Reynolds Averaging of DNS Data

− 6

∆²(x²₁+ x²₂+ x²₃)



(4.20)

In the current study a Gaussian filter is used to filter the DNS data. This filter was chosen as the filtering operation is commutative with the differential operation.

The DNS data was filtered for a range of filter widths ∆ (i.e. ∆ = 0.4δ_th to

∆ = 2.4δ_th increasing insteps of 0.4δ_th). This range of filter widths has been chosen as it allows for the study of filters where there is partial resolution of the flame (i.e. η < ∆ < δ_th) and to cases where there is no resolution of the flame (i.e. ∆ > δ_th). The higher end of the range also coincides roughly to the integral length scale, and if the filter width ∆ is greater than integral length scale the resulting filtered data will have lost all the information of large scales of flow, therefore this condition was used to set the upper boundary for the filter range.

4.3.2 Reynolds Averaging of DNS Data

All Reynolds and Favre averages were carried out by ensemble averaging the relevant quantities in the x₂− x³ plane at a given x₁ location. In order to check for the statistical convergence, the averaged quantities were evaluated using a spatially distinct half of the available sample size in the x₂− x³ plane and were compared to the corresponding quantities obtained using the full sample size available in the same plane, and good qualitative and quantitative agreements were found between the results obtained based on half and full sample sizes [40].

The results corresponding to the full sample size will be presented in chapter 7 and 8 for the sake of conciseness.

4.4 Summary

In this chapter a review of the requirements for the DNS were presented along with the numerical methodology that was used. Additionally more information on the numerical methodology can be found in appendix A. This was followed by

a discussion of the main parameters of interest in the current work, i.e. the Lewis number and the turbulent Reynolds number. Increasing (decreasing) turbulent Reynolds number (Lewis number) was shown to result in increased flame area generation. Moreover explanations to the increased flame area generation are presented in §.4.2.1 and §.4.2.2. Approaches available for LES filtering are then presented, and in this work a Gaussian filter in physical space is used which results in the analysis shown chapters 5, 6, 7 and 8. Similarly the methodology to Reynolds average the DNS data is presented, which was used in chapters 7 and 8 to analyse the curvature and strain rate terms of the generalised FSD transport equation respectively. In the following chapter, analysis of algebraic FSD models using filtered DNS data is presented. Additionally, two algebraic models that currently exist in literature are modified based on the current DNS database.

Algebraic Modelling of FSD in the context of LES

One of the ways to implement the FSD based reaction rate closure in LES would be to evaluate the generalised FSD using an algebraic model. Thus in this chapter a summary of existing algebraic models for the generalised FSD in the context of LES has been provided. In addition to this the performance of a sub-set of models known as the power-law based algebraic FSD models are assessed using a priori DNS analysis. In addition to this a new power-law model is proposed based on the current analysis of DNS data.

5.1 Generalised Algebraic Flame Surface Den-sity Models

Many models are currently available for the generalised FSD. Additionally it is possible to use algebraic models originally proposed for related quantities such as the wrinkling factor Ξ and turbulent flame speed S_T. Models proposed for the wrinkling factor Ξ can be used to predict for the generalised FSD using the following relationship:

Ξ = Σ_gen

|∇¯c| (5.1)

A model for Ξ was suggested by Angelberger et al. [2] and it can be written in the context of generalised FSD in the following manner:

Σ_gen = [1 + aΓ(u⁰_∆/S_L)]|∇¯c| (5.2) where a = 1.0, u⁰_∆ =

q

2˜k_∆/3, ˜k_∆ = (ug_iu_i − ˜u_iu˜_i)/2 is the subgrid turbulent kinetic energy and the efficiency function Γ takes the same form as in Eq. 5.2:

Γ = 0.75exp

 −1.2 (u⁰_∆/S_L)^0.3

  ∆ δ_z

2/3

(5.3)

One of the first models to be proposed for the wrinkling factor was done by Weller et al. [159], which can be written to express the generalised FSD in the following manner:

Σ_gen = [1 + 2˜c(Θ− 1)]|∇¯c| (5.4) where Θ = 1 + 0.62p(u⁰_∆/S_L)Re_η and Re_η = u⁰_∆η/ν with τ_η and ρ₀ denote Kol-mogorov scale and unburned gas density respectively. Colin et al. [53] proposed a model for the wrinkling factor Ξ which takes the same form as Eq. 5.2 but the model constant a is give as:

a = 2βln(2)/[3cms(Re^1/2_t − 1)] (5.5) where Re_t = ρ₀u⁰l/µ₀ (where µ₀ is the unburned viscosity and l is the turbulence integral length scale), β = 1.0 and cms = 0.28 and Γ is given by Eq. 5.3. By modifying the model proposed by Colin et al. [53], Charlette et al. [49] proposed an alternative model by reducing the input parameters from three to two (i.e.

u⁰_∆/S_L and ∆/δ_z):

Σ_gen =



1 + min ∆

δ_z, Γ_∆ u⁰_∆ S_L

β1

|∇¯c| (5.6)

where the model parameters takes the following form:

Γ_∆= [((f_u^−a1 + f_∆^−a1))^b1] (5.7a)

f_u = 4 27C_k 110

1/2

 18C_k 55



(u⁰_∆/S_L)² (5.7b)

f_∆= 27C_kπ^4/3 110

  ∆ δ_z



− 1

^1/2

(5.7c)

f_Re= 9 55

 exp

−1.5C^kπ^4/3 Re⁻¹_∆

^1/2

Re^1/2_∆ (5.7d)

a₁ = 0.60 + 0.2 exp



−0.1 u⁰_∆ S_L



− 0.2 exp



−0.01 ∆ δ_z



(5.7e)

b₁ = 1.4; β₁ = 0.5; C_k = 1.5; Re_∆= u⁰_δ(∆/ν) (5.7f) A dynamic algebraic model for Σ_gen was proposed by Knikker et al. [89] which uses a power-law approach:

Σgen = ∆ η_i

β_k

|∇¯c| (5.8)

where the inner cut off scale η_i is taken to be η_i = 3δ_z and β_k is estimated based on a dynamics formulation given as: β_k = [log \h|∇¯c|i − logh∇ˆ¯ci]/ log γ where ˆ¯c denotes progress variable at the test filter level γ∆. A model for the wrinkling factor Ξ using the power-law approach was proposed by Fureby [58]. This model can be expressed in terms of the generalised FSD in the following form.

Σ_gen =

 Γu⁰_∆

S_L

D−2

|∇¯c| (5.9)

where Γ is given by Eq. 5.3 and D is the fractal dimension defined using the following parameterisation:

D = 2.05

(u⁰_∆/S_L) + 1 + 2.35

(S_L/u⁰_∆) + 1 (5.10)

A power-law based model was proposed by Chakraborty and Klein [33] for the generalised FSD which takes the following form:

Σ_gen =|∇¯c|

( exp

−Θ∆

ηi

 +



1− f exp

−Θ∆

ηi

  ∆ ηi

D−2)

(5.11)

where Θ and f are model constants and η_i is the inner cut off scale which is parameterised as η_i = [0.345Ka⁻²exp(−Ka) + 6.41Ka^−1/2× [1 − exp(−Ka)]]δz, where Ka is the Karlovitz number. The inner cut off scale was formulated using a DNS database where a case of corrugated flamelets regime and thin reaction zones regime were used. The term D in the Eq. 5.11 is the fractal dimension, which was shown to be dependent on the regime of combustion by Chakraborty and Klein [33], which resulted in the following expression for D: D = (1/3)erf (2Ka).

The model by Chakraborty and Klein [33] was modified in the current study to incorporate the effects of Lewis number and local turbulent Reynolds number, ad-ditionally the usage of localised Karlovitz number instead of the global Karlovitz number was explored. This analysis is presented in the following section.