Algorithm for the determination of the sheet cavity shapeshape

In this section the numerical procedure for the determination of the sheet cavity shape based on the partially non-linear model is introduced. There are two boundary conditions that are to be fulfilled on the cavity surface: the kinematic and the dynamic boundary condition. Additionally, the cavitation inception must be detected and the cavity closure must be modelled. The approach used here at the cavity closure is based on a closed model. For this approach, the geometry grid is not being re-meshed and a point must be found downstream (or upstream, in case of face cavitation) of the cavitation inception with the smallest possible cavity thickness value.

Chapter 4. Sheet cavitation model The overall solution algorithm for the steady cavitating flow consists of the following main steps:

(1) In the first steady step the linear system of equations (3.16) is solved once with-out the sheet cavitation model in order to compute the pressure distribution on the bodies submerged in the flow domain.

(2) Now the criterionC_p,v ≤ −σ_v is applied so that a first guess can be made concern-ing the cavity length at each radial stripe of the body. For this, a detachment and a reattachment point are defined on each radial stripe according to the following iterative procedure:

Detachment point: There are two approaches that are implemented for the detection of the detachment point. In the first approach, the midpoint of the leading edge panel is defined as the detachment point. The second approach is based on the Villat-Brilloin condition. Starting at the midpoint of the leading edge panel the point on the pressure side (back cavitation) or suction side (face cavitation) has to be found where the pressure is less than or equal to the vapour pressure (i.e. C_p,v ≤ −σ_v). For the points upstream (back cavitation) or downstream (face cavitation) of this point, the pressure should be greater than the vapour pressure, i.e. C_p,v >−σ_v.

Reattachment point: Starting from the estimated detachment point, one searches for the first point on the suction (back cavitation) or pressure side (face cavi-tation) where the pressure exceeds the vapour pressure, i.e. C_p,v >−σ_v. The point upstream (back cavitation) or downstream (face cavitation) of this point where the condition C_p,v ≤ −σ_v holds is then defined as the reattachment point.

(3) After the first guess of the cavity length, the linear system of equations(4.19)with cavitating panels is set up and solved and new velocity and pressure distributions are calculated. Additionally, the cavity thickness is computed on each cavitating panel of the body by solving the linear system of equations(4.30).

(4) Based on the cavity thickness distribution at the detachment and reattachment points, a new guess for the cavity shape is made. For this, the detachment and reattachment points for each radial stripe are adjusted according to a prescribed scheme.

Detachment point: If the detachment point is defined as a fixed point that is located at the leading edge, its location is not modified. If the Villat-Brilloin criterion is used (which demands that cavitation has a non-negative thickness at the detachment point and that the pressure on the non-cavitating part of the

Chapter 4. Sheet cavitation model

body is greater than the vapour pressurep_v), the detachment point is adjusted according to the following case distinctions:

(i) If the cavity thickness at the detachment point is positive, but the pressure on the non-cavitating part upstream (back cavitation) or downstream (face cavitation) of the detachment point is less than or equal to the vapour pressure, then the detachment point is moved upstream (back cavitation) or downstream (face cavitation), respectively.

(ii) If the cavity thickness at the detachment point is negative, then the de-tachment point is moved downstream (back cavitation) or upstream (face cavitation), respectively.

(iii) Else, the detachment point location is not modified.

Reattachment point: The procedure used for the estimation of the reattachment point reads as follows. Firstly, calculate the sum of the cavity thickness at the reattachment points for all radial stripes:

Nk

X

k=1

|η_(l^cav_r,k)|. (4.31)

If the sum (4.31) is smaller than a given tolerance threshold ₁, i.e when it holds:

Nk

X

k=1

|η_(l^cav_r,k)| ≤₁, (4.32) then the reattachment points are not displaced. The tolerance threshold is de-fined inpanMARE as₁ = ₁₀¹

Nk

P

k=1

min⁴_i=1(d_i|P anel_(lr,k)), whered_iis the length of the edgeion the reattachment panel,

If the sum (4.31) is greater than the given tolerance threshold ₁, then the reattachment points are adjusted according to the following procedure:

(i) If the cavity thickness at the reattachment point at stripekis positive and greater than a given bound ₂ = 2₁/N_k, then the reattachment point is moved one panel downstream in case of back cavitation or upstream in case of face cavitation.

(ii) If the cavity thickness at the reattachment point at stripekis negative and smaller than the given bound−₂, then the reattachment point is moved one panel upstream in case of back cavitation or downstream in case of face cavitation.

(iii) If the cavity thickness at the reattachment point at stripe k is within the bound[−₂, ₂], the reattachment point is not changed.

Chapter 4. Sheet cavitation model Subsequently, the linear system of equations(4.19) is solved again to modify the unknown value of the strengthsµonS_B andσ^cav onS_BC. Supplementary, the cav-ity thickness is calculated again from the linear system of equations(4.30)with the new locations of the detachment and reattachment point.

The procedure above is repeated until the criterion for the reattachment and detach-ment point location are fulfilled.

(5) After the determination of the new cavity shape propeller characteristics are calcu-lated.

(6) For the next steady iteration steps, the solution obtained from the previous steady iteration step is used as an initial guess for the cavity length and the steps (3)-(5) are repeated.