3.1 The scientific context

3.1.1 Computational fluid mechanics: modeling or not before discretizing ?

A typical continuous solution of the Navier-Stokes equations at sufficiently large values of the Reynolds number is governed by a wide spectrum of temporal and spatial scales closely connected with the turbulent nature of the flow. The term deterministic chaos employed by Frisch in his enlightening book 39 is certainly conveying most adequately the difficulty in analyzing and simulating this kind of flows. The broadness of the turbulence spectrum is directly controlled by the Reynolds number defined as the ratio between the inertial forces and the viscous forces. This number is not only useful to determine the transition from a laminar to a turbulent flow regime, it also indicates the range of scales of fluctuations that are present in the flow under consideration. Typically, for the velocity field and far from solid walls, the ratio between the largest scale (the integral length scale) and the smallest one (Kolmogorov scale) is proportional to $R{e}_t^{3/4}$ per dimension, where $R{e}_t^{3/4}$ is the turbulent Reynolds number, based on the length and velocity scales of the largest turbulent eddies. In addition, for internal flows, viscous effects near the solid walls yield a scaling proportional to $R{e}_{\tau }$ per dimension, where $R{e}_{\tau }$ is the friction Reynolds number. The smallest scales play a crucial role in the dynamics of the largest ones, which implies that an accurate framework for the computation of turbulent flows must take into account all the scales, which can lead to unrealistic computational costs in real-world applications. Thus, the usual practice to deal with turbulent flows is to choose between an a priori modeling (in most situations) or not (low Re number and rather simple configurations) before proceeding to the discretization step, followed by the simulation itself. If a modeling phase is on the agenda, then one has to choose again among the above-mentioned variety of approaches. The different simulation options and their date of availability for high-Reynolds-number applications are illustrated in Fig. 1: simulation of turbulent flows can be achieved either by directly solving the Navier-Stokes equations (DNS) or by first applying to the equations a statistical averaging (RANS), a spatial filtering (LES), or a combination of these two operators (hybrid RANS/LES). The new terms brought about by the operator have to be modeled. From a computational point of view, the RANS approach is the least demanding, which explains why historically it has been the workhorse in both the academic and the industrial sectors, and it remains the standard approach nowadays for industrial design, except for very specific applications. It has permitted quite a substantial progress in the understanding of various phenomena such as turbulent combustion or heat transfer. Its inherent inability to provide a time-dependent information has led to promote in the last decade the recourse to either LES or DNS to supplement if not replace RANS. By simulating the large scale structures while modeling the smallest ones, assumed more isotropic, LES proved to be quite a breakthrough to fully take advantage of the increasing power of computers to study complex flow configurations. At the same time, DNS was gradually applied to geometries of increasing complexity (channel flows with values of $R{e}_{\tau }$ multiplied by 45 during the last 30 years, jets, turbulent premixed flames, among many others), and proved to be a formidable tool to (i) improve our knowledge on turbulent flows and (ii) test (i.e., validate or invalidate) and improve the modeling hypotheses inherently associated to the RANS and LES approaches. From a numerical point of view, due to the steady nature of the RANS equations, numerical accuracy is generally not ensured via the use of high-order schemes, but rather on careful grid convergence studies. In contrast, the high computational cost of LES or DNS makes necessary the use of highly-accurate numerical schemes in order to optimize the use of computational resources.

To the noticeable exception of the hybrid RANS-LES modeling, which is not yet accepted as a reliable tool for industrial design, as mentioned in the preamble of the Go4hybrid European program 6, a turbulence model represents turbulent mechanisms in the same way in the whole flow. Thus, depending on its intrinsic strengths and weaknesses, accuracy will be a rather volatile quantity, strongly dependent on the flow configuration. For instance, RANS is perfectly suited to attached boundary layers, but exhibits severe limitations in massively-separated flow regions. Therefore, the turbulence modeling and industrial design communities waver between the desire to continue to rely on the RANS approach, which is unrivaled in terms of computational cost, but is still not able to accurately represent all the complex phenomena; and the temptation to switch to LES, which outperforms RANS in many situations, but is prohibitively expensive in high-Reynolds number wall-bounded flows. In order to account for the limitations of the two approaches and to combine them for significantly improving the overall performance of the models, the hybrid RANS-LES approach has emerged during the last two decades as a viable, intermediate way, and we are definitely inscribing our project in this innovative field of research, with an original approach though, based on temporal filtering (Hybrid temporal LES, HTLES) rather than spatial filtering, and a systematic and progressive validation process against experimental data produced by the team.

3.2 Research directions

4.1 Aeronautics

Cagire is presently involved in studies mainly related to:

• The combustion chamber wall: the modelling, the simulation and the experimentation of the flow around a multiperforated plate representative of a real combustion chamber wall are the three axes we have been developing during the recent period. The continuous improvement of our in-house test facility Maveric is also an important ingredient to produce our own experimental validation data for isothermal flows. For non-isothermal flows, our participation in the EU funded program Soprano will be giving us access to non-isothermal data produced by Onera.
• The flow around airfoils: the modelling of the turbulent boundary layer has been for almost a century a key issue in the aeronautics industry. However, even the more advanced RANS models face difficulties in predicting the influence of pressure gradients on the development of the boundary layer. A main issue is the reliability of the modelling hypotheses, which is crucial for less conservative design. One of the technological barriers is the prediction of the flow in regimes close to the edge of the flight domain (stall, buffeting, unsteady loads) when the boundary layer is slowed down by an adverse pressure gradient. This is the subject of the CIFRE PhD thesis of Gustave Sporschill, started in 2018, in collaboration with Dassault Aviation.
• Impinging jets: because of their high heat transfer efficiency, turbulent impinging jets are commonly used in a large variety of applications, and in particular blade cooling systems. Understanding the underlying physics of the mechanisms at play is of prime interest and is still an open question. Additionally, this configuration remains a challenging test case for turbulence models since it embraces many flow features despite a relatively simple geometry, and causes strong discrepancies between standard turbulence closures. Reynolds stress transport models have been shown to be promising candidates but still suffer from a lack of validation regarding this flow configuration. Such models are the subject of a collaboration with Onera.
• Atmospheric reentry problem: When a body enters the atmosphere with a high velocity, most of its movement is driven by a hypersonic flow surrounding the body. Integrity of the body is maintained by a shield that is progressively ablated. The sharp control of the movement can be driven only provided a very good knowledge of the surrounding hypersonic flow and of the interaction between the ablated shield and the surrounding hypersonic flow. Within the Seigle project, the team is involved in the simulation of the interaction of a droplet (representing the ablated body) and a hypersonic flow. In the Asturies project, the aim is to study the improvement on the shock/turbulence interaction by using higher order RANS models.

4.2 Energy

• The prediction of heat transfer in fluid and solid components is of major importance in power stations, in particular, nuclear power plants. Either for the thermohydraulics of the plenum or in the study of accidental scenarii, among others, the accurate estimation of wall heat transfer, mean temperatures and temperature fluctuations are necessary for the evaluation of relevant thermal and mechanical design criteria. The PhD thesis (CIFRE EDF) of G. Mangeon, defended in February 2020, was dedicated to the development of relevant RANS models for these industrial applications 11.
• Moreover, the prediction of unsteady hydrodynamic loadings is a key point for operating and for safety studies of PWR power plants. Currently, the static loading is correctly predicted by RANS computations but when the flow is transient (as, for instance, in Reactor Coolant Pumps, due to rotor/stator interactions, or during operating transients) or in the presence of large, energetic, coherent structures in the external flow region, the RANS approach is not sufficient, whereas LES is still too costly for a wide use in industry. This issue was the main focus of the PhD thesis (CIFRE EDF) of Vladimir Duffal, defended in November 2020.
• For the design of high temperature solar receiver for concentrated solar power plants, flows are characterized by strong variations of the fluid properties, such that, even in the forced convection regime, they significantly deviate from isothermal flows, with a possible tendency to relaminarize, which can significantly reduce heat transfer. A better understanding and modeling of the physical mechanisms observed in turbulent flows with strong temperature gradients are important and are the focus of a collaboration with the LaTeP laboratory of UPPA 12.
• Thermal storage is interesting to decorrelate the production of heat or cold from its use whether for direct operation as for a heat network (smoothing of heat supply to meet intermittent needs) or for power generation (phase shift between heat generation and power generation). The challenge is to study, via CFD, the dynamic and thermal behavior of the storage during the loading, resting and discharge phases. The PhD thesis of Alexis Ferré, co-supervised by R. Manceau and S. Serra (LaTeP) started in November 2020.

4.3 Automotive propulsion

• The engine (underhood) compartment is a key component of vehicle design, in which the temperature is monitored to ensure the effectiveness and safety of the vehicle, and participates in 5 to 8% of the total drag and CO2 emissions. Dimensioning is an aerodynamic and aerothermal compromise, validated on a succession of road stages at constant speed and stopped phases (red lights, tolls, traffic jam). Although CFD is routinely used for forced convection, state-of-the-art turbulence models are not able to reproduce flows dominated by natural convection during stopped phases, with a Rayleigh number of the order of ${10}^{10}$, such that the design still relies on costly, full-scale, wind tunnel experiments. This technical barrier must be lifted, since the ambition of the PSA group is to reach a full digital design of their vehicles in the 2025 horizon, i.e., to almost entirely rely on CFD. This issue was the focus of the just-defended PhD thesis (CIFRE PSA) of S. Jameel, supervised by R. Manceau, and also a part of the ANR project MONACO_2025 described in section 8.4, in the framework of which S. Jameel will continue working in the team (post-doc, 18 months).
• The Power & Vehicles Division of IFPEN co-develops a CFD code, CONVERGE, to simulate the internal flow in a spark-ignition engine, in order to provide the automotive industry with tools to optimize the design of combustion engines. The RANS method, widely used in the industry, is not sufficiently reliable for quantitative predictions, and is only used as a tool to qualitatively compare different geometries. On the other hand, LES provides more detailed and accurate information, but at the price of a CPU cost unafordable for daily use in the industry. Therefore, IFPEN aims at developing the hybrid RANS/LES methodology, in order to combine the strengths of the two approaches. The PhD thesis of Hassan Afailal, co-supervised by Rémi Manceau, focused on this issue, was defended in December 2020. In the framework of the just-started collaborative project ASTURIES (E2S-UPPA/Inria/CEA/IFPEN), this collaboration with IFPEN will be pursued by the development of high-order methods in the CONVERGE code in order to make it possible to perform highly accurate and low-dissipative LES and hybrid RANS/LES in combustion engines.

5.1 New software

6.1 A methodology for simulating unsteady inert or reacting gaseous flows featuring large density variations

Participants: Pascal Bruel, Mariovane Donini.

This work aims at developing an efficient and versatile tool to simulate non-reacting and reacting flow fields in a zero Mach number framework. The formulation and implementation of the unsteady artificial compressibility approach in a rather standard finite volume framework has been carried out in a two-step way. In the first one, only divergence free velocity flow fields were considered, hence imposing quite a severe restriction on the class of reacting flows that could be dealt with. The preliminary results obtained during this first phase have been presented in 13. Then, during the second step, the approach has been generalised so as to deal also with non divergence free velocity flow fields, opening the way for a much broader field of application. The resulting time-accurate scheme was tested in five different cases: the Stokes’ second problem, the unsteady wake flow past a circular cylinder, the flow around a heated cylinder placed in a square enclosure, a steady Tsuji diffusion flame, and a flickering buoyant diffusion flame. The flow over the oscillating plate and past a circular cylinder case were chosen to put into evidence the basic properties of the implemented time-accurate approach, such as the ability to describe unsteady non-reactive flows and its convergence rate. An analysis of the influence of the artificial compressibility factor on the convergence rate was carried out. For the flow in the enclosure hosting a heated cylinder, a configuration that featured quite high density gradients, the predicted flow topology as well as the heat flux distribution along the enclosure walls matched very well those reported in the literature. Then, a comparison with experimental and numerical results was performed for a steady diffusion flame case. The simulated temperature profile and the flame shape were in good agreement with those reported in the literature, except in the region around the maximum temperature, where the reaction layer was not so well described, certainly because of the infinite-rate chemistry assumption considered here. The fifth case investigated the capability of the present approach to predict correctly the large scale unsteadiness of a flickering diffusion flame. An excellent agreement with the experimental results was observed in term of the fundamental frequency of the flame flickering. Qualitatively speaking, the predicted flame topology appeared to be in line with that observed experimentally. A journal paper gathering these results has been submitted.

6.2 Conjugate heat transfer with different fluid-solid physical properties: a new elliptic blending second-moment closure

Participants: Rémi Manceau, Gaëtan Mangeon.

Our approach to model the wall/turbulence interaction, based on Elliptic Blending, was successfully applied to flows with standard thermal boundary conditions at the walls 10. However, Conjugate Heat Transfer, which couples fluid and solid domains, are particularly challenging for turbulence models. We have developed an innovative model, the Elliptic Blending Differential Flux Model, to account for the influence of various wall thermal boundary conditions on the turbulent heat flux and the temperature variance. An assessment of this new model in Conjugate Heat Transfer has been performed for several values of fluid-solid thermal diffusivity and conductivity ratios. A careful attention is paid to the discontinuity of the dissipation rate associated with the temperature variance at the fluid-solid interface. The analysis is supported by successful comparisons with Direct Numerical Simulations. This work led to the publication of an article in Journal of Fluid Mechanics 11. This work constitutes the main part of the thesis of Gaetan Mangeon, defended in 2020 19.

6.3 Development of the EB-RSM model for jets in crossflow, impinging jets and adverse-pressure-gradient boundary layer

Participants: Rémi Manceau, Franck Mastrippolito, Pascal Bruel.

The EB-RSM, developed by the team, is a sophisticated Reynolds-stress model sensitized to the influence of the wall, implemented in several open-source, industrial and commercial codes (OpenFoam, Code$\_$Saturne, AETHER, CEDRE, EZNSS, StarCCM+). Although it has successfully been applied to many different configurations and showed improved results compared to standard industrial models (for instance, 48, 10, 58), some problems have been observed in particular situations, such that the model still requires some improvement.

• In the framework of the SOPRANO European project, it was observed that, when a flow is accelerated inside the holes of a multi-perforated plate, the model is able to predict the laminarization of the flow (not predicted by eddy-viscosity models), but the dissipation does not go properly to zero. This problem has been attributed to the appearance of a pseudo-laminar state 56 and remedied by the introduction of a bound on the turbulent-to-mean time-scale ratio.
• Another aspect concerns the improvement of the prediction of impinging jets, which are known to be wrongly predicted by Reynolds-stress models 62. In collaboration with A. Colombié, F. Chedevergne and E. Laroche from ONERA Toulouse, we have identified by extracting the budgets of the Reynolds stresses in a fine LES computation that this problem is due to the lack of representation of the pressure diffusion mechanisms far from the wall. A model has been proposed for this term that is under validation.
• The CIFRE PhD thesis of Gustave Sporschill (Dassault Aviation) is dedicated to the prediction of the flows around airfoils, and, more generally, boundary layers subject to adverse pressure gradient effects. Different Reynolds-stress models have been evaluated for such flows and showed improved predictions compared to the standard Spalart-Almaras model used routinely by Dassault. Among them, the EB-RSM gives the more accurate results 58 and therefore, is promising as concerns a future application of Reynolds-stress models in the aeronautical sector.

6.4 Eddy-viscosity models sensitized to buoyancy effects for predicting natural convection flows

Participants: Rémi Manceau, Saad Jameel.

Eddy-viscosity turbulence models have been sensitized to the effects of buoyancy, in order to improve the prediction in natural convection flows. The approach extends in a linear way the constitutive relations for the Reynolds stress and the turbulent heat flux, in order to account for the anisotropic influence of buoyancy. The novelty of this work involves the buoyancy extension applied to three very different eddy-viscosity models, which leads to encouraging results for the highly challenging case of the differentially heated vertical channel and the differentially heated cavity 42, 43. This models were developed during the PhD of Saad Jameel, defended in December 2020 18.

In the framework of the ANR MONACO project (post-doc of Saad Jameel), in collaboration with PSA Group, these models are implemented in the commercial code ANSYS Fluent and applied to real underhood configurations of vehicules in situations driven by natural convection.

6.5 Development of the HTLES approach

Participants: Rémi Manceau, Vladimir Duffal, Hassan Afailal, Franck Mastrippolito, Pascal Bruel, Puneeth Bikkanahally.

The HTLES (hybrid temporal LES) approach, developed by the team, has been improved by introducing shielding functions and an internal consistency constraint to enforce the RANS behavior in the near-wall regions 51. The influence of the underlying closure model was studied by applying HTLES to two RANS models: the k-$\omega$ SST and BL-v2/k models. The validation, carried out using two different solvers, Code_Saturne (collaboration with EDF) and Converge-CFD (collaboration with IFPEN), encompassed different type of internal flows: channel, periodic hill, internal combustion engines (in fixed position and with a moving piston), jet in crossflow 36, 21. The robustness of HTLES to grid coarsening and the accuracy of the predictions at a reduced numerical cost compared to LES was demonstrated. Notably, the capacity of HTLES to provide information on velocity and pressure temporal fluctuations at the wall was assessed, offering a cost-saving alternative to LES to predict unsteady loads. One of the major limitation of HTLES, i.e., the fact that it is based on statistically averaged quantities, which are challenging to estimate in non-stationary flows, has been removed, by approximating the statistical average by a Dynamic Temporal Filter. These developments were carried out in the framework of the European project SOPRANO; and during the PhD thesis of Vladimir Duffal (part of the ANR project MONACO), defended in November 2020 17; and during the PhD thesis of Al Hassan Afailal, defended in December 2020 16.

In the framework of the ANR project MONACO, the HTLES approach is extended to flows influenced by buoyancy. During the PhD thesis of Puneeth Bikkanahally, the model has been applied to cavity flows, and in particular the strongly stratified cavity of 57. Due to the co-existence of a turbulent, buoyant boundary layer and a laminar region in the center of the cavity, the shielding function based on the Kolmogorov scale developed during the thesis of Vladimir Duffal 16 deteriorates the results. The ongoing work is dedicated to the development of an alternative shielding function based on the resolution of an elliptic equation 29.

6.6 Compact WENO, positivity preserving stabilization of discontinuous Galerkin methods

Participants: Vincent Perrier.

Jointly with Alireza Mazaheri (NASA Langley) and Chi Wang Shu, we have developed a compact WENO stabilization that moreover ensures the positivity of physical quantities. The work was published in 52.

7.1 Bilateral contracts with industry

• EDF: "Advanced modelling of heat transfer for industrial configurations with or without accounting of the solid wall", contract associated to the PhD thesis of Gaëtan Mangeon
• EDF: "Hybrid RANS/LES modelling for unsteady loadings in turbulent flows", contract associated to the PhD thesis of Vladimir Duffal
• IFPEN: "3D simulation of non-reactive internal aerodynamics of spark-ignition engines using an hybrid RANS/LES method", contract associated to the PhD thesis of Hassan Al Afailal
• PSA: ""Turbulence modelling in the mixed and natural convection regimes in the context of automotive applications", contract associated to the PhD thesis of Saad Jameel.
• Dassault Aviation: "Improvement of the turbulence models”, contract associated to the PhD thesis of Gustave Sporschill.
• CEA: “Agile simulation of turbulent internal flows”, contract in the framework of the Asturies project.

7.2 Bilateral grants with industry

• EDF (Cifre PhD grant): "Advanced modelling of heat transfer for industrial configurations with or without accounting of the solid wall", PhD student: Gaëtan Mangeon
• EDF (Cifre PhD grant): "Hybrid RANS/LES modelling for unsteady loadings in turbulent flows", PhD student: Vladimir Duffal
• IFPEN (PhD grant): "3D simulation of non-reactive internal aerodynamics of spark-ignition engines using an hybrid RANS/LES method", PhD sutdent: Hassan Al Afailal
• PSA (Cifre PhD grant): "Turbulence modelling in the mixed and natural convection regimes in the context of automotive applications", PhD student: Saad Jameel.
• Dassault Aviation (Cifre PhD grant): "Improvement of the turbulence models. Application to the prediction of aerodynamic flows", PhD student: Gustave Sporschill.
• CEA: “CFD and experimental study of a thermocline-type thermal storage for an optimized design and data entry of component scale models in the framework of a multi-scale approach”, PhD student Alexis Ferré.
• CEA: “Development of Fast, Robust and Accurate numerical methods for turbulence models on Complex Meshes” (1/2 Grant), PhD student, Anthony Bosco.

8.1 International initiatives

8.2 International research visitors

8.3 European initiatives

8.4 National initiatives

8.5 Regional initiatives

9.1 Promoting scientific activities

9.2 Teaching - Supervision - Juries

9.3 Popularization

10.1 Major publications

• 1 articlePascalP. Bruel, SimonS. Delmas, JonathanJ. Jung and VincentV. Perrier. A low Mach correction able to deal with low Mach acousticsJournal of Computational Physics3782019, 723--759
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• 2 article StéphaneS. Dellacherie, JonathanJ. Jung, PascalP. Omnes and Pierre-ArnaudP.-A. Raviart. Construction of modified Godunov type schemes accurate at any Mach number for the compressible Euler system Mathematical Models and Methods in Applied Sciences November 2016
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• 3 articleJuan-LuisJ.-L. Florenciano and PascalP. Bruel. LES fluid-solid coupled calculations for the assessment of heat transfer coefficient correlations over multi-perforated wallsAerospace Science and Technology532016, 13
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• 4 articleErwinE. Franquet and VincentV. Perrier. Runge-Kutta discontinuous Galerkin method for the approximation of Baer and Nunziato type multiphase modelsJournal of Computational Physics23111February 2012, 4096-4141
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• 5 articleCh.C. Friess, R. Manceau and T.B.T. Gatski. Toward an equivalence criterion for Hybrid RANS/LES methodsComputers and Fluids1222015, 233-246
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• 6 article Jean-MarcJ.-M. Hérard and JonathanJ. Jung. An interface condition to compute compressible flows in variable cross section ducts Comptes Rendus Mathématiques February 2016
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• 7 articleR. Manceau. Recent progress in the development of the Elliptic Blending Reynolds-stress modelInt. J. Heat Fluid Fl.512015, 195-220URL: http://dx.doi.org/10.1016/j.ijheatfluidflow.2014.09.002
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• 8 articleGaëtanG. Mangeon, SofianeS. Benhamadouche, Jean-FrançoisJ.-F. Wald and RemiR. Manceau. Extension to various thermal boundary conditions of the elliptic blending model for the turbulent heat flux and the temperature varianceJournal of Fluid Mechanics905A1December 2020, 1-34
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• 9 articleYannY. Moguen, SimonS. Delmas, VincentV. Perrier, PascalP. Bruel and ErikE. Dick. Godunov-type schemes with an inertia term for unsteady full Mach number range flow calculationsJournal of Computational Physics281January 2015, 35
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10.2 Publications of the year

10.3 Other

10.4 Cited publications

• 21 articleAl HassanA. Afailal, JérémyJ. Galpin, AnthonyA. Velghe and RemiR. Manceau. Development and validation of a hybrid temporal LES model in the perspective of applications to internal combustion enginesOil & Gas Science and Technology - Revue d'IFP Energies nouvelles742019, 56
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• 22 articleDouglas ND. Arnold. An interior penalty finite element method with discontinuous elementsSIAM journal on numerical analysis1941982, 742--760
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• 23 articleDouglas ND. Arnold, FrancoF. Brezzi, BernardoB. Cockburn and L DonatellaL. Marini. Unified analysis of discontinuous Galerkin methods for elliptic problemsSIAM journal on numerical analysis3952002, 1749--1779
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• 24 articleCédricC. Augonnet, SamuelS. Thibault, RaymondR. Namyst and Pierre-AndréP.-A. Wacrenier. StarPU: A Unified Platform for Task Scheduling on Heterogeneous Multicore ArchitecturesConcurr. Comput. : Pract. Exper.232February 2011, 187--198URL: http://dx.doi.org/10.1002/cpe.1631
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• 25 articleF. Bassi, L. Botti, A. Colombo, D.A. DiD. Pietro and P. Tesini. On the flexibility of agglomeration based physical space discontinuous Galerkin discretizationsJournal of Computational Physics23112012, 45 - 65URL: http://www.sciencedirect.com/science/article/pii/S0021999111005055
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• 26 articleF. Bassi, A. Crivellini, S. Rebay and M. Savini. Discontinuous Galerkin solution of the Reynolds-averaged Navier-Stokes and k-omega turbulence model equationsComputers & Fluids344-52005, 507-540
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• 28 inproceedingsF. Bassi, S. Rebay, G. Mariotti, S. Pedinotti and M. Savini. A high-order accurate discontinuous finite element method for inviscid and viscous turbomachinery flowsProceedings of the 2nd European Conference on Turbomachinery Fluid Dynamics and ThermodynamicsTechnologisch Instituut, Antwerpen, Belgium1997, 99--109
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• 29 conference P. Bikkanahally, R. Manceau and F. Mastrippolito. Development of a hybrid RANS-LES model based on temporal filtering for natural convection flows 14th World Congress Comput. Mech. (WCCM), ECCOMAS Congress 2020, Paris (Virtual Congress), France 2021
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• 30 articleBernardoB. Cockburn, SuchungS. Hou and Chi-WangC.-W. Shu. The Runge-Kutta local projection discontinuous Galerkin finite element method for conservation laws. IV. The multidimensional caseMath. Comp.541901990, 545--581URL: http://dx.doi.org/10.2307/2008501
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• 31 articleBernardoB. Cockburn, San YihS. Lin and Chi-WangC.-W. Shu. TVB Runge-Kutta local projection discontinuous Galerkin finite element method for conservation laws. III. One-dimensional systemsJ. Comput. Phys.8411989, 90--113
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• 32 articleBernardoB. Cockburn and Chi-WangC.-W. Shu. TVB Runge-Kutta local projection discontinuous Galerkin finite element method for conservation laws. II. General frameworkMath. Comp.521861989, 411--435URL: http://dx.doi.org/10.2307/2008474
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• 33 articleBernardoB. Cockburn and Chi-WangC.-W. Shu. The Runge-Kutta discontinuous Galerkin method for conservation laws. V. Multidimensional systemsJ. Comput. Phys.14121998, 199--224URL: http://dx.doi.org/10.1006/jcph.1998.5892
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• 34 articleBernardoB. Cockburn and Chi-WangC.-W. Shu. The Runge-Kutta local projection $P^1$-discontinuous-Galerkin finite element method for scalar conservation laws'RAIRO Modél. Math. Anal. Numér.2531991, 337--361
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• 35 inproceedings S. ScottS. Colis. Discontinuous Galerkin methods for turbulence simulation Proceedings of the Summer Program Center for Turbulence Research 2002
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• 36 inproceedings VladimirV. Duffal, BenoîtB. De Laage De Meux and RemiR. Manceau. Development and validation of a hybrid RANS/LES approach based on temporal filtering ASME - JSME - KSME Joint Fluids Engineering Conference 2019 Proc. ASME - JSME - KSME Joint Fluids Engineering Conference 2019, Jul 2019, San Francisco, CA, United States. San Francisco, CA, United States July 2019
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• 37 articleMohamedM. Essadki, JonathanJ. Jung, AdamA. Larat, MilanM. Pelletier and VincentV. Perrier. A task-driven implementation of a simple numerical solver for hyperbolic conservation lawsESAIM: Proceedings and Surveys63January 2017, 228-247
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• 38 articleM. Feistauer and V. Kuċera. On a robust discontinuous Galerkin technique for the solution of compressible flowJ. Comput. Phys.22412007, 208--221URL: http://dx.doi.org/10.1016/j.jcp.2007.01.035
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• 39 book U. Frisch. Turbulence: The Legacy of AN Kolmogorov Cambridge University Press 1995
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• 40 article M. Giles. Non-Reflecting Boundary Conditions for Euler Equation Calculation The American Institute of Aeronautics and Astronautics Journal 42 12 1990
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• 41 articleRalfR. Hartmann and PaulP. Houston. Symmetric interior penalty DG methods for the compressible Navier-Stokes equations. I. Method formulationInt. J. Numer. Anal. Model.312006, 1--20
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• 42 inproceedings Syed Mohd SaadS. Jameel, RemiR. Manceau and VincentV. Herbert. A buoyancy extension for eddy-viscosity models for the natural convection regime 17th European Turbulence Conference (ETC-2019) Torino, Italy September 2019
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• 43 inproceedings Syed Mohd SaadS. Jameel, RemiR. Manceau and VincentV. Herbert. Sensitization of eddy-viscosity models to buoyancy effects for predicting natural convection flows HEFAT 2019 - 14th International Conference on Heat Transfer, Fluid Mechanics and Thermodynamics Proc. HEFAT 2019 - 14th International Conference on Heat Transfer, Fluid Mechanics and Thermodynamics Wicklow, Ireland July 2019
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• 44 inproceedings A. Jameson and M. Fatica. Using Computational Fluid Dynamics for Aerodynamics National Research Council Workshop on "The Future of Supercomputing" 2003
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• 45 articleClaesC. Johnson, AndersA. Szepessy and PeterP. Hansbo. On the convergence of shock-capturing streamline diffusion finite element methods for hyperbolic conservation lawsMath. Comp.541891990, 107--129URL: http://dx.doi.org/10.2307/2008684
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• 46 articleA. Klöckner, T. Warburton, J. Bridge and J.S.J. Hesthaven. Nodal discontinuous Galerkin methods on graphics processorsJournal of Computational Physics228212009, 7863 - 7882URL: http://www.sciencedirect.com/science/article/pii/S0021999109003647
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• 47 articleD.A.D. Knoll and D.E.D. Keyes. Jacobian-free Newton-Krylov methods: a survey of approaches and applicationsJournal of Computational Physics19322004, 357 - 397URL: http://www.sciencedirect.com/science/article/pii/S0021999103004340
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