**2. Physical Modelling**

The governing equations are given in the following section. The physical description of a turbulent flow with heat transport is based on the governing equation for mass, momentum and energy. The formulation used here is based on Anderson [22]. The continuity equation for a transient flow of a compressible fluid can be described as follows:

$$\frac{\partial \rho}{\partial t} + \nabla \cdot (\rho \mathbf{u}) = 0 \tag{1}$$

with the nabla-operator ∇ = *∂ <sup>∂</sup><sup>x</sup>* , *<sup>∂</sup> <sup>∂</sup><sup>y</sup>* , *<sup>∂</sup> ∂z* , the density *ρ*, the velocity field **u** and time *t*.

The conservation of momentum is given by

$$\frac{\partial \rho \mathbf{u}}{\partial t} + \nabla \cdot (\rho \mathbf{u} \mathbf{u}) = -\nabla P + \rho \mathbf{g} + $$
 
$$\nabla \cdot \left(2\mu\_{eff} \mathbf{S}(\mathbf{u})\right) - \nabla \left(\frac{2}{3} \mu\_{eff} (\nabla \cdot \mathbf{u})\right) \tag{2}$$

where *P* is the static pressure field and **g** is the gravitational acceleration. The effective viscosity *μeff* is the sum of molecular and turbulent viscosity. The rate of strain (deformation) tensor **S**(**u**) is defined as **S**(**u**) = <sup>1</sup> 2 ∇**<sup>u</sup>** + (∇ · **<sup>u</sup>**)*<sup>T</sup>* .

The conservation of energy in the fluid is defined in terms of the specific enthalpy *h* as

$$\begin{aligned} \frac{\partial(\rho h)}{\partial t} + \nabla \cdot (\rho \mathbf{u} h) + \frac{\partial(\rho k)}{\partial t} + \nabla \cdot (\rho \mathbf{u} k) - \frac{\partial P}{\partial t} &= \\ \nabla \cdot \left( \mathbf{a}\_{eff} \nabla h \right) + \rho \mathbf{u} \cdot \mathbf{g} \end{aligned} \tag{3}$$

where *<sup>k</sup>* <sup>=</sup> <sup>|</sup>**u**<sup>|</sup> 2 <sup>2</sup> is the specific turbulent kinetic energy. The effective thermal diffusivity *αeff* is defined as the sum of laminar and turbulent thermal diffusivity

$$\alpha\_{eff} = \frac{\rho \nu\_t}{Pr\_t} + \frac{\mu}{Pr} \tag{4}$$

where *Pr* is the Prandtl number, *Prt* is the turbulent Prandtl number and *ν<sup>t</sup>* is the turbulent kinematic viscosity.

To predict the effects of turbulence, the turbulent transport parameters require a turbulence model. In this work, the LES method is used to take turbulence into account. A RANS is clearly not suitable for the analysis of near wall turbulent transport phenomena. Since the use of a DNS significantly exceeds the computational effort of the LES, such a method is applied. Here, the large eddies, which contain most of the turbulent energy, are resolved by the conservation equations and only the small eddies are modeled. A filter function with a characteristic filter width of Δ = (Δ*x*Δ*ω*Δ*r*)(1/3) is applied to the conservation equations. This filter function splits up any field variable *φ* in a resolved *φ*ˆ and non-resolved (subgrid) part *φ* [23]. Following the Boussinesq approximation, the viscosity is replaced by an effective viscosity, which is the sum of molecular viscosity and the viscosity of the subgrid scales (eddy-viscosity), *ν*eff = *ν* + *ν*SGS. The subgrid scale viscosity can then be modeled as *ν*SGS = *C*kΔ <sup>√</sup>*k*SGS, where *<sup>C</sup>*<sup>k</sup> <sup>=</sup> 0.07 is a model constant and *<sup>k</sup>*SGS is the kinetic energy of the subgrid scale. The 0-equation WALE [24] (wall-adapting local eddy-viscosity) model calculates the kinetic energy of the subgrid scale using the following equation:

$$k\_{\rm SGS} = \left(\frac{\mathcal{C}\_{\rm w}^2 \Delta}{\mathcal{C}\_{\rm k}}\right)^2 \frac{\left(\mathcal{S}\_{\rm ij}^{\rm d} \mathcal{S}\_{\rm ij}^{\rm d}\right)^3}{\left(\left(\mathcal{S}\_{\rm ij} \overline{\mathcal{S}}\_{\rm ij}\right)^{5/2} + \left(\mathcal{S}\_{\rm ij}^{\rm d} \mathcal{S}\_{\rm ij}^{\rm d}\right)^{5/4}\right)^2} \,. \tag{5}$$

where *S*ij is the strain rate tensor of the resolved scale, *C*<sup>w</sup> = 0.325 and *C*<sup>k</sup> = 0.094 are model constants. This model takes into account the rotation of the flow field, so an additional damping function for *νSGS* in the near wall region is not necessary. Most of the turbulent energy is in the large eddies and these are difficult to model with a turbulence model because of their individual structure. Thus, the modelling part is very small compared to RANS.

OpenFOAM® (Open Source Field Operation and Manipulation) in the version of v1812 with the *buoyantPimpleFoam* solver was chosen for this study.
