Large Eddy Simulation of Stratified and Sheared Flames of a Premixed Turbulent Stratified Flame Burner Using a Flamelet Model with Heat Loss

This paper presents large eddy simulations (LES) of the Darmstadt turbulent stratified flame burner (TSF) at different operating conditions including detailed heat loss modeling. The target cases are a non-reacting and two reacting cases. Both reacting cases are characterized by stratification, while one flame additionally features shear. In the regime diagram for premixed combustion, the studied flames are found at the border separating the thin reaction zones regime and the broken reaction zones regime. A coupled level set/progress variable model is utilized to describe the combustion process. To account for heat loss, an enthalpy defect approach is adopted and reformulated to include differential diffusion effects. A novel power-law rescaling methodology is proposed to integrate the enthalpy defect approach into the level set/progress variable model which is extensively validated in two validation scenarios. It is demonstrated that the LES with the newly developed model captures the influence of heat loss well and that the incorporation of heat loss effects improves the predictions of the TSF-burner over adiabatic simulations, while reproducing the experimentally observed flame lift-off from the pilot nozzle.     

Large eddy simulation of turbulent premixed and stratified combustion using flame surface density model coupled with tabulation method

Large eddy simulations (LESs) are performed to investigate the Cambridge premixed and stratified flames, SwB1 and SwB5, respectively. The flame surface density (FSD) model incorporated with two different wrinkling factor models, i.e., the Muppala and Charlette2 wrinkling factor models, is used to describe combustion/turbulence interaction, and the flamelet generated manifolds (FGM) method is employed to determine major scalars. This coupled sub-grid scale (SGS) combustion model is named as the FSD-FGM model. The FGM method can provide the detailed species in the flame which cannot be obtained from the origin FSD model. The LES results show that the FSD-FGM model has the ability of describing flame propagation, especially for stratified flames. The Charlette2 wrinkling factor model performs better than the Muppala wrinkling factor model in predicting the flame surface area change by the turbulence. The combustion characteristics are analyzed in detail by the flame index and probability distributions of the equivalence ratio and the orientation angle, which confirms that for the investigated stratified flame, the dominant combustion modes in the upstream and downstream regions are the premixed mode and the back-supported mode, respectively.     

Large Eddy Simulation of Turbulent Premixed Swirling Flames Using Dynamic Thickened Flame with Tabulated Detailed Chemistry

A sub-grid scale (SGS) combustion model by combining dynamic thickened flame (DTF) with flamelet generated manifolds (FGM) tabulation approach (i.e. DTF-FGM) is developed for investigating turbulent premixed combustion. In contrast to the thickened flame model, the dynamic thickening factor of the DTF model is determined from the flame sensor, which is obtained from the normalized gradient of the reaction progress variable from the one-dimensional freely propagating premixed flame simulations. Therewith the DTF model can ensure that the thickening of the flame is limited to the regions where it is numerically necessary. To describe the thermo-chemistry states, large eddy simulation (LES) transport equations for two characteristic scalars (the mixture fraction and the reaction progress variable) and relevant sub-grid variances in the DTF-FGM model are presented. As to the evaluation of different SGS combustion models, another model by utilizing the combination of presumed probability density function (PPDF) and FGM (i.e. PPDF-FGM) is also described. LES of two cases with or without swirl in premixed regime of the Cambridge swirl burner flames are performed to evaluate the developed SGS combustion model. The predicted results are compared with the experimental data in terms of the influence of different LES grids, model sensitivities to the thickening factor, the wrinkling factor, and the PPDF of characteristic scalars, the evaluation of different modelling approaches for the sub-grid variances of characteristic scalars, and the predictive capability of different SGS combustion models. It is shown that the LES results with the DTF-FGM model are in reasonable agreement with the experimental data, and better than the results with the PPDF-FGM approach due to its ability to predict better in regions where flame is not resolved.     

Large-eddy Simulation of Pilot-assisted Pulverized-coal Combustion in a Weakly Turbulent Jet

Large-eddy simulation has been performed to investigate pilot-assisted pulverized-coal combustion in a weakly turbulent air jet. An advanced pyrolysis model, the chemical percolation devolatilization (CPD) model, has been incorporated into the LES framework to predict the local, instantaneous pyrolysis kinetics of coal particles during the simulation. Prediction on volatile species generation is thus improved, which provides an important initial condition for gas-phase volatile and solid-phase char combustion. For gas-phase combustion, the partially stirred reactor (PaSR) model is employed to model the combustion of volatile species, taking into account subgrid turbulence-chemistry interactions. For heterogeneous solid-phase char combustion, both the intrinsic chemical reaction on the internal surface of a char particle and the diffusion of gaseous oxidant through the film layer around the particle have been incorporated by using a kinetic/diffusion surface reaction model. The LES results show overall good agreements with experimental data. Sensitivity analysis has been performed to better understand the impact of parameter uncertainties on the LES results.     

Assessment of Finite Rate Chemistry Large Eddy Simulation Combustion Models

Large Eddy Simulations (LES) of a swirl-stabilized natural gas-air flame in a laboratory gas turbine combustor is performed using six different LES combustion models to provide a head-to-head comparative study. More specifically, six finite rate chemistry models, including the thickened flame model, the partially stirred reactor model, the approximate deconvolution model and the stochastic fields model have been studied. The LES predictions are compared against experimental data including velocity, temperature and major species concentrations measured using Particle Image Velocimetry (PIV), OH Planar Laser-Induced Fluorescence (OH-PLIF), OH chemiluminescence imaging and one-dimensional laser Raman scattering. Based on previous results a skeletal methane-air reaction mechanism based on the well-known Smooke and Giovangigli mechanism was used in this work. Two computational grids of about 7 and 56 million cells, respectively, are used to quantify the influence of grid resolution. The overall flow and flame structures appear similar for all LES combustion models studied and agree well with experimental still and video images. Takeno flame index and chemical explosives mode analysis suggest that the flame is premixed and resides within the thin reaction zone. The LES results show good agreement with the experimental data for the axial velocity, temperature and major species, but differences due to the choice of LES combustion model are observed and discussed. Furthermore, the intrinsic flame structure and the flame dynamics are similarly predicted by all LES combustion models examined. Within this range of models, there is no strong case for deciding which model performs the best.     

Effects of Spray and Turbulence Modelling on the Mixing and Combustion Characteristics of an n-heptane Spray Flame Simulated with Dynamic Adaptive Chemistry

Accurate modelling of spray combustion process is essential for efficiency improvement and emissions reduction in practical combustion engines. In this work, both unsteady Reynolds-averaged Navier-Stokes (URANS) simulations and large eddy simulations (LES) are performed to investigate the effects of spray and turbulence modelling on the mixing and combustion characteristics of an n-heptane spray flame in a constant volume chamber at realistic conditions. The non-reacting spray process is first simulated with URANS to investigate the effects of entrainment gas-jet model on the penetration characteristics and fuel vapor distributions. It is found that the droplet motion near the nozzle has significant influence on the fuel vapor distribution, while the liquid penetration length is controlled by the evaporation process and insensitive to gas-jet model. For the case considered, both URANS with the gas-jet model and large eddy simulations can properly predict the vapor penetration. For the combustion characteristics, it is found that LES yields better predictions in the global combustion characteristics. The URANS with gas jet model yields a comparable flame length and lift-off-length (LOL) to LES, but results in a larger ignition delay time compared to the experimental data. Another focus of this work is to qualify the convergence characteristics of the dynamic adaptive chemistry (DAC) method in these transient combustion simulations, where DAC is applied to reduce the mechanism locally and on-the-fly to accelerate chemistry calculations. The instantaneous flame structures and global combustion characteristics such as ignition delay time, flame lift-off length and emissions are compared between simulations with and without DAC. For URANS, good agreements are observed both on instantaneous flame structures and global characteristics. For LES, it is shown that the errors incurred by DAC are small for scatter distributions in composition space and global combustion characteristics, while they may significantly affect instantaneous flame structures in physical space. The study reveals that for DAC application in transient simulations, global or statistic information should be used to assess the accuracy, such as manifolds in composition space, conditional quantities and global combustion characteristics. For the cases investigated, a speed-up factor of more than two is achieved by DAC with a 92-species skeletal mechanism with less than 0.2 % and 3.0 % discrepancy in ignition delay and LOL, respectively.     

A 5-D Implementation of FGM for the Large Eddy Simulation of a Stratified Swirled Flame with Heat Loss in a Gas Turbine Combustor

Numerical simulations are foreseen to provide a tremendous increase in gas-turbine burners efficiency in the near future. Modern developments in numerical schemes, turbulence models and the consistent increase of computing power allow Large Eddy Simulation (LES) to be applied to real cold flow industrial applications. However, the detailed simulation of the gas-turbine combustion process remains still prohibited because of its enormous computational cost. Several numerical models have been developed in order to reduce the costs of flame simulations for engineering applications. In this paper, the Flamelet-Generated Manifold (FGM) chemistry reduction technique is implemented and progressively extended for the inclusion of all the combustion features that are typically observed in stationary gas-turbine combustion. These consist of stratification effects, heat loss and turbulence. Three control variables are included for the chemistry representation: the reaction evolution is described by the reaction progress variable, the heat loss is described by the enthalpy and the stratification effect is expressed by the mixture fraction. The interaction between chemistry and turbulence is considered through a presumed beta-shaped probability density function (PDF) approach, which is considered for progress variable and mixture fraction, finally attaining a 5-D manifold. The application of FGM in combination with heat loss, fuel stratification and turbulence has never been studied in literature. To this aim, a highly turbulent and swirling flame in a gas turbine combustor is computed by means of the present 5-D FGM implementation coupled to an LES turbulence model, and the results are compared with experimental data. In general, the model gives a rather good agreement with experimental data. It is shown that the inclusion of heat loss strongly enhances the temperature predictions in the whole burner and leads to greatly improved NO predictions. The use of FGM as a combustion model shows that combustion features at gas turbine conditions can be satisfactorily reproduced with a reasonable computational effort. The implemented combustion model retains most of the physical accuracy of a detailed simulation while drastically reducing its computational time, paving the way for new developments of alternative fuel usage in a cleaner and more efficient combustion.     

LES/PDF Modeling of Turbulent Premixed Flames with Locally Enhanced Mixing by Reaction

Large-eddy simulations (LES) combined with the transported probability density function (PDF) method are carried out for two turbulent piloted premixed methane-air jet flames (flame F1 and flame F3) to assess the capability of LES/PDF for turbulent premixed combustion. The conventionally used model for the sub-filter scale mixing time-scale (or the mixing frequency) fails to capture the premixed flames correctly. This failure is expected to be caused by the lack of the sub-filter scale premixed flame propagation property in the sub-filter scale mixing process when the local flame front is under-resolved. It leads to slower turbulent premixed flame propagation and wider flame front. A new model for specifying the sub-filter scale mixing frequency is developed to account for the effect of sub-filter scale chemical reaction on mixing, based on past development of models for the sub-filter scale scalar dissipation rate in premixed combustion. The new model is assessed in the two turbulent premixed jet flames F1 and F3. Parametric studies are performed to examine the new model and its sensitivity when combined with the different mixing models. Significantly improved performance of the new mixing frequency model is observed to capture the premixed flame propagation reasonably, when compared with the conventional model. The sensitivity of the flame predictions is found be relatively weak to the different mixing models in conjunction with the new mixing frequency model.     

Joint RANS/LES Approach to Premixed Flame Modelling in the Context of the TFC Combustion Model

We present an original timesaving joint RANS/LES approach to simulate turbulent premixed combustion. It is intended mainly for industrial applications where LES may not be practical. It is based on successive RANS/LES numerical modelling, where turbulent characteristics determined from RANS simulations are used in LES equations for estimation of the subgrid chemical source and viscosity. This approach has been developed using our TFC premixed combustion model, which is based on a generalization of the Kolmogorov’s ideas. We assume existence of small-scale statistically equilibrium structures not only of turbulence but also of the reaction zones. At the same time, non-equilibrium large-scale structures of reaction sheets and turbulent eddies are described statistically by model combustion and turbulence equations in RANS simulations or follow directly without modelling in LES. Assumption of small-scale equilibrium gives an opportunity to express the mean combustion rate (controlled by small-scale coupling of turbulence and chemistry) in the RANS and LES sub-problems in terms of integral or subgrid parameters of turbulence and the chemical time, i.e. the definition of the reaction rate is similar to that of the mean dissipation rate in turbulence models where it is expressed in terms of integral or subgrid turbulent parameters. Our approach therefore renders compatible the combustion and turbulent parts of the RANS and LES sub-problems and yields reasonable agreement between the RANS and averaged LES results. Combining RANS simulations of averaged fields with LES method (and especially coupled and acoustic codes) for simulation of corresponding nonstationary process (and unsteady combustion regimes) is a promising strategy for industrial applications. In this work we present results of simulations carried out employing the joint RANS/LES approach for three examples: High velocity premixed combustion in a channel, combustion in the shear flow behind an obstacle and the impinging flame (a premixed flame attached to an obstacle).     

Study of a Filtered Flamelet Formulation for Large Eddy Simulation of Premixed Turbulent Flames

Despite significant advances in the understanding and modelling of turbulent combustion, no general model has been proposed for simulating flames in industrial combustion devices. Recently, the increase in computational possibilities has raised the hope of directly solving the large turbulent scales using large eddy simulation (LES) and capturing the important time-dependant phenomena. However, the chemical reactions involved in combustion occur at very small scales and the modelling of turbulent combustion processes is still required within the LES framework. In the present paper, a recently presented model for the LES of turbulent premixed flames is presented, analysed and discussed. The flamelet hypothesis is used to derive a filtered source term for the filtered progress variable equation. The model ensures proper flame propagation. The effect of subgrid scale (SGS) turbulence on the flame is modelled through the flame-wrinkling factor. The present modelling of the source term is successfully tested against filtered direct numerical simulation (DNS) data of a V-shape flame. Further, a premixed turbulent flame, stabilised behind an expansion, is simulated. The predictions agree well with the available experimental data, showing the capabilities of the model for performing accurate simulations of unsteady premixed flames.     

Assessing the Predictive Capabilities of Combustion LES as Applied to the Sydney Flame Series

Large Eddy Simulation (LES) and flamelet-based combustion models were applied to four bluff-body stabilized nonpremixed and partially premixed flames selected from the Sydney flame series, based on Masri’s bluff-body test rig (University of Sydney). Three related non-reacting flow cases were also investigated to assess the performance of the LES solver. Both un-swirled and swirled cases were studied exhibiting different flow features, such as recirculation, jet precessing and vortex breakdown. Due to various fuel compositions, flow rates and swirl numbers, the combustion characteristics of the flames varied greatly. On six meshes with different blocking structure and mesh sizes, good prediction of flow and scalar fields using LES/flamelet approaches and known fuel and oxidizer mass fluxes was achieved. The accuracy of predictions was strongly influenced by the combustion model used. All flames were calculated using at least two modeling strategies. Starting with calculations of isothermal flow cases, simple single flamelet based calculations were carried out for the corresponding reacting cases. The combustion models were then adjusted to fit the requirements of each flame. For all flame calculations good agreement of the main flow features with the measured data was achieved. For purely nonpremixed flames burning attached to the bluff-body’s outer edge, flamelet modeling including strain rate effects provided good results for the flow field and for most scalars. The prediction of a partially premixed swirl flame could only be achieved by applying a flamelet-based progress variable approach.     

Chemical kinetics modeling and LES combustion model effects on a perfectly premixed burner

Chemical kinetics modeling and coupling with turbulent combustion models for compressible Large Eddy Simulations (LES) is a critical issue. Accurate flow predictions can only be guaranteed if the coupling is well mastered. In a first attempt to qualify the effect of each model, the case of a lean premixed swirled combustor with comprehensive measures is targeted (species mass fractions and temperature fields). For the investigation, two turbulent combustion models are considered. The first model relies on a presumed PDF approach coupled to a look-up chemistry table obtained with a reduced chemical scheme. The second model makes use of the thickened flame approach using the same reduced chemical scheme but with reaction rates computed explicitly as the computation advances. Then, to estimate kinetic schemes reduction effects, the first model is compared to a third one, with the same PDF approach, but coupled to a look-up chemistry table obtained with a complete chemical scheme. All LES are very close to each other. The main difference between the different predictions relies on CO mass fractions. Although they are all able to return good outlet mass fractions, CO values inside the flame are different depending on the model used. To cite this article: G. Albouze et al., C. R. Mecanique 337 (2009).     

Flow Physics of a Bluff-Body Swirl Stabilized Flame and their Prediction by Means of a Joint Eulerian Stochastic Field and Tabulated Chemistry Approach

In the frame of this work a transported joint scalar probability density function (PDF) method is combined with the flamelet generated manifolds (FGM) tabulated chemistry approach for large eddy simulation (LES) modeling of a three-dimensional turbulent premixed swirl burner. This strategy accounts for the turbulence-chemistry interaction at reasonable computational costs. At the same time, it allows the usage of detailed chemistry mechanisms for the creation of the chemical database. The simulation results obtained are comparatively assessed along with complementary measurements. Furthermore, transient and time-averaged data are used to provide insight into the flow physics of the bluff-body swirl stabilized flame considered. The sensitivity of the results to different modeling approaches regarding the predicted flame shape and its dynamics is also investigated, where the implemented approach is compared with the well-established artificially thickened flame (ATF) combustion model. Consequently, the investigation conducted in this work aims to provide a complete picture on the ability of the proposed combustion model to reproduce the flow conditions within complex bluff-body swirl stabilized flames.     

Large Eddy Simulation and Extended Dynamic Mode Decomposition of Flow-Flame Interaction in a Lean Premixed Low Swirl Stabilized Flame

In this paper, numerical studies are reported on the effect of flow-flame interaction at large and medium scales and its impact on flame stabilization in a lean premixed low swirl stabilized methane/air flame. The numerical simulations are based on a large eddy simulation (LES) approach with a three-scalar flamelet model with equations for mixture fraction and fuel mass fraction and the level-set G-equation to account respectively for stratification of the mixture, fuel leakage at the trailing edge of the flame, and tracking of the flame front. Distinct frequencies, associated with the flame stabilization process, are identified from point data of LES in the outer and inner shear layers of the burner induced flow field. To understand the effect of the spatial structures related to the observed flow frequencies, a dynamic mode decomposition (DMD) is performed. Based on the analysis of LES data, frequency specific coherent flow structures are extracted along with associated flame structures through an extended version of DMD. The inner shear layer generated vortices are associated with recurring frequency specific coherent structures of both flow and flame and contribute to the flame stabilization in the outer regions of the flow.     

LES Modeling of the Impact of Heat Losses and Differential Diffusion on Turbulent Stratified Flame Propagation: Application to the TU Darmstadt Stratified Flame

Common combustion chambers often exhibit turbulent flames propagating in partially-premixed mixtures. This propagation is generally governed by aerodynamics, unsteady mixing and chemical processes and may also be affected by conductive heat losses when the reactive zone develops close to the burner lips. The Filtered TAbulated Chemistry for Large Eddy Simulation (F-TACLES) model has been recently developed to include tabulated chemistry in Large Eddy Simulation (LES) of adiabatic stratified flames in flamelet regimes. The present article proposes a modeling approach to account for both differential diffusion and non-adiabatic effects on flame consumption speed following the F-TACLES formalism. The adiabatic F-TACLES model is first detailed using a generalized formalism for diffusive fluxes allowing either to account for differential diffusion or not. The F-TACLES model is then extended to non-adiabatic situations. A correction factor based on the non-adiabatic consumption rate is introduced to recover a realistic filtered flame consumption speed. The objective is here to tackle flame stabilization mechanisms when heat losses affect the reaction zone. The proposed approach is validated through the simulation of the unconfined stratified turbulent jet flame TSF-A for which stabilization process is affected by heat losses. Five simulations are performed for both adiabatic and non-adiabatic flow conditions comparing unity Lewis number and complex diffusion assumptions. The adiabatic F-TACLES model predicts a flame anchored at the burner lip disagreeing with experimental data. The non-adiabatic simulation exhibits local extinction due to heat losses near the burner exit. The flame is then lifted improving the comparison with experiments. Results also show a significant impact of molecular diffusion model on both mean flame consumption rate and angle.     

Modelling of a Premixed Swirl-stabilized Flame Using a Turbulent Flame Speed Closure Model in LES

This paper proposes a combustion model based on a turbulent flame speed closure (TFC) technique for large eddy simulation (LES) of premixed flames. The model was originally developed for the RANS (Reynolds Averaged Navier Stokes equations) approach and was extended here to LES. The turbulent quantities needed for calculation of the turbulent flame speed are obtained at the sub grid level. This model was at first experienced via an test case and then applied to a typical industrial combustor with a swirl stabilized flame. The paper shows that the model is easy to apply and that the results are promising. Even typical frequencies of arising combustion instabilities can be captured. But, the use of compressible LES may also lead to unphysical pressure waves which have their origin in the numerical treatment of the boundary conditions.     

Evaluations of SGS Combustion,Scalar Flux and Stress Models in a Turbulent Jet Premixed Flame

A newly developed fractal dynamic SGS (FDSGS) combustion model and a scale self-recognition mixed (SSRM) SGS stress model are evaluated along with other SGS combustion, scalar flux and stress models in a priori and a posteriori manners using DNS data of a hydrogen-air turbulent plane jet premixed flame. A posteriori tests reveal that the LES using the FDSGS combustion model can predict the combustion field well in terms of mean temperature distributions and peak positions in the transverse distributions of filtered reaction progress variable fluctuations. A priori and a posteriori tests of the scalar flux models show that a model proposed by Clark et al. accurately predicts the counter-gradient transport as well as the gradient diffusion, and introduction of the model of Clark et al. into the LES yields slightly better predictions of the filtered progress variable fluctuations than that of a gradient diffusion model. Evaluations of the stress models reveal that the LES with the SSRM model predicts the velocity fluctuations well compared to that with the Smagorinsky model.     

LES/CMC Simulations of Swirl-Stabilised Ethanol Spray Flames Approaching Blow-Off

Large Eddy Simulations (LES) with the Conditional Moment Closure (CMC) combustion model of swirling ethanol spray flames have been performed in conditions close to blow-off for which a wide database of experimental measurements is available for both flame and spray characterization. The solution of CMC equations exploits a three-dimensional unstructured code with a first order closure for chemical source terms. It is shown that LES/CMC is able to properly capture the flame structure at different conditions and agrees reasonably well with the measurements both in terms of mean flame shape and dynamic behaviour of the flame evaluated in terms of local extinctions and statistics of the lift-off height. Experimental measurements of the overall (liquid plus gaseous) mixture fraction, performed using the Laser-Induced Breakdown Spectroscopy technique, are also included allowing further assessment and validation of the numerical method. The sensitivity of the simulation results to the various boundary conditions is discussed.     

Resolution Requirements in Stochastic Field Simulation of Turbulent Premixed Flames

The spatial resolution requirements of the Stochastic Fields probability density function approach are investigated in the context of turbulent premixed combustion simulation. The Stochastic Fields approach is an attractive way to implement a transported Probability Density Function modelling framework into Large Eddy Simulations of turbulent combustion. In premixed combustion LES, the numerical grid should resolve flame-like structures that arise from solution of the Stochastic Fields equation. Through analysis of Stochastic Fields simulations of a freely-propagating planar turbulent premixed flame, it is shown that the flame-like structures in the Stochastic Fields simulations can be orders of magnitude narrower than the LES filter length scale. The under-resolution is worst for low Karlovitz number combustion, where the thickness of the Stochastic Fields flame structures is on the order of the laminar flame thickness. The effect of resolution on LES predictions is then assessed by performing LES of a laboratory Bunsen flame and comparing the effect of refining the grid spacing and filter length scale independently. The usual practice of setting the LES filter length scale equal to grid spacing leads to severe under-resolution and numerical thickening of the flame, and to substantial error in the turbulent flame speed. The numerical resolution required for accurate solution of the Stochastic Fields equations is prohibitive for many practical applications involving high-pressure premixed combustion. This motivates development of a Thickened Stochastic Fields approach (Picciani et al. Flow Turbul. Combust. X, YYY (2018) in order to ensure the numerical accuracy of Stochastic Fields simulations.     

Coupling tabulated chemistry with Large Eddy Simulation of turbulent reactive flows

A new modeling strategy is developed to introduce tabulated chemistry methods in the LES of turbulent premixed combustion. The objective is to recover the correct laminar flame propagation speed of the filtered flame front when the subgrid scale turbulence vanishes. The filtered flame structure is mapped by 1D filtered laminar premixed flames. Closure of the filtered progress variable and the energy balance equations are carefully addressed. The methodology is applied to 1D and 2D filtered laminar flames. These computations show the capability of the model to recover the laminar flame speed and the correct chemical structure when the flame wrinkling is completely resolved. The model is then extended to turbulent combustion regimes by introducing subgrid scale wrinkling effects on the flame front propagation. Finally, the LES of a 3D turbulent premixed flame is performed. To cite this article: R. Vicquelin et al., C. R. Mecanique 337 (2009).