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

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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.