Radiation equations are integral equations, and hence are more complex and
expensive to solve than differential equations. However, if the angular dependency
is assumed constant in the integral equation, radiation equations for a given
direction (
) reduce to a transport differential equation whose sole
differential term represents the transport of the radiative flux along
; the only other terms are source or sink terms. In order to take
advantage of this reduction, Ansys Polyflow allows you to discretize the space of the
domain into a defined number of solid or planar angles, which then act as radiative
directions. A partial differential equation (PDE) is then solved for each radiative
direction. Specifying more radiative directions results in better discretization,
but also makes the calculation more expensive. For example, if a 3D domain is
discretized into six directions (which only provides three directions in each
hemisphere), the simulation will basically require as many variables as a
single-mode viscoelastic model.
Each of the defined angles in the discretized domain has a separate variable field, which can be referred to as a directional mode. The directional modes are generally uncoupled; however, there are two mechanisms that can cause them to be coupled:
When scattering is taken into account, all of the directional modes are coupled.
Boundary conditions (such as those for radiation) can couple otherwise unrelated directional modes. Radiation boundary conditions make the radiative flux change direction in cases where the emissivity of the boundary is not set to 1, as a part of the radiative flux is reemitted as diffuse radiation.
For the DO model, the radiative transfer equation is written as
 | (13–12) |
| where |
| = | position vector |
| = | direction vector | |
| = | scattering direction vector | |
| = | absorption coefficient (in
units of  , where 0 corresponds to empty space) | |
| = | non-dimensional refractive index (where 1 corresponds to a vacuum) | |
| = | scattering coefficient (in
units of  , where 0 corresponds to no scattering) | |
| = | Stefan-Boltzmann constant
(5.6704  10-8
W/m2-K4)
| |
| = | radiation intensity, which
depends on position ( and direction  or 
| |
| = | local temperature | |
| = | phase function | |
| = | solid or planar angle |
The scattering function is represented by 
, which is only available in the Delta-Eddington form:
 | (13–13) |
In the previous equation, 
is the forward-scattering factor and 
is the Dirac delta function. The 
term essentially cancels a fraction 
of the out-scattering; therefore, for 
, the Delta-Eddington phase function will cause the radiation
intensity to behave as if there is no scattering at all. 
is the asymmetry factor. When the Delta-Eddington phase function
is used, you will specify nondimensional values for 
and 
.
For 3D simulations, Ansys Polyflow selects solid angles in a pseudo-optimal
way. The software attempts to best cover all the angles of the domain with
(nearly) equally spaced radiative directions. As the problem of optimally
placing an arbitrary number of points on a sphere is unsolved in three
dimensions, Ansys Polyflow has adopted distributions that correspond to either
the vertices, the mid-edge nodes, or the nodes central to the faces of two
Platonic solids: a cube and an icosahedron. Such distributions allow you to
divide your 3D domain into 6, 8, 12, 14, 20, or 30 discrete radiative
directions; you are only required to select which one of these numbers you would
like to use for the discretization (which is then designated as 
).
To be clear, the method employed by Ansys Polyflow to discretize the solid
angles does not correspond to simply dividing the latitude and longitude into
equal angular slices, as such a scheme would bias the radiative directions
toward the "north" and "south" poles (that is, toward the positive and negative
directions).
Because of scattering and boundary conditions, a coupled resolution approach is generally used in Ansys Polyflow. By default, the number of radiative directions is set to 8 in order to limit the problem size.
For 2D simulations, the angular discretization is more straightforward: Ansys Polyflow divides the planar domain into equal angular sectors, in accordance with the number of directions you specify. For 2D flows, the number of directions must be an even integer between 4–30.
At domain boundaries, Equation 13–12
requires that the incident radiation intensity (
) be calculated specifically for each incident radiative
direction 
. The radiative direction is considered to be incident if
, where 
represents the outward normal vector for the boundary. The
discontinuous behavior of the boundary conditions as a function of the normal is
one of the major reasons why radiation domains need to be fixed.
Ansys Polyflow allows you to specify the following types of boundary conditions at domain boundaries:
diffuse gray wall (DGW) You can specify that the boundary behaves as a diffuse gray wall of temperature
, where the incident radiative heat flux
(
) is treated as follows: a fraction of the flux is
absorbed by the wall and converted to heat; a fraction is transmitted
through the wall; and the rest is reemitted within the domain,
contributing to the radiative heat flux 
that is emitted away from the wall in all directions
as an outward radiation intensity 
. The following are the equations used for the DGW
boundary condition: 
(13–14)
(13–15)
(13–16)
In the previous equations, you can specify the values for the following parameters:
represents the transmittance of the wall, and
is defined for individual boundaries of the domain. The
transmittance determines the fraction of the original incident
radiative heat flux that passes through the boundary (and is
thereby lost to the sub-task), and ranges from 0 (for a
completely opaque boundary) to 1 (for a completely transparent
boundary, where the directional radiative heat flux is
continuous in every direction). Physically, 
corresponds to the ratio of the area / length
of "holes" versus the total area / length of the boundary. 
represents the emissivity of the wall, and is
defined for individual boundaries of the domain. The emissivity
determines the fraction of the remaining radiative heat flux
(that is, the flux that has not been transmitted through the
boundary) that is absorbed by the boundary. The emissivity value
ranges from 0 (for perfect reflection) to 1 (for a black body).
represents the refractive index of the medium
adjacent to the wall (that is, the ratio of the velocity of
radiation in a vacuum to the velocity of radiation in the
medium). The defined value is applied to all of the domain
boundaries, and affects how much radiation is emitted by the
boundaries as a result of absorption.
insulated / symmetry
You can set the boundary to act as an insulated wall. Such a boundary condition is the equivalent of the DGW boundary condition where the emissivity is set to 0 (that is, no heating of the boundary) and the transmittance is set to 0 (that is, completely opaque). All of the incident radiative energy is reflected away from the insulated wall in all directions.
A boundary that behaves like an insulated wall is the equivalent of a symmetry boundary. Note that such a symmetry condition is only with respect to internal radiation, and does not apply to the flow.
interface
The interface boundary condition imposes continuity of the "irradiance" field through the boundary. In Ansys Polydata, each radiative direction
has an associated scalar field of radiation intensity
values; the entire set of these fields constitute the irradiance field.
The interface condition allows you to connect different sub-tasks that
model internal radiation without any losses due to absorption or
reflection. The sub-tasks on either side of an interface boundary must
have same number of radiative directions 
.
Important: Ansys Polyflow does not model specular radiation, so all reflected radiation is diffuse.
A diffuse gray wall boundary can be modeled inside a domain for any boundary
that is explicitly designated as a 
dimensional PMesh, where 
is one less than the number of dimensions for the domain (for
example, the PMesh must be 2D for a 3D domain). See PMeshes for information about PMeshes.
As is the case for the DGW boundaries described in the previous section, you can specify values for the transmittance, emissivity, and refractive index for the internal boundary. Internal boundaries are different than domain boundaries, however, in the following ways:
You have the option of setting different emissivity values for the faces on each of the two sides of the internal boundary (that is, the positive normal face and the opposing face), so that you can model a boundary where the absorption loss can vary depending on the direction of the incident radiative flux. The Graphics Display window identifies the positive normal face of the boundary via direction darts.
You have the option of setting different refractive indices for the faces on each of the two sides of the internal boundary (that is, the positive normal face and the opposing face), so that you can model the intersection of different media. As a consequence, different amounts of radiative energy (due to the absorption) will be emitted on either side of the boundary. The Graphics Display window identifies the positive normal face of the boundary via direction darts.
The transmittance of the internal boundary does not represent a loss for the sub-task as a whole, as this fraction of the incident radiative heat flux is continuous on both sides of the boundary.
When the transmittance is less than 1, the values of the radiative heat flux on either side of DGW internal boundary will be discontinuous (though the discontinuities are smoothed out for graphic postprocessing). If you have a selected a PMesh as part of a DGW sub-model, it will exhibit this discontinuity by default; if transparent behavior is expected, you must be sure to explicitly set the transmittance value to 1.