5.3 Assessment of the Scaling Laws in Reacting Flows
5.3.2 Effects of Inflow Reynolds Number and Turbulence Intensity
param-5.3. ASSESSMENT OF THE SCALING LAWS IN REACTING FLOWS 66 eters can vary between the cases. In Secs. 5.3.2 - 5.3.4 they are determined such that they provide a best fit between DNS data and the scaling relationships for each case. In Sec. 5.3.5, we analyze the sensitivity of the results to the exact value of the different modelling parameters and find correlations for their predictive modelling.
5.3.2 Effects of Inflow Reynolds Number and Turbulence
the turbulence further downstream of the respective droplet layer. This is caused by the larger characteristic undulation length of the smallest eddy (see Table 5.4), as will be demonstrated below. The internal group combustion regime [118] is also identified for cases 2 and 3: droplets in the first layer are enveloped by individual flames and group combustion occurs from the second layer onwards. Figure 5.21 also shows that a larger Kolmogorov length scale broadens the wake structures and a higher inflow Reynolds numbers tends to favour more slender wakes and weaker interactions between the droplets in the same layer. A thin boundary layer at the droplet surface can also be identified in Fig. 5.21. The flow in the boundary layer around dispersed particles has been investigated by Eaton [36]. This is not the focus of this study and the boundary layer mixture fraction profiles is presented in Sec.
5.1.2.
Figure 5.22 provides a more quantitative picture for mixture fraction
distribu-Inter-droplet space 1
0 0.1 0.2 0.3 0.4
0 1 2 3 4 5 6
f
xˆ fmax DNS
fmax,NDZ fmax,KSZ
0 0.1 0.2 0.3 0.4
0 1 2 3
f
xˆ fmax DNS
fmax,NDZ fmax,KSZ
0 0.1 0.2 0.3 0.4
0 1 2 3 4
f
xˆ fmax DNS
fmax,NDZ fmax,KSZ
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 = 0.9 DNS xˆ = 0.9 fNDZ xˆ = 1.6 DNS xˆ = 1.6 fNDZ xˆ = 2.2 DNS xˆ = 2.2 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 = 0.6 DNS xˆ = 0.6 fNDZ xˆ = 1.3 DNS xˆ = 1.3 fNDZ 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 x
ˆ = 0.8 DNS xˆ = 0.8 fNDZ x
ˆ = 1.5 DNS xˆ = 1.5 fNDZ x
ˆ = 2.1 DNS xˆ = 2.1 fNDZ
0 0.05 0.1 0.15
0 0.1 0.2 0.3 0.4 0.5
f
r/rc
xˆ = 2.2 DNS xˆ = 2.2 fKSZ xˆ = 2.5 DNS xˆ = 2.5 fKSZ xˆ = 2.8 DNS xˆ = 2.8 fKSZ
(a) case 1
0 0.05 0.1 0.15
0 0.1 0.2 0.3 0.4 0.5
f
r/rc
xˆ = 1.8 DNS xˆ = 1.8 fKSZ xˆ = 2.1 DNS xˆ = 2.1 fKSZ xˆ = 2.4 DNS xˆ = 2.4 fKSZ
(b) case 2
0 0.05 0.1 0.15
0 0.1 0.2 0.3 0.4 0.5
f
r/rc
xˆ = 1.9 DNS xˆ = 1.9 fKSZ xˆ = 2.3 DNS xˆ = 2.3 fKSZ xˆ = 2.7 DNS xˆ = 2.7 fKSZ
(c) case 3
Figure 5.22: 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 1, 2 and 3
5.3. ASSESSMENT OF THE SCALING LAWS IN REACTING FLOWS 68 tions in the axial and radial directions in inter-droplet space 1 for cases 1, 2 and 3.
Referring to case 1 (Fig. 5.22, top), the scaling law, fmax,N DZ (Eq. (2.38)), holds for the axial variation of maximum mixture fraction in the near droplet zone that can be identified for regions within ˆx <1.2. This is also consistent with the results from the non-reacting case in Sec. 5.2 (ˆx ≈ 1). There, the Kolmogorov scale zone is also assumed to start directly from the plane with ˆx ≈ 1 onwards. This section will attempt to determine the transition region between the two zones and a more accurate location of the Kolmogorov scale zone will be quantified. Now the model, fmax,KSZ (Eq. (2.40)), is employed to compare with the highest mixture fraction at each plane perpendicular to the flow direction and it holds for the Kolmogorov scale zone in regions with ˆx > 2.5. Hence, the transition between the two zones occurs within 1.2 < x <ˆ 2.5. In order to model the mixture fraction and its conditional scalar dissipation and PDF for the whole inter-droplet space, a transition location as opposed to a transition region needs to be identified. The axial locations of the intersection of the two scaling laws could mark such a transition point. The scal-ing laws for the near droplet zone should be valid upstream of this location while the mixing characteristics can be approximated with scaling laws derived for the Kolmogorov zones downstream of this location. Therefore, the transition location in inter-droplet space 1 for case 1 is set to ˆx ≈ 2.1. For cases 2 and 3, the transi-tion region between the near droplet zone and the Kolmogorov scale zone is located roughly within 1 <x <ˆ 2.3 and 1.2<x <ˆ 2.3, respectively. Compared with case 1 (1.2<x <ˆ 2.5), the transition region does not change much. Similar to case 1, the transition locations where fmax,N DZ and fmax,KSZ intersect are located at ˆx≈2 and
ˆ
x ≈ 2.2, respectively. The radial variations are also presented for the three cases.
For ˆx < 1, the scaling law for the near droplet zone, fN DZ (Eq. (2.37)), holds, see Fig. 5.22 (middle row). Then, the accuracy starts to diminish and is poor further downstream. Beyond the upper bounds of the transient region, the scaling law for the Kolmogorov scale zone, fKSZ (Eq. (2.39)), holds.
Figure 5.23 presents the modelling for mixture fraction distributions in inter-droplet space 2 for cases 1, 2 and 3. It needs to be assessed whether evaporation and chemical reactions upstream of the second droplet layer affects the transition location and the validity of the scaling laws in the second inter-droplet space. Consistent with the visualization from Fig. 5.21, the area of the near droplet zone expands due to the turbulence decay for the three cases. For cases 1 and 3, the scaling laws for the Kolmogorov scale zone, fKSZ and fmax,KSZ (Eqs. (2.39) and (2.40)), still hold beyond ˆx ≈ 2.3. For case 2, the whole inter-droplet space spans across
Inter-droplet space 2
0 0.1 0.2 0.3 0.4 0.5
0 1 2 3
f
xˆ fmax DNS
fmax,NDZ fmax,KSZ
0 0.1 0.2 0.3 0.4 0.5
0 0.5 1 1.5 2
f
xˆ fmax DNS
fmax,NDZ
0 0.1 0.2 0.3 0.4 0.5
0 1 2 3
f
xˆ fmax DNS
fmax,NDZ fmax,KSZ
0 0.05 0.1 0.15 0.2 0.25 0.3
0 0.1 0.2 0.3 0.4 0.5
f
r/rc
xˆ = 0.6 DNS xˆ = 0.6 fNDZ xˆ = 1.3 DNS xˆ = 1.3 fNDZ xˆ = 2.1 DNS xˆ = 2.1 fNDZ
0 0.05 0.1 0.15 0.2 0.25 0.3
0 0.1 0.2 0.3 0.4 0.5
f
r/rc
xˆ = 0.5 DNS xˆ = 0.5 fNDZ xˆ = 1.0 DNS xˆ = 1.0 fNDZ xˆ = 1.6 DNS xˆ = 1.6 fNDZ
0 0.05 0.1 0.15 0.2 0.25 0.3
0 0.1 0.2 0.3 0.4 0.5
f
r/rc x
ˆ = 0.7 DNS xˆ = 0.7 fNDZ x
ˆ = 1.4 DNS xˆ = 1.4 fNDZ x
ˆ = 2.1 DNS xˆ = 2.1 fNDZ
0 0.05 0.1 0.15
0 0.1 0.2 0.3 0.4 0.5
f
r/rc
xˆ = 1.8 DNS xˆ = 1.8 fKSZ xˆ = 2.1 DNS xˆ = 2.1 fKSZ xˆ = 2.4 DNS xˆ = 2.4 fKSZ
(a) case 1
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 2
0 0.05 0.1 0.15
0 0.1 0.2 0.3 0.4 0.5
f
r/rc
xˆ = 1.7 DNS xˆ = 1.7 fKSZ xˆ = 2.0 DNS xˆ = 2.0 fKSZ xˆ = 2.4 DNS xˆ = 2.4 fKSZ
(c) case 3
Figure 5.23: Comparisons of mixture fraction distributions in axial and radial di-rections in inter-droplet space 2 with the scaling laws, fN DZ, fmax,N DZ, fKSZ and fmax,KSZ (Eqs. (2.37-2.40)), for cases 1, 2 and 3
0 < x <ˆ 2 and the Kolmogorov scale zone does not appear. The scaling laws for the near droplet zone, fN DZ and fmax,N DZ (Eqs. (2.37) and (2.38)), present good agreement with the axial variation of maximum mixture fraction and radial variation of mixture fraction on each plane perpendicular to the flow direction for the whole inter-droplet space. Yet again, the transition region is located within 1 < x <ˆ 2.3 and the transition location is approximately at ˆx≈2.2 in inter-droplet space 2 that is similar to that in inter-droplet space 1. Therefore, based on the comparisons among cases 1, 2 and 3, it can be stated that the characteristic regions of the near droplet zone and Kolmogorov scale zone are independent of inflow Reynolds number, turbulence intensity, the existence of a single or internal group combustion regimes and the positions in the droplet arrays.
Scalar dissipation and PDF modelling for the near droplet zone was investigated
5.3. ASSESSMENT OF THE SCALING LAWS IN REACTING FLOWS 70 in [159]. It is not the major novelty of the present work and is therefore only pre-sented in Appendix B using the present setup. Here, the validity of scaling laws for the Kolmogorov scale zone given by Nf,KSZ (Eq. (2.42)), Pf,KSZ (Eq. (2.44)) and Pf,G (Eq. (2.45)) will be evaluated. The conditional dissipation of mixture fraction and the mixture fraction PDF are also computed beyond the location where the two scaling laws, fmax,N DZ and fmax,KSZ, intersect. The results from Fig. 5.24 indi-cate that the scaling law that is derived based on the mixture fraction distribution, Nf,KSZ (Eq. (2.42)) (see Sec. 3), agrees well with the DNS results of the scalar dis-sipation in the Kolmogorov scale zone of inter-droplet space 1 independent of inflow Reynolds number and turbulence intensity. This can also be seen in inter-droplet space 2 for cases 1 and 3, as shown in Fig. 5.25. Case 3 shows the same level of scalar dissipation as case 1 but higher than case 2. This is consistent with the results from Fig. 5.22: Case 2 presents lower mixture fraction gradients in both axial and radial directions and this is caused by different levels of turbulent energy and its dissipa-tion. Figures 5.24 and 5.25 also show modelling results for the PDF for the three cases. For the smaller inflow Reynolds number of the cases 1 and 2 (Re∞ ≤ 8.6), the PDF can be better estimated by the corrected Gaussian PDF function given by Pf,G (Eq. (2.45)). It is also seen, however, that the quality of modelling deteriorates
Inter-droplet space 1
0 2 4 6 8 10
0 0.02 0.04 0.06
Nf (1/s)
f KSZ, DNS
Nf,KSZ
0 2 4 6 8 10
0 0.02 0.04 0.06
Nf (1/s)
f KSZ, DNS
Nf,KSZ
0 2 4 6 8 10
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 2
0 10 20 30 40 50 60 70 80
0 0.02 0.04 0.06
f
KSZ, DNS Pf,KSZ Pf,G
(c) case 3
Figure 5.24: 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 1, 2 and 3
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.