Eddy Viscosity Formulation Research Articles

The Role of Turbulence in an Intense Tropical Cyclone: Momentum Diffusion, Eddy Viscosities, and Mixing Lengths

Abstract Improved representation of turbulent processes in numerical models of tropical cyclones (TCs) is expected to improve intensity forecasts. To this end, the authors use a large-eddy simulation (with 31-m horizontal grid spacing) of an idealized category 5 TC to understand the role of turbulent processes in the inner core of TCs and their role on the mean intensity. Azimuthally and temporally averaged budgets of the momentum fields show that TC turbulence acts to weaken the maximum tangential velocity, diminish the strength of radial inflow into the eye, and suppress the magnitude of the mean eyewall updraft. Turbulent flux divergences in both the vertical and radial directions are shown to influence the TC mean wind field, with the vertical being dominant in most of the inflowing boundary layer and the eyewall (analogous to traditional atmospheric boundary layer flows), while the radial becomes important only in the eyewall. The validity of the downgradient eddy viscosity hypothesis is largely confirmed for mean velocity fields, except in narrow regions which generally correspond to weak gradients of the mean fields, as well as a narrow region in the eye. This study also provides guidance for values of effective eddy viscosities and vertical mixing length in the most turbulent regions of intense TCs, which have rarely been measured observationally. A generalized formulation of effective eddy viscosity (including the Reynolds normal stresses) is presented. Significance Statement This study uses a turbulence-resolving simulation of a category 5 tropical cyclone to understand the role of turbulence in intense storms. Results show that turbulence clearly modulates storm structure and intensity. This study provides guidance for the values of turbulent quantities (which are usually parameterized in comparatively coarse operational TC forecast models) in scarcely observed regions of intense storms. Furthermore, a complete formulation of the effective eddy viscosities is proposed, incorporating contributions from typically ignored Reynolds normal stress terms.

Numerical and experimental study of two-vehicle overtaking process

The overtaking process between two simplified Ahmed body models is studied using a set of static experiments to provide a comparison basis for the numerical study. Both steady-state (static) and transient (dynamic) simulations are performed. There are notable differences of the drag and side force coefficients between the predicted and experimental results in the wake effects on the overtaking model as well as under a situation when two models are parallel for the static tests. These differences are attributed to the SST [Formula: see text] model, which is a two-equation Reynolds-averaged Navier-Stokes equation (RANS) turbulence model with an assumption of isotropic eddy viscosity, employed in the simulation. A more advanced turbulence model that simulates large-eddy organized motion in a turbulent flow and which uses an anisotropic eddy viscosity formulation in the near-wall flow subregion would increase the prediction accuracy for the overtaking simulation. A comparison of the results for the steady-state and the transient simulations at various relative velocity ratios ([Formula: see text]) shows that the quasi-steady approach is suited to simulate the dynamic two-vehicle overtaking process when the condition [Formula: see text] is met.

A parameter-free LES model for anisotropic mesh adaptivity

Balancing accuracy and computational cost is a challenge in the modelling of turbulent flows. A widely used method for turbulence modelling is large–eddy simulation (LES). LES allows one to describe large scale flow features at a reasonable computational cost compared to the more accurate direct numerical simulation (DNS), making it a popular choice for engineering applications. One strategy to balance accuracy and cost with LES is through the use of mesh adaptivity, which allows the degrees of freedom in a problem to be reduced by changing spatial discretisation. However, mesh adaptivity can affect accuracy when using the standard Smagorinsky LES model with an implicit filter, considering that the parameter Cs is highly dependent on the filter width, which depends on mesh resolution. This work is aimed to develop an LES model that does not require any user–defined parameters and is suitable for mesh adaptivity with implicit filter. In this study we introduce a parameter–free LES model incorporating an anisotropic eddy–viscosity formulation combined with anisotropic mesh adaptivity. In our model, the parameter Cs in the eddy–viscosity formulation of the Smagorinsky model, is replaced by a function that evaluates the relative location of turbulence fluctuations in each element with respect to the turbulence spectrum inertial range. The anisotropic formulation of the eddy–viscosity allows for the application of an appropriate filter width in different directions, improving accuracy. Additionally, the mesh adaptivity algorithm assesses the local turbulence fluctuations via local Reynolds number and vortex identification criteria. This assessment leads to the refinement of regions with higher turbulence fluctuations down to the smallest scale limit in the inertial range in the corresponding direction, and also leads to the coarsening of regions without turbulence fluctuation up to largest scale limit in the inertial range. This method is tested using a flow past a sphere test case. The results are compared both qualitatively and quantitatively to results obtained with the standard Smagorinsky model, and demonstrate the better performance of our method with lower computational cost. This allows us to simulate large Reynolds number cases and compare our quantitative results to experimental results found in the literature, emphasising that our method produces good results at reasonable computational cost.

Large eddy simulation of two separated hypersonic shock/turbulent boundary layer interactions

We present a detailed statistical description of turbulence in the high-fidelity Large Eddy Simulation (LES) of hypersonic shock-and-turbulent boundary layer interactions with strong separation. Our high-fidelity LES method uses a 4th-order, bandwidth-optimized Weighted Essentially Non-Oscillatory (WENO) scheme combined with a robust shock-capturing method and dynamic mixed model formulations for the closure of subgrid scale terms. Our mixed models, which contain a scale-similar term in addition to the purely dissipative one, are shown to accurately and significantly improve separation prediction compared to purely dissipative closures, such as the dynamic eddy viscosity formulation.

Wall-Modeled Large Eddy Simulation of High Speed Flows

The use of wall-modeled large-eddy simulation (WMLES) is explored in the context of compressible flows with a focus on cold-wall boundary layers and flows with shock-induced separation. It is observed that for cold-wall flows, a “mixed” scaling for the length scale appearing in the eddy viscosity formulation outperforms the classical semilocal scaling for obtaining predictions of heat flux and skin friction. A few shock/boundary-layer interaction (SBLI) cases are examined in some detail, and model modifications are proposed to overcome identified deficiencies. It is shown that using WMLES the low-frequency characteristics of SBLI at high Reynolds number can be quantitatively captured. A dynamically switched version of the equilibrium model is proposed; this shows promise for relatively inexpensive simulations at these conditions.

Deep structured neural networks for turbulence closure modeling

Despite well-known limitations of Reynolds-averaged Navier–Stokes (RANS) simulations, this methodology remains the most widely used tool for predicting many turbulent flows due to computational efficiency. Machine learning is a promising approach to improve the accuracy of RANS simulations. One major area of improvement is using machine learning models to represent the complex relationship between the mean flow field gradients and the Reynolds stress tensor. In the present work, modifications to improve the stability of previous optimal eddy viscosity approaches for RANS simulations are presented and evaluated. The optimal eddy viscosity is reformulated with a non-negativity constraint, which promotes numerical stability. We demonstrate that the new formulation of the optimal eddy viscosity improves conditioning of RANS equations for a periodic hills test case. To demonstrate the suitability of this proportional/orthogonal tensor decomposition for use in a physics-informed data-driven turbulence closure, we use two neural networks (structured on this specific tensor decomposition, which is incorporated as an inductive bias into the network design) to predict the newly reformulated linear and non-linear parts of the Reynolds stress tensor. Injecting these network model predictions for the Reynolds stresses into RANS simulation improves predictions of the velocity field, even when compared to a sophisticated (state of the art) physics-based turbulence closure model. Finally, we apply shapley additive explanations values to obtain insights from the learned representation for inner workings of the neural network used to predict the optimal eddy viscosity from the input feature data.

Friction factor from velocity profiles in smooth turbulent channel flows

Fluid dynamics plays an important role in many Renewable energies studies, i.e. wind and tidal turbines, wave energy geothermal and solar power, …). Friction factor f is an important parameter for the determination of pressure drop in different processes and systems. In this study, we use DNS data of turbulent smooth channels to evaluate different methods. First, a recalibration of Dean’s correlation (Dean, 1978) is proposed. The aim of the study is to obtain accurate wall friction factors from velocity profiles. On the one hand, we obtained two implicit analytical relations based on the law-of-the-wall: a logarithmic friction relation similar to that of pipes and a linear-logarithmic friction relation. On the other hand, we obtained f from the computation of the average velocity. It is first calculated from the law-of-the-wall and allows a good prediction of f for Reτ > 395 but presents a gap for low Reτ which is related to inaccurate velocities. Low-Reynolds number effect in channel flows has been previously observed in different experimental and computational studies. In order to provide suitable friction factor values, it is important to predict velocities accurately on the overall channel height. We used therefore a more appropriate method which consists to use for y+ < 20 the momentum equation with an eddy viscosity formulation (Absi, 2019) and the log-wake law for high y+ values. This method provides accurate friction factor values and allows good agreement with DNS data.

Predicting fully-developed channel flow with zero-equation model

A new zero-equation model (ZEM) is devised with an eddy-viscosity formulation using a stress length variable which the structural ensemble dynamics (SED) theory predicts. The ZEM is distinguished by obvious physical parameters, quantifying the underlying flow domain with a universal multi-layer structure. The SED theory is also utilized to formulate an anisotropic Bradshaw stress-intensity factor, parameterized with an eddy-to-laminar viscosity ratio. Bradshaw’s structure function is employed to evaluate the kinetic energy of turbulence k and turbulent dissipation rate epsilon . The proposed ZEM is intrinsically plausible, having a dramatic impact on the prediction of wall-bounded turbulence.

Modeling the third-order velocity structure function in the scaling range at finite Reynolds numbers

A model for the third-order velocity structure function, S3, is proposed for closing the transport equation of the second-order velocity structure function, S2, in the scaling range of finite Reynolds number homogeneous and isotropic turbulence (HIT). The model is based on a gradient type hypothesis with an eddy-viscosity formulation. The present model differs from previous ones in that no assumptions are made with regard to the behaviors of S2 and S3 in the scaling range. This allows S3 to be modeled whether the intermittency of the energy dissipation ɛ (as modeled by the intermittency phenomenology) is considered or not. In both cases, the model predicts the same (infinite Reynolds number) asymptotic behavior for S2. This corresponds to the K41 prediction, i.e., S2 ∼ r2/3. The model yields good agreement against direct numerical simulation data for forced HIT at all scales of motion except in the transition region between the dissipative and scaling ranges. The introduction of a simple bridging function in the model improves significantly the agreement in this region. Furthermore, the model illustrates the effect of the finite Reynolds number on the scaling range and shows that this effect is responsible for the deviation from a power-law behavior.

Anisotropic SST turbulence model for shock-boundary layer interaction

Menter SST k-ω is a Reynolds-averaged Navier–Stokes based two-equation turbulence model routinely used in industry for predicting aerodynamic flows. It shows excellent performance for low-speed flows, but gives inconsistent predictions for high-speed shock-induced separated flows. The model assumption of using a constant value of 0.31 for the structure parameter contradicts experimental observations. The model is also unable to predict Reynolds stress anisotropy generated by shock waves. In this work, we augment the SST model with quadratic eddy viscosity formulation of an explicit algebraic Reynolds stress model. A new relation for the structure parameter is proposed, making it a function of the local strain-rates and is no longer a constant in the regions of shock/turbulent boundary layer interaction (SBLI). Additional shock-physics is introduced using (Sinha et al., 2003) shock-unsteadiness model and an upper limit to the value of structure parameter is set in regions of shock waves. The new model, termed as SUQ-SST, is validated using a number of SBLI test cases ranging from supersonic to hypersonic speeds and near-incipient to fully-separated flows. Results show that the modifications do not alter the boundary layer prediction capability of the SST model. On the other hand, the new model gives significant improvement in predicting Reynolds stress anisotropy, flow separation, and surface properties in a wide range of SBLI flows.

A Simple Nonlinear Eddy Viscosity Model for Geophysical Turbulent Flows

Eddy viscosity models in turbulence modeling can be mainly classified as linear and nonlinear models. Linear formulations are simple and require less computational resources but have the disadvantage that, those can’t predict actual flow pattern in complex geophysical flows where streamline curvature and swirling motion are predominant. A constitutive equation of Reynolds stress anisotropy is adopted for the formulation of eddy viscosity including all the possible higher order terms quadratic in the mean velocity gradients and a simplified model is developed for actual oceanic flows where only the vertical velocity gradients are important. The simplified formulation is used for the study of natural convection flow in a vertical water column and the results are compared with the observational data and predictions of other existing turbulence models. The developed formulation can be incorporated in other computational fluid dynamics codes for the flow analysis in various engineering applications. The model predictions of marine turbulence and other related data (e.g. sea surface temperature, surface heat flux and vertical temperature profile) can be utilized in determining the effective siting for the Ocean Thermal Energy Conversion (OTEC) plants and in particular for the development of tidal energy projects.

A K–L model with improved realizability for turbulent mixing

Turbulent mixing, induced by Rayleigh–Taylor (RT), Richtmyer–Meshkov (RM), and Kelvin–Helmholtz (KH) instabilities, broadly occurs in both practical astrophysics and inertial confined fusion problems. The Reynolds-averaged Navier–Stokes models remain the most viable approach for the solution of these practical flows. The commonly used mixing models based on the standard eddy viscosity formulation are shown to be capable of accurately predicting the global mixing zone width. However, we find that this approach will become non-realizable for local flow characteristics in the case of a large mean strain rate, including yielding the negative normal stress and the unphysically large turbulence kinetic energy (TKE) in the presence of shocks. This can affect the numerical robustness in calculating turbulent statistics and give rise to highly inaccurate predictions for complex mixings. To overcome this problem, a realizable K–L mixing model is developed, extended from the standard K–L model given by our recent works. A new eddy viscosity formulation is used and modified from the work by Shih et al. to reproduce the growth rate of the KH mixing. This new model yields similar results as the standard model for canonical RT, RM, and KH mixings. However, for complex mixing problems, the present model gives a significant improvement in physically capturing the turbulence characteristics, e.g., predicting the non-negative normal stress for RT mixing with the initial tilted interface and the appropriate TKE when shock interacts with the mixing zone for spherical implosion.

Assessment of large-scale forcing in isotropic turbulence using a closed Kármán–Howarth equation

We attempt to model the effects of large-scale forcing on the statistical behavior of small scales in isotropic turbulence. More specifically, the effect of large-scale forcing on the second-order velocity structure function, S2, in the region beyond the dissipative range, is analyzed via the transport equation for S2 where a closure model for S3, the third-order velocity structure function, is introduced. The model [L. Djenidi and R. A. Antonia, Fluid Turbulence Applications in Both Industrial and Environmental Topics, Marseille, 9–11 July, 2019, https://fab60.sciencesconf.org/] is based on a gradient type with an eddy-viscosity formulation and has the following expression: S3=−CS3(S2)2ϵ¯∂S2∂r, where ϵ¯ is the mean rate of the turbulent kinetic energy dissipation, r is the spatial increment, and CS3 is a constant. The closed S2-transport equation is further exploited to derive a model for S2 for scales beyond the dissipative range. The model for S2 takes the form S2=CK(ϵrr)23 with ϵr=ϵ¯1−Br/Reλ1/2, where CK is a constant, Reλ is the Taylor microscale Reynolds number, and the function Br accounts for the effect of the large scales. The numerical solutions of the S2 equation and the predictions based on the model for S2 agree very well with direct numerical simulation data for steady-state forced homogeneous and isotropic turbulence. The solutions of the S2-transport equation without large-scale forcing show that S2 behaves like (ϵ¯r)2/3. When forcing is applied, S2 deviates from this behavior. However, increasing the Reynolds number tends to restore this behavior over an increasing range of scales. This is also observed in the predictions of the model for S2.

Impact of Prandtl numbers on turbulence modeling

A low-Reynolds number (LRN) turbulence model is devised that includes the near-wall turbulence modeling and turbulent Prandtl numbers σ(k,ε,t). A secondary source term added in the -equation accounts substantially for the anisotropic production in near-wall regions. Hence, improved predictions of adverse pressure-gradient flows, involving separation and reattachment are obtained with a considerable reduction in the turbulent kinetic energy and length scale magnitudes. The eddy-viscosity formulation responds to both rotational and irrotational strains to better predict nonequilibrium flows. The current model employs no wall function/distance parameter to bridge the near-wall integration as required by conventional k-ε models. Model predictions are compared with direct numerical simulation (DNS) and experimental data; a good correlation is achieved.

Influence of eddy viscosity parameterisation on the characteristics of turbulence and wind flow: Assessment of steady RANS turbulence model

Eddy viscosity-based Reynolds-averaged Navier–Stokes (RANS) turbulence models remain a standard approach for accurately predicting the characteristics of turbulence and wind flow in computational wind engineering. RANS turbulence models have been widely studies; however, various types of deficiencies and numerical errors remain when they are used in numerical simulations. For overcoming these drawbacks associated with bluff body aerodynamics, numerous researchers have proposed modifications for eddy viscosity-based RANS turbulence models. However, deficiencies remain challenging. The main objective of this study is to explore the influence of eddy viscosity parameterisation on the characteristics of turbulence and wind flow. This exploration will help urban planners and researchers to modify existing numerical techniques and avoid the uncertainties that are present in the RANS turbulence models. The anisotropic behaviour of the vorticity-to-strain ratio was investigated by varying the vorticity-to-strain ratio in the eddy viscosity formulation of a revised standard k–ε turbulence model. The results were validated using a wind tunnel experiment. A lower vorticity-to-strain ratio significantly reduced the uncertainty in predicting turbulent kinetic energy, particularly in the impingement regions around the building. However, varying the vorticity-to-strain ratio exerted only a minor effect on the wind flow parameters.

High-Resolution RANS Simulations of Flow Past a Surface-Mounted Cube Using Eddy-Viscosity Closure Models

AbstractWhile Reynolds-averaged simulations have found success in the evaluation of many canonical shear flows and moderately separated flows, their application to highly separated flows have shown notable deficiencies. This study aimed to investigate these deficiencies in the eddy-viscosity formulation of four commonly used turbulence models under separated flow in an attempt to aid in the improved formulation of such models. Analyses are performed on the flow field around a wall-mounted cube (WMC) at a Reynolds number of 40,000 based on the cube height, h, and freestream velocity, U0. While a common occurrence in industrial applications, this type of flow constitutes a complex structure exhibiting a large separated wake region, high anisotropy, and multiple vortex structures. As well, interactions between vortices developed off of different faces of the cube significantly alter the overall flow characteristics, posing a significant challenge for the commonly used industrial turbulence models. Comparison of mean flow characteristics show remarkable agreement between experimental values and turbulence models which are capable of predicting transitional flow. Evaluation of turbulence parameters show the general underestimation of Reynolds stress for transitional models, while fully turbulent models show this value to be overestimated, resulting in completely disparate representations of mean flow structures between the two classes of models (transitional and fully turbulent).

Eddy Viscosity and Velocity Profiles in Fully-Developed Turbulent Channel Flows

In this study, we present eddy viscosity formulations based on analytical solutions for turbulent kinetic energy (TKE) in fully-developed turbulent channel flows. Mean streamwise velocities are obtained by solving the momentum equation with the proposed analytical eddy viscosity formulations. Our results were validated by comparisons with DNS data of fully-developed turbulent channel flows for 300 < Reτ < 2000 (where Reτ denotes the friction Reynolds number defined by the friction velocity uτ, kinematic viscosity ν and the channel half-width δ). Our method allows accurate description of mean streamwise velocity profiles and turbulent shear stress outside the log-law layer, in the viscous and buffer layers and in the outer region. The aim of this study is to provide simple analytical tools for turbulent channel flow applications.

A semi-locally scaled eddy viscosity formulation for LES wall models and flows at high speeds

We show that the mean wall-shear stresses in wall-modeled large-eddy simulations (WMLES) of high-speed flows can be off by up to $$\approx 100\%$$ with respect to a DNS benchmark when using the van-Driest-based damping function, i.e., the conventional damping function. Errors in the WMLES-predicted wall-shear stresses are often attributed to the so-called log-layer mismatch, which, albeit also an error in wall-shear stresses $$\tau _\mathrm{w}$$ , is an error of about $$15\%$$ . The larger error identified here cannot be removed using the previously developed remedies for the log-layer mismatch. This error may be removed by using the semi-local scaling, i.e., $$l_\nu =\mu /\sqrt{\rho \tau _\mathrm{w}}$$ , in the damping function, where $$\mu $$ and $$\rho $$ are the local mean dynamic viscosity and density, respectively.

Effect of turbulence models' choice on the aero-thermal flow numerical validations within a ribbed trailing edge geometry

In this paper, the commercial ANSYS Fluent software was used to achieve a numerical survey of the effect of five (05) turbulence models’ formulation on the aero-thermal computational fluid dynamics validation of two 30:1 scaled models reproducing an original internal ribbed trailing edge prototype. Tests were conducted for stationary and rotation conditions, for Re = 10,000–40,000 and Ro = 0–0.23. Particle image velocimetry and thermochromic liquid crystal experimental data were employed to check the consistence computational fluid dynamics results qualitatively and quantitatively, aerodynamically and thermally, for various working conditions. Numerical predictions revealed that the choice of the turbulence model affects the accuracy of results. Concerning the shear stress transport k-w model, limiters defined in the eddy viscosity formulation induce a surplus estimation of the turbulence kinetic energy ( k) which leads to noticeable discrepancies in terms of velocity profiles and recirculation zones. Also, numerical calculations confirmed former experimental assumptions concerning origins of the aerodynamic structures and heat transfer's features, especially, those related to the increase of the cooling temperature balance efficiency, the appearance/disappearance of the horseshoe structures within the trailing edge region and velocities/boundary layers’ profile variations. The obtained results assist the understanding and the forecast of the flow field behavior, throughout the design process, by the assessment of the aerodynamic and thermal performances within the considered blade's cooling system.

The development and application of two-time-scale turbulence models for non-equilibrium flows

This study re-visits the largely overlooked, but highly promising, topic of multi-scale RANS modelling for non-equilibrium turbulent flows. The paper presents the development of RANS models which, at an effective-viscosity level, involve two transport equations for the turbulent kinetic energy; one for the energy of the large scale, energy-producing, eddies and one for that of the smaller eddies, which, through continuous break-up, transfer turbulence energy to the dissipative scales. Two transport equations for the transfer-rate of the turbulent kinetic energy are also necessary, one for the rate of energy transfer from the large to the smaller scales and one for the rate of transfer from the smaller to the dissipative scales, the latter being of course the dissipation rate of turbulence. The two-time-scale model of Hanjalic et al. (1980) has been the starting point of the current developments. The coefficients of the terms which appear in these models have been determined from asymptotic analyses of decaying grid turbulence, homogeneous shear flows and local-equilibrium boundary-layer flows. The models were subsequently further developed to become more responsive to strong non-equilibrium features, such as those caused by strong shear. It has been identified that in general, models which have been optimised to satisfy equilibrium flows (as is usually done) are unable to capture the strong changes the flows are subjected to due to highly non-equilibrium effects, thus needing further development. Within the two-time-scale framework, it has been found possible to overcome this problem, by exploiting the additional information available on the turbulence spectrum. In a further departure from the development of earlier effective-viscosity two-time scale models, here the eddy viscosity expression is made sensitive to the local strain rate, as was also the case in single-scale models developed by Craft et al. (1996).The original contribution of this research is the development of two widely tested two-time-scale linear-eddy-viscosity models, referred to here as NT1 and NT2. The difference between these two models is primarily in the eddy viscosity formulation. The two resulting models have been tested over nine test cases and their performance has been compared to those of the standard high-Reynolds number k−ɛ model, the SSG Reynolds stress transport model and the two-time-scale model of Hanjalic et al. (1980). The test cases comprise of homogeneous shear and normally strained flows, adverse-pressure-gradient, favourable-pressure-gradient and oscillatory boundary layer flows, fully developed oscillatory and ramp up pipe flows and steady and pulsated backward-facing step flows. The models proposed here show considerable promise. They perform well in all cases tested and provide clear improvements over a range of single- and multi-scale models tested in an earlier study (Klein et al., 2015), over a set of test cases which cover a wide range of physical flow phenomena. Moreover, these improvements are achieved without any significant increase in computational requirements in comparison to the computational needs of single-time-scale eddy-viscosity models.