A Study of Eddy Viscosity Coefficient in Numerical Tidal Simulation

Based on the fluid motion equations, the physical meaning of eddy viscosity coefficient and the rationality of the Boussinesq hypothesis are discussed in this paper. The effect of the coefficient on numerical stability is analyzed briefly. A semi-enclosed rectangular sea area, with an orthogonal spur dike, is applied in a 2-D numerical model to study the effect of horizontal eddy viscosity coefficient (A(H)), The computed result shows that A(H) has little influence on the tidal level and averaged flow velocity, but has obvious influence on the intensity and the range of return flow around near the spur dike. Correspondingly, a wind-driven current pool and an annular current are applied in a 3-D numerical model respectively to study the effect of vertical eddy viscosity coefficient (A(V)). The computed result shows that the absolute value of A(V) is inversely proportional to that of horizontal velocity, and the vertical gradient value of A(V) determines the vertical distribution of horizontal velocity, The distribution form of A(V) is theoretically recommended as a parabolic type, of which the maximum value appears at 0.5 H.

Modeling of the eddy viscosity by breaking waves

Breaking wave induced nearsurface turbulence has important consequences for many physical and biochemical processes including water column and nutrients mixing, heat and gases exchange across air-sea interface. The energy loss from wave breaking and the bubble plume penetration depth are estimated. As a consequence, the vertical distribution of the turbulent kinetic energy （TKE）, the TKE dissipation rate and the eddy viscosity induced by wave breaking are also provided. It is indicated that model results are found to be consistent with the observational evidence that most TKE generated by wave breaking is lost within a depth of a few meters near the sea surface. High turbulence level with intensities of eddy viscosity induced by breaking is nearly four orders larger than vw1（ = κu ＊wz）, the value predicted for the wall layer scaling close to the surface, where u ＊w is the friction velocity in water, κ with 0. 4 is the yon Kármán constant, and z is the water depth, and the strength of the eddy viscosity depends both on wind speed and sea state, and decays rapidly through the depth. This leads to the conclusion that the breaking wave induced vertical mixing is mainly limited to the near surface layer, well above the classical values expected from the similarity theory. Deeper down, however, the effects of wave breaking on the vertical mixing become less important.

Estimation of eddy viscosity on the South China Sea shelf with adjoint assimilation method

The eddy viscosity of the ocean is an important parameter indicating the small-scale mixing process in the oceanic interior water column. Ekman wind-driven current model and adjoint assimilation technique are used to calculate the vertical profiles of eddy viscosity by fitting model results to the observation data. The data used in the paper include observed wind data and ADCP data obtained at Wenchang Oil Rig on the SCS （the South China Sea） shelf in August 2002. Different simulations under different wind conditions are analyzed to explore how the eddy viscosity develops with varying wind field. The results show that the eddy viscosity endured gradual variations in the range of 10^-3 -10^-2 m^2 /s during the periods of wind changes. The mean eddy viscosity undergoing strong wind could rise by about 25% as compared to the value under weak wind.

A Simple Eddy Viscosity Model of Rough Turbulent Wave Boundary Layer

A one-layer time-invariant eddy viscosity model is specified to develop a mathematical model for describing the essential features of the turbulent wave boundary layer over a rough bed. The functional form of the eddy viscosity is evaluated based on computational results from a two-equation turbulence model in which the eddy viscosity varies with time and space. The present eddy viscosity model simplifies much of the mathematical complexity in many existing models. Predictions from the present model have been compared with a wide range of experimental data. It is found that the eddy viscosity model adopted in the present study is physically reasonable.

Effect of constant eddy viscosity assumption on optimization using a discrete adjoint method

This paper presents a thorough study of the effect of the Constant Eddy Viscosity(CEV)assumption on the optimization of a discrete adjoint-based design optimization system.First,the algorithms of the adjoint methods with and without the CEV assumption are presented,followed by a discussion of the two methods’solution stability.Second,the sensitivity accuracy,adjoint solution stability,and Root Mean Square(RMS)residual convergence rates at both design and offdesign operating points are compared between the CEV and full viscosity adjoint methods in detail.Finally,a multi-point steady aerodynamic and a multi-objective unsteady aerodynamic and aeroelastic coupled design optimizations are performed to study the impact of the CEV assumption on optimization.Two gradient-based optimizers,the Sequential Least-Square Quadratic Programming(SLSQP)method and Steepest Descent Method(SDM)are respectively used to draw a firm conclusion.The results from the transonic NASA Rotor 67 show that the CEV assumption can deteriorate RMS residual convergence rates and even lead to solution instability,especially at a near stall point.Compared with the steady cases,the effect of the CEV assumption on unsteady sensitivity accuracy is much stronger.Nevertheless,the CEV adjoint solver is still capable of achieving optimization goals to some extent,particularly if the flow under consideration is benign.

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 STABILIZED SUBGRID EDDY VISCOSITY METHOD BASED ON PRESSURE PROJECTION AND EXTRAPOLATED TRAPEZOIDAL RULE FOR THE TRANSIENT NAVIER-STOKES EQUATIONS

We consider a new subgrid eddy viscosity method based on pressure projection and extrapolated trapezoidal rule for the transient Navier-Stokes equations by using lowest equal-order pair of finite elements. The scheme stabilizes convection dominated problems and ameliorates the restrictive inf-sup compatibility stability. It has some attractive fea- tures including parameter free for the pressure stabilized term and calculations required for higher order derivatives. Moreover, it requires only the solutions of the linear system arising from an Oseen problem per time step and has second order temporal accuracy. The method achieves optimal accuracy with respect to solution regularity.

Dynamic modeling of the horizontal eddy viscosity coefficient for quasigeostrophic ocean circulation problems

This paper puts forth a simplified dynamic modeling strategy for the eddy viscosity coefficient parameterized in space and time.The eddy viscosity coefficient is dynamically adjusted to the local structure of the flow using two different nonlinear eddy viscosity functional forms to capture anisotropic dissipation mechanism,namely,(i)the Smagorinsky model using the local strain rate field,and(ii)the Leith model using the gradient of the vorticity field.The proposed models are applied to the one-layer and two-layer wind-driven quasigeostrophic ocean circulation problems,which are standard prototypes of more realistic ocean dynamics.Results show that both models capture the quasi-stationary ocean dynamics and provide the physical level of eddy viscosity distribution without using any a priori estimation.However,it is found that slightly less dissipative results can be obtained by using the dynamic Leith model.Two-layer numerical experiments also reveal that the proposed dynamic models automatically parameterize the subgrid-scale stress terms in each active layer.Furthermore,the proposed scale-aware models dynamically provide higher values of the eddy viscosity for smaller resolutions taking into account the local resolved flow information,and addressing the intimate relationship between the eddy viscosity coefficients and the numerical resolution employed by the quasigeostrophic models.

THE INFLUENCE OF EDDY VISCOSITY ON THE SUBTROPICAL PLANETARY-SCALE CIRCULATION IN SUMMER

The influence of eddy viscosity on the distribution of subtropical quasi-stationary planetary waves in summer is analysed theoretically.It is found that since the basic flow is very weak,the eddy viscosity may play an impor- tant role for the subtropical planetary-scale motion in summer. A linear,quasi-geostrophic,34-level spherical coordinate model is also utilized to calculate the differences of quasi-stationary planetary waves and of quasi-stationary disturbance pattern responding to forcing by topography and heat sources under the different eddy viscosities.The computed results show that the coefficient of eddy viscosity considerably influences the strength of the subtropical planetary-scale circulation in summer.

Tide-Induced Mixing in the Bottom Boundary Layer in the Western East China Sea

The East China Sea(ECS)boasts a vast continental shelf,where strong tidal motions play an important role in the substance transport and energy budget.In this study,the tide-induced mixing in the bottom boundary layer in the western ECS is analyzed based on records measured by moored acoustic Doppler current profilers from June to October 2014.Results show that the M_(2) tide is strong and shows a barotropic feature,whereas the O_(1) tide is much weaker.Based on the M_(2) tidal currents,the eddy viscosity in the bottom Ekman boundary layer is estimated with three schemes.The estimated eddy viscosity values vary within 10^(-4)–10^(-2)m^(2) s^(−1),reaching a maximum at approximately 5 m height from the bottom and decreasing exponentially with the height at all three stations.Moreover,the shear production of turbulent kinetic energy is calculated to quantify the mixing induced by different tidal constituents.The results show that the shear production of the M_(2) tide is much stronger than that of the O_(1) tide and shows a bottom intensified feature.

Two dimensional analytical solution for a partially vegetated compound channel flow

The theory of an eddy viscosity model is applied to the study of the flow in a compound channel which is partially vegetated. The governing equation is constituted by analyzing the longitudinal forces acting on the unit volume where the effect of the vegetation on the flow is considered as a drag force item, The compound channel is divided into 3 sub-regions in the transverse direction, and the coefficients in every region＇s differential equations were solved simultaneously. Thus, the analytical solution of the transverse distribution of the depth-averaged velocity for uniform flow in a partially vegetated compound channel was obtained. The results can be used to predict the transverse distribution of bed shear stress, which has an important effect on the transportation of sediment. By comparing the analytical results with the measured data, the analytical solution in this paper is shown to be sufficiently accurate to predict most hydraulic features for engineering design purposes.

A Boussinesq Equation-Based Model for Nearshore Wave Breaking

Based on the wave breaking model by Li and Wang (1999), this work is to apply Dally's analytical solution to the wave-height decay instead of the empirical and semi-empirical hypotheses of wave-height distribution within the wave breaking zone. This enhances the applicability of the model. Computational results of shoaling, location of wave breaking, wave-height decay after wave breaking, set-down and set-up for incident regular waves are shown to have good agreement with experimental and field data.

Improvement of Prandtl mixing length theory and application in modeling of turbulent flow in circular tubes

In order to correctly predict tube cross section time-smoothed velocity distribution, friction factor and mass transfer behavior, two models for turbulent flow in circular tubes based on classical Prandtl mixing length theory and a modified mixing length were established. The results show that the modified mixing length includes the introduction of a damping function for the viscous sublayer and the second-order derivative to approximate eddy velocity. The calculated dimensionless time-smoothed velocity from the model based on Prandtl mixing length is much better than the result from the concept of eddy viscosity. The calculated eddy viscosity from the model based on modified mixing length is much better than the result from the model based on the classical Prandtl mixing length theory. And the friction factor calculated from the model based on the modified mixing length agrees well with the reported empirical relationships.

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.

SHORT-AND RESONANT-RANGE INTERACTIONS BETWEEN SCALES IN TURBULENCE AND THEIR APPLICATIONS

Interactions between different scales in turbulence were studied starting from the incompressible Navier-Stokes equations. The integral and differential formulae of the short-range viscous stresses, which express the short-range interactions between contiguous scales in turbulence,were given. A concept of the resonant-range interactions between extreme contiguous scales was introduced and the differential formula of the resonant-range viscous stresses was obtained. The short- and resonant-range viscous stresses were applied to deduce the large-eddy simulation (LES) equations as well as the multiscale equations, which are approximately closed and do not contain any empirical constants or relations. The properties and advantages of using the multiscale equations to compute turbulent flows were discussed. The short-range character of the interactions between the scales in turbulence means that the multiscale simulation is a very valuable technique for the calculation of turbulent flows. A few numerical examples were also given.

Numerical Investigation on Two-dimensional Boundary Layer Flow with Transition

As a basic problem in many engineering applications, transition from laminar to turbulence still remains a difficult problem in computational fluid dynamics （CFD）. A numerical study of one transitional flow in two-dimensional is conducted by Reynolds averaged numerical simulation （RANS） in this paper. Turbulence model plays a significant role in the complex flows＇ simulation, and four advanced turbulence models are evaluated. Numerical solution of frictional resistance coefficient is compared with the measured one in the transitional zone, which indicates that Wilcox （2006） k-ω model with correction is the best candidate. Comparisons of numerical and analytical solutions for dimensionless velocity show that averaged streamwise dimensionless velocity profiles correct the shape rapidly in transitional region. Furthermore, turbulence quantities such as turbulence kinetic energy, eddy viscosity, and Reynolds stress are also studied, which are helpful to learn the transition＇s behavior.

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.

Suitability of artificial viscosity discontinuous Galerkin method for compressible turbulence

A discontinuous Galerkin method based on an artificial viscosity model is investigated in the context of the simulation of compressible turbulence. The effects of artificial viscosity on shock capturing ability, broadband accuracy and under-resolved instability are examined combined with various orders and mesh resolutions. For shock-dominated flows, the superior accuracy of high order methods in terms of discontinuity resolution are well retained compared with lower ones. For under-resolved simulations, the artificial viscosity model is able to enhance stability of the eighth order discontinuous Galerkin method despite of detrimental influence for accuracy. For multi-scale flows, the artificial viscosity model demonstrates biased numerical dissipation towards higher wavenumbers. Capability in terms of boundary layer flows and hybrid meshes is also demonstrated.It is concluded that the fourth order artificial viscosity discontinuous Galerkin method is comparable to typical high order finite difference methods in the literature in terms of accuracy for identical number of degrees of freedom, while the eighth order is significantly better unless the under-resolved instability issue is raised. Furthermore, the artificial viscosity discontinuous Galerkin method is shown to provide appropriate numerical dissipation as compensation for turbulent kinetic energy decaying on moderately coarse meshes, indicating good potentiality for implicit large eddy simulation.

A study on turbulence transportation and modification of Spalart–Allmaras model for shock-wave/turbulent boundary layer interaction flow

It is of great significance to improve the accuracy of turbulence models in shock-wave/ boundary layer interaction flow. The relationship between the pressure gradient, as well as the shear layer, and the development of turbulent kinetic energy in impinging shock-wave/turbulent bound- ary layer interaction flow at Mach 2.25 is analyzed based on the data of direct numerical simulation （DNS）. It is found that the turbulent kinetic energy is amplified by strong shear in the separation zone and the adverse pressure gradient near the separation point. The pressure gradient was non-dimensionalised with local density, velocity, and viscosity. Spalart Allmaras （S A） model is modified by introducing the non-dimensional pressure gradient into the production term of the eddy viscosity transportation equation. Simulation results show that the production and dissipation of eddy viscosity are strongly enhanced by the modification of S-A model. Compared with DNS and experimental data, the wall pressure and the wall skin friction coefficient as well as the velocity profile of the modified S-A model are obviously improved. Thus it can be concluded that the mod- ification of S-A model with the pressure gradient can improve the predictive accuracy for simulat- ing the shock-wave/turbulent boundary laver interaction.

TURBULENT FLOWS AROUND SAND DUNES IN ALLUVIAL RIVERS

The sand dunes are typical bed forms of natural alluvial rivers. In this article, a vertical 2-D Reynolds stress model is established for the simulation of turbulent flows around sand dunes, and water-sand boundary conditions are set with particular attention. By numerical simulations, the following conclusions can be drawn. （1） The flow resistance in rivers with sand dunes could be divided into the sand-grain resistance and the sand dune resistance, and the sand-grain resistance coefficient mainly depends on Reynolds number, relative sand grain roughness and sand dune steepness. This coefficient in rivers with sand dunes would be larger than that calculated in a flat riverbed, and the steeper the sand dunes, the larger the sand-grain resistance coefficient. （2） The sand dune resistance coefficient mainly depends on the relative sand dune height and sand dune steepness, the steeper the sand dunes, the larger the sand dune resistance coefficient. （3） For the flat riverbed, the turbulent eddy viscosity coefficient and the sediment diffusion coefficient are approximately identical, but for the sand dune riverbed, in the vertical position, where the sediment diffusion coefficient reaches its maximum, it would be higher than the turbulent eddy viscosity coefficient.