- Membre Senior de l'Institut Universitaire de France
- Rédacteur en chef du Journal "Computers and fluids"
- Membre du Comité de rédaction: Journal of Computational Physics, Journal of Turbulence, Journal of Scientific Computing
- Président du Conseil Scientifique et Technique de l'AFM
- Vice-Président du Conseil Scientifique de ERCOFTAC
Activités
Simulation et modélisation des écoulements turbulents
Aérodynamique, aéroacoustique, aérothermique
Méthodes numériques pour les équations de Navier-Stokes et de Boltzmann
Méthodes avancées: quantification des incertitudes et assimilation de données
Publications scientifiques au M2P2
2023
Mathis Pasquier, Stéphane Jay, Jérôme Jacob, Pierre Sagaut. A Lattice-Boltzmann-Based Modelling Chain for Traffic-Related Atmospheric Pollutant Dispersion at the Local Urban Scale. Building and Environment, 2023, 242, pp.110562. ⟨10.1016/j.buildenv.2023.110562⟩. ⟨hal-04190005⟩ Plus de détails...
Urban traffic-related air pollution is a major source of environmental and health damage and is difficult to quantify due to its inherent physical complexity. We construct a CFD-based simulation framework coupling an efficient numerical method for turbulent fluid flows with a microscopic traffic model and an emissions model to simulate road transport pollutant dispersion at the urban microscale. We improve the open-source Lattice-Boltzmann based CFD software OpenLB to overcome its original stability deficiencies for high Reynolds number flows. A stable recursive regularization procedure with a double distribution function approach is proposed to solve an advection diffusion equation for passive scalar transport at high Reynolds number. The code is successfully validated on three reference cases of increasing complexity and the traffic model SUMO along with a physical engine emissions model are coupled with OpenLB to simulate traffic-induced pollution from a road network in a realistic complex geometry. Transient flow features are analysed and the time-averaged concentration levels in different neighbourhoods of the considered geometry are evaluated: high concentration levels are observed close to the streets but also inside specific building infrastructures due to complex wind dynamics. Analyses of altitudinal concentration variations show that flow recirculations located close to traffic lights can drive pollutant over the buildings and increase concentration levels inside inner courtyards. Time-averaged concentration maps are constructed using both spatially uniform and non-uniform line sources and it is shown that using uniform sources leads to up to 20% local concentration overestimations inside the urban canopy.
Mathis Pasquier, Stéphane Jay, Jérôme Jacob, Pierre Sagaut. A Lattice-Boltzmann-Based Modelling Chain for Traffic-Related Atmospheric Pollutant Dispersion at the Local Urban Scale. Building and Environment, 2023, 242, pp.110562. ⟨10.1016/j.buildenv.2023.110562⟩. ⟨hal-04190005⟩
G. Farag, P. Boivin, P. Sagaut. Linear interaction approximation for shock/disturbance interaction in a Noble–Abel stiffened gas. Shock Waves, 2023, ⟨10.1007/s00193-023-01131-8⟩. ⟨hal-04097657⟩ Plus de détails...
When departure from the ideal gas equation of state is considered, the Noble-Abel stiffened gas model is an appealing and versatile candidate due to its simple form. The Linear Interaction Approximation formalism is extended to consider non-ideal gas effects introduced by this equation of state. Kovásznay decomposition and adequate definition of the energy of disturbances are provided in the context of this equation of state. Changes with respect to ideal gas are investigated on transfer functions, critical angle and compression factor. Those differences yield concrete effects on the damping and transfer of fluctuations across shock waves. Those changes are further illustrated by considering the interaction of an entropy spot with a Mach 3 stationary shock wave.
G. Farag, P. Boivin, P. Sagaut. Linear interaction approximation for shock/disturbance interaction in a Noble–Abel stiffened gas. Shock Waves, 2023, ⟨10.1007/s00193-023-01131-8⟩. ⟨hal-04097657⟩
Thomas Gianoli, Jean‐françois Boussuge, Pierre Sagaut, Jérôme de Laborderie. Development and validation of Navier–Stokes characteristic boundary conditions applied to turbomachinery simulations using the lattice Boltzmann method. International Journal for Numerical Methods in Fluids, 2023, 95 (4), pp.528-556. ⟨10.1002/fld.5160⟩. ⟨hal-04063964⟩ Plus de détails...
This article reports a procedure to implement as well as to validate non-reflecting boundary conditions applied for turbomachinery simulations, using Navier-Stokes characteristic boundary conditions in a compressible lattice Boltzmann solver. The implementation of both an inlet condition imposing total pressure, total temperature, and flow angles, as well as an outlet condition imposing a static pressure profile that allows the simulation to reach a simplified radial equilibrium, is described within the context of a lattice Boltzmann approach. The treatment at the boundaries relies on the characteristic methodology to derive conditions which are non-reflecting in terms of acoustics and is also compatible with turbulence injection at the inlet. These properties are evaluated on test cases of increasing complexity, ranging from a simple 2D periodic domain to an S-duct stage with turbulence injection.
Thomas Gianoli, Jean‐françois Boussuge, Pierre Sagaut, Jérôme de Laborderie. Development and validation of Navier–Stokes characteristic boundary conditions applied to turbomachinery simulations using the lattice Boltzmann method. International Journal for Numerical Methods in Fluids, 2023, 95 (4), pp.528-556. ⟨10.1002/fld.5160⟩. ⟨hal-04063964⟩
Journal: International Journal for Numerical Methods in Fluids
Georis Billo, Michel Belliard, Pierre Sagaut. Comparison of several interpolation methods to reconstruct field data in the vicinity of a finite element immersed boundary. Computers & Mathematics with Applications, 2022, 123, pp.123-135. ⟨10.1016/j.camwa.2022.08.002⟩. ⟨hal-04064030⟩ Plus de détails...
Thermal-hydraulics safety requirements for the second and third generation of nuclear reactors led to the development of innovative passive safety systems. In particular, new devices must be developed involving numerical simulations for turbulent two-phase flows around complex geometries. To reduce the time-consuming mesh generation phase when testing various geometries, we use a fictitious domain approach. More specifically, we choose the Penalized Direct Forcing Method to take into account inflow obstacles. Following a recent work, involving the resolution of the one-phase incompressible Navier-Stokes equations using a projection scheme and the Finite Element Method, this paper focuses on different techniques to recover data from the discrete immersed boundary and different ways to achieve order 2 in space via linear interpolation. Indeed, we investigate two data reconstruction approaches (one based on various weighted averaging, the other based on optimization) and compare their results for cylindrical and NACA0012 airfoil shapes: they provide similar accuracy but the weighting is much faster in terms of execution time. We also investigate three different interpolation types: unidirectional, multi-directional and a new hybrid between the two. The Taylor-Couette flow and the flow around a circular cylinder are used to carry out mesh convergence studies. Globally, order 2 in space is numerically assessed in both 2 and ∞ norms for all the interpolation types, which is consistent with theoretical expectations-even if the space convergence order is a bit higher for the multi-directional approach. For the flow around a circular cylinder, the values of aerodynamic coefficients and Strouhal number are in good agreement with the literature, especially when using directional interpolation. Finally, an industrial case, representative of passive safety systems, is presented to assess the robustness and capability of the method. The simulations tend to show that, here again, the directional interpolation offers the best behavior when dealing with complex geometries and relatively coarse meshes.
Georis Billo, Michel Belliard, Pierre Sagaut. Comparison of several interpolation methods to reconstruct field data in the vicinity of a finite element immersed boundary. Computers & Mathematics with Applications, 2022, 123, pp.123-135. ⟨10.1016/j.camwa.2022.08.002⟩. ⟨hal-04064030⟩
Journal: Computers & Mathematics with Applications
Shang-Gui Cai, Sajad Mozaffari, Jérôme Jacob, Pierre Sagaut. Application of immersed boundary based turbulence wall modeling to the Ahmed body aerodynamics. Physics of Fluids, 2022, 34 (9), pp.095106. ⟨10.1063/5.0098232⟩. ⟨hal-04065468⟩ Plus de détails...
This paper applies a recently developed immersed boundary-turbulence wall modeling approach to turbulent flows over a generic car geometry, known as the Ahmed body, under massive flow separation within a lattice Boltzmann solver. Although the immersed boundary method combined with hierarchical Cartesian grid offers high flexibility in automatic grid generation around complex geometries, the near-wall solution is significantly deteriorated compared to the body-fitted simulation, especially when coupled to wall models for turbulent flows at high Reynolds number. Enhanced wall treatments have been proposed in the literature and validated for attached flow configurations. In this work, the Ahmed body with a slant surface of angle 35 is considered where the flow separates massively over the slant surface and the vertical base. The large eddy simulation is performed with a Reynolds stress constraint near-wall. The eddy viscosity is computed dynamically by taking into account the actually resolved Reynolds stresses. It approaches the mixing length eddy viscosity in attached boundary layers and returns to the subgrid eddy viscosity in detached boundary layers. An explicit equilibrium wall model has also been proposed to accelerate the calculation. Comparison with the no-slip boundary condition on the separated surfaces shows that the near-wall treatments with the equilibrium wall model operate reasonably well on both attached and detached boundary layers.
Shang-Gui Cai, Sajad Mozaffari, Jérôme Jacob, Pierre Sagaut. Application of immersed boundary based turbulence wall modeling to the Ahmed body aerodynamics. Physics of Fluids, 2022, 34 (9), pp.095106. ⟨10.1063/5.0098232⟩. ⟨hal-04065468⟩
B. Bugeat, J.-Ch. Robinet, J.-C. Chassaing, P. Sagaut. Low-frequency resolvent analysis of the laminar oblique shock wave/boundary layer interaction. Journal of Fluid Mechanics, 2022, 942, pp.A43. ⟨10.1017/jfm.2022.390⟩. ⟨hal-04064039⟩ Plus de détails...
Resolvent analysis is used to study the low-frequency behaviour of the laminar oblique shock wave/boundary layer interaction (SWBLI). It is shown that the computed optimal gain, which can be seen as a transfer function of the system, follows a first-order low-pass filter equation, recovering the results of Touber & Sandham (J. Fluid Mech., vol. 671, 2011, pp. 417-465). This behaviour is understood as proceeding from the excitation of a single stable, steady global mode whose damping rate sets the time scale of the filter. Different Mach and Reynolds numbers are studied, covering different recirculation lengths L. This damping rate is found to scale as 1/L, leading to a constant Strouhal number St L as observed in the literature. It is associated with a breathing motion of the recirculation bubble. This analysis furthermore supports the idea that the low-frequency dynamics of the SWBLI is a forced dynamics, in which background perturbations continuously excite the flow. The investigation is then carried out for three-dimensional perturbations for which two regimes are identified. At low wavenumbers of the order of L, a modal mechanism similar to that of two-dimensional perturbations is found and exhibits larger values of the optimal gain. At larger wavenumbers, of the order of the boundary layer thickness, the growth of streaks, which results from a non-modal mechanism, is detected. No interaction with the recirculation region is observed. Based on these results, the potential prevalence of three-dimensional effects in the low-frequency dynamics of the SWBLI is discussed.
B. Bugeat, J.-Ch. Robinet, J.-C. Chassaing, P. Sagaut. Low-frequency resolvent analysis of the laminar oblique shock wave/boundary layer interaction. Journal of Fluid Mechanics, 2022, 942, pp.A43. ⟨10.1017/jfm.2022.390⟩. ⟨hal-04064039⟩
M. Nguyen, J. Boussuge, P. Sagaut, J. Larroya-Huguet. Large eddy simulation of a thermal impinging jet using the lattice Boltzmann method. Physics of Fluids, 2022, 34 (5), pp.055115. ⟨10.1063/5.0088410⟩. ⟨hal-03669901⟩ Plus de détails...
A compressible Hybrid Lattice Boltzmann Method solver is used to perform a wall-resolved Large eddy simulation of an isothermal axisymmetric jet issuing from a pipe and impinging on a heated flat plate at a Reynolds number of 23 000, a Mach number of 0.1, and an impingement distance of two jet diameters. The jet flow field statistics, Nusselt number profile (including the secondary peak), and shear stress profile were well reproduced. The azimuthal coherence of the primary vortical structures was relatively low, leading to no discernible temporal periodicity of the azimuthally averaged Nusselt number at the location of the secondary peak. While local unsteady near-wall flow separation was observed in the wall jet, this flow separation did not exhibit azimuthal coherence and was not found to be the only cause of the thermal spots blue, which lead to the secondary peak in the Nusselt number, as stream-wise oriented structures also played a significant role in increasing the local heat transfer.
M. Nguyen, J. Boussuge, P. Sagaut, J. Larroya-Huguet. Large eddy simulation of a thermal impinging jet using the lattice Boltzmann method. Physics of Fluids, 2022, 34 (5), pp.055115. ⟨10.1063/5.0088410⟩. ⟨hal-03669901⟩
Guanxiong Wang, Song Zhao, Pierre Boivin, Eric Serre, Pierre Sagaut. A new hybrid lattice-Boltzmann method for thermal flow simulations in low-Mach number approximation. Physics of Fluids, 2022, 34 (4), pp.046114. ⟨10.1063/5.0091517⟩. ⟨hal-03796386⟩ Plus de détails...
A new low-Mach algorithm for the thermal lattice Boltzmann method (LBM) is proposed aiming at reducing the computational cost of thermal flow simulations in the low Mach number limit. The well-known low Mach number approximation is adopted to accelerate the simulations by enlarging the time step through re-scaling the psuedoacoustic speed to the same order of the fluid motion velocity. This specific process is inspired by the similarity between the artificial compressibility method and the isothermal LBM and is further extended to its thermal counterpart. It must be emphasized that such low-Mach acceleration strategy is in a general form, thus can be easily applied to other compressible LB methods. The present method overcomes the drawback of the classical pressure gradient scaling method due to the pressure gradient changing. The new algorithm is validated by various well-documented academic test cases in laminar [one dimensional gravity column, 2D (two dimensional) rising thermal bubble, and 2D differentially heated square cavity] and turbulent [3D (three dimensional) Taylor–Green vortex and 3D heated cylinder] regimes. All the results show excellent agreement with the reference data and high computational efficiency.
Guanxiong Wang, Song Zhao, Pierre Boivin, Eric Serre, Pierre Sagaut. A new hybrid lattice-Boltzmann method for thermal flow simulations in low-Mach number approximation. Physics of Fluids, 2022, 34 (4), pp.046114. ⟨10.1063/5.0091517⟩. ⟨hal-03796386⟩
Gauthier Wissocq, Pierre Sagaut. Hydrodynamic limits and numerical errors of isothermal lattice Boltzmann schemes. Journal of Computational Physics, 2022, 450, pp.110858. ⟨10.1016/j.jcp.2021.110858⟩. ⟨hal-04064045⟩ Plus de détails...
With the aim of better understanding the numerical properties of the lattice Boltzmann method (LBM), a general methodology is proposed to derive its hydrodynamic limits in the discrete setting. It relies on a Taylor expansion in the limit of low Knudsen numbers. With a single asymptotic analysis, two kinds of deviations with the Navier-Stokes (NS) equations are explicitly evidenced: consistency errors, inherited from the kinetic description of the LBM, and numerical errors attributed to its space and time discretization. The methodology is applied to the Bhatnagar-Gross-Krook (BGK), the regularized and the multiple relaxation time (MRT) collision models in the isothermal framework. Deviation terms are systematically confronted to linear analyses in order to validate their expressions, interpret them and provide explanations for their numerical properties. The low dissipation of the BGK model is then related to a particular pattern of its error terms in the Taylor expansion. Similarly, dissipation properties of the regularized and MRT models are explained by a phenomenon referred to as hyperviscous degeneracy. The latter consists in an unexpected resurgence of high-order Knudsen effects induced by a large numerical prefactor. It is at the origin of over-dissipation and severe instabilities in the low-viscosity regime.
Gauthier Wissocq, Pierre Sagaut. Hydrodynamic limits and numerical errors of isothermal lattice Boltzmann schemes. Journal of Computational Physics, 2022, 450, pp.110858. ⟨10.1016/j.jcp.2021.110858⟩. ⟨hal-04064045⟩
Lincheng Xu, Eric Serre, Pierre Sagaut. A theoretical analysis of mass leakage at boundaries within the lattice Boltzmann method. Physics of Fluids, 2022, Physics of fluids, 34 (065113). ⟨hal-03683744⟩ Plus de détails...
Mass leakage at boundaries can be a critical issue for reliability of the lattice Boltzmann (LB) method based on Cartesian grids. Despite numerous work based on the LB method, the intrinsic macroscopic mechanisms causing mass leakage are still not fully charac- terised, but are essential to improve the mass conservation of LB simulations. In this paper, an original theoretical investigation of mass leakage at boundaries is proposed within the general LB framework. It is demonstrated that the mass leakage originates from the in- trinsic deficiency of the wall-cut LB links at boundary nodes in recovering macroscopic momenta. From a mesoscopic-level definition, i.e. the net loss of distribution functions during the streaming process, the local mass leakage at individual boundary nodes and its averaged value along smooth boundaries are mathematically expressed using macroscopic variables. The local mass leakage is shown to be dominated by terms proportional to the tangential momentum component. In contrast, the averaged mass leakage is shown to be contributed from various terms including the boundary curvature, the tangential momen- tum, and the gradients of density, momentum and momentum flux. Meanwhile, amplitude of the averaged mass leakage is theoretically estimated to be proportional to the local grid spacing, based on which a first-order accurate correction scheme is proposed. In addition, both the local and averaged mass leakage are demonstrated to be significantly dependent on boundary orientation with respect to the grid. The proposed theoretical analysis is assessed by performing numerical experiments. Two-dimensional weakly compressible flows through straight and curved moving channels are considered to estimate each term appearing in the theoretical analysis. The numerical results are in very good agreement with the proposed analysis, and the proposed mass correction scheme based on the av- eraged mass leakage effectively cures the mass leakage problems in the considered test cases.
Lincheng Xu, Eric Serre, Pierre Sagaut. A theoretical analysis of mass leakage at boundaries within the lattice Boltzmann method. Physics of Fluids, 2022, Physics of fluids, 34 (065113). ⟨hal-03683744⟩
Gauthier Wissocq, Thomas Coratger, Gabriel Farag, Song Zhao, Pierre Boivin, et al.. Restoring the conservativity of characteristic-based segregated models: application to the hybrid lattice Boltzmann method. Physics of Fluids, 2022, 34 (4), pp.046102. ⟨10.1063/5.0083377⟩. ⟨hal-03627520⟩ Plus de détails...
A general methodology is introduced to build conservative numerical models for fluid simulations based on segregated schemes, where mass, momentum and energy equations are solved by different methods. It is here especially designed for developing new numerical discretizations of the total energy equation, adapted to a thermal coupling with the lattice Boltzmann method (LBM). The proposed methodology is based on a linear equivalence with standard discretizations of the entropy equation, which, as a characteristic variable of the Euler system, allows efficiently decoupling the energy equation with the LBM. To this extent, any LBM scheme is equivalently written under a finite-volume formulation involving fluxes, which are further included in the total energy equation as numerical corrections. The viscous heat production is implicitly considered thanks to the knowledge of the LBM momentum flux. Three models are subsequently derived: a first-order upwind, a Lax-Wendroff and a third-order Godunov-type schemes. They are assessed on standard academic test cases: a Couette flow, entropy spot and vortex convections, a Sod shock tube, several two-dimensional Riemann problems and a shock-vortex interaction. Three key features are then exhibited: 1) the models are conservative by construction, recovering correct jump relations across shock waves, 2) the stability and accuracy of entropy modes can be explicitly controlled, 3) the low dissipation of the LBM for isentropic phenomena is preserved.
Gauthier Wissocq, Thomas Coratger, Gabriel Farag, Song Zhao, Pierre Boivin, et al.. Restoring the conservativity of characteristic-based segregated models: application to the hybrid lattice Boltzmann method. Physics of Fluids, 2022, 34 (4), pp.046102. ⟨10.1063/5.0083377⟩. ⟨hal-03627520⟩
Guanxiong Wang, Song Zhao, Pierre Boivin, Eric Serre, Pierre Sagaut. A new hybrid Lattice-Boltzmann method for thermal flow simulations in low-Mach number approximation. Physics of Fluids, 2022, Physics of fluids, 34 (046114). ⟨hal-03636905⟩ Plus de détails...
Guanxiong Wang, Song Zhao, Pierre Boivin, Eric Serre, Pierre Sagaut. A new hybrid Lattice-Boltzmann method for thermal flow simulations in low-Mach number approximation. Physics of Fluids, 2022, Physics of fluids, 34 (046114). ⟨hal-03636905⟩
Florian Renard, Gauthier Wissocq, Jean-François Boussuge, Pierre Sagaut. A linear stability analysis of compressible hybrid lattice Boltzmann methods. Journal of Computational Physics, 2021, 446, pp.110649. ⟨10.1016/j.jcp.2021.110649⟩. ⟨hal-03514639⟩ Plus de détails...
An original spectral study of the compressible hybrid lattice Boltzmann method (HLBM) on standard lattice is proposed. In this framework, the mass and momentum equations are addressed using the lattice Boltzmann method (LBM), while finite difference (FD) schemes solve an energy equation. Both systems are coupled with each other thanks to an ideal gas equation of state. This work aims at answering some questions regarding the numerical stability of such models, which strongly depends on the choice of numerical parameters. To this extent, several one- and two-dimensional HLBM classes based on different energy variables, formulations (primitive or conservative), collision terms and numerical schemes are scrutinized. Once appropriate corrective terms introduced, it is shown that all continuous HLBM classes recover the Navier-Stokes-Fourier behavior in the linear approximation. However, striking differences arise between HLBM classes when their discrete counterparts are analyzed. Multiple instability mechanisms arising at relatively high Mach number are pointed out and two exhaustive stabilization strategies are introduced: (1) decreasing the time step by changing the reference temperature T-r and (2) introducing a controllable numerical dissipation a via the collision operator. A complete parametric study reveals that only HLBM classes based on the primitive and conservative entropy equations are found usable for compressible applications. Finally, an innovative study of the macroscopic modal composition of the entropy classes is conducted. Through this study, two original phenomena, referred to as shear-to-entropy and entropy-to-shear transfers, are highlighted and confirmed on standard two-dimensional test cases. (C) 2021 Elsevier Inc. All rights reserved.
Florian Renard, Gauthier Wissocq, Jean-François Boussuge, Pierre Sagaut. A linear stability analysis of compressible hybrid lattice Boltzmann methods. Journal of Computational Physics, 2021, 446, pp.110649. ⟨10.1016/j.jcp.2021.110649⟩. ⟨hal-03514639⟩
Thomas Astoul, Gauthier Wissocq, Jean-François Boussuge, Alois Sengissen, Pierre Sagaut. Lattice Boltzmann method for computational aeroacoustics on non-uniform meshes: A direct grid coupling approach. Journal of Computational Physics, 2021, 447, pp.110667. ⟨10.1016/j.jcp.2021.110667⟩. ⟨hal-03514616⟩ Plus de détails...
The present study proposes an accurate lattice Boltzmann direct coupling algorithm, well suited for industrial purposes, making it highly valuable for aeroacoustic applications. It is indeed known that the convection of vortical structures across a grid refinement interface, where cell size is abruptly doubled, is likely to generate spurious noise that may corrupt the solution over the whole computational domain. This issue becomes critical in the case of aeroacoustic simulations, where accurate pressure estimations are of paramount importance. Consequently, any interfering noise that may pollute the acoustic predictions must be reduced. The proposed grid refinement algorithm differs from conventionally used ones, in which an overlapping mesh layer is considered. Instead, it provides a direct connection allowing a tighter link between fine and coarse grids, especially with the use of a coherent equilibrium function shared by both grids. Moreover, the direct coupling makes the algorithm more local and prevents the duplication of points, which might be detrimental for massive parallelization. This work follows our first study (Astoul et al. 2020 [1]) on the deleterious effect of non-hydrodynamic modes crossing mesh transitions, which can be addressed using an appropriate collision model: the hybrid recursive regularization. The grid coupling algorithm is assessed in the context of computational aeroacoustics and compared to a widely-used cell-vertex algorithm. The validation benchmark includes the simulation of (1) an acoustic pulse, (2) a vortex transport by a mean flow, and finally, (3) a turbulent circular cylinder wake flow at high Reynolds number. In the end, the proposed approach is proven to drastically reduced the spurious noise generated at grid interfaces, hence, paving the way for accurate and efficient aeroacoustic simulations based on lattice Boltzmann methods. (C) 2021 Elsevier Inc. All rights reserved.
Thomas Astoul, Gauthier Wissocq, Jean-François Boussuge, Alois Sengissen, Pierre Sagaut. Lattice Boltzmann method for computational aeroacoustics on non-uniform meshes: A direct grid coupling approach. Journal of Computational Physics, 2021, 447, pp.110667. ⟨10.1016/j.jcp.2021.110667⟩. ⟨hal-03514616⟩
T. Coratger, G. Farag, S. Zhao, Pierre Boivin, P. Sagaut. Large-eddy lattice-Boltzmann modeling of transonic flows. Physics of Fluids, 2021, 33 (11), pp.115112. ⟨10.1063/5.0064944⟩. ⟨hal-03424286⟩ Plus de détails...
T. Coratger, G. Farag, S. Zhao, Pierre Boivin, P. Sagaut. Large-eddy lattice-Boltzmann modeling of transonic flows. Physics of Fluids, 2021, 33 (11), pp.115112. ⟨10.1063/5.0064944⟩. ⟨hal-03424286⟩
Felix Marlow, Jérôme Jacob, Pierre Sagaut. A multidisciplinary model coupling Lattice-Boltzmann-based CFD and a Social Force Model for the simulation of pollutant dispersion in evacuation situations. Building and Environment, 2021, 205, pp.108212. ⟨10.1016/j.buildenv.2021.108212⟩. ⟨hal-03514660⟩ Plus de détails...
In closed rooms with limited convection human motion can considerably affect the airflow and thus the dispersion of pollutant. However, in Computational Fluid Dynamics (CFD) simulations on air quality and safety for human beings this effect is generally not considered, which is mainly due to a lack of a well-founded and detailed estimation of the human behavior and the high computational cost of taking into account moving objects in CFD meshes. This work addresses this issue by coupling multidisciplinary methods to allow for a more realistic simulation of pollutant dispersion by taking into account the influence of human movements. A Social Force Model predicts trajectory and speed of each person moving in a complex environment. A lattice Boltzmann-based CFD tool provides a Large Eddy Simulation of the unsteady turbulent airflow with pollutant dispersion and thermal effects. And an Actuator Line Model supplies the CFD tool with body forces that mimic the impact of moving objects on the airflow, thus, avoiding computationally expensive dynamic meshing. The capability of the coupled model is demonstrated on three realistic evacuation scenarios with various pollutant sources and a wide range of scales (dimension from 10 to 100 m, occupation from 10 to 6000 persons). The results allow to access instantaneous environmental parameters like pollutant concentration for each person during the course of the evacuation, enabling the assessment of associated health risks.
Felix Marlow, Jérôme Jacob, Pierre Sagaut. A multidisciplinary model coupling Lattice-Boltzmann-based CFD and a Social Force Model for the simulation of pollutant dispersion in evacuation situations. Building and Environment, 2021, 205, pp.108212. ⟨10.1016/j.buildenv.2021.108212⟩. ⟨hal-03514660⟩
Felix Marlow, Jérôme Jacob, Pierre Sagaut. A multidisciplinary model coupling Lattice-Boltzmann-based CFD and a Social Force Model for the simulation of pollutant dispersion in evacuation situations. Building and Environment, 2021, 205, pp.108212. ⟨10.1016/j.buildenv.2021.108212⟩. ⟨hal-03597658⟩ Plus de détails...
In closed rooms with limited convection human motion can considerably affect the airflow and thus the dispersion of pollutant. However, in Computational Fluid Dynamics (CFD) simulations on air quality and safety for human beings this effect is generally not considered, which is mainly due to a lack of a well-founded and detailed estimation of the human behavior and the high computational cost of taking into account moving objects in CFD meshes. This work addresses this issue by coupling multidisciplinary methods to allow for a more realistic simulation of pollutant dispersion by taking into account the influence of human movements. A Social Force Model predicts trajectory and speed of each person moving in a complex environment. A lattice Boltzmann-based CFD tool provides a Large Eddy Simulation of the unsteady turbulent airflow with pollutant dispersion and thermal effects. And an Actuator Line Model supplies the CFD tool with body forces that mimic the impact of moving objects on the airflow, thus, avoiding computationally expensive dynamic meshing. The capability of the coupled model is demonstrated on three realistic evacuation scenarios with various pollutant sources and a wide range of scales (dimension from 10 to 100 m, occupation from 10 to 6000 persons). The results allow to access instantaneous environmental parameters like pollutant concentration for each person during the course of the evacuation, enabling the assessment of associated health risks.
Felix Marlow, Jérôme Jacob, Pierre Sagaut. A multidisciplinary model coupling Lattice-Boltzmann-based CFD and a Social Force Model for the simulation of pollutant dispersion in evacuation situations. Building and Environment, 2021, 205, pp.108212. ⟨10.1016/j.buildenv.2021.108212⟩. ⟨hal-03597658⟩
Guanxiong Wang, Lincheng Xu, Eric Serre, Pierre Sagaut. Large temperature difference heat dominated flow simulations using a pressure-based lattice Boltzmann method with mass correction. Physics of Fluids, 2021, 33 (11), pp.116107. ⟨10.1063/5.0073178⟩. ⟨hal-03438869⟩ Plus de détails...
This paper addresses simulation of heat dominated compressible flows in a closed cavity using a pressure-based lattice Boltzmann (LB) method, in which thermal effects are modeled by applying a pressure-featured zero-order moment of distribution functions. A focus is made on the conservation of mass at boundary nodes, which is a challenging issue that significantly complicated by the density-decoupled zero-order moment here. The mass leakage at boundary nodes is mathematically quantified, which enables an efficient local mass correction scheme. The performance of this solver is assessed by simulating buoyancy-driven flows in a closed deferentially heated cavity with large temperature differences (non-Boussinesq) at Rayleigh numbers ranging from 103 to 107. Simulations show that mass leakage at solid walls in such configurations is a critical issue to obtain reliable solutions, and it eventually leads to simulations overflow when the cavity is inclined. The proposed mass correction scheme is, however, shown to be effective to control the mass leakage and get accurate solutions. Thus, associated with the proposed mass conservation scheme, the pressure-based LB method becomes reliable to study natural convection dominated flows at large temperature differences in closed geometries with mesh aligned boundaries or not
Guanxiong Wang, Lincheng Xu, Eric Serre, Pierre Sagaut. Large temperature difference heat dominated flow simulations using a pressure-based lattice Boltzmann method with mass correction. Physics of Fluids, 2021, 33 (11), pp.116107. ⟨10.1063/5.0073178⟩. ⟨hal-03438869⟩
Georis Billo, Michel Belliard, Pierre Sagaut. A Finite Element Penalized Direct Forcing Immersed Boundary Method for infinitely thin obstacles in a dilatable flow. Computers & Mathematics with Applications, 2021, 99, pp.292-304. ⟨10.1016/j.camwa.2021.08.005⟩. ⟨hal-03514671⟩ Plus de détails...
In the framework of the development of new passive safety systems for the second and third generations of nuclear reactors, the numerical simulations, involving complex turbulent two-phase flows around thin or massive inflow obstacles, are privileged tools to model, optimize and assess new design shapes. In order to match industrial demands, computational fluid dynamics tools must be the fastest, most accurate and most robust possible. To face this issue, we have chosen to solve the Navier-Stokes equations using a projection scheme for a mixture fluid coupled with an Immersed Boundary (IB) approach: the penalized direct forcing method - a technique whose characteristics inherit from both penalty and immersed boundary methods - adapted to infinitely thin obstacles and to a Finite Element (FE) formulation. Various IB conditions (slip, no-slip or Neumann) for the velocity on the IB can be managed by imposing Dirichlet values in the vicinity of the thin obstacles. To deal with these imposed Dirichlet velocities, we investigated two variants: one in which we use the obstacle velocity and another one in which we use linear interpolations based on discrete geometrical properties of the IB (barycenters and normal vectors) and the FE basis functions. This last variant is motivated by an increase of the accuracy/computation time ratio for coarse meshes. As a first step, concerning academic test cases for one-phase dilatable-fluid laminar flows, the results obtained via those two variants are in good agreement with analytical and experimental data. Moreover, when compared to each other, the linear interpolation variant increases the spatial order of convergence as expected. An industrial test case illustrates the advantages and drawbacks of this approach. In a shortcoming second step, to face two-phase turbulent fluid simulations, some methodology modifications will be considered such as adapting the projection scheme to low-compressible fluid and immersed wall-law boundary conditions.
Georis Billo, Michel Belliard, Pierre Sagaut. A Finite Element Penalized Direct Forcing Immersed Boundary Method for infinitely thin obstacles in a dilatable flow. Computers & Mathematics with Applications, 2021, 99, pp.292-304. ⟨10.1016/j.camwa.2021.08.005⟩. ⟨hal-03514671⟩
Journal: Computers & Mathematics with Applications
Georis Billo, Michel Belliard, Pierre Sagaut. A Finite Element Penalized Direct Forcing Immersed Boundary Method for infinitely thin obstacles in a dilatable flow. Computers & Mathematics with Applications, 2021, 99, pp.292-304. ⟨10.1016/j.camwa.2021.08.005⟩. ⟨hal-03596009⟩ Plus de détails...
In the framework of the development of new passive safety systems for the second and third generations of nuclear reactors, the numerical simulations, involving complex turbulent two-phase flows around thin or massive inflow obstacles, are privileged tools to model, optimize and assess new design shapes. In order to match industrial demands, computational fluid dynamics tools must be the fastest, most accurate and most robust possible. To face this issue, we have chosen to solve the Navier-Stokes equations using a projection scheme for a mixture fluid coupled with an Immersed Boundary (IB) approach: the penalized direct forcing method-a technique whose characteristics inherit from both penalty and immersed boundary methods-adapted to infinitely thin obstacles and to a Finite Element (FE) formulation. Various IB conditions (slip, no-slip or Neumann) for the velocity on the IB can be managed by imposing Dirichlet values in the vicinity of the thin obstacles. To deal with these imposed Dirichlet velocities, we investigated two variants: one in which we use the obstacle velocity and another one in which we use linear interpolations based on discrete geometrical properties of the IB (barycenters and normal vectors) and the FE basis functions. This last variant is motivated by an increase of the accuracy/computation time ratio for coarse meshes. As a first step, concerning academic test cases for one-phase dilatable-fluid laminar flows, the results obtained via those two variants are in good agreement with analytical and experimental data. Moreover, when compared to each other, the linear interpolation variant increases the spatial order of convergence as expected. An industrial test case illustrates the advantages and drawbacks of this approach. In a shortcoming second step, to face two-phase turbulent fluid simulations, some methodology modifications will be considered such as adapting the projection scheme to low-compressible fluid and immersed wall-law boundary conditions.
Georis Billo, Michel Belliard, Pierre Sagaut. A Finite Element Penalized Direct Forcing Immersed Boundary Method for infinitely thin obstacles in a dilatable flow. Computers & Mathematics with Applications, 2021, 99, pp.292-304. ⟨10.1016/j.camwa.2021.08.005⟩. ⟨hal-03596009⟩
Journal: Computers & Mathematics with Applications
Isabelle Cheylan, Julien Favier, Pierre Sagaut. Immersed boundary conditions for moving objects in turbulent flows with the lattice-Boltzmann method. Physics of Fluids, 2021, 33 (9), pp.095101. ⟨10.1063/5.0062575⟩. ⟨hal-03597108⟩ Plus de détails...
An immersed boundary method is coupled to a turbulent wall model and Large Eddy Simulation, within the Lattice-Boltzmann framework. The method is able to handle arbitrarily moving objects immersed in a high Reynolds number flow and to accurately capture the shear layer and near wall effects. We perform a thorough numerical study which validates the numerical method on a set of test-cases of increasing complexity, in order to demonstrate the application of this method to industrial conditions. The robustness and accuracy of the method are assessed first in a static laminar configuration, then in a mobile laminar case, and finally in a static and oscillating turbulent simulation. In all cases, the proposed method shows good results compared to the available data in the literature.
Isabelle Cheylan, Julien Favier, Pierre Sagaut. Immersed boundary conditions for moving objects in turbulent flows with the lattice-Boltzmann method. Physics of Fluids, 2021, 33 (9), pp.095101. ⟨10.1063/5.0062575⟩. ⟨hal-03597108⟩
Isabelle Cheylan, Julien Favier, Pierre Sagaut. Immersed boundary conditions for moving objects in turbulent flows with the lattice-Boltzmann method. Physics of Fluids, 2021, 33 (9), pp.095101. ⟨10.1063/5.0062575⟩. ⟨hal-03514710⟩ Plus de détails...
An immersed boundary method is coupled to a turbulent wall model and Large Eddy Simulation, within the Lattice-Boltzmann framework. The method is able to handle arbitrarily moving objects immersed in a high Reynolds number flow and to accurately capture the shear layer and near wall effects. We perform a thorough numerical study which validates the numerical method on a set of test-cases of increasing complexity, in order to demonstrate the application of this method to industrial conditions. The robustness and accuracy of the method are assessed first in a static laminar configuration, then in a mobile laminar case, and finally in a static and oscillating turbulent simulation. In all cases, the proposed method shows good results compared to the available data in the literature.
Isabelle Cheylan, Julien Favier, Pierre Sagaut. Immersed boundary conditions for moving objects in turbulent flows with the lattice-Boltzmann method. Physics of Fluids, 2021, 33 (9), pp.095101. ⟨10.1063/5.0062575⟩. ⟨hal-03514710⟩
Johan Degrigny, Shang-Gui Cai, Jean-François Boussuge, Pierre Sagaut. Improved wall model treatment for aerodynamic flows in LBM. Computers and Fluids, 2021, 227, pp.105041. ⟨10.1016/j.compfluid.2021.105041⟩. ⟨hal-03326170⟩ Plus de détails...
The article deals with an improved treatment of wall models for the simulation of turbulent flows in the framework of Immersed Wall Boundaries on Cartesian grids. The emphasis is put on the implementa-tion in a Lattice-Boltzmann Method solver without loss of generality, since the proposed approach can be used in Navier-Stokes-based solvers in a straightforward way. The proposed improved wall model im-plementation relies on the combination of several key elements, namely i) the removal of grid points too close to the solid surface and ii) an original computation of wall normal velocity gradient and iii) the interpolation scheme. The new method is successfully assessed considering URANS simulations focusing on steady solutions of the Zero Pressure Gradient turbulent flat plate boundary layer and the turbulent flow around a NACA0012 airfoil at several angles of attack.
Johan Degrigny, Shang-Gui Cai, Jean-François Boussuge, Pierre Sagaut. Improved wall model treatment for aerodynamic flows in LBM. Computers and Fluids, 2021, 227, pp.105041. ⟨10.1016/j.compfluid.2021.105041⟩. ⟨hal-03326170⟩
Elisa Buffa, Jérôme Jacob, Pierre Sagaut. Lattice-Boltzmann-based large-eddy simulation of high-rise building aerodynamics with inlet turbulence reconstruction. Journal of Wind Engineering and Industrial Aerodynamics, 2021, 212, pp.104560. ⟨10.1016/j.jweia.2021.104560⟩. ⟨hal-03596056⟩ Plus de détails...
Boltzmann-based Large-Eddy Simulation approach for wind load prediction on high-rise building is proposed and validated. An extension of the original incompressible Synthetic Eddy Method to reconstruct inlet turbulence is proposed within the Lattice-Boltzmann framework, including a low-noise frozen density variant. Extensive successful comparisons with experimental data are carried out, for both quantities defined on the building surface and in its wake. A detailed sensitivity analysis of the results with respect to inlet turbulence reconstruction, boundary conditions at the building surface and grid resolution is also provided. An almost unique set of comparisons with experimental data is presented, including mean and rms values, spectra, but also peak values of pressure at the building surface.
Elisa Buffa, Jérôme Jacob, Pierre Sagaut. Lattice-Boltzmann-based large-eddy simulation of high-rise building aerodynamics with inlet turbulence reconstruction. Journal of Wind Engineering and Industrial Aerodynamics, 2021, 212, pp.104560. ⟨10.1016/j.jweia.2021.104560⟩. ⟨hal-03596056⟩
Journal: Journal of Wind Engineering and Industrial Aerodynamics
H. Yoo, M. Bahlali, Julien Favier, Pierre Sagaut. A hybrid recursive regularized lattice Boltzmann model with overset grids for rotating geometries. Physics of Fluids, 2021, 33 (5), pp.057113. ⟨10.1063/5.0045524⟩. ⟨hal-03326134⟩ Plus de détails...
Simulating rotating geometries in fluid flows for industrial applications remains a challenging task for general fluid solvers and in particular for the lattice Boltzmann method (LBM) due to inherent stability and accuracy problems. This work proposes an original method based on the widely used overset grids (or Chimera grids) while being integrated with a recent and optimized LBM collision operator, the hybrid recursive regularized model (HRR). The overset grids are used to actualize the rotating geometries where both the rotating and fixed meshes exist simultaneously. In the rotating mesh, the fictitious forces generated from its non-inertial rotating reference frame are taken into account by using a second order discrete forcing term. The fixed and rotating grids communicate with each other through the interpolation of the macroscopic variables. Meanwhile, the HRR collision model is selected to enhance the stability and accuracy properties of the LBM simulations by filtering out redundant higher order non-equilibrium tensors. The robustness of the overset HRR algorithm is assessed on different configurations, undergoing mid-to-high Reynolds number flows, and the method successfully demonstrates its robustness while exhibiting the second order accuracy.
H. Yoo, M. Bahlali, Julien Favier, Pierre Sagaut. A hybrid recursive regularized lattice Boltzmann model with overset grids for rotating geometries. Physics of Fluids, 2021, 33 (5), pp.057113. ⟨10.1063/5.0045524⟩. ⟨hal-03326134⟩
H. Yoo, M. Bahlali, Julien Favier, Pierre Sagaut. A hybrid recursive regularized lattice Boltzmann model with overset grids for rotating geometries. Physics of Fluids, 2021, 33 (5), pp.057113. ⟨10.1063/5.0045524⟩. ⟨hal-03597721⟩ Plus de détails...
Simulating rotating geometries in fluid flows for industrial applications remains a challenging task for general fluid solvers and in particular for the lattice Boltzmann method (LBM) due to inherent stability and accuracy problems. This work proposes an original method based on the widely used overset grids (or Chimera grids) while being integrated with a recent and optimized LBM collision operator, the hybrid recursive regularized model (HRR). The overset grids are used to actualize the rotating geometries where both the rotating and fixed meshes exist simultaneously. In the rotating mesh, the fictitious forces generated from its non-inertial rotating reference frame are taken into account by using a second order discrete forcing term. The fixed and rotating grids communicate with each other through the interpolation of the macroscopic variables. Meanwhile, the HRR collision model is selected to enhance the stability and accuracy properties of the LBM simulations by filtering out redundant higher order non-equilibrium tensors. The robustness of the overset HRR algorithm is assessed on different configurations, undergoing mid-to-high Reynolds number flows, and the method successfully demonstrates its robustness while exhibiting the second order accuracy.
H. Yoo, M. Bahlali, Julien Favier, Pierre Sagaut. A hybrid recursive regularized lattice Boltzmann model with overset grids for rotating geometries. Physics of Fluids, 2021, 33 (5), pp.057113. ⟨10.1063/5.0045524⟩. ⟨hal-03597721⟩
M. Bahlali, H. Yoo, Julien Favier, Pierre Sagaut. A lattice Boltzmann direct coupling overset approach for the moving boundary problem. Physics of Fluids, 2021, 33 (5), pp.053607. ⟨10.1063/5.0044994⟩. ⟨hal-03326151⟩ Plus de détails...
We propose a new direct coupling scheme based on the overset technique to tackle moving boundary problems within the lattice Boltzmann framework. The scheme is based on the interpolation of distribution functions rather than moments, that is, macroscopic variables, and includes an additional hypothesis ensuring mass and momentum conservation at the interface nodes between fixed and moving grids. The method is assessed considering four test cases and considering both the vortical and the acoustic fields. It is shown that the direct coupling method results are in very good agreement with reference results on a configuration without any moving subdomain. Moreover, it is demonstrated that the direct coupling method provides an improvement of the accuracy of the lattice Boltzmann overset algorithm for aeroacoustics. In particular, a convected vortex test case is studied and reveals that the direct coupling approach leads to a better ability to conserve the vortex structure over time, as well as a reduction in spurious acoustic distorsions at the fixed/moving interface.
M. Bahlali, H. Yoo, Julien Favier, Pierre Sagaut. A lattice Boltzmann direct coupling overset approach for the moving boundary problem. Physics of Fluids, 2021, 33 (5), pp.053607. ⟨10.1063/5.0044994⟩. ⟨hal-03326151⟩
Shang-Gui Cai, Pierre Sagaut. Explicit wall models for large eddy simulation. Physics of Fluids, 2021, 33 (4), pp.041703. ⟨10.1063/5.0048563⟩. ⟨hal-03597083⟩ Plus de détails...
Algebraic explicit wall models covering the entire inner region of the turbulent boundary layer are proposed to reduce the computational effort for large eddy simulation of wall-bounded turbulent flows. The proposed formulas are given in closed forms with either logarithmicor power-function-based laws of the wall, allowing straightforward evaluation of the friction velocity on near wall grids independent of their locations in the turbulent boundary layer. The performance of the proposed models is demonstrated by the wall modeled large eddy simulation of a turbulent plane channel flow.
Shang-Gui Cai, Pierre Sagaut. Explicit wall models for large eddy simulation. Physics of Fluids, 2021, 33 (4), pp.041703. ⟨10.1063/5.0048563⟩. ⟨hal-03597083⟩
Jérémie Janin, Fabien Duval, Christophe Friess, Pierre Sagaut. A new linear forcing method for isotropic turbulence with controlled integral length scale. Physics of Fluids, 2021, 33 (4), pp.045127. ⟨10.1063/5.0045818⟩. ⟨hal-03326165⟩ Plus de détails...
Turbulence is a common feature to all flows that surround us. Despite its ubiquity, particularly in industrial flows, it is very difficult to provide a mathematical framework to the generation of turbulent eddies. Several techniques have been proposed which are able to reproduce the main features of turbulent flows, such as realistic pressure and velocity fluctuations, exhibiting proper space- and time-correlations. These techniques are usually first evaluated upon sustained homogeneous isotropic turbulence by introducing body forces to the Navier-Stokes equations. Among these techniques, Lundgren suggested a successful forcing, applied in physical space. The latter approach unfortunately lacks predicting the integral length scale of turbulence. The present study provides a forcing method based on a reconstruction approach which consists in building fluctuations with a turbulent synthetic velocity field based on a prescribed energy spectrum model. The proposed approach is assessed by performing large-eddy simulations of a sustained homogeneous isotropic turbulence in a triply periodic box of size L = 2pi. Properties of the new forcing technique are discussed, drawing on both spatial and time correlations and also on the shape of energy spectrum together with the level of resolved turbulent kinetic energy. A special attention is put on the control of resolved turbulent energy. In this framework, an efficient selective forcing technique is derived, making use of spectral space features. The results show that the proposed approach allows to drive efficiently the resolved kinetic energy towards its target value while preserving the integral length scale independent of the domain size. It is observed that the resulting longitudinal length scale is overestimated by 13%, while the two-time correlations are recovered when using stochastic frequencies.
Jérémie Janin, Fabien Duval, Christophe Friess, Pierre Sagaut. A new linear forcing method for isotropic turbulence with controlled integral length scale. Physics of Fluids, 2021, 33 (4), pp.045127. ⟨10.1063/5.0045818⟩. ⟨hal-03326165⟩
Florian Renard, Yongliang Feng, Jean-François Boussuge, Pierre Sagaut. Improved compressible hybrid lattice Boltzmann method on standard lattice for subsonic and supersonic flows. Computers and Fluids, 2021, 219, pp.104867. ⟨10.1016/j.compfluid.2021.104867⟩. ⟨hal-03326159⟩ Plus de détails...
A D2Q9 Hybrid Lattice Boltzmann Method (HLBM) is proposed for the simulation of both compressible subsonic and supersonic flows. This HLBM is an extension of the model of Feng et al. [1], which has been found, via different test cases, to be unstable for supersonic regimes. To circumvent this limitation, we propose:: (1) a new discretization of the lattice closure correction term that makes possible the simulation of supersonic flows, (2) a corrected viscous stress tensor that takes into account polyatomic gases, and (3) a novel discretization of the viscous heat production term fitting with the regularized formalism. The result is a hybrid method that resolves the mass and momentum equations with an LBM algorithm, and resolves the entropy-based energy equation with a finite volume method. This approach fully recovers the physics of the Navier-Stokes-Fourier equations with the ideal gas equation of state, and is valid from subsonic to supersonic regimes. It is then successfully assessed with both smooth flows and flows involving shocks. The proposed model is shown to be an efficient, accurate, and robust alternative to classic Navier-Stokes methods for the simulation of compressible flows.
Florian Renard, Yongliang Feng, Jean-François Boussuge, Pierre Sagaut. Improved compressible hybrid lattice Boltzmann method on standard lattice for subsonic and supersonic flows. Computers and Fluids, 2021, 219, pp.104867. ⟨10.1016/j.compfluid.2021.104867⟩. ⟨hal-03326159⟩
Jérémie Janin, Fabien Duval, Christophe Friess, Pierre Sagaut. A new linear forcing method for isotropic turbulence with controlled integral length scale. Physics of Fluids, American Institute of Physics, 2021, 33 (4), pp.045127. ⟨10.1063/5.0045818⟩. ⟨hal-03326165⟩ Plus de détails...
Turbulence is a common feature to all flows that surround us. Despite its ubiquity, particularly in industrial flows, it is very difficult to provide a mathematical framework for the generation of turbulent eddies. Several methods have been proposed which are able to reproduce realistic features for velocity fluctuations, exhibiting proper space- and time-correlations. Focusing on physical space forcing, these methods are usually first evaluated upon sustained homogeneous isotropic turbulence by introducing a body force to the Navier-Stokes equations. Since the pioneering work of Lundgren, these techniques usually experience difficulties in predicting the integral length scale. The present study provides a forcing through a reconstruction approach which consists in building velocity fluctuations with a prescribed energy spectrum model. The proposed approach is assessed by performing large-eddy simulations of a sustained homogeneous isotropic turbulence in a triply periodic box. Properties of this forcing technique are discussed, drawing on both spatial and time correlations and also on the shape of energy spectrum together with the level of resolved turbulent kinetic energy. A special attention is put on the control of resolved turbulent energy. In this framework, an efficient selective forcing technique is derived, making use of spectral space features. The results show that the proposed approach allows to drive efficiently the resolved kinetic energy toward its target value while preserving the integral length scale independent of the domain size. It is observed that the resulting longitudinal length scale is overestimated by 13%, while the two-time correlations are recovered when using stochastic frequencies.
Jérémie Janin, Fabien Duval, Christophe Friess, Pierre Sagaut. A new linear forcing method for isotropic turbulence with controlled integral length scale. Physics of Fluids, American Institute of Physics, 2021, 33 (4), pp.045127. ⟨10.1063/5.0045818⟩. ⟨hal-03326165⟩
S. Guo, Y. Feng, Pierre Sagaut. On the use of conservative formulation of energy equation in hybrid compressible lattice Boltzmann method. Computers and Fluids, 2021, 219, pp.104866. ⟨10.1016/j.compfluid.2021.104866⟩. ⟨hal-03597478⟩ Plus de détails...
Effect of density variations on mass conservation properties is widely recognized in the lattice Boltzmann method (LBM), thus non-conservative form of scalar transport equation was commonly adopted within the framework of hybrid LBM. Focusing on the compressible hybrid LBM, mass conservation and its effect on energy conservation equation are studied in this paper. Starting from the analysis on mass conservation law recovered by LBM, the consistency between conservative and non-conservative formulations of energy conservation equation based on various thermodynamic variables and lattice Boltzmann equation is addressed. Driven by the theoretical analysis, a set of modified consistent energy equations in entropy and internal energy form is derived to reduce the error terms and improve the consistency. The theoretical analysis and modified energy equations are intensively evaluated by several numerical test cases, e.g., the isentropic vortex convection, three-dimensional compressible Taylor-Green vortex and shock-vortex interaction.
S. Guo, Y. Feng, Pierre Sagaut. On the use of conservative formulation of energy equation in hybrid compressible lattice Boltzmann method. Computers and Fluids, 2021, 219, pp.104866. ⟨10.1016/j.compfluid.2021.104866⟩. ⟨hal-03597478⟩
S. Guo, Y. Feng, Pierre Sagaut. On the use of conservative formulation of energy equation in hybrid compressible lattice Boltzmann method. Computers and Fluids, 2021, 219, pp.104866. ⟨10.1016/j.compfluid.2021.104866⟩. ⟨hal-03326128⟩ Plus de détails...
Effect of density variations on mass conservation properties is widely recognized in the lattice Boltzmann method (LBM), thus non-conservative form of scalar transport equation was commonly adopted within the framework of hybrid LBM. Focusing on the compressible hybrid LBM, mass conservation and its effect on energy conservation equation are studied in this paper. Starting from the analysis on mass conservation law recovered by LBM, the consistency between conservative and non-conservative formulations of energy conservation equation based on various thermodynamic variables and lattice Boltzmann equation is addressed. Driven by the theoretical analysis, a set of modified consistent energy equations in entropy and internal energy form is derived to reduce the error terms and improve the consistency. The theoretical analysis and modified energy equations are intensively evaluated by several numerical test cases, e.g., the isentropic vortex convection, three-dimensional compressible Taylor-Green vortex and shock-vortex interaction.
S. Guo, Y. Feng, Pierre Sagaut. On the use of conservative formulation of energy equation in hybrid compressible lattice Boltzmann method. Computers and Fluids, 2021, 219, pp.104866. ⟨10.1016/j.compfluid.2021.104866⟩. ⟨hal-03326128⟩
Shang-Gui Cai, Johan Degrigny, Jean-François Boussuge, Pierre Sagaut. Coupling of turbulence wall models and immersed boundaries on Cartesian grids. Journal of Computational Physics, 2021, 429, pp.109995. ⟨10.1016/j.jcp.2020.109995⟩. ⟨hal-03597064⟩ Plus de détails...
An improved coupling of immersed boundary method and turbulence wall models on Cartesian grids is proposed, for producing smooth wall surface pressure and skin friction at high Reynolds numbers. Spurious oscillations are frequently observed on these quantities with most immersed boundary wall modeling methods, especially for the skin friction which is found to be very sensitive to the solid surface's position and orientation against the Cartesian grids. The problem originates from the irregularity of the wall distance on the stair-step grid boundaries where the immersed boundary conditions are applied. To reduce this directional error, several modifications are presented to enhance the near wall solution. First, the commonly used interpolation for the flow velocity is replaced by one for the friction velocity, which has much less variation near wall. The concept of using a fictitious point to retrieve flow fields in the wall normal direction is abandoned and the interpolation is performed in the wall parallel plane with existing fluid points. Secondly, the velocity gradients at the approximated boundary are computed with advanced schemes and the normal gradient of the tangential velocity is reconstructed from the wall laws. To further protect the near wall solution, the normal velocity gradient and the working viscosity from the Spalart-Allmaras turbulence model are enforced by their theoretical solutions in the interior fluid close to the wall. Additionally, various post-processing algorithms for reconstructing wall surface quantities and force integrations are investigated. Other related factors are also discussed for their effects on the results. The validity of present method has been demonstrated through numerical benchmark tests on a flat plate at zero pressure gradient, both aligned and inclined with respect to the grid, as well as aerodynamic cases of NACA 23012 airfoil and NASA trap wing.
Shang-Gui Cai, Johan Degrigny, Jean-François Boussuge, Pierre Sagaut. Coupling of turbulence wall models and immersed boundaries on Cartesian grids. Journal of Computational Physics, 2021, 429, pp.109995. ⟨10.1016/j.jcp.2020.109995⟩. ⟨hal-03597064⟩
Isabelle Cheylan, Song Zhao, Pierre Boivin, Pierre Sagaut. Compressible pressure-based Lattice-Boltzmann applied to humid air with phase change. Applied Thermal Engineering, 2021, pp.116868. ⟨10.1016/j.applthermaleng.2021.116868⟩. ⟨hal-03180596⟩ Plus de détails...
A new compressible pressure-based Lattice Boltzmann Method is proposed to simulate humid air flows with phase change. The variable density and compressible effects are fully resolved, effectively lifting the Boussinesq approximation commonly used, e.g. for meteorological flows. Previous studies indicate that the Boussinesq assumption can lead to errors up to 25%, but the model remains common, for compressible models often suffer from a lack of stability. In order to overcome this issue, a new pressure-based solver is proposed, exhibiting excellent stability properties. Mass and momentum conservation equations are solved by a hybrid recursive regularized Lattice-Boltzmann approach, whereas the enthalpy and species conservation equations are solved using a finite volume method. The solver is based on a pressure-based method coupled with a predictor-corrector algorithm, and incorporates a humid equation of state, as well as a specific boundary condition treatment for phase change. In particular, boundary conditions that handle mass leakage are also proposed and validated. Three test cases are investigated in order to validate this new approach: the Rayleigh-Bénard instability applied to humid air, the atmospheric rising of a condensing moist bubble, and finally the evaporation of a thin liquid film in a vertical channel. Results indicate that the proposed pressure-based Lattice-Boltzmann model is stable and accurate on all cases.
Isabelle Cheylan, Song Zhao, Pierre Boivin, Pierre Sagaut. Compressible pressure-based Lattice-Boltzmann applied to humid air with phase change. Applied Thermal Engineering, 2021, pp.116868. ⟨10.1016/j.applthermaleng.2021.116868⟩. ⟨hal-03180596⟩
Shang-Gui Cai, Johan Degrigny, Jean-François Boussuge, Pierre Sagaut. Coupling of turbulence wall models and immersed boundaries on Cartesian grids. Journal of Computational Physics, 2021, 429, pp.109995. ⟨10.1016/j.jcp.2020.109995⟩. ⟨hal-03326140⟩ Plus de détails...
An improved coupling of immersed boundary method and turbulence wall models on Cartesian grids is proposed, for producing smooth wall surface pressure and skin friction at high Reynolds numbers. Spurious oscillations are frequently observed on these quantities with most immersed boundary wall modeling methods, especially for the skin friction which is found to be very sensitive to the solid surface's position and orientation against the Cartesian grids. The problem originates from the irregularity of the wall distance on the stair-step grid boundaries where the immersed boundary conditions are applied. To reduce this directional error, several modifications are presented to enhance the near wall solution. First, the commonly used interpolation for the flow velocity is replaced by one for the friction velocity, which has much less variation near wall. The concept of using a fictitious point to retrieve flow fields in the wall normal direction is abandoned and the interpolation is performed in the wall parallel plane with existing fluid points. Secondly, the velocity gradients at the approximated boundary are computed with advanced schemes and the normal gradient of the tangential velocity is reconstructed from the wall laws. To further protect the near wall solution, the normal velocity gradient and the working viscosity from the Spalart-Allmaras turbulence model are enforced by their theoretical solutions in the interior fluid close to the wall. Additionally, various post-processing algorithms for reconstructing wall surface quantities and force integrations are investigated. Other related factors are also discussed for their effects on the results. The validity of present method has been demonstrated through numerical benchmark tests on a flat plate at zero pressure gradient, both aligned and inclined with respect to the grid, as well as aerodynamic cases of NACA 23012 airfoil and NASA trap wing.
Shang-Gui Cai, Johan Degrigny, Jean-François Boussuge, Pierre Sagaut. Coupling of turbulence wall models and immersed boundaries on Cartesian grids. Journal of Computational Physics, 2021, 429, pp.109995. ⟨10.1016/j.jcp.2020.109995⟩. ⟨hal-03326140⟩
Y. Feng, J. Miranda-Fuentes, Jérôme Jacob, Pierre Sagaut. Hybrid lattice Boltzmann model for atmospheric flows under anelastic approximation. Physics of Fluids, 2021, 33 (3), pp.036607. ⟨10.1063/5.0039516⟩. ⟨hal-03597258⟩ Plus de détails...
Lattice Boltzmann (LB) method for atmospheric dynamics is developed by considering the characteristics of the anelastic approximation. After introducing reference base state values in atmospheric flows, an LB model, with an external force term, has been constructed in anelastic framework. In the proposed anelastic LB model, mass and momentum conservation equations are solved by the LB method with a regularization procedure, and temperature field or scalar transport is simulated by finite volume method. The derived macroscopic governing equations from the anelastic model are analyzed and discussed in Chapman-Enskog asymptotic expansion. The anelastic LB model is assessed considering three benchmarks including a non-hydrostatic atmospheric inviscid convection, two-dimensional density currents, and inertia-gravity waves in stably stratified atmospheric layer. The validations demonstrate that the anelastic extension of the LB method can simulate atmospheric flows effectively and accurately. Besides, the proposed model offers a unified framework for both Boussinesq approximation and anelastic approximation, which is largely free of characteristic depth of atmospheric flows.
Y. Feng, J. Miranda-Fuentes, Jérôme Jacob, Pierre Sagaut. Hybrid lattice Boltzmann model for atmospheric flows under anelastic approximation. Physics of Fluids, 2021, 33 (3), pp.036607. ⟨10.1063/5.0039516⟩. ⟨hal-03597258⟩
Yongliang Feng, Johann Miranda‐fuentes, Shaolong Guo, Jérôme Jacob, Pierre Sagaut. ProLB: A Lattice Boltzmann Solver of Large‐Eddy Simulation for Atmospheric Boundary Layer Flows. Journal of Advances in Modeling Earth Systems, 2021, 13 (3), pp.e2020MS002107. ⟨10.1029/2020MS002107⟩. ⟨hal-03326123⟩ Plus de détails...
A large-eddy simulation tool is developed for simulating the dynamics of atmospheric boundary layers (ABLs) using lattice Boltzmann method (LBM), which is an alternative approach for computational fluid dynamics and proved to be very well suited for the simulation of low-Mach flows. The equations of motion are coupled with the global complex physical models considering the coupling among several mechanisms, namely basic hydro-thermodynamics and body forces related to stratification, Coriolis force, canopy effects, humidity transport, and condensation. Mass and momentum equations are recovered by an efficient streaming, collision, and forcing process within the framework of LBM while the governing equations of temperature, liquid, and vapor water fraction are solved using a finite volume method. The implementation of wall models for ABL, subgrid models, and interaction terms related to multiphysic phenomena (e.g., stratification, condensation) is described, implemented, and assessed in this study. An immersed boundary approach is used to handle flows in complex configurations, with application to flows in realistic urban areas. Applications to both wind engineering and atmospheric pollutant dispersion are illustrated.
Yongliang Feng, Johann Miranda‐fuentes, Shaolong Guo, Jérôme Jacob, Pierre Sagaut. ProLB: A Lattice Boltzmann Solver of Large‐Eddy Simulation for Atmospheric Boundary Layer Flows. Journal of Advances in Modeling Earth Systems, 2021, 13 (3), pp.e2020MS002107. ⟨10.1029/2020MS002107⟩. ⟨hal-03326123⟩
Journal: Journal of Advances in Modeling Earth Systems
Y. Feng, J. Miranda-Fuentes, Jérôme Jacob, Pierre Sagaut. Hybrid lattice Boltzmann model for atmospheric flows under anelastic approximation. Physics of Fluids, 2021, 33 (3), pp.036607. ⟨10.1063/5.0039516⟩. ⟨hal-03326143⟩ Plus de détails...
Lattice Boltzmann (LB) method for atmospheric dynamics is developed by considering the characteristics of the anelastic approximation. After introducing reference base state values in atmospheric flows, an LB model, with an external force term, has been constructed in anelastic framework. In the proposed anelastic LB model, mass and momentum conservation equations are solved by the LB method with a regularization procedure, and temperature field or scalar transport is simulated by finite volume method. The derived macroscopic governing equations from the anelastic model are analyzed and discussed in Chapman-Enskog asymptotic expansion. The anelastic LB model is assessed considering three benchmarks including a non-hydrostatic atmospheric inviscid convection, two-dimensional density currents, and inertia-gravity waves in stably stratified atmospheric layer. The validations demonstrate that the anelastic extension of the LB method can simulate atmospheric flows effectively and accurately. Besides, the proposed model offers a unified framework for both Boussinesq approximation and anelastic approximation, which is largely free of characteristic depth of atmospheric flows.
Y. Feng, J. Miranda-Fuentes, Jérôme Jacob, Pierre Sagaut. Hybrid lattice Boltzmann model for atmospheric flows under anelastic approximation. Physics of Fluids, 2021, 33 (3), pp.036607. ⟨10.1063/5.0039516⟩. ⟨hal-03326143⟩
G. Farag, T. Coratger, G. Wissocq, S. Zhao, Pierre Boivin, et al.. A unified hybrid lattice-Boltzmann method for compressible flows: Bridging between pressure-based and density-based methods. Physics of Fluids, 2021, 33 (8), pp.086101. ⟨10.1063/5.0057407⟩. ⟨hal-03324229⟩ Plus de détails...
A unified expression for high-speed compressible segregated consistent lattice Boltzmann methods, namely, pressure-based and improved density-based methods, is given. It is theoretically proved that in the absence of forcing terms, these approaches are strictly identical and can be recast in a unique form. An important result is that the difference with classical density-based methods lies in the addition of fourth-order term in the equilibrium function. It is also shown that forcing terms used to balance numerical errors in both original pressure-based and improved density-based methods can be written in a generalized way. A hybrid segregated efficient lattice-Boltzmann for compressible flow based on this unified model, equipped with a recursive regularization kernel, is proposed and successfully assessed on a wide set of test cases with and without shock waves.
G. Farag, T. Coratger, G. Wissocq, S. Zhao, Pierre Boivin, et al.. A unified hybrid lattice-Boltzmann method for compressible flows: Bridging between pressure-based and density-based methods. Physics of Fluids, 2021, 33 (8), pp.086101. ⟨10.1063/5.0057407⟩. ⟨hal-03324229⟩
Johan Degrigny, Shang-Gui Cai, Jean-François Boussuge, Pierre Sagaut. Improved wall model treatment for aerodynamic flows in LBM. Computers and Fluids, 2021, 227, pp.105041. ⟨10.1016/j.compfluid.2021.105041⟩. ⟨hal-03597146⟩ Plus de détails...
The article deals with an improved treatment of wall models for the simulation of turbulent flows in the framework of Immersed Wall Boundaries on Cartesian grids. The emphasis is put on the implementa-tion in a Lattice-Boltzmann Method solver without loss of generality, since the proposed approach can be used in Navier-Stokes-based solvers in a straightforward way. The proposed improved wall model im-plementation relies on the combination of several key elements, namely i) the removal of grid points too close to the solid surface and ii) an original computation of wall normal velocity gradient and iii) the interpolation scheme. The new method is successfully assessed considering URANS simulations focusing on steady solutions of the Zero Pressure Gradient turbulent flat plate boundary layer and the turbulent flow around a NACA0012 airfoil at several angles of attack.
Johan Degrigny, Shang-Gui Cai, Jean-François Boussuge, Pierre Sagaut. Improved wall model treatment for aerodynamic flows in LBM. Computers and Fluids, 2021, 227, pp.105041. ⟨10.1016/j.compfluid.2021.105041⟩. ⟨hal-03597146⟩
Jérôme Jacob, Lucie Merlier, Felix Marlow, Pierre Sagaut. Lattice Boltzmann Method-Based Simulations of Pollutant Dispersion and Urban Physics. Atmosphere, 2021, 12 (7), pp.833. ⟨10.3390/atmos12070833⟩. ⟨hal-03326148⟩ Plus de détails...
Mesocale atmospheric flows that develop in the boundary layer or microscale flows that develop in urban areas are challenging to predict, especially due to multiscale interactions, multiphysical couplings, land and urban surface thermal and geometrical properties and turbulence. However, these different flows can indirectly and directly affect the exposure of people to deteriorated air quality or thermal environment, as well as the structural and energy loads of buildings. Therefore, the ability to accurately predict the different interacting physical processes determining these flows is of primary importance. To this end, alternative approaches based on the lattice Boltzmann method (LBM) wall model large eddy simulations (WMLESs) appear particularly interesting as they provide a suitable framework to develop efficient numerical methods for the prediction of complex large or smaller scale atmospheric flows. In particular, this article summarizes recent developments and studies performed using the hybrid recursive regularized collision model for the simulation of complex or/and coupled turbulent flows. Different applications to the prediction of meteorological humid flows, urban pollutant dispersion, pedestrian wind comfort and pressure distribution on urban buildings including uncertainty quantification are especially reviewed. For these different applications, the accuracy of the developed approach was assessed by comparison with experimental and/or numerical reference data, showing a state of the art performance. Ongoing developments focus now on the validation and prediction of indoor environmental conditions including thermal mixing and pollutant dispersion in different types of rooms equipped with heat, ventilation and air conditioning systems.
Jérôme Jacob, Lucie Merlier, Felix Marlow, Pierre Sagaut. Lattice Boltzmann Method-Based Simulations of Pollutant Dispersion and Urban Physics. Atmosphere, 2021, 12 (7), pp.833. ⟨10.3390/atmos12070833⟩. ⟨hal-03326148⟩
Jérôme Jacob, Lucie Merlier, Felix Marlow, Pierre Sagaut. Lattice Boltzmann Method-Based Simulations of Pollutant Dispersion and Urban Physics. Atmosphere, 2021, 12 (7), pp.833. ⟨10.3390/atmos12070833⟩. ⟨hal-03326148⟩ Plus de détails...
Mesocale atmospheric flows that develop in the boundary layer or microscale flows that develop in urban areas are challenging to predict, especially due to multiscale interactions, multiphysical couplings, land and urban surface thermal and geometrical properties and turbulence. However, these different flows can indirectly and directly affect the exposure of people to deteriorated air quality or thermal environment, as well as the structural and energy loads of buildings. Therefore, the ability to accurately predict the different interacting physical processes determining these flows is of primary importance. To this end, alternative approaches based on the lattice Boltzmann method (LBM) wall model large eddy simulations (WMLESs) appear particularly interesting as they provide a suitable framework to develop efficient numerical methods for the prediction of complex large or smaller scale atmospheric flows. In particular, this article summarizes recent developments and studies performed using the hybrid recursive regularized collision model for the simulation of complex or/and coupled turbulent flows. Different applications to the prediction of meteorological humid flows, urban pollutant dispersion, pedestrian wind comfort and pressure distribution on urban buildings including uncertainty quantification are especially reviewed. For these different applications, the accuracy of the developed approach was assessed by comparison with experimental and/or numerical reference data, showing a state of the art performance. Ongoing developments focus now on the validation and prediction of indoor environmental conditions including thermal mixing and pollutant dispersion in different types of rooms equipped with heat, ventilation and air conditioning systems.
Jérôme Jacob, Lucie Merlier, Felix Marlow, Pierre Sagaut. Lattice Boltzmann Method-Based Simulations of Pollutant Dispersion and Urban Physics. Atmosphere, 2021, 12 (7), pp.833. ⟨10.3390/atmos12070833⟩. ⟨hal-03326148⟩
S. Zhao, G. Farag, Pierre Boivin, P. Sagaut. Toward fully conservative hybrid lattice Boltzmann methods for compressible flows. Physics of Fluids, 2020, 32 (12), pp.126118. ⟨10.1063/5.0033245⟩. ⟨hal-03087980⟩ Plus de détails...
S. Zhao, G. Farag, Pierre Boivin, P. Sagaut. Toward fully conservative hybrid lattice Boltzmann methods for compressible flows. Physics of Fluids, 2020, 32 (12), pp.126118. ⟨10.1063/5.0033245⟩. ⟨hal-03087980⟩
Thomas Astoul, Gauthier Wissocq, Jean-François Boussuge, Alois Sengissen, Pierre Sagaut. Analysis and reduction of spurious noise generated at grid refinement interfaces with the lattice Boltzmann method. Journal of Computational Physics, 2020, 418, pp.109645. ⟨10.1016/j.jcp.2020.109645⟩. ⟨hal-02960150⟩ Plus de détails...
The present study focuses on the unphysical effects induced by the use of non-uniform grids in the lattice Boltzmann method. In particular, the convection of vortical structures across a grid refinement interface is likely to generate spurious noise that may impact the whole computation domain. This issue becomes critical in the case of aeroacoustic simulations, where accurate pressure estimations are of paramount importance. The purpose of this article is to identify the issues occurring at the interface and to propose possible solutions yielding significant improvements for aeroacoustic simulations. More specifically, this study highlights the critical involvement of non-physical modes in the generation of spurious vorticity and acoustics. The identification of these modes is made possible thanks to linear stability analyses performed in the fluid core, and non-hydrodynamic sensors specifically developed to systematically emphasize them during a simulation. Investigations seeking pure acoustic waves and sheared flows allow for isolating the contribution of each mode. An important result is that spurious wave generation is intrinsically due to the change in the grid resolution (i.e. aliasing) independently of the details of the grid transition algorithm. Finally, the solution proposed to minimize spurious wave amplitude consists of choosing an appropriate collision model in the fluid core so as to cancel the non-hydrodynamic mode contribution regardless the grid coupling algorithm. Results are validated on a convected vortex and on a turbulent flow around a cylinder where a huge reduction of both spurious noise and vorticity are obtained.
Thomas Astoul, Gauthier Wissocq, Jean-François Boussuge, Alois Sengissen, Pierre Sagaut. Analysis and reduction of spurious noise generated at grid refinement interfaces with the lattice Boltzmann method. Journal of Computational Physics, 2020, 418, pp.109645. ⟨10.1016/j.jcp.2020.109645⟩. ⟨hal-02960150⟩
S. Guo, Yongliang Feng, Jérôme Jacob, F. Renard, Pierre Sagaut. An efficient lattice Boltzmann method for compressible aerodynamics on D3Q19 lattice. Journal of Computational Physics, 2020, 418, pp.109570. ⟨10.1016/j.jcp.2020.109570⟩. ⟨hal-02960161⟩ Plus de détails...
An efficient lattice Boltzmann (LB) model relying on a hybrid recursive regularization (HRR) collision operator on D3Q19 stencil is proposed for the simulation of three-dimensional high-speed compressible flows in both subsonic and supersonic regimes. An improved thermal equilibrium distribution function on D3Q19 lattice is derived to reduce the complexity of correcting terms. A simple shock capturing scheme and an upwind biased discretization of correction terms are implemented for supersonic flows with shocks. Mass and momentum equations are recovered by an efficient streaming, collision and forcing process on D3Q19 lattice. Then a non-conservative formulation of the entropy evolution equation is used, that is solved using a finite volume method. The proposed method is assessed considering the simulation of i) 2D isentropic vortex convection, ii) 3D non-isothermal acoustic pulse, iii) 2D supersonic flow over a bump, iv) 3D shock explosion in a box, v) 2D vortex interaction with shock wave, vi) 2D laminar flows over a flat plate at Ma of 0.5, 1.0 and 1.5.
S. Guo, Yongliang Feng, Jérôme Jacob, F. Renard, Pierre Sagaut. An efficient lattice Boltzmann method for compressible aerodynamics on D3Q19 lattice. Journal of Computational Physics, 2020, 418, pp.109570. ⟨10.1016/j.jcp.2020.109570⟩. ⟨hal-02960161⟩
M. Meldi, A. Mariotti, M. Salvetti, P. Sagaut. Numerical investigation of skewed spatially evolving mixing layers. Journal of Fluid Mechanics, 2020, 897, pp.A35. ⟨10.1017/jfm.2020.407⟩. ⟨hal-03251559⟩ Plus de détails...
The sensitivity of turbulent dynamics in spatially evolving mixing layers to small skew angles is investigated via direct numerical simulation. Angle is a measure of the lack of parallelism between the two asymptotic flows, whose interaction creates the turbulent mixing region. The analysis is performed considering a large range of values of the shear intensity parameter . This two-dimensional parameter space is explored using the results of a database of 18 direct numerical simulations. Instantaneous fields as well as time-averaged quantities are investigated, highlighting important mechanisms in the emergence of turbulence and its characteristics for this class of flows. In addition, a stochastic approach is used in which and are considered as random variables with a given probability distribution. The response surfaces of flow statistics in the parameter space are built through non-intrusive generalized polynomial chaos. It is found that variations of the parameter have a primary effect on the growth of the mixing region. A secondary effect associated with is observed as well. Higher values for the skew angle are responsible for a rapid increase in growth of the inlet structures, enhancing the development of the mixing region. The impact on the turbulence features and, in particular, on the Reynolds stress tensor is also significant. A modification of the normalized diagonal components of the Reynolds stress tensor due to is observed. In addition, the interaction between the parameters and is here the governing element.
M. Meldi, A. Mariotti, M. Salvetti, P. Sagaut. Numerical investigation of skewed spatially evolving mixing layers. Journal of Fluid Mechanics, 2020, 897, pp.A35. ⟨10.1017/jfm.2020.407⟩. ⟨hal-03251559⟩
Sylvia Wilhelm, Jérôme Jacob, Pierre Sagaut. A New Explicit Algebraic Wall Model for LES of Turbulent Flows Under Adverse Pressure Gradient. Flow, Turbulence and Combustion, 2020, ⟨10.1007/s10494-020-00181-7⟩. ⟨hal-02960184⟩ Plus de détails...
A new explicit algebraic wall law for the Large Eddy Simulation of flows with adverse pressure gradient is proposed. This new wall law, referred as adverse pressure gradient power law (APGPL), is developed starting from the power-law of Werner and Wengle (Turbulent Shear Flows, vol 8, Springer, New York, pp 155-168, 1993) in order to mimic an implicit non-equilibrium log-law based on Afzal's law (Afzal, IUTAM Symposium on Asymptotic Methods for Turbulent Shear Flows at High Reynolds Numbers, Kluwer Academic Publishers, Bochum, pp 95-118, 1996). No iterative method is needed for the evaluation of the wall shear stress from the APGPL contrary to the majority of models available in the literature. The APGPL model relies on the definition of three modes: the equilibrium power-law is used in regions of no or favourable pressure gradient, the APGPL is used in regions of adverse pressure gradient, and no wall model is used in separated flow regions. This model is assessed via Large Eddy Simulations of flows involving adverse pressure gradient and boundary layer separation using the Lattice Boltzmann Method on uniform nested grids. The flow around a clean and iced NACA23012 airfoil at Reynolds numberRe=1.88 x 10(6) and the flow over the LAGOON landing gear at Re=1.59x10(6) are considered. Results are found in good agreement with those obtained by the non-equilibrium log-law and experimental and numerical data available in the literature.
Sylvia Wilhelm, Jérôme Jacob, Pierre Sagaut. A New Explicit Algebraic Wall Model for LES of Turbulent Flows Under Adverse Pressure Gradient. Flow, Turbulence and Combustion, 2020, ⟨10.1007/s10494-020-00181-7⟩. ⟨hal-02960184⟩
Y. Feng, S. Guo, J. Jacob, P. Sagaut. Grid refinement in the three-dimensional hybrid recursive regularized lattice Boltzmann method for compressible aerodynamics. Physical Review E , 2020, 101 (6), pp.063302. ⟨10.1103/PhysRevE.101.063302⟩. ⟨hal-02892273⟩ Plus de détails...
Grid refinement techniques are of paramount importance for computational fluid dynamics approaches relying on the use of Cartesian grids. This is especially true of solvers dedicated to aerodynamics, in which the capture of thin shear layers require the use of small cells. In this paper, a three-dimensional grid refinement technique is developed within the framework of hybrid recursive regularized lattice Boltzmann method (HRR-LBM) for compressible high-speed flows, which is an efficient collide-stream-type method on a compact D3Q19 stencil. The proposed method is successfully assessed considering several test cases, namely, an isentropic vortex propagating through transition interface, shock-vortex interaction with intersection between grid refinement interface and shock corrugation, and transonic flows over three-dimensional DLR-M6 wing with seven levels of grid refinement.
Y. Feng, S. Guo, J. Jacob, P. Sagaut. Grid refinement in the three-dimensional hybrid recursive regularized lattice Boltzmann method for compressible aerodynamics. Physical Review E , 2020, 101 (6), pp.063302. ⟨10.1103/PhysRevE.101.063302⟩. ⟨hal-02892273⟩
Y. Feng, S. Guo, J. Jacob, P. Sagaut. Grid refinement in the three-dimensional hybrid recursive regularized lattice Boltzmann method for compressible aerodynamics. Physical Review E , 2020, 101 (6), pp.063302. ⟨10.1103/PhysRevE.101.063302⟩. ⟨hal-03228997⟩ Plus de détails...
Grid refinement techniques are of paramount importance for computational fluid dynamics approaches relying on the use of Cartesian grids. This is especially true of solvers dedicated to aerodynamics, in which the capture of thin shear layers require the use of small cells. In this paper, a three-dimensional grid refinement technique is developed within the framework of hybrid recursive regularized lattice Boltzmann method (HRR-LBM) for compressible high-speed flows, which is an efficient collide-stream-type method on a compact D3Q19 stencil. The proposed method is successfully assessed considering several test cases, namely, an isentropic vortex propagating through transition interface, shock-vortex interaction with intersection between grid refinement interface and shock corrugation, and transonic flows over three-dimensional DLR-M6 wing with seven levels of grid refinement.
Y. Feng, S. Guo, J. Jacob, P. Sagaut. Grid refinement in the three-dimensional hybrid recursive regularized lattice Boltzmann method for compressible aerodynamics. Physical Review E , 2020, 101 (6), pp.063302. ⟨10.1103/PhysRevE.101.063302⟩. ⟨hal-03228997⟩
G. Farag, S. Zhao, T. Coratger, Pierre Boivin, G. Chiavassa, et al.. A pressure-based regularized lattice-Boltzmann method for the simulation of compressible flows. Physics of Fluids, 2020, 32 (6), pp.066106. ⟨10.1063/5.0011839⟩. ⟨hal-02885427⟩ Plus de détails...
A new pressure-based Lattice-Boltzmann method (HRR-p) is proposed for the simulation of flows for Mach numbers ranging from 0 to 1.5. Compatible with nearest neighbor lattices (e.g. D3Q19), the model consists of a predictor step comparable to classical athermal Lattice-Boltzmann methods, appended with a fully local and explicit correction step for the pressure. Energy conservation-for which the Hermi-tian quadrature is not accurate enough on such lattice-is solved via a classical finite volume MUSCL-Hancock scheme based on the entropy equation. The Euler part of the model is then validated for the transport of three canonical modes (vortex, en-tropy, and acoustic propagation), while its diffusive/viscous properties are assessed via thermal Couette flow simulations. All results match the analytical solutions, with very limited dissipation. Lastly, the robustness of the method is tested in a one dimensional shock tube and a two-dimensional shock-vortex interaction.
G. Farag, S. Zhao, T. Coratger, Pierre Boivin, G. Chiavassa, et al.. A pressure-based regularized lattice-Boltzmann method for the simulation of compressible flows. Physics of Fluids, 2020, 32 (6), pp.066106. ⟨10.1063/5.0011839⟩. ⟨hal-02885427⟩
Gauthier Wissocq, Jean-François Boussuge, Pierre Sagaut. Consistent vortex initialization for the athermal lattice Boltzmann method. Physical Review E , 2020, 101 (4), ⟨10.1103/PhysRevE.101.043306⟩. ⟨hal-03229006⟩ Plus de détails...
A barotropic counterpart of the well-known convected vortex test case is rigorously derived from the Euler equations along with an athermal equation of state. Starting from a given velocity distribution corresponding to an intended flow recirculation, the athermal counterpart of the Euler equations are solved to obtain a consistent density field. The present initialization is assessed on a standard lattice Boltzmann solver based on the D2Q9 lattice. Compared to the usual isentropic initialization, a much lower spurious relaxation toward the targeted solution is observed, which is due to the spatial resolution rather than approximated macroscopic quantities. The amplitude of the spurious waves can be further reduced by including an off-equilibrium part in the initial distribution functions.
Gauthier Wissocq, Jean-François Boussuge, Pierre Sagaut. Consistent vortex initialization for the athermal lattice Boltzmann method. Physical Review E , 2020, 101 (4), ⟨10.1103/PhysRevE.101.043306⟩. ⟨hal-03229006⟩
Gauthier Wissocq, Jean-François Boussuge, Pierre Sagaut. Consistent vortex initialization for the athermal lattice Boltzmann method. Physical Review E , 2020, 101 (4), ⟨10.1103/PhysRevE.101.043306⟩. ⟨hal-02892501⟩ Plus de détails...
A barotropic counterpart of the well-known convected vortex test case is rigorously derived from the Euler equations along with an athermal equation of state. Starting from a given velocity distribution corresponding to an intended flow recirculation, the athermal counterpart of the Euler equations are solved to obtain a consistent density field. The present initialization is assessed on a standard lattice Boltzmann solver based on the D2Q9 lattice. Compared to the usual isentropic initialization, a much lower spurious relaxation toward the targeted solution is observed, which is due to the spatial resolution rather than approximated macroscopic quantities. The amplitude of the spurious waves can be further reduced by including an off-equilibrium part in the initial distribution functions.
Gauthier Wissocq, Jean-François Boussuge, Pierre Sagaut. Consistent vortex initialization for the athermal lattice Boltzmann method. Physical Review E , 2020, 101 (4), ⟨10.1103/PhysRevE.101.043306⟩. ⟨hal-02892501⟩
A Gineau, E. Longatte, D. Lucor, P. Sagaut. Macroscopic model of fluid structure interaction in cylinder arrangement using theory of mixture. Computers and Fluids, 2020, 202, pp.104499. ⟨10.1016/j.compfluid.2020.104499⟩. ⟨hal-03251640⟩ Plus de détails...
In the framework of the theory of mixture, the dynamic behaviour of solid cylinder bundles submitted to external hydrodynamic load exerted by surrounding viscous fluid flow is described. Mass conservation and momentum balance formulated on an elementary domain made of a given volume of mixture give rise to a system of coupled equations governing solid space-averaged displacement, fluid velocity and pressure provided that near-wall hydrodynamic load on each vibrating cylinder is expressed as a function of both fluid and solid space-averaged velocity fields. Then, the ability of the macroscopic model to reproduce over time an averaged flow surrounding vibrating cylinders in a large array in the context of small magnitude displacements is pointed out. Numerical solutions obtained on a two-dimensional configuration involving an array of several hundreds of cylinders subjected to an impulsional load are compared to those provided by averaged well-resolved microscopic-scale solutions. The relative error is less than 3% in terms of displacement magnitude and 5% for frequency delay. The proposed macroscopic model does not include any assumption on relative effect contributions to mechanical exchanges occurring in the full domain. Therefore it features interesting properties in terms of fluid solid interaction prediction capabilities. Moreover it contributes to a significant gain in terms of computational time and resources. Further developments are now required in order to extent the formulation to large magnitude displacements including three-dimensional effects. This could be recommended for investigations on fuel assembly vibration risk assessment in Pressure Water, Fast Breeder reactors at a whole core scale or any other large-scale mechanical system involving some kind of periodic geometry.
A Gineau, E. Longatte, D. Lucor, P. Sagaut. Macroscopic model of fluid structure interaction in cylinder arrangement using theory of mixture. Computers and Fluids, 2020, 202, pp.104499. ⟨10.1016/j.compfluid.2020.104499⟩. ⟨hal-03251640⟩
S. Guo, Yongliang Feng, Jérôme Jacob, F. Renard, Pierre Sagaut. An efficient lattice Boltzmann method for compressible aerodynamics on D3Q19 lattice. Journal of Computational Physics, 2020, 418, pp.109570. ⟨10.1016/j.jcp.2020.109570⟩. ⟨hal-03232070⟩ Plus de détails...
An efficient lattice Boltzmann (LB) model relying on a hybrid recursive regularization (HRR) collision operator on D3Q19 stencil is proposed for the simulation of three-dimensional high-speed compressible flows in both subsonic and supersonic regimes. An improved thermal equilibrium distribution function on D3Q19 lattice is derived to reduce the complexity of correcting terms. A simple shock capturing scheme and an upwind biased discretization of correction terms are implemented for supersonic flows with shocks. Mass and momentum equations are recovered by an efficient streaming, collision and forcing process on D3Q19 lattice. Then a non-conservative formulation of the entropy evolution equation is used, that is solved using a finite volume method. The proposed method is assessed considering the simulation of i) 2D isentropic vortex convection, ii) 3D non-isothermal acoustic pulse, iii) 2D supersonic flow over a bump, iv) 3D shock explosion in a box, v) 2D vortex interaction with shock wave, vi) 2D laminar flows over a flat plate at Ma of 0.5, 1.0 and 1.5.
S. Guo, Yongliang Feng, Jérôme Jacob, F. Renard, Pierre Sagaut. An efficient lattice Boltzmann method for compressible aerodynamics on D3Q19 lattice. Journal of Computational Physics, 2020, 418, pp.109570. ⟨10.1016/j.jcp.2020.109570⟩. ⟨hal-03232070⟩
Sylvia Wilhelm, Jérôme Jacob, Pierre Sagaut. A New Explicit Algebraic Wall Model for LES of Turbulent Flows Under Adverse Pressure Gradient. Flow, Turbulence and Combustion, 2020, 106 (1), pp.1-35. ⟨10.1007/s10494-020-00181-7⟩. ⟨hal-03231798⟩ Plus de détails...
A new explicit algebraic wall law for the Large Eddy Simulation of flows with adverse pressure gradient is proposed. This new wall law, referred as adverse pressure gradient power law (APGPL), is developed starting from the power-law of Werner and Wengle (Turbulent Shear Flows, vol 8, Springer, New York, pp 155-168, 1993) in order to mimic an implicit non-equilibrium log-law based on Afzal's law (Afzal, IUTAM Symposium on Asymptotic Methods for Turbulent Shear Flows at High Reynolds Numbers, Kluwer Academic Publishers, Bochum, pp 95-118, 1996). No iterative method is needed for the evaluation of the wall shear stress from the APGPL contrary to the majority of models available in the literature. The APGPL model relies on the definition of three modes: the equilibrium power-law is used in regions of no or favourable pressure gradient, the APGPL is used in regions of adverse pressure gradient, and no wall model is used in separated flow regions. This model is assessed via Large Eddy Simulations of flows involving adverse pressure gradient and boundary layer separation using the Lattice Boltzmann Method on uniform nested grids. The flow around a clean and iced NACA23012 airfoil at Reynolds numberRe=1.88 x 10(6) and the flow over the LAGOON landing gear at Re=1.59x10(6) are considered. Results are found in good agreement with those obtained by the non-equilibrium log-law and experimental and numerical data available in the literature.
Sylvia Wilhelm, Jérôme Jacob, Pierre Sagaut. A New Explicit Algebraic Wall Model for LES of Turbulent Flows Under Adverse Pressure Gradient. Flow, Turbulence and Combustion, 2020, 106 (1), pp.1-35. ⟨10.1007/s10494-020-00181-7⟩. ⟨hal-03231798⟩
Yongliang Feng, S. Guo, Jérôme Jacob, Pierre Sagaut. Solid wall and open boundary conditions in hybrid recursive regularized lattice Boltzmann method for compressible flows. Physics of Fluids, 2019, 31 (12), pp.126103. ⟨10.1063/1.5129138⟩. ⟨hal-02467965⟩ Plus de détails...
Complex geometries and open boundaries have been intensively studied in the nearly incompressible lattice Boltzmann method (LBM) framework. Therefore, only few boundary conditions for the high speed fully compressible LBM have been proposed. This paper deals with the definition of efficient boundary conditions for the compressible LBM methods, with the emphasis put on the newly proposed hybrid recursive regularized D3Q19 LBM (HRR-LBM) with applications to compressible aerodynamics. The straightforward simple extrapolation-based far-field boundary conditions, the characteristic boundary conditions, and the absorbing sponge layer approach are extended and estimated in the HRR-LBM for the choice of open boundaries. Moreover, a cut-cell type approach to handle the immersed solid is proposed to model both slip and no-slip wall boundary conditions with either isothermal or adiabatic behavior. The proposed implementations are assessed considering the simulation of (i) isentropic vortex convection with subsonic to supersonic inflow and outflow conditions, (ii) two-dimensional (2D) compressible mixing layer, (iii) steady inviscid transonic flow over a National Advisory Committee for Aeronautics (NACA) 0012 airfoil, (iv) unsteady viscous transonic flow over a NACA 0012 airfoil, and (v) three-dimensional (3D) transonic flows over a German Aerospace Center (DLR) F6 full aircraft configuration.
Yongliang Feng, S. Guo, Jérôme Jacob, Pierre Sagaut. Solid wall and open boundary conditions in hybrid recursive regularized lattice Boltzmann method for compressible flows. Physics of Fluids, 2019, 31 (12), pp.126103. ⟨10.1063/1.5129138⟩. ⟨hal-02467965⟩
Yongliang Feng, Pierre Boivin, Jérome Jacob, Pierre Sagaut. Hybrid recursive regularized lattice Boltzmann simulation of humid air with application to meteorological flows. Physical Review E , 2019. ⟨hal-02265484⟩ Plus de détails...
An extended version of the hybrid recursive regularized Lattice-Boltzmann model which incorporates external force is developed to simulate humid air flows with phase change mechanisms under the Boussinesq approximation. Mass and momentum conservation equations are solved by a regu-larized lattice Boltzmann approach well suited for high Reynolds number flows, whereas the energy and humidity related equations are solved by a finite volume approach. Two options are investigated to account for cloud formation in atmospheric flow simulations. The first option considers a single conservation equation for total water and an appropriate invariant variable of temperature. In the other approach, liquid and vapor are considered via two separated equations, and phase transition is accounted for via a relaxation procedure. The obtained models are then systematically validated on four well-established benchmark problems including a double diffusive Rayleigh Bénard convection of humid air, 2D and 3D thermal moist rising bubble under convective atmospheric environment as well as a shallow cumulus convection in framework of large-eddy simulation.
Yongliang Feng, Pierre Boivin, Jérome Jacob, Pierre Sagaut. Hybrid recursive regularized lattice Boltzmann simulation of humid air with application to meteorological flows. Physical Review E , 2019. ⟨hal-02265484⟩
G. Farag, Pierre Boivin, P. Sagaut. Interaction of two-dimensional spots with a heat releasing/absorbing shock wave: linear interaction approximation results. Journal of Fluid Mechanics, 2019, 871, pp.865-895. ⟨10.1017/jfm.2019.324⟩. ⟨hal-02142649⟩ Plus de détails...
The canonical interaction between a two-dimensional weak Gaussian disturbance (en-tropy spot, density spot, weak vortex) with an exothermic/endothermic planar shock wave is studied via the Linear Interaction Approximation. To this end, a unified framework based on an extended Kovasznay decomposition that simultaneously accounts for non-acoustic density disturbances along with a poloidal-toroidal splitting of the vorticity mode and for heat-release is proposed. An extended version of Chu's definition for the energy of disturbances in compressible flows encompassing multi-component mixtures of gases is also proposed. This new definition precludes spurious non-normal phenomena when computing the total energy of extended Kovasznay modes. Detailed results are provided for three cases, along with fully general expressions for mixed solutions that combine incoming vortical, entropy and density disturbances.
G. Farag, Pierre Boivin, P. Sagaut. Interaction of two-dimensional spots with a heat releasing/absorbing shock wave: linear interaction approximation results. Journal of Fluid Mechanics, 2019, 871, pp.865-895. ⟨10.1017/jfm.2019.324⟩. ⟨hal-02142649⟩
Isabelle Cheylan, Guillaume Fritz, Denis Ricot, Pierre Sagaut. Shape Optimization Using the Adjoint Lattice Boltzmann Method for Aerodynamic Applications. AIAA Journal, 2019, 57 (7), pp.2758-2773. ⟨10.2514/1.J057955⟩. ⟨hal-02468051⟩ Plus de détails...
The present work focuses on shape optimization using the lattice Boltzmann method applied to aerodynamic cases. The adjoint method is used to calculate the sensitivities of the drag force with respect to the shape of an object. The main advantage of the adjoint method is its cost, because it is independent from the number of optimization parameters. The approach used consists in developing a continuous adjoint of the primal problem discretized in space, time, and velocities. An adjoint lattice Boltzmann equation is thus found, which is solved using the same algorithms as in the primal problem. The test cases investigate new features compared to what exists in the literature, such as the derivation of the grid refinement models in the primal problem to obtain their adjoint counterparts, but also the derivation of a double-relaxation-time algorithm and the Ginzburg et al. interpolation at the wall ("Two-Relaxation-Time Lattice Boltzmann Scheme: About Parametrization, Velocity, Pressure and Mixed Boundary Conditions," Communications in Computational Physics, Vol. 3, No. 2, 2008, pp. 427-478). Regarding the unsteadiness of the primal problem, two methods differing in accuracy and computational effort are compared using a two-dimensional unsteady case. Finally, this first-of-a-kind adjoint solver is applied to a large-scale threedimensional turbulent case (the flow of air around a car at a speed of 130 km/h), which shows its usefulness in the industry.
Isabelle Cheylan, Guillaume Fritz, Denis Ricot, Pierre Sagaut. Shape Optimization Using the Adjoint Lattice Boltzmann Method for Aerodynamic Applications. AIAA Journal, 2019, 57 (7), pp.2758-2773. ⟨10.2514/1.J057955⟩. ⟨hal-02468051⟩
Gauthier Wissocq, Pierre Sagaut, Jean-François Boussuge. An extended spectral analysis of the lattice Boltzmann method: modal interactions and stability issues. Journal of Computational Physics, 2019, 380, pp.311-333. ⟨10.1016/j.jcp.2018.12.015⟩. ⟨hal-02176969⟩ Plus de détails...
An extension of the von Neumann linear analysis is proposed for the study of the discrete-velocity Boltzmann equation (DVBE) and the lattice Boltzmann (LB) scheme. While the standard technique is restricted to the investigation of the spectral radius and the dissipation and dispersion properties, a new focus is put here on the information carried by the modes. The technique consists in the computation of the moments of the eigenvectors and their projection onto the physical waves expected by the continuous linearized Navier-Stokes (NS) equations. The method is illustrated thanks to some simulations with the BGK (Bhatnagar-Gross-Krook) collision operator on the D2Q9 and D2V17 lattices. The present analysis reveals the existence of two kinds of modes: non-observable modes that do not carry any macroscopic information and observable modes. The latter may carry either a physical wave expected by the NS equations, or an unphysical information. Further investigation of modal interactions highlights a phenomenon called curve veering occurring between two observable modes: a swap of eigenvectors and dissipation rate is observed between the eigencurves. Increasing the Mach number of the mean flow yields an eigenvalue collision at the origin of numerical instabilities of the BGK model, arising from the error in the time and space discretization of the DVBE. (C) 2019 Elsevier Inc. All rights reserved.
Gauthier Wissocq, Pierre Sagaut, Jean-François Boussuge. An extended spectral analysis of the lattice Boltzmann method: modal interactions and stability issues. Journal of Computational Physics, 2019, 380, pp.311-333. ⟨10.1016/j.jcp.2018.12.015⟩. ⟨hal-02176969⟩
F. Muller, A. Burbeau, B.-J. Gréa, Pierre Sagaut. Minimum enstrophy principle for two-dimensional inviscid flows around obstacles. Physical Review E , 2019, 99 (2), ⟨10.1103/PhysRevE.99.023105⟩. ⟨hal-02176949⟩ Plus de détails...
Large-scale coherent structures emerging in two-dimensional flows can be predicted from statistical physics inspired methods consisting in minimizing the global enstrophy while conserving the total energy and circulation in the Euler equations. In many situations, solid obstacles inside the domain may also constrain the flow and have to be accounted for via a minimum enstrophy principle. In this work, we detail this extended variational formulation and its numerical resolution. It is shown from applications to complex geometries containing multiple circular obstacles that the number of solutions is enhanced, allowing many possibilities of bifurcations for the large-scale structures. These phase change phenomena can explain the downstream recombinations of the flow in rod-bundle experiments and simulations.
F. Muller, A. Burbeau, B.-J. Gréa, Pierre Sagaut. Minimum enstrophy principle for two-dimensional inviscid flows around obstacles. Physical Review E , 2019, 99 (2), ⟨10.1103/PhysRevE.99.023105⟩. ⟨hal-02176949⟩
F. Muller, A. Burbeau, B.-J. Gréa, Pierre Sagaut. Minimum enstrophy principle for two-dimensional inviscid flows around obstacles. Physical Review E , American Physical Society (APS), 2019, 99 (2), ⟨10.1103/PhysRevE.99.023105⟩. ⟨hal-02176949⟩ Plus de détails...
Large-scale coherent structures emerging in two-dimensional flows can be predicted from statistical physics inspired methods consisting in minimizing the global enstrophy while conserving the total energy and circulation in the Euler equations. In many situations, solid obstacles inside the domain may also constrain the flow and have to be accounted for via a minimum enstrophy principle. In this work, we detail this extended variational formulation and its numerical resolution. It is shown from applications to complex geometries containing multiple circular obstacles that the number of solutions is enhanced, allowing many possibilities of bifurcations for the large-scale structures. These phase change phenomena can explain the downstream recombinations of the flow in rod-bundle experiments and simulations.
F. Muller, A. Burbeau, B.-J. Gréa, Pierre Sagaut. Minimum enstrophy principle for two-dimensional inviscid flows around obstacles. Physical Review E , American Physical Society (APS), 2019, 99 (2), ⟨10.1103/PhysRevE.99.023105⟩. ⟨hal-02176949⟩
Lucie Merlier, Jérome Jacob, Pierre Sagaut. Lattice-Boltzmann large-eddy simulation of pollutant dispersion in complex urban environment with dense gas effect: Model evaluation and flow analysis. Building and Environment, 2019, 148, pp.634-652. ⟨10.1016/j.buildenv.2018.11.009⟩. ⟨hal-02176936⟩ Plus de détails...
The goal of this study is to assess the performance of an innovative Lattice Boltzmann (LB) - Large Eddy Simulation (LES) approach in simulating neutral and stratified pollutant dispersion in complex urban environments. Different simulations are performed for the central area of Paris, accounting for continuous neutral or non-neutral gas releases from a circular source located in both channeled or confined flows. Predicted concentrations are compared with detailed wind tunnel measurements from the MODITIC project (FFI, 2016). Results exhibit a good qualitative and quantitative agreement between numerical and experimental data for the different configurations studied. All the estimated quality metrics match acceptance criteria. In addition, it is shown that the new LBM LES approach is able to capture and highlight the key turbulent mechanisms underlying dispersion process in and above urban areas. Hence, being based on extensive and detailed simulations and quality assurance studies, this paper highlights that the developed approach is well suited to address urban dispersion issues, including accidental chemical releases and short term exposure problems. Such results are particularly valuable to support the design and use of fast response dispersion models.
Lucie Merlier, Jérome Jacob, Pierre Sagaut. Lattice-Boltzmann large-eddy simulation of pollutant dispersion in complex urban environment with dense gas effect: Model evaluation and flow analysis. Building and Environment, 2019, 148, pp.634-652. ⟨10.1016/j.buildenv.2018.11.009⟩. ⟨hal-02176936⟩
Shahram Khazaie, Xun Wang, Dimitri Komatitsch, Pierre Sagaut. Uncertainty quantification for acoustic wave propagation in a shallow water environment. Wave Motion, 2019, 91, pp.102390. ⟨10.1016/j.wavemoti.2019.102390⟩. ⟨hal-02467993⟩ Plus de détails...
Sound wave propagation in a shallow water environment is complex due to e.g. the uncertainties of sound speed profile being inhomogeneous and imprecisely measured, the bottom reflections, etc. The propagation and influence of several uncertainty parameters are quantified in this paper. A four-layer model, which can approximately represent a wide range of shallow water environments, is considered; six parameters representing sound speed profile and water depth are considered as random variables. We investigate how the wave field (pressure) in this model is influenced by these uncertainties. For this purpose, the sound field is computed for different realizations of the random variables, when the medium is excited with sources whose frequencies are appropriate, for example, for marine seismic exploration applications. Since classical Monte Carlo methods require a huge sample size to converge, we use three surrogate modeling techniques (Kriging, Polynomial Chaos, and Polynomial Chaos-based Kriging). The proposed methods require much smaller sample sizes, which makes the uncertainty quantification (UQ) possible. Wavelength-to-depth ratio (lambda/d) is introduced as the key parameter that defines the degree of interaction (reflection and transmission) of the sound waves with the boundaries of the shallow water waveguide. The results show that for small and large values of lambda/d, the wave field is more sensitive to the variations of the water depth and the velocity of the bottom layer, respectively. The robustness (precision) of the surrogate models is shown to decrease for lower values of lambda/d. The proposed UQ methodology can be used for more complicated underwater environments; it is even more advantageous because it can efficiently deal with a large number of model uncertainty parameters and identify the most influential ones.
Shahram Khazaie, Xun Wang, Dimitri Komatitsch, Pierre Sagaut. Uncertainty quantification for acoustic wave propagation in a shallow water environment. Wave Motion, 2019, 91, pp.102390. ⟨10.1016/j.wavemoti.2019.102390⟩. ⟨hal-02467993⟩
Benjamin Bugeat, Jean-Camille Chassaing, Jean-Christophe Robinet, Pierre Sagaut. 3D global optimal forcing and response of the supersonic boundary layer. Journal of Computational Physics, 2019, 398, pp.108888. ⟨10.1016/j.jcp.2019.108888⟩. ⟨hal-02462825⟩ Plus de détails...
3D optimal forcing and response of a 2D supersonic boundary layer are obtained by computing the largest singular value and the associated singular vectors of the global resolvent matrix. This approach allows to take into account both convective-type and component-type non-normalities responsible for the non-modal growth of perturbations in noise selective amplifier flows. It is moreover a fully non-parallel approach that does not require any particular assumptions on the baseflow. The numerical method is based on the explicit calculation of the Jacobian matrix proposed by Mettot et al. [1] for 2D perturbations. This strategy uses the numerical residual of the compressible Navier-Stokes equations imported from a finite-volume solver that is then linearised employing a finite difference method. Extension to 3D perturbations, which are expanded into modes of wave number, is here proposed by decomposing the Jacobian matrix according to the direction of the derivatives contained in its coefficients. Validation is performed on a Blasius boundary layer and a supersonic boundary layer, in comparison respectively to global and local results. Application of the method to a boundary layer at M = 4.5 recovers three regions of receptivity in the frequency-transverse wave number space. Finally, the energy growth of each optimal response is studied and discussed.
Benjamin Bugeat, Jean-Camille Chassaing, Jean-Christophe Robinet, Pierre Sagaut. 3D global optimal forcing and response of the supersonic boundary layer. Journal of Computational Physics, 2019, 398, pp.108888. ⟨10.1016/j.jcp.2019.108888⟩. ⟨hal-02462825⟩
Yongliang Feng, Pierre Boivin, Jérome Jacob, Pierre Sagaut. Hybrid recursive regularized thermal lattice Boltzmann model for high subsonic compressible flows. Journal of Computational Physics, 2019, 394, pp.82-99. ⟨hal-02142837⟩ Plus de détails...
A thermal lattice Boltzmann model with a hybrid recursive regularization (HRR) collision operator is developed on standard lattices for simulation of subsonic and sonic compressible flows without shock. The approach is hybrid: mass and momentum conservation equations are solved using a lattice Boltzmann solver, while the energy conservation is solved under entropy form with a finite volume solver. The defect of Galilean invariance related to Mach number is corrected by the third order equilibrium distribution function , supplemented by an additional correcting term and hybrid recursive regularization. The proposed approach is assessed considering the simulation of i) an isentropic vortex convection, ii) a two dimensional acoustic pulse and iii) non-isothermal Gaussian pulse with Ma number in range of 0 to 1. Numerical simulations demonstrate that the flaw in Galilean invari-ance is effectively eliminated by the compressible HRR model. At last, the compressible laminar flows over flat plate at Ma number of 0.3 and 0.87, Reynolds number of 10 5 are considered to validate the capture of viscous and diffusive effects.
Yongliang Feng, Pierre Boivin, Jérome Jacob, Pierre Sagaut. Hybrid recursive regularized thermal lattice Boltzmann model for high subsonic compressible flows. Journal of Computational Physics, 2019, 394, pp.82-99. ⟨hal-02142837⟩
Lucie Merlier, Jérôme Jacob, Pierre Sagaut. Lattice-Boltzmann Large-Eddy Simulation of pollutant dispersion in street canyons including tree planting effects. Atmospheric Environment, 2018, 195, pp.89-103. ⟨10.1016/j.atmosenv.2018.09.040⟩. ⟨hal-02114676⟩ Plus de détails...
This study assesses the performance of a large eddy simulation (LES) based on the lattice Boltzmann method (LBM) in predicting near field dispersion in street canyons with tree planting. Based on a benchmark test case benefiting from wind tunnel measurements (CODASC), this study qualitatively and quantitatively discusses the prediction of traffic-induced pollutant concentration with respect to several reference studies. It also analyses the physics of the flow and concentration fields. Although the problem might seem rather simple, the flow is highlighted to be strongly three dimensional and transient. These properties enhance pollutant dispersion in the empty street canyon but air flow velocity and turbulence intensity tend to decrease in tree crowns. This effect of trees increases both mean and peak concentration levels at pedestrian level, which may be problematic in cities with dense traffic. These results show that LBM-LES is particularly well suited to study dispersion problems towards the development of more breathable cities.
Lucie Merlier, Jérôme Jacob, Pierre Sagaut. Lattice-Boltzmann Large-Eddy Simulation of pollutant dispersion in street canyons including tree planting effects. Atmospheric Environment, 2018, 195, pp.89-103. ⟨10.1016/j.atmosenv.2018.09.040⟩. ⟨hal-02114676⟩
Jérôme Jacob, Orestis Malaspinas, Pierre Sagaut. A new hybrid recursive regularised Bhatnagar–Gross–Krook collision model for Lattice Boltzmann method-based large eddy simulation. Journal of Turbulence, 2018, pp.1 - 26. ⟨10.1080/14685248.2018.1540879⟩. ⟨hal-02114308⟩ Plus de détails...
A new Lattice Boltzmann collision model for large eddy simulation (LES) of weakly compressible flows is proposed. This model, referred to as the Hybrid Recursive Regularised Bhatnagar-Gross-Krook (HRR-BGK) model, is based on a modification of previously existing regularised collision models defined with the BGK Lattice Boltzmann method (LBM) framework. By hybridising the computation of the velocity gradient with an adequate Finite Difference scheme when reconstructing the non-equilibrium parts of the distribution function , a hyperviscosity term is introduced in the momentum equation, whose amplitude can be explicitly tuned via a weighting parameter. A dynamic version of the HRR-BGK is also proposed, in which the control parameter is tuned at each grid point and each time step in order to recover an arbitrarily fixed total dissipation. This new collision model is assessed for both explicit and implicit LES considering the flow around a circular cylinder at Re = 3900. The dynamic HRR-BGK is observed to yield very accurate results when equipped with Vreman's subgrid model to compute the target dissipation.
Jérôme Jacob, Orestis Malaspinas, Pierre Sagaut. A new hybrid recursive regularised Bhatnagar–Gross–Krook collision model for Lattice Boltzmann method-based large eddy simulation. Journal of Turbulence, 2018, pp.1 - 26. ⟨10.1080/14685248.2018.1540879⟩. ⟨hal-02114308⟩
Maria-Vittoria Salvetti, Marcello Meldi, Luca Bruno, Pierre Sagaut. Reliability of Large-Eddy Simulations: Benchmarking and Uncertainty Quantification. Direct and Large-Eddy Simulation X, 24, pp.15-23, 2018. ⟨hal-02115804⟩ Plus de détails...
Maria-Vittoria Salvetti, Marcello Meldi, Luca Bruno, Pierre Sagaut. Reliability of Large-Eddy Simulations: Benchmarking and Uncertainty Quantification. Direct and Large-Eddy Simulation X, 24, pp.15-23, 2018. ⟨hal-02115804⟩
Yong-Liang Feng, Shao-Long Guo, Wen-Quan Tao, Pierre Sagaut. Regularized thermal lattice Boltzmann method for natural convection with large temperature differences. International Journal of Heat and Mass Transfer, 2018, 125, pp.1379-1391. ⟨10.1016/j.ijheatmasstransfer.2018.05.051⟩. ⟨hal-02114047⟩ Plus de détails...
A new thermal lattice Boltzmann (LB) method is proposed for the simulation of natural convection with large temperature differences and high Rayleigh number. A regularization procedure is developed on LB equation with a third order expansion of equilibrium distribution functions, in which a temperature term is involved to recover the equation of state for perfect gas. A hybrid approach is presented to couple mass conservation equation, momentum conservation equations and temperature evolution equation. A simple and robust non-conservative form of temperature transport equation is adopted and solved by the finite volume method. A comparison study between classical Double Distribution Function (DDF) model and the hybrid finite volume model with different integration schemes is presented to demonstrate both consistency and accuracy of hybrid models. The proposed model is assessed by simulating several test cases, namely the two-dimensional non-Boussinesq natural convection in a square cavity with large horizontal temperature differences and two unsteady natural convection flows in a tall enclosure at high Rayleigh number. The present method can accurately predict both the steady and unsteady non-Boussinesq convection flows with significant heat transfer. For unsteady natural convection, oscillations with chaotic feature can be well captured in large temperature gradient conditions.
Yong-Liang Feng, Shao-Long Guo, Wen-Quan Tao, Pierre Sagaut. Regularized thermal lattice Boltzmann method for natural convection with large temperature differences. International Journal of Heat and Mass Transfer, 2018, 125, pp.1379-1391. ⟨10.1016/j.ijheatmasstransfer.2018.05.051⟩. ⟨hal-02114047⟩
Journal: International Journal of Heat and Mass Transfer
Romain Dupuis, Jean-Christophe Jouhaud, Pierre Sagaut. Surrogate Modeling of Aerodynamic Simulations for Multiple Operating Conditions Using Machine Learning. AIAA Journal, 2018, 56 (9), pp.3622-3635. ⟨10.2514/1.J056405⟩. ⟨hal-02113987⟩ Plus de détails...
This paper describes a methodology, called local decomposition method, which aims at building a surrogate model based on steady turbulent aerodynamic fields at multiple operating conditions. The various shapes taken by the aerodynamic fields due to the multiple operation conditions pose real challenges as well as the computational cost of the high-fidelity simulations. The developed strategy mitigates these issues by combining traditional surrogate models and machine learning. The central idea is to separate the solutions with a subsonic behavior from the transonic and high-gradient solutions. First, a shock sensor extracts a feature corresponding to the presence of discontinuities, easing the clustering of the simulations by an unsupervised learning algorithm. Second, a supervised learning algorithm divides the parameter space into subdomains, associated to different flow regimes. Local reduced-order models are built on each subdomain using proper orthogonal decomposition coupled with a multivariate interpolation tool. Finally, an improved resampling technique taking advantage of the subdomain decomposition minimizes the redundancy of sampling. The methodology is assessed on the turbulent two-dimensional flow around the RAE2822 transonic airfoil. It exhibits a significant improvement in terms of prediction accuracy for the developed strategy compared with the classical method of surrogate modeling.
Romain Dupuis, Jean-Christophe Jouhaud, Pierre Sagaut. Surrogate Modeling of Aerodynamic Simulations for Multiple Operating Conditions Using Machine Learning. AIAA Journal, 2018, 56 (9), pp.3622-3635. ⟨10.2514/1.J056405⟩. ⟨hal-02113987⟩
Jérôme Jacob, Pierre Sagaut. Wind comfort assessment by means of large eddy simulation with lattice Boltzmann method in full scale city area. Building and Environment, 2018, 139, pp.110 - 124. ⟨10.1016/j.buildenv.2018.05.015⟩. ⟨hal-02114339⟩ Plus de détails...
Large-eddy simulations based on the Lattice-Boltzmann method of the flow in a realistic, full scale urban area are performed to compare several wind comfort criteria. It is observed that popular criteria for pedestrian comfort lead to very different conclusions, due to the access to high spatio-temporal resolution data. Different mixed strategies based on the combination of several criteria are proposed and compared to enhance pedestrian wind comfort assessment in practical cases.
Jérôme Jacob, Pierre Sagaut. Wind comfort assessment by means of large eddy simulation with lattice Boltzmann method in full scale city area. Building and Environment, 2018, 139, pp.110 - 124. ⟨10.1016/j.buildenv.2018.05.015⟩. ⟨hal-02114339⟩
Antoine Briard, Benoît-Joseph Gréa, Vincent Mons, Claude Cambon, Pierre Sagaut, et al.. Advanced spectral anisotropic modelling for shear flows. Journal of Turbulence, 2018, 19 (7), pp.570-599. ⟨10.1080/14685248.2018.1478092⟩. ⟨hal-02112204⟩ Plus de détails...
In this work, the spectral modelling developed in MCS [Mons, Cambon, Sagaut. A spectral model for homogeneous shear-driven anisotropic turbulence in terms of spherically-averaged descriptors. J Fluid Mech. 2016;788:147-182] for shear-driven turbulence is further analysed and then improved. First, using self-similarity arguments, it is shown that the asymptotic kinetic energy exponential growth rate is independent of the large scales infrared slope sigma, with, unlike unstably stratified homogeneous turbulence where strongly depends on sigma. The MCS model relies on the truncation at the second order of the spectral two-point velocity correlation expansion into spherical harmonics. The expansion is here pursued at the next even order, the fourth one: the noteworthy consequence is that is decreased compared to MCS and is thus closer to values obtained in direct numerical simulations and experiments. Finally, some analytical considerations about odd-order contributions in the expansion of polarisation anisotropy are proposed.
Antoine Briard, Benoît-Joseph Gréa, Vincent Mons, Claude Cambon, Pierre Sagaut, et al.. Advanced spectral anisotropic modelling for shear flows. Journal of Turbulence, 2018, 19 (7), pp.570-599. ⟨10.1080/14685248.2018.1478092⟩. ⟨hal-02112204⟩
Jean-François Boussuge, Mathieu Catchirayer, J.-F Boussuge, Pierre Sagaut, Marc Montagnac, et al.. Extended integral wall-model for large-eddy simulations of compressible wall-bounded turbulent flows. Physics of Fluids, 2018, 30 (6), pp.065106. ⟨10.1063/1.5030859⟩. ⟨hal-02112710⟩ Plus de détails...
Wall-modeling is required to make large-eddy simulations of high-Reynolds number wall-bounded turbulent flows feasible in terms of computational cost. Here, an extension of the integral wall-model for large-eddy simulations (iWMLESs) for incompressible flows developed by Yang et al. ["Integral wall model for large eddy simulations of wall-bounded turbulent flows," Phys. Fluids 27(2), 025112 (2015)] to compressible and isothermal flows is proposed and assessed. The iWMLES approach is analogous to the von Karman-Pohlhausen integral method for laminar flows: the velocity profile is parameterized, and unknown coefficients are determined by matching boundary conditions obeying the integral boundary layer momentum equation. It allows non-equilibrium effects such as pressure gradient and convection to be included at a computing cost similar to analytical wall-models. To take into account density variations and temperature gradients, the temperature profile is also parameterized and the integral compressible boundary layer energy equation is considered. Parameterized profiles are based on the usual logarithmic wall functions with corrective terms to extend their range of validity. Instead of solving a set of differential equations as wall-models based on the thin boundary layer equation approach, a simple linear system is solved. The proposed wall-model is implemented in a finite-volume cell-centered structured grid solver and assessed on adiabatic and isothermal plane channel flows at several friction Reynolds and Mach numbers. For low Mach number cases, mean profiles, wall fluxes, and turbulent fluctuations are in agreement with those of Direct Numerical Simulation (DNS). For supersonic flows, the results are in good agreement with the DNS data, especially the mean velocity quantities and the wall friction, while standard analytical wall-models show their limits.
Jean-François Boussuge, Mathieu Catchirayer, J.-F Boussuge, Pierre Sagaut, Marc Montagnac, et al.. Extended integral wall-model for large-eddy simulations of compressible wall-bounded turbulent flows. Physics of Fluids, 2018, 30 (6), pp.065106. ⟨10.1063/1.5030859⟩. ⟨hal-02112710⟩
Sylvia Wilhelm, Jérôme Jacob, Pierre Sagaut. An explicit power-law-based wall model for lattice Boltzmann method–Reynolds-averaged numerical simulations of the flow around airfoils. Physics of Fluids, 2018, 30 (6), pp.065111. ⟨10.1063/1.5031764⟩. ⟨hal-02116210⟩ Plus de détails...
In this paper, an explicit wall model based on a power-law velocity profile is proposed for the simulation of the incompressible flow around airfoils at high Reynolds numbers. This wall model is particularly suited for the wall treatment involved in Cartesian grids. Moreover, it does not require an iterative procedure for the friction velocity determination. The validation of this power-law wall model is assessed for Reynolds-averaged Navier-Stokes simulations of the flow around a two-dimensional airfoil using the lattice Boltzmann approach along with the Spalart-Allmaras turbulence model. Good results are obtained for the prediction of the aerodynamic coefficients and the pressure profiles at two Reynolds numbers and several angles of attack. The explicit power-law is thus well suited for a simplified near-wall treatment at high Reynolds numbers using Cartesian grids.
Sylvia Wilhelm, Jérôme Jacob, Pierre Sagaut. An explicit power-law-based wall model for lattice Boltzmann method–Reynolds-averaged numerical simulations of the flow around airfoils. Physics of Fluids, 2018, 30 (6), pp.065111. ⟨10.1063/1.5031764⟩. ⟨hal-02116210⟩
Myriam Slama, Cédric Leblond, Pierre Sagaut. A Kriging-based elliptic extended anisotropic model for the turbulent boundary layer wall pressure spectrum. Journal of Fluid Mechanics, 2018, 840, pp.25 - 55. ⟨10.1017/jfm.2017.810⟩. ⟨hal-02115966⟩ Plus de détails...
The present study addresses the computation of the wall pressure spectrum for a turbulent boundary layer flow without pressure gradient, at high Reynolds numbers, using a new model, the Kriging-based elliptic extended anisotropic model (KEEAM). A space–time solution to the Poisson equation for the wall pressure fluctuations is used. Both the turbulence–turbulence and turbulence–mean shear interactions are taken into account. It involves the mean velocity field and space–time velocity correlations which are modelled using Reynolds stresses and velocity correlation coefficients. We propose a new model, referred to as the extended anisotropic model, to evaluate the latter in all regions of the boundary layer. This model is an extension of the simplified anisotropic model of Gavin (PhD thesis, 2002, The Pennsylvania State University, University Park, PA) which was developed for the outer part of the boundary layer. It relies on a new expression for the spatial velocity correlation function and new parameters calibrated using the direct numerical simulation results of Sillero et al. (Phys. Fluids, vol. 26, 2014, 105109). Spatial correlation coefficients are related to space–time coefficients with the elliptic model of He & Zhang (Phys. Rev. E, vol. 73, 2006, 055303). The turbulent quantities necessary for the pressure computation are obtained by Reynolds-averaged Navier–Stokes solutions with a Reynolds stress turbulence model. Then, the pressure correlations are evaluated with a self-adaptive sampling strategy based on Kriging in order to reduce the computation time. The frequency and wavenumber–frequency wall pressure spectra obtained with the KEEAM agree well with empirical models developed for turbulent boundary layer flows without pressure gradient.
Myriam Slama, Cédric Leblond, Pierre Sagaut. A Kriging-based elliptic extended anisotropic model for the turbulent boundary layer wall pressure spectrum. Journal of Fluid Mechanics, 2018, 840, pp.25 - 55. ⟨10.1017/jfm.2017.810⟩. ⟨hal-02115966⟩
