16 Uncertainty Treatment Applications: High-Enthalpy Flow Ground Testing
529
Heat Flux
The heat flux vector in Eq. (16.4) accounts for the energy transferred through
diffusion, thermal diffusion, and conduction, such that:
q =
j ∈S
ρ j h j V j + nk B T h
j ∈S
(k
h
j + k
e
j )V j −
m∈M
λ m ∇T m
(16.27)
where k h
T j
and k e
T j
are heavy particle and electron thermal diffusion ratios and
λ m = λ t
m + λ int
m is the effective thermal conductivity of the global energy mode
m ∈ M which can be split into translational λ t
m and internal λ int
m components. The
heavy particle translational thermal conductivity is obtained from the second order
Laguerre–Sonine polynomial approximation of the Chapman–Enskog expansion.
Finally, the thermal conductivity associated with the internal energy of mode m ∈
M , is given by the so-called Eucken corrections [37].
16.3.2.3 Chemistry and Internal Energy
Plasma flows are generally reactive: particles dissociate, recombine, and ionize.
Furthermore, molecules rotate and vibrate, and the electronic configuration of atoms
and molecules vary. To capture these aspects of atmospheric entry flows, the species
conservation equations depicted in Sect. 16.3.1.1 have to be modified by adding
additional terms to balance off the equations:
Species Continuity Equation
∂ρ i
∂t
+ ∇ · (ρ i v) + ∇ · (ρ i V i ) = ˙
w i , i ∈ S
(16.28)
Species Energy Conservation Equation
∂
∂t
(ρ i e i ) + ∇ · (ρ i e i v) + ∇ · (q i ) − j i · E’ + ρ i V i ·
∂
∂t
v + P i : ∇v = ΔE i , i ∈ S
(16.29)
The chemical production source term, ˙
w i , found in Eq. (16.28) accounts for
the production and destruction of individual species through elementary chemical
reactions. A rigorous derivation of the chemical production rate from kinetic theory
yields an expression of the form:
˙
w i = ˙
w
0
i + ˙
w
1
i
(16.30)
where ˙
w 0
i is the zero-order Maxwellian production rate and ˙
w 1
i is a first-order
perturbation [38].