Page 449 - Practical Design Ships and Floating Structures

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2.2 Boundary and Initial Conditwns
No-slip condition is imposed over the body surface and the free-stream condition away from the ship.
The no-slip condition implies that the fluid particles on the body surface are moving body velocity, say
V. and V,.

u=y; v=vy; w=o; -= ap 0 (7)
arl
On the body:
At infinity:

Free-stream characteristics are applied everywhere in the flow to define the initial condition.
Equations (13) below show the mathematical representation of this condition.

u=M,; v=O; w=O; +=l (9)

2.3 Equation of Motion

Applying Newton's second law to the ship with mass m under hydrodynamics forces - lift L and drag
D, one may obtain the dimensionless form of the equation of motion (10) that has to be solved for
initial conditions (1 1). The equation of motion must be solved simultaneously with the Navier-Stokes
equations to compute the ship velocity and ship position (y) that is used to regenerate the moving grid.

2m
y(0) = 0; y(0) = 0 (1 1); where : C, = - (12)
PD2
3 NUMEFUCAL SOLUTION
3.1 Beam- Warming Impkit Scheme

The Beam-Warming (1978) scheme is applied by using the Euler implicit method for time
integration - equation (13), that is a convenient choice due to both simplicity and because it does not
show restrictions to the step of integration.
n+l
e"+' Q" + Atz) + o(At)
=
at
The time derivative is then eliminated from (1 7) by using the governing equation (1):
+ FT')
Q"" = Q"