ABSTRACT Chaotic natural convection in differentially heated air-filled cavities of aspect ratios... more ABSTRACT Chaotic natural convection in differentially heated air-filled cavities of aspect ratios 1 and 4 with adiabatic horizontal walls is investigated by direct numerical simulation of the unsteady 2D Navier-Stokes equations. The unsteady equations in primitive variables in the Boussinesq approximation are integrated by pseudo-spectral algorithms combining spatial expansions in series of Chebyshev polynomials with a second-order finite-difference time-stepping scheme. Despite the fact that the routes to chaos are not the same, we show that, in this fully chaotic regime, the flows in both cavities are very similar in several respects, their time dynamics, the time-averaged flow structure and the turbulent statistics.
International Journal of Numerical Methods for Heat & Fluid Flow, 2004
External natural convection is rarely studied by numerical simulation in the literature due to th... more External natural convection is rarely studied by numerical simulation in the literature due to the fact that flow of interest takes place in an unbounded domain and that if a limited computational domain is used the corresponding outer boundary conditions are unknown. In this study, we propose outer boundary conditions for a limited computational domain and make the corresponding numerical implementation in the scope of a projection method combining spectral methods and domain decomposition techniques. Numerical simulations are performed for both steady natural convection about an isothermal cylinder and transient natural convection around a line‐source. An experiment is also realized in water using particle image velocimetry and thermocouples to make a comparison during transients of external natural convection around a platinum wire heated by Joule effect. Good agreement, observed between numerical simulations and experiments, validated the outer boundary conditions proposed and t...
Natural convection in a differentially heated horizontal cylinder is investigated numerically and... more Natural convection in a differentially heated horizontal cylinder is investigated numerically and analytically. Particular attention is paid to the structure of steady convection, the nature of the transients and the onset of unsteadiness for a range of Prandtl numbers extending from 0.7 to infinity. The numerical algorithm integrates the 2-D Navier-Stokes equations in velocity-pressure formulation with a Chebyshev-Fourier spatial approximation. A gradual shift from the conduction to the boundary layer regime is observed for increasing Rayleigh number and the steady flow structure becomes rapidly independent of Pr. Whereas classical scalings are obtained for the azimuthal velocity and the thermal boundary layer thickness, the dynamic boundary layer thickness is found to be independent of the Prandtl number. A simplified semi-analytical model derived from projecting the governing equations on the lowest Fourier modes is proposed, which explains this property. Its solutions are in goo...
The onset of convection in double diffusion with equal and opposite thermal and solutal buoyancy ... more The onset of convection in double diffusion with equal and opposite thermal and solutal buoyancy forces is studied. Numerical linear stability analyses and integration of the full Boussinesq equations are performed in an infinite vertical fluid layer and in closed rectangular cavities bounded by rigid walls. Detailed study of the subsequent nonlinear evolution is carried out for Le=1.2 and Pr=1. In the infinite vertical layer, the onset of convection is found to correspond to a subcritical circle pitchfork bifurcation. The finite-amplitude branch of steady states in turn loses stability to traveling waves via a supercritical drift pitchfork bifurcation. In a square cavity the bifurcation is transcritical and the full branch of stable and unstable solutions is constructed. With increasing cavity aspect-ratio, we observe alternating transcritical and pitchfork bifurcations, depending on the symmetry of the most unstable eigenvector.
The transition to unsteadiness and the dynamics of weakly turbulent natural convection, coupled t... more The transition to unsteadiness and the dynamics of weakly turbulent natural convection, coupled to wall or gas radiation in a differentially heated cubical cavity with adiabatic lateral walls, are studied numerically. The working fluid is air with small contents of water vapor and carbon dioxide whose infrared spectral radiative properties are modelled by the absorption distribution function model. A pseudo spectral Chebyshev collocation method is used to solve the flow field equations and is coupled to a direct ray tracing method for radiation transport. Flow structures are identified by means of either the proper orthogonal decomposition or the dynamic mode decomposition methods. We first retrieve the classical mechanism of transition to unsteadiness without radiation, characterized by counter-rotating streamwise-oriented vortices generated at the exit of the vertical boundary layers. Wall radiation through a transparent medium leads to a homogenization of lateral wall temperature...
Chaotic natural convection in a differentially heated air-filled cavity of aspect ratio 4 with ad... more Chaotic natural convection in a differentially heated air-filled cavity of aspect ratio 4 with adiabatic horizontal walls is investigated by direct numerical integration of the unsteady two-dimensional equations. Time integration is performed with a spectral algorithm using Chebyshev spatial approximations and a second-order finite-difference time-stepping scheme. Asymptotic solutions have been obtained for three values of the Rayleigh number based on cavity height up to 1010. The time-averaged flow fields show that the flow structure increasingly departs from the well-known laminar one. Large recirculating zones located on the outer edge of the boundary layers form and move upstream with increasing Rayleigh number. The time-dependent solution is made up of travelling waves which run downstream in the boundary layers. The amplitude of these waves grows as they travel downstream and hook-like temperature patterns form at the outer edge of the thermal boundary layer. At the largest Ra...
International Journal of Heat and Mass Transfer, 2003
Transient natural convection in liquid nitrogen around a heated wire is studied experimentally an... more Transient natural convection in liquid nitrogen around a heated wire is studied experimentally and numerically. A thin bronze wire of 40 μm in diameter and 5.1 cm in length heated by Joule effect is used in experiments and time evolutions of the wire temperature are measured through its electrical resistance. Numerical simulations of such flows are performed by using velocity–pressure formulation, spectral methods and domain decomposition technique. Experimental data and numerical results are compared and show that although the wire is extremely thin its thermal inertia plays an important role during early transients.Scaling analyses performed for the convection set-up and transients yield t∼q−1/2 and ΔT∼q3/4, which are confirmed by numerical simulations.For wires heated by Joule effect hybrid thermal conditions––combined Dirichlet and Neumann conditions are proposed on the wire surface and validated by numerical simulations. On the other hand flow conditions to be imposed on the outer artificial boundary are not well known and the answer to the question remains still open: two types of flow conditions are tested and yield different velocity fields.
International Journal for Numerical Methods in Fluids, 2002
Our contribution to the benchmark is multifold. In addition to providing accurate unsteady simula... more Our contribution to the benchmark is multifold. In addition to providing accurate unsteady simulations at the required Ra value of 3.4 × 105, we determine accurately the three first critical bifurcation points, investigate the supercritical regime, and study the differences between time‐averaged solutions and the corresponding base solution at Ra = 4× 105. We thereby establish the existence of, at least, 4 different branches of solutions and of 3 multiple unsteady periodic solutions for a Rayleigh value of 4 × 105. First appearance of quasi‐periodic flow is found at Ra about 5.0 × 105 and first appearance of chaotic solutions is found for 5.5 × 105 approximately. We investigate the differences between time‐averaged solutions and the corresponding base flow solution at Ra = 4 × 105. It is found that they exhibit symmetric feature and similar spatial distribution and that their amplitudes are proportional to the squared amplitude of periodic solutions.All these computations are carrie...
ABSTRACT The coupling between natural convection and gas and wall radiation is studied numericall... more ABSTRACT The coupling between natural convection and gas and wall radiation is studied numerically in a differentially heated cubical cavity filled with an air/CO2/H2O mixture. In order to solve coupled flow, heat transfer and radiation equations, we develop a 3D radiative transfer model based on the deterministic ray tracing method, coupled with a pseudo-spectral Chebyshev method for natural convection under Boussinesq approximation. Absorption Distribution Function (ADF) model is used to describe gas radiative properties. ...
20ème Congrès Français de Mécanique, 28 août/2 sept. 2011-25044 Besançon, France (FR), 2011
Nous étudions numériquement l'écoulement de convection naturelle dans un canal vertical soum... more Nous étudions numériquement l'écoulement de convection naturelle dans un canal vertical soumis à une stratification. Des conditions aux limites périodiques sont imposées dans la direction verticale. L'influence de la stratification et du nombre du Rayleigh est examinée. Un modèle peut être utilisé pour le régime faiblement nonlinéaire autour de la première bifurcation. Au-delà d'un nombre de Rayleigh plus élevé on observe l'apparition d'ondes solitaires.
ABSTRACT Numerical simulations are carried out of the flow between counter-rotating disks in cyli... more ABSTRACT Numerical simulations are carried out of the flow between counter-rotating disks in cylindrical containers of moderate aspect ratio. Interesting time-dependent behavior is found to result from interactions between primary instabilities.
ABSTRACT Chaotic natural convection in differentially heated air-filled cavities of aspect ratios... more ABSTRACT Chaotic natural convection in differentially heated air-filled cavities of aspect ratios 1 and 4 with adiabatic horizontal walls is investigated by direct numerical simulation of the unsteady 2D Navier-Stokes equations. The unsteady equations in primitive variables in the Boussinesq approximation are integrated by pseudo-spectral algorithms combining spatial expansions in series of Chebyshev polynomials with a second-order finite-difference time-stepping scheme. Despite the fact that the routes to chaos are not the same, we show that, in this fully chaotic regime, the flows in both cavities are very similar in several respects, their time dynamics, the time-averaged flow structure and the turbulent statistics.
International Journal of Numerical Methods for Heat & Fluid Flow, 2004
External natural convection is rarely studied by numerical simulation in the literature due to th... more External natural convection is rarely studied by numerical simulation in the literature due to the fact that flow of interest takes place in an unbounded domain and that if a limited computational domain is used the corresponding outer boundary conditions are unknown. In this study, we propose outer boundary conditions for a limited computational domain and make the corresponding numerical implementation in the scope of a projection method combining spectral methods and domain decomposition techniques. Numerical simulations are performed for both steady natural convection about an isothermal cylinder and transient natural convection around a line‐source. An experiment is also realized in water using particle image velocimetry and thermocouples to make a comparison during transients of external natural convection around a platinum wire heated by Joule effect. Good agreement, observed between numerical simulations and experiments, validated the outer boundary conditions proposed and t...
Natural convection in a differentially heated horizontal cylinder is investigated numerically and... more Natural convection in a differentially heated horizontal cylinder is investigated numerically and analytically. Particular attention is paid to the structure of steady convection, the nature of the transients and the onset of unsteadiness for a range of Prandtl numbers extending from 0.7 to infinity. The numerical algorithm integrates the 2-D Navier-Stokes equations in velocity-pressure formulation with a Chebyshev-Fourier spatial approximation. A gradual shift from the conduction to the boundary layer regime is observed for increasing Rayleigh number and the steady flow structure becomes rapidly independent of Pr. Whereas classical scalings are obtained for the azimuthal velocity and the thermal boundary layer thickness, the dynamic boundary layer thickness is found to be independent of the Prandtl number. A simplified semi-analytical model derived from projecting the governing equations on the lowest Fourier modes is proposed, which explains this property. Its solutions are in goo...
The onset of convection in double diffusion with equal and opposite thermal and solutal buoyancy ... more The onset of convection in double diffusion with equal and opposite thermal and solutal buoyancy forces is studied. Numerical linear stability analyses and integration of the full Boussinesq equations are performed in an infinite vertical fluid layer and in closed rectangular cavities bounded by rigid walls. Detailed study of the subsequent nonlinear evolution is carried out for Le=1.2 and Pr=1. In the infinite vertical layer, the onset of convection is found to correspond to a subcritical circle pitchfork bifurcation. The finite-amplitude branch of steady states in turn loses stability to traveling waves via a supercritical drift pitchfork bifurcation. In a square cavity the bifurcation is transcritical and the full branch of stable and unstable solutions is constructed. With increasing cavity aspect-ratio, we observe alternating transcritical and pitchfork bifurcations, depending on the symmetry of the most unstable eigenvector.
The transition to unsteadiness and the dynamics of weakly turbulent natural convection, coupled t... more The transition to unsteadiness and the dynamics of weakly turbulent natural convection, coupled to wall or gas radiation in a differentially heated cubical cavity with adiabatic lateral walls, are studied numerically. The working fluid is air with small contents of water vapor and carbon dioxide whose infrared spectral radiative properties are modelled by the absorption distribution function model. A pseudo spectral Chebyshev collocation method is used to solve the flow field equations and is coupled to a direct ray tracing method for radiation transport. Flow structures are identified by means of either the proper orthogonal decomposition or the dynamic mode decomposition methods. We first retrieve the classical mechanism of transition to unsteadiness without radiation, characterized by counter-rotating streamwise-oriented vortices generated at the exit of the vertical boundary layers. Wall radiation through a transparent medium leads to a homogenization of lateral wall temperature...
Chaotic natural convection in a differentially heated air-filled cavity of aspect ratio 4 with ad... more Chaotic natural convection in a differentially heated air-filled cavity of aspect ratio 4 with adiabatic horizontal walls is investigated by direct numerical integration of the unsteady two-dimensional equations. Time integration is performed with a spectral algorithm using Chebyshev spatial approximations and a second-order finite-difference time-stepping scheme. Asymptotic solutions have been obtained for three values of the Rayleigh number based on cavity height up to 1010. The time-averaged flow fields show that the flow structure increasingly departs from the well-known laminar one. Large recirculating zones located on the outer edge of the boundary layers form and move upstream with increasing Rayleigh number. The time-dependent solution is made up of travelling waves which run downstream in the boundary layers. The amplitude of these waves grows as they travel downstream and hook-like temperature patterns form at the outer edge of the thermal boundary layer. At the largest Ra...
International Journal of Heat and Mass Transfer, 2003
Transient natural convection in liquid nitrogen around a heated wire is studied experimentally an... more Transient natural convection in liquid nitrogen around a heated wire is studied experimentally and numerically. A thin bronze wire of 40 μm in diameter and 5.1 cm in length heated by Joule effect is used in experiments and time evolutions of the wire temperature are measured through its electrical resistance. Numerical simulations of such flows are performed by using velocity–pressure formulation, spectral methods and domain decomposition technique. Experimental data and numerical results are compared and show that although the wire is extremely thin its thermal inertia plays an important role during early transients.Scaling analyses performed for the convection set-up and transients yield t∼q−1/2 and ΔT∼q3/4, which are confirmed by numerical simulations.For wires heated by Joule effect hybrid thermal conditions––combined Dirichlet and Neumann conditions are proposed on the wire surface and validated by numerical simulations. On the other hand flow conditions to be imposed on the outer artificial boundary are not well known and the answer to the question remains still open: two types of flow conditions are tested and yield different velocity fields.
International Journal for Numerical Methods in Fluids, 2002
Our contribution to the benchmark is multifold. In addition to providing accurate unsteady simula... more Our contribution to the benchmark is multifold. In addition to providing accurate unsteady simulations at the required Ra value of 3.4 × 105, we determine accurately the three first critical bifurcation points, investigate the supercritical regime, and study the differences between time‐averaged solutions and the corresponding base solution at Ra = 4× 105. We thereby establish the existence of, at least, 4 different branches of solutions and of 3 multiple unsteady periodic solutions for a Rayleigh value of 4 × 105. First appearance of quasi‐periodic flow is found at Ra about 5.0 × 105 and first appearance of chaotic solutions is found for 5.5 × 105 approximately. We investigate the differences between time‐averaged solutions and the corresponding base flow solution at Ra = 4 × 105. It is found that they exhibit symmetric feature and similar spatial distribution and that their amplitudes are proportional to the squared amplitude of periodic solutions.All these computations are carrie...
ABSTRACT The coupling between natural convection and gas and wall radiation is studied numericall... more ABSTRACT The coupling between natural convection and gas and wall radiation is studied numerically in a differentially heated cubical cavity filled with an air/CO2/H2O mixture. In order to solve coupled flow, heat transfer and radiation equations, we develop a 3D radiative transfer model based on the deterministic ray tracing method, coupled with a pseudo-spectral Chebyshev method for natural convection under Boussinesq approximation. Absorption Distribution Function (ADF) model is used to describe gas radiative properties. ...
20ème Congrès Français de Mécanique, 28 août/2 sept. 2011-25044 Besançon, France (FR), 2011
Nous étudions numériquement l'écoulement de convection naturelle dans un canal vertical soum... more Nous étudions numériquement l'écoulement de convection naturelle dans un canal vertical soumis à une stratification. Des conditions aux limites périodiques sont imposées dans la direction verticale. L'influence de la stratification et du nombre du Rayleigh est examinée. Un modèle peut être utilisé pour le régime faiblement nonlinéaire autour de la première bifurcation. Au-delà d'un nombre de Rayleigh plus élevé on observe l'apparition d'ondes solitaires.
ABSTRACT Numerical simulations are carried out of the flow between counter-rotating disks in cyli... more ABSTRACT Numerical simulations are carried out of the flow between counter-rotating disks in cylindrical containers of moderate aspect ratio. Interesting time-dependent behavior is found to result from interactions between primary instabilities.
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