上海交大船舶原理耐波性ppt

The free-surface conditions 2.10 and 2.11 are non-linear. Assumption: the structure has no forward speed and the current is zero Linear theory: the velocity potential is proportional to the wave amplitude. Transfer the free-surface conditions (by a Taylor expansion)
Under the wave crest the fluid
velocity is in the wave propagation direction. Beneath a wave trough the fluid velocity is opposite to the wave propagation direction.
) where
is a constant
phase angle
+ : waves propagating along the negative x-axis
- : waves propagating along the positive x-axis

: the velocity of the wave form (the phase velocity c) cf. the fluid velocity, see Eq. 2.1 The group velocity (the energy propagation velocity) is a third type of velocity used in describing wave features.
fluid particle in space. Free surface definition:
: wave elevation
A fluid particle on the free-surface is assumed to stay on the free-surface. DF/Dt = 0
Obtain expressions for waves propagating in an arbitrary direction
relative to the x-axis by writing the x-, y- and t-dependence of the velocity potential as cos . The z-dependence is .
The linear theory assumes the velocity potential and fluid velocity to be
constant from the mean free-surface to the free-surface level. The horizontal velocity distribution shown in Fig.2.2 for the low under a wave rest is consistent with linear theory. The difference between the horizontal velocity at the wave trough and the analytical fictitious velocity at z=0 is small compared with the velocity itself.
Fig.2.3 shows how the pressure varies with depth both under a wave
crest and a wave trough.
The ‘hydrostatic’ pressure ‘
’ should cancel the dynamic pressure
C: an arbitrary function of time -> constant. Assumed that the only external force field is gravity, C is related to the
atmospheric pressure or the ambient pressure (equation 2.11).
is 0.5
for deep water waves (half the phase velocity)
Example: A wave maker in a model basin that generates harmonically
oscillating waves. It takes for the wave front to reach the ‘beach’ at the end of the model basin, we should use the group velocity to analysis the time.
Kinematic boundary conditions For fixed body,
expresses impermeability (no fluid enters or leaves the body surface).
For moving body,
Velocity U may be different for different points on the body surface.
the mathematical analysis of irrotational fluid motion. The complete mathematical problem of finding a velocity potential consists of the solution of the Laplace equation with relevant boundary conditions on the fluid.
The expressions for waves propagating along the positive x-axis, and
afterwards renaming the (x’, y’)-coordinates (x, y), we have obtained expressions for waves propagating in an arbitrary direction .

the Laplace equation

the sea bottom condition
Assumption : the velocity potential can be represented as a product of
functions which depend on just one independent variable. Use ‘separation of variables’ to solve the Laplace equation Equation 2.17 satisfies the Laplace equation:
- Substantial Derivative. V is the fluid velocity at the point (x,y,z) at time t.
Expresses the rate of change with time of the function F when following a
free-surface position
the mean free-surface at z=0
Keeping linear terms in the wave :
When the velocity potential
is oscillating harmonically in time with
Out of Phase
Fig.2.1. Wave elevation, pressure, velocity and acceleration in long-crested sinusoidal waves propagating along the positive x-axis (see Table 2.1).
i.e.
(2.10)
Kinematic Boundary Conditions
Dynamic free-surface condition
The water pressure = the constant atmospheric pressure Choose C as
on the free-surface. , and the equation holds with no fluid motion:
Contents
Basic Assumptions
Regular Wave Theory
Statistical Description of Waves
Wind
Current
Basic Assumptions
A velocity potential has no physical meaning itself, but is convenient in
Chapter 2
Velocity Potential
Fluid Mechanics
Bernoulli’s equation
basic information on waves, wind and current
evaluate sea loads and motions acting on ships and offshore structures.
circular frequency
Regular Wave Theory
A horizontal sea bottom
Linear wave theory
(Airy theory)
A free-surface of infinite horizontal extent

the free-surface condition
Dispersion Relation
Table 2.1 presents the results for both finite and infinite water depths.
Table 2.1 : The maximum values of the different physical variables do not happen at the same time. crest
the quantities A, B and
are arbitrary constants.
The dispersion relation between the wave number k and the circular
frequency
:
Equation 2.17 does not represent travelling (propagating) waves. x- and t-dependence is like cos (
at the free-suwenku.baidu.comface.
high-order error: the error is approximately proportional to
, where
the order The dynamic pressure half a wavelength down in the fluid is only 4% of its value at z=0. Cf. exp((2π/λ)(- λ/2))= exp(-π)=0.043
Quasi-static considerations:
Get a negative dynamic pressure under a wave trough. Get a positive dynamic pressure under the wave crest.
Trough
In-phase