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Drag measurements on long thin cylinders at small angles and high Reynolds numbers

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Abstract

Measurements of the drag caused by turbulent boundary layer mean wall shear stress on cylinders at small angles of attack and high length Reynolds numbers (8×106<Re L <6×107) are presented. The use of a full-scale, high-speed towing tank enabled the development of turbulent boundary layers on cylinders made of stainless steel, aluminum, titanium, and polyvinyl chloride. The diameter of all cylinders in this experiment was 12.7 mm; two cylinder lengths, 3.05 m and 6.10 m, were used, corresponding to aspect ratio values L/a=480 and 960, respectively. Materials of various densities were towed at critical angles, resulting in linear cylinder geometry for tow speeds ranging from 2.6 m/s to 20.7 m/s and angles between 0° and 12°. Towing angles were measured with digital photography, and streamwise drag was measured with a strut-mounted load cell at the tow point. The measured tangential drag was very sensitive to small increases in angle at all tow speeds. A momentum thickness length scale is proposed to scale the tangential drag coefficient. The effects of the cross-flow resulting from the small angles of tow have a significant effect on the tangential drag coefficient values. A scaling for the orthogonal force on the cylinders was determined and provides a correction to published normal drag coefficient values for pure cross-flow. The presence of the axial turbulent boundary layer has a significant effect on these orthogonal forces.

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Abbreviations

a :

Cylinder radius (mm)

A s :

Total cylindrical surface area (m2)

α :

Cylinder angle of attack (°)

β, ϕ, γ:

Coordinates in cross-sectional plane of cylinder

C d :

Tangential drag coefficient=\( \frac{{\ifmmode\expandafter\bar\else\expandafter\=\fi{\tau }_{{\text{w}}} }} {{\frac{1} {2}\rho U^{2}_{{\text{o}}} }} \)

C n :

Orthogonal drag coefficient=\( \frac{{F\gamma }} {{\frac{1} {2}\rho {\left( {U_{{\text{o}}} \sin \alpha } \right)}^{2} A_{{\text{s}}} }} \)

d :

Cylinder diameter (mm)

δ :

Maximum boundary layer thickness (mm)

F x :

Steady state (measured) horizontal force exerted by the fluid on the cylinder (N)

F y :

Steady state vertical force exerted by the fluid on the cylinder (N)

L :

Cylinder length (m)

ν :

Kinematic viscosity (m2/s)

p :

Static pressure (N/m2)

P β , P γ :

Pressure force components (N)

θ :

Momentum thickness length scale (mm)

ρ :

Fluid density (kg/m3)

r :

Radial coordinate (mm)

R x , R y :

Reaction force components in moving coordinate system (N)

Re D :

Reynolds number based on diameter=\( \frac{{U_{{\text{0}}} \sin {\left( \alpha \right)}d}} {\nu } \)

Re L :

Reynolds number based on length=\( \frac{{U_{{\text{0}}} L}} {\nu } \)

Re θ :

Reynolds number based on momentum thickness length scale=\( \frac{{U_{{\text{0}}} \theta }} {\nu } \)

S :

Axial shear force (N)

σ :

Circumferential wall shear stress (N/m2)

T β , T γ :

Circumferential wall shear force components (N)

τ w :

Local axial wall shear stress (N/m2)

\(\bar \tau _{\text{w}} \) :

Spatially averaged axial wall shear stress (N/m2)

u(r, ϕ):

Temporal mean streamwise velocity (m/s)

\( \ifmmode\expandafter\bar\else\expandafter\=\fi{u}{\left( r \right)} \) :

Circumferentially averaged streamwise velocity (m/s)

u τ :

Spatially averaged friction velocity=\( {\sqrt {\frac{{\ifmmode\expandafter\bar\else\expandafter\=\fi{\tau }_{{\text{w}}} }} {\rho }} }\;{\left( {{\text{m}} \mathord{\left/ {\vphantom {{\text{m}} {\text{s}}}} \right. \kern-\nulldelimiterspace} {\text{s}}} \right)} \)

U 0 :

Tow speed (m/s)

W net :

Cylinder weight in water (N)

x, y:

Cartesian coordinates of moving reference frame (m)

ξ :

Coordinate along cylinder axis (m)

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Acknowledgements

The authors wish to acknowledge funding for this effort from the Office of Naval Research codes 321 MS and 333, and the NAVSEA Newport and Carderock In-House Laboratory Independent Research Programs. The authors are grateful to Professor Timothy Wei at Rutgers University for his many helpful discussions.

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Authors and Affiliations

  1. Naval Sea Systems Command Newport, 1176 Howell Street, Newport, RI, 02841, USA

    William L. Keith, Kimberly M. Cipolla & David R. Hart

  2. Naval Sea Systems Command Carderock, 9500 MacArthur Blvd, West Bethesda, MD, 20817, USA

    Deborah A. Furey

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  1. William L. Keith
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  2. Kimberly M. Cipolla
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  3. David R. Hart
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  4. Deborah A. Furey
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Correspondence to Kimberly M. Cipolla.

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Keith, W.L., Cipolla, K.M., Hart, D.R. et al. Drag measurements on long thin cylinders at small angles and high Reynolds numbers. Exp Fluids 38, 759–769 (2005). https://doi.org/10.1007/s00348-005-0959-7

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  • DOI: https://doi.org/10.1007/s00348-005-0959-7

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