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Poster presented at the 86. Annual Meeting of DPG and DPG-Frühjahrstagung (DPG Spring Meeting) of the Matter and Cosmos Section (SMuK) - Dresden, Germany - 20.03. - 24.03.2023

The Rayleigh number $Ra$ dependence of the Nusselt number $Nu$ in turbulent Rayleigh--B\'enard convection is numerically investigated for a moderate and low Prandtl number, $Pr=0.7$ and $0.021$, respectively. Here we specifically address the case of a Boussinesq fluid in a planar configuration with smooth horizontal walls and notionally infinite aspect ratio. Numerical simulations up to $Ra=10^{16}$ for $Pr=0.7$ and up to $Ra=8\times10^{13}$ for $Pr=0.021$ are made feasible on state-of-the-art workstations by utilising the stochastic one-dimensional turbulence (ODT) model. The ODT model parameters were estimated once for two combinations $(Pr,Ra)$ in the classical regime and kept fixed afterwards in order to address the predictive capabilities of the model. The ODT results presented exhibit various effective Nusselt number scalings $Nu\propto Ra^b$. The exponent changes from $b\approx1/3$ to $b\approx1/2$ when the $Ra$ number increases beyond the critical value $Ra_*\simeq6\times10^{14}$ ($Pr=0.7$) and $Ra_*\simeq6\times10^{11}$ ($Pr=0.021$), respectively. This is consistent with the literature. Furthermore, our results suggest that the transition to the ultimate regime is correlated with a relative enhancement of the temperature-velocity cross-correlations in the bulk of the fluid as hypothesised by Kraichnan, R. H., Phys. Fluids, 5, 1374 (1962)., Comment: Preprint

Direct numerical simulation (DNS), mostly used in fundamental turbulence research, is limited to low turbulent intensities due the current and future computer resources. Standard turbulence models, like RaNS (Reynolds averaged Navier-Stokes) and LES (Large Eddy Simulation), are applied to flows in engineering, but they miss small scale effects, which are frequently of importance, see e.g. the whole area of reactive flows, flows with apparent Prandtl or Schmidt number effects, or even wall bounded flows. A recent alternative to these standard approaches is the one-dimensional turbulence (ODT) model, which is limited to 1D sub-domains. In two papers we will provide a generalized filter strategy, called XLES (extended LES), including a formal theory (part I) and one special approach in the XLES family of models, called ODTLES (in part II (see Glawe et al. (2015))). ODTLES uses an ODT sub-grid model to describe all turbulent scales not represented by XLES, which leaves the larger scales to be simulated in 3D. This allows a turbulence modeling approach with a 3D resolution mainly independent of the turbulent intensity. Thus ODTLES is able to compute highly turbulent flows in domains of moderate complexity affordably and including the full range of turbulent and diffusive scales. The convergence of XLES to DNS is shown and the unconventional XLES advection approach is investigated in basic numerical tests. In part II, highly turbulent channel and duct flow results are discussed and show the future potential of XLES and ODTLES.