Criterion II: Conditionally Averaged FSD Behaviour101

Criterion II allows for the assessment of model performance with respect to the variations of mean Σ_gen conditionally averaged ¯c. The variation of Σ_gen condi-tionally averaged on ¯c is shown for ∆ = 8∆_m = 0.8δ_th for cases A-E and E-J in Figs. 5.4 and 5.6, respectively. This filter width has been used to show a case where ∆ < η_i which corresponds to case where partial resolution of flame is available. The behaviour of the models at this filter width (∆ = 8∆_m = 0.8δ_th) for the current database can be summarised as follows:

1. The FSDW model tends to overpredict the value of Σ_gen for flames with higher Ret(e.g., cases D and E). However, the extent of this overprediction is relatively lower in the low Ret cases (e.g., cases A and B). In cases E-J the model FSDW consistently overpredicts the conditional mean value of Σ_gen for all cases. The FSDW model also predicts a skewed shape, which fails to capture the trend predicted by DNS. This skewness appears towards the burned gas side i.e. ¯c > 0.6, whereas the peak of Σ_gen obtained from DNS is roughly at location given by ¯c≈ 0.6.

2. The model FSDK tends to underpredict the value of Σ_gen for all the cases

FSDA FSDC FSDW FSDCH FSDK FSDF FSDNEW MFSDF

−50 0 50

Error(%)

(a) Case A

FSDA FSDC FSDW FSDCH FSDK FSDF FSDNEW MFSDF

−50 0 50

Error(%)

(b) Case B

FSDA FSDC FSDW FSDCH FSDK FSDF FSDNEW MFSDF

−50 0 50

Error(%)

(c) Case C

FSDA FSDC FSDW FSDCH FSDK FSDF FSDNEW MFSDF

−50 0 50

Error(%)

(d) Case D

FSDA FSDC FSDW FSDCH FSDK FSDF FSDNEW MFSDF

−100 0 100

Error(%)

(e) Case E

Figure 5.2: Percentage error of the model prediction from hΣ^geni obtained from DNS for LES filter widths ∆ = 4∆_m = 0.4δ_th ( ), ∆ = 8∆_m = 0.8∆_th ( ),

∆ = 12∆_m = 1.2δ_th ( ), ∆ = 16∆_m = 1.6δ_th ( ), ∆ = 20∆_m = 2.0δ_th ( ),

∆ = 24∆m = 2.4δth ( ) for: (a-e) corresponding to cases A-E, respectively. The DNS grid size is given by ∆_m

shown in Figs. 5.4 and 5.6.

3. Models FSDA, FSDC, FSDCH, FSDF and FSDNEW all tend to capture the

FSDA FSDC FSDW FSDCH FSDK FSDF FSDNEW MFSDF

−100 0 100

Error(%)

(a) Case F

FSDA FSDC FSDW FSDCH FSDK FSDF FSDNEW MFSDF

−50 0 50

Error(%)

(b) Case G

FSDA FSDC FSDW FSDCH FSDK FSDF FSDNEW MFSDF

−50 0 50

Error(%)

(c) Case H

FSDA FSDC FSDW FSDCH FSDK FSDF FSDNEW MFSDF

−50 0 50

Error(%)

(d) Case I

FSDA FSDC FSDW FSDCH FSDK FSDF FSDNEW MFSDF

−50 0 50

Error(%)

(e) Case J

Figure 5.3: Percentage error of the model prediction from hΣ^geni obtained from DNS for LES filter widths ∆ = 4∆_m = 0.4δ_th ( ), ∆ = 8∆_m = 0.8∆_th ( ),

∆ = 12∆_m = 1.2δ_th ( ), ∆ = 16∆_m = 1.6δ_th ( ), ∆ = 20∆_m = 2.0δ_th ( ),

∆ = 24∆m = 2.4δth ( ) for: (a-e) corresponding to cases F-J, respectively.

Σ_gen variation with ¯c obtained from DNS data. The prediction of MFSDF model remains comparable to that of the FSD model for ∆ = 8∆_m= 0.8δ_th.

In order to examine the models behaviour at a filter width where the flame is unresolved, the filter widht ∆ = 24∆_m = 2.4δ_th. The model predictions con-ditionally averaged on ¯c are shown for ∆ = 24∆m = 2.4δth in Figs. 5.5 and 5.7.

The behaviour of the model predictions can be summarised as follows:

1. Models FSDW, FSDA and FSDC all overpredict the value of Σ_gen, and the overprediction increases with increasing Re_t. The FSDCH model cap-tures the behaviour of Σ_gen for small values of Re_t (e.g. cases A-C), but it overpredicts the value of Σ_gen for higher Re_t cases (e.g. cases D-E).

2. In cases F-J the model FSDW predicts a peak at ¯c > 0.6, whereas the peak value of conditionally averaged Σ_gen from DNS occurs at ¯c≈ 0.5 for all the cases. Additionally the models FSDW, FSDA, FSDC and FSDCH tend to overpredict the conditionally averaged value of Σgen and the level of overprediction increases with decreasing Lewis number (i.e. cases F-J, Fig.

5.7).

3. The FSDF, FSDK, FSDNEW and MFSDF models predict Σ_gen satisfacto-rily throughout the flame brush. Additionally at this filter width the models FSDF and MFSDF predict almost identical predictions.

Unlike the other models the behaviour of model FSDK improves with increas-ing ∆, which is consistent with observations made in the context of Figs. 5.2 and 5.3. Moreover the predictions from FSDW remains skewed towards the burned products due to the ¯c dependence of Ξ (i.e. Ξ = 1 + 1.24˜cp(u⁰_∆/S_L)Re_η). The models FSDW, FSDA and FSDC underestimate the destruction of flame surface area in the thin reaction zones regime by ignoring Σ_gen destruction arising due to [D_c(∇. ~N )²]_s which eventually leads to the overpredictions of Σ_gen. The efficiency function Γ in the FSDCH model is parameterised to capture the turbulent flame speed behaviour for both high and low Damk¨ohler combustion [49], and hence this model performs satisfactorily with respect to criterion 1 and 2 for cases with Le≈ 1.0 flames considered here. As the efficiency function proposed by Charlette et al. [49] was not parameterised to account for Le effects, the performance of FS-DCH worsens with decreasing Le and this model starts to overpredict for Le 1 flames.

The use of local Karlovitz number Ka_∆ and turbulent Reynolds number Re_t∆ enables local variation of D, and a satisfactory performance of the FS-DNEW model with respect to criteria 1 and 2 indicates that Eq. 5.16 cap-tures the local Reynolds and Karlovitz number dependence of D for the present definitions of these numbers (i.e. Ka_∆ = 6.66

q(k)˜_∆/S_L

3/2

(∆/δ_z)^−1/2 and Re_t∆ = 4.0(ρ₀u⁰_∆∆/µ₀).

5.2.2.3 Criterion III: Correlation Coefficients

The FSD predicted by the models should have the correct resolved strain rate and curvature dependence in the context of LES and thus the correlation coefficient between the FSD obtained from DNS and from the model prediction should remain as close to unity as possible. The variation of the correlation coefficients between the model prediction and generalised FSD Σ_gen obtained from DNS in the range of filtered reaction progress variable 0.1 ≤ c ≤ 0.9 are shown in Fig.

5.8 and 5.9 for different filter widths. The regions corresponding to 0.1 < ¯c and

¯

c > 0.9 have been ignored since the correlation coefficients have little physical significance in these regions due to small values of Σ_gen obtained from both DNS and model predictions. Figs 5.8 and 5.9 indicate that the correlation coefficients decrease with increasing ∆ due to increased unresolved subgrid wrinkling, which makes the local variation of Σ_gen different from |∇¯c|. The extent of the deviation of the correlation coefficients from unity increases with decreasing (increasing) Le (Re_t) for a given value of ∆. Figs. 5.8 and 5.9 indicates that the models FSDA, FSDC, FSDCH, FSDF, MFSDF, FSDK, FSDNEW and FSDW have comparable correlation coefficients, which deviate considerably from unity for large values of

∆. This indicates that algebraic models may not be able to adequately predict the local strain rate and curvature dependencies of Σ_gen, especially in the thin reaction zones regime. Hence a transport equation for FSD might need to be solved to account for the local strain rate and curvature effects on Σ_gen [29, 35, 67, 68, 73].

0 0.5 1

Figure 5.4: Variation of the mean values of Σgen× δ^z conditional on ¯c across the flame brush for ∆ = 8∆_m = 0.8δ_th according to DNS ( ) and model ( ) for cases A-E.

0 0.5 1

Figure 5.5: Variation of the mean values of Σgen× δ^z conditional on ¯c across the flame brush for ∆ = 24∆_m = 2.4δ_thaccording to DNS ( ) and model ( ) for cases A-E.

0 0.5 1

Figure 5.6: Variation of the mean values of Σ_gen× δz conditional on ¯c across the flame brush for ∆ = 8∆m = 0.8δth according to DNS ( ) and model ( ) for cases F-J.

0 0.5 1

Figure 5.7: Variation of the mean values of Σ_gen× δz conditional on ¯c across the flame brush for ∆ = 24∆m = 2.4δthaccording to DNS ( ) and model ( ) for cases F-J.

5.3 Summary

The performance of several power-law based algebraic models of Σ_gen have been assessed in context of LES for a large range of filter width, using a priori based DNS database of varying turbulent Reynolds and Lewis numbers. It was found that the fractal dimension increases with decreasing Lewis number and increasing turbulent Reynolds number. However the inner cut off scale was shown to be unaffected by turbulent Reynolds and Lewis numbers and additionally the inner cut off scale was shown to be approximately equal to the thermal flame thickness (i.e. η_i ≈ δth) for all the DNS cases. A new parameterisation was proposed for the fractal dimension D which took into account the variation of turbulent Reynolds number, Karlovitz number and Lewis number. This new D parameterisation was used to propose a power-law based model for Σ_gen, which was found to perform comparably or better than the existing algebraic models, for the current DNS database.

The algebraic models for turbulent flame speed S_T may also be used to predict for the generalised FSD [63, 111, 112, 166] using the following relationship: Σ_gen = Ξ|∇¯c| = (ST/S_L)|∇¯c|. These model were however were not used in the current analysis as the relationship S_T/S_L = Ξ = A_T/A_L can only be applied for cases with Le = 1.0 [38, 40, 42], and many of these models have been proposed based on the assumptions which are strictly valid in the context of RANS.

FSDA FSDC FSDW FSDCH FSDK FSDF FSDNEW MFSDF 0

0.5 1

Corr. Coeff.

(a) Case A

FSDA FSDC FSDW FSDCH FSDK FSDF FSDNEW MFSDF 0

0.5 1

Corr. Coeff.

(b) Case B

FSDA FSDC FSDW FSDCH FSDK FSDF FSDNEW MFSDF 0

0.5 1

Corr. Coeff.

(c) Case C

FSDA FSDC FSDW FSDCH FSDK FSDF FSDNEW MFSDF 0

0.5 1

Corr. Coeff.

(d) Case D

FSDA FSDC FSDW FSDCH FSDK FSDF FSDNEW MFSDF 0

0.5 1

Corr. Coeff.

(e) Case E

Figure 5.8: Correlation coefficients between modelled and actual values of Σ_gen in the ¯c range of 0.1 ≤ ¯c ≤ 0.9 for LES filter widths ∆ = 4∆m = 0.4δ_th ( ),

∆ = 8∆m = 0.8∆th ( ), ∆ = 12∆m = 1.2δth ( ), ∆ = 16∆m = 1.6δth ( ),

∆ = 20∆_m = 2.0δ_th ( ), ∆ = 24∆_m = 2.4δ_th ( ) for: (a-e) corresponding to cases A-E, respectively.

FSDA FSDC FSDW FSDCH FSDK FSDF FSDNEW MFSDF 0

0.5 1

Corr. Coeff.

(a) Case F

FSDA FSDC FSDW FSDCH FSDK FSDF FSDNEW MFSDF 0

0.5 1

Corr. Coeff.

(b) Case G

FSDA FSDC FSDW FSDCH FSDK FSDF FSDNEW MFSDF 0

0.5 1

Corr. Coeff.

(c) Case H

FSDA FSDC FSDW FSDCH FSDK FSDF FSDNEW MFSDF 0

0.5 1

Corr. Coeff.

(d) Case I

FSDA FSDC FSDW FSDCH FSDK FSDF FSDNEW MFSDF 0

0.5 1

Corr. Coeff.

(e) Case J

Figure 5.9: Correlation coefficients between modelled and actual values of Σ_gen in the ¯c range of 0.1 ≤ ¯c ≤ 0.9 for LES filter widths ∆ = 4∆m = 0.4δ_th ( ),

∆ = 8∆m = 0.8∆th ( ), ∆ = 12∆m = 1.2δth ( ), ∆ = 16∆m = 1.6δth ( ),

∆ = 20∆_m = 2.0δ_th ( ), ∆ = 24∆_m = 2.4δ_th ( ) for: (a-e) corresponding to cases F-J, respectively.

Statistical Behaviour of the Generalised FSD Transport

The LES generalised FSD Σ_gen is an unclosed quantity in the context of LES, and it is closed either by using an algebraic expression or by solving a mod-elled transport equation alongside other conservation equations. In the previous chapter the a priori analysis of algebraic models of the generalised FSD was dis-cussed. The algebraic closure is valid when the generation rate of flame surface area is assumed to be in equilibrium with its destruction rate. Under unsteady conditions it is often advantageous to solve a modelled transport equation of the generalised FSD Σ_gen. In this chapter the behaviour of the unclosed terms of the FSD transport equation are examined through scaling and by a priori analysis of the DNS database. This analysis then forms the basis for the remaining two chapters of this thesis where the modelling of the curvature and strain rate terms is shown.

6.1 Generalised FSD Transport Equation

The exact transport equation for the generalised FSD Σ_gen is given as [29, 35, 48, 66, 67, 68], which is repeated here from Eq. 2.91 in order to present the naming

convention that is used:

∂Σ_gen

∂t + ∂(˜u_jΣ_gen)

∂x_j =− ∂{[(ui)_s− ˜ui]Σ_gen}

∂x_i

| {z }

T1

+



(δ_ij − NⁱN_j)∂u_i

∂x_j



s

Σ_gen

| {z }

T2

+



S_d ∂N_i

∂xi



s

Σ_gen

| {z }

T3

−∂(S_dN_i)_sΣ_gen

∂xj

| {z }

T4

(6.1)

The terms on the left hand side of 6.1 denote transient and mean advection effects respectively. The unclosed terms T₁, T₂, T₃ and T₄ are known as subgrid convection term, tangential strain rate term, curvature term and propagation term respectively [29, 39, 66, 68, 73].

In order to model for the various unclosed terms of the generalised FSD trans-port equation the statistical behaviour of these terms must be examined, but firstly an order of magnitude analysis was used. To carry out the order of mag-nitude analysis, it is assumed that the Favre filtered velocity in the i^th direction is scaled with a reference velocity scale u_ref i.e.

˜

u_i ∼ u^ref (6.2)

The gradients of filtered quantities are scaled according to ∆⁻¹, the gradients of the sub grid scale quantities are scaled according to δ_L⁻¹ (e.g. Σ_gen ∼ 1/δ^L) and time t scales as t ∼ ∆/u^ref. Based on the above scaling analysis the lo-cal turbulent Reynolds number may be slo-caled as Re_∆ ∼ (u⁰∆∆)/(S_Lδ_L), the local Damk¨ohler number and Karlovitz number are taken to scale as Da_∆ ∼ (∆S_L)/(u⁰_∆δ_L) and the local Karlovitz number can be expressed as Ka_∆ ∼ (u⁰_∆/S_L)^3/2(∆/δ_L)^−1/2. The transient term of the generalised FSD transport equa-tion may then be scaled in the following manner:

∂Σ_gen

∂t ∼ 1

δ_L × u_ref δ_L ×δ_L

∆ × 1

Re^1/2_∆ Da^1/2_∆

∼ u_ref S_L × S_L

δ_L² δ_L

∆

(6.3)

Similarly the mean advection term may be scaled in the following manner:

∂

∂x_i(˜uiΣgen)∼ SL

δ_L² × 1

Re^1/2_∆ Da^1/2_∆ ×uref

S_L (6.4)

The subgrid convection term T₁ can be split into terms arising from heat release and due to subgrid turbulent velocity fluctuation. The heat release contribution of the T1 term maybe scaled as follows:

T₁ due to Heat release: − ∂

∂x_i{[(uⁱ)_s− ˜uⁱ]Σ_gen} ∼ S_L

∆δ_L

∼ SL

δ_L² δL

∆

∼ S_L

δ_L² × 1 Re^1/2_∆ Da^1/2_∆

(6.5)

and the contribution due to turbulent velocity fluctuation can be scaled as:

T₁ due to turbulent velocity fluctuation:

− ∂

∂x_i{[(uⁱ)_s− ˜uⁱ]Σ_gen} ∼ u⁰_∆

∆δ_L

∼ u⁰_∆δ_L S_L∆ ×S_L

δ²_L

∼ SL

δ²_L 1 Da_∆

(6.6)

The tangential strain rate term T₂ can be split as follows:

(a_T)_sΣ_gen = S_m+ S_hr+ S_sg (6.7) where S_m, S_hr and S_sg are the mean, heat release and the subgrid contributions of the T₂ term. The mean component of the tangential term S_m can be scaled in

the following manner:

Sm = [δij − (NⁱNj)_s]∂ ˜ui

∂x_jΣgen ∼ uref

∆δ_L ∼ uref

S_L × SL

δ_L² × 1

Re^1/2_∆ Da^1/2_∆ (6.8) and the heat release contribution of the strain rate term S_hr can be scaled as:

S_hr ∼ S_L

δ_L∆ ∼ S_L δ_L² × δ_L

∆

∼ S_L

δ_L² × 1 Da^1/2_∆ Re^1/2_∆

(6.9)

If the subgrid velocity gradients are scaled using δ_L one obtains the following scaling for S_sg:

Reactive contribution of S_sg: T₂ ∼ S_L

δ_L² (6.10)

The non-reacting contribution of S_sg can be scaled as:

S_sg ∼ u⁰_∆

∆δ_L ∼ u⁰_∆ S_L × δL

∆ × SL

δ_L²

∼ S_L δ²_LDa_∆

(6.11)

The curvature term, T₃, (S_d∇. ~N )_sΣ_gen can be split as follows:

(S_d∇. ~N )_sΣ_gen= C_mean+ C_sg (6.12)

where C_meanis the resolved curvature term and C_sg is the subgrid curvature term.

The resolved curvature term C_mean can be scaled as:

C_mean = (S_d)_s∂(N_i)_s

∂x_i Σ_gen

∼ S_L

∆ 1 δ_L

∼ SL

δ²_L

1 Da^1/2_∆ Re^1/2_∆

(6.13)

The subgrid curvature term C_sg can be scaled as:

C_sg ∼ S_L

δ²_L (6.14)

Finally the propagation term T₄ can be scaled in the following manner:

T₄ =− ∂

∂x_i[(S_d∇. ~N )_sΣ_gen]∼ 1

∆ S_L δ_L ∼ S_L

δ²_L

1

Re^1/2_∆ Da^1/2_∆ (6.15) Therefore from the above analysis the following order of magnitude scaling can be prescribed to the terms of generalised FSD transport equation:

∂Σ_gen

∂t ∼ Ou_ref S_L × S_L

δ²_L δ_L

∆ (6.16)

∂

∂x_i(˜u_iΣ_gen)∼ O u_ref u⁰_∆

1

Da_∆ × S_L δ_L²



∼ O S_L

δ_L² × 1

Re^1/2_∆ Da^1/2_∆ × u_ref SL

! (6.17)

T₁ contribution due to heat release:

∼ O S_L δ²_L × 1

Da_∆

 (6.18)

T₁ contribution due to turbulent velocity fluctuation:

∼ O S_L δ²_L × 1

Da_∆

 (6.19)

S_m ∼ O u_ref S_L × S_L

δ_L² × 1 Re^1/2_∆ Da^1/2_∆

!

(6.20)

S_hr ∼ O S_L δ_L² × 1

Da_∆



(6.21)

Reactive contribution of: S_sg ∼ O S_L δ_L²



(6.22)

Nonreating contribution of: Ssg ∼ O S_L δ²_L × 1

Da_∆



(6.23)

Cmean∼ O S_L δ_L²

1 Re^1/2_∆ Da^1/2_∆

!

(6.24)

Csg ∼ O S_L δ_L²



(6.25)

T₄ ∼ O SL

δ_L²

1 Re^1/2_∆ Da^1/2_∆

!

(6.26) The above scaling analysis was used, alongside, with a priori analysis of the current DNS database. Furthermore this analysis can be used to propose models in the context of LES, which can be seen in chapters 7 and 8. In the following section the behaviour of the unclosed terms of the FSD transport equation are looked into using filtered DNS data.

In document Development of flame surface density closure for turbulent premixed flames based on a priori analysis of direct numerical simulation data (Page 136-153)