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Cavitation Models
Roman Samulyak, Yarema Prykarpatskyy
Center for Data Intensive Computing
Brookhaven National Laboratory
U.S. Department of Energy
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 ( ( ) ) 0
2( , )
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/pulse
2 microseconds
I x
t
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.