arXiv:1201.0100v1 [physics.flu-dyn] 30 Dec 2011 A multi-layer model for turbulent kinetic energy in pipe flows Xi Chen 1 , Fazle Hussain 2 , Zhen-Su She 1* 1 State Key Laboratory for Turbulence and Complex Systems and Department of Mechanics and Aerospace Engineering, College of Engineering, Peking University, Beijing 100871, P.R. China 2 Department of Mechanical Engineering, University of Houston, Houston, TX 77204-4006, USA (Dated: February 13, 2019) A multi-layer model of an energy length function is developed by employing recent results of the authors [1]. The theory predicts the complete, mean streamwise turbulent kinetic-energy profile (MKP), in good agreement with empirical data for a wide range of Reynolds numbers (Re ). In particular, a critical Reτ around 5000 is predicted, beyond which a scaling anomaly appears and MKP develops a second peak whose location varies as Re 1/2 τ /κ (κ is the Karman constant) and kinetic energy peak value as 3.5Re 0.06 τ . Development of an analytical theory for the predic- tion of turbulent mean velocity profile (MVP) and mean streamwise kinetic-energy profile (MKP) for the entire pipe flows has persisted as an elusive goal for over a cen- tury. Recently, notable progress has been made in the former [1, 2], but very little in the latter. An interesting observation in MKP is the second kinetic energy peak located in the log layer at high Re [3], see Fig.1, which signals a new outer structure [3, 4], but not revealed by the existing models [5–7]. To describe such an outer peak, a centerline velocity scaling is newly proposed [8], how- ever, a complete solution for MKP is still missing. A vi- able theory should provide both MVP and MKP leading to an integrated understanding of mean fields and tur- bulent fluctuations - not achieved in any of the current turbulence theories. We extend a newly developed statistical mean-field theory based on the symmetries of ”order parameter ” [9], which are defined, for wall-bounded turbulent flows, as a set of characteristic lengths. The first example of such length is the mixing length [10], ℓ + m =  −〈uv〉 + /S + (1) where the mean shear S + = dU + /dy + , Reynolds stress W + = −〈 uv〉 + , and + denotes wall units normalization, which is shown [1] to displays a set of distinct dilation invariance in each of the commonly know layers such as viscous sublayer, buffer layer, bulk zone and core zone. A quantitative multi-layer model for the mixing length is derived [1], utilizing empirical data, as ℓ + m =0.967  y + 9.8  3/2  1+( y + 9.8 ) 4  1/8  1+( y + 44.7 ) 4  -1/4 (1-r m ) mZc(1-r) ( 1+( r 0.67 ) -2 ) 1/4 , (2) where variables, y + and r, are used to denote the dis- tance to the wall and to the center, respectively, with Z c = (1 + 0.67 2 ) 1/4 ≈ 1.097 (m = 4 for channel and 5 for pipe). This four-layer model was shown to give an accurate description for MVP over a wide range of Re (for Re τ > 5000) in Princeton pipe. Here, we introduce, in a similar way, a new length func- tion, called energy length, relating the streamwise kinetic FIG. 1: (color online). MKP as a function of the normalized distance from wall. The numerical (DNS) data points (solid) are from [11]; the experimental (EXP) data points (open) are from [3]; and the lines are from the present model (dashed line for Reτ = 10 6 ). energy to mean shear: ℓ + u =  〈uu〉 + /S + (3) Phenomenologically, both ℓ + m and ℓ + u are the sizes of en- ergy containing eddies [2], which are responsible for mo- mentum transfer: ℓ m corresponds to the vertical size of the characteristic eddy, and ℓ u to the longitudinal. In addition to all layers seen in ℓ m [1], the kinetic energy length here discovers and quantifies a new layer beyond the buffer layer, which is identified to be the so-called meso-layer [12]. The multi-layer formula of ℓ + u is ℓ + u = ρ u y +  1+( y + y + sub ) p1  γ 1 p 1  1+( y + y + buf ) p2  γ 2 p 2  1+( y + y + meso ) p3  γ 3 p 3 (1-r m ) m ˜ Zc(1-r) ( 1+( ˜ rc r ) p4 ) 1 p 4 , (4) In this five-layer model (Eq. 4), in addition to the coef- ficient ρ u , there are three sets of physical parameters - namely scaling (γ 1 , γ 2 , γ 3 ), layer thickness (y + sub , y + buf , y + meso ,˜ r c ) and transition sharpness (p 1 , p 2 , p 3 , p 4 ) with ˜ Z c = (1 + ˜ r p4 c ) 1/p4 . Although many, they are determined layer by layer, and are not free parameters. Among them,