Periodic Filtering as a subgrid-scale model for LES of laminar separation bubble flows

Laminar separation bubbles develop over many blades and airfoils at moderate angles of attack and Reynolds numbers ranging from 10⁴ to 10⁵. More accurate simulation tools are necessary to enable higher fidelity design optimisation for these airfoils and blades as well as to test flow control schemes. Following previous investigators, an equivalent problem is formulated by imposing suitable boundary conditions for flow over a flat plate which allows to use a high accuracy spectral solver. Large eddy simulation (LES) of such a flow were performed at drastically reduced resolution to assess the accuracy of several LES modelling approaches: the explicit dynamic Smagorinsky model, implicit LES, and the truncated Navier–Stokes approach (TNS). To mimic dissipation that occurs in implicit LES, the solution on a coarse mesh is filtered at every time step and two different filter strengths are used. In the TNS approach, the solution is filtered periodically, every few hundred time steps. The performance of each approach is evaluated against benchmark direct numerical simulation (DNS) data focusing on pressure and skin friction distributions, which are critical to airfoil designers. TNS results confirm that periodic filtering can act as an apt substitute for explicit subgrid-scale models, whereas filtering at every time step demonstrates the dependence of implicit LES on details of numerics.     

A velocity-estimation subgrid model constrained by subgrid scale dissipation

Purely dissipative eddy-viscosity subgrid models have proven very successful in large-eddy simulations (LES) at moderate resolution. Simulations at coarse resolutions where the underlying assumption of small-scale universality is not valid, warrant more advanced models. However, non-eddy viscosity models are often unstable due to the lack of sufficient dissipation. This paper proposes a simple modeling approach which incorporates the dissipative nature of existing eddy viscosity models into more physically appealing non-eddy viscosity SGS models. The key idea is to impose the SGS dissipation of the eddy viscosity model as a constraint on the non-eddy viscosity model when determining the coefficients in the non-eddy viscosity model. We propose a new subgrid scale model (RSEM), which is based on estimation of the unresolved velocity field. RSEM is developed in physical space and does not require the use of finer grids to estimate the subgrid velocity field. The model coefficient is determined such that total SGS dissipation matches that from a target SGS model in the mean or least-squares sense. The dynamic Smagorinsky model is used to provide the target dissipation. Results are shown for LES of decaying isotropic turbulence and turbulent channel flow. For isotropic turbulence, RSEM displays some level of backward dissipation, while yielding as good results as the dynamic Smagorinsky model. For channel flow, the results from RSEM are better than those from the dynamic Smagorinsky model for both statistics and instantaneous flow structures.     

What does Finite Volume-based implicit filtering really resolve in Large-Eddy Simulations?

The study illustrated in this paper completes the topics initially investigated in Ref. [42], the aim being here to analyze the role of the integral-based Finite Volume (FV) discretizations in Large Eddy Simulations that exploit the implicit filtering approach. Specifically, a theoretical study on the effective shape and length of three-dimensional filters induced by some FV-based flux reconstructions is the object of this paper. For any integral-based flux reconstruction, one gets always an approximation of the top-hat filter kernel. This is not the case of the filters induced by the differential-based Finite Difference operators, such as those reported and analyzed in Refs.  and . Considering the sub-filter resolution parameter Q = Δ_eff/h, being Δ_eff the effective filter width and h the computational grid size, allows us discerning the effective measure of the approximate built-in top-hat filter. The induced shape and width is analyzed by means of a modified wavenumber-like analysis that is developed in the 3D Fourier space. Several evaluation criteria applied on different schemes are considered and the differences in terms of either velocity or flux interpolations on staggered or non-staggered grids are analyzed. Conclusions are reported that, depending on the using of either the integral or the differential form of the filtered equations, the induced numerical filter is or is not a congruent approximation of the exact top-hat transfer function for some value Q. The need of a suitable estimation of the sub-filter parameter Q is assessed from several real LES computations, that make use of the new integral-based version of the eddy-viscosity dynamic modeling presented in Ref. [42]. In fact, it is shown that the test-filtering length has to be carefully chosen as a function of the FV-based induced filter.     

Analysis of a dynamic model for subfilter scalar dissipation rate in large eddy simulation based on the subfilter scalar variance transport equation

Accurate prediction of non-premixed turbulent combustion using large eddy simulation (LES) requires detailed modelling of the mixing between fuel and oxidizer that occurs at scales smaller than the LES filterwidth. The small-scale mixing process can be quantitatively characterized by two related variables, the subfilter scalar variance and the subfilter scalar dissipation rate. A recently proposed alternative dynamic modelling procedure for the subfilter scale dissipation rate, designed for use with transport equation based models for subfilter scalar variance, is analysed in this work. This new dynamic non-equilibrium modelling approach produces a nonlinear interaction between variance and dissipation rate predictions that makes it difficult to isolate the performance of any single modelling component in a conventional LES simulation. To gain a better understanding of the new model, a three-part study is undertaken here. The first part of the study uses a priori analysis to examine some novel aspects of the model’s computation and guide its practical implementation. In the second part of the study, detailed a posteriori analysis of the model is performed. This analysis suggests that the dynamic estimate of the dissipation rate model coefficient helps to compensate for over-prediction of variance production rates and improves the accuracy of variance prediction. However, improved modelling of the variance production term, which in turn depends on the accuracy of models for the subfilter scalar flux, is necessary to allow both the scalar variance and dissipation rate to be predicted accurately. Therefore, the third part of the study examines the effect of the scalar flux model on the predictions of the dynamic non-equilibrium model. Use of a mixed model for the fluxes, rather than a gradient-diffusion-only model, is found to improve variance predictions in some cases.     

Simulation of diesel spray combustion using LES and a multicomponent vapourisation model

Diesel spray and combustion in a constant-volume engine cylinder was simulated by a large eddy simulation (LES) approach coupling with a multicomponent vapourisation (MCV) modelling. The simulation focused on the inclusion of the interaction between fuel spray and gas-phase turbulence flow at the sub-grid scale. The LES was based on the dynamic structure sub-grid model, and an additional source term was added to the filtered momentum equation to account for the effect of drop motion on the gas-phase turbulence. The multicomponent drop vapourisation modelling was based on the continuous thermodynamics approach using a gamma distribution to describe the complex diesel fuel composition and was capable of predicting a more complex drop vapourisation process. The effect of gas-phase turbulence flow on the fuel drop vapourisation process was evaluated through the solution of the gas-phase moments of the distribution in the present LES framework. A non-evaporative spray in a constant-volume engine cylinder was first simulated to examine the behaviours of LES, in comparison with a Reynolds-averaged Navier–Stokes (RANS) simulation based on the RNG k–? model. More realistic diesel spray structures and improved agreement on liquid penetration length with the corresponding experimental data were predicted by the LES, using a grid resolution close to that of RANS. A more comprehensive simulation of diesel spray and combustion in cylindrical combustor was also performed. Predicted distributions of soot particles were compared to the experimental image, and improved agreement with the experimental data was also observed by using the present LES and MCV models. Consequently, results of the present models proved that improved overall performance of the fuel spray simulation can be achieved by the LES without a significant increase in the computational load compared to the RANS.     

Impact of subgrid fluid turbulence on inertial particles subject to gravity

Two-phase turbulent flows with the dispersed phase in the form of small, spherical particles are increasingly often computed with the large-eddy simulation (LES) of the carrier fluid phase, coupled to the Lagrangian tracking of particles. To enable further model development for LES with inertial particles subject to gravity, we consider direct numerical simulations of homogeneous isotropic turbulence with a large-scale forcing. Simulation results, both without filtering and in the a priori LES setting, are reported and discussed. A full (i.e. a posteriori) LES is also performed with the spectral eddy viscosity. Effects of gravity on the dispersed phase include changes in the average settling velocity due to preferential sweeping, impact on the radial distribution function and radial relative velocity, as well as direction-dependent modification of the particle velocity variance. The filtering of the fluid velocity, performed in spectral space, is shown to have a non-trivial impact on these quantities.     

On the application of a new formulation of nonlinear eddy viscosity based on anisotropy to numerical ocean models

The present paper explores the derivation of an alternative nonlinear eddy viscosity formulation based on Reynolds stress anisotropy and its implementation to numerical ocean models. This formulation takes into account the vorticity in addition to the mean strain rate. The proposed formulation does not include the stability function method which is a common approach in eddy viscosity calculations used in the present state-of-the-art numerical ocean models. Instead, it depends on the second invariant of anisotropy. Initially, the performance of the formulation is checked through a simple channel flow simulation. Consequently for model calibration, an idealised experiment of mixed-layer entrainment into stably stratified flow has been simulated and compared to empirical data. For sensitivity studies related to shear and stable stratification, concept of steady-state Richardson number is applied for homogeneous shear layer. Finally, the performance of the new formulation is tested by implementing it into one-dimensional General Ocean Turbulence Model. Furthermore, a realistic oceanic test case of a storm has been investigated considering different physical processes for the Fladenground Experiment (FLEX’ 76) in the northern North Sea and the results have been compared to the measured data. The main results signify that the overall performance of the nonlinear eddy viscosity model with a different value of steady-state Richardson number is as good as the Mellor–Yamada model in terms of predictability, and the eddy viscosity and diffusivity profiles follow the principle of law of the wall. Additionally, the present formulation does not require computing the stability functions and the ease of implementation into numerical ocean models gives the present formulation an upper hand over the existing formulations in the field of turbulence modelling in oceanography.     

Numerically accurate computational techniques for optimal estimator analyses of multi-parameter models

Modelling unclosed terms in partial differential equations typically involves two steps: First, a set of known quantities needs to be specified as input parameters for a model, and second, a specific functional form needs to be defined to model the unclosed terms by the input parameters. Both steps involve a certain modelling error, with the former known as the irreducible error and the latter referred to as the functional error. Typically, only the total modelling error, which is the sum of functional and irreducible error, is assessed, but the concept of the optimal estimator enables the separate analysis of the total and the irreducible errors, yielding a systematic modelling error decomposition. In this work, attention is paid to the techniques themselves required for the practical computation of irreducible errors. Typically, histograms are used for optimal estimator analyses, but this technique is found to add a non-negligible spurious contribution to the irreducible error if models with multiple input parameters are assessed. Thus, the error decomposition of an optimal estimator analysis becomes inaccurate, and misleading conclusions concerning modelling errors may be drawn. In this work, numerically accurate techniques for optimal estimator analyses are identified and a suitable evaluation of irreducible errors is presented. Four different computational techniques are considered: a histogram technique, artificial neural networks, multivariate adaptive regression splines, and an additive model based on a kernel method. For multiple input parameter models, only artificial neural networks and multivariate adaptive regression splines are found to yield satisfactorily accurate results. Beyond a certain number of input parameters, the assessment of models in an optimal estimator analysis even becomes practically infeasible if histograms are used. The optimal estimator analysis in this paper is applied to modelling the filtered soot intermittency in large eddy simulations using a dataset of a direct numerical simulation of a non-premixed sooting turbulent flame.     

Large eddy simulation of a light gas stratification break-up by an entraining turbulent fountain

The erosion process of a stably stratified light gas layer by a vertical turbulent fountain of denser fluid inside a generic containment – for which experimental reference data are available – is studied computationally using large eddy simulation (LES). In addition, various Reynolds averaged Navier–Stokes (RANS) models are applied aiming at a comparative assessment of different computational approaches for the considered case. With the LES methodology included into the present modelling study, a novelty to date is established for fountain-stratification interaction inside generic containments. The high Reynolds number RANS models applied in the framework of this study include both the realisable k–? eddy viscosity model (EVM) as well as the basic Reynolds stress model (RSM). Furthermore, we show that certain regimes of the present configuration can be predicted using an analytically derived scaling approach. Various data beyond the experimentally obtained ones are computationally provided in order to facilitate the calibration of less costly statistical turbulence models and lumped parameter codes, since the presently considered configuration is regarded to be a valuable small-scale equivalent for containment flow applications.     

Proposed stochastic parameterisations of subgrid turbulence in large eddy simulations of turbulent channel flow

Stochastic and deterministic subgrid parameterisations are developed for the large eddy simulation (LES) of a turbulent channel flow with friction-velocity-based Reynolds number of Re_τ = 950 and centreline-based Reynolds number of Re₀ = 20,580. The subgrid model coefficients (eddy viscosities) are determined from the statistics of truncated reference direct numerical simulations (DNSs). The stochastic subgrid model consists of a mean-field shift, a drain eddy viscosity acting on the resolved field and a stochastic backscatter force of variance proportional to the backscatter eddy viscosity. The deterministic variant consists of a net eddy viscosity acting on the resolved field, which represents the net effect of the drain and backscatter. LES adopting the stochastic and deterministic models is shown to reproduce the time-averaged kinetic energy spectra of the DNS within the resolved scales.     

Evaluation of deconvolution modelling applied to numerical combustion

A possible modelling approach in the large eddy simulation (LES) of reactive flows is to deconvolve resolved scalars. Indeed, by inverting the LES filter, scalars such as mass fractions are reconstructed. This information can be used to close budget terms of filtered species balance equations, such as the filtered reaction rate. Being ill-posed in the mathematical sense, the problem is very sensitive to any numerical perturbation. The objective of the present study is to assess the ability of this kind of methodology to capture the chemical structure of premixed flames. For that purpose, three deconvolution methods are tested on a one-dimensional filtered laminar premixed flame configuration: the approximate deconvolution method based on Van Cittert iterative deconvolution, a Taylor decomposition-based method, and the regularised deconvolution method based on the minimisation of a quadratic criterion. These methods are then extended to the reconstruction of subgrid scale profiles. Two methodologies are proposed: the first one relies on subgrid scale interpolation of deconvolved profiles and the second uses parametric functions to describe small scales. Conducted tests analyse the ability of the method to capture the chemical filtered flame structure and front propagation speed. Results show that the deconvolution model should include information about small scales in order to regularise the filter inversion. a priori and a posteriori tests showed that the filtered flame propagation speed and structure cannot be captured if the filter size is too large.     

Large Eddy Simulation composition equations for single-phase and two-phase fully multicomponent flows

The Large Eddy Simulation (LES) equations for multicomponent (MC) fuel single-phase (SP) flow and two-phase (TP) flow with phase change are derived from the Direct Numerical Simulation (DNS) equations by filtering the DNS equations using a top-hat filter. Additional to the equations solved for single-component (SC) fuels, composition equations enter the formulation. The species composition is represented through a Probability Distribution Function (PDF), and DNS equations for the PDF moments are solved to find the composition. The TP filtered equations contain three categories of subgrid-scale (SGS) terms: (1) SGS–flux terms, (2) filtered source terms (FSTs) and (3) terms representing the ‘LES assumptions’. For SP flows no FSTs exist. The SGS terms in the LES equations must be either shown negligible or modeled. It is shown that for the composition equations, two equivalent forms of the DNS equations lead to two non-equivalent forms of the LES equations. Criteria are proposed to select the form best suited for LES. These criteria are used in conjunction with evaluations based on a DNS database portraying mixing and phase change, and lead to choosing one of the LES forms which satisfies all criteria. It is shown that the LES assumptions lead to additional SGS terms which require modeling. Further considerations are made for reactive flows.     

方柱绕流大涡模拟

采用有限体/有限元混合格式、非结构网格和大涡模拟方法求解可压缩的N-S方程,对Re=22 000的方柱绕流进行数值模拟,并对不同的边界条件进行详细的分析比较.通过对以往研究经验的总结和利用精细的边界条件,使得采用二阶精度的数值格式和较稀疏的网格仍然得到了令人满意的计算结果,甚至优于以往采用密网格的模拟结果.     

A new mixed subgrid-scale model for large eddy simulation of turbulent drag-reducing flows of viscoelastic fluids

A mixed subgrid-scale(SGS) model based on coherent structures and temporal approximate deconvolution(MCT) is proposed for turbulent drag-reducing flows of viscoelastic fluids. The main idea of the MCT SGS model is to perform spatial filtering for the momentum equation and temporal filtering for the conformation tensor transport equation of turbulent flow of viscoelastic fluid, respectively. The MCT model is suitable for large eddy simulation(LES) of turbulent dragreducing flows of viscoelastic fluids in engineering applications since the model parameters can be easily obtained. The LES of forced homogeneous isotropic turbulence(FHIT) with polymer additives and turbulent channel flow with surfactant additives based on MCT SGS model shows excellent agreements with direct numerical simulation(DNS) results. Compared with the LES results using the temporal approximate deconvolution model(TADM) for FHIT with polymer additives, this mixed SGS model MCT behaves better, regarding the enhancement of calculating parameters such as the Reynolds number.For scientific and engineering research, turbulent flows at high Reynolds numbers are expected, so the MCT model can be a more suitable model for the LES of turbulent drag-reducing flows of viscoelastic fluid with polymer or surfactant additives.     

Employing Taylor and Heisenberg subfilter viscosities to simulate turbulent statistics in LES models

A turbulent subfilter viscosity for Large Eddy Simulation (LES) based on the Taylor statistical diffusion theory is proposed. This viscosity is described in terms of a velocity variance and a time scale, both associated to the inertial subrange. This new subfilter viscosity contains a cutoff wavenumber k_c, presenting an identical form (differing by a constant) to the Heisenberg subfilter viscosity. Therefore, both subfilter viscosities are described in terms of a sharp division between large and small wavenumbers of a turbulent flow and, henceforth, Taylor and Heisenberg subfilter viscosities are in agreement with the sharp Fourier filtering operation, frequently employed in LES models. Turbulent statistics of different orders, generated from atmospheric boundary layer simulations employing both Taylor and Heisenberg subfilter viscosities have been compared with observations and results provided by other simulations. The comparison shows that the LES model utilizing the approaches of Taylor and Heisenberg reproduces these turbulent statistics correctly in different vertical regions of a planetary convective boundary layer (CBL).     

On discretization errors and subgrid scale model implementations in large eddy simulations

We analyze the impact of discretization errors on the performance of the Smagorinsky model in large eddy simulations (LES). To avoid difficulties related to solid boundaries, we focus on decaying homogeneous turbulence. It is shown that two numerical implementations of the model in the same finite volume code lead to significantly different results in terms of kinetic energy decay, time evolutions of the viscous dissipation and kinetic energy spectra. In comparison with spectral LES results, excellent predictions are however obtained with a novel formulation of the model derived from the discrete Navier–Stokes equations. We also highlight the effect of discretization errors on the measurement of physical quantities that involve scales close to the grid resolution.     

Experimental study of velocity-scalar filtered joint density function for LES of turbulent combustion

The velocity-scalar filtered joint density function (FJDF) used in large eddy simulation (LES) of turbulent combustion is experimentally studied. Measurements are made in the fully developed region of an axisymmetric turbulent jet using an array consisting of three X-wires and resistance-wire temperature sensors. Filtering in the cross-stream and streamwise directions is realized by using the array and by invoking Taylor’s hypothesis, respectively. The means of the FJDF conditional on the subgrid-scale (SGS) turbulent kinetic energy and the SGS scalar variance at a given location range from close to joint normal to bimodal with the peaks separated in both velocity and scalar spaces, which correspond to qualitatively different mixing regimes. For close to joint normal FJDFs, the SGS fields are well mixed. For bimodal FJDFs, the conditionally filtered scalar diffusion and dissipation strongly depend on the SGS velocity and scalar, consistent with a combination of diffusion layers and plane strain in the SGS fields, which is similar to the counter-flow model for laminar flamelets. The results suggest that in LES, both mixing regimes could potentially be modeled accurately. The velocity field affects the SGS variance and the filtered scalar dissipation rate primarily by changing the degree of nonequilibrium of the SGS scalar and the SGS time scale, respectively. This study further demonstrates the importance of including velocity in mixing models.     

Generalised scale-by-scale energy-budget equations and large-eddy simulations of anisotropic scalar turbulence at various Schmidt numbers

A necessary condition for the accurate prediction of turbulent flows using large-eddy simulation (LES) is the correct representation of energy transfer between the different scales of turbulence in the LES. For scalar turbulence, transfer of energy between turbulent length scales is described by a transport equation for the second moment of the scalar increment. For homogeneous isotropic turbulence, the underlying equation is the well-known Yaglom equation. In the present work, we study the turbulent mixing of a passive scalar with an imposed mean gradient by homogeneous isotropic turbulence. Both direct numerical simulations (DNS) and LES are performed for this configuration at various Schmidt numbers, ranging from 0.11 to 5.56. As the assumptions made in the derivation of the Yaglom equation are violated for the case considered here, a generalised Yaglom equation accounting for anisotropic effects, induced by the mean gradient, is derived in this work. This equation can be interpreted as a scale-by-scale energy-budget equation, as it relates at a certain scale r terms representing the production, turbulent transport, diffusive transport and dissipation of scalar energy. The equation is evaluated for the conducted DNS, followed by a discussion of physical effects present at different scales for various Schmidt numbers. For an analysis of the energy transfer in LES, a generalised Yaglom equation for the second moment of the filtered scalar increment is derived. In this equation, new terms appear due to the interaction between resolved and unresolved scales. In an a-priori test, this filtered energy-budget equation is evaluated by means of explicitly filtered DNS data. In addition, LES calculations of the same configuration are performed, and the energy budget as well as the different terms are thereby analysed in an a-posteriori test. It is shown that LES using an eddy viscosity model is able to fulfil the generalised filtered Yaglom equation for the present configuration. Further, the dependence of the terms appearing in the filtered energy-budget equation on varying Schmidt numbers is discussed.     

Subgrid-scale scalar flux modelling based on optimal estimation theory and machine-learning procedures

New procedures are explored for the development of models in the context of large eddy simulation (LES) of a passive scalar. They rely on the combination of the optimal estimator theory with machine-learning algorithms. The concept of optimal estimator allows to identify the most accurate set of parameters to be used when deriving a model. The model itself can then be defined by training an artificial neural network (ANN) on a database derived from the filtering of direct numerical simulation (DNS) results. This procedure leads to a subgrid scale model displaying good structural performance, which allows to perform LESs very close to the filtered DNS results. However, this first procedure does not control the functional performance so that the model can fail when the flow configuration differs from the training database. Another procedure is then proposed, where the model functional form is imposed and the ANN used only to define the model coefficients. The training step is a bi-objective optimisation in order to control both structural and functional performances. The model derived from this second procedure proves to be more robust. It also provides stable LESs for a turbulent plane jet flow configuration very far from the training database but over-estimates the mixing process in that case.