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Register a new userCoherent Solutions and Transition to Turbulence in Two-Dimensional Rayleigh-B{e}nard Convection
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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.
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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.
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.
Multi-fluid models have recently been proposed as an approach to improving the representation of convection in weather and climate models. This is an attractive framework as it is fundamentally dynamical, removing some of the assumptions of mass-flux convection schemes which are invalid at current model resolutions. However, it is still not understood how best to close the multi-fluid equations for atmospheric convection. In this paper we develop a simple two-fluid, single-column model with one rising and one falling fluid. No further modelling of sub-filter variability is included. We then apply this model to Rayleigh-B{e}nard convection, showing that, with minimal closures, the correct scaling of the heat flux (Nu) is predicted over six orders of magnitude of buoyancy forcing (Ra). This suggests that even a very simple two-fluid model can accurately capture the dominant coherent overturning structures of convection.
We investigate the structures of the near-plate velocity and temperature profiles at different horizontal positions along the conducting bottom (and top) plate of a Rayleigh-B{e}nard convection cell, using two-dimensional (2D) numerical data obtained at the Rayleigh number Ra=10^8 and the Prandtl number Pr=4.4 of an Oberbeck-Boussinesq flow with constant material parameters. The results show that most of the time, and for both velocity and temperature, the instantaneous profiles scaled by the dynamical frame method [Q. Zhou and K.-Q. Xia, Phys. Rev. Lett. 104, 104301 (2010) agree well with the classical Prandtl-Blasius laminar boundary layer (BL) profiles. Therefore, when averaging in the dynamical reference frames, which fluctuate with the respective instantaneous kinematic and thermal BL thicknesses, the obtained mean velocity and temperature profiles are also of Prandtl-Blasius type for nearly all horizontal positions. We further show that in certain situations the traditional definitions based on the time-averaged profiles can lead to unphysical BL thicknesses, while the dynamical method also in such cases can provide a well-defined BL thickness for both the kinematic and the thermal BLs.
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.
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