AI 中文总结
本文质疑Boussinesq近似能否正确描述Rayleigh-Bénard对流的统计定常状态，通过能量分析推导标度关系，发现三维近似尚可而二维完全失效，并建议用弱可压缩DNS验证。
AI 中文摘要
人们普遍认为，Rayleigh-Bénard对流（RBC）的统计定常状态可以用Boussinesq近似的统计定常解来描述。基于对机械能方程的分析，我们认为这一观点并不成立，因为该近似隐含给出的由压缩和膨胀引起的内能向动能平均转化的表达式，并不能算作一种近似。因此，将定常状态下的平均动能耗散与Rayleigh数和Nusselt数联系起来的精确Boussinesq表达式是否有效，是值得怀疑的。利用部分超出Boussinesq近似的假设，我们推导了三维和二维系统中定常状态下平均动能耗散的标度关系。三维系统的关系式与Boussinesq表达式类似，但包含一个未知的前置因子，而二维系统的关系式则完全不同。我们的论证表明，三维情况下Boussinesq近似的统计定常解能较好地再现各统计量之间的标度关系，尽管严格意义上它们不应被视为近似；而二维情况下的解则完全失效。我们建议通过执行弱可压缩RBC的直接数值模拟（DNS）并与Boussinesq RBC的DNS进行比较来研究这一问题。
英文摘要
It is generally believed that statistically stationary states of Rayleigh-Bénard convection (RBC) can be described by statistically stationary solutions to the Boussinesq approximation. Based on an analysis of the mechanical energy equation, we argue that this belief is not justified, because the expression for mean conversion of internal energy into kinetic energy by compressions and expansions, which is implicitly given by the approximation, does not qualify as an approximation. In consequence, it is questionable whether the exact Boussinesq expression relating the mean kinetic energy dissipation in a stationary state to the Rayleigh and Nusselt numbers is valid. Using assumptions that partly lie outside the Boussinesq approximation, we derive scaling relations for the mean kinetic energy dissipation in a stationary state in three and two dimensions. The relation for the three-dimensional system is similar to the Boussinesq expression, but includes an unknown prefactor, while the relation for the two-dimensional system is completely different. Our arguments suggest that statistically stationary solutions to the Boussinesq approximation in three dimensions, reproduce scaling relations between various statistical quantities quite well, although they should not be regarded as approximations in a strict sense, while solutions in two dimensions completely fail. We suggest that this should be investigated by performing DNS of weakly compressible RBC and compare with DNS of Boussinesq RBC.
CommentsI discovered a logical error in the paper
