Multiple states in turbulent large-aspect-ratio thermal convection: What determines the number of rolls?

  • Shishkina, Olga (Max Planck Institute for Dynamics and Self-Or)
  • Lohse, Detlef (Physics of Fluids Group, Max Planck Center Tw)
  • Wang, Qi (Physics of Fluids Group, Max Planck Center Tw)
  • Verzicco, Roberto (University of Rome `Tor Vergata')

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Recent findings suggest that wall-bounded turbulent flow can take different statistically stationary turbulent states, with different transport properties, even for the very same values of the control parameters. What state the system takes depends on the initial conditions. Here we analyze the multiple states in the large-aspect ratio ($\Gamma$) two-dimensional turbulent Rayleigh--B\'enard flow with no-slip plates and horizontally periodic boundary conditions as a model system. We determine the number $n$ of convection rolls, their mean aspect ratios $\Gamma_r = \Gamma /n$, and the corresponding transport properties of the flow (i.e., the Nusselt number $Nu$ and Reynolds number $Re$), as a function of the control parameters Rayleigh ($Ra$) and Prandtl number ($Pr$). The effective scaling exponent $\beta$ in $Nu \sim Ra^\beta$ is found to depend on the realized state and thus $\Gamma_r$, with a larger value for the smaller $\Gamma_r$. By making use of Poincar\'e--Friedrichs inequalities, one can derive that for two-dimensional Rayleigh--B\'enard convection with no-slip boundary conditions at the plates, $\Gamma_r$ can take values only within a quite restricted range, $0.6<\Gamma_r<1.7$ (see section V-B in Ref.\ [1]). For specific values of $Ra$ and $Pr$, this estimate can be refined, which leads to a more precise $\Gamma_r$ window for the realizable turbulent states, see Ref.\ [2]. The theoretical results are in excellent agreement with our numerical finding $2/3 \le \Gamma_r \le 4/3$, where the lower threshold is approached for the larger $Ra$ [2]. We currently plan to extend the method to Taylor--Couette turbulence, where we also expect to find multiple states with Taylor number dependent roll aspect ratios $\Gamma_r$, as experimentally found in Ref.\ [3] for a strongly turbulent case.