Ballistics, hydrodynamics - tensile cavitation in tissue and fluids

[tuoni]

I have made decent progress with my ballistics model, but I have run into trouble and I'm not quite sure how to solve it. This particular problem deals with penetration and tensile cavitation. Based on the Held equation for cavitation in jet penetration (still haven't found any sites discussing this equation), an approximation for cavity area would be as follows:

A_{h(\tau)} = \delta A_{x} \cdot \frac{\rho_{p}v^{2}}{2} \cdot \frac{1}{p_{m(\tau)}k_{m}^{2}}

k_{m} = 1 + \sqrt{\frac{\rho_{p}}{\rho_{m}}}

Ah(τ) -- cavitation area; tensile failure (m^2)
Ax -- deformed projectile area (m^2)
v -- penetration velocity; velocity of nose relative to medium (m/s)
ρp -- density of projectile (kg/m^3)
ρm -- density of medium (kg/m^3)
pm(τ) -- tensile yield strength (Pa)
km -- density constant (k/1)

Thus the cavitation potential of a bullet is primarily dependent on the deformation of the projectile, with greater deformation creating greater cavitation. The second term looks like fluid stress, but pressure exerted on the medium as opposed to exerted on the penetrator, so (ρv^2)/2 is the radial pressure pushing away the medium as the bullet penetrates the medium in front of it, i.e. the nose pushing the medium outwards. The final term is completely unknown to me, and I cannot recognise it from anywhere, but I looks like it characterises the strength of the medium and how much cavitation occurs.

Does anyone have any clues as to the derivation/explanation of the final term?

Aside from that, I'm also wondering about cavitation in water (and other fluids). The only difference is that fluids do not have static strength (pm), but other than that I think the equation should be the same for fluids, I just don't know what could be form of the final term.

Using the following seems pretty reasonable (results seem nice), but I don't think it's correct:

\frac{1}{a_{g}p_{g}k_{m}^{2}}

ag -- gravitational acceleration (m/s^2)
pg -- static pressure of fluid (Pa)

Any ideas on how to characterise the cavitation in fluids? I think it should be pretty close to the original equation, as the phenomena are actually quite similar and the mechanics isn't too different.

 

[Studiot]

If I understand you correctly you are modelling the passage of a solid object, such as a bullet, through a quasi fluid material - the tissue.

Would it not make more sense to describe the effect in the tissue by the dynamics of a wake, rather than modelling the projectile as the front of a jet?
Cavities form naturally in wakes, which is what you seem to be asking.

The flow regime for both jets and wakes is obtained by solving the

Momentum Integral Equation

I think your equation is derived from this.

However jets are often discussed in texts because they are one of the very few cases where we have obtained analytical solutions to this equation.

 

[tuoni]

If I understand you correctly you are modelling the passage of a solid object, such as a bullet, through a quasi fluid material - the tissue.

Would it not make more sense to describe the effect in the tissue by the dynamics of a wake, rather than modelling the projectile as the front of a jet?
Cavities form naturally in wakes, which is what you seem to be asking.

The flow regime for both jets and wakes is obtained by solving the

Momentum Integral Equation

I think your equation is derived from this.

However jets are often discussed in texts because they are one of the very few cases where we have obtained analytical solutions to this equation.

I had to look into that a little, and it does seem interesting, but I haven't been able to find any real examples/experiments/equations, so unfortunately I won't be able to use it. Good to know about the momentum integral theorem, but I doubt I'll be able to use it myself in the near future, seems complicated enough.