Trans RINA, Vol 153, Part B2, Intl J Small Craft Tech, 2011 Jul-Dec


model based on a Volume-of-Fluid (VOF) approach is applied, which assumes that the two phases share a common velocity and pressure


field. Effectively the


method treats both phases in the computational domain as one fluid with variable properties. The additional transport equation is solved for the VOF-fraction c in every cell, with values between 0 and 1 indicating a cell which is filled with both fluids. The free surface interface is assumed to be represented by a value of c=0.5.


dt 


d cd   0 unc dS


(5)


The density  and molecular viscosity  are calculated from volume fraction c and the fluid properties as shown below. If a cell is filled with both fluids, they are assumed to share the same velocity and pressure.


cc cc


 112   112 


(6) (7)


Convection terms are discretised using second order schemes whilst the temporal scheme is of first order (Implicit Euler). An adaptive time-stepping scheme is used to keep the average Courant number in the flow domain between 1.5 and 2.0. This usually leads to time step sizes between 0.01s and 0.1s in the unsteady RANSE simulation.


3.2 RIGID BODY MOTION


To take into account the effects of hydro- and aerodynamic forces acting on the yacht, RVPP makes use of a 6-degree-of-freedom (DOF) body motion module which is embedded in the global RANSE iteration.


The translation and rotation resulting from the forces acting on the body are determined by integrating the equation of linear and angular momentum. The equation of linear momentum may be written as:


mdv  F dt


(8)


Here m stands for the mass of the investigated body, v is the linear velocity of the centre of mass and F is the resultant force. The contributions of external and internal forces to the resulting force F is listed below:


FF g m FExt Hydro (9)


where g denotes the gravity vector which is positive in downward direction and Fext may be any kind of external force applied. Typical examples for this application are the sail force and an additional rudder force vector.


The flow force FHydro is the resultant force of the flow field acting on the body. It is determined from the RANS


equations by integrating viscous wall shear stresses and pressure field over the body’s boundary faces.


Fn τ  i


ii i  HydropS i (10)


Here pi is the pressure acting on the face of a control volume whilst ni is the normal vector of the individual control volume face. The viscous stresses are denoted i and the surface of the control volume face is iS .


In general, the equation of the angular momentum may be written as follows:


TIT ω ω TIT ω M d


  


11 dt


(11)


with I representing the tensor of the moment of inertia of the investigated body using the body fixed coordinate system,  the angular velocity vector of the rigid body and M the resultant moment acting on the body. T is the transformation matrix from the body-fixed coordinate system, which has its origin at the centre of gravity of the investigated body, into the global coordinate system. The resultant moment M can be summarized as follows:


MMEx t HydroM  Here MExt represent Mx x Ex  F


external forces which may be expressed as: 


t CE G Ext (12) the trimming moment due to (13)


with xCE representing the vector to the centre of effort of the external force and xG being the location of the centre of gravity. The dynamic contribution of the flow force to the fluid flow moment MHydro may be expressed as stated below. Here xi stands for the control volume face centre vectors.


Mx i i


i G x n τ i  Hydro pS     ii (14)


The translation and rotation of the body is determined by integrating the equation of linear and angular momentum.


The simulation uses two different coordinate systems. First, an arbitrary non-moving Cartesian coordinate system with the vertical axis normal to the free surface in design conditions in which the RANSE equations are resolved.


Second, a moving boat-fixed Cartesian reference frame attached to the centre of gravity with positive x-axis points forward. The motions of the rigid body are resolved in this coordinate system.


©2011: The Royal Institution of Naval Architects


B-85


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