Governing equations and models formulation
3.5 Diffusion and reactions terms
(3.80) and by identification, it comes,
bmk = amk
2 (Tm+ T ) + bmk (3.81)
Further noting hm0k= hm0k′− bmkTmand hm0 = XN k=1
Ykhm0k, the total non-chemical energy of the mixture becomes,
E = hm0 + XN k=1
YkbmkT −P ρ +u2
2
= hm0 + CpT −P ρ +u2
2
= hm0 + P
ρ (γ − 1)+u2
2 (3.82)
which is a form equivalent to that of calorically perfect gas.
It shall be noted that hm0kcan be precomputed and tabulated over the intervals spanning T06T 6 Tm, and we have thus,
hk = h0k+ bmk (T − Tm) (3.83)
which is extremely fast to compute during the simulation. The mixture internal energy is computed using,
ρε = P
γ − 1 (3.84)
Cpk
T
Tm T Tm+1
amkT + bmk JANAF Cpk
∆T bmk
Figure 3.2: Approximation of species Cpk and enthalpy hk
Further details on the QCMF approach are given in §.4.3.4.
3.5 Diffusion and reactions terms
All systems presented up to now (QCMF and FCMF (§.3.2), QCVF (§.3.3.1), FCVF (§.3.3.2)) were pre-sented for inviscid flows. In combustion, diffusion phenomena and source terms are of prime importance
and need to be accounted for in the equation sets. The different diffusion fluxes and source terms are presented in this section for all models, along with the calculation of the different transport coefficients.
3.5.1 Diffusion fluxes for gaseous species
Species diffusion occurs at a molecular level and is one of the main driving mechanisms of premixed combustion. Reactants are already mixed at the macroscopic level, but it is only when species are mixed at a molecular level that combustion occurs. The mass diffusive fluxes estimation for the different equation sets considered is presented here.
Mass fraction formulations (FCMF and QCMF)
Species diffusion is modelled using species diffusion velocities given by,
−∇ · Jk = −∇ · (ρYkVk) (3.85)
where Vk is the diffusion velocity of species k, and Jk represents the diffusive mass flux. This term is present at the right-hand side of the species mass transport equations. The global mass conservation is ensured by,
XN k=1
YkVk = 0 (3.86)
To obtain the diffusion velocities values, an exact system of N × N equations has to solved which proves overly expensive in practical applications. Species diffusion velocities are therefore approximated by the Hirschfelder and Curtis [128] equation representing the best first-order closure of the full system and reads,
YkVk= −DkWk
W ∇Xk (3.87)
where Dk is the species diffusion coefficient. Pressure and temperature gradients effects (Soret effect) on the species diffusion velocities are neglected [107].
An outstanding issue of the Hirschfelder and Curtis approximation is that it does not satisfy global mass conservation when species Schmidt numbers are not equal. Two solutions can be used to overcome this issue,
◦ The solution of the N − 1 first species transport equations is computed and the last species mass fraction is recovered by using the fact that the species mass fractions sum is always unity. This means that all diffusion errors are absorbed by one species, which is usually N2, as it is the flame diluent in air. However, this is dangerous and should only be used for very diluted flames
◦ A correction term is added to the species diffusion velocity to enforce the conservation of global mass [238]. The diffusion velocity of species k is then expressed by,
Jk= −ρ where Vck is a correction velocity computed as,
Vck =
The second solution is retained in the present work, and as such the diffusive mass fluxes are given by,
∇ · Jk = −∇ · ρDk and the estimation of the diffusion coefficient Dk is discussed in §. 3.5.3.
QCVF formulation
In the QCVF formulation, two equations are considered for each species, the first one being the species mass conservation which is similar to the mass fraction model, therefore the diffusive term presented above is also used. The main difference arises in the treatment of the species volume fraction advection equations where the diffusion fluxes formulation has to be modified. The addition of diffusion terms to this equation set represents one of the novelty of this work as, to the author knowledge, the 5-equation model of Allaire et al. [7] was only used for inviscid non-reacting simulations so far. A similar comment can be made regarding the source term addition as discussed in §.3.5.4.
By expanding the species mass conservation equation, as presented in Eq. 3.22, but accounting for the diffusive fluxes, we have,
ρk
which can be rearranged as,
∂zk By remembering that species density are materially advected (Eq.3.23), we end up with,
∂zk
∂t + u · ∇zk= − 1
ρk∇ · Jk (3.93)
and by identification with the volume fraction transport equation accounting for species diffusion given by,
The volume fraction diffusive fluxes in the QCVF model are thus the mass diffusive fluxes scaled by the species density.
FCVF formulation
Similarly to the QCVF model, the FCVF formulation features two transport equations per species, one similar to the mass fraction models transporting species mass, and the second transporting species den-sity. The treatment of the partial density transport is discussed here.
By adding a diffusive term to the partial density transport, Eq.3.24becomes,
where by expanding the first relation and using Eq.3.95, we end up with,
(∇ · Jk)ρ = 0 (3.97)
The species density are not diffusing, but are materially advected.
3.5.2 Diffusion fluxes for the momentum and energy equations
Both momentum and energy equations are common to all approaches highlighted earlier. Several diffusive processes have to be taken into account in these equations, such as momentum dissipation by viscous stresses, energy diffusion due to viscous stresses work, temperature diffusion (heat fluxes) or even species diffusion (enthalpy fluxes).
Momentum diffusion
Momentum is dissipated by viscous stresses (∇ · τ ) estimated by the viscous stress tensor τ calculated as follows when considering a Newtonian fluid,
τ = β (∇ · u) I + 2µS (3.98)
where β denotes the second coefficient of viscosity, µ is the dynamic viscosity, I represents the identity tensor (Iij = δij, where δij denotes the Kronecker function) and S stands for the symmetric strain rate tensor given by,
S = 1 2
∇u + (∇u)T
(3.99) and (∇u)T is the transpose of the dyadic ∇u. In index notation, it is given by,
Sij = 1 2
∂ui
∂xj
+∂uj
∂xi
(3.100)
By separating the isotropic and deviatoric parts of the viscous stress tensor, it can be rewritten as, τ =
β +2
3µ
(∇ · u) I + 2µS −2
3µ (∇ · u) I (3.101)
The second coefficient of viscosity is evaluated using Stoke’s hypothesis, where the bulk viscosity estimated by β + 2
3µ = 0. This assumption is valid a priori only for dilute mono-atomic gas and at low Mach number. In other situations, this can lead to incorrect results [26], as it was shown to have a significant impact on the shock induced vortex generation as well as on the smoothing of situations with complex shock interactions. On the other hand, it is a complex parameter to calculate accurately as it requires the use of the kinetic theory of gases [93]. Nonetheless, this assumption will be used throughout this work.
The viscous stress tensor is finally defined as, τ = −2
3µ (∇ · u) I + µ
∇u + (∇u)T
(3.102)
Energy diffusion fluxes
Energy is dissipated through viscous stresses work, temperature diffusion and energy transport by molec-ular diffusion. These processes are as follows,
◦ Viscous fluxes work given by τ · u
◦ Heat conduction modelled by Fourier’s law
qc= −λ∇T (3.103)
where λ is the heat conductivity of the mixture
◦ The molecular diffusion induced energy dissipation is estimated by (neglecting Dufour effect),
qd= XN k=1
hkJk (3.104)
This last term is often neglected but Cook [68] highlights its importance in ensuring the consistency between mass and energy transport equations. It is therefore used in this work.
3.5.3 Transport properties for gaseous species
Transport properties have to be estimated for individual species. Mixture rules are then applied to obtain mixture properties. It can be noted that numerous mixing rules exist, and there is no clear choice about which one is best in a given situation. In this work, Wilke’s mixing rule is considered [294].
Species dynamic viscosities are calculated at a given temperature using Sutherland’s law,
µk= µkref
where T0k is the reference temperature (different for each species), µkref is the dynamic viscosity at the reference temperature and Sk is the Sutherland constant.
The mixture viscosity is estimated as,
µ =
where the inter-collisional parameter Φkj is given by,
Φkj = 1
Similarly to species viscosities, species thermal conductivities could be estimated using kinetic theory [93], but this is much more complex and this level of accuracy is not needed for the considered applications.
In this work, the mixture thermal conductivity is computed from the mixture dynamic viscosity using the assumption of a constant mixture Prandtl (P r) number,
λ =µCp
P r (3.108)
This assumption has been found to be valid in most air-flames due to the large amount of diluent (N2) present.
The diffusion coefficient used in the Hirschfelder and Curtis [128] law, is an effective diffusion coefficient of species k in the mixture. The simplest way of evaluating it consists in deriving it from assumed species Lewis (Lek) or Schmidt (Sck) numbers [106]. Smooke and Giovangigli [263] were the first to propose the constant Lewis strategy considered in this work. Since, Sck = µ
ρDk
= ν Sc
= LekP r, the effective diffusion coefficient can be estimated as,
Dk = µ ρSck = ν
Sck (3.109)
where Sck is an a priori specified constant Schmidt number.
3.5.4 Gaseous chemical kinetics
Chemical reactions are taken into account in the equations by adding a source term in both the mass and energy conservation equations. This last term depends on the form of the energy retained. Some authors use total chemical energy (sum of chemical, sensible and kinetic energy), and as such do not need an energy source term in the energy conservation equation. However, in this work, the total non-chemical energy has been used (sum of sensible and kinetic energy), and a source term accounting for the net heat release of chemical reactions is needed.
Mass fraction formulation
The chemical source term described in this section applies to the simulation of detailed chemistry effects accounting for N species reacting through M reactions. Chemical mechanisms can be written as,
XN k=1
νk′jMk ⇄ XN k=1
νk′′jMk for j = 1, · · · , M (3.110)
where Mk denotes the name of species k, νk′j and νk′′j are the stoichiometric coefficients of species k in reaction j. The mass conservation through chemical reactions enforces,
XN k=1
νk′jWk= XN k=1
νk′′jWk (3.111)
The rate of production/destruction of species k, ˙ωk is the sum of the rates ˙ωkj produced by the M reactions,
˙ωk= XM j=1
˙ωkj = Wk
XM j=1
νk′′j− ν′kj
Qj (3.112)
where Qj is the rate of progress of reaction j.
The rate of progress of reaction j is computed as follows,
Qj= Kfj
YN k=1
[Xk]ν′kj − Krj
YN k=1
[Xk]νkj′′ (3.113)
where Kfj and Krj are the forward and backward reactions rate constants of reaction j. The modelling of these two terms represents a central problem in combustion.
Some reactions require a third-body to proceed, an example of such reaction is,
H + OH + M ⇒ H2O + M (3.114)
In these reactions, the molar concentration of the third-body species is computed by,
[M ] = XN k=1
αkj[Xk] (3.115)
where the coefficients αkj are referred to as third-body efficiencies of species k in reaction j. Unless otherwise specified, a third-body efficiency is assumed to be unity.
Both reaction rate constants are related together through an equilibrium constant defined by,
Krj = KfjKjeq (3.116)
and computed as,
where P0 = 1 [atm], ∆Hj0 and ∆Sj0 refer to the enthalpy (sum of sensible and chemical) and entropy changes during reaction j, and are given by,
∆Hj0=
where sk is the entropy of species k computed using JANAF tables [59].
In its simplest formulation, the forward rate constant is modelled by an empirical Arrhenius law given by, where Afj is the pre-exponential factor, βj the temperature exponent and Eaj the activation energy.
Appendix Bpresents the chemical mechanisms used throughout this work.
The net heat release due to chemical reactions is simply computed as the sum of the production rate of each species times its formation enthalpy. It reads,
˙ωT = XN k=1
∆h0fk˙ωk (3.121)
Volume fraction formulations
Similarly to the analysis conducted in §.3.5.1, the energy source term does not change from the FCMF formulation when using the QCVF or FCVF models. For the same reason, the volume fraction source term for the QCVF model reads,
( ˙ωk)z= 1 ρk
˙ωk (3.122)
and, for the FCVF model we have,
( ˙ωk)ρ= 0 (3.123)