Rowan-Gollan-PhD-Thesis - Mechanical Engineering - University of ...

22 Background for Aeroshell Modelling Chapter 2
and dissociation and recombination affect the average energy in the various pools of internal
energy.
The first work to include a general model of thermochemical nonequilibrium in a finite-
volume method was that of Candler [62]. Candler gave a description of a multi-temperature
formulation and presented calculations using a two-temperature model for hypersonic air
flows: a temperature, Ttr, for the translational and rotational modes and a separate temper-
ature for the vibrational and electronic modes, Tve. Later Walters et al. [50] generalised the
multi-temperature models for various flux schemes and included the work in a production
code. Walpot [63] presented work in 2002 that included up-to-date physical modelling for the
the various energy exchange processes.
Before concluding this section, it would be remiss of me to ignore the recent comments
of Giordano [9] in relation to the use of the Born-Oppenheimer approximation. The Born-
Oppenheimer approximation underpins multi-temperature modelling because the modelling
implicitly relies on the fact that the various energy modes are separable. Giordano calls the use
of the Born-Oppenheimer approximation into question and goes on to argue that there is no
physical and rational basis to support a separation of energy modes in hypersonic flows. The
repercussions are that heat flux estimates should be viewed with some suspicion. While I agree
with Giordano’s reasoning and share his concerns about the state of aerothermodynamics with
its wildly varying heat predictions, the pragmatic part of me reconciles the use of a flawed
multi-temperature model for heat transfer estimates as better than no estimate at all.
Non-Boltzmann gases
At certain conditions toward the more rarefied limit, the gas mixture is so far from equilibrium
that a Boltzmann distribution of states, even separated into various temperatures, is a poor
assumption. This nonequilibrium effect is captured naturally in rarefied gas simulations using
the direct simulation Monte Carlo (DSMC) method. However, if we wish to use continuum
solvers, then some attempt should be made to account for this. Candler et al. [64] performed
detailed simulations of nitrogen dissociation in flow about a 10 cm sphere using a finite-volume
method. Their work used ten conservation equations for the mass of nitrogen species in various
vibrational states. They accounted for vibrational transitions of up to four quantum levels and
for the rates of transitions they used either the model of Schwartz, Slawsky and Herzfeld [65]
or the forced harmonic oscillator model of Adamovich et al. [66]. The work of Candler et
al. concluded that there are indeed non-Boltzmann distributions present in the shock layer at
low density conditions and that multiple-quantum transitions are an important mechanism for
dissociation.
A further conclusion to draw is that this detailed state-to-state vibrational modelling is
too computationally expensive for practical aerothermal analyses on present-day computers.
Candler et al. needed to highly optimise the grid to reduce the computational burden. If we
are striving towards computational aerothermodynamic analysis as part of the design process,
then the modelling of non-Boltzmann gases on highly optimised grids presently seems too
burdensome.