We report a combined experimental and numerical study of the effect of boundary layer (BL) fluctuations on the scaling properties of the mean temperature profile $\unicode[STIX]{x1D703}(z)$ and temperature variance profile $\unicode[STIX]{x1D702}(z)$ in turbulent Rayleigh–Bénard convection in a thin disk cell and an upright cylinder of aspect ratio unity. Two scaling regions are found with increasing distance $z$ away from the bottom conducting plate. In the BL region, the measured $\unicode[STIX]{x1D703}(z)$ and $\unicode[STIX]{x1D702}(z)$ are found to have the scaling forms $\unicode[STIX]{x1D703}(z/\unicode[STIX]{x1D6FF})$ and $\unicode[STIX]{x1D702}(z/\unicode[STIX]{x1D6FF})$, respectively, with varying thermal BL thickness $\unicode[STIX]{x1D6FF}$. The functional forms of the measured $\unicode[STIX]{x1D703}(z/\unicode[STIX]{x1D6FF})$ and $\unicode[STIX]{x1D702}(z/\unicode[STIX]{x1D6FF})$ in the two convection cells agree well with the recently derived BL equations by Shishkina et al. (Phys. Rev. Lett., vol. 114, 2015, 114302) and by Wang et al. (Phys. Rev. Fluids, vol. 1, 2016, 082301). In the mixing zone outside the BL region, the measured $\unicode[STIX]{x1D703}(z)$ remains approximately constant, whereas the measured $\unicode[STIX]{x1D702}(z)$ is found to scale with the cell height $H$ in the two convection cells and follows a power law, $\unicode[STIX]{x1D702}(z)\sim (z/H)^{\unicode[STIX]{x1D716}}$, with the obtained values of $\unicode[STIX]{x1D716}$ being close to $-1$. Based on the experimental and numerical findings, we derive a new equation for $\unicode[STIX]{x1D702}(z)$ in the mixing zone, which has a power-law solution in good agreement with the experimental and numerical results. Our work demonstrates that the effect of BL fluctuations can be adequately described by the velocity–temperature correlation functions and the new BL equations capture the essential physics.
Finite difference methods are traditionally used for modelling the time domain in numerical weather prediction (NWP). Time-spectral solution is an attractive alternative for reasons of accuracy and efficiency and because time step limitations associated with causal, CFL-like critera are avoided. In this work, the Lorenz 1984 chaotic equations are solved using the time-spectral algorithm GWRM. Comparisons of accuracy and efficiency are carried out for both explicit and implicit time-stepping algorithms. It is found that the efficiency of the GWRM compares well with these methods, in particular at high accuracy. For perturbative scenarios, the GWRM was found to be as much as four times faster than the finite difference methods. A primary reason is that the GWRM time intervals typically are two orders of magnitude larger than those of the finite difference methods. The GWRM has the additional advantage to produce analytical solutions in the form of Chebyshev series expansions. The results are encouraging for pursuing further studies, including spatial dependence, of the relevance of time-spectral methods for NWP modelling.
We report on Rayleigh-Bénard convection with strongly varying fluid properties experimentally and theoretically. Using pressurized sulfur-hexafluoride (SF 6) above its critical point, we are able to make measurements at mean temperatures (T m) and pressures (P m) along Prandtl-number isolines in the (T, P) parameter space. This allows us to keep the mean Rayleigh-(Ra m) and Prandtl number (Pr m) constant while changing the temperature dependences of the fluid properties independently, e.g., probing the liquidlike or gaslike region that are left and right of the supercritical isochore. Hence, non-Oberbeck-Boussinesq (NOB) effects can be measured and analyzed cleanly. We measure the temperature at midheight (T c) as well as the global vertical heat flux. We observe a significant heat transport enhancement of up to 112% under strong NOB conditions. Furthermore, we develop a theoretical model for the global vertical heat flux based on ideas of Grossmann and Lohse (GL) in OB systems, adjusted for nonconstant fluid properties. In this model, the NOB effects influence the boundary layer and hence T c , but the change of the heat flux is predominantly due to a change of the fluid properties in the bulk, in particular the heat capacity c p and density ρ. Predictions from our model are consistent with our experimental results as well as with previous measurements carried out in pressurized ethane and cryogenic helium.
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