Figure 7.9 presents the instantaneous heat release rate and temperature fields for more dilute sprays (C2, C3) and higher inflow temperature (C4), i.e. higher
pre-7.5. EFFECTS OF DROPLET EQUIVALENCE RATIO AND
PRE-EVAPORATION 110
0.02<C<0.5
0 1 2 3 4
0 0.2 0.4 0.6 0.8
Nf
f/f–
d
KSZ,DNS Nf,KSZ
0 20 40 60 80 100 120
0 0.2 0.4 0.6 0.8
f/f–
d
KSZ,DNS P*f,KSZ Pf,KSZ Beta Gaussian
0.5<C<0.83
0 50 100 150
0 0.2 0.4 0.6
Nf
f/f–
d
KSZ,DNS Nf,KSZ
fst
0 10 20 30 40 50 60
0 0.2 0.4 0.6
f/f–
d
KSZ,DNS P*f,KSZ Pf,KSZ Beta Gaussian fst
0.83<C<0.99
0 10 20 30 40 50 60
0 0.1 0.2 0.3
Nf
f/f–
d
KSZ,DNS Nf,KSZ
fst
(a) Scalar dissipation
0 10 20 30 40 50 60
0 0.1 0.2 0.3
f/f–
d
KSZ,DNS P*f,KSZ Pf,KSZ Beta Gaussian
fst
(b) PDF
Figure 7.8: Modelling of mixture fraction conditional scalar dissipation and PDF in the Kolmogorov scale zone within the pre-heat zone (0.02 < C < 0.5), the homogeneous reaction zone (0.5< C < 0.83) and the droplet burning zone (0.83<
C < 0.99) for case C1.
evaporation rate. The reduction of droplet equivalence ratio from 5 to 2.5, as realized in C2, does not induce significant changes and the spray combustion regime is still identified as the external group combustion. A further reduction of the droplet equivalence ratio (φd = 1, C3) leads to single droplet and internal group combus-tion regimes in which the flames envelop the isolated droplets and droplet clusters, respectively. Comparison of cases C1 and C4 demonstrates that a higher inflow
temperature, i.e. higher pre-evaporation rate, not only leads to a higher flame tem-perature, but also to a higher heat release rate in the thermal decomposition region.
Additionally, the single droplet and group combustion regimes tend to appear just behind the thermal decomposition region also due to the higher pre-evaporation rate, which may result in increasing corrugation of the leading flame front.
T∞=400K,φd=2.5T∞=400K,φd=1T∞=600K,φd=5
(a) Heat release rate (b) Temperature
Figure 7.9: Instantaneous heat release rate, dQ, and temperature,T, fields for cases C2 (the first row), C3 (the second row) and C4 (the third row).
Figure 7.10 presents conditionally averaged fuel vapour mass fraction, YF, the oxygen mass fraction,YO, and heat release rate,dQ, for cases C2-C4. Similar to the analysis of C1 in Fig. 7.4, different zones are distinguished for mixing modelling.
Consistent with case C1 (φd = 5), the pre-heat zone is still approximately located within 0.02 < C < 0.5 for the more dilute spray cases C2 (φd = 2.5) and C3 (φd= 1). Smaller equivalence ratios postpone the appearance of the droplet burning zone which starts at C ≈ 0.86 and C ≈ 0.89 for C2 and C3, respectively. For C4,
7.5. EFFECTS OF DROPLET EQUIVALENCE RATIO AND
PRE-EVAPORATION 112
T∞=400K,φd=2.5
0 0.002 0.004 0.006 0.008 0.01
0 0.2 0.4 0.6 0.8 1 0 0.05 0.1 0.15 0.2 0.25
YF YO
C C-YF C-YO
0 0.2 0.4 0.6 0.8 1
0 0.2 0.4 0.6 0.8 1 dQ×103 (J/s)
C C-dQ
T∞=400K,φd=1
0 0.002 0.004 0.006 0.008
0 0.2 0.4 0.6 0.8 1 0 0.05 0.1 0.15 0.2 0.25
YF YO
C C-YF C-YO
0 0.1 0.2 0.3
0 0.2 0.4 0.6 0.8 1 dQ×103 (J/s)
C C-dQ
T∞=600K,φd=5
0 0.01 0.02 0.03
0 0.2 0.4 0.6 0.8 1 0 0.05 0.1 0.15 0.2 0.25
YF YO
C C-YF C-YO
(a) C−YF, YO
0 1 2 3 4
0 0.2 0.4 0.6 0.8 1 dQ×103 (J/s)
C C-dQ
(b) C−dQ
Figure 7.10: The reaction progress variable conditional fuel vapour mass fraction, YF, and oxygen mass fraction, YO, and heat release rate, dQ, for cases C2 (the first row), C3 (the second row), C4 (the third row).
higher inflow temperature leads to a higher pre-evaporation rate, and the respective onsets of the homogeneous reaction and droplet burning zones are shifted towards the unburned mixture to C≈0.3 and C ≈0.8.
Figures7.11-7.13evaluate the scaling laws for mixture fraction conditional scalar dissipation and its PDF for cases C2-C4. For each case, the statistics in the pre-heat, homogeneous reaction and droplet burning zones are compared with the scaling
0.02<C<0.5
0 1 2 3
0 0.2 0.4 0.6
Nf
f/f–
d
KSZ,DNS Nf,KSZ
0 30 60 90 120 150
0 0.2 0.4 0.6
f/f–
d
KSZ,DNS P*f,KSZ Pf,KSZ Beta Gaussian
0.5<C<0.86
0 20 40 60 80 100
0 0.1 0.2 0.3 0.4 0.5
Nf
f/f–
d
KSZ,DNS Nf,KSZ
fst
0 10 20 30 40 50 60
0 0.1 0.2 0.3 0.4 0.5
f/f–
d
KSZ,DNS P*f,KSZ Pf,KSZ Beta Gaussian fst
0.86<C<0.99
0 5 10 15 20 25
0 0.1 0.2
Nf
f/f–
d
KSZ,DNS Nf,KSZ
fst
(a) Scalar dissipation
0 10 20 30 40 50 60
0 0.1 0.2
f/f–
d
KSZ,DNS P*f,KSZ Pf,KSZ Beta Gaussian
fst
(b) PDF
Figure 7.11: Modelling of mixture fraction conditional scalar dissipation and PDF in the Kolmogorov scale zone within the pre-heat zone (0.02 < C < 0.5), the homogeneous reaction zone (0.5< C < 0.86) and the droplet burning zone (0.86<
C <0.99) for case C2.
laws. The scaling law provides good predictions for scalar dissipation. The PDF model, Pf,KSZ∗ , captures the variation of the PDF for the mixture fraction higher than the (zonal) mixture fraction mean in the pre-heat and homogeneous reaction zones. This holds for all cases. Similar to C1, the apparent deviation in the droplet burning zone can be found around the stoichiometric value of mixture fraction, fst, due to the variation of droplet evaporation rate. Figure 7.13 also shows that the
7.5. EFFECTS OF DROPLET EQUIVALENCE RATIO AND
PRE-EVAPORATION 114
0.02<C<0.5
0 0.5 1 1.5 2
0 0.1 0.2 0.3 0.4
Nf
f/f–
d
KSZ,DNS Nf,KSZ
0 50 100 150 200
0 0.1 0.2 0.3 0.4
f/f–
d
KSZ,DNS P*f,KSZ Pf,KSZ Beta Gaussian
0.5<C<0.89
0 2 4 6 8 10
0 0.05 0.1 0.15 0.2
Nf
f/f–
d
KSZ,DNS Nf,KSZ
0 20 40 60 80 100
0 0.05 0.1 0.15 0.2
f/f–
d
KSZ,DNS P*f,KSZ Pf,KSZ Beta Gaussian
0.89<C<0.99
0 5 10 15 20 25 30
0 0.05 0.1 0.15 0.2
Nf
f/f–
d
KSZ,DNS Nf,KSZ
fst
(a) Scalar dissipation
0 10 20 30 40 50 60
0 0.05 0.1 0.15 0.2
f/f–
d
KSZ,DNS P*f,KSZ Pf,KSZ Beta Gaussian fst
(b) PDF
Figure 7.12: Modelling of mixture fraction conditional scalar dissipation and PDF in the Kolmogorov scale zone within the pre-heat zone (0.02 < C < 0.5), the homogeneous reaction zone (0.5< C < 0.89) and the droplet burning zone (0.89<
C < 0.99) for case C3.
Gaussian shape function provides preferable estimates for higher pre-evaporation rates in all the three zones. In case of lower pre-evaporation rates (see Figs. 7.11 and 7.12), the Gaussian shape function works well in the pre-heat zone and the β-PDF gives better agreement in the droplet burning zone, especially for mixture fraction lower than the average value of the Kolmogorov scale zone.
0.02<C<0.3
0 10 20 30
0 0.2 0.4 0.6 0.8
Nf
f/f–
d
KSZ,DNS Nf,KSZ
0 10 20 30 40 50
0 0.2 0.4 0.6 0.8
f/f–
d
KSZ,DNS P*f,KSZ Pf,KSZ Beta Gaussian
0.3<C<0.8
0 10 20 30 40 50 60
0 0.2 0.4 0.6
Nf
f/f–
d
KSZ,DNS Nf,KSZ
0 10 20 30 40 50
0 0.2 0.4 0.6
f/f–
d
KSZ,DNS P*f,KSZ Pf,KSZ Beta Gaussian
0.8<C<0.99
0 10 20 30 40 50 60
0 0.1 0.2 0.3
Nf
f/f–
d
KSZ,DNS Nf,KSZ
fst
(a) Scalar dissipation
0 10 20 30 40
0 0.1 0.2 0.3
f/f–
d
KSZ,DNS P*f,KSZ Pf,KSZ Beta Gaussian
fst
(b) PDF
Figure 7.13: Modelling of mixture fraction conditional scalar dissipation and PDF in the Kolmogorov scale zone within the pre-heat zone (0.02 < C < 0.3), the homogeneous reaction zone (0.3 < C < 0.8) and the droplet burning zone (0.8 <
C <0.99) for case C4.
7.6 Effect of Droplet Stokes Number
Based on the definition in [39], different Stokes numbers can be realized by varying turbulence intensities (cases C5 and C6) and droplet size (case C7). The droplet Stokes numbers are varied by one order of magnitude (1∼ 10). Similar to C1, the spray combustion mode can be characterized as external group combustion due to
7.6. EFFECT OF DROPLET STOKES NUMBER 116 0.02< C <0.99
d0=50µm,St0=11.62
0 10 20 30 40 50
0 0.2 0.4 0.6 0.8
Nf
f/f–
d
KSZ,DNS
Nf,KSZ f
st
0 10 20 30 40 50 60
0 0.2 0.4 0.6 0.8
f/f–
d
KSZ,DNS P*f,KSZ Pf,KSZ Beta Gaussian fst
d0=50µm,St0=1.16
0 10 20 30 40 50
0 0.2 0.4 0.6 0.8
Nf
f/f–
d
KSZ,DNS
Nf,KSZ fst
0 10 20 30 40 50 60
0 0.2 0.4 0.6 0.8
f/f–
d
KSZ,DNS P*f,KSZ Pf,KSZ Beta Gaussian fst
d0=75µm,St0=8.74
0 5 10 15 20 25
0 0.2 0.4 0.6 0.8 1
Nf
f/f–
d
KSZ,DNS Nf,KSZ
fst
(a) Scalar dissipation
0 20 40 60 80 100
0 0.2 0.4 0.6 0.8 1
f/f–
d
KSZ,DNS P*f,KSZ Pf,KSZ Beta Gaussian fst
(b) PDF
Figure 7.14: Modelling of mixture fraction conditional scalar dissipation and PDF in the Kolmogorov scale zone within 0.02< C < 0.99 for cases C5 (the first row), C6 (the second row) and C7 (the third row).
the high droplet equivalence ratio (φd = 5). The contour plots look similar to Fig.
7.3. The separation into the pre-heat, homogeneous reaction and droplet burning zones is not repeated here and the statistics are extracted from 0.002 < C < 0.99 for the three cases. The effect of droplet Stokes number on the applicability of the scaling laws is presented in Fig. 7.14. Consistent with the results from C1 in Fig.
7.6, the scalar dissipation can be estimated well by Eq. (2.42). Yet again, Pf,KSZ∗
approximately predicts the DNS statistics for the mixture fraction values that are higher than the average value but the agreement is inferior to the β-PDF shape function for smaller mixture fraction values. Some mismatch can be seen around the stoichiometric value. For a larger droplet size, these discrepancies tend to reduce.
7.7 Quantitative Analysis for the Deviation in the Flame Region
The scaling laws given by Eqs. (2.42), (2.44) and (3.32) indicate the modelling of the LES filtered scalar dissipation and PDF. However, the average values of the key parameters cannot represent the needed input value for the entire flame region well. Therefore, the convex variation of scalar dissipation and the secondary peak of the PDF in the flame region cannot be captured by the scaling laws (see Fig. 7.8). An alternative modelling procedure can be considered assuming local homogeneity as follows: 1) the key parameters from each computational cell (e.g.
an LES cell) are collected and their conditional moments are computed by taking the conditional average based on the cell values within the homogeneous region. This procedure is equivalent to CMC modelling of the conditionally averaged quantities where local homogeneity of the conditional moments is assumed in the entire CMC cell with ∆CM C >∆LES, 2) the conditional moment of scalar dissipation, hNf |fi, is computed by the scaling law with the input of the conditionally averaged mean value of each input parameter and the PDF is calculated by the quasi-steady state relationship with hNf | fi. To validate this procedure in CP-DNS, the DNS data is suggested to compare with the mixture fraction conditionally averaged scalar dissipation and PDF in the Kolmogorov scale zone that are given by
hNf,KSZ |fi= 1
A1(f −f1)7/3
hJmif(hfdif −f2) hρ|fihUdifhε|fi
−1/3
, (7.2)
Pf,KSZ = chJmif (hfdif −f2)
hρ|fihNf,KSZ |fi, (7.3)
Pf,KSZ∗ = 2chD|fihx1 |fi/hUdif qhN
f,KSZ|fi 2hD|fi
, (7.4)
where h· | fi represents the parameters conditioned on mixture fraction, h· if indi-cates the average value of the parameters related to droplets that dominating the
7.7. QUANTITATIVE ANALYSIS FOR THE DEVIATION IN THE FLAME
REGION 118
mixture fraction value. Here, hx1 |fi is modelled as hx1 |fi= hfdif
f ∆x. (7.5)
Figure 7.15 shows the new modelling results of scalar dissipation and PDF for case C1. The DNS data are extracted from the representative reaction zone (0.5<
0.5< C <0.99
0 20 40 60 80 100
0 0.02 0.04 0.06 0.08 0.1
Nf
f KSZ,DNS
〈 Nf,KSZ|f 〉 fst
(a) Scalar dissipation
0 10 20 30 40
0 0.02 0.04 0.06 0.08 0.1
f
KSZ,DNS P*f,KSZ Pf,KSZ Beta Gaussian
fst
(b) PDF
Figure 7.15: Modelling of mixture fraction conditional scalar dissipation and PDF in the reaction zone (0.5< C <0.99) for case C1.
C < 0.99) which includes the homogeneous reaction zone and the droplet burning zone. The scaling law, hNf,KSZ | fi (Eq. (7.2)), tends to capture the convex variation of scalar dissipation in the flame region and present an almost perfect match for mixture fractions of high probability. Both PDF models, Pf,KSZ (Eq.
(7.3)) andPf,KSZ∗ (Eq. (7.4)), provide satisfactory estimates for the secondary peak of the PDF around the stoichiometric value and again agree well with the DNS data for mixture fractions higher than the zonal mean. Pf,KSZ even presents a good agreement for the mixture fraction of the highest PDF which is lower than the zonal mean. The scaling laws seem superior to the Gaussian shape or β-PDF functions for modelling mixture fraction PDF.
The variations of the key parameters in the scaling laws with mixture fraction are presented in Figure 7.16. The parameters related to droplets, hJmif, hfdif and hUdif, change in similar trends that their peak values which are around 150% of their smallest values emerge around f ≈fst. Due to the large variation of temperature, hρ | fi decreases by a factor of 3 and hD | fi increases by a factor of 7 within f < fst, respectively. The variation of conditional turbulent energy dissipation shows a different behaviour. A peak arises inbetween 0 < f < fst and a trough
emerges around the stoichiometric value. The extremely high energy dissipation for high mixture fractions of near-zero PDF arises because the evaporation induces additional velocity fluctuations near the droplet.
0.5< C <0.99
1.0e-08 1.1e-08 1.2e-08 1.3e-08 1.4e-08 1.5e-08 1.6e-08
0 0.02 0.04 0.06 0.08 0.1
〈 Jm〉f (kg/s)
f KSZ,DNS
fst
0.2 0.22 0.24 0.26 0.28 0.3
0 0.02 0.04 0.06 0.08 0.1
〈 fd〉f
f KSZ,DNS
fst
0.7 0.8 0.9 1 1.1 1.2
0 0.02 0.04 0.06 0.08 0.1
〈 Ud〉f
f KSZ,DNS
fst
0.1 0.2 0.3 0.4 0.5 0.6 0.7
0 0.02 0.04 0.06 0.08 0.1
〈ρ|f 〉
f KSZ,DNS
fst
0e+00 1e-04 2e-04 3e-04 4e-04 5e-04 6e-04 7e-04 8e-04
0 0.02 0.04 0.06 0.08 0.1
〈 D|f 〉
f KSZ,DNS
fst
0 1000 2000 3000 4000 5000
0 0.02 0.04 0.06 0.08 0.1
〈ε|f 〉
f KSZ,DNS
fst
Figure 7.16: The variations of the key parameters against mixture fraction in the reaction zone (0.5< C <0.99) for case C1.
7.8. SUMMARY 120
7.8 Summary
Carrier-phase DNS are performed for turbulent combustion in moderately dense sprays. This chapter evaluates the applicability of the scaling laws for mixture frac-tion condifrac-tional scalar dissipafrac-tion and its PDF that were calibrated by fully resolved DNS of droplet arrays (see Chapter 5) but are now applied to spray flames where droplets move freely subject to turbulent advection. Due to the point particle as-sumption and insufficient resolution of the quasi-laminar wake near the droplet, the CP-DNS data are selected from within the Kolmogorov scale zone for the assess-ment of scaling laws for turbulent micro-mixing in inter-droplet space. The scaling law, Nf,KSZ (Eq. (2.42)), provides a comprehensive approximation for the scalar dissipation independent of DNS sampling space, droplet equivalence ratio, droplet pre-evaporation rate, spray combustion regimes, droplet Stokes number, droplet size and large turbulence scales. As the Lagrangian point particle model does not al-low for any reliable description of mixture fraction in the vicinity of the droplet, an alternative PDF model, Pf,KSZ∗ (Eq. (3.32)) is suggested for comparison with CP-DNS data. Due to the interactions of droplet evaporation fields, the agreement between the DNS data and the scaling law, Pf,KSZ∗ , is acceptable for mixture frac-tions higher than the mean value within the Kolmogorov scale zone. Note that the model, Pf,KSZ, which is derived based on the accurate boundary conditions at the droplet surface and demonstrated by the fully resolved DNS of droplet array combustion may be the preferential solution for LES sub-grid modelling. Since the input model parameters change strongly due to evaporation and combustion, a con-vex variation of scalar dissipation and a secondary peak of the PDF emerge around the stoichiometric value of mixture fraction. This behaviour cannot be captured well by the scaling laws with average value of the key parameters as input, however, can be better predicted if the conditionally averaged mean of each key parameter is used instead as input. A corrected Gaussian shape function or a β-PDF may provide a preferred PDF shape especially for mixture fractions lower than the zonal averaged mean. It is concluded that the scaling laws have the potential to provide improved sub-grid closures for mixture fraction based combustion models such as flamelet or CMC approaches in the LES or RANS context. This is certainly true for the mod-elling of conditional scalar dissipation but the best suitable PDF model needs to be carefully selected dependent on the mixture fraction value and its relative position to the zonal mean.
Conclusion & Outlook
8.1 Conclusion
The objective of the thesis is to provide physical sub-grid closures for mixture frac-tion condifrac-tional scalar dissipafrac-tion and PDF that are a pre-requisite for LES-flamelet or LES-CMC computations of turbulent spray combustion. The scaling laws which describe the effects of evaporation and turbulence on the scalar dissipation and PDF in inter-droplet space are derived following the suggestions of Klimenko and Bilger [67]. Fully resolved DNS is performed for droplet array combustion where the ran-dom character of droplet is neglected to form a basis to assess the scaling laws and to calibrate the unknown scaling constants in the scaling laws. Carrier-phase DNS where droplets move freely subject to turbulent advection is conducted to further demonstrate the validity of the scaling laws for turbulent micro-mixing.
Fully resolved DNS is performed for the combustion of droplets arranged in 3-dimensional infinite periodic arrays. The existing [67] and newly extended scaling laws are employed to model the key sub-grid scale quantities, such as mixture frac-tion distribufrac-tion and its condifrac-tional scalar dissipafrac-tion and PDF. The inter-droplet space is split into two zones. With the assumption of Le = 1, the transition lo-cation between the near droplet zone and the Kolmogorov scale zone is located at ˆ
x ≈ 2−2.3. This is independent of the inflow integral length scale, droplet size, positions within the droplet array, inflow Reynolds number, inflow turbulence in-tensity, inter-droplet distance, droplet combustion regime and different phases of the evaporation process. The mixture fraction conditional scalar dissipation and its PDF of the Kolmogorov scale zone can be modelled downstream of the transition location. The scaling law for scalar dissipation that is derived based on the mixture fraction distribution,Nf,KSZ (Eq. (2.42)), is applicable for all the investigated cases.
121
8.1. CONCLUSION 122 However, the modelling of the PDF is dependent on the inflow Reynolds number and inter-droplet distance. For higher Reynolds numbers and larger inter-droplet dis-tances (Re∞≥11 andrc= 20d), the PDF model,Pf,KSZ (Eq. (2.44)), derived from the assumption of quasi-steady state provides a good approximation for the PDF because of the very weak interactions between the droplets in the same layer. In case of more significant interactions, Pf,KSZ approximates the mixture fraction dis-tribution well for values larger than the average value of the Kolmogorov scale zone only. For smaller mixture fraction values the corrected Gaussian PDF function, Pf,G (Eq. (2.45)), may provide more reasonable approximations. Suitable fitting functions for modelling constants inherent in the scaling laws are suggested. The modelling results of the characteristic mixing quantities using the fitting functions are reported, and it is shown that the scaling laws provide acceptable predictive tools if the fitting functions are used. It is also noted that the current setup using regular droplet arrays affects turbulence scales larger than the droplet spacing and hinders the mixing modelling for very dense droplet packings (here rc = 5d). The random character of the droplets’ location that would be observed for real spray flames will then not be negligible and again, carrier-phase DNS will be needed to assess the effect of these larger scales on mixture preparation in inter-droplet space.
Carrier-phase DNS is then performed to assess the validity of scaling laws for sub-grid scale scalar characteristics in turbulent evaporating sprays. The scaling laws for the Kolmogorov scale zone that are calibrated by fully resolved DNS are used to model scalar dissipation and the PDF. The errors introduced in CP-DNS due to the point particle assumption and insufficient resolution of the quasi-laminar wake are estimated by a comparison of the CP-DNS with results from fully resolved DNS. The data from the CP-DNS approximates the fully resolved DNS rather well for smaller mixture fractions. However, if an approximation for the entire mixture fraction space is required, the scaling laws provide a qualitatively better estimate than CP-DNS. The scaling laws can be applicable to a much larger inter-droplet region than the CP-DNS data suggest. CP-DNS data should be selected beyond the insufficiently resolved quasi-laminar wake for the assessment of scaling laws. Scalar dissipation can be approximated well by Nf,KSZ (Eq. (2.42)) for a range of Stokes numbers, droplet number densities and turbulent intensities. Similar to the fully resolved DNS, the PDF model, Pf,KSZ (Eq. (2.44)), agrees well with the DNS data for mixture fractions higher than the average value of the Kolmogorov scale zone.
CP-DNS data must be analysed with care in particular for cases with dense droplet loadings with liquid volume fractions of the order of or larger than 0.1% as there
the cells associated with the near droplet zone can cover more than 70% of the computational domain.
The analysis in CP-DNS is extended to turbulent spray combustion. Following the conclusions from the study on purely evaporating sprays, the DNS statistics need to be taken beyond the unresolved quasi-laminar wake of the droplet for the assessment of the scaling laws. The scaling law, Nf,KSZ (Eq. (2.42)), provides a comprehensive approximation for the scalar dissipation independent of DNS sam-pling space, droplet equivalence ratio, droplet pre-evaporation rate, spray combus-tion regimes, droplet Stokes number, droplet size and large turbulence scales. As the Lagrangian point particle model does not allow for any reliable description of mixture fraction in the vicinity of the droplet, an alternative PDF model, Pf,KSZ∗ (Eq. (3.32)) is suggested for comparison with CP-DNS data. Due to the interac-tions of droplet evaporation fields, the agreement between the DNS data and the scaling law, Pf,KSZ∗ , is acceptable for mixture fractions higher than the mean value within the Kolmogorov scale zone. Note that the model,Pf,KSZ (Eq. (2.44)), which is derived based on the accurate boundary conditions at the droplet surface and demonstrated to be a good model by comparison with the fully resolved DNS may still be favourable for LES sub-grid modelling. Since the determining parameters change violently due to evaporation and combustion, a convex variation of scalar dis-sipation and a secondary peak of the PDF emerge around the stoichiometric value of mixture fraction. This behaviour cannot be captured well by the scaling laws with average value of the key parameters as input, however, can be predicted well with the conditionally averaged mean of each key parameter as input. A corrected Gaussian shape function or aβ-PDF may provide a preferred PDF shape especially for mixture fractions lower than the zonal averaged mean. It is concluded that the scaling laws have the potential to provide improved sub-grid closures for mixture fraction based combustion models such as flamelet or CMC approaches in the LES or RANS context. This is certainly true for the modelling of conditional scalar dis-sipation but the best suitable PDF model needs to be carefully selected dependent on the mixture fraction value and its relative position to the zonal mean.
8.2 Outlook
There are some suggestions for the future work:
(1) Turbulence energy dissipation is one of the most import parameters for the scaling laws. The neglect of turbulence modulation in moderately dense sprays is
8.2. OUTLOOK 124 questionable. The large inertia, evaporation and enveloping flame of droplets are highly likely to affect the turbulent characteristics and this can be evaluated by CP-DNS where droplets move subject to the turbulence advection. The commonly used turbulence models for LES of single phase or dilute spray combustion need to be validated. Some corrections for the existing models or some new sub-grid turbulence models may be required.
(2) Some existing CP-DNS studies (e.g. [47]) found the standard Bilger mixture fraction is not uniquely correlated to the reacting species and temperature, and they proposed a new definition of mixture fraction for flamelet formulation. This contradicts the results from the fully resolved DNS of droplet array combustion (not shown here) although the scales of droplet arrays are limited. There is a premise for these CP-DNS studies that they accepted the accuracy of the standard Lagrangian point-particle model for spray combustion and indiscriminately used all the CP-DNS data for analysis. However, the current study uses fully resolved DNS to demonstrate that a large fraction of CP-DNS cells are polluted by the unresolved quasi laminar wakes of droplets and the movement of the droplets between CFD cells which leads to an additional spatial extent of the wake zone. The validity of the standard Bilger mixture fraction for spray combustion needs to be reassessed by future DNS studies.
(3) CP-DNS seems more reliable for larger average droplet spacings independent of the droplet equivalence ratio. The assumption to neglect the volume of droplet and the evaporation source model tends to be compromised for a smaller droplet size with smaller droplet distance. It is also found smaller droplet size and smaller inter-droplet distance tend to generate multi-correlations between the standard mixture fraction and reacting species. There may be a limitation of droplet size and inter-droplet distance for CP-DNS of spray combustion. Future DNS study can attempt to explore it.
(4) In CP-DNS or LES, the scaling laws for the near droplet zone and Kol-momogorov scale zone provide alternative solutions to distribute the evaporation source term into local CFD cells. The newly computed evaporation rate, mixing and combustion characteristics can be evaluated by the fully resolved DNS.
(5) The future study can employ scaling laws as LES sub-grid models to simulate practical applications of turbulent spray combustion and the applicability of the scaling laws can be evaluated.
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