Scaling of self-similar turbulent boundary layers under adverse pressure gradient

We present a theoretical and numerical study of the self-similarity turbulent boundary layers (SSTBLs) under adverse pressure gradient. Two relations are emphasized in theoretical extension. The first one explains the necessary relation to establish the same TBL with different choices of scaling, called coordinate transformation relation. The second relation is the equivalent requirement on the outer scaling of the mean velocity profile, named as scaling equivalence relation. Numerical verification is implemented with a set of ideal SSTBL velocity profiles that exhibit no Reynolds number effect by numerically solving the generalized governing equations of SSTBL under specified external conditions and self-similar scalings. Existing scalings in literature and two presently designed scalings are examined based on self-similar requirement to show their fulfillment of the two relations. Large-eddy simulations (LESs) of SSTBL are performed by directly specifying the pressure gradient parameter beta corresponding to different scalings. We show once the coordinate transformation relation is satisfied, the similar fully developed SSTBL can be settled with different scalings. Further detailed quantitative analysis confirms the validity of the two conditions. Two velocity profile models of SSTBL are evaluated with LES results. The former is based on the numerical solution of the SSTBL equations, and the latter is based on the similarity between the outer region of TBL and the laminar boundary layer. Both models work reasonably well for present LES.