Discretization of 1D Equations#
The governing equations outlined in the governing equations must be discretized in order to obtain solutions for 1D flame configurations. Various discretization methods are used for different terms in the equations, and their details are outlined here.
For this discussion, consider a discretized 1D domain. Cantera uses a non-uniform node spacing for the 1D domain.
Finite difference 1D domain#
The 1D solver sweeps over all points in a domain and computes a residual for each grid point and each equation. The solver attempts to find the combination of the variables that minimizes the residual vector.
The solution vector
This residual vector has entries in a form like what is shown below.
This is a vector-value function of the solution vector.
For each governing equation, there is a default boundary condition applied at the left and right boundaries. During the sweep, the final evaluations are done for the domain boundary points, which allows for the default residual to be augmented or overwritten by a boundary condition that is different from the default boundary condition.
Continuity Equation#
The continuity equation is:
Discretization#
Because the continuity equation is first order and the boundary condition is applied at
the right boundary in the 1D domain, the first derivative approximation uses a
one-sided form which uses a stencil that pulls from points to right of a grid point.
For consistency, the radial momentum term, (
There are three 1D flame configurations: free flames, strained flames, and burner stabilized flames. The free flames have a constant mass flow rate that is set by the user. The burner stabilized flames solve for a mass flow rate that achieves a desired temperature at a point in the domain. The strained flames have a mass flow rate that is set by the user and varies with axial position.
Strained flames#
The discretized equation in residual form (all terms moved to one side) at the interior points in the domain for strained flames is given below.
Free flames#
The discretized equation in residual form at the interior points in domain for free flames is given below.
Where
Unstrained flames#
The discretized equation in residual form at the interior points in the domain for unstrained flames is given below.
This is a zero gradient condition for the mass flow i.e. the mass flow rate is constant.
Boundary Conditions#
The boundary conditions are expressed in a residual form in the Cantera solver. This allows for complex expressions to be used at the boundaries. The solver will attempt to find the solution such that the residual at the boundary goes to zero.
Right Boundary#
At the right boundary, there are a few default boundary conditions depending on the type of flow.
Strained Flames#
For strained flames, the default right boundary condition is a zero mass flow rate.
At the right boundary (
Unstrained Flames#
For unstrained flames, the default right boundary condition is a zero axial velocity gradient.
At the right boundary (
Left Boundary#
There is no imposed boundary condition at the left boundary because only one boundary condition can be enforced for a first-order differential equation. As such, the residual equation at the left boundary is the same as the interior points.
At the left boundary (
Radial Momentum Equation#
The radial momentum equation is:
Discretization#
The term
uses upwinding.The second derivative term
uses a central difference approximation.
The discretized equation in residual form (all terms moved to one side) at the interior points in the domain is given below.
Upwinding#
The upwinding formula for the radial velocity derivative term
(
Where the value of j
. If the axial velocity is negative,
the value of j+1
. A positive velocity means that the flow is moving
left-to-right.
Second Derivative Term#
For the second derivative term (the shear term in the momentum equation), a three-point central difference formula is used. The term being discretized is:
Let j+1
and j-1
points, the derivative is estimated
using j+1/2
and j-1/2
(halfway between the grid points around point j).
The values of j-1/2
and j+1/2
are estimating using a central
difference formula:
The outer discretization uses a central difference between the j+1/2
and j-1/2
locations.
Where the values of
and,
The difference between these two values is:
Substituting these values into the central difference formula gives:
This formula is used to discretize the second derivative term in the radial momentum
equation. It takes this form due to the construction of a conservative scheme, meaning
that point j
sees the same value of the flux ( j+1
sees on its left side. This is a conservative scheme because
the fluxes are balanced at each point.
Boundary Conditions#
At the right boundary, the default boundary condition is a zero radial velocity.
At the right boundary (
Expressing the boundary residual in this form will drive the Newton root finding
algorithm to find the value of
The same boundary condition is used at the left boundary.
At the left boundary (
Energy Equation#
The steady-state energy equation is described by:
Discretization#
The term
uses upwinding.The second derivative term
uses a central difference approximation.
These terms are discretized in the same way as was described above for the momentum
equation for the upwinded term (
The discretized equation in residual form (all terms moved to one side) at the interior points in the domain is given below.
The enthalpy gradient term uses upwinding.
Boundary Conditions#
At the right boundary, the default boundary condition is a zero temperature. Keep in
mind that this is just the default boundary condition that is applied by the interior
domain, and will be modified by boundary objects. As discussed earlier for example,
a boundary object that specifies a fixed temperature will augment this residual by
subtracting the desired boundary temperature from this residual, which will make
the solver drive the boundary temperature to the desired temperature value in the
course of minimizing the residual
At the right boundary (
Expressing the boundary residual in this form will drive the Newton root finding
algorithm to find the value of
The same boundary condition is used at the left boundary.
At the left boundary (
Species Equation#
The species equation is given by:
Discretization#
The term
uses upwinding.The diffusive mass flux term
uses a conservative central difference formula.
For the interior points in the domain, moving all terms to the right-hand-side, the discretized equation is:
This discretization can be seen in Kee et al. [2003] , equation 16.106.
Boundary Conditions#
At the right boundary, the default boundary condition is a zero species mass-flux. This is different from a zero species mass fraction condition, because species may diffuse towards the boundaries during the solution process.
At the right boundary (
The same boundary condition is used at the left boundary.
At the left boundary (