16 research outputs found
Full self-similar solutions of the subsonic radiative heat equations
We study the phenomenon of diffusive radiative heat waves (Marshak waves)
under general boundary conditions. In particular, we derive full analytic
solutions for the subsonic case, that include both the ablation and the shock
wave regions. Previous works in this regime, based on the work of [R. Pakula
and R. Sigel, Phys. Fluids. 443, 28, 232 (1985)], present self-similar
solutions for the ablation region alone, since in general, the shock region and
the ablation region are not self-similar together. Analytic results for both
regions were obtained only for the specific case in which the ratio between the
ablation front velocity and the shock velocity is constant. In this work, we
derive a full analytic solution for the whole problem in general boundary
conditions. Our solution is composed of two different self-similar solutions,
one for each region, that are patched at the heat front. The ablative region of
the heat wave is solved in a manner similar to previous works. Then, the
pressure at the front, which is derived from the ablative region solution, is
taken as a boundary condition to the shock region, while the other boundary is
described by Hugoniot relations. The solution is compared to full numerical
simulations in several representative cases. The numerical and analytic results
are found to agree within in the ablation region, and within in
the shock region. This model allows better prediction of the physical behavior
of radiation induced shock waves, and can be applied for high energy density
physics experiments
3D-2D transition in mode-I fracture microbranching in a perturbed hexagonal close-packed lattice
Mode-I fracture exhibits microbranching in the high velocity regime where the
simple straight crack is unstable. For velocities below the instability,
classic modeling using linear elasticity is valid. However, showing the
existence of the instability and calculating the dynamics post-instability
within the linear elastic framework is difficult and controversial. The
experimental results give several indications that the microbranching
phenomenon is basically a three-dimensional phenomenon. Nevertheless, the
theoretical effort has been focused mostly in two-dimensional modeling. In this
work we study the microbranching instability using three-dimensional atomistic
simulations, exploring the difference between the 2D and 3D models. We find
that the basic 3D fracture pattern shares similar behavior with the 2D case.
Nevertheless, we exhibit a clear 3D-2D transition as the crack velocity
increases, while as long as the microbranches are sufficiently small, the
behavior is pure 3D-behavior, while at large driving, as the size of the
microbranches increases, more 2D-like behavior is exhibited. In addition, in 3D
simulations, the quantitative features of the microbranches, separating the
regimes of steady-state cracks (mirror) and post-instability (mist-hackle) are
reproduced clearly, consistent with the experimental findings.Comment: 9 pages, 11 figure
Modeling of Supersonic Radiative Marshak waves using Simple Models and Advanced Simulations
We study the problem of radiative heat (Marshak) waves using advanced
approximate approaches. Supersonic radiative Marshak waves that are propagating
into a material are radiation dominated (i.e. hydrodynamic motion is
negligible), and can be described by the Boltzmann equation. However, the exact
thermal radiative transfer problem is a nontrivial one, and there still exists
a need for approximations that are simple to solve. The discontinuous
asymptotic approximation, which is a combination of the asymptotic
and the discontinuous asymptotic diffusion approximations, was tested in
previous work via theoretical benchmarks. Here we analyze a fundamental and
typical experiment of a supersonic Marshak wave propagation in a low-density
foam cylinder, embedded in gold walls. First, we offer a
simple analytic model, that grasps the main effects dominating the physical
system. We find the physics governing the system to be dominated by a simple,
one-dimensional effect, based on the careful observation of the different
radiation temperatures that are involved in the problem. The model is completed
with the main two-dimensional effect which is caused by the loss of energy to
the gold walls. Second, we examine the validity of the discontinuous asymptotic
approximation, comparing to exact simulations with good accuracy.
Specifically, the heat front position as a function of the time is reproduced
perfectly in compare to exact Boltzmann solutions.Comment: 14 pages, 8 figure
Self-similar solution of the subsonic radiative heat equations using a binary equation of state
Radiative subsonic heat waves, and their radiation driven shock waves, are
important hydro-radiative phenomena. The high pressure, causes hot matter in
the rear part of the heat wave to ablate backwards. At the front of the heat
wave, this ablation pressure generates a shock wave which propagates ahead of
the heat front. Although no self-similar solution of both the ablation and
shock regions exists, a solution for the full problem was found in a previous
work. Here, we use this model in order to investigate the effect of the
equation of state (EOS) on the propagation of radiation driven shocks. We find
that using a single ideal gas EOS for both regions, as used in previous works,
yields large errors in describing the shock wave. We use the fact that the
solution is composed of two different self-similar solutions, one for the
ablation region and one for the shock, and apply two ideal gas EOS
(binary-EOS), one for each region, by fitting a detailed tabulated EOS to power
laws at different regimes. By comparing the semi-analytic solution with a
numerical simulation using a full EOS, we find that the semi-analytic solution
describes both the heat and the shock regions well
The Discontinuous Asymptotic Telegrapher's Equation () Approximation
Modeling the propagation of radiative heat-waves in optically thick material
using a diffusive approximation is a well-known problem. In optically thin
material, classic methods, such as classic diffusion or classic , yield
the wrong heat wave propagation behavior, and higher order approximation might
be required, making the solution harder to obtain. The asymptotic
approximation [Heizler, {\em NSE} 166, 17 (2010)] yields the correct particle
velocity but fails to model the correct behavior in highly anisotropic media,
such as problems that involve sharp boundary between media or strong sources.
However, the solution for the two-region Milne problem of two adjacent
half-spaces divided by a sharp boundary, yields a discontinuity in the
asymptotic solutions, that makes it possible to solve steady-state problems,
especially in neutronics. In this work we expand the time-dependent asymptotic
approximation to a highly anisotropic media, using the discontinuity jump
conditions of the energy density, yielding a modified discontinuous
equations in general geometry. We introduce numerical solutions for two
fundamental benchmarks in plane symmetry. The results thus obtained are more
accurate than those attained by other methods, such as Flux-Limiters or
Variable Eddington Factor.Comment: 18 pages, 13 figure
Micro-branching in mode-I fracture in a randomly perturbed lattice
We study mode-I fracture in lattices with noisy bonds. In contrast to
previous attempts, by using a small parameter that perturbs the force-law
between the atoms in perfect lattices and using a 3-body force law, simulations
reproduce the qualitative behavior of the beyond steady-state cracks in the
high velocity regime, including reasonable micro-branching. As far as the
physical properties such as the structure factor , the radial or angular
distributions, these lattices share the physical properties of perfect lattices
rather than that of an amorphous material (e.g., the continuous random network
model). A clear transition can be seen between steady-state cracks, where a
single crack propagates in the midline of the sample and the regime of unstable
cracks, where micro-branches start to appear near the main crack, in line with
previous experimental results. This is seen both in a honeycomb lattice and a
fully hexagonal lattice. This model reproduces the main physical features of
propagating cracks in brittle materials, including the behavior of velocity as
a function of driving displacement and the increasing amplitude of oscillations
of the electrical resistance. In addition, preliminary indications of power-law
behavior of the micro-branch shapes can be seen, potentially reproducing one of
the most intriguing experimental results of brittle fracture
The Time-Dependent Asymptotic Approximation for the Transport Equation
In this study a spatio-temporal approach for the solution of the
time-dependent Boltzmann (transport) equation is derived. Finding the exact
solution using the Boltzmann equation for the general case is generally an open
problem and approximate methods are usually used. One of the most common
methods is the spherical harmonics method (the approximation), when the
exact transport equation is replaced with a closed set of equations for the
moments of the density, with some closure assumption. Unfortunately, the
classic closure yields poor results with low-order in highly
anisotropic problems. Specifically, the tails of the particle's positional
distribution as attained by the approximation, are inaccurate compared to
the true behavior. In this work we present a derivation of a linear closure
that even for low-order approximation yields a solution that is superior to the
classical approximation. This closure, is based on an asymptotic
derivation, both for space and time, of the exact Boltzmann equation in
infinite homogeneous media. We test this approximation with respect to the
one-dimensional benchmark of the full Green function in infinite media. The
convergence of the proposed approximation is also faster when compared to
(classic or modified) approximation.Comment: 33 pages, 8 figure
Radiation drive temperature measurements in aluminium via radiation-driven shock waves: Modeling using self-similar solutions
We study the phenomena of radiative-driven shock waves using a semi-analytic
model based on self similar solutions of the radiative hydrodynamic problem.
The relation between the hohlraum drive temperature and the
resulting ablative shock is a well-known method for the estimation of the
drive temperature. However, the various studies yield different scaling
relations between and , based on different simulations.
In [T. Shussman and S.I. Heizler, Phys. Plas., 22, 082109 (2015)] we have
derived full analytic solutions for the subsonic heat wave, that include both
the ablation and the shock wave regions. Using this self-similar approach we
derive here the relation for aluminium, using the
detailed Hugoniot relations and including transport effects. By our
semi-analytic model, we find a spread of eV in the
curve, as a function of the temperature profile's
duration and its temporal profile. Our model agrees with the various
experiments and the simulations data, explaining the difference between the
various scaling relations that appear in the literature.Comment: 28 pages, 12 figure
Asymptotic Approximation in Radiative Transfer Problems
We study the validity of the time-dependent asymptotic approximation in
radiative transfer of photons. The time-dependent asymptotic is an
approximation which uses the standard equations with a closure that is
based on the asymptotic solution of the exact Boltzmann equation for a
homogeneous problem, in space and time. The asymptotic approximation for
radiative transfer requires careful treatment regarding the closure equation.
Specifically, the mean number of particles that are emitted per collision
() can be larger than one due to inner or outer
radiation sources and the coefficients of the closure must be extended for
these cases. Our approximation is tested against a well-known radiative
transfer benchmark. It yields excellent results, with almost correct particle
velocity that controls the radiative heat-wave fronts.Comment: 18 pages, 5 figure
Key to understanding supersonic radiative Marshak waves using simple models and advanced simulations
This article studies the propagation of supersonic radiative Marshak waves.
These waves are radiation dominated, and play an important role in inertial
confinement fusion and in astrophysical and laboratory systems. For that
reason, this phenomenon has attracted considerable experimental attention in
recent decades in several different facilities. The present study integrates
the various experimental results published in the literature, demonstrating a
common physical base. A new simple semi-analytic model is derived and presented
along with advanced radiative hydrodynamic implicit Monte Carlo direct
numerical simulations, which explain the experimental results. This study
identifies the main physical effects dominating the experiments,
notwithstanding their different apparatuses and different physical regimes.Comment: 33 pages, 17 figure