Hydrodynamic Paradox

Since nearly 300 years d'Alembert's paradox is known. Wikipedia:
"In fluid dynamics, d'Alembert's paradox is a contradiction reached in 1752 by french mathematician Jean le Rond d'Alembert. D'Alembert proved - for incompressible and inviscid potential flow - the drag force is zero on a body moving with constant velocity relative to the fluid".

This is contradictary to our experience since bodies moved in water always need force to overcome surface friction because of viscosity of water. In reality water is not incompressible and inviscid. So you can't postulate "potential flow". But there exists a possibility to overcome viscosity, if you can reduce or even minimize the surface of a body which is in contact with water. In this case you should assume water a nonviscid fluid. Cavitation offers this possibility.

Look at the following picture. It shows a rotational ellipsoid-shaped body of 20 mm diameter and 237 mm length. The body has a convertible head. The ellipsoid-shaped head can be exchanged against a cylinder shaped head of 5 mm diameter (right side above the model). The material consists of brass and aluminum.

The model possessed a central bore hole for a guidance wire. This wire (stainless steel) was spaned through the whole water basin and the adjacend acceleration equipment 30 mm beneath the water surface. The body moves from the left side to the right side.


    v = 60 m/s

   v = 69 m/s



    v = 72 m/s

    v = 78 m/s

The figures show the model with a cavitation bubble released at the cylinder shaped head. With growing velocity the bubble becomes larger. At a velocity of ca. 78 m/s the length of the bubble has reached the length of the model.

The analysis delivered drag coefficients of 0.75 - 0.83 for the whole velocity range of 60 -106 m/s. Only two trials showed drag coefficients of 0.46 at velocities of 78 m/s. In these experiments the length of the model nearly coincided with the length of the cavitation bubble. In this case the drag coefficient was ca. 35 % lower then the drag coefficient of the cylindrical head (Lit.9).

The experiments proved that a part of the momentum required to open the cavity can be recovered at the heck where the collapsing water delivers a momentum in direction of the motion. In the case of the described experiments this was ca. one third of the whole resistance.

Taking the small dimensions of the used models into account it seems realistic to achieve drag reductions of 80 - 90 % for submarines.

home                                                                      back                                                                             more