Zienkiewicz O.C., Taylor R.L. Vol. 3. The finite - tiera.ru

8.3 Dif®culties in modelling surface waves
The main defects of the simple surface-wave model described above are the following:
1. inaccuracy when the wave height becomes large. The equations are no longer valid
when becomes large, and for very large , the waves will break, which introduces
energy loss.
2. lack of modelling of bed friction. This will be discussed below.
3. lack of modelling of separation at re-entrant corners. At re-entrant corners there is
a singularity in the velocity of the form 1= r
p . The velocities become large, and
physically the viscous e€ects, neglected above, become important. They cause
retardation, ¯ow separation and eddies. This e€ect can only be modelled in an
approximate way.
Now the response can be determined for a given excitation frequency, as discussed in
Chapter 17 of Volume 1.
8.4 Bed friction and other effects
The Che zy bed friction term is non-linear and if it is included in its original form it
makes the equations very di cult to solve. The usual procedure is to assume that
its main e€ect is to damp the system, by absorbing energy, and to introduce a
linear term, which in one period absorbs the same amount of energy as the Che zy
term. The linearized bed friction version of Eq. (8.2) is
r T …Hr †‡ !2
g
ÿ i!M ˆ 0 or
@
@x i
H @
@x i
‡ !2
g
ÿ i!M ˆ 0 …8:11†
where M is a linearized bed friction coe cient, which can be written as
M ˆ 8u max=3 C 2 H, C is the Che zy constant and u max is the maximum velocity at
the bed at that point. In general the results for will now be complex, and iteration
has to be used, since M depends upon the unknown u max. From the ®nite element
point of view, there is no longer any need to separate the `sti€ness' and `mass'
matrices. Instead, Eq. (8.11) is weighted using the element shape function and the
entire complex element matrix is formed. The matrix right-hand side arises from
whatever exciting forces are present. The re-entrant corner e€ect and wave-absorbing
walls and permeable breakwaters can also be modelled in a similar way, as both of
these introduce a damping e€ect, due to viscous dissipation. The method is explained
in reference 15, where an example showing ¯ow through a perforated wall in an
o€shore structure is solved.
8.5 The short-wave problem
The short-wave problem 245
Short-wave di€raction problems are those in which the wavelength is much smaller
than any of the dimensions of the problem. Such problems arise in surface waves
on water, acoustics and pressure waves, electromagnetic waves and elastic waves.
The methods described in this chapter will solve the problems, but the requirement