Cavitation Models

Roman Samulyak, Yarema Prykarpatskyy

Center for Data Intensive Computing

Brookhaven National Laboratory

U.S. Department of Energy

rosamu@bnl.gov, yarpry@bnl.gov

Talk outline

• Modeling of the equation of state for two-phase fluid systems

• Numerical simulation of the interaction of mercury with a proton pulse in a thimble (BNL AGS and CERN ISOLDE experiments)

• Conclusions• Future research

The system of equations of compressible hydrodynamics

2

2

0

0

1( ) 12 ( ( ) ) 02

( , )

tv v v Pt

v v ev v e Pv

tP P e

• The system is solved using the front tracking technique for free surfaces

Isentropic two phase EOS model for cavitating liquid

• Approach: connect thermodynamically consistently different models for different phases

• Pure liquid is described by the stiffened polytropic EOS model (SPEOS)

• Pure vapor is described by the polytropic EOS model (PEOS)

• An analytic model is used for the mixed phase• SPEOS and PEOS reduced to an isentrope and connected

by the model for liquid-vapor mixture• All thermodynamic functions depend only on density

The EOS • does not take into account drag forces, viscous and surface

tension forces• does not have full thermodynamics

Current work on improved models will be discussed in section Future Research

Simulation of the Mercury Splash

Schematic of the experiment

Mercury splash (thimble): experimental data

Mercury splash at t = 0.88, 1.25 and 7 ms after proton impact of 3.7 e12 protons

Mercury splash (thimble): numerical simulation

240 480 609 1.014t s t s t s t ms

123.7 10 /I x protons pulse

Mercury splash (thimble): numerical simulation1217 10 /I x protons pulse

200 515 810 1.2t s t s t s t ms 1217 10 /I x protons pulse

Increasing the spot size of the proton beam results in a decrease of the splash velocities

1217 10 protons/pulse2 microseconds

I xt

Approximation from experimental data

Numerical simulations

The splash velocity of mercury depends linearly on the proton intensity 2 microsecondst

Conclusions

• Numerical simulations show a very good agreement with experimental data at early time.

• The lack of full thermodynamics in the EOS leads to some disagreements with experiments for the time evolution of the velocity. Can be corrected by the energy deposition.

• Equation of states needs additional physics (better mechanism of mass transfer, surface tension, viscosity etc.)

Experimental data on the evolution of the explosion velocity (from Adrian Fabich’s thesis)

• Further work on the EOS modeling for cavitating and bubbly flows. a) Direct method: study a liquid-vapor gas mixture as a system of one phase domains separated by free surfaces using FronTier’s code interface tracking capability

Future Research

a) b) c)Direct numerical simulation of the pressure wave propagation in a bubbly liquid: a) initial density; red: mercury, blue: gas bubbles, b) initial pressure; the pressure is 500 bar at the top and 1 bar at the bottom, c) pressure distribution at time 100 microseconds; pressure is 6 bar at the bottom.

Future problem: develop a nonlocal Riemann solver for the propagation of free interfaces which takes into account the mass transfer due to phase transitions

b) Homogeneous method: use a continuous description of the liquid-vapor gas mixture by means of homogeneous equation of state models and the Rayleigh-Plesset equation for the average gas bubble size evolution

2

3 3

3 1 2 1 1 11 0,2 2

where is the effective damping coefficient is the Weber number

is the cavitation number is the pressure term

tt t D t p

D

p

RR R R CR We R R R

We

C

Future problem: include terms describing the mass transfer due to phase transitions

• Further studies of the muon collider target issues. Studies of the cavitation phenomena in a magnetic field.

• Studies of hydrodynamic aspects of the cavitation induced erosion in the SNS target.