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Direct Numerical Simulation of Multiphse Flows with Phase
Transitions
T. Lu, R. Samulyak, Z. Xu, and J. Glimm
An accurate description of cavitation and wave propagation in
cavitating and bubbly fluids is a key problem in modeling and simulation
of hydrodynamic processes in a variety of applications ranging from
marine engineering to high-energy physics. We are interested in the
study of cavitation and bubbly fluids occurring in high speed liquid
jets such as diesel jets in fuel injectors and liquid mercury targets
that interact with high intensity proton pulses. Such targets are key
components of advanced accelerators such as the Spallation Neutron
Source (http://www.sns.gov) and Muon Collider/Neutrino Factory (www.cap.bnl.gov/mumu).
Another important application is the simulation of cavitation in a
high-speed liquid lithium of helium jet in a device proposed for the
plasma disruption mitigation in tokamaks.
Modeling of the cavitation is a complex multiscale and multiphysics
problem involving the description of thermodynamic properties of liquids
under strong external fields and nonlinear wave phenomena in a
multiphase system. In the direct numerical simulation of bubbly flows, a
liquid-vapor or liquid-non-dissolvable-gas mixture is represented as a
system of one-phase domains (vapor bubbles, for instance) separated by
free interfaces. The FronTier code is capable of tracking simultaneously
a large number of interfaces and resolving their topological changes
(the breakup and merger of droplets) in two- and three-dimensional
spaces. Though computationally intensive, such an approach is
potentially very accurate in treating important effects in bubbly flows
including bubble oscillations, heat transfer, drag, viscosity, and
surface tension. The method makes it possible to resolve spatial scales
smaller than the typical distance between bubbles, and to model some
non-equilibrium thermodynamic features such as finite critical tension
in cavitating liquids. The direct method has been validated through the
comparison of numerical simulations with theoretical predictions and
classical experiments on linear (sound) and nonlinear (shock) waves in
bubbly fluids [1-3] (see Figure 1) and applied to the study of the Muon
Collider and Spallation Neutron Source targets.
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1. Phase velocity and attenuation rate of linear waves
in bubbly water.
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Numerical simulation of cavitation presents an
additional level of complexity compared to the simulation of wave
phenomena in fluids containing small non-dissolvable gas bubbles.
The problem is associated with the dynamic creation and collapse of
bubbles in the computational domain. The corresponding numerical
models and software routines have been developed and implemented in
the FronTier code. The phase transition rate is proportional to the
deviation of the vapor pressure from the saturated pressure
where α is the condensation coefficient.
The Riemann problem for phase transition has been studied using the
method of viscous profiles. The transient waves in realistic
(thermal conductive) phase transitions with Riemann data have been
demonstrated, in particular, by the analytical solution for the
linear waves, as shown in Figure 2.
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Figure
2. Analytical solution for the transient linear pressure
wave induced by the temperature difference in the Riemann
data. |
To account for the phase transition induced mass transfer
across the liquid-vapor interface, a numerical scheme has been developed in
the frame of front tracking. A non-local Riemann solver governing the
evolution of interfaces has been implemented and a numerical technique has
been introduced to account for the thin thermal layer. The algorithm has
been validated and applied to physical problems such as the jet breakup in
diesel engines [2].
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| Figure 3. 2D numerical simulation of
cavitation in the mercury jet after the interaction with a
proton pulse depositing 100 J/g of energy into the jet.
Density distribution on the jet cross-section is shown: red
is mercury, light blue is the rarefied gas in cavitation
bubbles, and dark blue is the ambient gas. Time frames are
20 (left) and 40 (right) microseconds. |
References
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[1] Lu, T., Samulyak, R., Prykarpatskyy, Y, Glimm, J., Xu, Z., and Kim,
M.N. Comparison of heterogeneous and homogenized numerical models of
cavitation. Int. J. Multiscale Comp. Eng. 4(3) (2006).
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[2] Xu, Z., Kim, M., Oh, W., Glimm, J., Samulyak, R., Li., X., Lu, T.,
and Tzanos, C. Atomization of a high speed jet. Int. J. Multiscale Comp.
Eng. Accepted, 2006.
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[3] Xu, Z., Lu, T., Samulyak, R., Glimm, J., and Ji, X.M. Dynamic phase
boundaries for compressible fluids. SIAM J. Computing. Submitted, 2006.
Last Modified: January 31, 2008
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Claire Lamberti
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