*2.1. The Compressible Navier–Stokes Equation*

The Navier–Stokes equations describe the motion of fluids and model the conservation of mass, momentum, and energy. The non-dimensional compressible equations in conservative form can be written as:

$$\frac{\partial \rho}{\partial t} = -\nabla \cdot \mathbf{m}\_{\prime} \tag{1}$$

$$\frac{\partial m\_i}{\partial t} + \sum\_{j=1}^3 \frac{\partial}{\partial \mathbf{x}\_j} \left( m\_i v\_j + p \delta\_{i\bar{j}} \right) - \frac{1}{\text{Re}} \sum\_{j=1}^3 \frac{\partial}{\partial \mathbf{x}\_j} \pi\_{i\bar{j}} = 0 \qquad \mathbf{i} = 1, 2, 3 \tag{2}$$

$$\frac{\partial \rho \varepsilon}{\partial t} + \nabla \cdot \left[ \left( \varepsilon + \frac{p}{\rho} \right) \mathbf{m} \right] - \frac{1}{Re} \sum\_{j=1}^{2} \frac{\partial}{\partial \mathbf{x}\_j} \left( \sum\_{i=1}^{2} \tau\_{ij} \upsilon\_i - \frac{1}{\gamma - 1} \frac{\mu}{Pr} T \right) = 0 \tag{3}$$

where the conserved quantities are the density *ρ*, the momentum **m** = *ρ***v**, and the total energy density *e*, given by the sum of kinetic and internal energy density:

$$\mathbf{e} = \frac{1}{2} |\mathbf{v}|^2 + \frac{p}{\rho(\gamma - 1)}. \tag{4}$$

where **v** = (*v*1, *v*2, *v*3)*<sup>T</sup>* is the velocity vector, *δij* is the Kronecker delta, *Re* is the reference Reynolds number, and *Pr* the reference Prandtl number. *γ* stands for the isentropic expansion factor, given by the heat capacity ratio of the fluid, and *T* denotes the temperature. Viscous effects are described by the shear stress tensor:

$$\pi\_{ij} = \mu \left( \frac{\partial v\_i}{\partial x\_j} + \frac{\partial v\_j}{\partial x\_i} \right) \tag{5}$$

and the dynamic viscosity *μ*.

To close the system, we use the ideal gas law as the equation of state, which yields the following relation:

$$p = \rho RT.\tag{6}$$

where *R* represents the gas constant.
