# Review of Convective Heat Transfer Modelling in CFD Simulations of Fire-Driven Flows
## 3 Main Modelling Approaches
### Law of the Wall  
This approach employs the use of wall functions [45] in order to try to mimic the sudden change of transport (i.e., from molecular to turbulent) in the vicinity of surfaces without explicitly resolving the viscous sublayer (i.e., the smallest length scales). The approach presented below is one of the available methods for modelling convective heat transfer employed by the Fire Dynamics Simulator (FDS) code [23; 46]. The main theory follows dimensional analysis based on the idea that shear at the wall is constant. The friction velocity, \(u_{\tau}=\sqrt{\tau_{w}/\rho}\), is an important scaling variable in the near-wall region, from which the non-dimensional streamwise velocity, \(u^{+}=u/u_{\tau}\), and non-dimensional wall-normal stress distance, \(y^{+}=y/\delta_{v}\), can be defined where \(\delta_{v}=v/u_{\tau}=\mu/(\rho u_{\tau})\) is the viscous length scale.  
The law of the wall [47] for velocity can then be approximated as:  
* Smooth wall: \[u^{+}=y^{+}\quad\quad for\quad\quad y^{+}<11.81\] (17) \[u^{+}=\frac{1}{\kappa}ln(y^{+})+B\quad\quad for\quad\quad y^{+} \geq 11.81,\] where \(\kappa=0.41\) is the von K\(\acute{a}\)rm\(\acute{a}\) constant and \(B=5.2\). The viscous and inertial stresses are important in regions (i.e., called buffer layer) where \(5<y^{+}<30\). Following [48], the solution in this region is approximated by matching the viscous region and log regions at \(y^{+}=11.81\).
* Rough wall: \[u^{+}=\frac{1}{\kappa}ln(\frac{y}{s})+\widetilde{B}(s^{+}),\] (18) where \(s^{+}=s/\delta_{v}\) is the roughness length in viscous units and \(s\) is the dimensional roughness. The parameter \(\widetilde{B}\) can be estimated based on a piecewise function as: \[\widetilde{B}=\begin{cases}B+\frac{1}{\kappa}ln(s^{+})&for\quad s^{+}\geq 5.83 \\ \widetilde{B}_{max}&for\quad 5.83\leq s^{+}<30\\ B_{2}&for\quad s^{+}\geq 30,\end{cases}\] (19)where \(\widetilde{B}_{max}=9.5\) and the parameter \(\widetilde{B}\) varies with \(s^{+}\) but reaches a constant value of \(B_{2}=8.5\) in the fully rough limit [46].  
By analogy to the near-wall model for velocity, the non-dimensional temperature is defined as [46]:  
\[T^{+}=\frac{T_{g}-T_{w}}{T_{\tau}}, \tag{20}\]  
where \(T_{g}\) is the gas phase temperature of the first grid cell next to the wall and the \(T_{w}\) is the wall temperature.  
The law of the wall model for \(T^{+}\) is expressed as:  
\[\begin{split} T^{+}=Pr\ y^{+}&\quad for\quad y^{ +}\leq 11.81\\ T^{+}=\frac{Pr_{t}}{\kappa}ln(y^{+})+B_{T}&\quad for \quad y^{+}\geq 11.81,\end{split} \tag{21}\]  
where \(Pr\) and \(Pr_{t}\) are the molecular and turbulent Prandtl numbers, respectively.  
The temperature scale, \(T_{\tau}\), is defined as:  
\[T_{\tau}=\frac{q_{c}^{{}^{\prime\prime}}}{\rho c_{p}u_{\tau}}, \tag{22}\]  
where \(q_{c}^{{}^{\prime\prime}}\) is the convective heat flux at the wall, while \(\rho\), \(c_{p}\) and \(u_{\tau}\) are the fluid density, the specific heat and the friction velocity, respectively.  
The parameter \(B_{T}\), which is the integration constant from the relation between velocity and temperature gradients, can be determined experimentally as, e.g., [49]:  
\[B_{T}=(3.85 Pr^{1/3}-1.3)^{2}+2.12\ ln(Pr). \tag{23}\]  
Finally, the convective heat transfer coefficient can then be calculated as:  
\[h=\frac{q_{c}^{{}^{\prime\prime}}}{T_{g}-T_{w}}=\frac{\rho c_{p}u_{\tau}}{T^{+ }}. \tag{24}\]  
The law of the wall type of models may be more appropriate for well-resolved LES calculations in practical fire applications as they still require grid sizes that are not too coarse through evaluation of the \(y^{+}\) values (i.e., the non-dimensional distance from the wall to the first mesh node).  
Examples of modelling convective heat transfer using the law of the wall exist in literature, e.g., [50].