海岸动力学英文PPT课件Coastal Hydrodynamics_2.3

p g z g H co k h sz h ck o x s t 2co ks hh
The pressure equation contains two terms: the hydrostatic pressure (静水压强)
& the dynamic pressure (动水压强)

z
zh
0
1
g t
z0
K.F.S.B.C. L.B.C.

t
z
z0
x ,z ,t x c,zt
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Chapter 2
2. Solution
D LGB F BDB S CECB
C
Velocity Potential
a c gc oko h s k s h z h h sikn x (t)
Velocity components decrease exponentially
with depth.
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Chapter 2
Water particle velocities in a progressive wave
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Chapter 2
Acceleration components
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accelerations.
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Chapter 2
Water particle trajectory
The displacement of the water particle can be found by integrating the velocity with respect to time.
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Chapter 2
Velocity components The velocity components can be found by substituting the solution of velocity potential into the definition of potential function.
ekh/2 ekh/2
1
small kh
1
kh kh
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Chapter 2
Regions for valid approximation:
If the largest error incurred by using the asymptotes rather than the actual value of the function is restricted within 5%, then limits for three regions are denoted as:
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Chapter 2
In shallow water, the orbital semi-axes reduce
to
A HT g
4 h
B H 1 z0 2 h
The horizontal excursion of a water particle is
a constant distance from the surface to the
The acceleration components are as follows.
d u uH 2co k(z s h )sikn x (t)
dt t 2 sik n)h h(
d w w H 2sik n (z h h )co k s x t)(
in different de来自ths21/31Chapter 2
The ball makes no net advance in the direction of wave motion.
The reason lies in that the ball moves roughly in a circular path with a diameter approximately equal to the wave height. After each wave passes, the ball returns to nearly its original position.
the shallow water kh 10
intermediate depth 10kh
the deep water
kh
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Chapter 2
Dispersion relationship for deep water waves: A deep water wave is a wave whose wavelength is very small compared with the water depth.
acoks x(t)
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Chapter 2
Dispersion relationship
Substituting the velocity potential and the
surface elevation into the K.F.S.B.C yields
2 gktankhh
LgT2 tanh2h 2 L
The horizontal velocity component has the same phase as the elevation of the free surface.
The horizontal and vertical components are 90º out of phase.
cgTtanh2h 2 L
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Chapter 2
Dispersion relationship in shallow & deep water
Asymptotic forms of hyperbolic functions:
function
coshkh
sinhkh tanhkh
large kh
The dispersion relationship is:
2 gk2h
or
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c gh
Chapter 2
3. Characteristics Water particle velocity components Water particle trajectory Pressure field Energy and energy propagation
u H co k (z s h h )co k s x t)( x T sik n )h h(
w H sik n (z h h )sikn x ( t) z T sik n)h h(
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Chapter 2
Velocity components are harmonic functions of x and t.
of small amplitude waves 4. Standing waves
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Chapter 2
Summary of the two-dimensional periodic water wave boundary value problem
G.D.E.
2 0
hz, x
The dispersion relationship is:

2 0

gk
or
L0
g T2
2
1.56T2
or
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c0

gT
2
Chapter 2
Dispersion relationship for shallow water waves: A shallow water wave is a wave whose wavelength is very large compared with the water depth.
bottom. The vertical excursion increases
linearly with elevation, being zero at the
bottom and being H on the surface.
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Chapter 2
Water particle orbits
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Chapter 2
1. Linearization Suppose that the wave is a small amplitude
wave, namely H<<L or H<<h. Use the Taylor series expansion to relate the
boundary conditions at the unknown elevation to the still water level. Take the pressure on the free surface to be zero.
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Chapter 2
In deep water, the wave orbits become true circles with radius of
AB Hekz0 2
The orbits decay in diameter exponentially with depth. The rate of decrease is dependent on the value of wave number.
dt t 2 sik n)h h(
The maximum vertical accelerations occur as the
horizontal velocities are extremes and the same is
true for the vertical velocities and the horizontal