Precise measurements of the wavelength at the onset of Rayleigh-Benard convection in a long rectangular duct

Abstract

WHEN a fluid layer is heated uniformly from below, the state of rest becomes unstable when the critical Rayleigh number Ra” (the non-dimensional temperature gradient) exceeds a certain value. At the onset of convection, the shape of the convective motion depends on the nature and the geometry of the boundaries. In the case of a rectangular channel, various theoretical works have been achieved; they all show that convection appears under the form of rolls aligned with their axis perpendicular to the axis of the channel [l-5]. The results concerning the onset of free convection in an infinite rectangular duct with four rigid boundaries, two perfectly conducting horizontal plates and two insulating lateral walls have been given elsewhere [S]. There the stability of the following cases was studied : (i) infinite longitudinal rolls [with their axis parallel to the axis of the channel (V, = 0 in Fig. l)]. (ii) finite transverse rolls (V, = 0), (iii) 3-dim. rolls (V,, V,, V, # 0). Three-dimensional perturbations lead to the lowest critical Rayleigh numbers. At the onset of convection, the existence of a y-component of the velocity leads to a z-symmetrical ycurvature of the trajectories of fluid particles. The dependence of the critical wavenumber (related to the size of convective rolls at the critical point), on the width-to-height ratio is very characteristic of the 3-dim. structure: in the case of longitudinal rolls as’ is obviously equal to zero; for finite transverse rolls, it decreases monotonically when the width-to-height ratio increases (curve a, Fig. 2) and for three-dimensional rolls one obtains the particular curve bon Fig. 2. This characteristic shape does not depend on the nature of the horizontal walls since it has been sketched as well as for ‘free-free’ boundary conditions [3], and for ‘rigid-rigid’ ones [5]. Concerning the experimental point of view, Oertel and Biihler [6] quoted,via a differential interferometry technique, the presence of a three-dimensional convective motion in a duct with a length-to-width ratio (y) of 2.5 and Rayleigh numbers not less than 1.5 times the critical Rayleigh number. In this work we intend to measure the wavelength of convective cells as a function of the width-to-height ratio in a very long apparatus (y 2 17.5) and as close as possible to the threshold of Rayleigh-Bthrard convection. For that, we use the same experimental apparatus and the same shadowgraph method described earlier [7] in a paper about the influence of a superimposed basic flow on the shape of thermoconvective rolls in a long duct.