312 research outputs found
Eulerian-Lagrangian method for simulation of cloud cavitation
We present a coupled Eulerian-Lagrangian method to simulate cloud cavitation
in a compressible liquid. The method is designed to capture the strong,
volumetric oscillations of each bubble and the bubble-scattered acoustics. The
dynamics of the bubbly mixture is formulated using volume-averaged equations of
motion. The continuous phase is discretized on an Eulerian grid and integrated
using a high-order, finite-volume weighted essentially non-oscillatory (WENO)
scheme, while the gas phase is modeled as spherical, Lagrangian point-bubbles
at the sub-grid scale, each of whose radial evolution is tracked by solving the
Keller-Miksis equation. The volume of bubbles is mapped onto the Eulerian grid
as the void fraction by using a regularization (smearing) kernel. In the most
general case, where the bubble distribution is arbitrary, three-dimensional
Cartesian grids are used for spatial discretization. In order to reduce the
computational cost for problems possessing translational or rotational
homogeneities, we spatially average the governing equations along the direction
of symmetry and discretize the continuous phase on two-dimensional or
axi-symmetric grids, respectively. We specify a regularization kernel that maps
the three-dimensional distribution of bubbles onto the field of an averaged
two-dimensional or axi-symmetric void fraction. A closure is developed to model
the pressure fluctuations at the sub-grid scale as synthetic noise. For the
examples considered here, modeling the sub-grid pressure fluctuations as white
noise agrees a priori with computed distributions from three-dimensional
simulations, and suffices, a posteriori, to accurately reproduce the statistics
of the bubble dynamics. The numerical method and its verification are described
by considering test cases of the dynamics of a single bubble and cloud
cavitaiton induced by ultrasound fields.Comment: 28 pages, 16 figure
A finite Toda representation of the box-ball system with box capacity
A connection between the finite ultradiscrete Toda lattice and the box-ball
system is extended to the case where each box has own capacity and a carrier
has a capacity parameter depending on time. In order to consider this
connection, new carrier rules "size limit for solitons" and "recovery of
balls", and a concept "expansion map" are introduced. A particular solution to
the extended system of a special case is also presented.Comment: 20 pages, 9 figure
Solvable difference equations similar to the Newton-Raphson iteration for algebraic equations
It is known that difference equations generated as the Newton-Raphson
iteration for quadratic equations are solvable in closed form, and the solution
can be constructed from linear three-term recurrence relations with constant
coefficients. We show that the same construction for four-term recurrence
relations gives the solution to the initial value problem of difference
equations similar to the Newton-Raphson iteration for cubic equations. In many
cases, the solution converges to a root of the cubic equation and the
convergence rate is quadratic.Comment: 13 page
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