Symmetry based scaling of the multi-point statistics of a turbulent Couette flow extended by wall-transpiration

本研究的最终目标是加深我们对基于李氏对称性的湍流剪切流的认识，并将我们对多点相关方程（MPCE）的认识扩展到更基本的概率密度函数（PDF） Ludgren-Monin-Novikov方程。我们对典型剪切流的某些对称性是活跃的或被打破的事实的理解必须完全修正，因为物理机制，如壁面蒸腾似乎打破了平均速度的某些对称性，然而，同时又产生了更高阶相关性的新对称性。本提案的目的是通过分析有蒸腾和没有蒸腾的Couette湍流，特别是PDF方法，对基于对称性的湍流理论的部分内容进行修正，从而缩小这一差距。后一种流动是理想的，因为某些对称性可以自由地打开和关闭。为此，我们需要理解湍流标度律是基于李对称群的解。基于这一理论，本申请人为平面剪切流的平均速度生成了各种标度定律（2000,2001），除平面Couette情况外，所有这些都得到了无疑的验证。在2010年取得了实质性的进展，目前的申请人推导了MPCE的李氏对称性的扩展集。这极大地修正了我们对湍流统计的理解，最重要的是，还提供了计算更高相关性的缺失环节，包括经典的近壁对数区域，这一点得到了很好的验证。在前体方案中，我们甚至预测了一个新的对数中心区域标度律，用于恒定壁蒸腾的湍流泊泽维尔流，并在大规模DNS数据中得到了令人信服的验证，包括所有相关应力。尽管如此，该理论的各种问题仍未解决：(i)最近对湍流库埃特流的大规模DNS很好地验证了额外的新的统计对称性，尽管对于平均速度，后者仅在库埃特流中出现活跃，原因尚不清楚。（ii）较高的相关性选择性地依赖于这些新的统计对称性，例如，对于完全平行的剪切流，这种对称性仅在11分量中开启，而对于非平行流，即由于壁面蒸腾，这些对称性对于所有秒矩都是关键的。最后，也是最重要的一点，所有的对称性在MPCE和更中心的PDF方程中都有对应的对称性。PDF的对称不变解必须服从PDF的非负性约束，这对解和群参数的值都有很大的约束。反过来，这会对缩放律参数施加约束，例如对数律参数$\kappa$，最终目标是获得它们的第一原理约束。

The ultimate goal of the present proposal is to deepen our knowledge on turbulent shear flows based on Lie symmetries, and to extend our knowledge of the multi-point correlation equations (MPCE) to the companion more fundamental probability density function (PDF) Ludgren-Monin-Novikov equations.Our understanding of the fact that certain symmetries of canonical shear flow are active or broken has to be completely revised as physical mechanisms such as wall transpiration appears to break certain symmetries for the mean velocity, however, at the same time gives rise to new symmetries for higher order correlations.The present proposal aims in closing this gap by revising parts of the symmetry based turbulence theory by analysing the turbulent Couette flow with and without transpiration supplemented by related large Reynolds number DNS particularly focussing on the PDF approach. The latter flow is ideal in the sense that certain symmetries may be freely switched on and off.For this we need to comprehend that turbulent scaling laws are Lie symmetry group based solutions. Based on this theory the present applicant generated various scaling laws for the mean velocity of plane shear flows (2000,2001), all of which, except the plane Couette case, have since then been undoubtedly validated.A substantial progress was gained in 2010 where the present applicant derived an extended set of Lie symmetries of the MPCE. This significantly revised our understanding of turbulence statistics and, most important, also delivered the missing link to compute higher correlations, which was nicely validated including the classical near-wall log-region.In the precursor proposal even a new logarithmic centre region scaling law was forecasted for a turbulent Poiseuille flow with constant wall transpiration and convincingly validated including all related stresses against large scale DNS data.Still, various issues of the theory are unresolved: (i) recent large-scale DNS for the turbulent Couette flow nicely validated additional new statistical symmetries though for the mean velocity the latter only appear active for the Couette flow the reason being unknown. (ii) the higher correlations selectively rely on these new statistical symmetries, e.g. for fully parallel shear flows this symmetry is only switched on for the 11-component while for non-parallel flows, i.e due to wall transpiration, these symmetry is pivotal for all second moments.Finally, and most important, all symmetries have their counterparts both in the MPCE and in the more central PDF equations. Symmetry-invariant solutions for the PDF have to obey the non-negativity restriction of PDFs, which poses a significant constraint onto the solution and hence on the values of the group parameters. In turn, this inflicts constraints onto the scaling law parameter such as log-law parameter $\kappa$ with the eventual goal to obtain first principle constraints for them.