Mathematical model contains the following sub models.

### Flow model

Mathematical model, that governs the flow, comprise

$$\begin{aligned}&\frac{\partial \rho }{\partial t}+\frac{\partial \rho u_j}{\partial x_j}=0, \end{aligned}$$

(1)

$$\begin{aligned}&\frac{\partial \rho u_i}{\partial t}+\frac{\partial \rho u_j u_i}{\partial x_j} = \rho g_i-\frac{\partial p}{\partial x_i}+\frac{\partial }{\partial x_j}\left( \mu \frac{\partial \ u_i}{\partial x_j}+\mu \frac{\partial u_j}{\partial x_i}-\frac{2}{3}\mu \frac{\partial u_k}{\partial x_k}\ \delta _{ij}\right) +S_i, \end{aligned}$$

(2)

$$\begin{aligned}&\frac{\partial \rho E}{\partial t}+\frac{\partial \left( \left( \rho E+p\right) u_j\right) }{\partial x_j} = \frac{\partial u_i\tau _{ji}}{\partial x_j}+\frac{\partial }{\partial x_j}\left( k_1 \frac{\partial T}{\partial x_j}\right) + S_E. \end{aligned}$$

(3)

In Eq. (2), it is assumed that \(S_i=0\). i.e., there is no extra source contributing to govern the flow. In Eq. (3), \(k_1\) stands for thermal conductivity. These equations are respectively known as continuity, momentum or momentum transport and energy equations. The general transport equation for any specie \(\phi \) is given by

$$\begin{aligned} \frac{\partial }{\partial t}\left( \rho \phi \right) +\frac{\partial }{\partial x_j}\left( \rho u_j \phi \right) = \frac{\partial }{\partial x_j}\left( \Gamma \frac{\partial \phi }{\partial x_j}\right) +S_\phi . \end{aligned}$$

(4)

### Turbulence model

It has been established that there is no single turbulence model suitable for every type of turbulent flow. The \(k-\epsilon \) turbulence model works well in free-stream regions of flow which fails near the wall boundaries while \(k-\omega \) model behaves quite well in rather opposite conditions like in the near-wall regions and is not suitable for free-stream flows. This situation leads to the design of a hybrid model, like the SST \(k-\omega \) model which is switched to the \(k-\epsilon \) model in the free stream regions and is switched to the \(k-\omega \) model near the wall regions to capture the wall effects precisely [17]. In our study, firstly we tested the realizable \(k-\epsilon \) model which failed. We did not test the \(k-\omega \) model in view of the above arguments. Secondly we tested the SST \(k-\omega \) model which solves the following two transport equations, Eqs. (5) and (6),

$$\begin{aligned} \frac{\partial \rho k}{\partial t}+\frac{\partial \rho \ u_i\ k}{\partial x_i}= & {} \frac{\partial }{\partial x_j}\left( \Gamma _k\ \frac{\partial k}{\partial x_j}\right) -Y_k+G_k+S_k, \end{aligned}$$

(5)

$$\begin{aligned} \frac{\partial \rho \ \omega }{\partial t}+\frac{\partial \rho \ u_i\ \omega }{\partial x_i}= & {} \frac{\partial }{\partial x_j}\left( \Gamma _{\omega }\ \frac{\partial \omega }{\partial x_j}\right) -Y_{\omega }+G_{\omega }+D_{\omega }+S_{\omega }. \end{aligned}$$

(6)

This model worked well. Thirdly we tested the Transition SST model which comprises the above SST \(k-\omega \) model and two additional equations one for the intermittency \(\gamma \) given by Eq. (7) and the other for the transition momentum thickness Reynolds number \(R{\tilde{e}}_{\theta t}\) given by Eq. (8). This model was developed by Menter et al. [18].

$$\begin{aligned} \frac{\partial \rho \ \gamma }{\partial t} + \frac{\partial \rho \ U_j\ \gamma }{\partial x_j}= & {} P_{\gamma 1} + P_{\gamma 2} - E_{\gamma 1} - E_{\gamma 2} + \frac{\partial }{\partial x_j} \left[ \left( \mu + \frac{\mu _t}{\sigma _\gamma }\right) \frac{\partial \gamma }{\partial x_j}\right] , \end{aligned}$$

(7)

$$\begin{aligned} \frac{\partial \rho \ R{\tilde{e}}_{\theta t}}{\partial t} + \frac{\partial \rho \ U_j\ R{\tilde{e}}_{\theta t}}{\partial x_j}= & {} P_{\theta t} + \frac{\partial }{\partial x_j} \left[ \sigma _{\theta t}\left( \mu + \mu _t\right) \frac{\partial R{\tilde{e}}_{\theta t}}{\partial x_j}\right] . \end{aligned}$$

(8)

This model did not give any significant improvement over the SST \(k-\omega \) model and the results obtained by the SST \(k-\omega \) and Transition SST models were quite comparable. Therefore, either of these two models could be used. However, we decided to use the 4-equation model namely the Transition SST model for more reliability, robustness and accuracy.

### Chemical species transport

The transport equation for species produced due to chemical reaction is

$$\begin{aligned} \frac{\partial }{\partial t}\left( \rho Y_i\right) +\nabla \cdot \left( \rho {{\varvec{v}}} Y_i\right) = -\nabla \cdot {{\varvec{J}}}_i+R_i+S_i. \end{aligned}$$

(9)

### Spray breakup model

For spray breakup, we use Kelvin–Helmholtz and Rayleigh–Taylor (KHRT) model designed by Beale and Reitz [19]. This model is a modified version of the KH and RT models and consists of two steps, the primary breakup and the secondary breakup. The KH model is used to predict the primary breakup of the intact liquid core of a diesel jet while to predict the secondary breakup of the individual drops, the KH model is used in conjunction with the RT model. The model equations are

$$\begin{aligned} \Omega _{KH} \sqrt{\frac{a^3\ \rho _l}{\sigma }}= & {} \frac{0.34\ +\ 0.38\ We_g^{1.5}}{(1+1.4T^{0.6})(1+Oh)}, \end{aligned}$$

(10)

$$\begin{aligned} \frac{\Lambda _{KH}}{a}= & {} \frac{9.02\left( 1+0.4\ T^{0.7}\right) \left( 1+0.45\ Oh^{0.5}\right) }{\left( 1\ +\ 0.87\ We_g^{1.67}\right) ^{0.6}}, \end{aligned}$$

(11)

$$\begin{aligned} \Omega _{RT}= & {} \sqrt{\frac{2}{3}\frac{\left[ (\rho _p-\rho _g)(-g_t)\right] ^{\frac{3}{2}}}{(\rho _p+\rho _g)\ \sqrt{3\sigma }}}, \end{aligned}$$

(12)

$$\begin{aligned} K_{RT}= & {} \sqrt{\frac{(\rho _p-\rho _g)(-g_t)}{3\sigma }}. \end{aligned}$$

(13)

### Turbulence–chemistry interaction

In non-premixed flames, turbulence is mainly responsible for the air–fuel mixture preparation. The rate at which this mixture is prepared dictates the rate at which combustion will take place. This rate of mixing of the reactants is determined by the characteristics of turbulence and is much less than the rate at which reactions take place. Therefore, it is quite reasonable to assume that the combustion is limited by the rate at which the air–fuel mixture is prepared and not by the rate of reactions determined kinetically. Similarly, in premixed flames, the rate of mixing of the cold reactants and hot products is very much less than the rate of reactions. These are the cases when the combustion is mixing-limited and so the chemical kinetic rates can be neglected.

Eddy dissipation model, designed by Magnussen and Hjertager [20], is based on the eddy breakup model of Spalding [21]. The eddy dissipation model uses the rate of dissipation of eddies containing reactants and products to determine the reaction rate as the rate of eddy dissipation determines the rate of mixing. In this way, the model works under the assumption that the reactions are mixing controlled as only mixed fuel and oxidizer are burnt. Therefore, for the problem under consideration, this model is more suitable for the turbulence–chemistry interaction. In this model, the source term \(R_{i,r}\) for *i*-th specie from the reaction *r* is

$$\begin{aligned} R_{i,r} = min(R_1,R_2), \end{aligned}$$

(14)

where

$$\begin{aligned} R_1 = v^\prime _{i,r}\ M_{w,i}\ A\ \rho \ \frac{\epsilon }{k}\ min_{\mathcal {R}}\left( \frac{Y_{\mathcal {R}}}{v^\prime _{{\mathcal {R}},r}\ M_{w,{\mathcal {R}}}}\right) , \end{aligned}$$

(15)

and

$$\begin{aligned} R_2 = v^\prime _{i,r}\ M_{w,i}\ A\ B\ \rho \ \frac{\epsilon }{k}\ \frac{\sum _P\ Y_P}{\sum ^N_j v^{\prime \prime }_{j,r}\ M_{w,j}}. \end{aligned}$$

(16)

In the above equations, *N* denotes the number of species, \(Y_{\mathcal {R}}\) and \(Y_P\) denote respectively the mass fractions of reactant \({\mathcal {R}}\) and product *P*, *A* is a constant whose value is equal to 4 and *B* is a constant whose value is equal to 5.

### NO model

Oxides of nitrogen are mainly formed in the high-temperature burned gases, i.e., products of combustion, through chemical reactions involving nitrogen and oxygen atoms and molecules, which do not reach chemical equilibrium [22]. The main constituent of nitric oxides (NOx) emissions from diesel engines is nitric oxide NO or thermal NO. This thermal mechanism requires a temperature above 1800 K for its activation [23].

The species transport equation for thermal *NO* is given by

$$\begin{aligned} \frac{\partial \rho \ Y_{NO}}{\partial t}+\nabla \cdot \left( \rho \ \mathbf {v}\ Y_{NO}\right) = \nabla \cdot \left( \rho \ \mathbf{D }\nabla Y_{NO}\right) +S_{NO}. \end{aligned}$$

(17)

In the above equation, \(Y_{NO}\) denotes the gas phase *NO* mass fraction , \({\mathcal {D}}\) denotes the effective diffusion coefficient. The source term of thermal NO, the \(S_{NO}\), is determined from the extended Zeldovich mechanism [24]. The principal reactions governing the formation of thermal NO from molecular nitrogen are as follows:

$$\begin{aligned} N_2 + O\rightleftharpoons & {} NO + N,\end{aligned}$$

(18)

$$\begin{aligned} N + O_2\rightleftharpoons & {} NO + O,\end{aligned}$$

(19)

$$\begin{aligned} N + OH\rightleftharpoons & {} NO + H. \end{aligned}$$

(20)

### Soot model

In the current study, we use Moss–Brookes soot model to determine the soot production during the combustion process. This model was designed by Brookes and Moss [25]. It has less empiricism and theoretically, it provides better accuracy as compared to the existing one-step Khan and Greeves [26] and two-step Tesner [27] models. This model solves transport equations for the soot mass fraction and the normalized radical nuclei concentration. These equations are

$$\begin{aligned}&\frac{\partial \rho \ Y_{soot}}{\partial t}+\nabla \cdot \left( \rho \ \mathbf {v}\ Y_{soot}\right) = \nabla \cdot \left( \frac{\mu _t}{\sigma _{soot}}\ \nabla Y_{soot}\right) +\frac{dM}{dt}, \end{aligned}$$

(21)

$$\begin{aligned}&\frac{\partial \rho \ b^*_{nuc}}{\partial t}+\nabla \cdot \left( \rho \ \mathbf {v}\ b^*_{nuc}\right) = \nabla \cdot \left( \frac{\mu _t}{\sigma _{nuc}}\ \nabla b^*_{nuc}\right) +\frac{1}{N_{norm}}\frac{dN}{dt}. \end{aligned}$$

(22)

In the above equations \(Y_{soot} \) represents the mass fraction of soot, \(\sigma _{soot} \) denotes the turbulent Prandtl number for soot, *M* denotes the mass concentration of soot, \(\sigma _{nuc} \) denotes the turbulent Prandtl number for nuclei transport, \(b^*_{nuc}\) denotes the normalized radical nuclei concentration.

In our study, we tested the one-step and two-step models, but these did not work while the Moss–Brookes model worked well.