coupled with the law of wall is capable of giving a satisfying estimation for the bed shear stress. Moreover, the piping phenomena due to the great difference of pressure between the upstream and downstream parts of pipelines and the vortex structure around submarine pipelines are also simulated successfully, which are believed to be the important factor that lead to the onset of local scour.

Turbulence and cavitation models for time-dependent turbulent cavitating flows

Cavitation typically occurs when the fluid pressure is lower than the vapor pressure at a local thermodynamic state, and the flow is frequently unsteady and turbulent. To assess the state-of-the-art of computational capabilities for unsteady cavitating flows, different cavitation and turbulence model combinations are conducted. The selected cavitation models include several widely-used models including one based on phenomenological argument and the other utilizing interface dynamics. The k-e turbulence model with additional implementation of the filter function and density correction function are considered to reduce the eddy viscosity according to the computed turbulence length scale and local fluid density respectively. We have also blended these alternative cavitation and lustrate that the eddy viscosity turbulence treatments, to ilnear the closure region can significantly influence the capture of detached cavity. From the experimental validations regarding the force analysis, frequency, and the cavity visualization, no single model combination performs best in all aspects. Furthermore, the implications of parameters contained in different cavitation models are investigated. The phase change process is more pronounced around the detached cavity, which is better illustrated by the interfacial dynamics model. Our study provides insight to aid further modeling development.

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.

Numerical Evaluation of Two k-εTurbulence Model for Predicting Flow and Solidification in Continuous Casting Slab

A steady three-dimensional fluid flow and solidification model was built based on CFD software by high-Reynolds-number and Lam-Bremhorst low-Reynolds-number k-ε model.During the simulation,the fixed-grid enthalpy-porosity technique was used to represent the solidification,and Darcy law was adopted to simulate the flow in mushy region.The prediction for steel flow and solidification was evaluated by the comparison of two turbulence models.It is found that both Lam-Bremhorst low-Reynolds-number and high-Reynolds-number k-ε models predict the same trend of the steel flow and temperature distribution.However,due to the effect of turbulent flow on heat transfer,the low-Reynolds-number turbulence model predicts longer penetration depth of molten steel in sub-mold region,less shell growth and higher shell surface temperature at the narrow face compared with standard k-ε model.

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.

Numerical prediction of inner turbulent flow in conical diffuser by using a new five-point scheme and DLR k-ε turbulence model

The internal turbulent flow in conical diffuser is a very complicated adverse pressure gradient flow.DLR k-ε turbulence model was adopted to study it.The every terms of the Laplace operator in DLR k-ε turbulence model and pressure Poisson equation were discretized by upwind difference scheme.A new full implicit difference scheme of 5-point was constructed by using finite volume method and finite difference method.A large sparse matrix with five diagonals was formed and was stored by three arrays of one dimension in a compressed mode.General iterative methods do not work wel1 with large sparse matrix.With algebraic multigrid method(AMG),linear algebraic system of equations was solved and the precision was set at 10-6.The computation results were compared with the experimental results.The results show that the computation results have a good agreement with the experiment data.The precision of computational results and numerical simulation efficiency are greatly improved.

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.

PARAMETER ESTIMATION OF ENGINEERING TURBULENCE MODEL

A parameter estimation algorithm is introduced and used to determine the parameters in the standard k-epsilon two equation turbulence model (SKE). It can be found from the estimation results that although the parameter estimation method is an effective method to determine model parameters, it is. difficult to obtain a set of parameters for SKE to suit all kinds of separated flow and a modification of the turbulence model structure should be considered. So, a new nonlinear k-e two-equation model (NNKE) is put forward in this paper and the corresponding parameter estimation technique is applied to determine the model parameters. By implementing the NNKE to solve some engineering turbulent flows, it is shown that NNKE is more accurate and versatile than SKE. Thus, the success of NNKE implies that the parameter estimation technique may have a bright prospect in engineering turbulence model research.

Turbulence Models of Hydrodynamic Lubrication

The main theoretical turbulence models for application to hydrodynamic lubrication problems were briefly reviewed, and the course of their development and their fundamentals were explained. Predictions by these models on flow fields in turbulent Couette flows and shear-induced countercurrent flows were compared to existing measurements, and Zhang & Zhang's combined k-ε model was shown to have surpassingly satisfactory results. The method of application of this combined k-ε model to high speed journal bearings and annular seals was summarized, and the predicted results were shown to be satisfactory by comparisons with existing experiments of journal bearings and annular seals.

REVISED k-ε TURBULENCE MODEL IN ELECTROMAGNETIC CONTINUOUS CASTING OF MELT

The research is motivated by the ongoing the electromagnetic continuous casting of molten metal. The revised k-ε model considering the effect of magnetic field application was derived. The specific model equations for the electromagnetic braking were used to calculate the velocity distribution in the continuous casting mold of steel. The results show that the revised k-ε model considering the effect of magnetic field application tends to suppress the production of turbulence and difference between the conventional and revised k-e model is small.

End-to-end differentiable learning of turbulence models from indirect observations

The emerging push of the differentiable programming paradigm in scientific computing is conducive to training deep learning turbulence models using indirect observations.This paper demonstrates the viability of this approach and presents an end-to-end differentiable framework for training deep neural networks to learn eddy viscosity models from indirect observations derived from the velocity and pressure fields.The framework consists of a Reynolds-averaged Navier–Stokes(RANS)solver and a neuralnetwork-represented turbulence model,each accompanied by its derivative computations.For computing the sensitivities of the indirect observations to the Reynolds stress field,we use the continuous adjoint equations for the RANS equations,while the gradient of the neural network is obtained via its built-in automatic differentiation capability.We demonstrate the ability of this approach to learn the true underlying turbulence closure when one exists by training models using synthetic velocity data from linear and nonlinear closures.We also train a linear eddy viscosity model using synthetic velocity measurements from direct numerical simulations of the Navier–Stokes equations for which no true underlying linear closure exists.The trained deep-neural-network turbulence model showed predictive capability on similar flows.