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Register a new userHigh Rayleigh number variational multiscale large eddy simulations of Rayleigh-B{e}nard Convection
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The variational multiscale (VMS) formulation is used to develop residual-based VMS large eddy simulation (LES) models for Rayleigh-B{e}nard convection. The resulting model is a mixed model that incorporates the VMS model and an eddy viscosity model. The Wall-Adapting Local Eddy-viscosity (WALE) model is used as the eddy viscosity model in this work. The new LES models were implemented in the finite element code Drekar. Simulations are performed using continuous, piecewise linear finite elements. The simulations ranged from $Ra = 10^6$ to $Ra = 10^{14}$ and were conducted at $Pr = 1$ and $Pr = 7$. Two domains were considered: a two-dimensional domain of aspect ratio 2 with a fluid confined between two parallel plates and a three-dimensional cylinder of aspect ratio $1/4$. The Nusselt number from the VMS results is compared against three dimensional direct numerical simulations and experiments. In all cases, the VMS results are in good agreement with existing literature.
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If a fluid flow is driven by a weak Gaussian random force, the nonlinearity in the Navier-Stokes equations is negligibly small and the resulting velocity field obeys Gaussian statistics. Nonlinear effects become important as the driving becomes stronger and a transition occurs to turbulence with anomalous scaling of velocity increments and derivatives. This process has been described by V. Yakhot and D. A. Donzis, Phys. Rev. Lett. 119, 044501 (2017) for homogeneous and isotropic turbulence (HIT). In more realistic flows driven by complex physical phenomena, such as instabilities and nonlocal forces, the initial state itself, and the transition to turbulence from that initial state, are much more complex. In this paper, we discuss the Reynolds-number-dependence of moments of the kinetic energy dissipation rate of orders 2 and 3 obtained in the bulk of thermal convection in the Rayleigh-B{e}nard system. The data are obtained from three-dimensional spectral element direct numerical simulations in a cell with square cross section and aspect ratio 25 by A. Pandey et al., Nat. Commun. 9, 2118 (2018). Different Reynolds numbers $1 lesssim {rm Re}_{ell} lesssim 1000$ which are based on the thickness of the bulk region $ell$ and the corresponding root-mean-square velocity are obtained by varying the Prandtl number Pr from 0.005 to 100 at a fixed Rayleigh number ${rm Ra}=10^5$. A few specific features of the data agree with the theory but the normalized moments of the kinetic energy dissipation rate, ${cal E}_n$, show a non-monotonic dependence for small Reynolds numbers before obeying the algebraic scaling prediction for the turbulent state. Implications and reasons for this behavior are discussed.
For two-dimensional Rayleigh-B{e}nard convection, classes of unstable, steady solutions were previously computed using numerical continuation (Waleffe, 2015; Sondak, 2015). The `primary steady solution bifurcates from the conduction state at $Ra approx 1708$, and has a characteristic aspect ratio (length/height) of approximately $2$. The primary solution corresponds to one pair of counterclockwise-clockwise convection rolls with a temperature updraft in between and an adjacent downdraft on the sides. By adjusting the horizontal length of the domain, (Waleffe, 2015; Sondak, 2015) also found steady, maximal heat transport solutions, with characteristic aspect ratio less than $2$ and decreasing with increasing $Ra$. Compared to the primary solutions, optimal heat transport solutions have modifications to boundary layer thickness, the horizontal length scale of the plume, and the structure of the downdrafts. The current study establishes a direct link between these (unstable) steady solutions and transition to turbulence for $Pr = 7$ and $Pr = 100$. For transitional values of $Ra$, the primary and optimal heat transport solutions both appear prominently in appropriately-sized sub-fields of the time-evolving temperature fields. For $Ra$ beyond transitional, our data analysis shows persistence of the primary solution for $Pr = 7$, while the optimal heat transport solutions are more easily detectable for $Pr = 100$. In both cases $Pr = 7$ and $Pr = 100$, the relative prevalence of primary and optimal solutions is consistent with the $Nu$ vs. $Ra$ scalings for the numerical data and the steady solutions.
The shape of velocity and temperature profiles near the horizontal conducting plates in turbulent Rayleigh-B{e}nard convection are studied numerically and experimentally over the Rayleigh number range $10^8lesssim Ralesssim3times10^{11}$ and the Prandtl number range $0.7lesssim Prlesssim5.4$. The results show that both the temperature and velocity profiles well agree with the classical Prandtl-Blasius laminar boundary-layer profiles, if they are re-sampled in the respective dynamical reference frames that fluctuate with the instantaneous thermal and velocity boundary-layer thicknesses.
We present mesoscale numerical simulations of Rayleigh-B{e}nard convection in a two-dimensional concentrated emulsion, confined between two parallel walls, heated from below and cooled from above, under the effect of buoyancy forces. The systems under study comprise finite-size droplets, whose concentration $Phi_0$ is varied, ranging from the dilute limit up to the point where the emulsion starts to be packed and exhibits non-Newtonian rheology. We focus on the characterisation of the convective heat transfer properties close to the transition from conductive to convective states. The convective flow is confined and heterogeneous, which causes the emulsion to exhibit concentration heterogeneities in space $phi_0(y)$, depending on the location in the wall-to-wall direction ($y$). With the aim of assessing quantitatively the heat transfer efficiency of such heterogeneous systems, we resort to a side-by-side comparison between the concentrated emulsion system and a single-phase (SP) system, whose local viscosity $eta^{mbox{SP}}(y)$ is suitably constructed from the shear rheology of the emulsion. Such comparison highlights that a suitable degree $Lambda$ of coarse-graining needs to be introduced in the local viscosity $eta_{Lambda}^{mbox{SP}}(y)$, in order for the single-phase system to attain the same heat transfer efficiency of the emulsion. Specifically, it is shown that a quantitative matching between the two systems is possible whenever the coarse-graining is performed over a scale of the order of the droplet size.
Results from direct numerical simulation for three-dimensional Rayleigh-Benard convection in samples of aspect ratio $Gamma=0.23$ and $Gamma=0.5$ up to Rayleigh number $Ra=2times10^{12}$ are presented. The broad range of Prandtl numbers $0.5<Pr<10$ is considered. In contrast to some experiments, we do not see any increase in $Nu/Ra^{1/3}$, neither due to $Pr$ number effects, nor due to a constant heat flux boundary condition at the bottom plate instead of constant temperature boundary conditions. Even at these very high $Ra$, both the thermal and kinetic boundary layer thicknesses obey Prandtl-Blasius scaling.
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