# Numerical Insight into the Kelvin-Helmholtz Instability Appearance in Cavitating Flow
## 3 Numerical Procedure
### Reynolds-Averaged Navier-Stokes (RANS) Equations  
The Navier-Stokes equations are a set of partial differential equations, describing the conservation of momentum and mass for a viscous fluid flow motion. A direct numerical simulation, in which the Navier-Stokes equations are solved without a turbulence model, requires a huge amount of computing power because the whole range of spatial and temporal scales of the turbulence must be resolved. As this is not practical, averaging of variables is used. The idea behind averaging is the Reynolds decomposition, an instantaneous quantity decomposition into its time-averaged \(\Phi\) and fluctuating quantities \(\varphi\)(t) with zero mean value. Reynolds-averaged Navier-Stokes equations will be used in the form, that is generally accepted in the commercially available computational fluid dynamics (CFD) solvers. Here, the overline indicates a time-averaged variable, while the tilde indicates a density-weighted averaged or Favre-averaged variable [10]. The Favre-averaging method was selected because in the present case turbulent fluctuations lead to sizeable density fluctuations. The continuity equation is as follows, where \(\rho\) is the fluid density, \(t\) time, and \(\mathbf{U}\) velocity vector, defined as \(\mathbf{U}=[U,V,W]^{\mathrm{T}}\):  
\[\frac{\partial\overline{\rho}}{\partial t}+\
abla\cdot\left(\overline{\rho} \overline{\mathbf{U}}\right)=0 \tag{1}\]  
The momentum equations in the \(x\) and \(y\) directions are as follows, where \(P\) is pressure, \(\mu\) dynamic viscosity, \(u^{\prime}\), \(v^{\prime}\) and \(w^{\prime}\) velocity fluctuations of velocity vector \(\mathbf{U}\), while \(S_{M\mathrm{x}}\) and \(S_{My}\) represent momentum source terms in \(x\) and \(y\) directions, respectively  
\[\begin{split}\frac{\partial(\overline{\rho}\overline{U})}{ \partial t}+\
abla\cdot\left(\overline{\rho}\overline{U}\overline{\mathbf{U}} \right)&=-\frac{\partial\overline{\rho}}{\partial x}+\
abla \cdot\left(\mu\
abla\overline{U}\right)+\left[-\frac{\partial\left(\overline {\rho}\overline{u^{\prime}}\overline{v^{\prime}}\right)}{\partial x}-\frac{ \partial\left(\overline{\rho}\overline{u^{\prime}}\overline{v^{\prime}} \right)}{\partial y}-\frac{\partial\left(\overline{\rho}\overline{u^{ \prime}}\overline{u^{\prime}}\overline{v^{\prime}}\right)}{\partial z}\right] +S_{M\mathrm{x}}\\ \frac{\partial(\overline{\rho}\overline{V})}{\partial t}+\
abla \cdot\left(\overline{\rho}\overline{V}\overline{U}\overline{\mathbf{U}}\right)& =-\frac{\partial\overline{\rho}}{\partial y}+\
abla\cdot\left(\mu \
abla\overline{V}\right)+\left[-\frac{\partial\left(\overline{\rho} \overline{u^{\prime}}\overline{v^{\prime}}\right)}{\partial x}-\frac{ \partial\left(\overline{\rho}\overline{v^{\prime}}\overline{v^{\prime}} \right)}{\partial y}-\frac{\partial\left(\overline{\rho}\overline{v^{\prime}} \overline{u^{\prime}}\overline{v^{\prime}}\right)}{\partial z}\right]+S_{M \mathrm{x}}\end{split} \tag{2}\]  
For the accurate representation of cavitating flow details, some authors perform compressible flow simulations [11; 12; 13]. In the compressible case, the energy conservation equation is added to the above set of continuity and momentum equations. Also, equations of state of vapor and liquid are introduced. On the other hand, some authors have shown, that sufficient flow representation accuracy was obtained in the incompressible flow case, much decreasing the computational load requirements (for example [2; 14]).