Appendix: Time Integration
The run control file variables that control the time integration of the system of governing equations are discussed in the following subsections.
Integration Method
The time-integration method used in Stream is specified with the run control file variable timeIntegrator.
Three methods are available. The first method is the standard first-order backward differencing method. This method can be
chosen by using the value BDF. If the timeIntegrator variable is not specified in the vars file, the value
BDF is chosen by default. This method is commonly used for marching simulations to a steady-state condtion, but is
not generally acceptable for unsteady flows due to the requirement for excessively small time steps in order to reduce
numerical dissipation. For unsteady flows, Stream offers two second-order differencing methods. The first of these methods
is the second-order backward differencing method, which can be chosen by using the value BDF2. The second is the
Crank-Nicolson method, which can chosen by using the value CN. An example of setting the time integrator in a vars
file is shown below.
timeIntegrator: BDF2
Blended Crank-Nicholson Method
While both the BDF2 and CN are formally second-order accurate methods, CN has been found in our experience to
be less dissipative than BDF2. However, the pure CN scheme is not always robust for complex flows unless very small
timesteps are used, which can lead to prohibitive run times. To mitigate this problem, STREAM provides a blended
CN/BDF scheme, which is activated by a blending factor using the run control file variable CNBDFBlend, when using the
CN time itegrator. This blending factor can be set anywhere between 0 (resulting scheme is 100% BDF) and 1 (resulting
scheme is 100% CN). If this variable is not provided, the value defaults to 1. In practice, it has been found that blending
factor above 0.9 can help stabilize the CN scheme at higher time step, while still providing slightly lower dissipation
characteristics than BDF2. An example of using the blending factor is shown below.
CNBDFBlend: 0.9
For finite-rate chemistry simulations, Stream uses a second-order Strang-splitting procedure, which is not compatible with
the specification of BDF2 for the timeIntegrator variable. For unsteady DES and LES simulations with Stream, one
should always use the CN scheme with the highest stable value of the variable CNBDFBlend as possible.
Time Step Selection
The time step for a simulation can be set in two different ways. To
run a simulation with a single constant time step value, use the run
control file variable timeStep in the following manner:
timeStep: 1.0e-01
If one is interested in only a steady-state solution, often the best
approach is to simply eliminate the temporal terms from the governing
equations by specifying a large time step value, say 1.0e+30, which is
the default value for this variable. One can then control the iterative
process by only using the relaxation factors specified via the variables
for the individual governing equations. This is the preferred method for
steady-state flows. As a measure of last resort however, one can use a
finite time step to perform temporal relaxation, which can often be
effective at stabilizing a convergence path where iterative relaxation
alone is not sufficient.
A second method of setting the time step is also available, whereby one can specify several
so-called ramps in which the time step remains at a fixed value for a
certain number of time steps before moving to a new ramp value. At the
end of this process the time step will take on the value specified by
the variable timeStep. Consider the following vars file specification:
timeStep: 1.0e-01
timeStepRamp: <ramp0=[n=1000,dt=1.0e-04],ramp1=[n=100,dt=1.0e-03],
ramp2=[n=10,dt=1.0e-02]>
In this case, the simulation will start off by performing 1000 time
steps at \(dt= 10^{-4} s\), followed by 100 time steps at
\(dt= 10^{-3} s\), followed by 10 time steps at
\(dt= 10^{-2} s\), and then assume the constant value of
\(dt= 10^{-1} s\) for the remainder of the simulation. It is
important to note that the time step ramp will be active if the time
step number in the simulation is lower than the total number of time
steps in the ramp. So, for example, using the case above, if one ran a
simulation of 1000 time steps and then stopped, upon restart the time
step ramp would proceed to immediately do 100 time steps at
\(dt= 10^{-3} s\), followed by 10 time steps at
\(dt= 10^{-2} s\) and then use the constant value of
\(dt= 10^{-1} s\) for the remainder of the restart simulation. On the
other hand, if one had run a simulation with 2000 time steps and then
stopped, upon restart, the time step ramp would no longer be active,
having stopped at time step 1110. One may use an arbitrary number of
ramps. In addition, the names of the ramps (ramp0, ramp1, …) are not
important. One can choose any names. Only the order of the ramps is
important.
Number of Time Steps
The number of time steps to run in the simulation is specified by the run control file
variable numTimeSteps, as follows:
numTimeSteps: 1001
Once a simulation is initiated, it will not terminate until the number
of specified time steps is run. If one wishes to terminate the solution
at an arbitrary point in an orderly manner before the maximum number of
time steps specified by numTimeSteps has been executed, one can use
the so-called touch stop utility, by simply executing the
command touch stop while in the directory from which the simulation
was initiated. This will create a file called stop in the directory.
When the code detects the presence of this file, it will terminate in an
orderly manner, writing both output and restart files at the last time
step executed. The code will automatically delete the file stop so that
it will not linger around and cause problems with future runs from the
same directory.
Time Step Convergence
To achieve the proper temporal accuracy of a time integration scheme, one must converge the system of equations within each time step to the level required to eliminate the iterative (SIMPLE) or corrective (PIMPLE) error. The topic of convergence estimation is discussed in more detail here. Convergence within any time step is controlled using the run control file convergence tolerance and maximum iteration variables. Two type of convergence specification are available. If one wishes to have a single absolute tolerance value for all governing equations, the following specification should be used (default values shown):
convergenceTolerance: 1.0e-30
maxIterationsPerTimeStep: 50
Using this form, if the total residuals (the residual values
printed to standard output) at an iteration all become lower than the
value specified by convergenceTolerance, the code will automatically
advance to the next time step after that iteration. If such a
convergence level is not obtained within the maximum number of
iterations specified by the value of maxIterationsPerTimeStep, the code
will in any case advance to the next time step. Note that the variable
convergenceTolerance can only be effectively used if residuals have been
normalized using reference values, as discussed here. Common
practice is normally to forego the usage of convergenceTolerance by
setting its value to a small number, say \(10^{-30}\), and simply
using the variable maxIterationsPerTimeStep to control convergence within
the time step. This is done generally to avoid the complication of
having to compute reference values that provide meaningful
nondimensionalization of all total residuals simultaneously. In
addition, for engineering geometries, it is often impossible to achieve
a consistent convergence tolerance limit on all residuals at every time
step. For steady-state simulations, one would typically run a fixed
number of time steps by setting values like the following:
convergenceTolerance: 1.0e-30
maxIterationsPerTimeStep: 1
numTimeSteps: 500
To control convergence within the time step in a more refined manner, one can use the run control file absolute and relative convergence tolerance variables as follows (all possible specifiable tolerances shown):
convergenceAbsoluteTolerance: <default=1.0e-03, momentum=1.0e-04, pressure=1.0e-05,
energy=1.0e-06, k=1.0e-07, omega=1.0e-03, epsilon=1.0e-04, species=1.0e-02>
convergenceRelativeTolerance: <default=1.0e-03, momentum=1.0e-04, pressure=1.0e-04,
energy=1.0e-04, k=1.0e-04, omega=1.0e-04, epsilon=1.0e-04, species=1.0e-04>
maxIterationsPerTimeStep: 100
In the above options, the default value is first assigned to all governing equations. Subsequent entries for each of the specific governing equations are then specified to override the default value, if desired. If one does not specify a default value, the default value for the default value is \(10^{-30}\). One need only specify override tolerance values for the specific equations of interest. For example, one could specify an absolute convergence tolerance of \(10^{-3}\) for all equations, but a tolerance of \(10^{-4}\) for pressure, as follows:
convergenceAbsoluteTolerance: <default=1.0e-03, pressure=1.0e-04>
One may use convergenceAbsoluteTolerance and convergenceRelativeTolerance
either separately or together. For example, to specify that convergence
is to be determined by the satisfaction of only relative tolerances, one
would specify:
convergenceRelativeTolerance: <default=1.0e-03, pressure=1.0e-04>
This specification is equivalent to the following:
convergenceAbsoluteTolerance: <default=1.0e-30>
convergenceRelativeTolerance: <default=1.0e-03, pressure=1.0e-04>
whereby the small default absolute convergence tolerance guarantees that
absolute convergence is never satisfied, and only relative convergence
determines convergence within the time step. Convergence within a time step is
declared when all the active governing equations are individually converged. Convergence for any
governing equation is declared when either the absolute tolerance
or the relative tolerance is satisfied. If all active governing
equations have not converged within the maximum number of iterations
specified by the value of maxIterationsPerTimeStep, the code will
automatically advance to the next time step.