Research Papers: Fundamental Issues and Canonical Flows

Understanding Tow Tank Measurements of Total Drag for 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 March 27, 2013; final manuscript received December 9, 2013; published online January 27, 2014. Assoc. Editor: Mark F. Tachie.

J. Fluids Eng 136(3), 031205 (Jan 27, 2014) (10 pages) Paper No: FE-13-1197; doi: 10.1115/1.4026237 History: Received March 27, 2013; Revised December 09, 2013

Both the experimental measurements and numerical computations tell us that thin circular cylinders own turbulent boundary layer (TBL) characteristics which are dissimilar from planar geometries once the TBL thickness (δ) exceeds the cylinder radius. But the exceedingly long cylinders that serve as acoustic array systems have resorted to tow tank measurements due to the experimental complexities of axial sag or the required simplifications for efficient predictions. One key measurement in the tow tank experiments is the total drag, where the respective average tangential coefficients have been assessed relative to various scaled cylinder lengths, such as the length-based Reynolds number. We now know that the skin friction whose axial summation provides the total drag is governed by the radius-based Reynolds number as well. Herein, we revisit the many tow tank experiments to isolate the measurement discrepancies attributed to these two distinct Reynolds numbers. Once separated, these measurements provide a vital answer to the transformation of the TBL spatial growth to a temporal one (no additional net δ growth). This final stage is readily identifiable by a streamwise constant skin friction as given by near-wall flow homogeneity. This understanding lends subsequent evaluation of the TBL momentum thickness at the cylinder trailing edge. The present process involves applying several semiempirical expressions formed from the experimental and numerical evidence that supply the spatially evolving TBL characteristics along thin cylinders. Besides gaining an enhanced understanding of the TBL behavior, these semiempirical expressions are further clarified to form a final set that are helpful for design engineers of tow array devices in the military and oceanographic communities.

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Fig. 2

Sketch of the axisymmetric boundary layer characteristics of the spatially evolving state

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Fig. 1

Spatially underdeveloped skin friction along long thin cylinders

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Fig. 3

Axial growth of the TBL thicknesses along the thin cylinder

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Fig. 4

Axial evolution of the skin friction coefficient along a thin cylinder (a) Cf versus δ/a, and (b) Cf versus Rex

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Fig. 7

Axial distributions of the skin friction coefficient along a thin cylinder at fixed length (L/a = 2340) for four sets <CT, Rea>

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Fig. 8

Axial distribution of the skin friction coefficient along a thin cylinder at fixed lengths for four sets <CT, Rea> (a) L/a = 4680, and (b) L/a = 7010

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Fig. 9

Axial distribution and TBL transition to the temporal state along a thin cylinder at fixed length L/a = 4140 (2a = 36.8 mm) (a) (δ/a)t versus Rex, and (b) (δ/a)t versus Rea

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Fig. 10

Estimated TBL thickness at conversion to the temporal state from tow tank tests of four long thin cylinders (15.9 ≤ 2a (mm) ≤ 50.8) (a) δt/a versus Rea with 2640 ≤ L/a ≤ 11,405, and (b) δt/L versus ReL with 1920 ≤ L/a ≤ 7010

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Fig. 5

Average tangential drag coefficients from tow tank measurements in terms of ReL (a) Cipolla and Keith [12], and (b) Cipolla et al. [13]

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Fig. 6

Corrected average tangential drag coefficients in terms of ReL and Rea (a) Fig. 5(a), and (b) Fig. 5(b)

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Fig. 11

Estimates of the TBL thickness at initial conversion to the temporal state [0.89 ≤ 2a (mm) ≤ 50.8 with 1920 < L/a < 11,405].

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Fig. 12

Predictions of the TBL δ-growth and conversion to the temporal state (a) comparison to the DPIV observations of Cipolla et al. [13], and (b) conversion dependence on < L,a > in terms of the Reynolds numbers

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Fig. 13

Estimates of trailing momentum thickness as evaluated from tow tank datasets [13,24] with 1920 ≤ L/a ≤ 11,405 (a) θ/a versus ReL, and (b) Reθ versus Rea

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Fig. 14

Power-law fits for ratio θ/a and subsequent predictions of Reθ variability in terms of Rea and ReL



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