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

Section 2.2 Modelling of atmospheric-entry flows 25
Another geometric modelling technique is the tangent-slab approximation. In this model,
the distance between the shock and the vehicle is treated as an infinite slab. Analytical methods
can be used if the important radiation is considered to occur in the cross-shock layer direction
only. This method can give good results near the stagnation region (Anderson Jr. [7]). A number
of researchers [74,79,80] have used this model with reasonable success to calculate the radiative
heat transfer to the stagnation point of a hypersonic vehicle. It is known that the tangent-slab
model is quite erroneous away from the stagnation region (Elbert and Cinnella [69]).
A more complete approach to the geometric problem of radiation transfer is to integrate
over all solid angles allowing for emission and absorption from all gas elements in the path.
This kind of modelling takes account of the gas as a full participating medium and is the most
detailed way of accounting for the actual physics of radiative transfer through gases. In Shah’s
thesis a three-dimensional technique was developed [81], which has come to be known as the
Discrete Radiative Transfer Model. This has been applied to a sphere of radius 1.524 m in an
axisymmetric version of the Discrete Radiative Transfer Model by Elbert and Cinnella [69].
Also, Karl [82] used this model when computing the radiating flow field of an argon plasma.
Hartung [67] used the Modified Differential Approach which is an example of a flux method
to model three-dimensional radiative transport. The idea is to separate the angular depen-
dence of intensity and the spatial dependence (Oran and Boris [78]). In this way, the governing
integro-differential equations for radiative transfer can be reduced to a system of differential
equations, hence the name Modified Differential Approach.
Calculations of radiating flow fields about aeroshells
The discussion up to this point of modelling radiating flows looked at the details of calculating
the radiative properties and the radiation transport. Now we turn our attention to calculations
of radiating flow fields which use a combination of the models and methods just discussed. We
restrict our attention to calculations for air and the Titan atmosphere, and, in keeping with the
theme thus far, the review focuses on calculations based on the finite-volume method.
Hartung et al. [83] used their modern multi-band model and the tangent-slab approxima-
tion to compute the radiative heat transfer based on the nonequilibrium flow field computed
by the flow solver LAURA (Gnoffo [84]). The radiation calculation was applied as a post-
processing step to the nonequilibrium flow field calculation. In 1994, Hartung et al. [85] ex-
tended their work in air flows by loosely coupling the flow solver and radiation transport
calculation: for nonequilibrium flows a radiation calculation was applied every 200 iterations
of flow field calculation while for near equilibrium flows, only every 1 000–5 000 iterations.
They concluded that the correct coupling of radiation and flow field calculation is important
for nonequilibrium situations.
In 2001, Sakai and Sawada [86] presented calculations of radiating air flow which simulated
a ballistic range experiment on 0.5 cm and 0.7 cm cylinders. The radiation transport calculation
was fully coupled to the implicit flow solver scheme. The spectral detail was modelled with
the PRG approximation [71] and the radiation transport was calculated using the tangent-slab
approximation. The largest calculation of hypersonic radiating flow in air, to my knowledge,