Non-Oberbeck-Boussinesq Effects in Gaseous Rayleigh-Bénard Convection

Abstract

Non-Oberbeck-Boussinesq (NOB) effects are measured experimentally and calculated theoretically for strongly turbulent Rayleigh-Bénard convection of ethane gas under pressure where the material properties strongly depend on the temperature. Relative to the Oberbeck-Boussinesq case we find a decrease of the central temperature as compared to the arithmetic mean of the top-and bottom-plate temperature and an increase of the Nusselt number. Both effects are of opposite sign and greater magnitude than those for NOB convection in liquids like water. DOI: 10.1103/PhysRevLett.98.054501 PACS numbers: 47.27.ÿi, 47.20.Bp Turbulent convection in a fluid heated from below and cooled from above (Rayleigh-Bénard convection) is an important model system in fluid dynamics [1]. The induced temperature difference across a sample is represented by the Rayleigh number Ra gL 3 = ( is the thermal expansion coefficient, g the acceleration of gravity, L the sample height, the thermal diffusivity, and the kinematic viscosity). The problem usually is analyzed within an approximation due to Oberbeck [2] and Boussinesq [3] (OB), where it is assumed that all fluid properties are constant within the entire sample except for the density where it induces the buoyancy force. Here we address, both experimentally and theoretically, the nature of deviations from this approximation.A central aspect of this system is an understanding of the boundary layers (BLs) near the top and bottom plates. At modest Ra they remain laminar while the fluid interior is turbulent, and their instabilities impact the Nusselt number Nu (the effective thermal conductivity eff normalized by the diffusive thermal conductivity ) [4,5]. At much larger Ra it was suggested [6 -8] that the BLs become turbulent as well, that they then no longer influence Nu, and that an asymptotic (''ultimate'' or ''Kraichnan'') regime is achieved where Nu Ra 1=2 . In the present Letter we show, by comparing theoretical calculations with new experimental measurements, that the laminar BLs are also significantly influenced by non-Oberbeck-Boussinesq (NOB) effects. These effects can be described well by an extension of the Prandtl-Blasius boundary-layer theory [9,10].Though NOB effects in turbulent Rayleigh-Bénard convection were measured already 15 years ago [11,12], a quantitative comparison between OB and NOB convection was only done recently [13]. However, such study was restricted to NOB effects in liquids like water and glycerol. It found a decrease of Nu and an increase of the center temperature T c as compared to the OB case. The latter could be explained quantitatively by an extension of the Prandtl-Blasius BL theory [13]. Extending this theory to gases is considerably more challenging, as then also the density depends on temperature (beyond the OB dependence), leading to a density boundary layer. Moreover, all material properties such as , , the shear viscosity , and the specific heat c P depend on both temperature and density. Here we shall show that nevertheless an extension of th...

New perspectives in turbulent Rayleigh-Bénard convection

“…While in [171,172] the deviation of the centre temperature from the algebraic mean, T ref = (T bottom + T top )/2, is observed, Burnishev et al [61] report a high-Rayleigh-number NOB convection with symmetric temperature profiles and a scaling Nu ∼ Ra β which is comparable to the experiments in the OB limit at same Ra (see sect. 6).…”

“…The anelastic approximation (5) is a first weak compressibility effect which is for example incorporated in atmospheric convection that exceeds layers of height H ≈ 2 km. Stronger compressibility effects and thus large variations of the fluid properties can arise when the working fluids are operated in the vicinity of their critical points as done in helium [171], in ethane [172], or in SF 6 [61].…”

Horizontal structures of velocity and temperature boundary layers in two-dimensional numerical turbulent Rayleigh-Bénard convection

“…As the GL theory is based on the assumption that the BL thickness scales inversely proportional to the square root of the Reynolds number according to Prandtl's 1904 theory, the validity of Prandtl-Blasius BL flow needs to be tested also locally. Note that comparison of the mean bulk temperature calculated using the Prandtl-Blasius theory with that measured in both liquid and gaseous nonOberbeck-Boussinesq RB convection shows very good agreement [16][17][18] . In addition, the kinematic BL thickness evaluated by solving the laminar Prandtl-Blasius BL equations was found to agree well with that obtained in the direct numerical simulation (DNS) 19 .…”

Lattice Boltzmann methods for thermal flows: Continuum limit and applications to compressible Rayleigh–Taylor systems

“…On the other hand, the limit Z → 0 corresponds to the so-called Oberbeck-Boussinesq approximation, where both stratification and compressibility are vanishingly small. The latter is, by far, the most studied convection configuration, even though some important applications for astrophysics [45,46] and recently also for laboratory set-up [47][48][49] cannot neglect compressible modes. It is possible to show [35] that in the Boussinesq approximation, the dependency from the polytropic index disappear (as it must obviously do) while it remains a possible effect induced by the adiabatic gradient (usually small on laboratory experiments, but not necessarily on atmospheric scales).…”