# Development of Attached Cavitation at Very Low Reynolds Numbers from Partial to Super-Cavitation
## 2 Experimental Study Overview
### Numerical Highlights and Comparison with Experimental Data  
The topology of the single-phase flow and its nature as a function of the Reynolds number have been studied numerically [28]. The computation was performed with the Computational Fluid Dynamics code StarCCM+ (version 13.02). It uses a finite volume approach. The computation is three-dimensional; the numerical domain is similar to the experiment with an extrusion of \(10h_{throat}\) upstream of the inlet of the test section and an extrusion of \(20h_{throat}\) downstream of the outlet of the test section. The boundary conditions are a uniform inlet velocity and a uniform static pressure outlet. The domain is meshed with a structured grid. A grid convergence test was made at \(Re=1500\) with up to \(7.2\times 10^{6}\) cells. The final mesh contains \(1.1\times 10^{6}\) cells, with \(51\) points in the cross-stream direction and a near-wall cell size of \(5\times 10^{-3}h_{throat}\). The equations that are solved are the incompressible Navier-Stokes equations for a Newtonian fluid with constant properties. The spatial discretization for the convective flux is the second-order upwind scheme. Unsteady simulations with the Euler first-order implicit scheme have been performed. The main results are plotted in Figure 4. The experimental (obtained in single-phase flows) and numerical pressure loss coefficients fairly collapse, which is a clue in favor of the quality of the simulations (Figure 4a).  
Figure 4: Main results of the single-phase numerical simulations. (**a**) Pressure loss coefficient \(K_{p}\) as a function of the Reynolds number \(Re\). Blue squares represent experimental values while black squares represent numerical simulations. The dashed line is a line of equation \(K_{p}\propto Re^{-1}\). (**b**) Dimensionless pressure profile \(P^{*}=P/\frac{1}{2}\rho V_{in}^{2}\) with respect to axial coordinate \(X^{*}=X/h_{throat}\). (**c**) Dimensionless velocity profile \(V^{*}=V/V_{in}\) with respect to \(X^{*}\). For both images (**b**,**c**): \(Re=1200\), \(\sigma_{1}=6.57\) and \(\sigma_{2}=2.58\); Blue line is along the horizontal line \(Z^{*}=0.5\) and \(Y^{*}=0\); Red line is a broken line following the bottom of the vein, \(0.5\) mm above it at \(Y^{*}=0\).  
The main results of this numerical study are the following:  
* The flow remains laminar for \(Re\lesssim 2000\). A primary flow separation is present in the wake of the throat, on the divergent wall (\(X^{*}>0\) and \(Z^{*}<0\)), first along each lateral wall (\(Y^{*}\simeq\pm 0.5\)) for \(Re\geqslant 350\), and on all along the width for \(Re\geqslant 650\). This corresponds to the negative values of the velocity profile in Figure 4c.
* A secondary laminar boundary layer separation emerges at a Reynolds number \(Re\geqslant 1100\) at the top wall of the test section (\(X^{*}>0\) and \(Z^{*}\simeq 1\)).
* Finally, the critical cavitation numbers that lead to a minimal absolute pressure equal to the vapor pressure can be estimated numerically with the simulations. For instance, here, for \(Re=1200\), \(\sigma_{1}=6.57\) and \(\sigma_{2}=2.58\), one can notice that the minimal pressure is close to zero and that this occurs at the throat, on the bottom wall (see Figure 4b). According to the simulation, cavitation of the oil is thus possible.