Classical Smagorinsky Model Research Articles

Validation of an OpenFOAM®-based solver for the Euler equations with benchmarks for mesoscale atmospheric modeling

Within OpenFOAM, we develop a pressure-based solver for the Euler equations written in conservative form using density, momentum, and total energy as variables. Under simplifying assumptions, these equations are used to describe non-hydrostatic atmospheric flow. For the stabilization of the Euler equations and to capture sub-grid processes, we consider two Large Eddy Simulation models: the classical Smagorinsky model and the one equation eddy-viscosity model. To achieve high computational efficiency, our solver uses a splitting scheme that decouples the computation of each variable. The numerical results obtained with our solver are validated against numerical data available in the literature for two classical benchmarks: the rising thermal bubble and the density current. Through qualitative and quantitative comparisons, we show that our approach is accurate. This paper is meant to lay the foundation for a new open-source package specifically created for the quick assessment of new computational approaches for the simulation of atmospheric flows at the mesoscale level.

Numerical analysis of a corrected Smagorinsky model

AbstractThe classical Smagorinsky model's solution is an approximation to a (resolved) mean velocity. Since it is an eddy viscosity model, it cannot represent a flow of energy from unresolved fluctuations to the (resolved) mean velocity. This model has recently been corrected to incorporate this flow and still be well‐posed. Herein we first develop some basic properties of the corrected model. Next, we perform a complete numerical analysis of two algorithms for its approximation. They are tested and proven to be effective.

A new idea to predict reshocked Richtmyer–Meshkov mixing: constrained large-eddy simulation

Abstract

Large-Eddy Simulations of Stratified Atmospheric Boundary Layers: Comparison of Different Subgrid Models

The development and assessment of subgrid-scale (SGS) models for large-eddy simulations of the atmospheric boundary layer is an active research area. In this study, we compare the performance of the classical Smagorinsky model, the Lagrangian-averaged scale-dependent (LASD) model, and the anisotropic minimum dissipation (AMD) model. The LASD model has been widely used in the literature for 15 years, while the AMD model was recently developed. Both the AMD and the LASD models allow three-dimensional variation of SGS coefficients and are therefore suitable to model heterogeneous flows over complex terrain or around a wind farm. We perform a one-to-one comparison of these SGS models for neutral, stable, and unstable atmospheric boundary layers. We find that the LASD and the AMD models capture the logarithmic velocity profile and the turbulence energy spectra better than the Smagorinsky model. In stable and unstable boundary-layer simulations, the AMD and LASD model results agree equally well with results from a high-resolution reference simulation. The performance analysis of the models reveals that the computational overhead of the AMD model and the LASD model compared to the Smagorinsky model is approximately 10% and 30% respectively. The LASD model has a higher computational and memory overhead because of the global filtering operations and Lagrangian tracking procedure, which can result in bottlenecks when the model is used in extensive simulations. These bottlenecks are absent in the AMD model, which makes it an attractive SGS model for large-scale simulations of turbulent boundary layers.

Subgrid-scale stress parameterization for anisotropic turbomachinery flow as deduced from stereoscopic particle image velocimetry measurements

The present study searches for parametric subgrid-scale models for the turbomachinery flow based on single-plane stereoscopic particle image velocimetry measurements. A group of deformation tensors are used as the predictors to parameterize the subgrid-scale stress tensor as the response. Missing terms in the rotation rate tensor are completed with a novel approach which is based on geometrical relations between deformation and subgrid-scale tensors. The optimal relations for parametric subgrid-scale models are shaped after minimizing the square of error between modeled and real SGS stresses. The most superior and inferior one-predictor models are based on strain rate and rotation rate tensors, respectively. The two-predictor models show the most efficient prediction of subgrid-scale stress where the correlation coefficient is up to 43% larger than that of the classical Smagorinsky model. The three- to five-predictor models are inefficient since they give minor enhancement of correlation coefficient with a considerably larger computational cost.

Large eddy simulation of flow through a periodic array of urban-like obstacles using a canopy stress method

Large-eddy simulation (LES) of a turbulent flow through an array of building-like obstacles is an idealized model to study transport of contaminants in the urban atmospheric boundary layer (urban ABL). A reasonably accurate LES prediction of turbulence in such an urban ABL must resolve a significant proportion of the small but energetic eddies in the roughness sublayer, which remains prohibitive even though computational power has been increased significantly. Recently, some researchers also reported a high level of inaccuracy in turbulence prediction if LES were coupled with an adaptive mesh refinement technique in order to optimize the cost of resolving the roughness sublayer. In this article, we present a turbulence closure methodology for LES of urban ABLs in which the roughness elements are represented through the canopy stress method, and the subgrid scale stress is modeled through a dynamically adaptive eddy viscosity method. Unlike the classical Smagorinsky model that considers only the ‘strain portion’ of the velocity gradient tensor, we consider both the ‘strain tensor’ and the ‘rotation tensor’ to compute the eddy viscosity. This allows us to dynamically adapt the rate of energy dissipation to the scale of the energetic eddies in the roughness sublayer. Without employing a mesh conforming to the urban roughness elements, the effect of such solid bodies are represented in the LES model through a canopy stress method in which the loss of pressure and the sink of momentum due to the interaction between eddies and roughness elements are parameterized using the instantaneous velocity field. Simulation results of the proposed canopy stress method is compared with that of a conventional Computational Fluid Dynamics (CFD) method employing a block-structured mesh conforming around the roughness elements. For urban flow simulations, the results demonstrate that the proposed canopy stress model is accurate in predicting vertical profiles of mean and variance, as well as the temporal intermittency of coherent structures.

A Modulated-Gradient Parametrization for the Large-Eddy Simulation of the Atmospheric Boundary Layer Using the Weather Research and Forecasting Model

The performance of the modulated-gradient subgrid-scale (SGS) model is investigated using large-eddy simulation (LES) of the neutral atmospheric boundary layer within the weather research and forecasting model. Since the model includes a finite-difference scheme for spatial derivatives, the discretization errors may affect the simulation results. We focus here on understanding the effects of finite-difference schemes on the momentum balance and the mean velocity distribution, and the requirement (or not) of the ad hoc canopy model. We find that, unlike the Smagorinsky and turbulent kinetic energy (TKE) models, the calculated mean velocity and vertical shear using the modulated-gradient model, are in good agreement with Monin-Obukhov similarity theory, without the need for an extra near-wall canopy model. The structure of the near-wall turbulent eddies is better resolved using the modulated-gradient model in comparison with the classical Smagorinsky and TKE models, which are too dissipative and yield unrealistic smoothing of the smallest resolved scales. Moreover, the SGS fluxes obtained from the modulated-gradient model are much smaller near the wall in comparison with those obtained from the regular Smagorinsky and TKE models. The apparent inability of the LES model in reproducing the mean streamwise component of the momentum balance using the total (resolved plus SGS) stress near the surface is probably due to the effect of the discretization errors, which can be calculated a posteriori using the Taylor-series expansion of the resolved velocity field. Overall, we demonstrate that the modulated-gradient model is less dissipative and yields more accurate results in comparison with the classical Smagorinsky model, with similar computational costs.

Large-eddy simulations of forced isotropic turbulence with viscoelastic fluids described by the FENE-P model

A new subgrid-scale (SGS) model developed for large-eddy simulations (LES) of dilute polymer solutions, described by the finitely extensible nonlinear elastic constitutive equation closed with the Peterlin approximation, is presented. In this distortion similarity model (DSIM) the filtered conformation tensor evolution equation is based on the self-similarity of the polymer stretching terms, and on a global equilibrium of the trace of the conformation tensor, which is proportional to the elastic energy stored in the polymer molecules, while the SGS stresses are modelled with the classical Smagorinsky model. The DSIM closure is assessed in direct numerical simulations (DNS) of forced isotropic turbulence using classical a priori tests, and in a posteriori (LES) showing very good agreement with all the exact (filtered DNS) results. The DSIM model is simple to implement and computationally inexpensive and represents a major step forward in the numerical simulation of turbulent flows of Newtonian fluids with polymer additives.

LES Predictions of Self-Sustained Oscillations in Homogeneous Density Round Free Jet

The paper presents a detailed LES analysis of turbulent round jets dominated by the mechanism of Kelvin-Helmholtz (K-H) instability and the so-called self-sustained regime, which is characterised by large velocity fluctuations, reminiscent of the behaviour of excited jets. It is shown that the occurrence of this regime is largely conditioned by the type and parameters of the inlet jet velocity profile, i.e., the shear layer momentum thickness 𝜃, turbulence intensity Ti. A high order numerical code based on the combined pseudo-spectral / compact difference methods is used in the simulations. Analysis is performed for the Reynolds number Re = 1 × 104 with 𝜃 characterised by R/𝜃 = 16, 20, 24, 28 and 32 (with R - jet radius) and for Ti = 10−2, 10−3, 10−4. Two inlet velocity profiles are used in the simulations: hyperbolic tangent and Blasius. Comparisons focus on the axial velocity profiles and the spectra of the velocity signals. It is shown that in the self-sustained regime the results obtained with the Blasius profile are significantly closer to the experimental data. Sensitivity tests of the self-sustained regime on the sub-grid modelling are performed based on four well known models: classical and dynamic Smagorinsky, the filtered structure function model of Ducros et al. (JFM, 1996) and the relatively new model proposed by Vreman (PoF, 2004). It is shown that in the case of the classical Smagorinsky model an excess of sub-grid dissipation prevents the appearance of self-sustained velocity oscillations and in effect gives results significantly different from the remaining models. On the other hand, when the jets are dominated by K-H instability all the models lead to very similar solutions.

Comment on “A self-adjusting flow dependent formulation for the classical Smagorinsky model coefficient” [Phys. Fluids 25, 055102 (2013)

In this comment, we show that the novel formulation of the Smagorinsky model coefficient, proposed by Ghorbaniasl et al. [Phys. Fluids 25, 055102 (2013)], satisfies neither the Galilean invariance nor the coordinate invariance, thus does not have physical significance. We conclude that their model is just to provide a coordinate-system-dependent dissipation term.

Response to “Comment on ‘A self-adjusting flow dependent formulation for the classical Smagorinsky model coefficient’” [Phys. Fluids 25, 099101 (2013)

First Page

A self-adjusting flow dependent formulation for the classical Smagorinsky model coefficient

In this paper, we propose an efficient formula for estimating the model coefficient of a Smagorinsky model based subgrid scale eddy viscosity. The method allows vanishing eddy viscosity through a vanishing model coefficient in regions where the eddy viscosity should be zero. The advantage of this method is that the coefficient of the subgrid scale model is a function of the flow solution, including the translational and the rotational velocity field contributions. Furthermore, the value of model coefficient is optimized without using the dynamic procedure thereby saving significantly on computational cost. In addition, the method guarantees the model coefficient to be always positive with low fluctuation in space and time. For validation purposes, three test cases are chosen: (i) a fully developed channel flow at ${\mathop{\rm Re}\nolimits} _\tau = 180,\,395$ Re τ=180,395, (ii) a fully developed flow through a rectangular duct of square cross section at ${\mathop{\rm Re}\nolimits} _\tau = 300$ Re τ=300, and (iii) a smooth subcritical flow past a stationary circular cylinder, at a Reynolds number of ${\mathop{\rm Re}\nolimits} = 3900$ Re =3900, where the wake is fully turbulent but the cylinder boundary layers remain laminar. A main outcome is the good behavior of the proposed model as compared to reference data. We have also applied the proposed method to a CT-based simplified human upper airway model, where the flow is transient.

High-order Large Eddy Simulations of Confined Rotor-Stator Flows

In many engineering and industrial applications, the investigation of rotating turbulent flow is of great interest. In rotor-stator cavities, the centrifugal and Coriolis forces have a strong influence on the turbulence by producing a secondary flow in the meridian plane composed of two thin boundary layers along the disks separated by a non-viscous geostrophic core. Most numerical simulations have been performed using RANS and URANS modelling, and very few investigations have been performed using LES. This paper reports on quantitative comparisons of two high-order LES methods to predict a turbulent rotor-stator flow at the rotational Reynolds number Re=400000. The classical dynamic Smagorinsky model for the subgrid-scale stress (Germano et al., Phys Fluids A 3(7):1760-1765, 1991) is compared to a spectral vanishing viscosity technique (S\'everac & Serre, J Comp Phys 226(2):1234-1255, 2007). Numerical results include both instantaneous data and postprocessed statistics. The results show that both LES methods are able to accurately describe the unsteady flow structures and to satisfactorily predict mean velocities as well as Reynolds stress tensor components. A slight advantage is given to the spectral SVV approach in terms of accuracy and CPU cost. The strong improvements obtained in the present results with respect to RANS results confirm that LES is the appropriate level of modelling for flows in which fully turbulent and transition regimes are involved.

A novel dynamic coherent eddy model and its application to LES of a turbulent jet with free surface

We has developed a novel dynamic coherent eddy model, in which the coherent structure—Q criterion introduced by Hunt et al (1988)—is taken into account in the subgrid-scale turbulent viscosity based on the eddy viscosity model. One proposed method is to combine the resolved-scale velocity-gradient tensor in the classical Smagorinsky model with Q criterion weighted. A kind of dynamic procedure which was averaged in a general process at temporal direction based on the autocorrelations of the characteristic resolved scales of turbulence was taken for the coefficient of subgrid model. The model is implemented in the σ-coordinate and the filtered Navier-Stokes equations are solved by the operator splitting method. The validation was explored to 2-D turbulent slot jet both in ambient environment and in regular waves. The prediction of the present model was compared with the experiment data, including the averaged velocity profiles, the velocity fluctuations and the Reynolds stress. The model performance is shown to be satisfactory.

Computational study of conjugate heat transfer in T-junctions

In this work we focus on the numerical prediction of temperature fluctuations induced in solid materials through turbulent mixing processes. As test case we use the mixing of two streams of different temperature in a T-junction. Due to the turbulent mixing of the two streams temperature fluctuations occur which are also transferred to the solid walls in contact with the fluid. Such fluctuations in the solid material may lead to thermal fatigue and are therefore relevant for the lifetime management of components used in nuclear power plants (NPP). We investigate the mixing in T-junctions made of different materials and having different pipe wall thicknesses. The temperature difference between the streams in the main and side branch is set to 75 °C and the mass flow rate in the main pipe is three times larger than in the side branch. In a first step we perform a set of simulations by using different formulations of the large-eddy simulation (LES) subgrid scale model, i.e. classical Smagorinsky model and dynamic procedure, to identify the influence of the modeled subgrid scales on the simulation results. The comparison between available experimental data and the numerical results reveals a good agreement when using the dynamic procedure. In a second step we address the temperature fluctuations in the solid wall subject to the wall thickness. The influence of the wall thickness is represented as a damping effect on the temperature fluctuations in radial direction in the pipe material. This study shows the capability of LES to predict thermal fluctuations in turbulent mixing.

Gas−Solid Turbulent Flow in a Circulating Fluidized Bed Riser: Numerical Study of Binary Particle Systems

Numerical simulations were performed of a turbulent gas-particle multiphase flow in a circulating fluidized bed riser using a hard-sphere discrete particle model (DPM) for the particle phase and the Navier−Stokes equations for the gas phase, where the subgrid scale stresses (SGS) were modeled with the SGS model proposed by Vreman.(1) The model enables the simulation of systems with an arbitrary particle size distribution. In this work, binary mixtures of particles with different diameters were studied. From the numerical results it is found that the particles display size distribution in both horizontal and vertical directions. Small particles have a higher vertical particle velocity than the large particles. With increasing superficial gas velocity, the vertical particle velocity is increased. The average particle velocity and concentration vary both in the horizontal and vertical directions. Finally, the numerical results are compared with the experimental and numerical results of Mathiesen et al.(2). It is found that the simulation results with Vreman’s SGS model agree better with the literature data than those obtained with a classical Smagorinsky(3) model, especially in the zone close to the wall.

Quality of dynamic variational multiscale models on distorted grids

The present paper focuses on a dynamic version of the variational multiscale model and investigates its performance in large eddy simulations (LES) of turbulent channel flow at Reynolds numbers (based on friction velocity) of 180, 395, and 590. The dynamic procedure was implemented for the variational multiscale model on the basis of the classical Smagorinsky model and the wall-adapting local eddy-viscosity (WALE) model. One base regular grid and three different distorted grids are employed in the evaluation. The dynamic multiscale models are compared with the direct numerical simulation (DNS) data and the conventional dynamic models. Next, the error components involved in LES are separated and the grid irregularity effect on the shortcoming of the subgrid scale models and the error of numerical approximation is assessed. A main outcome is that the dynamic multiscale models in comparison with the dynamic Smagorinsky and the dynamic WALE are less affected by grid distortion, leading to better behavior on the irregular grids.

ON THE FLOW PAST RECTANGULAR CYLINDERS: PHYSICAL ASPECTS AND NUMERICAL SIMULATION

This paper concerns with unsteady incompressible flows around rectangular cylinders with side ratio varying from 1 to 10. Phenomenological aspects are discussed and numerical simulations are performed using a SIMPLEC finite volume code. A third-order QUICK scheme is employed for the advective terms in the Navier-Stokes equations, while a second-order fully implicit method is used for the time discretization. For validation purpose, preliminary simulations are carried out at Re = 300. Afterwards, the flow patterns and the wake periodic features are examined at Re = 1,000, 5,000, and 22,000, for which turbulent effects should not be neglected. In some of those cases, large-eddy simulation (LES) is employed, using the classical sub-grid Smagorinsky model. Important physical mechanisms determining vortex shedding frequency are placed in evidence. The present predictions are compared with numerical and experimental results from other works and a good agreement is reached.

Inertial consistent subgrid model for large-eddy simulation based on the lattice Boltzmann method

The recently introduced inertial-range (IR) consistent Smagorinsky model and the classical Smagorinsky model are applied to the large-eddy simulation (LES) of decaying homogeneous isotropic turbulence based on the lattice Boltzmann method (LBM), which is implemented using the 19-velocity D3Q19 lattice model. The objectives of this study are to examine the effectiveness of the LES-LBM technique for study of turbulence and to extend and validate the efficiency of the inertial-range consistent Smagorinsky model for lattice Boltzmann fluid dynamics. The LES-LBM results are compared with the direct numerical simulation data as well as experimental data. The time evolution of the kinetic energy and the decay exponents of the dissipation rate, the velocity derivative skewness, and instantaneous energy spectra are analyzed. The dependency of behavior of the model coefficients on the ratio of grid width Δ and the Kolmogorov scale η is examined numerically. The results demonstrate that the LES-LBM in conjunction with the IR consistent Smagorinsky model can be used to simulate turbulence more satisfactorily than the standard Smagorinsky model.

A study of time correlations in lattice Boltzmann-based large-eddy simulation of isotropic turbulence

As one of statistical characteristics of turbulent flow, space-time correlation is important when studies contribute to understanding of the mechanism of sound radiation and wave scattering by turbulence. This paper combines large-eddy simulations (LES) with the lattice Boltzmann method (LBM) to study the effects of subgrid models on time correlations in decaying isotropic turbulence. The performance of the inertial-range (IR) consistent model and the classical Smagorinsky model is evaluated in terms of two-time Eulerian velocity correlations at the LBM framework. The present study compares the correlations evaluated by LES and the direct numerical simulations based on the lattice Boltzmann equation and Navier–Stokes (NS) equations, respectively. It shows that the lattice Boltzmann subgrid models yield an underestimation of the magnitude of the time correlation on different time intervals and on a variety of wavenumber modes, similar to NS subgrid models. It is also observed that the calculated decorrelation time scale is in good agreement with the theoretical analysis for the Eulerian time microscale and scales nearly as τ(k)∝(Vk)−1. It appears that τ(k) may be either underestimated or overestimated depending on the shape of initial energy spectrum to some extent. Furthermore, compared to the classical Smagorinsky model with different initial spectra, the IR consistent subgrid model shows promising performance on the prediction of the time-space correlations in turbulent flows.