5.3 Assessment of the Scaling Laws in Reacting Flows
5.3.3 Effect of Inter-droplet Distance
Inter-droplet space 2
0 1 2 3 4
0 0.02 0.04 0.06
Nf (1/s)
f KSZ, DNS
Nf,KSZ
0 1 2 3 4
0 0.02 0.04 0.06
Nf (1/s)
f KSZ, DNS
Nf,KSZ
0 10 20 30 40 50 60 70 80
0 0.02 0.04 0.06
f
KSZ, DNS Pf,KSZ Pf,G
(a) case 1
0 10 20 30 40 50 60 70 80
0 0.02 0.04 0.06
f
KSZ, DNS Pf,KSZ Pf,G
(b) case 3
Figure 5.25: Comparisons of mixture fraction conditional scalar dissipation and PDF in the Kolmogorov scale zone of inter-droplet space 2 with the scaling laws,Nf,KSZ (Eq. (2.42)),Pf,KSZ (Eq. (2.44)) and Pf,G (Eq. (2.45)), for cases 1, 2 and 3
for larger mixture fraction values and Pf,KSZ (Eq. (2.44)) approximates the PDF better for mixture fraction values larger than the Kolmogorov zone averaged mean.
This is different for case 3: here, the inflow Reynolds number is higher (Re∞= 17.2) and the PDF model given byPf,KSZ approximates the DNS data better. For higher inflow Reynolds numbers and relatively large inter-droplet distances, the wakes are too slim and the droplets in the same layer weakly interact with each other. Ar-eas of pure inflow fluid persist leading to non-zero probabilities for f ≈ 0. For larger Reynolds number, the most likely mixture fraction value shifts towards zero, resulting in strong asymmetry that cannot be approximated by a Gaussian shape.
This statement may be only applied to the evaporation with combustion in droplet arrays, but some care should be taken in droplet clouds because the movement of droplets also leads to the interactions between different droplets. Therefore, the corrected Gaussian PDF function given byPf,G may be preferred for modelling the PDF. This will be examined in the carrier-phase DNS context in Chapters 6and 7.
5.3. ASSESSMENT OF THE SCALING LAWS IN REACTING FLOWS 72 fraction and temperature contour plots in a plane containing the droplets for the three cases. Cases 1 and 4 differ in the inter-droplet distance. In contrast to case 1, the internal group combustion regime establishes itself for case 4 and tends to switch to external group combustion which involves a single flame enveloping all droplets [118] (see Fig. 5.26(a)). A further reduction of the inter-droplet distance and of
0.24 0.48 0.72
0.00 0.95 f
0.24 0.48 0.72
0.00 0.95 f
0.24 0.48 0.72
0.00 0.95 f
1200 1800 2400
600 3000 T
(a) Case 4
1200 1800 2400
600 3000 T
(b) Case 5
1200 1800 2400
600 3000 T
(c) Case 6
Figure 5.26: Instantaneous mixture fraction and temperature fields in a plane con-taining the droplets for cases 4, 5 and 6. The white curves represent the iso-contour lines of the stoichiometric mixture fraction (f = 0.0645).
the inflow Reynolds number leads to external group combustion, as shown for cases 5 and 6 (see Figs. 5.26(b) and 5.26(c)). It can be concluded that lower Reynolds numbers and smaller inter-droplet distances lead to external group combustion of droplet arrays. This is consistent with the results in [152].
Figure 5.27 shows the modelling results for the mixture fraction distribution in inter-droplet space 1. The scaling law for the near droplet zone expressed by fmax,N DZ (Eq. (2.38)) can model the distribution of the highest mixture fraction on each plane perpendicular to the flow direction for ˆx < 1 for case 4. Then, the transition extends until ˆx ≈ 2.3 and after that the scaling law for the Kolmogorov scale zone given by fmax,KSZ (Eq. (2.40)) holds. The radial variation within ˆx <1 can be well predicted by the near droplet zone model given by fN DZ (Eq. (2.37)).
The Kolmogorov scale zone model expressed by fKSZ (Eq. (2.39)) can agree well with the radial variation for ˆx >2.3. The transition location is located at ˆx≈2. For
Inter-droplet space 1
0 0.1 0.2 0.3 0.4 0.5 0.6
0 1 2 3
f
xˆ fmax DNS
fmax,NDZ fmax,KSZ
0 0.1 0.2 0.3 0.4 0.5 0.6
0 1 2 3
f
xˆ fmax DNS
fmax,NDZ fmax,KSZ
0 0.1 0.2 0.3 0.4 0.5 0.6
0 1 2 3
f
xˆ fmaxDNS
fmax,NDZ fmax,KSZ
0 0.1 0.2 0.3 0.4 0.5 0.6
0 0.1 0.2 0.3 0.4 0.5
f
r/rc
xˆ = 0.7 DNS xˆ = 0.7 fNDZ xˆ = 1.4 DNS xˆ = 1.4 fNDZ xˆ = 2.1 DNS xˆ = 2.1 fNDZ
0 0.1 0.2 0.3 0.4 0.5 0.6
0 0.1 0.2 0.3 0.4 0.5
f
r/rc
xˆ = 0.9 DNS xˆ = 0.9 fNDZ xˆ = 1.5 DNS xˆ = 1.5 fNDZ xˆ = 2.3 DNS xˆ = 2.3 fNDZ
0 0.1 0.2 0.3 0.4 0.5 0.6
0 0.1 0.2 0.3 0.4 0.5
f
r/rc
xˆ = 0.5 DNS xˆ = 0.5 fNDZ fNDZ xˆ = 1.2 DNS xˆ = 1.2 xˆ = 1.9 DNS xˆ = 1.9 fNDZ
0 0.05 0.1 0.15 0.2 0.25
0 0.1 0.2 0.3 0.4 0.5
f
r/rc
xˆ = 1.7 DNS xˆ = 1.7 fKSZ xˆ = 2.1 DNS xˆ = 2.1 fKSZ xˆ = 2.4 DNS xˆ = 2.4 fKSZ
(a) case 4
0 0.05 0.1 0.15
0 0.1 0.2 0.3 0.4 0.5
f
r/rc
No Kolmogorov scale zone
(b) case 5
0 0.05 0.1 0.15 0.2 0.25
0 0.1 0.2 0.3 0.4 0.5
f
r/rc
xˆ = 1.7 DNS xˆ = 1.7 xˆ = 2.0 DNS xˆ = 2.0 xˆ = 2.3 DNS xˆ = 2.3
fKSZ fKSZ fKSZ
(c) case 6
Figure 5.27: Comparisons of mixture fraction distributions in axial and radial di-rections in inter-droplet space 1 with the scaling laws, fN DZ, fmax,N DZ, fKSZ and fmax,KSZ (Eqs. (2.37-2.40)), for cases 4, 5 and 6
case 6, the near droplet zone models expressed by fN DZ and fmax,KSZ capture the mixture fraction distribution for almost the entire inter-droplet space. Meanwhile, similar to case 4, the scaling laws for the Kolmogorov scale zone given by fKSZ and fmax,KSZ capture the radial and axial variations from ˆx≈2.3 onwards and the scaling laws, fmax,KSZ and fmax,KSZ, intersect at ˆx ≈ 2. This is consistent with the results from case 1 and reveal that the characteristic regions of the near droplet zone and Kolmogorov scale zone are independent of inter-droplet distance and the droplet combustion regime. However, for case 5, the near droplet zone expands such that fN DZ and fmax,N DZ hold even beyond ˆx ≈ 2.3 where fmax,KSZ cannot capture the trend of mixture fraction variation in the axial direction. This may be because the large turbulent eddies are affected by the droplet array structure, and the largest effective turbulence scales are limited by a scale comparable to the
5.3. ASSESSMENT OF THE SCALING LAWS IN REACTING FLOWS 74 inter-droplet separation (cf. [20, 54] and Sec. 5.1). Due to the small inter-droplet distance (rc/d= 5), the turbulent scale separation may not be large enough for the turbulent flow structures to survive. Even larger turbulent structures would move the droplet cloud as a whole and not contribute to inter-droplet mixing. It is noted that for this specific case the setup influences the DNS results, but does not as such invalidate the scaling laws themselves. For very dense sprays random dispersion of the droplets due to turbulence needs to be included and this will be investigated in Chapters 6and 7 using carrier-phase DNS.
Figure5.28 shows the results in inter-droplet space 2 of the droplet arrays. For cases 4 and 6, the inter-droplet distance spans across 0 <x <ˆ 2 and the scaling laws for the near droplet zone give good agreement with the mixture fraction distribution in the whole droplet space. For case 5, the same can be observed as for inter-droplet space 1: the near-inter-droplet zone extents beyond ˆx ≈ 2.3 and fmax,KSZ (Eq.
(2.40)) cannot perfectly capture the mixture fraction values for ˆx >2.3.
Inter-droplet space 2
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7
0 1 2
f
xˆ fmax DNS
fmax,NDZ
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7
0 1 2 3
f
xˆ fmax DNS
fmax,NDZ fmax,KSZ
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7
0 0.5 1 1.5 2
f
xˆ fmaxDNS
fmax,NDZ
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7
0 0.1 0.2 0.3 0.4 0.5
f
r/rc x
ˆ = 0.5 DNS x ˆ = 0.5 fNDZ x
ˆ = 1.3 DNS x ˆ = 1.3 fNDZ x
ˆ = 2.1 DNS x ˆ = 2.1 fNDZ
(a) case 4
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7
0 0.1 0.2 0.3 0.4 0.5
f
r/rc
xˆ = 1.0 DNS xˆ = 1.0 fNDZ xˆ = 1.7 DNS xˆ = 1.7 fNDZ xˆ = 2.6 DNS xˆ = 2.6 fNDZ
(b) case 5
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7
0 0.1 0.2 0.3 0.4 0.5
f
r/rc
xˆ = 0.5 DNS xˆ = 0.5 xˆ = 1.2 DNS xˆ = 1.2 xˆ = 1.8 DNS xˆ = 1.8
fNDZ fNDZ fNDZ
(c) case 6
Figure 5.28: Comparisons of mixture fraction distributions in axial and radial di-rections in inter-droplet space 2 with the scaling laws, fN DZ (Eq. (2.37)), fmax,N DZ
(Eq. (2.38)) and fmax,KSZ (Eq. (2.40)), for cases 4, 5 and 6
Since the Kolmogorov scale zone only appears in inter-droplet space 1 for cases 4 and 6, mixture fraction conditional scalar dissipation and its PDF are modelled and presented in Fig. 5.29. Similar to the above results from case 1 (cf. Fig. 5.24), the scalar dissipation can be approximately predicted by the scaling law that is derived
from the model for mixture fraction distribution,Nf,KSZ (Eq. (2.42)), and the PDF can be better estimated by the corrected Gaussian PDF function,Pf,G (Eq. (2.45)).
Yet again, the PDF model expressed by Pf,KSZ (Eq. (2.44)) that is derived from
Inter-droplet space 1
0 5 10 15
0 0.02 0.04 0.06 0.08 0.1 Nf (1/s)
f
KSZ, DNS Nf,KSZ
0 5 10 15
0 0.02 0.04 0.06 0.08 0.1 Nf(1/s)
f
KSZ, DNS Nf,KSZ
0 5 10 15 20 25
0 0.02 0.04 0.06 0.08 0.1
f
KSZ, DNS Pf,KSZ Pf,G
(a) case 4
0 5 10 15 20 25
0 0.02 0.04 0.06 0.08 0.1
f
KSZ, DNS Pf,KSZ Pf,G
(b) case 6
Figure 5.29: Comparisons of mixture fraction conditional scalar dissipation and PDF in the Kolmogorov scale zone of inter-droplet space 1 with the scaling laws,Nf,KSZ (Eq. (2.42)),Pf,KSZ (Eq. (2.44)) and Pf,G (Eq. (2.45)), for cases 4 and 6
the assumption of quasi-steady-state gives satisfactory estimates for the mixture fraction values that are higher than the average value of the Kolmogorov scale zone.