Fig 1.
Illustration of the advection, diffusion and linear transport configuration studied in the article.
The solid domain is shown in gray, the fluid domain
is shown in blue. The model is: pure diffusion in the solid; advection-diffusion in the fluid; linear transport in the absorbing and scattering fluid. The boundary
of the system is the union of
, shown in grey, where Dirichlet boundary condition are formulated (the density η is prescribed), and
, shown in red, where the diffusion flux is null. The incident distribution function f obeying linear transport is prescribed over the whole boundary
. See a practical implementation of this generic configuration in Figs 5 and 6.
Fig 2.
A diffusive/linear-transport path in an infinite uniform static fluid for which the density is known outside a rectangular cuboid.
N such paths are sampled to estimate . Each path starts from the observation position
and returns a Monte Carlo weight equal to the value
at the location
of the first brownian-walk interruption occurring outside the parallelepiped.
is estimated as the average of these N weights. Blue sub-paths are brownian walks. Red sub-paths are multiple-scattering linear-transport paths. All algorithmic details are provided in the text of Sect 2. Simulation examples are provided in Appendix A and Table 1.
Table 1.
Simulation results for the diffusion/linear-transport algorithm of Sect 2 with the parameters of A.
is the estimation of
that is provided by N = 105 Monte Carlo samples and σ is its associated statistical uncertainty as provided by the standard error (take
for a 0.9995 confidence). T is the computation time recorded when performing the N = 105 Monte Carlo samples,
is the required number of Monte Carlo samples in order to reach
and
is the associated computation time. Computations were made using a x86i_64 Intel(R) Core(TM) i9-9880H CPU 2.30GHz.
Fig 3.
Sampling of confined brownian walks to estimate the density η at point .
(a) Using the RWS method: the next positions are sampled on the largest sphere inscribed in the domain until the sampled position is on the boundary, where η is known. In order to end the algorithm, the studied domain’s surface is thickened by a small value ε in which the position is projected on the border. (b) Using the proposition made in Sect 4, that is compatible with ray tracing. All the spheres have the same radius
. When getting close to a boundary, the displacement is truncated through the knowledge of the distance to the surface in this direction, which make the spheres distorded.
Fig 4.
Example of ray tracing to evaluate the length of the next step in the approximate random walk based on Patankar scheme: starting from the current location, two rays are drawn in the velocity direction v and its opposite, and two others in a random diffusion direction u (and its opposite) in the plane orthogonal to v.
These four rays enable to compute ).
Fig 5.
Stack of 4 kelvin cells and surface representation of the fluid flow for an homogeneous inlet fluid flow on the bottom right face and an outlet at the top left face.
Fluid velocity at inlet face is ; maximum velocity is
. Pore size is
and strand thickness is
. This configuration is an emblematic example to study heat transfer (see Appendix F for the correspondences with such problems).
Fig 6.
a) Representation of the deployed porous media submitted to an incident isotropic distribution function fi with gaussian spatial profile at the inlet face.
The gaussian profile ranges between and
. b) Representation of a typical path in the 3D geometry, which is constituted by three type of subpaths: advective subpaths (within the fluid) are in green, conductive subpaths (within the solid) in blue and linear transport subpaths (between interfaces, with possible absorption and scattering in the fluid) in red.
Fig 7.
Evolution of the averaged outlet density and computation time for an increasing number of kelvin cells.
Loading time corresponds to the time required to generate and copy the geometry in the RAM memory as well as to build the acceleration grid, whilst the solving time is the time required to sample 104 coupled random paths in the system. Diffusion coefficients are , particles speed is
, surface absorptivity is
, single-scattering and absorption frequencies are
, the single-scattering phase function is isotropic, viz.
. Loading and computation times are obtained with an Intel(R) Core(TM) i9-9880H CPU @ 2.30GHz CPU.
Table 2.
Results of Algorithms 1, 2, and 5 for random samples on an analytic boundary value problem
in a unit cube for various values of position
and step length
.
Monte Carlo results are shown in column along with their corresponding standard error σ.
Table 3.
Results of Algorithms 1, 3, and 5 for random samples on an analytic boundary value problem
in a unit cube for various values of position
and step length
.
Monte Carlo results are shown in column along with their corresponding standard error σ.
Table 4.
Correspondences between the model addressed in the body of the article and heat transfer problems.
Fig 8.
2D-slice of an isolated duct structure used as a heat exchanger between a thermal source at and an incoming cold Poiseuille circulation at
.
Retained values for the simulation are: ,
, where ϕ is the ratio of inner duct radius over outer radius,
, where
and
are the thermal conductivity for the solid and fluid respectively, and
, where aF is the thermal diffusivity for the fluid.
Fig 9.
Poiseillle duct results. Comparison of the temperature obtained with Comsol Multiphysics and the Monte Algorithm for the configuration defined in Fig 8.
In these simulations, .
Fig 10.
Comparison of the temperature obtained with ANSYS Fluent and the Monte Algorithm for the stack of 4 Kelvin cells presented in Fig 5.
Results for locations on the center line going down the flow; the inlet face is at adimensionned position 0 and the outlet at adimensionned position 1. In order to obtain a curve showing high variations of temperature in the fluid, the solid’s and fluid’s thermal diffusivities are respectively set at and
. The inlet velocity field is uniform at
.