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Research Papers: Fundamental Issues and Canonical Flows

# Self-Similarity Analysis of Turbulent Wake Flows

[+] Author and Article Information
Tie Wei

Department of Mechanical Engineering,
New Mexico Institute of Mining and Technology,
Socorro, NM 87801
e-mail: tie.wei@nmt.edu

Contributed by the Fluids Engineering Division of ASME for publication in the JOURNAL OF FLUIDS ENGINEERING. Manuscript received April 8, 2016; final manuscript received December 1, 2016; published online March 16, 2017. Assoc. Editor: Oleg Schilling.

J. Fluids Eng 139(5), 051203 (Mar 16, 2017) (6 pages) Paper No: FE-16-1223; doi: 10.1115/1.4035633 History: Received April 08, 2016; Revised December 01, 2016

## Abstract

This paper investigates the self-similarity properties in the far downstream of high Reynolds number turbulent wake flows. The growth rate of the wake layer width, $dδ/dx$; the decaying rate of the maximum velocity defect, $dUs/dx$; and the scaling for the maximum mean transverse (across the stream) velocity, Vmax, are derived directly from the self-similarity of the continuity equation and the mean momentum equation. The analytical predictions are validated with the experimental data. Using an approximation function for the mean axial flow, the self-similarity analysis yields approximate solutions for the mean transverse velocity, V, and the Reynolds shear stress, $T=−〈uv〉$. Close relations among the shapes of U, V, and T are revealed.

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## Figures

Fig. 1

Schematic of a statistically two-dimensional stationary wake flow (statistics do not vary in the third, z−, direction). Thedominant direction of mean flow is in the axial direction, x, and the cross-stream coordinate is y. Mean velocity along thecenterline is denoted as Uc, where the velocity defect is maximum, Us=U∞−Uc. Wake half-width, δ=y0.5, is defined as U(y0.5)=U∞−0.5Us.

Fig. 2

Transverse distribution of the mean flow in the far-wake of turbulent wake flows. (a) The mean axial velocity defect profile, f≡(U∞−U)/Us. (b) Mean transverse velocity and Reynolds shear stress. Experimental data are from Ref. [2] (WCM), Ref. [16] (Hickey), Ref. [17] (Liu), Ref. [15] (GH at two downstream locations, x = 1 m and x = 0.58 m), and Ref. [11] (OW). Note that the Reynolds shear stress should be antisymmetric. GH (x = 0.58 m) data at positive y agree well with the two other data, but not in the negative y.

Fig. 3

The mean axial velocity defect, f, along with curves of Gaussian function e−aη2. Curves with three different constants a=0.6,0.7,0.8 are plotted. The curve with a = 0.7 seems to fit the data better.

Fig. 4

Experimental data of g/(ηf). The difference between the values at x=0.34 m, and the rest is attributed to the near-wake effect.

Fig. 5

Scaling of the maximum mean transverse velocity, Vmax, in the far downstream of turbulent wake flow

Fig. 6

Approximation function for the mean transverse velocity, g=V/Vmax≈−2ηe−aη2

Fig. 7

Approximation function for the Reynolds shear stress, h=T/Us2≈0.092ηe−aη2

Fig. 8

(a) Relation between V and U. Solid symbols are data of V/Vmax, and open symbols are data of −2ηf. (b) Relation between T and U. Solid symbols are data of T/Us2, open symbols 0.092ηf. (c) Relation between T and V. Solid symbols are data of −T/Us2 and open symbols (V/Us)(U∞/Us).

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