A simple and stable scale-similarity model for large Eddy simulation: Energy balance and existence of weak solutions

In averaging the Navier-Stokes equations, the problem of closure arises. Scale-similarity models address closure by (roughly speaking) extrapolation from the (known) resolved scales to the (unknown) unresolved scales. In a posteriori tests, scale-similarity models are often the most accurate but can prove to be unstable when used in a numerical simulation. In this report, we consider the scale-similarity model given by

. We prove it is stable (solutions satisfy an energy inequality) and deduce from that the existence of weak solutions of the model.     

A mixing model providing statistics of the conditional scalar dissipation rate

Joint composition probability density function (PDF) methods are used for the numerical simulation of turbulent reactive flows. Here, other than in classical Reynolds averaged Navier–Stokes (RANS) or large eddy simulation (LES) approaches, the highly non-linear chemical source term appears in closed form. On the other hand, mixing models are required for the closure of the molecular diffusion term. In the present work, the joint statistics of the scalar and the scalar dissipation rate provided by the parameterized scalar profile (PSP) mixing model are validated. The goal is to combine the PDF method with a flamelet approach, where the scalar dissipation rate plays a crucial role in determining the contribution of the chemical source term. (© 2008 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Numerical simulation of cavitation around a two-dimensional hydrofoil using VOF method and LES turbulence model

In this paper simulation of cavitating flow over the Clark-Y hydrofoil is reported using the large eddy simulation (LES) turbulence model and volume of fluid (VOF) technique. We applied an incompressible LES modelling approach based on an implicit method for the subgrid terms. To apply the cavitation model, the flow has been considered as a single fluid, two-phase mixture. A transport equation model for the local volume fraction of vapour is solved and a finite rate mass transfer model is used for the vapourization and condensation processes. A compressive volume of fluid (VOF) method is applied to track the interface of liquid and vapour phases. This simulation is performed using a finite volume, two phase solver available in the framework of the OpenFOAM (Open Field Operation and Manipulation) software package. Simulation is performed for the cloud and super-cavitation regimes, i.e., σ = 0.8, 0.4, 0.28. We compared the results of two different mass transfer models, namely Kunz and Sauer models. The results of our simulation are compared for cavitation dynamics, starting point of cavitation, cavity’s diameter and force coefficients with the experimental data, where available. For both of steady state and transient conditions, suitable accuracy has been observed for cavitation dynamics and force coefficients.     

Numerical solution of turbulent jet flows

The work deals with numerical modelling of several turbulent 3D jet flows: steady impinging jet, steady free jet in cross–flow, synthetic free jet (unsteady) and synthetic impinging jet (unsteady). The numerical method is based on artificial compressibility method with dual time extension for unsteady cases. Space discretization uses cell–centered finite volume method with third order accurate upwind approximation for convection, the time discretisations are implicit. Turbulence is modelled using two–equation eddy viscosity models and by explicit algebraic Reynolds stress model (EARSM by Wallin and Hellsten). The results of first three cases are compared with measurements. (© 2008 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Large eddy simulation of an ethylene-air turbulent premixed V-flame

Large eddy simulation (LES) using a dynamic eddy viscosity subgrid scale stress model and a fast-chemistry combustion model without accounting for the finite-rate chemical kinetics is applied to study the ignition and propagation of a turbulent premixed V-flame. A progress variable c-equation is applied to describe the flame front propagation. The equations are solved two dimensionally by a projection-based fractional step method for low Mach number flows. The flow field with a stabilizing rod without reaction is first obtained as the initial field and ignition happens just upstream of the stabilizing rod. The shape of the flame is affected by the velocity field, and following the flame propagation, the vortices fade and move to locations along the flame front. The LES computed time-averaged velocity agrees well with data obtained from experiments.     

The effect of modeling of velocity fluctuations on prediction of collection efficiency of cyclone separators

The effect of modeling of velocity fluctuations on the prediction of collection efficiency of cyclone separators has been numerically investigated using the Reynolds stress turbulence model (RSTM) and large eddy simulation (LES). The Eulerian–Lagrangian modeling approach of CFD code Fluent 6.3.26 has been employed to simulate the three dimensional, unsteady turbulent gas–solid flows in a Stairmand high efficiency cyclone. The simulated results have been compared with experimental observations available in the literature. The analysis of results shows that the RSTM and the LES have adequately predicted the mean flow field. Results of the present study demonstrate that the LES has good performance on prediction of fluctuating flow field and collection efficiency for each and every particle size. However, the performance of the RSTM is found poor in terms of prediction of velocity fluctuations and collection efficiency, especially for small particles. This relates to the precessing of the vortex core phenomenon, which is resolved more accurately by LES as compared to the RSTM simulation. The results suggest that the prediction of collection efficiency, especially for small particles is greatly influenced by the simulation of velocity fluctuations in cyclones.     

An Approach to Model Partially Premixed Turbulent Combustion withProbability Density Function (PDF) Methods

In turbulent combustion one distinguishes between premixed, non-premixed and partially premixed combustion. While laminar flamelet models proved to be extremely valuable for a wide range of non-premixed flame simulations, similar approaches are more problematic in the partially premixed regime. Here the laminar flamelet concept for non-premixed turbulent combustion simulations is generalized for the partially premixed regime. Similar as in the unsteady flamelet approach, the joint statistics of a progress variable, mixture fraction and scalar dissipation rate is used to obtain the joint statistics of the compositions from pre-computed flame tables. The required distribution is computed with a joint PDF method and the main differences between the new approach and previous ones, are the pre-computed tables and the way the evolution of the progress variable is calculated. Instead of evolving 1D flamelets, steady 2D solutions of burning flamelets propagating into unburned mixtures with varying mixture fraction are considered. The location of a fluid particle in this 2D laminar flame is defined by its mixture fraction and a burning time, which are modeled for each computational particle used in the PDF method. Numerical experiments of a turbulent lifted diffusion flame and a premixed Bunsen flame demonstrate that this approach can be employed for a wide range of applications. (© 2006 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Notes on modelling of environmental transport in wetland

This paper presents an effort towards a basic model for environmental transport of momentum, heat and mass transfer in the wetland. To smear out the discontinuity between the two phases of water and solid in the wetland, the continuum models distinctively applying for the water body and solid frame are transformed via the technique of phase average to give equations for a virtual single-phase flow in the entire domain of the wetland. Then to filter out the vortex and fluctuation common in the wetland flow, the operation of large eddy simulation (LES) is applied to yield a basic model for practical simulation. With reference to the modelling of flows in porous media and turbulent flows, closure relations are presented for the correlation terms due to the phase average and large eddy simulation.     

On the well-posedness and conservation laws of a family of multiscale deconvolution models for the magnetohydrodynamics equations

We present a mathematical study of a large eddy simulation (LES) model for the incompressible magnetohydrodynamics equations. The classical closure problem arising for LES models is solved with the multiscale deconvolution technique developed by Dunca in [11]. We prove the model admits unique, regular weak solutions and provide a mathematical study of the modeling error.     

Numerical simulation of methane partial oxidation in the burner and combustion chamber of autothermal reformer

This study aims to model the methane partial oxidation process in the burner and combustion chamber of autothermal reactor. The numerical simulation based on this model offers a powerful tool that can assist in reactor design and optimization and scale up of the process saving expensive pilot work. The steady-state governing equations were solved using the SIMPLE algorithm and the effect of turbulence on the mean flow field was accounted for using the RNG k–ε model. A two-step reaction mechanism was used for the gas combustion with CO as the intermediate species. The reaction rates were modeled using an Eddy-Dissipation Model. In terms of the geometrical model, a 3D model for burner was developed while an axis-symmetric model for the combustion chamber was implemented to reduce the computational costs. The model formulated was validated against a currently operating autothermal reactor and then has been used to investigate different aspects of these reactors. Results show that effect of oxygen to methane ratio is more than that of feed temperature. It is demonstrated that a 60% increase in O₂/CH₄ ratio causes a 15.4% decrease and 42.7% increase in H₂/CO ratio and methane conversion, respectively. In contrast, a 60% increase in feed temperature does not have a significant effect on the process.     

Comparisons between thermodynamic and one-dimensional combustion models of spark-ignition engines

A one-dimensional combustion model, employing a constant eddy diffusivity and a one-step chemical reaction, has been developed and applied to study the flame propagation in a spark-ignition engine. Calculations have been made at 1600 and 4200 rev min⁻¹ under fuel rich conditions and compared with available engine pressure data. One- and two-zone thermodynamic models have also been developed and applied to study the combustion process in the engine. The thermodynamic models have been compared with the one-dimensional model results and comparisons include the average mixture temperature, the temperatures of the burned and unburned gases and the flame surface area. These comparisons indicate that the one-dimensional model predictions are very sensitive to the eddy diffusivity and reaction rate data. The two-zone thermodynamic model predicts, first, a monotonically increasing flame surface area with time and, then, a monotonically decreasing surface area, whereas the one-dimensional model always predicts a monotonically increasing flame surface area. The average mixture temperature predicted by the one-zone thermodynamic model is higher than those of the two-zone and one-dimensional models during the compression stroke, while that of the one-dimensional model is higher than the temperatures predicted by the one- and two-zone models during the expansion stroke. The one-dmensional model predicts an accelerating flame even when the front approaches the cold cylinder wall. This yields a faster fuel consumption rate than those predicted by the one- and two-zone thermodynamic models which predict smoother burned fuel mass profiles.     

Finite element analysis of laminar and turbulent flows using LES and subgrid-scale models

Numerical simulations of laminar and turbulent flows in a lid driven cavity and over a backward-facing step are presented in this work. The main objectives of this research are to know more about the structure of turbulent flows, to identify their three-dimensional characteristic and to study physical effects due to heat transfer. The filtered Navier–Stokes equations are used to simulate large scales, however they are supplemented by subgrid-scale (SGS) models to simulate the energy transfer from large scales toward subgrid-scales, where this energy will be dissipated by molecular viscosity. Two SGS models are applied: the classical Smagorinsky’s model and the Dynamic model for large eddy simulation (LES). Both models are implemented in a three-dimensional finite element code using linear tetrahedral elements. Qualitative and quantitative aspects of two and three-dimensional flows in a lid-driven cavity and over a backward-facing step, using LES, are analyzed comparing numerical and experimental results obtained by other authors.     

On dispersion of settling particles from an elevated source in an open-channel flow

Longitudinal dispersion of suspended particles with settling velocity in a turbulent shear flow over a rough-bed surface is investigated numerically when the settling particles are released from an elevated continuous line-source. A combined scheme of central and four-point upwind differences is used to solve the steady turbulent convection–diffusion equation and the alternating direction implicit (ADI) method is adopted for the unsteady equation. It is shown how the mixing of settling particles is influenced by the ‘log-wake law’ velocity and the corresponding eddy diffusivity when the initial distribution of concentration is regarded as a line-source. The concentration profiles for the steady-state conditions agree well with the existing experimental data and some other numerical results when the settling velocity is zero. The behaviours of iso-concentration lines in the vertical plane for different releasing heights are studied in terms of the relative importance of convection, eddy diffusion and settling velocity.     

Extended flamelet model and improved interaction of chemistry and turbulence for modeling partially premixed combustion with a joint PDF method

Turbulent combustion is commonly categorized into premixed, non-premixed and partially premixed combustion. For nonpremixed combustion simulations the laminar flamelet concept proved to be very valuable while for the more complex case of partially premixed combustion this model shows considerable deficiencies. Here, the classical laminar flamelet approach is extended to the partially premixed combustion regime. For that, the joint statistics of mixture fraction, scalar dissipation rate and a progress variable, calculated with a joint probability density function (PDF) method, is used to get the statistics of the compositions and of the chemical energy source term from pre-processed flame tables. This approach can be compared with the unsteady flamelet concept; the main differences consists of the way the progress variable evolution is computed and in the pre-computed flame tables. The progress variable describes the point of time a fluid parcel is consumed by a flame front. The fluid parcels are represented by computational particles, which are used for PDF methods. The pre-computed flame tables are computed from steady solutions 2D stabilized flames propagating into an unburnt mixture with varying mixture fraction. The corresponding position of a fluid particle in such a 2D laminar flame is determined by its mixture fraction and a burning time; both to be modeled for each computational particle in the PDF simulation. Numerical experiments of turbulent diffusion jet flames demonstrate that this approach can be employed for challenging test cases. (© 2008 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Approximating the larger eddies in fluid motion V: Kinetic energy balance of scale similarity models

We first review a classical scale-similarity model used to simulate the motion of large eddies in a turbulent flow. The kinetic energy balance of this model is very unclear in theory. Experiments with it often have reported that an additional Smagorinski type subgridscale term is needed. This term is not benign; it can alter significantly the predicted long term dynamics of the large eddies. However, we also show that the principal of scale-similarity (introduced in 1980 by Bardina, Ferziger and Reynolds) can also give rise to other scale similarity models which have the correct kinetic energy balance.     

Effects of plane strain on inhomogeneous turbulence

The influence of the mean plane strain on the turbulence transportation is investigated by large eddy simulation (LES) in the shearless turbulence mixing layer. It is found that the mean strains enhance the turbulent fluctuations in the mixing region. Compression in the inhomogeneous direction can greatly increase the transport of turbulent kinetic energy by triple correlation terms, while stretching in the inhomogeneous direction decreases the turbulence transportation. The gradient diffusion models for turbulent transportation are evaluated and it is found that the intermittency consideration can improve the prediction ability of the gradient-type models for the triple correlation terms. Project supported by the Sino-French Laboratory in Beijing and the National Natural Science Foundation of China (Grant No. 19572041).     

A dynamic subgrid-scale model for the large eddy simulation of stratified flow

A new dynamic subgrid-scale (SGS) model, including subgrid turbulent stress and heat flux models for stratified shear flow is proposed by using Yoshizawa’s eddy viscosity model as a base model. Based on our calculated results, the dynamic subgrid-scale model developed here is effective for the large eddy simulation (LES) of stratified turbulent channel flows. The new SGS model is then applied to the large eddy simulation of stratified turbulent channel flow under gravity to investigate the coupled shear and buoyancy effects on the near-wall turbulent statistics and the turbulent heat transfer at different Richardson numbers. The critical Richardson number predicted by the present calculation is in good agreement with the value of theoretical analysis     

Computations of transport of pollutant in shallow water

The focus of this paper is to simulate the transport of a passive pollutant by a flow modelled by the two-dimensional shallow water equations. Considering the friction terms, new model for simulating the steady and unsteady transport of pollutant is established. Then the adaptive semi-discrete central-upwind scheme based on central weighted essentially non-oscillatory reconstruction is utilized for simulating the two-dimensional steady and unsteady transport of pollutant. The non-oscillatory behavior and accuracy of the scheme are demonstrated by the numerical result.     

Asymptotic behaviour of commutation errors and the divergence of the Reynolds stress tensor near the wall in the turbulent channel flow

The derivation of the space averaged Navier–Stokes equations for the large eddy simulation (LES) of turbulent incompressible flows introduces two groups of terms which do not depend only on the space averaged flow field variables: the divergence of the Reynolds stress tensor and commutation errors. Whereas the former is studied intensively in the literature, the latter terms are usually neglected. This note studies the asymptotic behaviour of these terms for the turbulent channel flow at a wall in the case that the commutation errors arise from the application of a non‐uniform box filter. To perform analytical calculations, the unknown flow field is modelled by a wall law (Reichardt law and 1/αth power law) for the mean velocity profile and highly oscillating functions model the turbulent fluctuations. The asymptotics show that near the wall, the commutation errors are at least as important as the divergence of the Reynolds stress tensor. Copyright © 2006 John Wiley & Sons, Ltd.     

Finite element error analysis of a zeroth order approximate deconvolution model based on a mixed formulation

A suitable discretization for the Zeroth Order Model in Large Eddy Simulation of turbulent flows is sought. This is a low order model, but its importance lies in the insight that it provides for the analysis of the higher order models actually used in practice by the pioneers Stolz and Adams [N.A. Adams, S. Stolz, On the approximate deconvolution procedure for LES, Phys. Fluids 2 (1999) 1699-1701; N.A. Adams, S. Stolz, Deconvolution methods for subgrid-scale approximation in large eddy simulation, in: B.J. Geurts (Ed.), Modern Simul. Strategies for Turbulent Flow, Edwards, Philadelphia, 2001, pp. 21-44] and others. The higher order models have proven to be of high accuracy. However, stable discretizations of them have proven to be tricky and other stabilizations, such as time relaxation and eddy viscosity, are often added. We propose a discretization based on a mixed variational formulation that gives the correct energy balance. We show it to be unconditionally stable and prove convergence.