528
A. del Val et al.
16.3.2.2 Transport Phenomena
Closure of the transport fluxes is derived through a multiscale Chapman–Enskog
perturbative solution of the Boltzmann equation as explained in Sec. 16.3.1. The
fluxes considered are the stress tensor, the diffusion fluxes, and the heat flux which
play a major role in hypersonic flows.
Stress Tensor
P = p I − η
∇v + (∇v)
T
−
κ −
2
3
η
∇ · v I
(16.24)
where η and κ are the dynamic (shear) and bulk (volume) mixture viscosities. In an
expansion or compression of a gas, the work done by the pressure alters immediately
the translational energy of the molecules. Bulk viscosity arises from a time-lag
necessary to reequilibrate the translational and internal energies through inelastic
collisions. This term is nearly always neglected in hypersonic flow calculations
under the assumption that κ/η << 1. However, in [36], Giovangigli et al. have
shown that this assumption is not always valid for larger polyatomic gases.
The dynamic viscosity η is obtained from the first Laguerre–Sonine polynomial
approximation of the Chapman–Enskog expansion [27].
Diffusion Fluxes
Species diffusion velocities can be obtained from the multicomponent diffusion
coefficient matrix D ij by
V i = −
j ∈S
D ij (d j + k
h
j ∇ln T h + k
e
j ∇ln T e )
(16.25)
where d j are the species specific driving forces defined as:
d j =
∇p j
nk B T h
−
y j p
nk B T h
∇ lnp − κ j E
(16.26)
with κ j = x j q j /k B T h − y j q/k B T h , q the mixture charge and E the electric field.
The multicomponent diffusion coefficient matrix D ij is a function of the species
binary collision integrals and compositions [8].
A. del Val et al.
16.3.2.2 Transport Phenomena
Closure of the transport fluxes is derived through a multiscale Chapman–Enskog
perturbative solution of the Boltzmann equation as explained in Sec. 16.3.1. The
fluxes considered are the stress tensor, the diffusion fluxes, and the heat flux which
play a major role in hypersonic flows.
Stress Tensor
P = p I − η
∇v + (∇v)
T
−
κ −
2
3
η
∇ · v I
(16.24)
where η and κ are the dynamic (shear) and bulk (volume) mixture viscosities. In an
expansion or compression of a gas, the work done by the pressure alters immediately
the translational energy of the molecules. Bulk viscosity arises from a time-lag
necessary to reequilibrate the translational and internal energies through inelastic
collisions. This term is nearly always neglected in hypersonic flow calculations
under the assumption that κ/η << 1. However, in [36], Giovangigli et al. have
shown that this assumption is not always valid for larger polyatomic gases.
The dynamic viscosity η is obtained from the first Laguerre–Sonine polynomial
approximation of the Chapman–Enskog expansion [27].
Diffusion Fluxes
Species diffusion velocities can be obtained from the multicomponent diffusion
coefficient matrix D ij by
V i = −
j ∈S
D ij (d j + k
h
j ∇ln T h + k
e
j ∇ln T e )
(16.25)
where d j are the species specific driving forces defined as:
d j =
∇p j
nk B T h
−
y j p
nk B T h
∇ lnp − κ j E
(16.26)
with κ j = x j q j /k B T h − y j q/k B T h , q the mixture charge and E the electric field.
The multicomponent diffusion coefficient matrix D ij is a function of the species
binary collision integrals and compositions [8].