# Drag crisis

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.

## History

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 supervisor 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]

## Explanation

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 using Large Eddy Simulation(LES) that includes the dynamic displacements of the structure (DLES) [11]. These calculations also demonstrate the importance of the blockage ratio present for intrusive fittings in pipe flow and wind-tunnel tests.

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.

## References

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)

[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.

[12] Bartran, D., 2018, "The Drag Crisis and Thermowell Design", J. Press. Ves. Tech. 140(4), 044501, Paper No: PVT-18-1002. DOI: 10.1115/1.4039882.