Logarithmic corrections to scaling in turbulent thermal convection

We use an analytic toy model of turbulent convection to show that most of the scaling regimes are spoiled by logarithmic corrections, in a way consistent with the most accurate experimental measurements available nowadays. This sets a need for the search of new measurable quantities which are less prone to dimensional theories.Comment: Revtex, 24 pages, 7 figure

Momentum transport and torque scaling in Taylor-Couette flow from an analogy with turbulent convection

We generalize an analogy between rotating and stratified shear flows. This analogy is summarized in Table 1. We use this analogy in the unstable case (centrifugally unstable flow v.s. convection) to compute the torque in Taylor-Couette configuration, as a function of the Reynolds number. At low Reynolds numbers, when most of the dissipation comes from the mean flow, we predict that the non-dimensional torque $G=T/\nu^2L$, where $L$ is the cylinder length, scales with Reynolds number $R$ and gap width $\eta$, $G=1.46 \eta^{3/2} (1-\eta)^{-7/4}R^{3/2}$. At larger Reynolds number, velocity fluctuations become non-negligible in the dissipation. In these regimes, there is no exact power law dependence the torque versus Reynolds. Instead, we obtain logarithmic corrections to the classical ultra-hard (exponent 2) regimes: $$G=0.50\frac{\eta^{2}}{(1-\eta)^{3/2}}\frac{R^{2}}{\ln[\eta^2(1-\eta)R^ 2/10^4]^{3/2}}.$$ These predictions are found to be in excellent agreement with available experimental data. Predictions for scaling of velocity fluctuations are also provided.Comment: revTex, 6 Figure

Scaling in large Prandtl number turbulent thermal convection

We study the scaling properties of heat transfer $Nu$ in turbulent thermal convection at large Prandtl number $Pr$ using a quasi-linear theory. We show that two regimes arise, depending on the Reynolds number $Re$. At low Reynolds number, $Nu Pr^{-1/2}$ and $Re$ are a function of $Ra Pr^{-3/2}$. At large Reynolds number $Nu Pr^{1/3}$ and $Re Pr$ are function only of $Ra Pr^{2/3}$ (within logarithmic corrections). In practice, since $Nu$ is always close to $Ra^{1/3}$, this corresponds to a much weaker dependence of the heat transfer in the Prandtl number at low Reynolds number than at large Reynolds number. This difference may solve an existing controversy between measurements in SF6 (large $Re$) and in alcohol/water (lower $Re$). We link these regimes with a possible global bifurcation in the turbulent mean flow. We further show how a scaling theory could be used to describe these two regimes through a single universal function. This function presents a bimodal character for intermediate range of Reynolds number. We explain this bimodality in term of two dissipation regimes, one in which fluctuation dominate, and one in which mean flow dominates. Altogether, our results provide a six parameters fit of the curve $Nu(Ra,Pr)$ which may be used to describe all measurements at $Pr\ge 0.7$.Comment: RevTex, 8 Figure

Statistical mechanics of two-dimensional Euler flows and minimum enstrophy states

A simplified thermodynamic approach of the incompressible 2D Euler equation is considered based on the conservation of energy, circulation and microscopic enstrophy. Statistical equilibrium states are obtained by maximizing the Miller-Robert-Sommeria (MRS) entropy under these sole constraints. The vorticity fluctuations are Gaussian while the mean flow is characterized by a linear $\bar{\omega}-\psi$ relationship. Furthermore, the maximization of entropy at fixed energy, circulation and microscopic enstrophy is equivalent to the minimization of macroscopic enstrophy at fixed energy and circulation. This provides a justification of the minimum enstrophy principle from statistical mechanics when only the microscopic enstrophy is conserved among the infinite class of Casimir constraints. A new class of relaxation equations towards the statistical equilibrium state is derived. These equations can provide an effective description of the dynamics towards equilibrium or serve as numerical algorithms to determine maximum entropy or minimum enstrophy states. We use these relaxation equations to study geometry induced phase transitions in rectangular domains. In particular, we illustrate with the relaxation equations the transition between monopoles and dipoles predicted by Chavanis and Sommeria [J. Fluid. Mech. 314, 267 (1996)]. We take into account stable as well as metastable states and show that metastable states are robust and have negative specific heats. This is the first evidence of negative specific heats in that context. We also argue that saddle points of entropy can be long-lived and play a role in the dynamics because the system may not spontaneously generate the perturbations that destabilize them.Comment: 26 pages, 10 figure

Scaling laws prediction from a solvable model of turbulent thermal convection

A solvable turbulent model is used to predict both the structure of the boundary layer and the scaling laws in thermal convection. The transport of heat depends on the interplay between the thermal, viscous and integral scales of turbulence, and thus, on both the Prandtl number and the Reynolds numbers. Depending on their values, a wide variety of possible regimes is found, including the classical 2/7 and 1/3 law, and a new $4/13=0.308$ law for the Nusselt power law variation with the Rayleigh number.Comment: RevTex4, 4 page

Fast Numerical simulations of 2D turbulence using a dynamic model for Subgrid Motions

We present numerical simulation of 2D turbulent flow using a new model for the subgrid scales which are computed using a dynamic equation linking the subgrid scales with the resolved velocity. This equation is not postulated, but derived from the constitutive equations under the assumption that the non-linear interactions of subgrid scales between themselves are equivalent to a turbulent viscosity.The performances of our model are compared with Direct Numerical Simulations of decaying and forced turbulence. For a same resolution, numerical simulations using our model allow for a significant reduction of the computational time (of the order of 100 in the case we consider), and allow the achievement of significantly larger Reynolds number than the direct method.Comment: 35 pages, 9 figure