# Review of Convective Heat Transfer Modelling in CFD Simulations of Fire-Driven Flows
## 2 Newton's Law and CFD Equations
### CFD Governing Equations  
Within the context of fire modelling, most CFD codes (e.g., [22; 23]) solve the filtered Navier-Stokes equations, along with transport equations for species mass fractions and sensible enthalpy, using Favre-filtered quantities. An overview of the main governing equations [24; 25; 26] (i.e., focusing on the gas phase only), often employed by CFD codes, is presented below. * Continuity: \[\frac{\partial\overline{\rho}}{\partial t}+\
abla\cdot(\overline{\rho}\overline {u})=0\] (2) \begin{table}
\begin{tabular}{c c} \hline \hline
**Application** & \(h\) \\  & **(W/(m\({}^{2}\cdot\) K))** \\ \hline
**Natural convection** & \\ Gases & 2–25 \\ Liquids & 100–1000 \\ \hline
**Forced convection** & \\ Gases & 25–250 \\ Liquids & 50–20,000 \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Typical values of the convective heat transfer coefficient [14]. * Momentum: \[\frac{\partial(\overline{\rho}\widetilde{u})}{\partial t}+\
abla\cdot(\overline{ \rho}\widetilde{u}\widetilde{u})=-\
abla\overline{p}+\
abla\cdot\left[\mu_{ eff}\bigg{(}\
abla\widetilde{u}+(\
abla\widetilde{u})^{T}-\frac{2}{3}(\
abla \cdot\widetilde{u})I\bigg{)}\right]+\overline{\rho}g\] (3)
* Chemical species: \[\frac{\partial(\overline{\rho}\widetilde{Y}_{k})}{\partial t}+\
abla\cdot( \overline{\rho}\widetilde{u}\widetilde{Y}_{k})=\
abla\cdot\left[\overline{ \rho}\bigg{(}D_{k}+\frac{v_{sss}}{Sc_{t}}\bigg{)}\
abla\widetilde{Y}_{k} \right]+\overline{\omega_{k}^{{}^{\prime\prime}}},\ \ (k=1,...,N_{s}-1)\] (4)
* Sensible enthalpy: \[\frac{\partial(\overline{\rho}\widetilde{u}_{s})}{\partial t}+\
abla\cdot( \overline{\rho}\widetilde{u}\widetilde{u}_{s})=\frac{\overline{D}p}{Dt}+ \
abla\cdot\left[\overline{\rho}\bigg{(}\alpha+\alpha_{sgs}\bigg{)}\
abla \widetilde{h}_{s}\right]-\
abla\cdot\overline{q_{r}^{{}^{\prime\prime}}}+ \overline{q_{c}^{{}^{\prime\prime}}},\] (5) where \(\rho\) is the density, \(u\) is the velocity, \(p\) is the pressure, \(\mu_{eff}=\mu+\mu_{sgs}\) is the effective dynamic viscosity, \(\mu\) is the molecular viscosity, \(\mu_{sgs}=\rho v_{sgs}\) is the sub-grid scale viscosity, \(l\) is the identity tensor, \(g\) is the gravitational acceleration, \(Y_{k}\) is the species mass fraction, \(D_{k}\) is the species mass diffusivity, \(Sc_{t}\) is the turbulent Schmidt number, \(\dot{\omega}_{k}^{{}^{\prime\prime}}\) is the species reaction rate, \(N_{s}\) is the number of chemical species (i.e., typically solving for \(N_{s}-1\) species and obtaining \(N_{2}\) from mass conservation), \(h_{s}\) is the sensible energy, \(\alpha\) is the thermal diffusivity, \(Pr_{t}\) is the turbulent Prandtl number, \(\dot{q}_{r}^{{}^{\prime\prime}}\) is the radiative heat flux, \(\dot{q}_{c}^{{}^{\prime\prime}}=\Delta H_{c}\ \cdot\dot{\omega}_{F}^{{}^{ \prime\prime}}\) is the heat release rate per unit volume due to combustion and \(\Delta H_{c}\) is the heat of combustion of the fuel. The sub-grid scale thermal diffusivity is calculated as:  
\[\alpha_{sgs}=\frac{\mu_{sgs}}{\rho Pr_{t}}. \tag{6}\]  
The assumption of a unity Lewis number is often employed by CFD codes (i.e., \(Le=\alpha/D_{k}=1\)), effectively using the effective thermal diffusivity for the diffusion in both the species (i.e., Equation (4)) and sensible energy (i.e., Equation (5)) equations. Differential diffusion effects are often neglected (i.e., \(D_{k}=D\)) since turbulent transport is more dominant than molecular diffusion for typical fire-related scenarios. Finally, the turbulent Prandtl and Schmidt numbers are also often assigned the same value (i.e., \(Sc_{t}=Pr_{t}\)). The use of sub-models for closure of some of the terms involved in the governing equations (i.e., \(\mu_{sgs}\) related to turbulence, \(\overline{\dot{\omega}_{k}^{{}^{\prime\prime}}}\) related to combustion and \(\overline{\dot{q}_{r}^{{}^{\prime\prime}}}\) related to radiation) is required.