**A. Sergent****, Y. Duguet, P. Le Quéré, Y. Fraigneau (P2I), L. Pastur (AERO), B. Podvin (AERO), C. Tenaud (AERO), **M. Belkadi, L. Cadet, A. Castillo, Z. Gao, C. Garnier, L. Oteski, E. Saikali, H.-L. Tran, Ya. Wang, Yi. Wang

We study flows whose motion is due to temperature or mass gradients. Using direct numerical simulations (DNS), Large Eddy Simulation (LES) or (semi-) analytical approaches, we address the issues of bifurcation tracking and route to chaos, mixing, turbulent flows and interface modelling in a semi-open environment. Comparison with experimental data is widely used in the validation of physical models and in the choice of numerical and analytical modelling.

### Transition to chaos in differentially heated convection flows

This topic aims at the numerical determination of successive bifurcations leading to a chaotic regime. The air flows considered verify the Boussinesq hypothesis and are forced by a finite temperature difference between two walls parallel to gravity.

For a bi-periodic flow between plates, the development of transverse instabilities is limited by the frequency lengths. In highly horizontally confined geometry, the successive bifurcations leading to chaos have been described as well as the associated flow. 3D spectral DNS have shown the appearance of structures composed of transverse main rolls connected by oblique counter-rotating secondary rolls. At higher Rayleigh numbers, a chaotic flow appears through the competition of two mechanisms: a cascade of period doubling leading to intermittency and a spatial modulation mechanism of the rolls (see figure below).

*Doubling-period cascade for a computation domain restricted to a single structure. Left: bifurcation diagram obtained by tracking the local maxima of a time series of temperature at a point (0.038, 0.097, 0.983). Right: temporal evolution of the flow in the vertical median plane at Ra=12 380 for three separate moments of the base oscillation period (T=28)[Gao et al., PRE (2015)].*

The development of transverse instabilities is studied using linear analysis (Arnoldi) that allows the most unstable wavelengths to be calculated in order to establish a computational domain adapted to the description of the scenario of the appearance of the chaotic regime in slightly confined geometries (PhD thesis of Z. Gao 2010-2013, funding ED SMAER UPMC, collab. S. Xin at CETHIL and L. Tuckerman at PMMH).

### Mixing in two-dimensional convective flows

In the case of a differentially heated two-dimensional convective flow, for an aspect ratio close to 2, the transition to chaos is more complex and involves several branches of quasiperiodic regimes with different symmetry properties. The aim of the study is to gain a detailed understanding of the homogenization of a passive non-diffusive scalar within a model convective flow between two vertical plates heated at different temperatures (see figure below).

*(Left) Lagrangian trajectories of passive tracers within a differentially heated two-dimensional cavity with an aspect ratio of 2, Ra=1.625.10 ^{8}. Periodic regime. (Right) Mixing rate as a function of the Rayleigh number for different post-processing parameters (Oteski et al., JFM (2014)).*

The approach, derived from Hamiltonian systems, consists of an analysis of the geometry of the phase space associated with the motion of a passive tracer. The analysis reveals that, when the Rayleigh number increases beyond the first Hopf bifurcation, the system begins to mix partially via the destabilization of homoclinic trajectories (thesis L. Oteski 2012-2015, funding from the EADS Foundation grant, ED SMEMAG). The mixing slowly evolves towards a homogeneous mixing as the material barriers (KAM cores) resonate with the frequency of the flow. The non-hyperbolic nature of the mixing is evidenced by the algebraic decay over time of the variance of the concentration.

### Turbulent convection: Rayleigh-Bénard convection and radiation-convection coupling

The Rayleigh-Bénard turbulent convection is characterized by strong interactions between small and large scales, the latter forming a large-scale circulation (LSC), subject to intermittent reversals. Based on DNS performed over very long physical times, we studied the dynamics of reversals in a square cavity using two complementary approaches (see figure below): (i) characterization of the dynamics of the most energetic structures by POD analysis leading to the development of reduced models based on the first 3 or 5 POD modes (collab. B. Podvin), (ii) identification of a characteristic cycle of standard reversals by statistical analysis after separation of the standard and aborted reversals (PhD thesis of Castillo Castellanos 2013-2017, financing ED SMAER UPMC, collab. Mr Rossi IJLRA). The same approach is used to analyze with the SUNFLUIDH code how the forcing of small scales by roughness modifies the spatiotemporal dynamics of the LSC (PhD thesis of M. Belkadi, EMP financing Algiers 2016-2019, ED SMAER UPMC, collab. B. Podvin, Y. Fraigneau). A comparison with experimental results will be carried out as part of a PEPS CNRS Energie 2018 (collab. F. Chilla and J. Salort Lab. Phys. ENS Lyon).

*Average reversal cycle for (Ra = 5 10 ^{7}, Pr = 3): three successive steps: relaxation (orange), acceleration (green) and accumulation (blue). (Left) Temporal evolution of the angular momentum L_{2D}, the kinetic energy E_{kin}, and the potential available energy E_{apot} for two successive reversals. (Right) Special moments (i-x) of the generic cycle: reference height fields y_{r} with respect to the equivalent stratified state and current lines (conditional means)[Castillo et al., JFM (2016)].*

The use of a radiation model of a semi-transparent medium by the discrete ordinate method combined with a compact real gas model has made it possible to develop a massively parallel finite volume code (ROCOCO code, APP 2017 repository) for the convection-radiation coupling, thus giving access to flows with a high level of turbulence (PhD thesis of L. Cadet 2013-2015, ED SI-MMEA U La Rochelle funding, collab. P. Joubert Lasie and D. Lemonnier and D. Saury PPRIME). After validation on test cases from the literature, the influence of parietal emissivities on turbulent convection in a differentially heated cavity was demonstrated in the case of a convection/parietal radiation coupling. Then direct numerical simulation results for a semi-transparent gas were compared with experimental data. The radiation module has been introduced in SUNFLUIDH. This work continues with Yi Wang's thesis (financed by ED SI-MMEA U La Rochelle, 2016-2019, collab. P. Joubert Lasie, D. Lemonnier and D. Saury PPRIME) for greenhouse gas injections (forced plumes) in confined environments.

### Convection in a semi-confined environment: chimneys and binary plumes in a cooled cavity

The vertical channel is a simplified model of a natural convection open flow such as ventilated cavities or chimneys. Numerical modelling of this type of flows can be carried out either by simulating the channel and its external environment (at a prohibitive cost in 3D), or by considering the channel alone. The main difficulty lies in the definition of the cavity / environment interfaces.

Several sets of boundary conditions exist in the literature. A comparison benchmark organized by the SFT thermal community illustrates the wide dispersion of results. However, there is no reference solution to assess the quality of the approximation obtained, while the comparison between numerical and experimental results remains difficult. We have established reference numerical solutions for a chimney immersed in an infinite isothermal environment at rest modeled by a large cavity. The reference solution fields highlight the complexity of the input/output flows at the interfaces. A new set of boundary conditions has been proposed as part of the Boussinesq approximation, approaching the reference solutions more satisfactorily (PhD thesis of C. Garnier 2011-2014, ENS funding, ED SMAER UPMC).

The use of hydrogen (e. g. fuel cells) presents significant risks due to the flammability of the air-hydrogen mixture. To model a typical accident situation, we place ourselves in the case of a mixture of slightly turbulent helium-air injected locally into a ventilated cavity, for which a stable homogeneous zone is established at the top of the cavity. We address this problem through LES / DNS simulations (see figure below).

*Effect of the size variation of the external computing domain on the velocity field in the median vertical plane. Left: LES in small domain, middle: LES with a computation domain including an external volume, right: PIV measurements[Saikali et al., ICHS (2018)].*

In comparison with experimental results obtained at CEA, we have shown the requirement to model part of the external environment (PhD thesis of H.-L. Tran 2009-2013 and of E. Saikali 2014-2017, CEA financing, collab. C. Tenaud (AERO), Y. Fraigneau (P2i), G. Bernard-Michel CEA Saclay (DEN/DM2S/STMF). The project continues with the establishment of new interface boundary conditions in a Low Mach formulation (post-doc of A. Castillo-Castellanos, funded by Labex LASIPS 2017-18). In the longer term, the objective of this work is to characterize the dispersion, mixing and entrainment properties of this type of flow, in comparison with the experimental results and theoretical models used by industrials (PhD thesis of Ya. Wang, CEA funding, 2017-2020, collab. Y. Fraigneau (P2i), G. Bernard-Michel CEA Saclay DEN).