Improved bounds on horizontal convection
Improved bounds on horizontal convection
复制标题
改进水平对流的界限
DOI：
10.1017/jfm.2019.850
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发表时间：
2019
影响因子：
3.7
通讯作者：
W. Young
中科院分区：
文献类型：
--
作者：
C. Rocha;Thomas Bossy;Stefan G. Llewellyn Smith;W. Young
对于水平对流问题，基于熵产生的Nusselt数从上至上由$C，Ra^{1/3}$作为水平对流Rayleigh数$Ra\right tarrow\inty$，对于某个常数$C$(Siggers等人，《流体力学》，第517卷，2004年，第55-70页)。我们通过使用Wentzel-Kramers-Brillouin方法来解决$Ra\right tarrow\inty$极限中的变分问题，并给出了实现最终$Ra^{1/3}$标度的解决方案，重新审视了导致这一‘终极机制’的变分论点。正如预期的那样，优化流动有厚度为Ra-1/3的边界层压在非均匀加热表面上，但变分解也有沿壁面随波长的快速振荡变化。由于变分问题的精确解，对于无滑移和无应力边界条件，常数$C$比先前估计的值小了2.5$和1.6美元。这一$C$的适度减少表明，Siggers等人使用的不平等。(《流体机械》，第517卷，2004年，第55-70页)令人惊讶地准确。
For the problem of horizontal convection the Nusselt number based on entropy production is bounded from above by $C\,Ra^{1/3}$ as the horizontal convective Rayleigh number $Ra\rightarrow \infty$ for some constant $C$ (Siggers et al., J. Fluid Mech., vol. 517, 2004, pp. 55–70). We re-examine the variational arguments leading to this ‘ultimate regime’ by using the Wentzel–Kramers–Brillouin method to solve the variational problem in the $Ra\rightarrow \infty$ limit and exhibiting solutions that achieve the ultimate $Ra^{1/3}$ scaling. As expected, the optimizing flows have a boundary layer of thickness ${\sim}Ra^{-1/3}$ pressed against the non-uniformly heated surface; but the variational solutions also have rapid oscillatory variation with wavelength ${\sim}Ra^{-1/3}$ along the wall. As a result of the exact solution of the variational problem, the constant $C$ is smaller than the previous estimate by a factor of $2.5$ for no-slip and $1.6$ for no-stress boundary conditions. This modest reduction in $C$ indicates that the inequalities used by Siggers et al. (J. Fluid Mech., vol. 517, 2004, pp. 55–70) are surprisingly accurate.