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

由壁蒸腾扩展的湍流库埃特流的多点统计的基于对称的缩放

• 批准号: 267513790
• 负责人: Professor Dr.-Ing. Martin Oberlack
• 金额: --
• 依托单位: Fachgebiet Strömungsdynamik
• 依托单位国家: 德国
• 项目类别: Research Grants
• 财政年份: 2015
• 资助国家: 德国
• 起止时间: 2014-12-31 至 2022-12-31
• 项目状态: 已结题
• 来源: https://gepris.dfg.de/gepris/projekt/267513790?language=en
• 关键词: Symmetry; based; scaling; multi; point

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.

本提案的最终目标是加深我们对基于Lie对称性的湍流剪切流的认识，并将我们对多点相关方程（MPCE）的了解扩展到更基本的概率密度函数（PDF）Ludding-Monin-诺维科夫方程。我们对正则剪切流的某些对称性是活跃的或被破坏的这一事实的理解必须被完全修正，因为物理机制，由于壁面蒸发似乎打破了平均速度的某些对称性，然而，同时又产生了高阶相关性的新对称性，本建议旨在通过分析有和无蒸发的湍流Couette流，特别是通过PDF方法补充，来修正基于对称性的湍流理论的部分，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Lie对称群的解。基于这一理论，本申请人推导出了平面剪切流平均速度的各种标度律（2000、2001），除平面Couette情况外，所有这些律此后都得到了毫无疑问的验证。2010年取得了实质性进展，本申请人推导出了MPCE的Lie对称性的扩展集。这显著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。在前体提案中，甚至对具有恒定壁蒸发的湍流Poiffille流预测了一个新的对数中心区域标度律，并令人信服地验证了包括大尺度DNS数据的所有相关应力。该理论的各种问题尚未解决：（i）最近对湍流库埃特流的大规模DNS很好地验证了附加的新的统计对称性，尽管对于平均速度，后者只对库埃特流有效，原因尚不清楚。(ii)更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分量时开启，而对于非平行流，即由于壁蒸发，这些对称性对于所有二阶矩都是关键的。 PDF的对称不变解必须服从PDF的非负性限制，这对解以及群参数的值构成了重要的约束。反过来，这会对标度律参数（如对数律参数$\kappa$）施加约束，最终目标是获得它们的第一原理约束。