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

Abstract

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 large-aspect ratio ($Γ$) two-dimensional turbulent Rayleigh--Bénard flow with no-slip plates and horizontally periodic boundary conditions as model system. We determine the number $n$ of convection rolls, their mean aspect ratios $Γ_r = Γ/n$, and the corresponding transport properties of the flow (i.e., the Nusselt number $Nu$), as function of the control parameters Rayleigh ($Ra$) and Prandtl number. The effective scaling exponent $β$ in $Nu \sim Ra^β$ is found to depend on the realized state and thus $Γ_r$, with a larger value for the smaller $Γ_r$. By making use of a generalized Friedrichs inequality, we show that the elliptical instability and viscous damping determine the $Γ_r$-window for the realizable turbulent states. The theoretical results are in excellent agreement with our numerical finding $2/3 \le Γ_r \le 4/3$, where the lower threshold is approached for the larger $Ra$. Finally, we show that the theoretical approach to frame $Γ_r$ also works for free-slip boundary conditions.