In fluid dynamics, drag crisis (also Eiffel paradox[1] or Lukyanov–Eiffel paradox[2]) is a phenomenon in which drag coefficient drops off suddenly as Reynolds number increases. This has been well studied for round bodies like spheres and cylinders. The drag coefficient of a sphere will change rapidly from about 0.5 to 0.2 at a Reynolds number in the range of 300000. This corresponds to the point where the flow pattern changes, leaving a narrower turbulent wake. The behavior is highly dependent on small differences in the condition of the surface of the sphere.

External video
Drag crisis. Lukyanov–Eiffel paradox, YouTube video

Contents

HistoryEdit

The drag crisis was first identified in 1905 by a Russian student G.I.Lukyanov[3] in experiments on wind tunnel of Moscow University. The superviser of the experiments, Zhukovsky correctly guessed that this paradox can be explained by ``detachment of streamlines at different points of the sphere at different velocities``[4].

Later the paradox was independently discovered in experiments by G.Eiffel[5] and Maurain[6]. Gustave Eiffel is known as a man who designed and built the Eiffel Tower, and the Statue of Liberty. Upon his retirement, he built the first wind tunnel in a lab located at the basis of the Eiffel Tower, to investigate wind loads on structures and early aircraft. In a series of test he found that the force loading experienced an abrupt decline at a critical Reynolds number.

A clear explanation of the paradox from the point of view of boundary-layer theory is due to German fluid dynamicist L.Prandtl[7].

ExplanationEdit

This transition is associated with a transition from laminar to turbulent boundary layer flow adjacent to the object in question. In the case of cylindrical structures this transition is associated with a transition from well organized vortex shedding to randomized shedding behavior for super-critical Reynolds numbers, eventually returning to well organized shedding at the post-critical Reynolds number with a return to elevated drag force coefficients.

The super-critical behavior can be described semi-empirically using statistical means or by sophisticated computational fluid dynamics software (CFD) that takes into account the fluid-structure interaction for the given fluid conditions.

The critical Reynolds number is a function of turbulence intensity, upstream velocity profile, and wall-effects (velocity gradients). The semi-empirical descriptions of the drag crisis are often described in terms of a Strouhal bandwidth and the vortex shedding is described by broad-band spectral content.

ReferencesEdit

  1. ^ Birkhoff, Garrett (2015). Hydrodynamics: A study in logic, fact, and similitude. Princeton University Press. p. 41.
  2. ^ Drag crisis. Lukyanov–Eiffel paradox
  3. ^ A table with drag coefficient data for Lukyanov's experiments can be found on page 73 in Zhukovsky, N.Ye. (1938). Collected works of N.Ye.Zukovskii.
  4. ^ Op.cit, p. 72.
  5. ^ Eiffel G. Sur la résistance des sphères dans l'air en mouvement, 1912
  6. ^ Toussaint, A. (1923). Lecture on Aerodynamics (PDF). NACA Technical Memorandum No. 227. p. 20.
  7. ^ Prandtl L. Der Luftwiderstand von Kugeln (1914)

Additional readingEdit

[1] Fung, Y.C., (1960), "Fluctuating Lift and Drag Acting on a Cylinder in a Flow at Supercritical Reynolds Numbers," J. Aerospace Sci., 27 (11), pp. 801-814.

[2] Roshko, A. (1961) "Experiments on the flow past a circular cylinder at very high Reynolds number," J. Fluid Mech., 10, pp. 345-356.

[3] Jones,G.W. (1968) "Aerodynamic Forces on Stationary and Oscillating Circular Cylinder at High Reynolds Numbers," ASME Symposium on Unsteady Flow, Fluids Engineering Div. , pp. 1-30.

[4] Jones,G.W., Cincotta, J.J., Walker, R.W. (1969) "Aerodynamic Forces on Stationary and Oscillating Circular Cylinder at High Reynolds Numbers," NASA Report TAR-300, pp. 1-66.

[5] Achenbach, E. Heinecke, E. (1981) "On vortex shedding from smooth and rough cylinders in the range of Reynolds numbers 6x103 to 5x106," J. Fluid Mech. 109, pp. 239-251.

[6] Schewe, G. (1983) "On the force fluctuations acting on a circular cylinder in crossflow from subcritical up to transcritical Raynolds numbers," J. Fluid Mech., 133, pp.265-285.

[7] Kawamura, T., Nakao, T., Takahashi, M., Hayashi, T., Murayama, K., Gotoh, N., (2003), "Synchronized Vibrations of a Circular Cylinder in Cross Flow at Supercritical Reynolds Numbers", ASME J. Press. Vessel Tech., 125, pp. 97-108, DOI:10.1115/1.1526855.

[8] Zdravkovich, M.M. (1997), Flow Around Circular Cylinders, Vol.I, Oxford Univ. Press. Reprint 2007, p.188.

[9] Zdravkovich, M.M. (2003), Flow Around Circular Cylinders, Vol. II, Oxford Univ. Press. Reprint 2009, p.761.

[10] Bartran, D. (2015) "Support Flexibility and Natural Frequencies of Pipe Mounted Thermowells," ASME J. Press. Vess. Tech., 137, pp.1-6 , DOI:10.1115/1.4028863

[11] Botterill, N. ( 2010) "Fluid structure interaction modelling of cables used in civil engineering structures," PhD dissertation (http://etheses.nottingham.ac.uk/11657/), University of Nottingham.

External linksEdit