Applicability of Boussinesq approximation in a turbulent fluid with constant properties

The equations of motion describing buoyant fluids are often simplified using a set of approximations proposed by J. Boussinesq one century ago. To resume, they consist in assuming constant fluid properties, incompressibility and conservation of calories during heat transport. Assuming fulfilment of the first requirement (constant fluid properties), we derive a set of 4 criteria for assessing the validity of the two other requirements in turbulent Rayleigh-B\'enard convection. The first criterion $\alpha \Delta \ll 1$ simply results from the incompressibility condition in the thermal boundary layer ($\alpha$ and $\Delta$ are the thermal expansion coefficient and the temperature difference driving the flow). The 3 other criteria are proportional or quadratic with the density stratification or, equivalently with the temperature difference resulting from the adiabatic gradient across the cell $\Delta_{h}$. Numerical evaluations with air, water and cryogenic helium show that most laboratory experiments are free from such Boussinesq violation as long as the first criterion is fulfilled. In ultra high Rayleigh numbers ($Ra>10^{16}$) experiments in He, one of the stratification criteria, scaling with $\alpha \Delta_{h}$, could be violated. This criterion garanties that pressure fluctuations have a negligible influence both on the density variation and on the heat transfer equation through compression/expansion cycles. Extrapolation to higher $Ra$ suggests that strong violation of Boussinesq approximation could occur in atmospheric convection.Comment: Submitted to Phys.Fluids (oct 2007

Energy cascade and the four-fifths law in superfluid turbulence

The 4/5-law of turbulence, which characterizes the energy cascade from large to small-sized eddies at high Reynolds numbers in classical fluids, is verified experimentally in a superfluid 4He wind tunnel, operated down to 1.56 K and up to R_lambda ~ 1640. The result is corroborated by high-resolution simulations of Landau-Tisza's two-fluid model down to 1.15 K, corresponding to a residual normal fluid concentration below 3 % but with a lower Reynolds number of order R_lambda ~ 100. Although the K\'arm\'an-Howarth equation (including a viscous term) is not valid \emph{a priori} in a superfluid, it is found that it provides an empirical description of the deviation from the ideal 4/5-law at small scales and allows us to identify an effective viscosity for the superfluid, whose value matches the kinematic viscosity of the normal fluid regardless of its concentration.Comment: 6 pages, 7 figure

Investigation of intermittency in superfluid turbulence

International audienceThis paper reports new experimental and simulation velocity data for superfluid steady turbulence above 1 K. We present values for the scaling exponent of the absolute value of velocity-increment structure functions. In both experiments and simulations, they evidence that intermittency occurs in superfluid flows in a quite comparable way to classical turbulence. In particular, the deviation from Kolmogorov 1941 keeps the same strength as we cross the superfluid transition. To the best of our knowledge, this is the first confirmation of the superfluid 4He experimental results from Maurer et al. EPL 1998 and the first numerical evidence of intermittency in superfluid turbulence

Evidence of a boundary layer instability at very high Rayleigh number

In 1997, a Rayleigh-B\'enard experiment evidenced a significant increase of the heat transport efficiency for Rayleigh numbers larger than $Ra \sim 10^{12}$ and interpreted this observation as the signature of the Kraichnan's ``Ultime Regime'' of convection. According to Kraichnan's 1962 prediction, the flow boundary layers above the cold and hot plates -in which most of the fluid temperature drop is localized- become unstable for large enough $Ra$ and this instability boosts the heat transport compared to the other turbulent regimes. Using the same convection cell as in the 1997 experiment, we show that the reported heat transport increase is accompanied with enhanced temperature fluctuations of the bottom plate, which was heated at constant power levels. Indeed, for $Ra < 10^{12}$, the bottom plate fluctuations can simply be accounted from those in the bulk of the flow. In particular, they share the same spectral density at low frequencies, as if the bottom plate was following the slow temperature fluctuations of the bulk, modulo a constant temperature drop across the bottom boundary layer. Conversely, to account for the plate's temperature fluctuations at higher $Ra$, we no-longuer can ignore the fluctuations of the temperature drop across the boundary layer. The negative skewness of fluctuations at high $Ra$ supports the picture of a boundary layer instability. These observations provide new evidence that the transition reported in 1997 corresponds to the triggering of the Ultimate Regime of convection.Comment: Submitted for publicatio

Comment on "Turbulent heat transport near critical points: Non-Boussinesq effects" (cond-mat/0601398)

In a recent preprint (cond-mat/0601398), D. Funfschilling and G. Ahlers describe a new effect, that they interpret as non-Boussinesq, in a convection cell working with ethane, near its critical point. They argue that such an effect could have spoiled the Chavanne {\it et al.} (Phys. Rev. Lett. {\bf 79} 3648, 1997) results, and not the Niemela {\it et al.} (Nature, {\bf 404}, 837, 2000) ones, which would explain the differences between these two experiments. We show that:-i)Restricting the Chavanne's data to situations as far from the critical point than the Niemela's one, the same discrepancy remains.-ii)The helium data of Chavanne show no indication of the effect observed by D. Funfschilling and G. Ahlers.Comment: comment on cond-mat/060139

Thixotropic behaviour of paving grade bitumens under dynamic shear

A material exhibits a thixotropic behaviour if its apparent viscosity decreases in time under stress, and if it progressively recovers its initial viscosity when the stress is released. Methods for characterizing thixotropy for fluids are relatively well known, the situation is more difficult for viscoelastic materials. For bituminous binders, it corresponds to a decrease of material stiffness under cyclic loading by modification of its internal structure and to a recovery of this stiffness after rest. This property has been highlighted by simple shear tests, carried out using a controlled stress rheometer. An experimental procedure has been defined for studying the thixotropic behaviour of binders in dynamic mode. Then, it has allowed describing the corresponding curves of complex modulus versus the set stress and the rest period at different conditioning times of samples. This study has also permitted to demonstrate that stiffness variations are linked to re-arrangement of molecular structure and not to cracks formation and healing. Finally, based on specific indicators, it has been possible to compare different bituminous binders in order to estimate the ability of materials to restore their internal structures

Vorticity scattering measurements in a superfluid inertial round jet

International audienceThe aim of this proceeding paper is twofold. First, we present a newly developed cryogenic testing facility where a steady high Reynolds liquid helium inertial round jet flow is generated allowing to address classical turbulence issues, such as statistical intermittency, and quantum turbulence when the facility is operating in superfluid helium. Secondly we present the first spatial Fourier vorticity modes measurements made both above and below the superfluid transition at different nozzle velocities. These preliminary results were obtained by probing the vorticity flow-field with the ultrasonic scattering technique

Prandtl and Rayleigh numbers dependences in Rayleigh Bénard convection

International audienceUsing low-temperature gaseous helium close to the critical point, we investigate the Prandtl-number dependence of the effective heat conductivity (Nusselt number) for a 1/2 aspect ratio Rayleigh-Bénard cell. Very weak dependence is observed in the range 0.7 < Pr < 21; 2 × 10^8 < Ra < 2 × 10^10: the absolute value of the average logarithmic slope δ = (∂ln Nu/∂ln Pr)Ra is smaller than 0.03. A bimodality of Nu, with 7% difference between the two sets of data, is observed, which could explain some discrepancies between precise previous experiments in this range