Eddy Viscosity Concept Research Articles

Wave–current interaction at an angle 2: theory

A theoretical model is presented to describe the current velocity within and outside a wave–current boundary layer and to quantify the associated bed shear stresses for wave–current interaction at an arbitrary angle. The model assumes that the wave-induced bed shear stress in the combined flow varies sinusoidally with time while the current-induced bed shear stress follows an eddy viscosity concept. The computational procedure is given for both depth-averaged and reference-point current-based computations. Unlike other analytical models proposing different coefficients for different problems, the present model uses the same set of equations and coefficients to describe wave–current flows at any angle under any flow regime. This model agrees better with both field measurements and laboratory data than existing theoretical models. The model indicates that the wave–current interaction angle is not significant for wave–current flow properties.

Large eddy simulations without explicit eddy viscosity models

The classical approach to performing large eddy simulations of turbulent flows is to employ in the governing equations subgrid scale models based on eddy viscosity concepts. The development of the similarity model about 30 years ago led to the realisation that subgrid scale models that do not use eddy viscosity concepts are possible and have many desirable properties. In this article, we review modelling approaches belonging to this category and discuss their advantages and the limitations.

Calculation of High-Lift Cascades in Low Pressure Turbine Conditions Using a Three-Equation Model

A three-equation model has been applied to the prediction of separation-induced transition in high-lift low-Reynolds-number cascade flows. Classical turbulence models fail to predict accurately laminar separation and turbulent reattachment, and usually overpredict the separation length, the main reason for this being the slow rise of the turbulent kinetic energy in the early stage of the separation process. The proposed approach is based on solving an additional transport equation for the so-called laminar kinetic energy, which allows the increase in the nonturbulent fluctuations in the pretransitional and transitional region to be taken into account. The model is derived from that of Lardeau et al. (2004, “Modelling Bypass Transition With Low-Reynolds-Number Non-Linear Eddy-Viscosity Closure,” Flow, Turbul. Combust., 73, pp. 49–76), which was originally formulated to predict bypass transition for attached flows, subject to a wide range of freestream turbulence intensity. A new production term is proposed, based on the mean shear and a laminar eddy-viscosity concept. After a validation of the model for a flat-plate boundary layer, subjected to an adverse pressure gradient, the T106 and T2 cascades, recently tested at the von Kármán Institute, are selected as test cases to assess the ability of the model to predict the flow around high-lift cascades in conditions representative of those in low-pressure turbines. Good agreement with experimental data, in terms of blade-load distributions, separation onset, reattachment locations, and losses, is found over a wide range of Reynolds-number values.

복단면 개수로흐름에서 홍수터 식생의 흐름저항을 반영한 1차원 모형

본 연구에서는 식생이 포설된 홍수터를 포함하는 복단면 개수로 흐름의 수위를 예측하기 위하여 유효 전단응력기법에 근거한 일차원 모형을 제시하였다. 제안된 모형은 주수로와 홍수터 접합부에서 발생하는 운동량 교환효과를 와점성계수 개념을 이용하여 반영할 수 있다. 주수로와 홍수터의 경계에서 발생하는 경계와점성계수는 3차원 레이놀즈 응력 모형을 이용하여 결정하였다. 경계와점성계수의 영향을 파악하기 위하여 민감도분석을 수행하였다. 경계와점성계수의 변화에 대하여 배수곡선의 변화는 크지 않았으나, 홍수터에서 부담하는 유량은 경계와점성계수에 비례하여 증가하였다. 마지막으로 식생된 홍수터의 식생밀도 및 침수비에 따른 경계와점성계수의 변화를 살펴보았다. 계산된 경계와점성계수는 식생밀도와 침수비에 비례하며, 대략<TEX>$(2-5){\times}\;10^{-4}$</TEX> 정도의 값을 갖는 것으로 나타났다. In this study, the 1D apparent shear stress model for vegetated compound open-channel flows was suggested. To consider the effect of momentum exchange between main channel and floodplain, the eddy viscosity concept was used in the present model. The interfacial eddy viscosity in the interface of main channel and floodplain was determined from the 3D Reynolds stress model. The evaluated interfacial eddy viscosity appears to be good agreement with those proposed previously. To investigate the effect of interfacial eddy viscosity, sensitive analysis was carried out. the computed backwater profiles are nearly identical with respect to the value of the interfacial eddy viscosity. However, the discharge conveyed by the floodplain changes is proportional to the interfacial eddy viscosity. Finally, the changes of the interfacial eddy viscosity due to the vegetation density and vegetation height were examined. The computed results of interfacial eddy viscosity are in proportion to the vegetation density and vegetation height, and the interfacial eddy viscosity has a range of <TEX>$(2-5)\;{\times}\;10^{-4}$</TEX>.

Statistical Eulerian Diffusion Approach of Four-Way-Coupled Multiphase Systems

Volume-fraction weighted and Reynolds averaged momentum transport equations are solved in an Euler/Euler approach to simulate numerically the turbulent, dispersed two- phase °ow in a two-dimensional channel and a three-dimensional conic di®user °ow. Particular attention is given to the modelling of turbulent di®usion and particle wall interaction, assuming local equilibrium but introducing individual terms for particle/°uid drag interaction, particle collisions and trajectory crossings. These in°uences have been quanti¯ed in terms of partial viscosities, a restitution power and a turbulence structure parameter. Boussinesq approxima- tions have been used for each phase and their interaction, whose formulation was provided in the framework of the eddy-viscosity modelling concept.

Shock-capturing Boussinesq-type model for nearshore wave processes

This paper describes the formulation and validation of a nearshore wave model for tropical coastal environment. The governing Boussinesq-type equations include the conservative form of the nonlinear shallow-water equations for shock capturing. A Riemann solver supplies the inter-cell flux and bathymetry source term, while a Godunov-type scheme integrates the evolution variables in time. The model handles wave breaking through momentum conservation with energy dissipation based on an eddy viscosity concept. The computed results show very good agreement with laboratory data for wave propagation over a submerged bar, wave breaking and runup on plane beaches as well as wave transformation over fringing reefs. The model accurately describes transition between supercritical and subcritical flows as well as development of dispersive waves in the processes.

Toward advanced subgrid models for Lattice-Boltzmann-based Large-eddy simulation: Theoretical formulations

This paper addresses the issue of developing advanced subgrid model for large-eddy simulations (LES) of turbulent flows based on Lattice Boltzmann methods (LBM). Most of already existing subgrid closures used in LES-LBM are straightforward extensions of the most crude model developed within the Navier–Stokes equations, namely the Smagorinsky eddy-viscosity model. In a first part, it is shown how to obtain an improved eddy-viscosity subgrid model for LBM. The original implementation of the Inertial-Range Consistent Smagorinsky model proposed by Dong and Sagaut for the D3Q19 scheme is used as an illustration. In a second step, an original extension of the Approximate Deconvolution Method proposed by Adams and Stolz for Navier–Stokes simulation is proposed. This new LBM-LES approach does not rely on the eddy-viscosity concept and is written directly within the LBM framework. It is shown that it can be implemented thanks to a trivial modification of the existing LBM solvers for Direct Numerical Simulation.

Cascades, thermalization, and eddy viscosity in helical Galerkin truncated Euler flows

The dynamics of the truncated Euler equations with helical initial conditions are studied. Transient energy and helicity cascades leading to Kraichnan helical absolute equilibrium at small scales, including a linear scaling of the relative helicity spectrum are obtained. Strong helicity effects are found using initial data concentrated at high wave numbers. Using low-wave-number initial conditions, the results of Cichowlas et.al. [Phys. Rev. Lett. 95, 264502 (2005)] are extended to helical flows. Similarities between the turbulent transient evolution of the ideal (time-reversible) system and viscous helical flows are found. Using an argument in the manner of Frisch et. al. [Phys. Rev. Lett. 101, 144501 (2008)], the excess of relative helicity found at small scales in the viscous run is related to the thermalization of the ideal flow. The observed differences in the behavior of truncated Euler and (constant viscosity) Navier-Stokes are qualitatively understood using the concept of eddy viscosity. The large scales of truncated Euler equations are then shown to follow quantitatively an effective Navier-Stokes dynamics based on a variable (scale dependent) eddy viscosity.

Modelling flow transition in a hypersonic boundary layer with Reynolds-averaged Navier-Stokes approach

Based on Reynolds-averaged Navier-Stokes approach, a laminar-turbulence transition model is proposed in this study that takes into account the effects of different instability modes associated with the variations in Mach numbers of compressible boundary layer flows. The model is based on k-ω-γ three-equation eddy-viscosity concept with k representing the fluctuating kinetic energy, ω the specific dissipation rate and the intermittency factor γ. The particular features of the model are that: 1) k includes the non-turbulent, as well as turbulent fluctuations; 2) a transport equation for the intermittency factor γ is proposed here with a source term set to trigger the transition onset; 3) through the introduction of a new length scale normal to wall, the present model employs the local variables only avoiding the use of the integral parameters, like the boundary layer thickness δ, which are often cost-ineffective with the modern CFD (Computational Fluid Dynamics) methods; 4) in the fully turbulent region, the model retreats to the well-known k-ω SST (Shear Stress Transport) model. This model is validated with a number of available experiments on boundary layer transitions including the incompressible, supersonic and hypersonic flows past flat plates, straight/flared cones at zero incidences, etc. It is demonstrated that the present model can be successfully applied to the engineering calculations of a variety of aerodynamic flow transition.

Direct numerical simulation of turbulence in a square annular duct

Direct numerical simulation (DNS) is performed to investigate the fully developed turbulence in a straight square annular duct. The mean flow field and the turbulent statistics are compared with existing experiments and numerical results. The comparisons and the analysis of the DNS data led to the discovery of the turbulent boundary layers of concave and convex 90° corners, a corner flow similarity and the scaling characteristics of corner turbulence. Analysis of the mean streamwise velocity near the concave and convex 90° corners resulted in establishing the ‘law-of-the-corner’ formulations. Comparing these formulations with the ‘law-of-the-wall’ relation, both damping and enhancing mechanisms analytically represented by the van Driest damping function, and the enhancement function were revealed for the concave and convex corner turbulence. The investigation captures the distinctive turbulence-driven secondary flows for both convex and concave 90° corners, and a corner flow similarity rule is discovered, which is associated with the pattern of these secondary flows. A turbulence energy spectrum analysis provides the distinctive features of the fully developed turbulence in the wall and corner regions. The validity of the turbulence eddy viscosity concept is evaluated based on these turbulence energy spectra. The turbulence-driven secondary-flow generation mechanisms are investigated by analysing the anisotropy of the Reynolds stresses.

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.

Anisotropic Organised Eddy Simulation for the prediction of non-equilibrium turbulent flows around bodies

Abstract The unsteady turbulent flow around bodies at high Reynolds number is predicted by an anisotropic eddy-viscosity model in the context of the Organised Eddy Simulation (OES). A tensorial eddy-viscosity concept is developed to reinforce turbulent stress anisotropy, that is a crucial characteristic of non-equilibrium turbulence in the near-region. The theoretical aspects of the modelling are investigated by means of a phase-averaged PIV in the flow around a circular cylinder at Reynolds number 1.4 × 10 5 . A pronounced stress–strain misalignment is quantified in the near-wake region of the detached flow, that is well captured by a tensorial eddy-viscosity concept. This is achieved by modelling the turbulence stress anisotropy tensor by its projection onto the principal matrices of the strain-rate tensor. Additional transport equations for the projection coefficients are derived from a second-order moment closure scheme. The modification of the turbulence length scale yielded by OES is used in the Detached Eddy Simulation hybrid approach. The detached turbulent flows around a NACA0012 airfoil (2-D) and a circular cylinder (3-D) are studied at Reynolds numbers 10 5 and 1.4 × 10 5 , respectively. The results compared to experimental ones emphasise the predictive capabilities of the OES approach concerning the flow physics capture for turbulent unsteady flows around bodies at high Reynolds numbers.

Prediction of laminar, transitional and turbulent flow regimes, based on three-equation k-ω turbulence model

AbstractA recently developed transitional model for boundary-layer flows has been examined on a flat plate and the well-known S809 wind turbine blade. Proposed numerical model tries to simulate streamwise fluctuations, induced by freestream turbulence, in pre-transitional boundary-layer flows by introducing an additional transport equation for laminar kinetic energy term. This new approach can be used for modeling of transitional flows which are exposed to both the freestream turbulence intensity and streamwise pressure gradient, which are known as the most dominant factors in occurrence of transition. Computational method of this model is based on the solution of the Reynolds averaged Navier-Stokes (RANS) equations and the eddy-viscosity concept. The model includes three transport equations of laminar kinetic energy, turbulent kinetic energy and dissipation rate frequency. The present model is capable of predicting either natural or bypass transitional mechanisms, which may occur in attached boundary-layer flows. In addition, the model can simulate transition in the separated free shear layers and the subsequent turbulent re-attachment to form a laminar separation bubble. Flat plate was exposed to different freestream turbulence intensities and streamwise pressure gradients. Wind turbine blade was examined under two different Reynolds numbers, with one of them suitable for the occurrence of laminar separation bubbles on its surfaces. To evaluate the performance of this new model in resolving transitional boundary-layer flows, final results have been compared to those obtained through application of conventional turbulence models. Comparison of final results for the flat plate and the S809 aerofoil with available experimental data show very close agreements.

A method for estimating wall friction in turbulent wall-bounded flows

We describe a simple method for estimating turbulent boundary layer wall friction using the fit of measured velocity data to a boundary layer model profile that extends the logarithmic profile all the way to the wall. Two models for the boundary layer profile are examined, the power-series interpolation scheme of Spalding and the Musker profile which is based on the eddy viscosity concept. The performance of the method is quantified using recent experimental data in zero pressure gradient flat-plate turbulent boundary layers, and favorable pressure gradient turbulent boundary layers in a pipe, for which independent measurements of wall shear are also available. Between the two model profiles tested, the Musker profile performs much better than the Spalding profile. Results show that the new procedure can provide highly accurate estimates of wall shear with a mean error of about 0.5% in friction velocity, or 1% in shear stress, an accuracy that is comparable to that from independent direct measurements of wall shear stress. An important advantage of the method is its ability to provide accurate estimates of wall shear not only based on many data points in a velocity profile but also very sparse data points in the velocity profile, including only a single data point such as that originating from a near-wall probe.

Explicit small-scale velocity simulation for high- Re turbulent flows. Part II: Non-homogeneous flows

Two level simulation (TLS) approach based on the explicit reconstruction of the small-scale velocity fields, is extended to study non-homogeneous turbulent flows on coarse grids. Assumptions needed for the simplified treatment of the 3D small-scale governing equation on lines are analyzed using DNS data of well-developed turbulent channel flow. Fully coupled simulations of turbulent channel flow and temporally developed mixing layers suggest that explicit accounting of the small-scale velocity in the large-scale equations allows to correctly predict the large-scale turbulent dynamics without using a concept of eddy-viscosity.

Dynamics of Air Flow in Sewer Conduit Headspace

Pressurization in sanitary sewer conduit atmosphere is modeled using computational fluid dynamics techniques. The modeling approach considers both turbulent and laminar flow regimes. The turbulent model takes into consideration the turbulence-driven secondary currents associated with the sewer headspace and hence the Reynolds equations governing the air flow field are closed with an anisotropic closure model which comprises the use of the eddy viscosity concept for the turbulent shear stresses and semiempirical relations for the turbulent normal stresses. The resulting formulations are numerically integrated. The turbulent model outputs are verified with experimental data reported in the literature. Satisfactory agreement is obtained between numerical simulations and experimental data. Mathematical formulas and curves as functions of longitudinal pressure gradient, wastewater velocity, and sewer headspace geometry are developed for the cross-sectional average streamwise velocity.

Eddy viscosity and flow properties of the solar wind: Co-rotating interaction regions, coronal-mass-ejection sheaths, and solar-wind/magnetosphere coupling

The coefficient of magnetohydrodynamic (MHD) eddy viscosity of the turbulent solar wind is calculated to be νeddy≈1.3×1017cm2∕s: this coefficient is appropriate for velocity shears with scale thicknesses larger than the ∼106 km correlation length of the solar-wind turbulence. The coefficient of MHD eddy viscosity is calculated again accounting for the action of smaller-scale turbulent eddies on smaller scale velocity shears in the solar wind. This eddy viscosity is quantitatively tested with spacecraft observations of shear flows in co-rotating interaction regions (CIRs) and in coronal-mass-ejection (CME) sheaths and ejecta. It is found that the large-scale (∼107km) shear of the CIR fractures into intense narrow (∼105km) slip zones between slabs of differently magnetized plasma. Similarly, it is found that the large-scale shear of CME sheaths also fracture into intense narrow slip zones between parcels of differently magnetized plasma. Using the solar-wind eddy-viscosity coefficient to calculate vorticity-diffusion time scales and comparing those time scales with the ∼100-h age of the solar-wind plasma at 1AU, it is found that the slip zones are much narrower than eddy-viscosity theory says they should be. Thus, our concept of MHD eddy viscosity fails testing. For the freestream turbulence effect in solar-wind magnetosphere coupling, the eddy-viscous force of the solar wind on the Earth's magnetosphere is rederived accounting for the action of turbulent eddies smaller than the correlation length, along with other corrections. The improved derivation of the solar-wind driver function for the turbulence effect fails to yield higher correlation coefficients between measurements of the solar-wind driver and measurements of the response of the Earth's magnetosphere.

Models for the Turbulent Diffusion Terms of Shallow Water Equations

The analysis of three different approximations of the turbulent diffusion terms, widely used to simulate shallow water flows, is carried out both analytically and experimentally. Based on the eddy viscosity concept, the terms are solved for steady, uniform, turbulent flow in a simplified geometry, which may represent a tidal estuary or a compound channel. It is shown that, although the three approximations are identical in constant depth, they behave differently if strong depth gradients exist and, consequently, the transverse velocity profile obtained varies depending on the turbulence term used. It is also shown that the relative depth (flood plain depth-to-main channel depth ratio) has an important influence on the lateral momentum transfer and, consequently, depending on the approximation adopted, different dimensionless eddy viscosity coefficient λ values must be used to best fit the experimental data. A tentative relationship between the relative depth and the dimensionless eddy viscosity coefficient is presented for each expression. Comparison of analytical results with experimental data shows that not all the widely used expressions for the turbulence terms can adequately represent the velocity profile if strong depth gradients exist.

Momentum Exchange in Straight Uniform Compound Channel Flow

Transverse exchange of momentum between the channel and the floodplain in straight uniform compound channel flow is considered in this paper. This process results in the so-called “kinematic effect,” a lowering of the total discharge capacity of a compound channel compared to the case where the channel and the floodplain are considered separately. The mechanisms responsible for the momentum exchange are considered. The transverse shear stress in the mixing region is modeled using a newly developed effective eddy viscosity concept, that contains: (1) the effects of horizontal coherent structures moving on an uneven bottom, taking compression and stretching of the vortices into account and (2) the effects of the three-dimensional bottom turbulence. The model gives a good prediction of the transverse profiles of the streamwise velocity and the transverse shear stress of the flood channel facility experiments. Characteristic features of the lateral profile of the eddy viscosity are also well predicted qualita...

STATISTICAL CHARACTERISTICS OF UNSTEADY REYNOLDS-AVERAGED NAVIER–STOKES SIMULATIONS

ABSTRACT The statistical characteristics of unsteady Reynolds-averaged Navier–Stokes (RANS) turbulence models, utilizing the eddy-viscosity concept, are investigated. Detailed assessment is made to the original Launder-Spalding k –ϵ model as well as a nonlinear modification utilizing multiple scales from both turbulent variables and the local mean strain rate. The purpose of the study is to shed light on the issues of unsteady RANS computations, not to recommend a “good” model. The models are tested for flow past a square cylinder, employing progressively refined resolutions. The comparison between the length scale imposed by the turbulence model and the length scale of the spatial resolution helps understand the fidelity of the simulations. If the eddy viscosity is low, the flow field can be numerically underresolved in RANS computations, and consequently the solutions exhibit noticeable sensitivity to numerics such as discretization schemes and grid distribution. Under such situations, in practical computations, grid-independent solutions will be difficult to attain with conventional numerical schemes used in RANS simulations. It is demonstrated that in order to appreciate the statistical behavior of a turbulence model, both time-averaged and time-dependent results need be examined.