5.9.2 Physics
This section describes how to construct and solve a species/scalar transport equation. It consists of two subsections:
Scalar Transport Equation
In species module, Simerics-MP can solve the transport equation for an arbitrary, user-defined scalar in the same way as it solves the transport equation for a scalar such as (but not limited to) the mass fraction in a multi-component flow. For an arbitrary scalar, 
, the general transport equation has the form:
|
where 
, 
and 
are the user-specified diffusion coefficient, turbulent Schmidt number, and source term for the scalar 
, respectively. In Simerics-MP, 
is assumed to be isotropic. It can be a directly specified value or a user defined function. It can also be indirectly determined through a specified Schmidt number (it is in turn either a specified value or a user defined function). The turbulent Schmidt number, 
, is a user-specified constant with one as the default value. The source term 
can be a constant or a user-defined function in the form of per-volume source or the total source in the computation domain.
It may be noted that by selecting the module Species, only one scalar equation is added. For 
-scalars, the module needs to be selected 
times. And each species is required to be assigned a unique name.
It should be pointed out that equation 5.497 is a general scalar equation. It can be solved alone for a scalar transport, or as an addition to any or all of the standard modules. Since the diffusion and source terms are determined by user inputs (either constant values or user-defined functions), the general scalar transport equation offers a great deal of freedom for users to form meaningful physical models of their own. For instance, it can be used to develop new physical models such as turbulence and combustion models. It can also be used in reduced forms that only consist of some of the terms in the equation. The followings are two such examples:
Poisson and Laplace Equation
In the steady-state mode, if the convective flux is not solved or remains constant, equation 5.497 is then reduced to a diffusion only problem:
|
Furthermore, if the turbulent diffusion is ignored (
,or,
), and 
is a constant, equation 5.497 becomes a Poisson equation:
|
And when 
, the scalar equation 5.497 further added to a Laplace equation.
Among many applications, if 
is replaced by the volume charge density (
), and 
is the permittivity (
), then equation 5.497 can be applied to calculate electric potential (
) in an electric field:
|
Convective Transport
Without the diffusion term (
, and 
, or 
), equation 5.497 is then reduced to:
|
equation 5.497 can be used to model the transport of the phase volume fractions (
) in multiphase flows, in which the phases are immiscible (see Multiphase module):
|
when 
, and ⃗
, then equation 5.497 represents the Euler equations for non-viscous flows:
|
In summary, the Species module provides a powerful tool to reproduce the models in Simerics-MP as well as to develop customized physical models.
Boundary Conditions
For a user-defined scalar, it can be any physical quantity. The boundary conditions, therefore, may not be defined in the same way as the flow boundary conditions. For example, a flow inlet boundary may mean entirely something else for the scalar 
. As a result, for the general scalar transport equation, all the defined boundary types can be applied for the physical boundaries upon the user's choice. In this section, instead of assigning specific types of boundary conditions to the physical boundaries (inlet, outlet, wall and symmetry), only the available boundary types and formulations are described.
Assuming that 
is a unit vector normal to the local boundary surface, we have the general expression of the per unit area mass flux as:
|
if the advection and diffusion are both present at the boundary.
For the scalar transport, 
represents the flux per unit area leaving/entering the physical domain at a boundary. Depending on applications, the following common boundary conditions are derived from this general formulation:
Zero Flux
In this case, the flux per unit area across (normal to) the boundary is specified to zero. Clearly, with the condition of zero flux, 
, the convective and diffusive fluxes must exactly balance:
|
In other words, if one term is zero, the other term must also be zero. For example, at a solid (wall) boundary, the normal velocity to the surface is zero, 
, though 
may not be zero. To satisfy the constraint in equation 5.497, the gradient of the scalar at the boundary must be zero, 
.
In Simerics-MP, at a wall, zero flux is the default boundary condition for the scalar 
.
Specified Value
Specified value is a boundary condition under which the value of the scalar at the boundary, 
, is directly determined by a user input value of 
:
|
In Simerics-MP, at a flow inlet, specified constant value is the default boundary condition for 
.
Symmetry
For a symmetry boundary condition, zero normal-to-boundary gradient is applied for the scalar 
:
|
In Simerics-MP, at a flow symmetry boundary, symmetry is also the default boundary condition for 
.
Outlet
In Simerics-MP, outlet is used as a boundary condition at an opening where the flow is expected to exit (or enter) the domain. For a specified pressure outlet, or Resistor or Capacitor in the flow, this is the default condition for the scalar 
.
At an outlet boundary, the required input is the specified value for scalar 
. The actual boundary condition applied for 
depends on the flow conditions:
Flow Leaving the Domain: When the flow is leaving the computational domain either from an outlet or inlet (reversed flow), zero-gradient is assumed at the boundary:
|
Flow Entering the Domain: When the flow is entering the computational domain either from an inlet or outlet (reversed flow), the specified value is applied for the boundary:
|
Convective Flux
At a boundary, the convective flux of 
per unit area (
) can be determined as a function of the external ambient value of the scalar (
) and an exchange coefficient (
):
|
where 
and 
are user input parameters. Note that the exchange coefficient 
has the unit of 
. From the known convective flux 
, the boundary value of 
is obtained from equation 5.497.
Specified Scalar Flux
In this boundary condition, the flux of the scalar can be specified in two ways:
Flux per Area: In equation 5.497, the scalar flux per unit area, 
, is specified by a user input (either as a constant value or user-defined function):
|
Then from equation 5.497, 
is obtained based on the flow conditions.
Total Flux: In this option, the total scalar flux is known through a user input (either as a constant value or user-defined function):
|
where 
is the specified total scalar flux and 
is the total boundary area. Then 
is obtained from equation 5.497 based on the flow conditions.