OrcaFlex Manual - Orcina

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Theory, Environment Theory
stream function theory of Rienecker and Fenton (1981). This method is also known as Fourier approximation wave
theory.
The problem is to find a stream function which:
1. satisfies Laplace's equation δ 2 ψ/δx 2 + δ 2 ψ/δz 2 = 0, which means that the flow is irrotational,
2. is zero at the seabed, that is ψ(x,0) = 0,
3. is constant at the free surface z = η(x), say ψ(x,η) = -Q and
4. satisfies Bernoulli's equation ½ [ (δψ/δx) 2 + (δψ/δz) 2 ] + η = R, where R is a constant.
In these equations all variables have been non-dimensionalised with respect to water depth d and gravity g.
By standard methods, equations (1) and (2) are satisfied by a stream function of the form
ψ(x,z) = B0 z + ∑ Bj [sinh (jkz) / cosh (jk)] cos (jkx)
where k is the wave number which is as yet undetermined, and the summation is from j = 1 to N. The constant N is
said to be the order of the stream function. The problem now is to find coefficients Bj and k which satisfy equations
(3) and (4).
Implementing stream function theory requires numerical solution of complex non-linear equations. The number of
these equations increases as N increases and there is a short pause in the program while these equations are solved.
For most waves the default value will suffice. However, for nearly breaking waves the solution method sometimes
has problems converging. If this is the case then it might be worth experimenting with different values.
Accuracy of method
Because the method is a numerical best fit method it does not suffer from the truncation problems of the Stokes' 5th
and cnoidal theories. For these methods, power series expansions are obtained and then truncated at an arbitrary
point. If the terms which are being ignored are not small then these methods will give inaccurate answers. In theory,
Dean's method should cope well in similar circumstances as it is finding a best fit to the governing equations. This
means that stream function wave theory is very robust. In very shallow water Fenton believes that his high order
cnoidal wave theory is best, although we would recommend stream function theory here. It is possible that, by their
very nature, Stokes' 5th and the cnoidal theories may give inaccurate results if applied to the wrong waves. In all
circumstances the stream function method, if it converges, will give sensible results. Hence it can be used as a coarse
check on the applicability of other theories. That is if your preferred wave theory gives significantly different results
from Dean's, applied to the same wave, then it is probably wrong!
Stokes' 5th
The engineering industry's standard reference on 5th order Stokes' wave theory is Skjelbreia and Hendrickson
(1961). This paper presents a 5th order Stokes' theory with expansion term ak where a is the amplitude of the
fundamental harmonic and k = 2π / L is the wave number. The length a has no physical meaning and by choosing ak
as expansion parameter, convergence for very steep waves cannot be achieved. Fenton (1985) gives a 5th order
Stokes' theory based around an expansion term kH/2 and demonstrates that it is more accurate than Skjelbreia and
Hendrickson's theory. Thus it is Fenton's theory which is implemented in OrcaFlex. It is worth noting that the linear
theory of Airy is a 1 st order Stokes' theory.
Assuming that the user supplies wave train information comprising water depth, wave height and wave period then
the wave number k must be computed before the theory can be applied. In order to do this a non-linear implicit
equation in terms of k is solved using Newton's method. This equation is known as the dispersion relationship. Once
k is known, a number of coefficients are calculated and these are used for power series expansions in order to find
the surface profile and wave kinematics.
Accuracy of method
Inherent in the method is a truncation of all terms of order greater than 5. Thus if the terms which are discarded are
significant then this theory will give poor results. See Ranges of applicability for the waves for which Stokes' 5th
theory is valid, but essentially this is a deep water, steep wave theory.
Cnoidal theory
This is a steady periodic water wave theory designed to be used for long waves in shallow water. The Stokes' 5th
order theory is invalid in such water as the expansion term is large and the abandoned terms due to truncation are
significant. The high-order cnoidal theory of Fenton (1979) has been regarded as the standard reference for many