**Participants : **A. Sergent, D. Băltean-Carlès, J. Chergui (P2I), V. Daru (AERO), I. Delbende, Y. Fraigneau (P2I), D. Juric (TSF), P. Le Quéré, C.T. Pham, B. Podvin (AERO), C. Weisman.* Doctorants : L. Cadet, A. Castillo, Z.L. Gao, C. Garnier, L. Ma, H.L. Tran.*

Several fundamental models of convection flows are analyzed numerically, theoritically or experimentally, such as buoyant flows, thermal convection coupled with acoustic effects or with evaporation (Leidenfrost liquid cylinder, moving contact line). Topics as instabilities, bifurcations scenarios, turbulence, or boundary conditions modelling are tackled.

#### Korteweg-De Vries solitons in Leidenfrost liquid cylinder

When a drop of volatile liquid is deposited on a very hot surface, it can levitate above its own vapor. This effect is called the Leidenfrost effect. Leidenfrost drops are limited in volume, and beyond a critical volume, chimneys appear inside the liquid puddle. However we show experimentally that this volume limitation can be circumvented by creating large liquid volumes using curved substrates. By considering long straight channels with curved bottom, we show that solitary surface waves can propagate along the channel and we identify them as Korteweg-de Vries solitons of negative amplitude, for which capillary effects dominate owing to an effective reduction of gravity. Their theoretical properties can be recovered both analytically and experimentally. [C.-T. Pham in collaboration with S. Perrard and L. Deike at University Paris Diderot].

#### Moving contact line under evaporation

Understanding the dynamics of a moving contact line in the presence of evaporation is crucial, for instance for coating processes using drying solutions. The problem is complex since it involves both hydrodynamic and evaporative singularities at the contact lines. We have porposed a model for a moving contact line under evaporation in partial or complete wetting situations, taking into account the divergent evaporative flux near the contact line. Analytical calculations together with numerical simulations lead to a generalization of the Cox-Voinov wetting laws that relate the apparent macroscopic contact angle to the speed of the contact line. In the case of complete wetting, a disjunction pressure term due to van der Waals interactions between the substrate and the liquid is considered, and the existence of a precursor film is shown. Its length and thickness are computed together with the dynamics of an evaporating wetting droplet, which depend on Hamaker constant and evaporative flux [collab. C.-T. Pham with F. Lequeux at ESPCI and L. Limat at University Paris Diderot].

#### Thermal convection and thermoacoustics

Standing wave thermoacoustic engines consist of a resonating tube, closed at one end, with a load at the other end, inside which heat exchangers and a stack made of parallel plates are placed. One heat exchanger is connected to a hot source, the other to a cold source. The combined effect of pressure fluctuations and oscillating heat exchange in the boundary layers near the stack plates provide a heat engine effect. A multiple scale analysis allows for the global compressible flow problem to be reduced to a dynamically incompressible problem in the active cell, with boundary conditions obtained from linear acoustics in the resonator, providing a proper asymptotic approximation in the low Mach number limit, under the assumption that the flow sweeps a length comparable with the stack length (collab. D. Băltean-Carlès, P. Le Quéré, C. Weisman with L. Bauwens, University of Calgary). Direct numerical simulations of the flow in the active cell, coupled with an exact solution of one-dimensional acoustics in the resonators, are performed using a two-dimensional, time-dependent finite volume code. If a sufficiently large temperature difference is imposed between the heat exchangers, initial pressure perturbations are amplified, the fluid starts oscillating and amplitudes grow, up to the point when the engine reaches a stationary periodic regime (Algerian thesis of O. Hireche). The influence of the load model on the wave saturation is studied, yielding a specific load range for saturation at levels comparable with experiments (Ph.D. thesis of L. Ma). The critical temperature for the complete simplified engine is studied as a function of the load and of the position of the active cell inside the resonator. Due to the temperature difference between the hot and cold heat exchangers, a longitudinal temperature/density gradient exists, which associated to the acceleration of the oscillating flow results in a strong instability, present only during part of the cycle. The vortex dynamics associated with cavities and step-like cross-section changes also show interesting features, notably interplays between temperature gradients and accelerations.

*Amplification and saturation of a wave by thermoacoustic effect: Left :time history of acoustic pressure at active cell location. Right: Snapshot of streamlines and temperature (color) between heater and stack [Ma et al. JASA 2015]*

Rayleigh streaming refers to the second order mean velocity that is generated by viscous effects related with the interaction of an acoustic wave with a solid surface. In thermoacoustic devices, acoustic streaming results in a convected heat flow that can reduce the total efficiency. Because the power density of a thermoacoustic device is roughly proportional to the square of the acoustic pressure amplitude, the study of high acoustic amplitude phenomena is important for the field of thermoacoustics.

*Acoustic streaming in a closed two-dimensional channel: streamlines and isotherms of the associated mean temperature field (upper half of the domain)*

The streaming flow can be linear (slow streaming) or nonlinear (fast streaming) and the two regimes are characterized with a non dimensional number ReNL, that quantifies the influence of fluid inertia on acoustic streaming. Numerical simulations of compressible Navier-Stokes equations in closed waveguides are performed, using high resolution finite difference schemes developed at LIMSI (collab. D. Băltean-Carlès, V. Daru and C. Weisman). Two geometries are investigated: closed two-dimensional channels and cylindrical axisymmetric waveguides. A plane standing wave is excited inside the guide and the associated acoustic streaming is investigated by averaging the solution over the fundamental acoustic period. As expected, the streaming velocity agrees reasonably well with the slow streaming theory for small ReNL, but deviates significantly from predictions for fast streaming, when ReNL>1. The numerical results are compared with experimental results obtained by LDV (collab. Ida Reyt, H. Bailliet, J.-C. Valière from Institut Pprime, Poitiers). Both experimental and numerical results show that the centers of the outer streaming cells are pushed towards the acoustic velocity nodes until counter-rotating additional vortices are generated near the acoustic velocity antinodes. The evolution of the mean temperature field within the resonator is also investigated in order to understand the coupling with the streaming flow. When increasing ReNL the mean temperature field becomes two-dimensional as a consequence of the balance between heat convection through acoustic streaming and heat conduction.

**Turbulent Rayleigh-Bénard convection: large scale circulation and reversals**

Large scale intermittency in turbulent Rayleigh-Bénard flow is investigated in a 2D square cell using a spectral DNS code and Proper Orthogonal Decomposition technique. Two types of reversals involving corner flow growth and pattern rotation have been evidenced. Three principal modes were identified: a single-roll, large-scale circulation, a quadrupolar flow, and a horizontal double-roll symmetry-breaking mode. Analysis of the interaction coefficients between the spatial modes leads us to suggest a three-dimensional model for the reversals, based on the interaction of the three principal modes with the addition of noise [collab. B. Podvin and A. Sergent].

#### Differentially heated vertical channel: bifurcations and route to chaos

The onset of chaos in the flow of air inside a vertical differentially heated channel as the Rayleigh number is increased is studied numerically (Ph.D thesis of Z. Gao) using a spectral DNS code developed at LIMSI. The first instability is a supercritical circle pitchfork bifurcation leading to steady 2D co-rotating rolls. A Ginzburg-Landau equation is derived analytically for the flow around this first bifurcation. In two dimensions, rolls become unstable via a Hopf bifurcation. As Ra is further increased, the flow becomes quasi-periodic then temporally chaotic for a limited range of Rayleigh numbers, beyond which the flow returns to a steady state through a spatial modulation instability. In three dimensions, the rolls instead undergo another pitchfork bifurcation to 3D structures, which consist of transverse rolls connected by counter-rotating vorticity braids (see figure). The flow becomes then time-dependent through a Hopf bifurcation, energy being exchanged between the rolls and the braids. Chaotic behavior subsequently occurs through two competing routes: a sequence of period-doubling bifurcations leading to intermittency or a spatial pattern modulation reminiscent of the Eckhaus instability [collab. B. Podvin, P. Le Quéré and A. Sergent at LIMSI, S. Xin at CETHIL and L. Tuckerman at PMMH].

*Flow inside an air-filled differentially heated channel at Ra=11500 (Criterion-Q isocontour colored by the vertical vorticity showing transverse rolls connected by counter-rotating vorticity braids) [Gao et al. PRE 2013]*

#### Buoyant convection modeling: chimney

For buoyant flows in open or semi-infinite configurations, prescription of the appropriate boundary conditions on the frontiers of the computational domain is a key issue for the relevance and the accuracy of numerical simulations. This issue has been the subject of numerous studies, which have generally focused on the outlet boundary, the issue of the inlet boundary condition being deemed less difficult. If this is the case for forced convection configuration, it is less trivial for situations where the driven force of the flow depends on the conditions inside the computational domain. In that case what happens on the inlet boundary is unknown and therefore cannot be imposed a priori. A vertical channel (chimney) is considered as a prototype configuration of such flows. A benchmark organized within the French thermal science community (SFT) has revealed a significant scatter amongst numerical solutions, which questions the well-defined character of the problem. This issue was addressed by investigating the kernel modes of the associated discrete Stokes operator combined linearly with additional conditions or by imposing directly pressure through dedicated boundary conditions. However no clear conclusion can be established from comparisons with experimental results (Ph.D thesis of C. Garnier).

#### Buoyant convection modeling: starting buoyant jet

Knowledge of the dispersion and mixing mechanisms of hydrogen in confined air-filled cavities at low Mach number is an important safety issue for all hydrogen-based systems, since fire or explosion may result from specific concentration distributions. Due to the complexity of performing measurements with hydrogen, most of the experimental studies of the buoyant jet resulting from hydrogen leakage in air are performed using helium as a light gas. Coexistence within the cavity of pure helium and pure air regions causes significant variations of the fluid properties (non-Boussinesq effects) which make numerical convergence difficult. A 2D unsteady numerical model has been developed (Ph.D thesis of H.L. Tran) for binary mixtures. The model was validated by comparison with experimental results of laminar starting plumes of glycerol-water solution. It is currently applied in the case of a benchmark configuration simulating hydrogen leakage in a cavity [collab. P. Le Quéré and A. Sergent with G. Bernard-Michel at CEA].

#### Buoyant convection coupled with radiation

The well-documented discrepancy between numerical and experimental estimates of the thermal stratification in high-Ra air-filled differentially heated cavities has motivated a joint research program to integrate both wall and medium radiation into physical models of natural convection. This project, called COCORACOPHA2 for « Couplage Convection-Rayonnement Pour l’Habitat 2 », is supported by a « Action Incitative » from the CNRS Energie program. It brings together five research teams from CETHIL (Lyon), LEPTAB (Univ. La Rochelle), PPRIME (ENSMA, Poitiers), MSME (UPEMLV), and LIMSI-CNRS. In order to solve the stratification paradox, we have first established reference solutions by means of three-dimensional (3D) spectral direct numerical simulations of a buoyancy-driven flow (RaH=1.5×109) for two configurations: an idealized cavity (perfectly adiabatic cavity, PAC) and an Intermediate Realistic Cavity (IRC) based on experimental temperature distributions (Salat, 2004) at the top and bottom walls. Using a LES approach, it was proven that the complete set of experimental temperature distributions at the walls (Full Realistic Cavity) is needed to recover full agreement between numerical and experimental results. In particular, thermal boundary conditions at the front and rear walls are crucial for a good agreement. Finally a 3D numerical code coupling convection, conduction and wall radiation has been developed, and is able to reproduce the experimental temperature distribution inside the cavity.