Eddy-viscosity Turbulence Model Research Articles

A stable algorithm for Reynolds stress turbulence modelling with applications to rectilinear and circular tidal flows

This paper employs one-point, linear eddy viscosity and differential second-moment (DSM) turbulence closures to predict the turbulent characteristics of both rectilinear and circular tidal flows. The numerical scheme is based on a finite volume approach applied to a non-staggered grid such that all flow variables are stored at one and the same set of nodes. Numerical stability is maintained through the implementation of apparent viscosities and source term linearization, which are essential if eddy viscosity terms are absent. A stable algorithm is devised for the Reynolds stresses which includes a non-linear velocity smoothing in order to stabilise the numerical scheme during flow reversal and relaminarization. Favourable agreement with the experimental rectilinear tidal data of Schroder (Tech. Rep. GK55 87/E/16, GKSS-Forshungszentrum Geesthacht, 1983) and McClean (Turbulence and Sediment Transport Measurements in a North Sea Tidal Inlet (the Jade), Springer, New York, 1987, p. 436) is reported. Numerical calculations of circular tidal flows are also presented which were motivated by the preliminary investigations of Davies and Jones (Int, j. numer meth. fluids, 12, 17 (1991)) and Davies (Continental Shelf Res., 11, 1313 (1991)), who employed the one-equation, k-l, eddy viscosity turbulence model to simulate rectilinear and circular tidal flows.

Buoyancy-driven displacement ventilation flows: Evaluation of two eddy viscosity turbulence models for prediction

A computational fluid dynamics code has been used to model buoyancy-driven displacement ventilation flows. The results of using two eddy viscosity turbulence models are compared with those of mathematical theory and salt bath experiments carried out at the University of Cambridge. The work highlights some of the difficulties involved in modelling buoyancy-driven flows and identifies a preferable turbulence model for predicting such flows.

RNG Modeling of Turbulent Heat Flux and Its Application to Wall Shear Flows.

In a previous paper[Phys.Fluids 9(1997)], we proposed an improved renormalization group(RNG)theory based on conditional iterative-averaging for turbulent flow with mean shear to derive an eddy-viscosity type turbulence model. In the present paper, we have applied this iterative-averaging RNG theory to the governing equation for a thermal field, and derived a thermal eddy diffusivity representation relevant to the turbulent heat flux. Furthermore, we have also obtained the RNG-based equation for the turbulent Prandtl number Prt, with a hogh Reynolds-number limit as Prt=0.79. The present RNG-based model is applied to the calculations of wall turbulent heat transfer. Agreement with the experiment and the direct simulation data is generally very satisfactory.

Eddy Viscosity Transport Equations and Their Relation to the k-ε Model

A formalism will be presented which allows transforming two-equation eddy viscosity turbulence models into one-equation models. The transformation is based on Bradshaw’s assumption that the turbulent shear stress is proportional to the turbulent kinetic energy. This assumption is supported by experimental evidence for a large number of boundary layer flows and has led to improved predictions when incorporated into two-equation models of turbulence. Based on it, a new one-equation turbulence model will be derived from the k-ε model. The model will be tested against the one-equation model of Baldwin and Barth, which is also derived from the k-ε model (plus additional assumptions) and against its parent two-equation model. It will be shown that the assumptions involved in the derivation of the Baldwin-Barth model cause significant problems at the edge of a turbulent layer.

Numerical Simulation of Drag Reducing Turbulent Flow in Annular Conduits

Inadequate transport of rock cuttings during drilling of oil and gas wells can cause major problems such as excessive torque, difficulty to maintain the desired orientation of the drill string, and stuck or broken pipe. The problem of cuttings transport is aggravated in highly inclined wellbores due to the eccentricity of the annulus which results in nonuniformity of the flowfield within the annulus. While optimum cleaning of the borehole can be achieved when the flow is turbulent, the added cost due to the increased frictional losses in the flow passages may be prohibitive. A way around this problem is to add drag-reducing agents to the drilling fluid. In this way, frictional losses can be reduced to an acceptable level. Unfortunately, no model is available which can be used to predict the flow dynamics of drag-reducing fluids in annular passages. In this paper, a numerical model is presented which can be used to predict the details of the flowfield for turbulent annular flow of Newtonian and non-Newtonian, drag-reducing fluids. A one-layer turbulent eddy-viscosity model is proposed for annular flow. The model is based on the mixing-length approach wherein a damping function is used to account for near wall effects. Drag reduction effects are simulated with a variable damping parameter in the eddy-viscosity expression. A procedure for determining the value of this parameter from pipe flow data is discussed. Numerical results including velocity profiles, turbulent stresses, and friction factors are compared to experimental data for several cases of concentric and eccentric annuli.

Computations of periodic turbulent boundary layers with moderate adverse pressure gradient

In this computational study a two-equation eddy-viscosity model of turbulence is used to predict the development of boundary layers with periodic variation of the free-stream velocity and time-mean adverse pressure gradient. Transport equations are solved for q, the square root of the turbulence kinetic energy and ζ, its dissipation rate. Comparisons with reliable experimental data on such flows verified that the model can satisfactorily reproduce the time-average prime variables as well as their amplitudes. The predicted boundary-layer integral parameters were also found to be in good agreement with comparable data in the literature. Additional comparisons are made with the computed development of the same flow at constant pressure. The effects of the pressure gradient isolated and studied in this way were generally found to be similar to those of nonperiodic flow in similar conditions.

On the sensitization of turbulence models to rotation and curvature

Empirical alterations of eddy-viscosity turbulence models to account for system rotation and streamline curvature are discussed. Except in a narrow class of flows, the streamline curvature itself is an inadequate entry into a model, because it is not Galilean-invariant. We propose a measure of the extra influence on the turbulence which is invariant, fully defined in three dimensions, and unifies rotation and curvature effects. This is at the expense of involving higher derivatives than the “traditional” (non-invariant) terms do. It is closely related to an idea of Knight & Saffman [1]. Its interpretation is straightforward, at least in two dimensions, and coincides with the classical measures in a few “building-block” flows such as rotating shear flow or a curved boundary layer. It can be used in one-equation, two-equation, and similar models in conjunction with empirical functions which depend on the model; an example is given.

Prediction of turbulent transitional phenomena with a nonlinear eddy-viscosity model

Abstract This paper describes a new nonlinear eddy-viscosity model of turbulence designed with a view to predicting flow far from equilibrium, including transition. The scheme follows earlier UMIST practice in adopting a cubic relation between the stress and the strain/ vorticity tensors but broadens the range of flows to which the model applies by including a third transport equation for an anisotropy parameter of the stress field. Applications are shown for transition on a flat plate at different levels of free-stream turbulence, for the normal impingement of a turbulent jet on a flat plate, and for the flow around a turbine blade. The model is shown to generate much more realistic predictions than what is said to be the best of the linear eddy-viscosity schemes.

Renormalization group theory for turbulence: Eddy-viscosity type model based on an iterative averaging method

The renormalization group (RNG) theory of turbulence is often used for the forced Navier–Stokes equation in order to investigate turbulence models in Fourier space. The strong point of this kind of theory is the ability to construct turbulence models with the aid of the Kolmogorov −5/3 power law for the energy spectrum. In this paper, we have made use of an iterative averaging method proposed by McComb (1990), which does not have the misleading ε-expansion technique developed by Yakhot and Orszag (1986), then applied this method to the derivation of an eddy-viscosity type turbulence model. Using the exact Navier–Stokes equation excluding artificial external forces, we have obtained the eddy-viscosity type turbulence model which is equivalent to the Boussinesq postulate, and its model constant Cμ is determined from only a Kolmogorov constant α.

96/05354 Development and application of a cubic eddy-viscosity model of turbulence

96/05354 Development and application of a cubic eddy-viscosity model of turbulence

A Comparison of Some Recent Eddy-Viscosity Turbulence Models

The performance of recently developed eddy-viscosity turbulence models, including the author’s SST model, is evaluated against a number of attached and separated adverse pressure gradient flows. The results are compared in detail against experimental data, as well as against the standard k-ε model. Grid convergence was established for all computations. The study involves four different, state-of-the-art finite difference (finite volume) codes.

NUMERICAL PREDICTION OF SEMI-CONFINED JET IMPINGEMENT AND COMPARISON WITH EXPERIMENTAL DATA

The standard k–ε eddy viscosity model of turbulence in conjunction with the logarithmic law of the wall has been applied to the prediction of a fully developed turbulent axisymmetric jet impinging within a semi-confined space. A single geometry with a Reynolds number of 20,000 and a nozzle-to-plate spacing of two diameters has been considered with inlet boundary conditions based on measured profiles of velocity and turbulence. Velocity, turbulence and heat transfer data have been obtained using laser–Doppler anemometry and liquid crystal thermography respectively. In the developing wall jet, numerical results of heat transfer compare to within 20% of experiment where isotropy prevails and the trends in turbulent kinetic energy are predicted. However, stagnation point heat transfer is overpredicted by about 300%, which is attributed directly to the turbulence model and inapplicability of the wall function.

Modelling mass transfer entrance lengths in turbulent pipe-flow with applications to small cathodes for measuring local mass transfer rates

Pipe-wall mass transfer, in the developing concentration boundary layer region, under fully developed hydrodynamic conditions is simulated with a low-Reynolds number,k−e, eddy viscosity turbulence model. The predictions are in good agreement with the electrochemical measurements of Berger and Hau, in the rangeRe=104 to 105, forSc=2244, and of Son and Hanratty, in the rangeRe=1×104–5×104, forSc=2400, in both the developing boundary layer region and the fully developed region. The application of small cathodes embedded in a larger active cathode to measure local mass transfer rates is also simulated. The size of the electrode and the thickness of the electrical insulation around the small electrode give rise to errors that increase as the insulation thickness increases and the electrode size decreases.

Compressibility and pressure-gradient correction assessment for turbulent hypersonic flows

NOWLEDGE of the flow development in hypersonic windtunnel nozzles is of primary importance to the design of such nozzles as well as to the characterization of the flowfield in the test section. In the perfect-gas regime, reliable numerical methods have been developed for both of the design work and the flow analysis, but there still exist some deficiencies in predicting the turbulent hypersonic nozzle wall boundary layers. This problem is believed to be a consequence of the deficiencies in the compressible eddy-viscosity turbulence models currently adopted in engineering applications. Specifically, to improve the accuracy of such calculations, the importance of compressibility and pressure-gradient effects should be investigated, since the hypersonic wind-tunnel nozzle flow is characterized by the existence of an increasing main flow Mach number and a varying pressure gradient in the streamwise direction. A numerical study of that nature is presented in this work. Several explicit compressibility and pressure-gradient corrections are introduced to two widely used eddy-viscosity turbulence models for the calculations of hypersonic wind-tunnel nozzle flows. The results are compared with those obtained with the standard models to evaluate the effects of the corrections, and are compared with measurements to verify the accuracy of the calculations and to see the improvements gained. Numerical Modeling The study was carried out by solving three-dimensional Reynolds-averaged Navier-Stokes equations. They are the transport equations to describe the conservation of mass, momentum, and energy with the assumption of a perfect gas. When two-equation turbulence models are employed, the system has two more equations for the turbulent kinetic energy k and its dissipation rate €. In the present work the turbulence is closed by four alternative eddy-viscosity models, among which the Baldwin-Lomax algebraic model1 and the k-€ model of Chien 2 are chosen as the baseline models for the purpose of evaluating the compressibility and pressure-gradient corrections. The other two are their modified versions, respectively, with the modifications outlined below.

Development and application of a cubic eddy-viscosity model of turbulence

Many quadratic stress-strain relations have been proposed in recent years to extend the applicability of linear eddy-viscosity models at modest computational cost. However, comparison shows that none achieves much greater width of applicability. This paper, therefore, proposes a cubic relation between the strain and vorticity tensor and the stress tensor, which does much better than a conventional eddy-viscosity scheme in capturing effects of streamline curvature over a range of flows. The flows considered range from simple shear at high strain rates and pipe flow, to flows involving strong streamline curvature and stagnation.

Performance of eddy-viscosity-based turbulence models in three-dimensional turbulent interaction

We examine several eddy-viscosity turbulence models for their ability to reproduce the experimental data in a strong three-dimensional external-shock-wave-turbulent-boundary-layer interaction. The four models investigated are the zero-equation Baldwin-Lomax model, the one-equation models of Baldwin and Barth and of Spalart and Allmaras, and the two-equation k-e closure.

Renormalization group based κ − ε turbulence model for flows in a duct with strong curvature

This paper reports the outcome of our research for resolving the very severe disagreement between the computed and measured results of the 90 deg duct flow with strong curvature. Three kinds of turbulence models (standard κ − ε eddy-viscosity turbulence model, RNG κ − ε model and RNG κ − ε model with C μ modification) have been used to predict the abovementioned flow. The results demonstrate that RNG model and the revised model can provide satisfactory computed results. Both of the mean velocities and Reynolds shear stresses are in quantitative agreement with the experimental data. We feel that the RNG κ − ε model suggested in this paper can be a useful turbulence model for practical engineering and scientific computations.

A numerical study of wave-breaking in stratified flow over obstacles

This paper describes results of numerical computations of stratified flow of finite depth D over a variety of obstacles using the time-dependent Navier Stokes equations with stability-dependent eddy viscosity turbulence models. Most computations were performed with the Froude number F h in the range 0.5 ⩽ F h ⩽ 2 and the parameter K = D /( πF h ) in the range 1 ⩽ K ⩽ 10. Here F h = U / Nh , where U is the free stream velocity, N is the buoyancy frequency and h is the height of the body. The domain and boundary conditions correspond to those of a towing tank experiment and the initial conditions were ‘impulsive start’. Critical Froude numbers for wave-breaking over two-dimensional obstacles are compared with theoretical predictions and experimental results. Preliminary results with mixing length and one-equation turbulence models suggest that differences in the representation of flows with breaking waves with such models are small. The occurrence of ‘merged flow’, in which the breaking region joins with the secondary separation zone, and which has been seen in experiments, is found in the computations in three dimensions, but not those in two.

Flowfield surveys and computations of a crossing-shock wave/boundary-layer interaction

A joint experimental and computational study has been performed to investigate the flowfield structure created by two crossing oblique shock waves interacting with a turbulent boundary layer. Such an interaction is of practical importance in the design of high-speed sidewall-compression inlets. The interaction is created by a test model, consisting of two sharp fins mounted at 15-deg angle of attack to a flat plate, placed in a Mach 3.85 freestream flow with a unit Reynolds number of 76 X 106/m. Two computational solutions, one using a Baldwin-Lomax algebraic turbulent eddy viscosity model and one using a modified K-£ (Rodi) turbulence model, are compared with experimental flowfield data obtained from a fast-response five-hole probe. Both the experiment and the computations show that the flowfield is dominated by a large, low-Mach-number, low-total-pressure separated region located on the interaction centerline. A comparison of the results shows significant differences between experiment and computations within this separated region. Outside the separated region, the experiment and computations are in good agreement. Additionally, the comparison shows that both turbulence models provide similar results, with neither model being clearly superior in predicting the flowfield.

Depth-Averaged Open-Channel Flow Model

A general mathematical model is developed to solve unsteady, depth-averaged equations. The model uses boundary-fitted coordinates, includes effective stresses, and may be used to analyze sub- and super critical flows. The time differencing is accomplished using a second-order accurate Beam and Warming approximation, while the spatial derivatives are approximated by second-order accurate central differencing. The equations are solved on a nonstaggered grid using an alternative-direction-implicit scheme. To enhance applicability, the equations are solved in transformed computational coordinates. The effective stresses are modeled by incorporating a constant eddy-viscosity turbulence model to approximate the turbulent Reynolds stresses. As is customary, the stresses due to depth-averaging are neglected. Excluding recirculating flows, it is observed that in most cases the effective stresses do not significantly affect the converged solution. The model is used to analyze a wide variety of hydraulics problems including flow in a channel with a hydraulic jump, flow in a channel contraction, flow near a spur-dike, flow in a 180° channel bend, and a dam-break simulation. For each of these cases, the computed results are compared with experimental data. The agreement between the computed and experimental results is satisfactory.