论文标题
雷利 - 贝纳德与旋转平面couette在2D中的确切关系
Exact relations between Rayleigh-Bénard and rotating plane Couette flow in 2D
论文作者
论文摘要
Rayleigh-Bénard对流（RBC）和Taylor-Couette流（TCF）是两个范式流体动力学系统，尽管它们的几何形状不同和强迫，但由于它们的许多相似性而经常一起讨论。通常，这些类比需要近似，但是在可以建立TCF变为旋转平面流动（RPC）精确关系的大半径的极限。当流量仅限于两个空间自由度时，有一个确切的规范，将RPC中的三个速度分量映射到两个速度成分和RBC中的一个温度场。使用此过程，我们推断了两个流之间的几个关系：（i）对流的瑞利数字$ ra $ ra $ re $ re $和rotation $r_Ω$ rpc流中的$r_Ω$数字与$ ra = re^re^2r_Ω（1-r_Ω）$相关。 （ii）热量和角动量传输的不同（1-r_Ω）$不同，解释了为什么即使$ ra $和$r_Ω$之间的关系具有此对称性，即使角动量传输不在$r_Ω= 1/2 $左右对称。这种关系导致预测的$r_Ω$，它最大化了角动量传输，与完整3D系统的现有数值模拟非常吻合。 （iii）两个流中的一个变量满足最大原理，即田地的极值出现在墙壁上。因此，在此二维设置中，剪切流\ emph {不能}中的回流事件发生。 （iv）对于轴向和径向速度组件上的自由滑移边界条件，对RBC的先前严格分析意味着，RPC中的方位角动量传输的限制在上面的限制在$ re re^{5/6} $中，其标度指数小于预期的$ re^1 $。
Rayleigh-Bénard convection (RBC) and Taylor-Couette Flow (TCF) are two paradigmatic fluid dynamical systems frequently discussed together because of their many similarities despite their different geometries and forcing. Often these analogies require approximations, but in the limit of large radii where TCF becomes rotating plane Couette flow (RPC) exact relations can be established. When the flows are restricted to two spatial degrees of freedom there is an exact specification that maps the three velocity components in RPC to the two velocity components and one temperature field in RBC. Using this, we deduce several relations between both flows: (i) The Rayleigh number $Ra$ in convection and the Reynolds $Re$ and rotation $R_Ω$ number in RPC flow are related by $Ra= Re^2 R_Ω(1-R_Ω)$. (ii) Heat and angular momentum transport differ by $(1-R_Ω)$, explaining why angular momentum transport is not symmetric around $R_Ω=1/2$ even though the relation between $Ra$ and $R_Ω$ has this symmetry. This relationship leads to a predicted value of $R_Ω$ that maximizes the angular momentum transport that agrees remarkably well with existing numerical simulations of the full 3D system. (iii) One variable in both flows satisfy a maximum principle i.e., the fields' extrema occur at the walls. Accordingly, backflow events in shear flow \emph{cannot} occur in this two-dimensional setting. (iv) For free slip boundary conditions on the axial and radial velocity components, previous rigorous analysis for RBC implies that the azimuthal momentum transport in RPC is bounded from above by $Re^{5/6}$ with a scaling exponent smaller than the anticipated $Re^1$.
