In combustion flows like many other flows, chaotic and random behavior of flow which known as turbulence appears above a certain number of Reynolds number.

In order to consider this phenomenon and effects of instabilities in flows there is a need to
define a suitable turbulence model. This condition takes place when viscous forces are not
sufficient to resist instabilities in flow, hence Reynolds number appears to evaluate this
situation. By defining a critical Reynold number π
_{π}_{πΆπππ‘}, laminar and turbulent regimes are
separated from each other. So, calculation of turbulence is important in combustion flows.

There are three main groups to calculate turbulence as mentioned below:

β’ Reynolds-average Navier-stokes (RANS) equations: Reynolds decomposes any flow properties to two parts as mean flow and fluctuation flow. This model focuses mainly on mean flow and effects of mean flow properties and implements time averaging of Navier-stokes equations. Extra terms appear in time-averaged (or Reynolds averaged) flow equations due to interactions among turbulent fluctuations. Extra terms are modelled with turbulence model such as π β π and Reynolds stress models.

β’ Large eddy simulation (LES): intermediate form of turbulence calculation that follows large eddies behavior and filter unsteady Navier-stokes equations. This model passes large eddies and rejects small eddies.

β’ Direct numerical simulation (DNS): calculate all mean flow and turbulent velocity fluctuations. Unsteady Navier-stokes equation are computed on spatial grid. So, it is fine enough to consider small time step and resolve period of fastest fluctuations.

In following subchapters, two kinds of averaging method, Reynolds averaging, and Favre averaging for RANS turbulence calculation are introduced. Furthermore, π β π turbulence model is discussed in order to estimate Reynolds stresses and scalar transport terms for turbulent flows computation in RANS. Finally, characteristic scales in turbulent flows are described.

**3.2.1 ** **Reynolds averaging **

For every single fluid property of π, Reynolds divided it to two composition as mean value and fluctuation value. It is known as Reynold decomposition of Reynolds averaging of that flow variable and is shown as follows:

π(π‘) = πΜ
+ π^{β²}(π‘) (3.18)

Where time average value is:

πΜ
= ^{1}

βπ‘β«_{π‘}^{π‘}^{0}^{+βπ‘}π(π‘)ππ‘

0 (3.19)

Selecting βπ‘ is wisely to be large enough compared to period of random turbulence fluctuation and small enough compared to slow variations time constant in ordinary unsteady flows.

Also, average value of fluctuation π^{β²} is zero.

π^{β²}

Μ
Μ
Μ
(π‘) = ^{1}

βπ‘β«_{π‘}^{π‘}^{0}^{+βπ‘}π^{β²(π‘)}ππ‘

0 = 0 (3.20)

In combustion flows, due to high and strong heat generation, density varies as a function of
position. Furthermore, by turbulent flow, fluctuation of density is observable. So, it is practical
to show density in form of Reynolds decomposition π = πΜ
+ π^{β²}.

This Reynolds decomposition can be generalized to other variables like π’_{π},π,π,π_{π},β in reacting
flows (π’_{π} = π’Μ
+ π’_{π} _{π}^{β²} , etc.).

By implementing Reynolds averaging of π’_{π} and π in continuity equation (3.2) it would result
to following equation.

ππΜ

ππ‘+^{π(π}^{Μ
π’}^{Μ
Μ
Μ
)}^{π}

ππ₯_{π} +^{π(π}^{Μ
Μ
Μ
Μ
Μ
Μ
Μ
Μ
)}^{β²}^{π’}^{β²}^{π}

ππ₯_{π} = 0 (3.21)

Thus, density fluctuations show additional term by using Reynolds averaging system. This
additional term ^{π(π}^{Μ
Μ
Μ
Μ
Μ
Μ
Μ
Μ
)}^{β²}^{π’}^{β²}^{π}

ππ₯π has to be modelled since it comes from correlation between velocity and density fluctuations in reacting flow. Other additional terms also can be derived from momentum, species and other transport equations which need to be modelled. Momentum transport equations (3.3) by implementing Reynolds decomposition is expressed as below.

π

In purpose of reducing these number of additional terms, it has been proposed to use a density-weighted averaging procedure named as Favre averaging by Favre[96] and Jones and Whitelaw[97].

**3.2.2** **Favre averaging **

As mentioned above, Favre averaging is used in order to simplify averaging procedure by
means of reducing additional terms. It is known as mass- weighted or density- weighted
averaging. Same as Reynolds averaging procedure, Favre averaging, or Favre decomposition
defined for any single fluid property of π as two parts of mean value πΜ and fluctuation value
π^{β²β²}.

π = πΜ + π^{β²β²} (3.24)

Where:

πΜ =^{ππ}^{Μ
Μ
Μ
Μ
}

πΜ (3.25)

So, by substituting eq. (3.25) in eq. (3.24) it results,
π =^{ππ}^{Μ
Μ
Μ
Μ
}

πΜ
+ π^{β²β²} (3.26)

It should be considered that mass averaged is only for velocity components and thermal variables. Fluid properties like density and pressure are treated as before.

In contrast to Reynolds decomposition shown in equation (3.20), mean value of fluctuations
π^{β²β²} is not zero (πΜ
Μ
Μ
Μ
β 0). ^{β²β²}

In Favre decomposition, it can be shown that ππΜ
Μ
Μ
Μ
Μ
Μ
= 0 same as ^{β²β²} ππΜ
Μ
Μ
Μ
Μ
= 0 for Reynolds ^{β²}
decomposition. Also, it can be result that π^{β²} shows a turbulent fluctuation of property π, while
π^{β²β²} include effect of density fluctuation rather than fluctuation of property π.

By implementing π and π’ in convective term of continuity equation with Favre averaging it can be shown that:

ππ’ = π(π’Μ + π’^{β²β²}) = ππ’Μ + ππ’^{β²β²} (3.27)
By time averaging it will be:

ππ’

Μ
Μ
Μ
Μ
= πΜ
π’Μ + ππ’Μ
Μ
Μ
Μ
Μ
= πΜ
π’Μ ^{β²β²} (3.28)
So, continuity equation (3.2) will be as follows:

ππΜ

ππ‘+^{π(π}^{Μ
π’}^{Μ)}^{π}

ππ₯_{π} = 0 (3.29)

In contrast to Reynolds averaging form of continuity equation, above equation is in same form of original continuity equation, but with this difference that mean velocity here is density-weighted Favre-average velocity.

So, it can be observed that Favre averaging procedure, reduces similarly the number of additional terms which are product of fluctuation in other transport equations. But it should be considered that results from Favre averaging need to be converted to time averaging to compare with experimental result.

Other Favre averaging equations to be used in combustion modelling for transport equations are as follows.

Momentum equation (3.3) by implementing Favre averaging is expressed as:

π

Here viscosity fluctuations are neglected. Also, practically, second viscous term in equation (3.31) is too small compared to other term that can be neglected.

Furthermore, Boussinesq expressed turbulent Reynolds stresses ππ’Μ
Μ
Μ
Μ
Μ
Μ
Μ
Μ
Μ
Μ
Μ
^{β²β²}_{π}π’^{β²β²}_{π} term in equation
Where π
is turbulent kinetic energy and expressed in three directions as follows:

π
=^{1}

2β^{3}_{π=1}π’Μ^{β²β²}_{π}π’^{β²β²}_{π} = ^{1}

2(π’Μ + π’^{β²β²}_{1}^{2} Μ + π’^{β²β²}_{2}^{2} Μ ) ^{β²β²}_{3}^{2} (3.33)

Species conservation equation (3.5) by implementing Favre averaging is:
Where πΜΜ_{π} is Favre-averaged reaction rate of production or consumption of species k. By using
gradient diffusion assumption, turbulent species will be:

ππ^{β²β²}_{π}π’^{β²β²}_{π}

Μ
Μ
Μ
Μ
Μ
Μ
Μ
Μ
Μ
Μ
Μ
= ^{π}^{π‘}

π_{ππ}

ππΜπ

ππ₯π (3.35)

Where: π_{π}_{π} is turbulent Schmidt number for species k.

Finally, by substitution of equation (3.35) in (3.34) species equations will be:

π

and π_{β} is turbulent Prandtl number.

In order to calculate π_{π‘} in momentum, species and enthalpy equations, proper turbulent model
should be selected. These models are classified based on number of additional transport
equations need to be computed along with RANS flow equations. Some turbulent model
examples are: Mixing length model with zero extra transport equation, Spalart-Allmaras model
with one extra transport equation, π
β π, π
β π and Algebraic stress models which use two
extra transport equations and Reynold stress model that uses seven extra transport equations.

In following subchapter, π β π model is discussed as one of the frequent models in RANS turbulence calculations.

**3.2.3** πΏ β πΊ turbulence model

In this model, two additional transport equations are used in order to describe turbulence due to consideration effects of transport of turbulence properties by convection and diffusion and production and destruction of turbulence. First equation is for turbulent kinetic energy π as

P_{π
}= βπΜ
π’Μ^{β²β²}_{π}π’^{β²β²}_{π}^{ππ’}^{Μ}^{π}

ππ₯π (3.41)

Also πΜ
π’Μ^{β²β²}_{π}π’^{β²β²}_{π} was defined as Boussinesq equation (3.31).

Another extra equation is for rate of dissipation of turbulent kinetic energy π as follows.

π

So, Turbulence viscosity π_{π‘} will be:

π_{π‘} = πΜ
πΆ_{π’}^{π
}^{2}

π (3.44)

Standard values for constants in equations (3.40), (3.42) and (3.44) are obtained from data fitting over a wide range of turbulent flows are mentioned below.

πΆ_{π’} = 0.09 ; π_{π
} = 1.00 ; π_{π} = 1.30 ; πΆ_{1π} = 1.44 ; πΆ_{2π} = 1.92 (3.45)