Research Papers: Fundamental Issues and Canonical Flows

On the Axisymmetric Turbulent Boundary Layer Growth Along Long Thin Circular Cylinders

[+] Author and Article Information
Stephen A. Jordan

Naval Undersea Warfare Center,
Newport, RI 02841
e-mail: stephen.jordan@navy.mil

Contributed by the Fluids Engineering Division of ASME for publication in the JOURNAL OF FLUIDS ENGINEERING. Manuscript received October 19, 2013; final manuscript received December 23, 2013; published online March 10, 2014. Assoc. Editor: Sharath S. Girimaji.

J. Fluids Eng 136(5), 051202 (Mar 10, 2014) (11 pages) Paper No: FE-13-1627; doi: 10.1115/1.4026419 History: Received October 19, 2013; Revised December 23, 2013

Even after several decades of experimental and numerical testing, our present-day knowledge of the axisymmetric turbulent boundary layer (TBL) along long thin circular cylinders still lacks a clear picture of many fundamental characteristics. The main issues causing this reside in the experimental testing complexities and the numerical simplifications. An important characteristic that is crucial for routine scaling is the boundary layer length scales, but the downstream growth of these scales (boundary layer, displacement, and momentum thicknesses) is largely unknown from the leading to trailing edges. Herein, we combine pertinent datasets with many complementary numerical computations (large-eddy simulations) to address this shortfall. We are particularly interested in expressing the length scales in terms of the radius-based and axial-based Reynolds numbers (Rea and Rex). Although the composite dataset gave an averaged shape factor H = 1.09 that is substantially lower than the planar value (H = 1.27), the shape factor distribution along the cylinder axis actually begins at the flat plate value then decays logarithmically to near unity. The integral length scales displayed power-law evolutions with variable exponents until high Rea (Rea > 35,000) where both scales then mimic streamwise consistency. Beneath this threshold, their streamwise growth is much slower than the flat plate (especially at low-Rea). The boundary layer thickness grew according to an empirical expression that is dependent on both Rea and Rex where its streamwise growth can far exceed the planar turbulent flow. These unique characteristics rank the thin cylinder axisymmetric TBL as a separate canonical flow, which was well documented by the previous investigations.

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Fig. 1

Sketch of turbulent boundary layer growth along a long thin cylinder at some time after an impulsive start

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Fig. 2

Sketches of the axisymmetric boundary layer characteristics for a momentum integral analysis of the thin cylinder: (a) spatial growth and (b) temporal growth

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Fig. 3

Sketch of inflow production procedure for turbulent boundary layer spatial growth along long thin cylinders

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Fig. 4

Final downstream distribution of the inflow parameters (transient phase) and corresponding axial velocity profile (in wall units) for LES case Rea = 620, Reθ = 3140, a+ = 38, and δi/a = 27; (a) Inflow parameters and (b) axial velocity

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Fig. 5

Spectral energies in the outer layers of a temporally matured turbulent boundary layer [12]; a=4.45cm, Rea=7475, and a+=353; (a) Streamwise fluctuations and (b) azimuthal fluctuations

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Fig. 6

Growth of the integral length scales (δ*,θ) under a spatially evolving turbulent boundary layer (Rea = 586, 7475, 11,213) and composite of experimental and numerical results defining the H factor (H=1.09); (a) Spatial growth and (b) composite

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Fig. 7

Compilation of experimental and numerical results defining a variable shape factor according a power-law; θ/a = 0.905(δ*/a)1.045

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Fig. 8

Experimental determinations [12,12] of the displacement thickness (δ*) spatial growth using the radius and axial distances as length scaling; (a) Radius length scaling and (b) axial length scaling

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Fig. 9

Spatial growth rates of the displacement thickness (δ*) using axial length scaling and composite of rates (ε) from experimental measurements [12] and present numerical computations; (a) Axial length scaling and (b) composite

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Fig. 10

Growth of the boundary layer thickness (Reδ and δ/a) relative to the momentum thickness (Reθ) and axial length (x/a) where the closed symbols are experimental measurements, open symbols and lines are computations; (a) Point evaluations and (b) spatial growth

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Fig. 11

Spatial growth rates of the boundary layer thickness (δ) using axial length scaling and composite of rates (γ) from experimental measurements [4,5,8-5,8] and numerical computations ([11], present); (a) Axial length scaling and (b) composite

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Fig. 12

Dependence of the power-law coefficient (ξ) on δ/a; previous experimental and present numerical results are the solid and open symbols, respectively

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Fig. 13

Turbulent boundary growth according to Eq. (18); (a) comparison among the experimental measurements [8] and axi-BLM computation [11] for 2a = 0.475 cm and Uo = 20 m/s and (b) effect by increasing Rea

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Fig. 14

Comparisons between experimental measurements, White's analysis, and present Eqs.(17) and (18) for axial growth the displacement and TBL thicknesses along the long thin cylinder; (a) TBL thickness and (b) displacement and TBL thicknesses



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