Two-equation Turbulence Closure Model Research Articles

RANS Modelling of a NACA4412 Wake Using Wind Tunnel Measurements

Wake analysis plays a significant role in wind-farm planning through the evaluation of losses and energy yield. Wind-tunnel tests for wake studies have high costs and are time-consuming. Therefore, computational fluid dynamics (CFD) emerges as an efficient alternative. An especially attractive approach is based on the solution of the Reynolds-averaged Navier–Stokes (RANS) equations with two-equation turbulence closure models. The validity of this approach and its inherent limitations, however, remain to be fully understood. To this end, detailed wind-tunnel experiments in the wake of a NACA4412 wing section profile are compared with CFD results. Two- and three-dimensional RANS simulations are carried out for a range of angles of attack up to stall conditions at a chord- and inflow-based Reynolds number of Rec=4×105. Here, we aim to investigate the wake characteristics and self-similar behaviour, both from the experimental and numerical perspectives. The measurements are carried out by means of hot-wire anemometry capturing the wake pattern in several planes. The sensitivity of the CFD model to different configurations of the setup and the considerations required for reliable simulation are discussed. The agreement between CFD, experiments, and the literature is fairly good in many aspects, including the self-similar behaviour and wake parameters, as well as the flow field. Comparison of experiments with URANS/RANS data indicates that the latter is an adequate methodology to characterize wings and their wakes once the CFD setup is designed appropriately and the limitations due to discretization and turbulence modelling are considered.

Radiused Edge Blade Tip for a Wider Operating Range in Wells Turbine

The narrow operating range of a Wells turbine restricts its energy extraction capability from the ocean waves. In this work, the concept of a radiused edge tip blade (RETB) was introduced to overcome such an issue. The RET modifies the tip and changes the tip leakage flow behaviour. The flow through the turbine annulus was simulated numerically and compared with the existing experimental results of the reference turbine. Three-dimensional Reynolds-averaged Navier–Stokes equations with a two-equation turbulence closure model available in ANSYS CFX 14.5 was used for the simulations. The computational domain was discretized with unstructured tetrahedral elements, and the grid independence study gave an optimal grid. The RETB altered the tip leakage flow characteristics and delayed the stall inception. The RETB enhanced the relative operating range by 25% and peak torque by 37%.

NEW METHODS FOR STABILIZING RANS TURBULENCE MODELS WITH APPLICATION TO LARGE SCALE BREAKING WAVES

Recently, Larsen and Fuhrman (2018) have shown that seemingly all commonly used (both k-omega and k-epsilon variants) two-equation RANS turbulence closure models are unconditionally unstable in the potential flow beneath surface waves, helping to explain the wide-spread over-production of turbulent kinetic energy in CFD simulations, relative to measurements. They devised and tested a new formally stabilized formulation of the widely used k-omega turbulence model, making use of a modified eddy viscosity. In the present work, three new formally-stable k-omega turbulence model formulations are derived and tested in CFD simulations involving the flow and dynamics beneath large-scale plunging breaking waves.Recorded Presentation from the vICCE (YouTube Link): https://youtu.be/T2fFRgq3I8E

Instability of the realizable k– ε turbulence model beneath surface waves

Recent research has proved that most widely used two-equation turbulence closure models are unconditionally unstable in regions of nearly potential flow having finite strain, as commonly found beneath non-breaking surface waves. In this work, we extend such analysis to consider the popular realizable k–ε turbulence model. It is proved that this model, unlike all others thus far analyzed, is only conditionally unstable in such regions due to the addition of viscosity in the ε dissipation term. A method for formally stabilizing the model in the problematic regions is likewise developed. The results of the analysis, using both standard and stabilized turbulence closures, are confirmed via numerical simulations of progressive surface wave trains using a computational fluid dynamics model. Important qualitative differences are likewise demonstrated in simulations involving spilling breaking waves. (In the above, k is the turbulent kinetic energy density, and ε is the turbulence dissipation rate.)

Discontinuous Galerkin discretization for two-equation turbulence closure model

Accurate representation of vertical turbulent fluxes is crucial for numerical ocean modeling, both in global and coastal applications. The state-of-the-art approach is to use two-equation turbulence closure models which introduces two dynamic equations to the system. Solving these equations numerically, however, is challenging due to the strict requirement of positivity of the turbulent quantities (e.g., turbulence kinetic energy and its dissipation rate), and the non-linear source terms that may render the numerical system unstable. In this paper, we present a Discontinuous Galerkin (DG) finite element discretization of the Generic Length Scale (GLS) equations designed to be incorporated in a DG coastal ocean model, Thetis. To ensure numerical stability, the function space for turbulent quantities must be chosen carefully. In this work, we propose to use zeroth degree elements for the turbulent quantities and linear discontinuous elements for the tracers and velocity. The spatial discretization is completed with a positivity preserving semi-implicit time integration scheme. We validate the implementation with standard turbulence closure model benchmarks and an idealized estuary simulation. Finally, we use the full three-dimensional model to simulate the Columbia River plume. The results confirm that the coupled model generates realistic vertical mixing, and remains stable under strongly stratified conditions and strong tidal forcing. River plume characteristics are well captured.

Numerical Simulation of Turbulent Flow and Pollutant Dispersion in Urban Street Canyons

In this study, we have developed a numerical model based on an open source Computational Fluid Dynamics (CFD) package OpenFOAM, in order to investigate the flow pattern and pollutant dispersion in urban street canyons with different geometry configurations. In the new model, the pollutant transport driven by airflow is modeled by the scalar transport equation coupling with the momentum equations for airflow, which are deduced from the Reynolds Averaged Navier-Stokes (RANS) equations. The turbulent flow calculation has been calibrated by various two-equation turbulence closure models to select a practical and efficient turbulence model to reasonably capture the flow pattern. Particularly, an appropriate value of the turbulent Schmidt number has been selected for the pollutant dispersion in urban street canyons, based upon previous studies and careful calibrations against experimental measurements. Eventually, the numerical model has been validated against different well-known laboratory experiments in regard to various aspect ratios (a relationship between the building height and the width of the street canyon), and different building roof shapes (flat, shed, gable and round). The comparisons between the numerical simulations and experimental measurements show a good agreement on the flow pattern and pollutant distribution. This indicates the ability of the new numerical model, which can be applied to investigate the wind flow and pollutant dispersion in urban street canyons.

Numerical investigation into the effect of burner swirl direction on furnace and superheater heat absorption for a 620 MWe opposing wall-fired pulverized coal boiler

The paper investigates the effects of pulverized coal swirl burners’ secondary air swirl directions on furnace water-wall, platen superheater and high temperature superheater heat absorption by using numerical simulations. The resultant effect on the carbon-monoxide and unburnt carbon levels was also investigated to establish the furnace performance. The conventional RANS approach was used along with a two-equation turbulence closure model to simulate the multispecies gas phase flow. Fuel particles were modeled as a discrete phase. The gas phase combustion was simulated using the eddy dissipation finite rate kinetics model and the surface reactions using a diffusion-kinetics limited model. To establish the validity of the modelling approach an actual 620 MWe boiler furnace and its superheaters were modelled, and the results compared to processed plant data. Once the model showed that it can predict the plant performance with reasonable accuracy it was used to investigate the performance changes for various burner swirl arrangements. Six swirl arrangement cases were simulated to determine the effect on the furnace performance indicators. It was found that by swirling all the burners per wall in the same direction that the furnace heat absorption could be increased by up to 6.67% and the unburned carbon fraction reduced from 0.029% to 0.022% when compared to the current boiler swirl configuration results, which has a staggered swirl direction arrangement. Furthermore, it was shown that the enhanced mixing in the volume adjacent to the burners also reduced the carbon-monoxide levels. The simulations demonstrated, for the given modelling methodology, that there is indeed interaction between adjacent burners which influences the combustion rate and in turn heat absorption in the water-cooled furnace walls and platen superheater for the modelled boiler.

High-performance ocean energy harvesting turbine design – Detailed flow analysis with blade leaning strategy

Oscillating water column wave energy converter is having low efficiency because of its principal component, a bidirectional turbine. An analysis of the internal flow of the turbine gives an idea of improving the performance through optimization of geometrical parameters. In this study, an impulse turbine of 0.3 m diameter with fixed guide vanes is numerically simulated by solving three-dimensional incompressible steady Reynolds averaged Navier-stokes equation with two-equation turbulence closure model. This study shows that the numerical results very well match with the experimental results. The detailed flow physics demonstrates that different types of losses occur in this type of turbine and shows that the downstream diverging path of the rotor and guide vane is responsible for low performance. In this study, the effect of guide vane lean, as well as the combined rotor and guide vane lean on the performance of the turbine, has been discussed in detail and found to increase the efficiency of the turbine.

On the over-production of turbulence beneath surface waves in Reynolds-averaged Navier–Stokes models

In previous computational fluid dynamics studies of breaking waves, there has been a marked tendency to severely over-estimate turbulence levels, both pre- and post-breaking. This problem is most likely related to the previously described (though not sufficiently well recognized) conditional instability of widely used turbulence models when used to close Reynolds-averaged Navier–Stokes (RANS) equations in regions of nearly potential flow with finite strain, resulting in exponential growth of the turbulent kinetic energy and eddy viscosity. While this problem has been known for nearly 20 years, a suitable and fundamentally sound solution has yet to be developed. In this work it is demonstrated that virtually all commonly used two-equation turbulence closure models are unconditionally, rather than conditionally, unstable in such regions. A new formulation of the $k$–$\unicode[STIX]{x1D714}$ closure is developed which elegantly stabilizes the model in nearly potential flow regions, with modifications remaining passive in sheared flow regions, thus solving this long-standing problem. Computed results involving non-breaking waves demonstrate that the new stabilized closure enables nearly constant form wave propagation over long durations, avoiding the exponential growth of the eddy viscosity and inevitable wave decay exhibited by standard closures. Additional applications on breaking waves demonstrate that the new stabilized model avoids the unphysical generation of pre-breaking turbulence which widely plagues existing closures. The new model is demonstrated to be capable of predicting accurate pre- and post-breaking surface elevations, as well as turbulence and undertow velocity profiles, especially during transition from pre-breaking to the outer surf zone. Results in the inner surf zone are similar to standard closures. Similar methods for formally stabilizing other widely used closure models ($k$–$\unicode[STIX]{x1D714}$ and $k$–$\unicode[STIX]{x1D700}$ variants) are likewise developed, and it is recommended that these be utilized in future RANS simulations of surface waves. (In the above $k$ is the turbulent kinetic energy density, $\unicode[STIX]{x1D714}$ is the specific dissipation rate, and $\unicode[STIX]{x1D700}$ is the dissipation.)

Topology optimization of turbulent flows

The aim of this work is to present a fast and viable approach for taking into account turbulence in topology optimization of complex fluid flow systems, without resorting to any simplifying assumptions in the derivation of discrete adjoints. Topology optimization is an iterative gradient-based design process which minimizes an objective and satisfies a set of selected design constraints by distributing material in a design domain. The gradients are obtained using adjoint sensitivity analysis which requires solutions of a forward state problem and an additional adjoint problem. In the presented article the forward solver is based on finite volume discretized Reynolds-averaged Navier–Stokes equations coupled with either one- or two-equation turbulence closure models, and the adjoint solver is obtained via automatic differentiation. The presented approach is demonstrated on the optimization of several 2D and 3D examples including a detailed comparison to designs and sensitivities obtained with different turbulence models and under a frozen turbulence assumption. The results demonstrate the importance of exact sensitivity analysis and open new possibilities for the design of large scale multiphysics problems involving turbulent flows.

The nonlinear effects of the eddy viscosity and the bottom friction on the Lagrangian residual velocity in a narrow model bay

The nonlinear effects of the eddy viscosity and the bottom friction on the Lagrangian residual velocity (LRV) are studied numerically in a narrow model bay. Three groups of the experiments with different eddy viscosity and different forms of the bottom friction are designed and are carried out in the three kinds of the topography. When the eddy viscosity is obtained from a two-equation turbulence closure model, the pattern of the LRV is more complex than that of the time invariant eddy viscosity case and the intensity is from more than 1.3 times to one order smaller than that of the linear eddy viscosity condition. The LRV are also acquired when the eddy viscosity varies from the flood-averaged one to the ebb-averaged one. It is found that when the flood-averaged eddy viscosity is bigger than the ebb-averaged eddy viscosity (flood-dominated asymmetry), the direction of the breadth-averaged LRV and the 3D LRV is nearly opposite to that when the eddy viscosity asymmetry is reverse (ebb-dominated asymmetry). However, the intensity of the LRV for the ebb-dominated case decreases toward the flood-dominated case as the ratio of the maximum depth in the deep channel and the minimum depth on the shoal increases. The different forms of the bottom friction also play a role in the LRV. The structures of the 3D LRV and the depth-integrated LRV are simpler, and the intensity of the LRV is two times smaller when the linear bottom friction is used than those when the quadratic bottom friction is used.

Experimental Study and Simulation of a Small-Scale Horizontal-Axis Wind Turbine

The advancement of wind energy as an alternative source to hydrocarbons depends heavily on research activities in turbulence modeling and experimentation. The velocity deficit behind wind turbines affects the power output and efficiency of a wind farm. Being able to simulate the wake dynamics of a wind turbine effectively can result in optimum spacing, longer wind turbine life, and shorter payback on the wind farm investment. Two-equation turbulence closure models, such as k–ε and k–ω, are used extensively to predict wind turbine performance and velocity deficit profiles. The application of the Reynolds stress model (RSM) turbulence closure method has been limited to few studies where the rotor is modeled as an actuator disk (AD). The computational cost associated with RSM has made it challenging for simulations where the rotor is discretized directly; however, with advances in computer speed and power coupled with parallel computing architecture, RSM may be a better turbulence closure option. In this research, wind tunnel experiments were conducted, using hot-wire anemometry, to measure the velocity deficit profiles at different wake locations behind a small-scale, three-bladed, horizontal-axis wind turbine (HAWT). Experiments were also performed with two and three HAWTs in series to evaluate the change in velocity deficit and turbulence intensity (TI). High-speed imaging with an oil-based mist captured the vortices produced at the blade tips and showed the vortices dissipated approximately three rotor diameters downstream. Computational fluid dynamics (CFD) simulations were performed to predict the velocity deficit at wake locations matching the experiments. The Reynolds stress model was applied to a fully discretized rotor with a tower and nacelle included in the simulation. A steady-state moving reference frame (MRF) model was created with the computational domain subdivided into rotating and stationary domains. The MRF results were used as an initial condition for time-accurate rigid body motion (RBM) simulations. The RBM CFD simulations showed excellent agreement with experimental measurements for velocity deficit after properly accounting for experimental boundary effects. Isosurfaces of the Q-criterion highlighted the vortices produced at the blade tips and were consistent with high-speed images.

On the modeling of wave-enhanced turbulence nearshore

A high resolution k–ω two-equation turbulence closure model, including surface wave forcing was employed to fully resolve turbulence dissipation rate profiles close to the ocean surface. Model results were compared with observations from Surface Wave Instrument Floats with Tracking (SWIFTs) in the nearshore region at New River Inlet, North Carolina USA, in June 2012. A sensitivity analysis for different physical parameters and wave and turbulence formulations was performed. The flux of turbulent kinetic energy (TKE) prescribed by wave dissipation from a numerical wave model was compared with the conventional prescription using the wind friction velocity. A surface roughness length of 0.6 times the significant wave height was proposed, and the flux of TKE was applied at a distance below the mean sea surface that is half of this roughness length. The wave enhanced layer had a total depth that is almost three times the significant wave height. In this layer the non-dimensionalized Terray scaling with power of −1.8 (instead of −2) was applicable.

Flow and transport simulation of Madeira River using three depth-averaged two-equation turbulence closure models

This paper describes a numerical simulation in the Amazon water system, aiming to develop a quasi-three-dimensional numerical tool for refined modeling of turbulent flow and passive transport of mass in natural waters. Three depth-averaged two-equation turbulence closure models, k˜−ε˜,k˜−w˜, and k˜−ω˜ , were used to close the non-simplified quasi-three dimensional hydrodynamic fundamental governing equations. The discretized equations were solved with the advanced multi-grid iterative method using non-orthogonal body-fitted coarse and fine grids with collocated variable arrangement. Except for steady flow computation, the processes of contaminant inpouring and plume development at the beginning of discharge, caused by a side-discharge of a tributary, have also been numerically investigated. The three depth-averaged two-equation closure models are all suitable for modeling strong mixing turbulence. The newly established turbulence models such as the k˜−ω˜ model, with a higher order of magnitude of the turbulence parameter, provide a possibility for improving computational precision.

Combined tidal and wind driven flows and residual currents

The effect of a residual current on the combined tidal and wind driven flow and the resulting bedload sediment transport in the ocean has been investigated, using a simple one dimensional two-equation turbulence closure model. Predictions of the combined tidal and wind driven flow with given residual currents are presented, showing that the residual current has a substantial effect on both the depth averaged mass transport and the mean bedload transport directions; in some cases the effect of the residual current is to almost reverse the mean bedload transport direction. The residual current affects the rotation of the flow due to the Coriolis effect in the lower part of the water column (the near-surface flow is wind dominated), causing a larger or smaller clockwise rotation of the depth averaged mass transport, depending on the direction of the residual current.

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 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.

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.