Analysis of Water Droplet Interaction with Turbulent Premixed and Spray Flames Using Carrier Phase Direct Numerical Simulations

Figures & data

Table 1. Thermo-physical properties of liquid n-heptane and water.

	n-heptane	water	unit
Density	684	999.9	$kg/m^3$
Molecular mass	100.0	18.02	$g/mol$
Latent heat of vaporization (at boiling temperature)	315.0	2258.0	$kJ/kg$
Specific heat at constant pressure	2296.45	4181.0	$J/kg/K$

Table 2. Simulation parameters for different cases: fuel droplet initial diameter $(a_d/{\delta }_{st}{)}_F$, water droplet initial diameter $(a_d/{\delta }_{st}{)}_W$, overall water loading $Y_W$. The p. means premixed, h. means heptane, d. means droplets and w. means water. These abbreviations are used throughout this work.

Flame configuration	$(a_d/{\delta }_{st}{)}_F$	$(a_d/{\delta }_{st}{)}_W$	$Y_W$
p.h.	-	-	0
p.h.+w.d.	-	0.02	0.1
p.h.+w.d.	-	0.04	0.1
h.d.	0.04	-	0
h.d.+w.d.	0.04	0.02	0.1
h.d.+w.d.	0.04	0.04	0.1

Figure 1. Iso-surface of progress variable $c$ ($1^{st}$ column) and temperature $T$ ($2^{nd}$ column) taken at $c=0.9$ and $T=0.9$ for premixed combustion ($1^{st}$ row) and spray combustion ($2^{nd}$ row) at $t=1.1t_{chem}$. In all cases, the water droplet size is $(a_d/{\delta }_{st}{)}_W=0.04$. The light blue dots are water droplets, while the red dots indicate n-heptane droplets (not to the scale).

Figure 2. Temperature field $T$ on the $x-y$ mid-plane, for premixed combustion (left column) and spray combustion (right column) cases without water injection ($1^{st}$ row), with $(a_d/{\delta }_{st}{)}_W=0.04$ ($2^{nd}$ row) and $(a_d/{\delta }_{st}{)}_W=0.02$ ($3^{rd}$ row). The white dots are the n-heptane droplets, while the pink dots indicate the water droplets (not to the scale). White iso-contours represent $c=0.1$, $0.5$ and $0.9$, respectively. The results are taken at $t/t_{chem}=4.1$.

Figure 3. Water vapour fields $Y_W^g$ on the $x-y$ mid-plane, for premixed combustion ($1^{st}$ column) and spray combustion ($2^{nd}$ column) cases with $(a_d/{\delta }_{st}{)}_W=0.04$ ($1^{st}$ row) and $(a_d/{\delta }_{st}{)}_W=0.02$ ($2^{nd}$ row). The white dots are the n-heptane droplets, while the pink dots indicate the water droplets (not to the scale). White iso-contours represent $c=0.1$, $0.5$ and $0.9$, respectively. The results are taken at $t/t_{chem}=4.1$.

Figure 4. Time evolution of water vapour concentration $Y_W^g$ within the flame ($0.1\le c\le 0.9$) for the cases with water injection. Water droplet size $(a_d/{\delta }_{st}{)}_W=0.04$ (left side) and $(a_d/{\delta }_{st}{)}_W=0.02$ (right side).

Figure 5. Turbulent burning velocity $S_T$ evolution in time, with $(a_d/{\delta }_{st}{)}_W=0.04$ (left side) and $(a_d/{\delta }_{st}{)}_W=0.02$ (right side).

Figure 6. Evolution of normalised burning rate per unit area $\mathrm{\Omega }/({\rho }_0{S}_L)$, with $(a_d/{\delta }_{st}{)}_W=0.04$ (left side) and $(a_d/{\delta }_{st}{)}_W=0.02$ (right side).

Figure 7. Evolution of the relative burning velocity ${\mathrm{\Omega }}^{droplets}/{\mathrm{\Omega }}^{ref}$, with $(a_d/{\delta }_{st}{)}_W=0.04$ (left side) and $(a_d/{\delta }_{st}{)}_W=0.02$ (right side).

Figure 8. Flame thickness ${\delta }_{SDF}$ evolution, with $(a_d/{\delta }_{st}{)}_W=0.04$ (left side) and $(a_d/{\delta }_{st}{)}_W=0.02$ (right side).

Figure 9. Surface density function $|\mathrm{\nabla }c|$ conditioned on progress variable $c$, with $(a_d/{\delta }_{st}{)}_W=0.04$ (left side) and $(a_d/{\delta }_{st}{)}_W=0.02$ (right side). Statistics are taken at $t=3.0t_{chem}$.

Figure 10. Evolution of flame area $A_c$ based on progress variable $c$, with $(a_d/{\delta }_{st}{)}_W=0.04$ (left side) and $(a_d/{\delta }_{st}{)}_W=0.02$ (right side).

Figure 11. Evolution of flame area $A_T$ based on temperature $T$, with $(a_d/{\delta }_{st}{)}_W=0.04$ (left side) and $(a_d/{\delta }_{st}{)}_W=0.02$ (right side).

Figure 12. Probability density function of gaseous equivalence ratio $PDF({\varphi }_g)$ inside the flame ($0.1\le c\le 0.9$) for the spray combustion cases at $t=3.0t_{chem}$, with $(a_d/{\delta }_{st}{)}_W=0.04$ (left side) and $(a_d/{\delta }_{st}{)}_W=0.02$ (right side).

Figure 13. Probability density function of the gaseous equivalence ratio $PDF({\varphi }_g)$ inside the flame ($0.1\le c\le 0.9$) for the laminar spray combustion cases at $t=3.0t_{chem}$, with $(a_d/{\delta }_{st}{)}_W=0.04$ (left side) and $(a_d/{\delta }_{st}{)}_W=0.02$ (right side).

Figure 14. Percentage of heat release nature $\mathrm{\%}$HR (left side) and combustion mode $\mathrm{\%}$ FI in the region given by $0.1<c<0.9$ (right side) for all spray combustion cases, at $t=3.0t_{chem}$. The L indicates laminar conditions.

Table 3. Flame thickness ${\delta }_{SDF}$, flame brush thickness ${\delta }_{FB}$ and fuel droplet mean residence time $t_{res}={\delta }_i/⟨u{⟩}_i$ (where $i$ stands for $SDF$ or $FB$ depending on whether the flame thickness or flame brush thickness is considered). $⟨u{⟩}_i$ stands for the averaged velocity of the fuel droplets within the flame $(SDF,$ $0.1\le c\le 0.9)$ or within the flame brush $(FB,$ $0.1\le \bar{c}\le 0.9)$. The statistics are taken for all the spray combustion cases at $t=3.0t_{chem}$. The L stands for laminar and in this case ${\delta }_{st}$ indicates the unstretched stoichiometric premixed flame thickness.

Case	${\delta }_{SDF}/{\delta }_{st}$	${\delta }_{FB}/{\delta }_{st}$	${\delta }_{SDF}/⟨u{⟩}_{SDF}\times \frac{S_L}{{\delta }_{st}}$	${\delta }_{FB}/⟨u{⟩}_{FB}\times \frac{S_L}{{\delta }_{st}}$
h.d.	1.62	5.78	0.39	1.86
h.d.+w.d. 0.04	1.82	6.54	0.45	2.14
h.d.+w.d. 0.02	2.64	7.96	0.73	2.68
h.d. L	1.53	2.19	0.75	1.17
h.d.+w.d. 0.04 L	1.71	2.25	0.88	1.28
h.d.+w.d. 0.02 L	2.76	2.84	1.82	1.94

Table 4. Flame thickness ${\delta }_{SDF}$, flame brush thickness ${\delta }_{FB}$ and water droplet mean residence time $t_{res}={\delta }_i/⟨u{⟩}_i$ (where $i$ stands for $SDF$ or $FB$ depending on whether the flame thickness or flame brush thickness is considered). $⟨u{⟩}_i$ stands for the averaged velocity of the water droplets within the flame $(SDF,$ $0.1\le c\le 0.9)$ or within the flame brush $(FB,$ $0.1\le \bar{c}\le 0.9)$. The statistics are taken for all the spray combustion cases at $t=3.0t_{chem}$. The L stands for laminar and in this case ${\delta }_{st}$ indicates the unstretched stoichiometric premixed flame thickness.

Case	${\delta }_{SDF}/{\delta }_{st}$	${\delta }_{FB}/{\delta }_{st}$	${\delta }_{SDF}/⟨u{⟩}_{SDF}\times \frac{S_L}{{\delta }_{st}}$	${\delta }_{FB}/⟨u{⟩}_{FB}\times \frac{S_L}{{\delta }_{st}}$
h.d.+w.d. 0.04	1.82	6.54	0.46	2.01
h.d.+w.d. 0.02	2.64	7.96	0.89	2.88
h.d.+w.d. 0.04 L	1.71	2.25	0.90	1.19
h.d.+w.d. 0.02 L	2.76	2.84	1.54	1.63

Figure 15. Flame index FI distribution conditioned on progress variable $c$ for all the turbulent (left side) and laminar (right side) spray combustion cases, at $t=3.0t_{chem}$.