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..

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.

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.

Calculation grid and turbulence model for numerical simulating pressure fluctuations in high-speed train tunnel

Calculation grid and turbulence model for numerical simulating pressure fluctuations in a high-speed train tunnel are studied through the comparison analysis of numerical simulation and moving model test.Compared the waveforms and peak-peak values of pressure fluctuations between numerical simulation and moving model test,the structured grid and the SST k-ωturbulence model are selected for numerical simulating the process of high-speed train passing through the tunnel.The largest value of pressure wave amplitudes of numerical simulation and moving model test meet each other.And the locations of the largest value of the initial compression and expansion wave amplitude of numerical simulation are in agreement with that of moving model test.The calculated pressure at the measurement point fully conforms to the propagation law of compression and expansion waves in the tunnel.

Comparison of the Reynolds-averaged Turbulence Models on Single Phase Flow Simulation in Agitated Extraction Columns

The flow field of liquid phase (water) of agitated extraction columns is simulated with the help of computational fluid dynamics (CFD). Four kinds of Reynolds-averaged turbulence models, i.e. the standard k-ε model, the RNG (renormalization group) k-s model, the realizable k-ε model and the Reynolds stress model, are compared in detail in order to judge which is the best model in terms of the accuracy, less CPU time and memory required. The performance of the realizable k-s model is obviously improved by reducing the model constant from C2 = 1.90 to C2 = 1.61. It is concluded that the improved realizable k-e model is the optimal model.

Basic Pattern in Atmospheric Turbulence Model

From the controlling equations of atmosphere motion, Prandtl's mixing length theory is used to derive the atmospheric turbulence models, such as Burgers equation model and Burgers-KdV equation model. And then the projective Riccati equations are applied to solve these atmospheric turbulence models, where much more patterns are obtained, including solitary wave pattern, singular pattern, and so on.

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.

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.

Numerical simulation of flow around square cylinder using different low-Reynolds number turbulence models

ABE-KONDOH-NAGANO,ABID,YANG-SHIH and LAUNDER-SHARMA low-Reynolds number turbulence models were applied to simulating unsteady turbulence flow around a square cylinder in different phases flow field and time-averaged unsteady flow field.Meanwhile,drag and lift coefficients of the four different low-Reynolds number turbulence models were analyzed.The simulated results of YANG-SHIH model are close to the large eddy simulation results and experimental results,and they are significantly better than those of ABE-KONDOH-NAGANO,ABID and LAUNDER-SHARMR models.The modification of the generation of turbulence kinetic energy is the key factor to a successful simulation for YANG-SHIH model,while the correction of the turbulence near the wall has minor influence on the simulation results.For ABE-KONDOH-NAGANO,ABID and LAUNDER-SHARMA models satisfactory simulation results cannot be obtained due to lack of the modification of the generation of turbulence kinetic energy.With the joint force of wall function and the turbulence models with the adoption of corrected swirl stream,flow around a square cylinder can be fully simulated with less grids by the near-wall.

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.

Comparison of Reynolds average Navier-Stokes turbulence models in numerical simulations of the DC arc plasma torch

Five turbulence models of Reynolds average Navier-Stokes(RANS),including the standard k-ω model,the RNG k-e model taking into account the low Reynolds number effect,the realizable k-ω model,the SST k-ω model,and the Reynolds stress model(RSM),are employed in the numerical simulations of direct current(DC)arc plasma torches in the range of arc current from 80 A to 240 A and air gas flow rate from 10 m^3 h^-1 to 50 m^3 h^-1.The calculated voltage,electric field intensity,and the heat loss in the arc chamber are compared with the experiments.The results indicate that the arc voltage,the electric field,and the heat loss in the arc chamber calculated by using the standard k-ω model,the RNG k-ωmodel taking into account the low Reynolds number effect,and the realizable k-ω model are much larger than those in the experiments.The RSM predicts relatively close results to the experiments,but fails in the trend of heat loss varying with the gas flow rate.The calculated results of the SST k-ω model are in the best agreement with the experiments,which may be attributed to the reasonable predictions of the turbulence as well as its distribution.

Turbulence Model Investigations on the Boundary Layer Flow with Adverse Pressure Gradients

In this paper, a numerical study of flow in the turbulence boundary layer with adverse and pressure gradients （APGs） is conducted by using Reynolds-averaged Navier-Stokes （RANS） equations. This research chooses six typical turbulence models, which are critical to the computing precision, and to evaluating the issue of APGs. Local frictional resistance coefficient is compared between numerical and experimental results. The same comparisons of dimensionless averaged velocity profiles are also performed. It is found that results generated by Wilcox （2006） k-co are most close to the experimental data. Meanwhile, turbulent quantities such as turbulent kinetic energy and Reynolds-stress are also studied.

Performances of Different Turbulence Models for Simulating Shallow Water Sloshing in Rectangular Tank

Liquid sloshing is a common phenomenon in the transportation of liquid-cargo tanks.Liquid waves lead to fluctuating forces on the tank walls.If these fluctuations are not predicted or controlled,for example,by using baffles,they can lead to large forces and momentums.The volume of fluid(VOF)two-phase numerical model in Open FOAM open-source software has been widely used to model the liquid sloshing.However,a big challenge for modeling the sloshing phenomenon is selecting a suitable turbulence model.Therefore,in the present study,different turbulence models were studied to determine their sloshing phenomenon prediction accuracies.The predictions of these models were validated using experimental data.The turbulence models were ranked by their mean error in predicting the free surface behaviors.The renormalization group(RNG)k-ε and the standard k–ω models were found to be the best and worst turbulence models for modeling the sloshing phenomena,respectively;moreover,the SST k-ω model and v2-f k-ε results were very close to the RNG k-εmodel result.

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.

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.

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.

A K-εTWO-EQUATION TURBULENCE MODEL FOR THE SOLID-LIQUID TWO-PHASE FLOWS

A two-equation turbulence model has been dereloped for predicting two-phase flow the two equations describe the conserration of turbulence kinetic energy and dissipation rate of that energy for the incompressible carrier fluid in a two-phase flow The continuity, the momentum, K and εequations are modeled. In this model,the solid-liquid slip veloeites, the particle-particte interactions and the interactions between two phases are considered,The sandy water pipe turbulent flows are sueeessfuly predicted by this turbulince model.

Comparative Experimental and Numerical Study of Wave Loads on A Monopile Structure Using Different Turbulence Models

This study numerically and experimentally investigates the effects of wave loads on a monopile-type offshore wind turbine placed on a 1:25 slope at different water depths as well as the effect of choosing different turbulence models on the efficiency of the numerical model.The numerical model adopts a two-phase flow by solving Unsteady Reynolds-Averaged Navier−Stokes(URANS)equations using the Volume Of Fluid(VOF)method and three differentk-ωturbulence models.Typical environmental conditions from the East China Sea are studied.The wave run-up and the wave loads applied on the monopile are investigated and compared with relevant experimental data as well as with mathematical predictions based on relevant theories.The numerical model is well validated against the experimental data at model scale.The use of different turbulence models results in different predictions on the wave height but less differences on the wave period.The baseline k-ωturbulence model and Shear-Stress Transport(SST)k-ωturbulence model exhibit better performance on the prediction of hydrodynamic load,at a model-scale water depth of 0.42 m,while the laminar model provides better results for large water depths.The SST turbulence model performs better in predicting wave run-up for water depth 0.42 m,while the laminar model and standard k-ωmodel perform better at water depth 0.52 m and 0.62 m,respectively.

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.

COMPUTATION OF TURBULENCE FLOW ON HYBRID GRIDS USING k-ω TURBULENCE MODEL AND OSHER SCHEME

An unstructured Reynolds-averaged Navier-Stokes flow solver using the finite volume method is studied. The spatial discretisation is based on the Osher approximate Riemann solvers. A two-equation turbulence model (k-ω model) is also developed for hybrid grids to compute the turbulence flow. The turbulence flow past NACA0012 airfoil and the double ellipsolids are computed, and the numerical results show that the above methods are very efficient.