# Sheet Cavitation

In order to predict sheet cavitation on the propeller blades, an approach introduced by Fine (1992) is used in *pan*MARE.
This approach is also based on the potential solution. In this approach the fluid domain is enclosed by the body and the cavitation
sheet, which together compose the fluid's boundaries. Two boundary conditions are introduced, which influence both the potential
solution as well as the thickness of the cavitation sheet.

##### Dynamic Boundary Condition

The dynamic boundary condition states, that the pressure inside the cavity as well as at its boundaries must be equal to vapour
pressure \(p_v\). Bernoulli's equation, which describes the pressure, is rearranged and intergrated along the flow line \(s\)
(as in: *s*treamline). This way, the cavity's potential \(\Phi_{\text{cav}}\) is determined:
$$
\Phi_\text{cav} = \Phi_\text{wet} + \int_{s_\text{cav}} \left[\frac{2}{\rho} \left(p_{ref}-p_v - \frac{\partial \Phi}{\partial t} \right)+ \vec{v}^2 + 2\vec{g}\vec{z} - \vec{v}_{\text{mot} \parallel s}^2 \right] \text{d}s
$$
Hereby, \(\Phi_{\text{wet}}\) denotes the potential on the wetted surface in front of the cavity and \(\vec{v}_{\text{mot} \parallel s}\)
is the motion velocity in the direction of integration (i.e. direction of the flow line). The calculated potential determined thus
is used as a known doublet strength at the cavity's location and the source strength is determined based on this.

##### Kinematic Boundary Condition

The kinematic boundary condition demands that the flow through the cavity's surface vanishes, i.e. that the substantial change of the cavity thickness corresponds to the flow velocity in direction of the surface normals: $$ \vec{v}_{\parallel S} \nabla \eta = \vec{v}_{\perp S} - \frac{\partial \eta}{\partial t} $$ Hereby, \(\vec{v}_{\parallel S}\) is the part of the flow velocity which is parallel to the body's surface, \(\vec{v}_{\perp S}\) denotes the perpendicular part and \(\eta\) is the cavity thickness. In order to determine the gradient of the cavity thickness a polynomial approach is applied using the neighbouring panels. This way, the gradient \(\nabla \eta\) is expressed using the cavity-adjacent panels.

Furthermore, finite differences are used to determine the time rate of change of the cavity thickness. Thus, the equation above only has the following unknowns: the cavity thicknesses of the examined panel and of its neighbours.

##### Solving Routine and Cavity Expansion

The rearranged kinematic boundary condition is solved in an additional system of equations after the potential solution in the *pan*MARE
solving routine. The dynamic boundary condition of the sheet cavitation is included in this process. Hereby, the non-cavitation panels are used
as known values without cavity thickness \(\left(\eta=0\right)\).

After the determination of the cavity thickness, the cavity's expansion on the body's surface is calculated using several factors:

- If a panel's pressure falls below the vapour pressure \(p_v\), this panel will start to cavitate.

- If a panel is already cavitating and if it is the last cavitating panel along a flow line, the cavitating area is either enlarged (cavity thickness \(\eta > 0\) ) or reduced ( \(\eta < 0\) ).

- If the cavity thickness is negative at the leading end of the cavitating area and the pressure is greater than \(p_v\), the leading end of the cavity area is moved towards the aft by one panel.

An additional iteration is required to calculate the change of the cavity's expansion. Therefore, the potential solution, including both cavitation boundary conditions and a subsequent adaptation of the cavity's current expansion, is iterated until the expansion finally converges.