Numerical Background
This page provides an overview of the numerical implementations used to model hydrofoils and wake in order to calculate the flow around a propeller. You can find additional information on the numerical background of panMARE on the pages of individual panMARE techniques and in various panMARE-related publications of the Institute for Fluid Dynamics and Ship Theory.
Numerical Implementation
The flow field is calculated by a low-order panel method in panMARE.
The body surface

It is assumed that the doublet strength
A collocation point is assigned to each panel, which is located at the panel's centre. For the body panels the collocation point is positioned on the inside of the body. The collocation point is the point at which the boundary conditions are evaluated. Thus an equation system is set in order to calculate the influence of the doublet
The integrals in the equations above can be calculated analytically. According to the Morino-Kutta condition, the
doublet strength on the wake panels equals the doublet strength difference of the body panels at the trailing edge.
Therefore, it is possible to unite the coefficients
Steady Case
In case of steady flow, the doublet strength of the wake does not change in time. Therefore, all wake panels which are dispatched
from a body panel at the trailing edge have the same doublet strength. Furthermore, the wake maintains its orientation in regard to
the body (except for its shape).
Unsteady Case
In unsteady cases, the body moves according to the specified body motion. While the body panels are moving, the wake panels remain at fixed
positions. Thus, the body is moving away from the wake panels in the unsteady calculation. For this reason, the wake is detatched from the body
during each time step. The wake is detatched just before the body moves. After the wake is detatched, new wake panels are inserted in the
resulting gap between the hydrofoil's trailing edge and the old panels. The "length" of these new panels is set corresponding to the time
increment. In order to consider the time rate of change of the wake's doublet strength, only the first wake panels at the trailing edge are
calculated and all remaining wake panels maintain their values from previous time steps. This results in the following readily changing coefficients:
Induced Potential and Pressure Calculation
In both steady and unsteady cases the determined doublet strengths are used to calculated the induced potential
Wake Sheet Orientation
In addition to the solution of the flow problem and the calculation of the flow velocities and acting pressure, the orientation of the hydrofoil's wake sheet is realigned to follow the lines of the flow. The wake sheet's initial orientation is set corresponding to the hydrofoil's motion at the start of the calculation. Hereby, a helical shape is initialised in cases of rotating bodies (e.g. propeller).
Theoretically, the wake sheet's length should be infinite. However, since this is not feasible from a numerical point of view, the wake sheet is modelled with a sufficiently large finite length. The modelled wake sheet's length is chosen in a way that the hydrofoil is not influenced significantly by the vortex at the sheet's end. This criterion is also applied whenever the wake sheet is shortened in unsteady cases, after new wake panels have been inserted.
During the calculation, the wake panels are oriented corresponding to the induced velocities and the gradient of the local perturbation of the underlying currents' potential. This way, the wake panels are oriented along the flow lines.
It should be noted that a large number of wake panels can lead to very long computing times. However, the computational effort can be reduced significantly, if a predefined shape of the wake sheet is used.