Les r´esultats de notre ´etude exp´erimentale sont pr´esent´es dans les chapitres sui- vants. Le chapitre 6 pr´esente l’´etude du syst`eme du point de vue de la m´ecanique des fluides. Nous y caract´erisons les deux ´echelles d’´ecoulement qui se forment dans la cuve, ainsi que leur interaction. Ce chapitre ne traite que des exp´eriences r´ealis´ees avec une condition limite inf´erieure isotherme (cuve n◦2). Le chapitre 7 ´elargit les r´esultats du chapitre 6. Nous y consid´erons les exp´eriences r´ealis´ees avec une condition limite inf´erieure adiabatique (cuve n◦1), et nous comparons les ph´enom`enes observ´es au cas isotherme. Nous reviendrons ´egalement dans ce chapitre sur les caract´eristiques de l’´ecoulement `a grande ´echelle, et nous verrons en particulier comment nous pouvons faire l’analogie entre nos exp´eriences et les ph´enom`enes convectifs g´en´er´es en convection de Rayleigh-B´enard dans le r´egime de turbulence dure. Enfin, dans le chapitre 8, nous appliquerons nos r´esultats `a la Terre, en voyant comment nos exp´eriences peuvent nous amener des informations sur l’´ecoulement en base de lithosph`ere oc´eanique.
Interaction of two scales of thermal
convection in viscous fluids
Sommaire
6.1 Introduction . . . . 86 6.2 Experimental setup . . . . 87 6.2.1 Description . . . 87 6.2.2 Fluids . . . 88 6.2.3 Observations and measurements . . . 88 6.2.4 Experimental conditions . . . 90 6.3 Large-scale convection . . . . 92 6.3.1 Description . . . 92 6.3.2 Flow near the vertical heated wall . . . 95 6.3.3 Stratified core . . . 99 6.3.4 Jet under the upper cold boundary layer . . . 100 6.4 Small-scale convection . . . 102 6.4.1 Instabilities under the upper boundary layer . . . 102 6.4.2 2D stationary or 3D time-dependent structures . . . 106 6.4.3 Spatial and temporal periodicity . . . 108 6.5 Conclusions . . . 110 6.6 Appendix A : Fluids properties . . . 111
V. Vidal and A. Davaille, Interaction of two scales of thermal convection in viscous fluids, submitted to Journal of Fluid Mechanics (2004)
Abstract
Dripping convective instabilities develop under a cold thermal boundary layer when the local Rayleigh number exceeds a critical value Rac (Howard, 1966). Their
interaction with a shear flow is studied experimentally in a cavity heated from one vertical wall and cooled from above. Rayleigh numbers range between 104 and 108, and Prandtl numbers are greater than 1000. Within this parameter range, a hot horizontal jet develops under the cold boundary. The instabilities dripping from the latter are therefore sheared by the flow and remain trapped in the jet, following a helicoidal path with axis parallel to the jet flow. For high jet velocities compared to the dripping velocity, 2D steady rolls prevail, while for low jet velocities, a 3D structure is observed. A phase diagram and scaling laws for the flow characteristics are determined.
6.1
Introduction
The interaction between two scales of convection is encountered in many buoyancy driven systems, from engineering to geophysics. Everytime a horizontal forced flow is cooled from above or heated from below, small-scale convective instabilities can develop from the horizontal boundary. In heat exchangers (e.g. reactors cooling), these instabilities can severely modify the heat flow. In geophysical fluids, clouds can organize in rolls in the atmosphere (Turner, 1973), and it has been suggested that the heat loss out of the solid mantle of our planet was in part governed by the interaction of dripping instabilities from its upper cooling surface with the large-scale motion of Plate Tectonics (e.g. Richter 1973 ; Parsons & McKenzie 1978 ; Doin et al. 1997). Here, we study the interaction of the two scales of convection which can develop in a rectangular cavity heated from a vertical wall and cooled from above and from below (Crambes, 2000; Vidal et al., 2003) : because of lateral heating, a roll develops over the whole cavity, while small dripping instabilities develop from the upper cold boundary.
Convection along a heated vertical wall has been well studied, especially for diffe- rentially end wall heated systems. The three-paper sequence of Cormack et al. (1974), Cormack et al. (1974) and Imberger (1974) adresses this problem, with a theoreti- cal, numerical and experimental approach respectively. At very low Rayleigh number (Ra < 1), heat conduction predominates (Imberger, 1974). The flow over most of the cavity is parallel and driven by the horizontal temperature gradient. For Ra > 1, convection predominates and the vertical temperature gradient increases (Imberger, 1974). For large enough Rayleigh numbers, a vertical boundary layer develops (Gill, 1966), as well as two horizontal jets along the upper and lower adiabatic boundaries, while most of the cavity bulk remains stagnant and thermally stratified (Bejan et al., 1981). As the Rayleigh number further increases, the steady flow along the vertical wall can become periodic, then chaotic, and finally reaches a turbulent state (Seki et al., 1978; Ravi et al., 1994). Our experiments will remain in the domain where horizontal jets develop. We shall use the latter to force a shear under the gravitationally unstable cold upper horizontal boundary.
Convective instabilities in presence of a forced shear have been studied exten- sively due to its numerous practical applications. Different configurations have been investigated : cylindrical tanks with a heated lower boundary and an upper boundary
rotating about a vertical axis (Ingersoll, 1966) ; inclined Rayleigh-B´enard cells (Spar- row & Husar, 1969; Lloyd & Sparrow, 1970; Hart, 1971; Busse & Clever, 2000) ; large aspect-ratio tanks with a heated lower boundary and an upper boundary moving ho- rizontally (Richter & Parsons, 1975; Houseman, 1983; Kincaid et al., 1996) or with an uneven temperature distribution imposed on the top boundary (Curlet, 1976). 2D numerical studies (Richter, 1973; Skilbeck & McKenzie, 1979; Houseman & McKenzie, 1982; Houseman, 1983) show that convective instabilities are then carried away by the shear flow. 3D laboratory studies, as well as theoretical analysis (Richter, 1973; Clever & Busse, 1977, 1991) and 3D numerical simulations (Hathaway & Sommerville, 1986; Domaradzki & Metcalfe, 1988; Clever & Busse, 1992), further show that the main effect of the forced shear is to suppress transverse convective instabilities and to organize the flow in longitudinal rolls parallel to the forced flow velocity, at least for Rayleigh num- bers up to 106. The heat transfer is also probably modified, although it is still not clear (Domaradzki & Metcalfe, 1988) whether it is increased (Richter 1973 ; Hathaway & Somerville 1986, for large shear) or decreased (Ingersoll 1966 ; Hathaway & Somerville 1986, for small shear) compared to the classical Rayleigh-B´enard situation at the same Rayleigh number.
In this study, we extend the Rayleigh number domain up to 108, and determine the morphology and characteristics of the convective instabilities, using Particle Image Ve- locimetry (PIV), direct visualization of isotherms and local temperature measurements. Section 6.2 describes the experimental setup. The large-scale motion forced by the ver- tical wall heating is analysed in section 6.3. Then section 6.4 focuses on the small-scale convective instabilities dripping out of the upper cold boundary and sheared by the jet. Conclusions are presented in section 6.5.