# Simulation of Aerosol Evolution within Background Pollution for Nucleated Vehicle Exhaust via TEMOM
## 2 Governing Equation  
### CFD Model  
The one-way couple was used between the continuous and discrete phase considering that the particle has a minimum mass. Thus, we solved the fluid dynamic equation and moment transport equation separately. To obtain the information of fluid flow, the RANS equation coupled with Renormalized Group (RNG) \(k\sim\omega\) turbulent model is utilized. The governing equations are described as follows:\[\begin{split}\frac{\partial\overline{u_{i}}}{\partial x_{i}}& =0\\ \rho\frac{\partial\overline{u_{i}}}{\partial t}+\frac{\partial}{ \partial x_{j}}\big{(}\rho\overline{u_{i}u_{j}}\big{)}&=-\frac{ \partial\overline{p}}{x_{i}}+\mu\frac{\partial^{2}\overline{u_{i}}}{\partial x _{j}\partial x_{i}}-\frac{\partial R_{ij}}{\partial x_{j}}+\overline{f_{i}} \end{split} \tag{1}\]  
in which  
\[R_{ij}=\mu_{t}\left(\frac{\partial\overline{u_{i}}}{\partial x_{j}}+\frac{ \partial\overline{u_{j}}}{\partial x_{i}}\right)-\frac{2}{3}\rho k\delta_{ij} \tag{2}\]  
where \(\overline{u_{i}}\) and \(\overline{u_{j}}\) are mean velocity in \(i\) th and \(j\)th direction coordinates, respectively; \(\rho\) is the fluid density; \(\mu\) is the laminar viscosity; \(\overline{f_{i}}\) is the body force in the \(i\)th direction coordinate, and \(\mu_{t}\) is the turbulent dynamic viscosity. The enthalpy \(h\) is calculated by solving the following energy equation in this study:  
\[\frac{\partial(\rho h)}{\partial t}+\frac{\partial}{\partial x_{i}}\big{(} \rho h\overline{u_{i}}\big{)}=\frac{\partial}{\partial x_{i}}\bigg{(}\frac{ k_{t}}{C_{p}}\frac{\partial h}{\partial x_{i}}\bigg{)} \tag{3}\]  
where \(k_{t}\) is the thermal conductivity and \(C_{p}\) is the specific heat at constant pressure. The temperature \(T\) is obtained by:  
\[h=C_{p}T \tag{4}\]  
the turbulence kinetic energy \(k_{t}\) and its rate of dissipation \(C_{p}\) are obtained by the equations below:  
\[\frac{\partial(\rho k)}{\partial t}+\frac{\partial}{\partial x_{i}}(\rho\kappa u _{i})=\frac{\partial}{\partial x_{j}}\Bigg{[}\bigg{(}\mu+\frac{\mu_{\tau}}{ \delta_{e}}\bigg{)}\frac{\partial k}{\partial x_{j}}\Bigg{]}+P_{k}+P_{b}-\rho \epsilon-Y_{M} \tag{5}\]  
\[\frac{\partial(\rho\epsilon)}{\partial t}+\frac{\partial}{\partial x_{i}}( \rho\epsilon u_{i})=\frac{\partial}{\partial x_{j}}\Bigg{[}\bigg{(}\mu+\frac {\mu_{\tau}}{\delta_{e}}\bigg{)}\frac{\partial\epsilon}{\partial x_{j}}\Bigg{]} +c_{e1}\frac{\epsilon}{k}(P_{k}+c_{e3}P_{b})-c_{e2}\rho\frac{\epsilon^{2}}{k} \tag{6}\]  
where \(P_{k}\) represents the production of turbulence kinetic energy due to the mean velocity gradients. \(P_{b}\) represents the production of turbulence kinetic energy due to buoyancy. \(Y_{M}\) represents the partition of fluctuating dilatation incompressible turbulence to the overall dissipation rate. The turbulent Prandtl number for \(k\), \(\delta_{k}\), is 1; the turbulent Prandtl number for \(\epsilon\), \(\delta_{e}\), is 1.30; and \(c_{e1}\) is 1.44, \(c_{e2}\) is 1.92; the turbulent viscosity, \(\mu_{t}\), is obtained by:  
\[\mu_{t}=c\rho\theta\ell=\rho c_{\mu}\frac{k^{2}}{\epsilon} \tag{7}\]  
where \(c_{\mu}\) is 0.09, \(\theta=k^{1/2}\), \(\ell=k^{3/2}/\epsilon\); and energy Prandtl number and Prandtl number are 0.85.