Wang Qi, Verzicco Roberto, Lohse Detlef, Shishkina Olga
Physics of Fluids Group and Max Planck Center for Complex Fluid Dynamics, MESA+ Institute and J. M. Burgers Centre for Fluid Dynamics, University of Twente, P.O. Box 217, 7500AE Enschede, Netherlands.
Department of Modern Mechanics, University of Science and Technology of China, Hefei 230027, China.
Phys Rev Lett. 2020 Aug 14;125(7):074501. doi: 10.1103/PhysRevLett.125.074501.
Wall-bounded turbulent flows 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∼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 shape of the rolls 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≤Γ_{r}≤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.
即使控制参数的值完全相同，壁面受限湍流也可以呈现出具有不同输运特性的不同统计稳态湍流状态。系统处于何种状态取决于初始条件。在此，我们将具有无滑移平板和水平周期性边界条件的大纵横比（Γ）二维湍流瑞利 - 贝纳德流中的多种状态作为模型系统进行分析。我们确定对流卷的数量(n)、它们的平均纵横比(\Gamma_{r}=\Gamma/n)以及相应的流动输运特性（即努塞尔数(Nu)），作为瑞利数（(Ra)）和普朗特数等控制参数的函数。发现努塞尔数(Nu\sim Ra^{\beta})中的有效标度指数(\beta)取决于所实现的状态，进而取决于(\Gamma_{r})，(\Gamma_{r})越小，(\beta)值越大。通过利用广义弗里德里希不等式，我们表明卷的椭圆形状和粘性阻尼决定了可实现湍流状态的(\Gamma_{r})窗口。理论结果与我们的数值发现(2/3\leq\Gamma_{r}\leq4/3)非常吻合，对于较大的(Ra)接近下限阈值。最后，我们表明用于确定(\Gamma_{r})的理论方法对于自由滑移边界条件也有效。
