Laws of Cavitation Bubbles

Gaining sufficiently high velocity   flow separates from the base of the cone generating an approximate rotational ellipsoidal cavity the so called "cavitation bubble" with a contour given by the equation :

Equation (1)
                                            

The ratio of maximum diameter of the bubble   D   versus its length   L   is given by the relation:

Equation (2)

                                                

where  σ  is defined as:

with   p_o     pressure at location of motion
         p_bl    pressure within the cavitation bubble
         q      dynamic pressure = ρ/2*v²
         ρ     density of water

Pressure   p_o   at location of movement is given by pressure in depth   T    plus atmospheric pressure on water surface    P_L  . Therefore   σ :

Equation (3)
                                                

with    p_L   pressure on the water surface,
           T   depth and
           g   gravitational acceleration.

According to Reichardt the drag of a cone with drag coefficient   cw_o   and the base diameter   d  equals the drag of a fictive body with the drag coefficient   σ   related to its maximum cross section area of the cavitation bubble with diameter   D :

By transformation results for the maximum bubble diameter  D   depending on cone diameter   d   , drag coeffcient  cw_o    and cavitation number  σ  :

Equation (4)
                                                

The force required to move the cone through water is given by the drag coefficient of the cone (Kegel), the cross section   F  of the cone, the density of water and the velocity   v  .

Equation (5)
                                                

The normal force on a cone (Kegel) moving with an angle of attack   α   versus direction of motion is calculated by:

Equation (6)
                                                

with       α               angle of attack
             ca_αo^cone     coefficient of lift per degree of α
(last picture in last chapter)

These 6 equations are sufficient for calculation of supercavitating objects.

For illustration we calculate an example:

A cylinder of diameter 25 cm is axially and horizontally moved through water in a depth of 5 m with velocity 100 m/s. Let us assume a pressure within the bubble of 0 bar then the cylinder produces a cavity with length 15 m and maximum diameter of 1.3 m.
The necessary force to move the cylinder would be  2*10⁵ Newton.