Troels Dyhr Pedersen.indd - Solid Mechanics

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Δt
v = u ⋅
Δx
where u is the speed of the gas, t is the time step and x is the cell size. Since u is the
velocity of the combustion products in the lab frame, the relevant speed is u1 – u2 . From
table 7 this speed is found to be the difference between 1533 m/s and 860 m/s, so it is 673
m/s). It cannot be expected that the speed reaches the theoretical limit in the simulation
but it may be used initially.
The internal solver used for the coupled complex chemistry in STAR-CD has a time step
of 1E-5 as default, but this is insufficient given the timescale of the chemical kinetics at
high temperatures and must be reduced to 1E-6 by setting constant 154 in the STAR-
GUIde to this value.
The time steps of the main solver are required to be small enough to allow the reaction
front to keep up with the wave front. If the steps are too large, a long distance is covered
by the gas between iterations. Hence the reactions will be separated from the wave by
that distance, which does not correspond to the real situation.
The time step size necessary to ensure convergence in the solver was found to be in the
order of 1E-4 to 1E-5 before the low temperature reactions begin, since the gas does not
move much initially. It must then be reduced to 1E-6 to ensure convergence during the
low temperature reactions. If the warning DIVODE (DIVerging Ordinary Differential
Equation) appear repeatedly between iterations, the time step size can be reduced further.
Assuming a time step of 1E-6 seconds, a gas speed of 673 m/s and a Courant number of
1, the cell size becomes:
−6
Δt
1⋅10
s
Δx
= u ⋅ = 673 m / s ⋅ = 0.
000673 m
v
1
A cell thickness of minimum 0.5 mm is thus appropriate to satisfy the Courant number
criterion. It was however considered too large for a proper resolution of the wave, so a
cell size of 0.1 mm was chosen instead and the time steps reduced correspondingly. It is
also possible to set a target for the Courant number in the solver, which will then adjust
the time step to keep this target value.
Boundary conditions
The end walls in the domain are solid. The side walls are set to symmetry. Wall friction is
negligible since the driving force provided by the pressure gradients are very large, and
heat transfer is likewise negligible due to the short time span.