Convective versus absolute instability in mixed Rayleigh–Bénard–Poiseuille convection

Abstract

Transition from convective to absolute instability in Rayleigh–Bénard convection in the presence of a uni-directional Poiseuille flow is studied. The evaluation of the long-time behaviour of the Green function in the horizontal plane allows the determination of regions of convective and absolute instability in the Rayleigh–Reynolds number plane as a function of Prandtl number. It is found that the mode reaching zero group velocity at the convective–absolute transition always corresponds to transverse rolls, while the system remains convectively unstable with respect to pure streamwise (longitudinal) rolls for all non-zero Reynolds numbers. Finally, the roll pattern within the entire wave packet and in particular near its centre is elucidated and possible connections between experiments and our findings are discussed.

Stabilization of absolute instability in spanwise wavy two-dimensional wakes

“…We note that the integrand in (2.8a) becomes singular when D(α, β, ω) = 0, implying that this gives dispersion relation of (2.1). The asymptotic solution of the integral (2.8a) is obtained using the method of steepest descent (Briggs 1964;Huerre & Monkewitz 1985Huerre & Rossi 1998) that was also adopted in Carriere & Monkewitz (1999) for the evaluation of an integral similar to (2.8a). Following the same procedure in Carriere & Monkewitz (1999), the asymptotic solution of (2.8a) for sufficiently large t is obtained as…”

Sidewall and thermal boundary condition effects on the evolution of longitudinal rolls in Rayleigh-Bénard-Poiseuille convection

“…11 showed that a transition from convective to absolute instability occurs for the transverse rolls while the longitudinal rolls always result from a convective instability and that this mode is the most amplified one in channels of infinite lateral extension. Those results could explain why longitudinal rolls are the preferred patterns observed in real flows in the region where the transverse rolls are not absolutely unstable.…”

Harmonic mechanical excitations of steady convective instabilities: A means to get more uniform heat transfers in mixed convection flows?

“…This instability is observed for sufficiently high Reynolds numbers (typically for Re>O (10) in air) and for Rayleigh numbers above a critical value varying between 1708 and 2000 when B>2 [9]. Carrière and Monkewitz (1999) [16] showed that this pattern is a convective instability of the basic conductive Poiseuille flow. Mergui et al (2011) [17] and Benderradji et al (2008) [18] demonstrated that these rolls are triggered in real channels of finite transverse aspect ratio just downstream the leading edge of the heated plate and near the vertical walls due to the presence of velocity and temperature boundary layers adjacent to these walls.…”

“…In the present DOE, the factor ε is preferred to Ra because it is well known that the main characteristics of the thermoconvective patterns in natural and mixed convection are related to ε (see [1,8,16,29] for instance in mixed convection flows). The Reynolds number range, 100≤Re≤300, was chosen, on the one hand, to avoid to be too close to the critical threshold between the longitudinal and wavy rolls at Re*≈70±30 and, on the other hand, to avoid too long wavy roll growth length, L g , beyond Re>300, since L g increases a lot when Re increases [10].…”