Practical Ship Hydrodynamics

10 Practical Ship Hydrodynamics
y + dy
u
v + v y dy
v
y
x x + dx
u + u x dx
Figure 1.1 Control volume to derive continuity equation in two dimensions
The continuity equation in three dimensions can be derived correspondingly to:
ux C vy C wz D 0
w is the velocity component in z direction.
The Navier–Stokes equations together with the continuity equation suffice
to describe all real flow physics for ship flows. The Navier–Stokes equations
describe conservation of momentum in the flow:
⊲ut C uux C vuy C wuz⊳ D f1 px C ⊲uxx C uyy C uzz⊳
⊲vt C uvx C vvy C wvz⊳ D f2 py C ⊲vxx C vyy C vzz⊳
⊲wt C uwx C vwy C wwz⊳ D f3 pz C ⊲wxx C wyy C wzz⊳
fi is an acceleration due to a volumetric force, p the pressure, the viscosity
and t the time. Often the volumetric forces are neglected, but gravity can
be included by setting f3 D g ⊲D9.81 m/s 2 ⊳ or the propeller action can
be modelled by a distribution of volumetric forces f1. The l.h.s. of the
Navier–Stokes equations without the time derivative describes convection,
the time derivative describes the rate of change (‘source term’), the last term
on the r.h.s. describes diffusion.
The Navier–Stokes equations in the above form contain on the l.h.s. products
of the velocities and their derivatives. This is a non-conservative formulation
of the Navier–Stokes equations. A conservative formulation contains
unknown functions (here velocities) only as first derivatives. Using the product
rule for differentiation and the continuity equation, the non-conservative formulation
can be transformed into a conservative formulation, e.g. for the first of
the Navier–Stokes equations above:
⊲u 2 ⊳x C ⊲uv⊳y C ⊲uw⊳z D 2uux C uyv C uvy C uzw C uwz
D uux C vuy C wuz C u⊲ux C vy C wz⊳
� �� �
D0
D uux C vuy C wuz
Navier–Stokes equations and the continuity equation form a system of coupled,
non-linear partial differential equations. An analytical solution of this system
is impossible for ship flows. Even if the influence of the free surface (waves)
is neglected, today’s computers are not powerful enough to allow a numerical