In this study three main numerical approaches are employed: (1) DNS to solve for the gas phase and the corresponding chemical reactions; (2) a Lagrangian description for tracking spray droplets, and (3) an Eulerian approach to model the processes controlling the evolution of the nanoparticles (growth, aggregation, coagulation...). Droplets, being noticeably smaller than the grid resolution, are modeled as point droplets with a variable diameter. All numerical models are integrated into the in-house DNS code called DINO, a Fortran90 code developed by our group during the last 10 years. A 6th-order central finite-difference approach is used for the spatial discretization, while a semi-implicit 3rd-order Runge–Kutta method is employed for temporal integration. The open-source library Cantera 2.4.0 (Goodwin et al.
2015) is used to compute all chemical reactions, thermodynamic terms, and molecular transport processes in the gas phase. More details about DINO can be found in particular in Abdelsamie et al. (
2016) and Chi et al. (
2017,
2018,
2020). In this code, the low Mach number formulation is implemented in order to get a highly efficient DNS solver, while adding source terms (written as
\(\varGamma\)) representing the coupling with the disperse phase (e.g., spray evaporation). The conservation equations for an ideal gas involving
\(N_{s}\) chemical components are implemented as follows:
$$\begin{aligned} \frac{\partial \rho }{\partial t} + \frac{\partial {(\rho u_i)}}{\partial x_i}&= \varGamma _m \end{aligned}$$
(1)
$$\begin{aligned} \frac{\partial (\rho u_i)}{\partial t} +\frac{\partial (\rho u_j u_i)}{\partial x_j}&= -\frac{\partial \tilde{p}}{\partial x_i} + \frac{\partial \tau _{ij} }{\partial x_j} + \varGamma _{ui}\quad (i=1,2,3)\, ,\end{aligned}$$
(2)
$$\begin{aligned} \frac{\partial T}{\partial t}+ u_j \frac{\partial T}{\partial x_j}&= \frac{1}{\rho C_p}\left[ \frac{\partial }{\partial x_j} \left( \lambda \frac{\partial T}{\partial x_j} \right) -\frac{\partial T}{\partial x_j} \sum _{k=1}^{N_s} \rho C_{p,k} Y_k \theta _{k,j} -\sum _{k=1}^{N_s} h_k \dot{\omega }_k+\varGamma _T\right] ,\end{aligned}$$
(3)
$$\begin{aligned} \rho \frac{\partial Y_k}{\partial t} + \rho u_j \frac{\partial Y_k}{\partial x_j}&= - \frac{\partial (\rho Y_k \theta _{kj})}{\partial x_j}+{\dot{\omega }_k}+{\varGamma _{m,k}} , \end{aligned}$$
(4)
This system of equations is closed by (1) the ideal gas law, and (2) the additional condition describing overall mass conservation:
$$\begin{aligned}&\rho = \frac{P\, W}{{\mathrm {R}} \,T} ,\end{aligned}$$
(5)
$$\begin{aligned}&\sum _{k=1}^{N_s} Y_k = 1. \end{aligned}$$
(6)
In these equations,
\(\rho\),
\(u_i\),
\(\tilde{p}\),
P,
T,
\(Y_k\),
\(N_s\),
\({\mathrm {R}}\) and
W are the density of the gas mixture,
i-th-component of flow velocity, fluctuation pressure, thermodynamic pressure, gas temperature,
k-th species mass fraction, number of species, ideal gas constant, and mixture mean molecular weight, respectively. In Eq. (
2)
\(\tau _{ij}\) is the viscous stress tensor,
$$\begin{aligned} \tau _{ij}=\mu \,\left( \left[ \frac{\partial u_i}{\partial x_j} +\frac{\partial u_j}{\partial x_i}\right] -\frac{2}{3}\delta _{ij} \left[ \frac{\partial u_l}{\partial x_l}\right] \right) , \end{aligned}$$
(7)
where
\(\delta _{ij}\) and
\(\mu\) are the Kronecker delta, and dynamic viscosity, respectively. In Eqs. (
3) and (
4),
\(C_p\),
\(h_k\),
\(\dot{\omega }_k\),
\(\lambda\), and
\(\theta _{k,j}\) represent the specific heat capacity at constant pressure, specific enthalpy, mass reaction rate, heat diffusion coefficient and
j-th component of the species molecular diffusion velocity, respectively. Additionally,
\(\varGamma _m\),
\(\varGamma _{u_i}\), and
\(\varGamma _{T}\) are the liquid source terms for mass, momentum, and temperature equations, respectively,
$$\begin{aligned} \varGamma _m&= -\frac{1}{\varOmega } \sum _d \alpha _d \frac{dm_d}{dt},\end{aligned}$$
(8)
$$\begin{aligned} \varGamma _{u_i}&= -\frac{1}{\varOmega } \sum _d \alpha _d \frac{d(m_d\,V_{i,d})}{dt},\end{aligned}$$
(9)
$$\begin{aligned} \varGamma _{T}&= -\frac{1}{\varOmega } \sum _d \alpha _d \frac{d(m_d\,C_{p,L}\,T_{d})}{dt}. \end{aligned}$$
(10)
Concerning the species transport equations, the source term is
$$\begin{aligned} \varGamma _{m,k} = \varGamma _{m}\,\delta _{k,f}, \end{aligned}$$
(11)
where
\(m_d=\rho _L\,\pi \,a_d^3/6\) is the mass of the liquid droplets with diameter
\(a_d\) and density
\(\rho _L\);
\(V_{i,d}\) is
i-th component of a droplet
d,
\(T_d\) is the temperature of a droplet
d,
\(C_{p,L}\) is the specific heat of a liquid droplet
d at constant pressure, and
\(\delta _{k,f}\) is 1 for the fuel species, 0 otherwise. The quantities
\(\alpha _d\) and
\(\varOmega\) are the interpolation weight and the volume of the Cartesian grid cell centered on the DNS node, respectively. In the code three different molecular diffusion models are available, with increasing level of accuracy, complexity, and computational cost: (1) unity Lewis numbers; (2) mixture-averaged diffusion velocities; (3) multicomponent diffusion velocities. In the present simulations, the intermediate approach (mixture-averaged approach) has been activated for all diffusion terms in DINO, as good compromise between accuracy and complexity.
A two-way coupling between gas and liquid phase is implemented via the exchange of mass, momentum and energy. The droplet equations rely on the model first introduced by Abramzon and Sirignano (
1989), taking into account the improvements suggested in Kitano et al. (
2014). The implemented equations describing droplet location, momentum, mass transfer, and heat transfer read as follows (Abdelsamie and Thévenin
2017,
2019):
$$\begin{aligned} \frac{{\mathrm {d}} {\mathbf{X}}_d}{{\mathrm {d}}t}&= {\mathbf{V}}_d, \end{aligned}$$
(12)
$$\begin{aligned} \frac{{\mathrm {d}} {\mathbf{V}}_d}{{\mathrm {d}}t}&= \frac{{\mathbf{{U}}}_\infty - {{{\mathbf{V}}}}_d }{\uptau _{v,d}}, \end{aligned}$$
(13)
$$\begin{aligned} \frac{{\mathrm {d}} a^2_d}{{\mathrm {d}}t}&= -\frac{a^2_d}{\uptau _{a,d}}, \end{aligned}$$
(14)
$$\begin{aligned} \frac{{\mathrm {d}} T_d}{{\mathrm {d}}t}&= \frac{1}{\uptau _{T,d}} \left[ {T}_\infty -T_d-\frac{{\mathrm {B}}_{T,d}\,L_v}{C_{p,f}^{F}\,W_F } \right] . \end{aligned}$$
(15)
In Eqs. (
12)–(
15), there are four different subscripts and superscripts:
\(\infty\),
F,
f, and
L, which are standing for variables in the far-field gaseous region, properties of fuel vapor in film region, mixture variable in film region, and liquid properties, respectively. The quantities
\({\mathbf{V}}_d\) and
\({\mathbf{U}}_\infty\) are the velocity of the
d-th droplet and of the surrounding gas at droplet location
\({\mathbf{X}}_d\). Also,
\(T_\infty\),
\(L_v\),
\(W_F\),
\(C_{p,f}^F\) and
\({\mathrm {B}}_{T,d}\) are mixture temperature in far-field, molar latent heat of droplet vaporization, molar mass of the fuel, specific heat of the fuel vapor in the film region and heat transfer number, respectively. The properties and variables in the film region are computed based on the one-third rule (Abramzon and Sirignano
1989; Wang and Rutland
2007) and have the subscript
f, as mentioned above. Motion and evaporation of the droplets are characterized by three characteristic time scales: momentum relaxation time (
\(\uptau _{v,d}\)), evaporation delay (
\(\uptau _{a,d}\)) and heating delay (
\(\uptau _{T,d}\)), given by:
$$\begin{aligned} \uptau _{v,d}&= \frac{\rho _L a^2_d}{18\, \mu _f\left( 1+\frac{1}{6}{\mathrm {Re}}_d^{2/3}\right) } , \end{aligned}$$
(16)
$$\begin{aligned} {\uptau _{a,d}}&= \frac{{\mathrm {Sc}}_d}{4{\mathrm {Sh}}_d}\frac{\rho _{_{L}}}{\mu _f}\frac{a^2_d}{{\mathrm {ln}}(1+{{\mathrm {B}}_{m,d}})}, \end{aligned}$$
(17)
$$\begin{aligned} {\uptau _{T,d}}&= \frac{{\mathrm {Pr}}_d}{6{\mathrm {Nu}}_d} \frac{C_{p,L}}{C_{p,f}}\frac{\rho _{_{L}} a_d^2}{{\mu }_f}\frac{{\mathrm {B}}_{T,d}}{{\mathrm {ln}}(1+{{\mathrm {B}}_{T,d}})}. \end{aligned}$$
(18)
In these equations, the characteristic time scales are computed as a function of various dimensionless numbers: the droplet Reynolds number,
\({\mathrm {Re}}_d\), the Spalding mass transfer number (
\({\mathrm {B}}_m\)) and the heat transfer number (
\({\mathrm {B}}_T\)),
$$\begin{aligned} {\mathrm {Re}}_d&= \frac{\rho _\infty \,a_d\,|{\mathbf{U}}_\infty -{\mathbf{V}}_d|}{\mu _f}, \end{aligned}$$
(19)
$$\begin{aligned} {\mathrm {B}}_{m,d}&= \frac{Y_{s,d}-Y_{F,\infty }}{1-Y_{s,d}}, \end{aligned}$$
(20)
$$\begin{aligned} Y_{s,d}&= \frac{W_F}{W_F+W_O\left( P_\infty /P_{{\mathrm {sat}},d} -1 \right) },\end{aligned}$$
(21)
$$\begin{aligned} {\mathrm {B}}_{T,d}&= (1+ {\mathrm {B}}_{m,d})^\phi -1, \end{aligned}$$
(22)
$$\begin{aligned} \phi&= \frac{C_{p,f}^F}{C_{p,f}}\frac{{\mathrm {Sh}}_d}{{\mathrm {Nu}}_d}\frac{{\mathrm {Pr}}_d}{{\mathrm {Sc}}_d}. \end{aligned}$$
(23)
Here,
\(Y_{s,d}\),
\(Y_{F, \infty }\),
\(W_O\),
\(P_\infty\) and
\(P_{{\mathrm {sat}},d}\) are the saturated vapor mass fraction, fuel mass fraction in far-field gas mixture, oxidizer molar mass, far-field pressure and saturated vapor pressure computed with the Clausius–Clapeyron equation:
$$\begin{aligned} P_{{\mathrm {sat}},d} = P_{{\mathrm {ref}}}\,\, {\mathrm {exp}} \left[ -\frac{L_v}{R} \left( \frac{1}{T_d} - \frac{1}{T_{{\mathrm {ref}}}} \right) \right] . \end{aligned}$$
(24)
In Eq. (
24),
\(P_{{\mathrm {ref}}}\), and
\(T_{{\mathrm {ref}}}\) are reference pressure and temperature, taken here as atmospheric pressure and boiling temperature of the fuel at this pressure, respectively, while
\(L_v\) is corrected using the Watson equation,
$$\begin{aligned} L_v=L_{v,s}\left( \frac{T_{\mathrm {cr}}-T_d}{T_{\mathrm {cr}}- T_{\mathrm {ref}}} \right) ^{0.38}. \end{aligned}$$
(25)
Here,
\(L_{v,s}\) and
\(T_{\mathrm {cr}}\) are the molar latent heat at reference temperature
\(T_{{\mathrm {ref}}}\), and critical temperature of the fuel, respectively. As shown in Eqs. (
22)–(
23) the heat transfer number depends on the fuel vapor to gas mixture specific heats (
\(C_{p,f}^F\),
\(C_{p,f}\)) at film region, Prandtl number (Pr), Schmidt number (Sc), Sherwood number (Sh), and Nusselt number (Nu), which are computed following Ref. Borghesi et al. (
2013):
$$\begin{aligned} {\mathrm {Nu}}_d&= 2 + \frac{{\mathrm {Nu}}_{d,0} -2}{F({\mathrm {B}}_{T,d})},\end{aligned}$$
(26)
$$\begin{aligned} {\mathrm {Sh}}_d&= 2 + \frac{{\mathrm {Sh}}_{d,0} -2}{F({\mathrm {B}}_{m,d})},\end{aligned}$$
(27)
$$\begin{aligned} {\mathrm {Nu}}_{d,0}&= 1 + (1+{\mathrm {Re}}_d \,{\mathrm {Sc}}_d)^{1/3}\, f({\mathrm {Re}}_d),\end{aligned}$$
(28)
$$\begin{aligned} {\mathrm {Sh}}_0&= 1 + (1+{\mathrm {Re}}_d \,{\mathrm {Pr}}_d)^{1/3}\, f({\mathrm {Re}}_d), \end{aligned}$$
(29)
$$\begin{aligned} F({\mathrm {B}}_d)&= (1+{\mathrm {B}}_d)^{0.7}\;{\mathrm {ln}} (1+{\mathrm {B}}_d)/{\mathrm {B}}_d, \end{aligned}$$
(30)
$$\begin{aligned} f({\mathrm {Re}}_d)&= {\mathrm {max}}\left(1; \,{\mathrm {Re}}_d^{0.077}\right). \end{aligned}$$
(31)
This coupling between the gas phase and the liquid phase is a combination of DNS (for the gas phase) and discrete particle simulation technique (DPS) for the disperse phase, and can thus overall be written DNS-DPS.
Concerning now the third modelling level, used to describe the evolution of the nanoparticles, the model developed by Kruis et al. (
1993) with improvements described in Panda and Pratsinis (
1995) and Weise et al. (
2015) has been implemented. This model can be summarized as follows.
$$\begin{aligned} \frac{dN}{dt}&= -\frac{1}{2}\,\beta \,N^2 + I, \end{aligned}$$
(32)
$$\begin{aligned} \frac{dA}{dt}&= -\frac{1}{\tau _s}\,(A\,-\,N \,a_s)+\, I\,a_0, \end{aligned}$$
(33)
$$\begin{aligned} \frac{d\mathcal {V}}{dt}&= I\,v_0. \end{aligned}$$
(34)
In these equations,
N is the nanoparticles’ concentration,
A is the total surface area concentration,
\(\mathcal {V}\) is the total volume concentration,
\(v_0\) is the monomer volume,
\(a_0\) is the monomer surface area, and
I is the nucleation rate. These equations are discretized using a sixth-order, central finite-difference stencil identical to that used for the gas-phase equations (Abdelsamie et al.
2016). Additionally, an eighth-order filter (Kennedy and Carpenter
1994) has been activated for the nanoparticle equations, in order to eliminate spurious oscillations caused by the low diffusivity of the nanoparticles, which cannot be eliminated by the high-order central stencil employed for discretization. The coagulation kernel,
\(\beta\) is computed as
$$\begin{aligned} \beta&= 4\,\pi \, d_c\, \mathcal {D}\left[ \frac{0.5\, d_c}{d_c + g\,\sqrt{2}} + \frac{\mathcal {D}\,\sqrt{2}}{0.5\, c \,d_c} \right] ^{-1}, \end{aligned}$$
(35)
$$\begin{aligned} g&= \frac{\left[ (d_c + L)^3 - (d_c^2 + L^2)^{1.5} \right] }{3\, L\, d_c} - d_c, \end{aligned}$$
(36)
$$\begin{aligned} L&= \frac{8\,\mathcal {D}}{\pi \,c}, \end{aligned}$$
(37)
$$\begin{aligned} c&= \sqrt{\frac{8\,k_b\,T}{\pi \, \rho _{n}\,V}}, \end{aligned}$$
(38)
$$\begin{aligned} \mathcal {D}&= \frac{k_b\,T}{3\, \pi \, \mu \, d_c}. \end{aligned}$$
(39)
where,
L is the mean free path,
\(k_b\) is the Boltzmann constant,
\(\rho _n\) is the particle density,
T is the gas temperature,
\(\mu\) is the viscosity of the gas,
c is the particle velocity, and
\(\mathcal {D}\) is the diffusion coefficient of the particles. The primary diameter
\(d_p\), aggregate diameter
\(d_a\), and collision diameter
\(d_c\) are computed as follows
$$\begin{aligned} d_p&= \left( \frac{6\,\mathcal {V}}{A} \right) ,\end{aligned}$$
(40)
$$\begin{aligned} d_a&= \left( \frac{6\,\mathcal {V}}{\pi \, N} \right) ^{1/3},\end{aligned}$$
(41)
$$\begin{aligned} d_c&= d_p \, n_p^{1/1.8},\end{aligned}$$
(42)
$$\begin{aligned} n_p&= \frac{6\,\mathcal {V}}{\pi \, N \, d_p^3}. \end{aligned}$$
(43)
In Eq.
33, the surface area of the completely fused particles
$$\begin{aligned} a_s = \left( \frac{\mathcal {V}}{N\,v_0}\right) ^{2/3}a_0, \end{aligned}$$
(44)
and the characteristic sintering time
\(\tau _s\) for titanium dioxide are computed similarly to Buesser et al. (
2011),
$$\begin{aligned} \tau _s= 7.44\, \times \, 10^{16}\, T \,d_p^4 \,{\mathrm {exp}}\left( \frac{31{,}000}{T}\right) . \end{aligned}$$
(45)