posted on Apr, 10 2006 @ 10:18 AM
As promised a basic mathematical description of the shape of a cavity behind a circular disc cavitator. This method can easily be put into software
to look at such effects of pressure, speed depth etc.
This is a very simple and is based upon:
OPTIMUM DESIGN OD A SUPERCAVITATING TORPEDO CONSIDERING OVERALL SIZE, SHAPE, AND STRUCTURAL CONFIGURATION
Edward Alyanak, Ramana Grandhi, Ravi Penmetsa
Department of Mechanical and Materials Engineering, Wright State University, Dayton, Ohio, USA
Found in International journal of solids and structures Vol 43 (2006)
NOTE: This is there work so please include the reference if you recite this anywhere.
“Cavitation is described by the cavitation number (supercavitation is characterised by very low cavitation numbers), it is a non-dimensional
quantity that represents the extent of cavitation.
It is given by:
sigma (cav no) = (P-Pcavity)/(0.5*rho*V^2)
P is the pressure outside the cavity
Pcavity represents the pressure inside the cavity
Rho is density of seawater
V = torpedo speed
For supercavitation to occur sigma must approach zero, thus Pcavity approaches P. Therefore if a torpedo is at a given depth the pressure P is
equivalent to the depth pressure P=rho*g*h (density * gravity * depth). As a result of this the depth pressure must be considered for supercavitating
torpedo structuaral survivability. Even at high dynamic pressures this argument still holds. For example, at a velocity of 120m/s, Pcavity is within
98% of P for a cavitation number of 0.01 and a depth of 600m.
The flow can be characterised by a torpedo speed V, the cavitation number, sigma, and the caviator diameter d. The drag coefficient can be determined
for a flat disc cavitator by the relation
CD=0.815(1+sigma) where the drag coefficient is defined by
Where D is the drag force and A is the cavitator area, in this case it is (pi*d^2)/4
Form these equations the drag on the torpedo can be defined as
The maximum cavity diameter dm can be defined by sigma
Dm = d(CD/(sigma-0.132*sigma^(8/7)))^0.5
From which cavity length Lc can be determined
Lc = dm ( (1.067*sigma^-0.658) – (0.52*sigma^0.465) )
The cavity shape can be determined using the above equations using the logvinovich principle for stationary cavities.
S-So/ Sk-So = t/tk* (2-(t/tk))
Where S is the cavity cross-sectional area at time t, So is the initatil cavity area, Sk is the maximum cavity area and tk is the time taken for the
area to grow from So to Sk. These quantise are defined as
So = (pi/4) *d^2
Sk = (pi/4)*dm^2
Assuming the x axis runs down the longitudinal axis of the cavity and is zero at the cavitator then
t/tk = 2x/Lc
This allows us to write the cavity radius Rc as a function of distance x from the cavitator
Rc(x) = ( 1/4 (2x/Lc(2- 2x/Lc)(dm^2-d^2) +d^2))^1/2
Hope you can understand all this.