# Review of Convective Heat Transfer Modelling in CFD Simulations of Fire-Driven Flows
## 3 Main Modelling Approaches
### Wall-Resolved Approach  
The wall-resolved approach implies the use of very fine grid sizes in order to resolve the viscous sub and transition layers (i.e., contribution from sub-grid scale quantities is zero). Within this approach, the convective heat fluxes can be calculated considering pure heat transfer by conduction (i.e., Fourier's law) in the first grid cells as:  
\[q_{c}^{\prime\prime}=\rho ac_{p}\frac{dT}{dn}, \tag{25}\]  
where \(\rho\) is the density, \(\alpha\) is the molecular thermal diffusivity, \(c_{p}\) is the specific heat, \(n\) denotes the direction normal to the surface and \(dT/dn\) is the temperature gradient calculated using the temperature in the first grid cell next to the wall and the wall temperature. The accuracy of the wall-resolved approach will be strongly dependant on the size of the computational grid and, typically, resolutions of the order of millimetres are needed in order to fully resolve the boundary layer. The accuracy of the convective heat transfer model has been reported to depend on the performance of the turbulence model in the near-wall region. In addition, discrepancies in the predictions of the total heat fluxes near the base of vertical flammable walls have been reported due to laminar flow structures which cannot be easily predicted by turbulence models [51].  
Examples of fine-grained LES simulations of scenarios involving turbulent wall fires in literature include, e.g., [51; 52; 53; 54], reporting on the heat feedback to the fuel surface (i.e., predicted convective and radiative heat fluxes). Additionally, an embedded flame heat flux method for simulations of quasi-steady state vertical flame spread has been reported. In this method, the convective and radiative heat fluxes during steady state were extracted from wall-resolved simulations and were used to improve the accuracy of coarse mesh simulations by embedding them as boundary conditions for the solid phase [55].  
* Special case: The convective heat fluxes calculated with Equation (25) will typically start to be under-predicted with increasing grid size. To compensate for the reduction in the resolved heat fluxes, use of the effective thermal diffusivity can be made for calculating the convecting heat fluxes: \[q_{c}^{\prime\prime}=\rho\alpha_{eff}c_{p}\frac{dT}{dn},\] (26) where \(\alpha_{eff}=\alpha+\alpha_{sgs}\) and \(\alpha_{sgs}\) are the effective and sub-grid scale thermal diffusivities, respectively. This approach considers sub-grid scale effects and attempts to model the convective heat transfer coefficient, \(h\), without the explicit need of an empirical correlation through the use of a turbulence model.  
The accuracy of this modelling approach will be dependant on the employed grid size as well as on the performance of the turbulence model near the wall. Ideally, the turbulence model should be able to predict vanishing turbulence close to the surface and to capture transitions from laminar to turbulent flows. Capturing these modelling aspects with turbulence models using constant coefficients will be more challenging as opposed to dynamic turbulence models. Towards the DNS limit, the sub-grid scale thermal diffusivity would tend to zero in this case and any heat transfer would be by pure conduction (i.e., Fourier's law). This requirement will be satisfied by the dynamic Smagorinsky model, since the model parameter will tend to zero as the grid size approaches the Kolmogorov length scale, in contrast to the Smagorinsky model using a constant coefficient which will introduce a fixed error [56].  
A convective heat transfer model, based on the concept of the effective thermal diffusivity, with a correction of the convective fluxes at coarse grids was recently reported in literature [11]. Additionally, the model has been applied to model convective heat transfer in upward flame spread of Single Burning Items (SBI) scenarios [12].