Practical Ship Hydrodynamics

48 Practical Ship Hydrodynamics
A complete vortex-lattice method (VLM) can be established on the basis of
the lifting-line method just described. The lifting-line model was used to find a
circulation 0 that corresponds to a given resultant flow direction at the lifting
line and is able to provide the predetermined (design) thrust. With a vortex
lattice instead of a lifting line, a model for the material blade is inserted. One
can now really investigate whether a given geometry corresponds to a desired
thrust, a task that is beyond the scope of a lifting-line theory.
Figure 2.9 shows a vortex-lattice system. The flow is generated by spanwise
and (dependent) streamwise line vortices. Control points are positioned
inside the loops of the vortex system. For steady flow, the vortex elements
in the wake have the same strength in each spanwise segment. The vertical
vortex lines then cancel each other and a semi-infinite horseshoe vortex results.
The most downstream control point is located at the trailing edge behind the
last streamwise vortex which is a very robust measure to enforce the Kutta
condition.
Collocation point
Vortex line segment
Figure 2.9 Allocation of vortex-lattice elements on propeller blade
The kinematic boundary condition (zero normal velocity in a blade-fixed
coordinate system) together with some basic relations between blade vortices
and trailing vortices is sufficient to calculate blade surface pressures and thus
propeller thrust and torque. Although the kinematic condition is fulfilled on a
zero thickness blade, the influence of the blade thickness is not excluded. The
thin wing theory provides a simple formula to derive a source system from the
slope of the section contours. This source system already enters the kinematic
conditions and serves to correct the angle of attack of the blade sections for
the displacement effect of the neighbouring blades.
In most applications a ‘frozen’ vortex wake is used, i.e. the trailing vortex
geometry is fixed from the start. A more or less empirical relation serves to
prescribe the pitch of the helical lines. Since surface friction effects are not
part of the solution, the forces and moments from the vortex lattice must be