Mathematical method in fluid dynamics
The Shvab–Zeldovich formulation is an approach to remove the chemical-source terms from the conservation equations for energy and chemical species by linear combinations of independent variables, when the conservation equations are expressed in a common form. Expressing conservation equations in common form often limits the range of applicability of the formulation. The method was first introduced by V. A. Shvab in 1948[1] and by Yakov Zeldovich in 1949.[2]
For simplicity, assume combustion takes place in a single global irreversible reaction
where
is the ith chemical species of the total
species and
and
are the stoichiometric coefficients of the reactants and products, respectively. Then, it can be shown from the law of mass action that the rate of moles produced per unit volume of any species
is constant and given by
where
is the mass of species i produced or consumed per unit volume and
is the molecular weight of species i.
The main approximation involved in Shvab–Zeldovich formulation is that all binary diffusion coefficients
of all pairs of species are the same and equal to the thermal diffusivity. In other words, Lewis number of all species are constant and equal to one. This puts a limitation on the range of applicability of the formulation since in reality, except for methane, ethylene, oxygen and some other reactants, Lewis numbers vary significantly from unity. The steady, low Mach number conservation equations for the species and energy in terms of the rescaled independent variables[3]
where
is the mass fraction of species i,
is the specific heat at constant pressure of the mixture,
is the temperature and
is the formation enthalpy of species i, reduce to
where
is the gas density and
is the flow velocity. The above set of
nonlinear equations, expressed in a common form, can be replaced with
linear equations and one nonlinear equation. Suppose the nonlinear equation corresponds to
so that
then by defining the linear combinations
and
with
, the remaining
governing equations required become
The linear combinations automatically removes the nonlinear reaction term in the above
equations.
Shvab–Zeldovich–Liñán formulation was introduced by Amable Liñán in 1991[4][5] for diffusion-flame problems where the chemical time scale is infinitely small (Burke–Schumann limit) so that the flame appears as a thin reaction sheet. The reactants can have Lewis number that is not necessarily equal to one.
Suppose the non-dimensional scalar equations for fuel mass fraction
(defined such that it takes a unit value in the fuel stream), oxidizer mass fraction
(defined such that it takes a unit value in the oxidizer stream) and non-dimensional temperature
(measured in units of oxidizer-stream temperature) are given by[6]
![{\displaystyle {\begin{aligned}\rho {\frac {\partial Y_{F}}{\partial t}}+\rho \mathbf {v} \cdot \nabla Y_{F}&={\frac {1}{Le_{F}}}\nabla \cdot (\rho D_{T}\nabla Y_{F})-\omega ,\\\rho {\frac {\partial Y_{O}}{\partial t}}+\rho \mathbf {v} \cdot \nabla Y_{O}&={\frac {1}{Le_{O}}}\nabla \cdot (\rho D_{T}\nabla Y_{O})-S\omega ,\\\rho {\frac {\partial T}{\partial t}}+\rho \mathbf {v} \cdot \nabla T&=\nabla \cdot (\rho D_{T}\nabla T)+q\omega \end{aligned}}}](https://backend.710302.xyz/https/wikimedia.org/api/rest_v1/media/math/render/svg/16c45af4b8af16f3163ad961dadcd832ff73683b)
where
is the reaction rate,
is the appropriate Damköhler number,
is the mass of oxidizer stream required to burn unit mass of fuel stream,
is the non-dimensional amount of heat released per unit mass of fuel stream burnt and
is the Arrhenius exponent. Here,
and
are the Lewis number of the fuel and oxygen, respectively and
is the thermal diffusivity. In the Burke–Schumann limit,
leading to the equilibrium condition
.
In this case, the reaction terms on the right-hand side become Dirac delta functions. To solve this problem, Liñán introduced the following functions
![{\displaystyle {\begin{aligned}Z={\frac {SY_{F}-Y_{O}+1}{S+1}},&\qquad {\tilde {Z}}={\frac {{\tilde {S}}Y_{F}-Y_{O}+1}{{\tilde {S}}+1}},\\H={\frac {T-T_{0}}{T_{s}-T_{0}}}+Y_{F}+Y_{O}-1,&\qquad {\tilde {H}}={\frac {T-T_{0}}{T_{s}-T_{0}}}+{\frac {Y_{O}}{Le_{O}}}+{\frac {Y_{F}-1}{Le_{F}}}\end{aligned}}}](https://backend.710302.xyz/https/wikimedia.org/api/rest_v1/media/math/render/svg/afadd80b4a9688b1555911309d9b325d7c63aae7)
where
,
is the fuel-stream temperature and
is the adiabatic flame temperature, both measured in units of oxidizer-stream temperature. Introducing these functions reduces the governing equations to
![{\displaystyle {\begin{aligned}\rho {\frac {\partial Z}{\partial t}}+\rho \mathbf {v} \cdot \nabla Z&={\frac {1}{Le_{m}}}\nabla \cdot (\rho D_{T}\nabla {\tilde {Z}}),\\\rho {\frac {\partial H}{\partial t}}+\rho \mathbf {v} \cdot \nabla H&=\nabla \cdot (\rho D_{T}\nabla {\tilde {H}}),\end{aligned}}}](https://backend.710302.xyz/https/wikimedia.org/api/rest_v1/media/math/render/svg/b89c38d178bcb54439344c1531ee53761ede26d7)
where
is the mean (or, effective) Lewis number. The relationship between
and
and between
and
can be derived from the equilibrium condition.
At the stoichiometric surface (the flame surface), both
and
are equal to zero, leading to
,
,
and
, where
is the flame temperature (measured in units of oxidizer-stream temperature) that is, in general, not equal to
unless
. On the fuel stream, since
, we have
. Similarly, on the oxidizer stream, since
, we have
.
The equilibrium condition defines[7]
![{\displaystyle {\begin{aligned}{\tilde {Z}}<{\tilde {Z}}_{s}:&\qquad Y_{F}=0,\,\,\,Y_{O}=1-{\frac {\tilde {Z}}{{\tilde {Z}}_{s}}}=1-{\frac {Z}{Z_{s}}},\\{\tilde {Z}}>{\tilde {Z}}_{s}:&\qquad Y_{O}=0,\,\,\,Y_{F}={\frac {{\tilde {Z}}-{\tilde {Z}}_{s}}{1-{\tilde {Z}}_{s}}}={\frac {Z-Z_{s}}{1-Z_{s}}}.\end{aligned}}}](https://backend.710302.xyz/https/wikimedia.org/api/rest_v1/media/math/render/svg/06adb3e31a2e06d5668a344d0b044d8ef876d01e)
The above relations define the piecewise function
![{\displaystyle Z={\begin{cases}{\tilde {Z}}/Le_{m},\quad {\text{if}}\,\,{\tilde {Z}}<{\tilde {Z}}_{s}\\Z_{s}+Le({\tilde {Z}}-{\tilde {Z}}_{s})/Le_{m},\quad {\text{if}}\,\,{\tilde {Z}}>{\tilde {Z}}_{s}\end{cases}}}](https://backend.710302.xyz/https/wikimedia.org/api/rest_v1/media/math/render/svg/87953a60e33d476c5506f7d4243e221ff8ef2582)
where
is a mean Lewis number. This leads to a nonlinear equation for
. Since
is only a function of
and
, the above expressions can be used to define the function
![{\displaystyle H={\tilde {H}}+{\begin{cases}(1/Le_{F}-1)-(1/Le_{O}-1)(1-{\tilde {Z}}/{\tilde {Z}}_{s}),\quad {\text{if}}\,\,{\tilde {Z}}<{\tilde {Z}}_{s}\\(1/Le_{F}-1)(1-{\tilde {Z}})/(1-{\tilde {Z}}_{s}),\quad {\text{if}}\,\,{\tilde {Z}}>{\tilde {Z}}_{s}\end{cases}}}](https://backend.710302.xyz/https/wikimedia.org/api/rest_v1/media/math/render/svg/6cc2671e8463e7f50270fb89fae8d596e44c1f02)
With appropriate boundary conditions for
, the problem can be solved.
It can be shown that
and
are conserved scalars, that is, their derivatives are continuous when crossing the reaction sheet, whereas
and
have gradient jumps across the flame sheet.
- ^ Shvab, V. A. (1948). Relation between the temperature and velocity fields of the flame of a gas burner. Gos. Energ. Izd., Moscow-Leningrad.
- ^ Y. B. Zel'dovich, Zhur. Tekhn. Fiz. 19,1199(1949), English translation, NACA Tech. Memo. No. 1296 (1950)
- ^ Williams, F. A. (2018). Combustion theory. CRC Press.
- ^ A. Liñán, The structure of diffusion flames, in Fluid Dynamical Aspects of Combustion Theory, M. Onofri and A. Tesei, eds., Harlow, UK. Longman Scientific and Technical, 1991, pp. 11–29
- ^ Liñán, A., & Williams, F. A. (1993). Fundamental aspects of combustion.
- ^ Linán, A. (2001). Diffusion-controlled combustion. In Mechanics for a New Mellennium (pp. 487-502). Springer, Dordrecht.
- ^ Linán, A., Orlandi, P., Verzicco, R., & Higuera, F. J. (1994). Effects of non-unity Lewis numbers in diffusion flames.