Nonlinear turbulence models for predicting strong curvature effects

Prediction of the characteristics of turbulent flows with strong streamline curvature, such as flows in turbomachines, curved channel flows, flows around airfoils and buildings, is of great importance in engineering applications and poses a very practical challenge for turbulence modeling. In this paper, we analyze qualitatively the curvature effects on the structure of turbulence and conduct numerical simulations of a turbulent Uduct flow with a number of turbulence models in order to assess their overall performance. The models evaluated in this work are some typical linear eddy viscosity turbulence models, nonlinear eddy viscosity turbulence models （NLEVM） （quadratic and cubic）, a quadratic explicit algebraic stress model （EASM） and a Reynolds stress model （RSM） developed based on the second-moment closure. Our numerical results show that a cubic NLEVM that performs considerably well in other benchmark turbulent flows, such as the Craft, Launder and Suga model and the Huang and Ma model, is able to capture the major features of the highly curved turbulent U-duct flow, including the damping of turbulence near the convex wall, the enhancement of turbulence near the concave wall, and the subsequent turbulent flow separation. The predictions of the cubic models are quite close to that of the RSM, in relatively good agreement with the experimental data, which suggests that these models may be employed to simulate the turbulent curved flows in engineering applications.     

Applications of URANS on predicting unsteady turbulent separated flows

Accurate prediction of unsteady separated turbulent flows remains one of the toughest tasks and a practi cal challenge for turbulence modeling. In this paper, a 2D flow past a circular cylinder at Reynolds number 3,900 is numerically investigated by using the technique of unsteady RANS （URANS）. Some typical linear and nonlinear eddy viscosity turbulence models （LEVM and NLEVM） and a quadratic explicit algebraic stress model （EASM） are evaluated. Numerical results have shown that a high-performance cubic NLEVM, such as CLS, are superior to the others in simulating turbulent separated flows with unsteady vortex shedding.     

SECOND-ORDER MOMENT MODEL FOR DENSE TWO-PHASE TURBULENT FLOW OF BINGHAM FLUID WITH PARTICLES

The USM-θmodel of Bingham fluid for dense two-phase turbulent flow was developed, which combines the second-order moment model for two-phase turbulence with the particle kinetic theory for the inter-particle collision. In this model, phases interaction and the extra term of Bingham fluid yield stress are taken into account. An algorithm for USM-θmodel in dense two-phase flow was proposed, in which the influence of particle volume fraction is accounted for. This model was used to simulate turbulent flow of Bingham fluid single-phase and dense liquid-particle two-phase in pipe. It is shown USM-θmodel has better prediction result than the five-equation model, in which the particle-particle collision is modeled by the particle kinetic theory, while the turbulence of both phase is simulated by the two-equation turbulence model. The USM-θmodel was then used to simulate the dense two-phase turbulent up flow of Bingham fluid with particles. With the increasing of the yield stress, the velocities of Bingham and particle decrease near the pipe centre. Comparing the two-phase flow of Bingham-particle with that of liquid-particle, it is found the source term of yield stress has significant effect on flow.     

Studies on gas turbulence and particle fluctuation in dense gas-particle flows

Particle fluctuation and gas turbulence in dense gas-particle flows are less studied due to complexity of the phenomena. In the present study, simulations of gas turbulent flows passing over a single particle are carried out first by using RANS modeling with a Reynolds stress equation turbulence model and sufficiently fine grids, and then by using LES. The turbulence enhancement by the particle wake effect is studied under various particle sizes and relative gas velocities, and the turbulence enhancement is found proportional to the particle diameter and the square of velocity. Based on the above results, a turbulence enhancement model for the particle-wake effect is proposed and is incorporated as a sub-model into a comprehensive two-phase flow model, which is then used to simulate dilute gas-particle flows in a horizontal channel. The simulation results show that the predicted gas turbulence by using the present model accounting for the particle wake effect is obviously in better agreement with the experimental results than the prediction given by the model not accounting for the wake effect. Finally, the proposed model is incorporated into another two-phase flow model to simulate dense gasparticle flows in a downer. The results show that the particle wake effect not only enhances the gas turbulence, but also amplifies the particle fluctuation.     

Swirling-strength based large eddy simulation of turbulent flow around single square cylinder at low Reynolds numbers

In view of the fact that large scale vortices play the substantial role of momentum transport in turbulent flows, large eddy simulation(LES) is considered as a better simulation model. However, the sub-grid scale(SGS) models reported so far have not ascertained under what flow conditions the LES can lapse into the direct numerical simulation. To overcome this discrepancy, this paper develops a swirling strength based the SGS model to properly model the turbulence intermittency, with the primary characteristics that when the local swirling strength is zero, the local sub-grid viscosity will be vanished. In this paper, the model is used to investigate the flow characteristics of zero-incident incompressible turbulent flows around a single square cylinder(SC)at a low Reynolds number range Re ∈ [103, 104]. The flow characteristics investigated include the Reynolds number dependence of lift and drag coefficients, the distributions of time-spanwise averaged variables such as the sub-grid viscosity and the logarithm of Kolmogorov micro-scale to the base of 10 at Re = 2 500 and 104, the contours of spanwise and streamwise vorticity components at t = 170. It is revealed that the peak value of sub-grid viscosity ratio and its root mean square(RMS) values grow with the Reynolds number. The dissipation rate of turbulent kinetic energy is larger near the SC solid walls.The instantaneous factor of swirling strength intermittency(FSI) exhibits some laminated structure involved with vortex shedding.     

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 kε 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 turbulence treatments,to illustrate that the eddy viscosity near 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 illus-trated by the interfacial dynamics model.Our study provides insight to aid further modeling development.     

Precise drag prediction of airfoil flows by a new algebraic model

We report the results of accurate prediction of lift(C L)and drag(C D)coefficients of two typical airfoil flows(NACA0012 and RAE2822)by a new algebraic turbulence model,in which the eddy viscosity is specified by a stress length(SL)function predicted by structural ensemble dynamics(SED)theory.Unprecedented accuracy of the prediction of C D with error of a few counts(one count is 10−4)and of C L with error under 1%-2%are uniformly obtained for varying angles of attack(AoA),indicating an order of magnitude improvement of drag prediction accuracy compared to currently used models(typically around 20 to 30 counts).More interestingly,the SED-SL model is distinguished with fewer parameters of clear physical meaning,which quantify underlying turbulent boundary layer(TBL)with a universal multi-layer structure,and is thus promising to be more easily generalizable to complex TBL.The use of the new model for the calibration of flow condition in experiment and the extraction of flow physics from numerical simulation data of aeronautic flows are discussed.     

Generation of linear and nonlinear waves in numerical wave tank using clustering technique-volume of fluid method

A two-dimensional(2D) numerical model is developed for the wave simulation and propagation in a wave flume.The fluid flow is assumed to be viscous and incompressible,and the Navier-Stokes and continuity equations are used as the governing equations.The standard k-ε model is used to model the turbulent flow.The NavierStokes equations are discretized using the staggered grid finite difference method and solved by the simplified marker and cell(SMAC) method.Waves are generated and propagated using a piston type wave maker.An open boundary condition is used at the end of the numerical flume.Some standard tests,such as the lid-driven cavity,the constant unidirectional velocity field,the shearing flow,and the dam-break on the dry bed,are performed to valid the model.To demonstrate the capability and accuracy of the present method,the results of generated waves are compared with available wave theories.Finally,the clustering technique(CT) is used for the mesh generation,and the best condition is suggested.     

Effect of wall boundary condition on CFD simulation of CFB risers

The effect of solid-phase wall boundary condition on the numerical simulation of gas-solid flow in CFB risers containing FCC particles was investigated using the two-fluid model incorporating the kinetic theory of granular flow.Both the Gidaspow drag model and the EMMS-based drag model were used.The Johnson and Jackson(1987) wall boundary condition was applied to describe the interaction between particles and wall.Based on the experimental system of Li and Kwauk(1994),parametric studies of specularity coefficient(φ=1.0,0.6,0.0005,0.00005,0) and particle-wall restitution coefficient(e_w=0.6,0.9,0.95,0.99,0.999) were performed to evaluate their effects on axial voidage profile,solids flux,meso-scale and heterogeneous structures.Simulation results showed that solid-phase wall boundary condition had little effect on axial voidage profile when the Gidaspow drag model was used.However,the specularity coefficient φ had a pronounced influence on flow behavior when the EMMS-based drag model was used,and a small specularity coefficient(φ=0.00005,0) could result in better agreement with experimental data.The particle-wall restitution coefficient e_w plays but a minor role in the holistic flow characteristics.     

Influence of variable viscosity and double diffusion on the convective stability of a nanofluid flow in an inclined porous channel

The influence of variable viscosity and double diffusion on the convective stability of a nanofluid flow in an inclined porous channel is investigated. The DarcyBrinkman model is used to characterize the fluid flow dynamics in porous materials. The analytical solutions are obtained for the unidirectional and completely developed flow.Based on a normal mode analysis, the generalized eigenvalue problem under a perturbed state is solved. The eigenvalue problem is then solved by the spectral method....     

Large eddy simulation of fuel injection and mixing process in a diesel engine

The large eddy simulation(LES) approach implemented in the KIVA-3V code and based on one-equation sub-grid turbulent kinetic energy model are employed for numerical computation of diesel sprays in a constant volume vessel and in a Caterpillar 3400 series diesel engine.Computational results are compared with those obtained by an RANS(RNG k-ε) model as well as with experimental data.The sensitivity of the LES results to mesh resolution is also discussed.The results show that LES generally provides flow and spray characteristics in better agreement with experimental data than RANS;and that small-scale random vortical structures of the in-cylinder turbulent spray field can be captured by LES.Furthermore,the penetrations of fuel droplets and vapors calculated by LES are larger than the RANS result,and the sub-grid turbulent kinetic energy and sub-grid turbulent viscosity provided by the LES model are evidently less than those calculated by the RANS model.Finally,it is found that the initial swirl significantly affects the spray penetration and the distribution of fuel vapor within the combustion chamber.     

Effect of the size and pressure on the modified viscosity of water in microchannels

The phenomenon that flow resistances are higher in micro scale flow than in normal flow is clarified through the liquid viscosity. Based on the experimental results of deionized water flow in fused silica microtubes with the inner radii of 2.5 μm, 5 μm, 7.5 μm, and 10 μm, respectively,the relationship between water flow velocity and pressure gradient along the axis of tube is analyzed, which gradually becomes nonlinear as the radius of the microtube decreases.From the correlation, a viscosity model of water flow derived from the radius of microtube and the pressure gradient is proposed. The flow results modified by the viscosity model are in accordance with those of experiments, which are verified by numerical simulation software and the Hagen–Poiseuille equation. The experimental water flow velocity in a fused silica microtube with diameter of 5.03 μm, which has not been used in the fitting and derivation of the viscosity model,is proved to be comsistent with the viscosity model, showing a rather good match with a relative difference of 5.56%.     

Vibration induced flow in hoppers: DEM 2D polygon model

A two-dimensional discrete element model （DEM） simulation of cohesive polygonal particles has been developed to assess the benefit of point source vibration to induce flow in wedge-shaped hoppers. The particle-particle interaction model used is based on a multi-contact principle. The first part of the study investigated particle discharge under gravity without vibration to determine the critical orifice size （Bc） to just sustain flow as a function of particle shape. It is shown that polygonal-shaped particles need a larger orifice than circular particles. It is also shown that Bc decreases as the number of particle vertices increases. Addition of circular particles promotes flow of polygons in a linear manner. The second part of the study showed that vibration could enhance flow, effectively reducing Bc. The model demonstrated the importance of vibrator location （height）, consistent with previous continuum model results, and vibration amplitude in enhancing flow.     

Modeling of drag reduction in turbulent channel flow with hydrophobic walls by FVM method and weakly-compressible flow equations

In this paper the effects of hydrophobic wall on skin-friction drag in the channel flow are investigated through large eddy simulation on the basis of weaklycompressible flow equations with the MacCormack's scheme on collocated mesh in the FVM framework. The slip length model is adopted to describe the behavior of the slip velocities in the streamwise and spanwise directions at the interface between the hydrophobic wall and turbulent channel flow. Simulation results are presented by analyzing flow behaviors over hydrophobic wall with the Smagorinky subgrid-scale model and a dynamic model on computational meshes of different resolutions. Comparison and analysis are made on the distributions of timeaveraged velocity, velocity fluctuations, Reynolds stress as well as the skin-friction drag. Excellent agreement between the present study and previous results demonstrates the accuracy of the simple classical second-order scheme in representing turbulent vertox near hydrophobic wall. In addition, the relation of drag reduction efficiency versus time-averaged slip velocity is established. It is also foundthat the decrease of velocity gradient in the close wall region is responsible for the drag reduction. Considering its advantages of high calculation precision and efficiency, the present method has good prospect in its application to practical projects.     

NUMERICAL SIMULATION OF COMPLICATED PIPE FLOW WITH k-εMODEL

In this paper, turbulence in a complicated pipe is simulated by using the k-ε model. The ladder-like mesh approximation is used to solve the problem of complicated boundary with the result of numerical simulation favorable. Two computational examples are given to validate the strong adaptability and stability of k-ε model.     

Numerical simulations of gas-particle flow behavior created by low-level rotary-winged aircraft flight over particle bed

The aerodynamics of gas-particle suspensions is simulated as an Euler-Euler two-fluid model in a revolving rotor over a particle bed. The interactions of collisions between the blade and particles and particle-particle interactions are modeled using the kinetic theory of granular flow(KTGF). The gas turbulence induced by the rotation of the rotor is modeled using the kg-εg model. The flow field of a revolving rotor is simulated using the multiple reference frame(MRF) method. The distributions of velocities, volume fractions, and gas pressure are predicted while the aircraft hovers at different altitudes.The gas pressure decreases from the hub to the tip of the blade, and it is higher at the pressure side than that at the suction side of the rotor. The turbulent kinetic energy of the gas increases toward the blade tip. The volume fraction of particles decreases as the hovering altitude increases. The simulated pressure coefficient is compared with that in experimental measurements.     

An investigation on internal thermal flow of non-Newtonian fluid

In the present paper an unsteady thermal flow of non-Newtonian fluid is investigated which is of the fiow into axisymmetric mould cavity. In the second part an unsteady thermal flow of upper-convected Maxwell fluid is studied, For the flow into mould cavity the constitutive equation of power-law fluid is used as a rheological model of polymer fluid. The apparent viscosity is considered as a function of shear rate and temperature. A characteristic viscosity is introduced in order to avoid the nonlinearity due to the temperature dependence of the apparent viscosity. As the viscosity of the fluid is relatively high the flow of the thermal fluid can be considered as a flow of fully developed velocity field. However, the temperature field of the fluid fiow is considered as an unsteady one. The governing equations are constitutive equation, momentum equation of steady flow and energy conservation equation of non-steady form. The present system of equations has been solved numerically by the splitting differen     

Direct modeling for computational fluid dynamics

All fluid dynamic equations are valid under their modeling scales, such as the particle mean free path and mean collision time scale of the Boltzmann equation and the hydrodynamic scale of the Navier–Stokes(NS) equations.The current computational fluid dynamics(CFD) focuses on the numerical solution of partial differential equations(PDEs), and its aim is to get the accurate solution of these governing equations. Under such a CFD practice, it is hard to develop a unified scheme that covers flow physics from kinetic to hydrodynamic scales continuously because there is no such governing equation which could make a smooth transition from the Boltzmann to the NS modeling. The study of fluid dynamics needs to go beyond the traditional numerical partial differential equations. The emerging engineering applications, such as air-vehicle design for near-space flight and flow and heat transfer in micro-devices, do require further expansion of the concept of gas dynamics to a larger domain of physical reality, rather than the traditional distinguishable governing equations. At the current stage, the non-equilibrium flow physics has not yet been well explored or clearly understood due to the lack of appropriate tools.Unfortunately, under the current numerical PDE approach, it is hard to develop such a meaningful tool due to the absence of valid PDEs. In order to construct multiscale and multiphysics simulation methods similar to the modeling process of constructing the Boltzmann or the NS governing equations, the development of a numerical algorithm should be based on the first principle of physical modeling. In this paper, instead of following the traditional numerical PDE path, we introduce direct modeling as a principle for CFD algorithm development. Since all computations are conducted in a discretized space with limited cell resolution, the flow physics to be modeled has to be done in the mesh size and time step scales.Here, the CFD is more or less a direct construction of discrete numerical evolution equations, where the mesh size and time step will play dynamic roles in the modeling process.With the variation of the ratio between mesh size and local particle mean free path, the scheme will capture flow physics from the kinetic particle transport and collision to the hydrodynamic wave propagation. Based on the direct modeling, a continuous dynamics of flow motion will be captured in the unified gas-kinetic scheme. This scheme can be faithfully used to study the unexplored non-equilibrium flow physics in the transition regime.     

Investigation of scale effect for the computation of turbulent flow around a circular cylinder

In order to investigate the scale effect of turbulent flow around a circular cylinder, two similarity numbers （criteria） based on turbulent kinetic and dissipation rates associ- ated with the fluctuation characteristics of turbulence wake are deduced by analyzing the Reynolds averaged NavierStokes equations （RANS）. The RNG k-s models and finite volume method are used to solve the governing equations and the second-order implicit time and upwind space discretization algorithms are used to discrete the governing equations. A numerical computation of flow parameters around a two-dimensional circular cylinder with Reynolds numbers ranging from 102 to l07 is accomplished and the result indicates that the fluctuation of turbulence flow along the center line in the wake of circular cylinder can never be changed with increasing Reynolds numbers when Re ≥ 3 × 10^6. This conclusion is useful for controlling the scale of numerical calculations and for applying model test data to engineering practice.