Two-equation Turbulence Closure Research Articles

A numerical and analytical study of the effect of aspect ratio on the behavior of a round thermal

A round thermal is formed when an element of buoyant fluid is released instantaneously into a quiescent ambient. Although the thermal spreading rate is of primary importance to mathematical modeling, the reported values in the literature vary greatly. To identify possible factors affecting the thermal spreading rate, we investigated the effect of different initial conditions numerically by solving the unsteady Reynolds-averaged Navier–Stokes equations with a two-equation turbulence closure. The initial aspect ratio (i.e. length-to-diameter ratio) of the thermal was varied between 0.125–4.0, and the initial density differences was changed from 1 to 10 %. Results show that the spreading rate is greatly affected by the initial aspect ratio, which also explains the variations in earlier reported values. Following the numerical study, an analytical model using buoyant vortex ring theory is developed to predict the spreading rate of a thermal. The predictions show good agreement with the results from both the numerical simulations and previous experimental studies. Another simple analytical model is also presented to approximate the thermal induced flow, and is validated using the numerical simulations.

An h-adaptive finite element method for turbulent heat transfer

A two-equation turbulence closure model (k-{omega}) using an h-adaptive grid technique and finite element method (FEM) has been developed to simulate low Mach flow and heat transfer. These flows are applicable to many flows in engineering and environmental sciences. Of particular interest in the engineering modeling areas are: combustion, solidification, and heat exchanger design. Flows for indoor air quality modeling and atmospheric pollution transport are typical types of environmental flows modeled with this method. The numerical method is based on a hybrid finite element model using an equal-order projection process. The model includes thermal and species transport, localized mesh refinement (h-adaptive) and Petrov-Galerkin weighting for the stabilizing the advection. This work develops the continuum model of a two-equation turbulence closure method. The fractional step solution method is stated along with the h-adaptive grid method (Carrington and Pepper, 2002). Solutions are presented for 2d flow over a backward-facing step.

Combined tidal and wind driven flows and bedload transport over a flat bottom

The combined tidal and wind driven flow and resulting sediment transport in the ocean over a flat bottom at intermediate water depth has been investigated, using a simple one dimensional two-equation turbulence closure model. This model has been verified against field measurements of a tidal flow in the Celtic Sea. The tidal velocity ellipses and the time series of the horizontal velocity components at given elevations above the bottom are well predicted through the water column although there are some deviations between the predicted and measured velocities near the bottom due to the uncertainty of the bottom roughness. For the combined tidal and wind driven flows the velocity profiles, turbulent kinetic energy profiles and surface particle trajectories are predicted for weak and strong winds. Furthermore, the bottom shear stress and the resulting bedload transport have been predicted; the parts of the particle trajectories in the close vicinity of the bottom where the bedload transport exists are displayed. Finally, the direction and magnitude of the surface drift, the depth-averaged mean velocity and the mean bedload transport are given, and the effect of the bottom roughness on the sea surface drift is investigated.

Evaluation of RANS/actuator disk modelling of wind turbine wake flow using wind tunnel measurements

Wake modelling plays a central role in wind farm planning through the evaluation of losses, the prediction of the energy yield, and the estimation of turbine loads. These models must be reasonably accurate – to minimise financial risk – and yet economical so that many configurations can be tested within reasonable time. While many such models have been proposed, an especially attractive approach is based on the solution of the Reynolds-averaged Navier-Stokes equations with two-equation turbulence closure and an actuator disk representation of the rotor. The validity of this approach and its inherent limitations however remain to be fully understood. To this end, detailed wind tunnel measurements in the wake of a porous disk (with similar aerodynamic properties as a turbine rotor) immersed in a uniform flow are compared with the predictions of several closures. Agreement with measurements is found to be excellent for all models. This unexpected result seems to derive from a fundamental difference in the turbulent nature of the homogeneous wind tunnel flow and that of the atmospheric boundary layer.

Evaluation of interphase drag models for the determination of gas hold-up of an air-water system in a spouted bed using CFD

The hydrodynamics of a dispersed air-water system within a spouted column with a concentric draft tube and a conical base is simulated using CFD based on a two-fluid Euler-Euler E-E modeling framework and k-e two-equation turbulence closure. The interaction between the dispersed gas phase and the continuous liquid phase is characterized by bubble-liquid interphase forces drag, turbulent dispersion and lift forces. The Ishii-Zuber drag model [1] and Grace adjusted drag model [2], the latter represented by: C_D^{Grace,dense} = υ _g^p C_D^{Grace}, are compared for their capability to match experimental gas hold-up. Numerical results of Reynolds-averaged Navier-Stokes equations with k-e two-equation turbulence closure model when compared with Pironti experimental data [3] indicated that both drag models, predicted the air hold-up within experimental errors. Furthermore, Ishii-Zuber liquid-gas drag model consistently provided better agreement with experimental results; it correctly determines the hold-up within 0.14%. Numerical agreement with adjusted Grace liquid-gas drag model, is exponent dependent 4 ≤ p ≤-0.5, turning down that the best computed hold-up is within 0.44% for p=0.5.

LES and URANS predictions of the hydrodynamic loads on a tension-leg platform

This paper reports on the prediction of the unsteady hydrodynamic forces that act on a floating Tension-Leg Platform (TLP) due to the action of a steady current. The results were obtained by solving the three-dimensional, time-dependent form of the equations governing conservation of mass and momentum. Movement of the free surface was tracked using the volume of fluid algorithm. The effects of turbulence were accounted for using two very different approaches: Large-Eddy Simulations (LES) and Unsteady Reynolds-Averaged Navier–Stokes (URANS). The latter approach utilized a two-equation turbulence closure that has been extended to capture the occurrence and consequences of vortex shedding from bluff bodies. The primary objective of the work was to explore the merits and de-merits of each modeling approach when applied to a large-scale structure of the type frequently encountered in practice. The test case chosen for this purposes of this assessment was the case of a conventional TLP in steady current at Reynolds numbers (based on column diameter) of 7.5×10 6 and 7.5×10 7. These values are representative of those encountered in deep-sea operations. Experiments and field observations have indicated that the resulting flows exhibit a number of complicated features due to the interactions between the shed vortices and the various structural components of the TLP. Many but not all of these features were captured by the present computations. In addition to a critical assessment of the two modeling approaches, the paper reports on a number of practical experiences gained in the course of conducting this study, including an assessment of the importance of allowing for the movement of the free surface (as opposed to adopting the usual solid-lid approximation) and an illustration of the effects of the current's angle of incidence on the computed hydrodynamic loads.

On the effect of structure-induced resistance and mixing on inflows into the Baltic Sea: A numerical model study

This study aims to estimate the additional mixing and dilution of dense bottom currents due to foundations of wind turbines in offshore wind farms projected in the region of the Western Baltic Sea. To some extent these offshore wind farms are planned to be build directly in the main pathways of dense bottom currents propagating into the Baltic Sea. This may have a significant effect for the Baltic Sea ecosystem. In the present study, cylindric structures are assumed for the underwater construction of the individual wind turbines, which are assembled in wind farms with typically 50–100 structures. A parameterisation of the additional mixing and friction due to a structure is developed as an extension of the k−ε two-equation turbulence closure model. Results of a high resolution Reynolds-Averaged Navier–Stokes (RANS) model of the local scale are used to calibrate this parameterisation for hydrostatic coastal ocean models. A Western Baltic Sea hydrodynamic model coupled to the extended turbulence closure model is applied in two different scenarios covering (i) weak and (ii) strong structure-induced mixing due to offshore wind farm distributions in accordance with the present (April 2009) planning situation. The scenarios are completed by two cases with unrealistically extensive wind farms simulating a theoretical future worst case scenario. By means of analysing annual model simulations, it is found that the impact of structure-induced mixing due to realistic wind farm distributions is comparably low with a typical decrease of bottom salinity in the range of 0.1−0.3psu. The annual mean bottom salinity at the outflow from the Arkona Sea through the Bornholm Channel into the direction of the Baltic Proper shows decreases due to mixing from a realistic wind farm distribution of only 0.02psu which is more than one order of magnitude smaller than the standard deviation of the bottom salinity change.

The influence of advection schemes and turbulence closure models on drag coefficient calculation around a circular cylinder at high Reynolds number

Different advection schemes and two-equation turbulence closure models based on eddy viscosity concept are used to compute the drag coefficient around a circular cylinder at high Reynolds number (106). The numerical results from these simulations are compared with each other and with experimental data in order to evaluate the performance of different combinations of advection scheme and two-equation turbulence model. The separate contributions from form drag and friction drag are also analyzed. The computational results show that the widely used standard k-ɛ turbulence closure is not suitable for such kind of study, while the other two-equation turbulence closure models produce acceptable results. The influence of the different advection schemes on the final results are small compared to that produced by the choice of turbulence closure method. The present study serves as a reference for the choice of advection schemes and turbulence closure models for more complex numerical simulation of the flow around a circular cylinder at high Reynolds number.

Quantifying the Contributions of Tidal Straining and Gravitational Circulation to Residual Circulation in Periodically Stratified Tidal Estuaries

Abstract This numerical modeling study quantifies for the first time the contribution of various processes to estuarine circulation in periodically stratified tidal flow under the impact of a constant horizontal buoyancy gradient. The one-dimensional water column equations with periodic forcing are first cast into nondimensional form, resulting in a multidimensional parameter space spanned by the modified inverse Strouhal number and the modified horizontal Richardson number, as well as relative wind speed and wind direction and the residual runoff. The along-tide momentum equation is then solved for the tidal-mean velocity profile in such a way that it is equated to the sum of the contributions of tidal straining (resulting from the temporal correlation between eddy viscosity and vertical shear), gravitational circulation (resulting from the depth-varying forcing by a constant horizontal buoyancy gradient), wind straining, and depth-mean residual flow (resulting from net freshwater runoff). This definition of tidal straining does not only account for tidal asymmetries resulting from horizontal buoyancy gradients but also from wind straining and residual runoff. For constant eddy viscosity, the well-known estuarine circulation analytical solution with polynomial residual profiles is directly obtained. For vertically parabolic and constant-in-time eddy viscosity, a new analytic solution with logarithmic residual profiles is found, showing that the intensity of the gravitational circulation scales with the horizontal Richardson number. For scenarios with realistic spatially and temporally varying eddy viscosity, a numerical water column model equipped with a state-of-the-art two-equation turbulence closure model is applied to quantify the individual contributions of the various processes to estuarine circulation. The fundamental outcome of this study is that, for irrotational flow with periodic stratification and without wind forcing and residual runoff, the tidal straining is responsible for about two-thirds and gravitational circulation is responsible for about one-third of the estuarine circulation, proportionally dependent on the horizontal Richardson number, and weakly dependent on the Strouhal number. This new and robust result confirms earlier estimates by H. Burchard and H. Baumert, who suggested that tidal straining is the major generation mechanism for estuarine turbidity maxima. However, a sensitivity analysis of the model results to details of the turbulence closure model shows some uncertainty with respect to the parameterization of sheared convection during flood. Increasing down-estuary wind straining and residual runoff reduce the quantitative contribution of tidal straining. For relatively small horizontal Richardson numbers, the tidal straining contribution to estuarine circulation may even be reversed by down-estuary wind straining.

A new second-order closure model for rough-wall turbulent flows using the Brinkman equation

A second-order turbulence closure is developed for the new rough-wall layer modeling approach using the Brinkman equation for turbulent flows over rough walls. In the proposed approach, we model the fluid dynamics of the volume averaged flow in the near-wall rough layer by using the Brinkman equation. The porosity can be calculated based on the volumetric characteristics of the roughness and the permeability is modeled. Interface stress jump conditions including the Reynolds stress components are also considered. The Reynolds-averaged Navier–Stokes equations are solved numerically above the near-wall rough layer, while a second-order turbulence closure is employed in all regions. The rough-wall second-order closure is developed by adopting an existing smooth-wall model. The computational results, including the skin friction coefficient, the log-law mean velocity, the roughness function, the Reynolds stresses, and the turbulent kinetic energy, are presented and compared with those obtained by using a previously developed two-equation turbulence closure. The results show that the new rough-wall layer modeling approach with the second-order turbulence closure model satisfactorily predicted the skin friction coefficient, the log-law mean velocity, the roughness function, and the Reynolds shear stress. However, the results for the Reynolds normal stresses are different from the measured data in the inner 20–60% of the boundary layer due to the interface stress jump conditions employed in the present rough-wall layer modeling approach.

Performance of two-equation turbulence closures in three-dimensional simulations of the Red Sea overflow

Mixing of overflows released from polar and marginal seas is a key process shaping the structure of the meridional overturning circulation. Ocean general circulation models have difficulty in resolving the overflows, and therefore they must rely on parameterizations. In this study, the performance of a set of turbulence closures in reproducing mixing of an overflow is quantified. We simulate the Red Sea overflow by employing standard k– ε, k– ω and Mellor–Yamada schemes with various stability functions, as well as a modified k– ε model that relies on the prescription of the turbulent Prandtl number rather than on stability functions. The simpler KPP mixing scheme and experiments without turbulent fluxes serve as useful references. To our knowledge, this is the first time that the performance of two-equation turbulence models has been examined so closely using data from an overflow. It is found that without turbulence closures, the hydrodynamic model has difficulty in reproducing the correct three-dimensional pathway of the Red Sea overflow, consisting of a distinct bifurcation into two diverging channels. All turbulence models capture the vertical structure of this overflow consisting of an interfacial layer, characterized by the density gradient, and a well-mixed bottom layer. Mean eddy diffusivity values from most closures are comparable those from observations. But we find that KPP leads to eddy diffusivity values that are too small while those from Mellor–Yamada with Galperin [Galperin, B., Kantha, L.H., Hassid, S., Rosati, A., 1988. A quasi-equilibrium turbulent energy model for geophysical flows. J. Atmos. Sci. 45, 55–62] stability functions are too large. Such high diffusivities lead to excessive mixing in the bottom layer of the overflow, ultimately resulting in a salinity deficit of approximately 1 psu in the product water mass. Salinity deviations between the models and observations are quantified using both data taken along the channels and two sections across the overflow. KPP and Mellor–Yamada with Galperin (1988) stability functions produce the largest deviations from the observations, while the modified k– ε exhibits the smallest deviations. The other four closures fall in between, showing results similar to one another. The performance of the Mellor–Yamada turbulence closure is improved considerably by using the stability functions by Kantha and Clayson [Kantha, L.H., Clayson, C.A., 1994. An improved mixed layer model for geophysical applications. J. Geophys. Res. 99 (December), 25235–25266], which allow for a stationary Richardson number of 0.21. In conclusion, we find that most turbulence closures lead to a satisfactory reproduction of the Red Sea overflow, within the temporal and spatial sampling uncertainties of the REDSOX data, provided that fairly high-resolution regional models are used.

Mixing in the surface boundary layer of a tropical freshwater reservoir

A combined observational-modeling study was conducted to investigate turbulence mixing, and the relation to surface forcing, in the surface boundary layer (SBL) of a tropical, high-altitude, freshwater reservoir. A suite of vertical profiles of temperature microstructure, collected at three different stations of one-day duration each, provided estimates of dissipation rates of turbulence kinetic energy, ϵ, and temperature variance, χ. Numerical simulations of ϵ and χ, using state-of-the-art, public domain, two-equation turbulence closure models, compared favorably with the observations and reproduced the dynamics of daytime wind mixing as well as the vertical and temporal turbulence structure during nighttime convective conditions. Two independent estimates of vertical eddy diffusivities in the stably stratified (daytime) SBL, computed from the microstructure measurements, agreed closely, and the near surface heat and buoyancy fluxes, computed from the diffusivities, were similar to those computed independently from surface meteorology. Model generated eddy diffusivities agreed closely with the observed values, except those generated by K profile parameterization (KPP) model simulations. The good agreement provides confidence that nutrient fluxes in the SBL may be accurately computed from the models when forced with regularly measured surface meteorological parameters. The consequences are important for estimation of daily primary productivity rates in the euphotic zone and the ability to predict algal blooms such as those observed in the present reservoir.

Validation Case Study: Prediction of Compressible Turbulent Mixing Layer Growth Rate

A recently developed validation metric is applied to the example problem of growth-rate prediction for a turbulent, compressible mixing layer. Emphasis is placed on the validation exercise as a well-defined process, which must include quantification of numerical error, evaluation of the level of confidence in the experimental data, and estimation of model accuracy. Available data for compressible mixing layers are reexamined, resulting in a new recommended data set for the study of mixing layer growth rate. Numerical solutions to the Favre-averaged Navier-Stokes equations with various two-equation turbulence closures and compressibility models are obtained and verified through formal solution verification procedures. The validation metric results are presented for each of the models, including local error estimates for detailed evaluation of model performance and global metrics that facilitate model performance comparisons.

Wind effect in turbulence parametrization

The action of wind blowing over a closed basin ultimately results in a steady shear-induced circulation pattern and in a leeward rising of the free surface—and a corresponding windward lowering—known as wind set-up. If the horizontal dimensions of the basin are large with respect to the average flow depth, the occurrence of local quasi-equilibrium conditions can be expected, i.e. the flow can be assumed to be locally driven only by the wind stress and by the opposing free surface gradient due to set-up. This wind-induced flow configuration shows a strong similarity with turbulent Couette–Poiseuille flow, the one dimensional flow between parallel plates generated by the simultaneous action of a constant pressure gradient and of the shear induced by the relative motion of the plates. A two-equation turbulence closure is then employed to perform a numerical study of turbulent Couette–Poiseuille flows for different values of the ratio of the shear stresses at the two walls. The resulting eddy viscosity vertical distributions are analyzed in order to devise analytical profiles of eddy viscosity that account for the effect of wind. The results of this study, beside allowing for a physical insight on the turbulence process of this class of flows, will allow for a more accurate description of the wind effect to be included in the formulation of quasi-3D and 3D models of lagoon hydrodynamics.

Turbulence characteristics in sharp open-channel bends

In spite of its importance, little is known about the turbulence characteristics in open-channel bends. This paper reports on an experimental investigation of turbulence in one cross section of an open-channel bend. Typical flow features are a bicellular pattern of cross-stream circulation (secondary flow) and a turbulence activity in the outer bend that is significantly less than in the equivalent straight uniform shear flow. Measured distributions are given of the turbulent kinetic energy, its production, the mixing coefficients, some parameters characterizing the turbulence structure, and the fourth-order correlations of the turbulent velocity fluctuations. The transport equation for the turbulent kinetic energy is evaluated term by term, on the basis of the measured data. The results show that the turbulence structure is different from straight uniform flow, in that the Reynolds stress tensor is more diagonally dominant. This is shown to be the main cause of the observed reduction of turbulence activity in the outer bend. The usual two-equation turbulence closure models include a transport equation for the turbulent kinetic energy, but they do not account for this modified turbulence structure. The departures of the measured turbulence structure from its equivalent in straight uniform shear flow are related to a curvature-flux-Richardson number Rf which includes the streamline curvature. Such a relation may be useful to improve simple turbulence closure models for curved open-channel flow.

Using sweep to extend the stall-free operational range in axial fan rotors

The paper discusses the use of sweep as a remedial strategy to control the aerodynamic limits in low-speed axial fan rotors. In this respect, the present work contributes to the understanding of the potential effect of blade lean on the shifting of the rotor stall margin. Numerical investigations have been undertaken on highly loaded fans of non-free vortex design, with the ideal total head rise coefficient typical of the industrial application range. Two rotors with identical nominal design parameters and, respectively, with 35° forward swept blades and unswept blades have been studied. The investigation has been carried out using an accurate in-house developed multilevel parallel finite element RANS solver, with the adoption of a non-isotropic two-equation turbulence closure. The pay-off derived from the sweep technology has been assessed with respect to the operating range improvement. To this end, the flow structure developing through the blade passages and downstream of the rotors, as well as loss distributions, have been analysed at design and near-peak pressure operating conditions. The analyses of three-dimensional flow structures showed that, sweeping forward the blade, the non-free vortex spanwise secondary flows are attenuated, and a control on the onset of stall is recovered. Moreover, the swept rotor features a reduced sensitivity to leakage flow effects. Consequently, it operates more efficiently approaching the throttling limit.

Turbulence Closure, Steady State, and Collapse into Waves

A new simple two-equation turbulence closure is constructed by hypothesizing that there is an extra energy sink in the turbulent kinetic energy (k) equation representing the transfer of energy from k to internal waves and other nonturbulent motions. This sink neither contributes to the buoyancy flux nor to dissipation, the nonturbulent mode being treated as inviscid. The extra sink is proportional to the squared ratio between the turbulent time scale t ; k/«, with turbulent dissipation rate «, and the buoyancy period T 5 2p/N. With a focus on high‐Reynolds number, spatially homogeneous, stably stratified shear flow away from boundaries, the turbulence is described by equations for a master length scale L ; k3/2/« and the master time scale t. It is assumed that the onset of the collapse of turbulence into nonturbulence occurs at t 5 T. The new theory is almost free of empirical parameters and compares well with laboratory and numerical experiments. Most remarkable is that the model predicts the turbulent Prandtl number, which is generally s 5 s 0/[1 2 (t/T) 2], with s 0 5 1/2, and hence is not a unique function of mean flow variables. Only in structural equilibrium ( 5 0) is the Prandtl t ˙ number a unique function of the gradient Richardson number Rg: s 5 s 0/(1 2 2Rg). These forms of the Prandtl number function immediately determine the flux Richardson number R f 5 Rg/s. Steady state occurs at 5 1/ s Rg 4 with R f 5 1/4, and within structural equilibrium the collapse of turbulence is complete at Rg 5 1/2.

Low Reynolds number turbulent flows around a dynamically shaped airfoil

A computational investigation for flows surrounding a dynamically shaped airfoil, at a chord Reynolds number of 78,800, is conducted along with a parallel experimental effort. A piezo-actuated flap on the upper surface of a fixed airfoil is adopted for active control. The actuation frequency focused on is 500 Hz. The computational framework consists of a multi-block, moving grid technique, the e n -based laminar–turbulent transition model, the two-equation turbulence closure, and a pressure-based flow solver. The moving grid technique, which handles the geometric variations in time, employs the transfinite interpolation scheme with a spring network approach. Comparing the experimental and computational results for pressure and velocity fields, implications of the detailed flap geometry, the flapping amplitude, turbulence modeling, and grid distributions on the flow structure are assessed. The effect of the flap movement on the separation location and vortex dynamics is also investigated.

Tidally Forced Internal Wave Mixing in ak–ε Model Framework Applied to Fjord Basins

Abstract A simple method for including tidally forced internal wave mixing in a two-equation turbulence closure framework, the k–e model, is presented. The purpose is to model the vertical mixing in the basin waters of stagnant sill fjords. An internal wave energy (IWE) term of the form aN, where N is the buoyancy frequency and a is a weighted average of the locally available IWE, is added to the equation for the turbulent kinetic energy (TKE). The total available IWE, F, is given by the energy flux from the barotropic tide to baroclinic internal waves, and a is determined assuming that all the available IWE is transferred to TKE, namely F = ∫ aN dV, where the integral is taken over the entire volume of the fjord. Comparisons with observations from two Scandinavian fjords show that the model successfully describes the vertical mixing in the basin water. Further, the buoyancy term in the equation for dissipation of TKE, e, is found to be irrelevant during stable stratification; that is, the empirical param...

Three-Dimensional Salinity Simulations of South San Francisco Bay

A conservative transport method and a two-equation turbulence closure model are added to an established three-dimensional hydrodynamic model. The resulting model is applied to simulate the development of the salinity field in South San Francisco Bay. All model parameters in the salinity simulation are identical to those used in a model calibration in which the model is shown to reproduce current meter data accurately. The 64-day time period studied is characterized by low freshwater input. During this period, the observed salinity data is reproduced well by the model. Because no model parameters are adjusted, the salinity simulation is considered to be a verification of the model. Once the model is verified, it is used to explore the importance of stratification and baroclinic pressure gradients on long-term transport. It is concluded that, under conditions of moderate density gradients, the density-driven flow and mild periodic stratification that results have a significant effect on scalar transport.