90
4 Physical Mechanisms Forming the Wave Spectrum in Deep Water
Tests of the collision integral calculation algorithm and assessment
of the calculation accuracy. Before further calculations the experiments
are made to test the aforementioned integration algorithm, using the quadrature formulas ( 4.9) described above. For this purpose the calculations of the
non-linear transfer integral for the JONSWAP spectrum are made using a different number of integrand points. Thus, the parameter n is assumed to be
equal to 2, 3, 4, 5, 7 and 8, respectively. The results of these calculations
are presented in Fig. 4.3. They show that with increasing n, the calculation
results converge quite rapidly to its accurate value. The numerical values become practically indiscernible for n > 5. For n = 4 the relative differences in
the energy range are not more than 15 per cent (of the corresponding value
at n = 8) sufficient for most practical calculations. The integration error is
estimated as 1-2 per cent with n = 7.
The accuracy for the typical JONSWAP spectrum is estimated by repeated calculations with double the number of grid points until the difference of the calculation results for two successive cases reaches the prescribed
value. By executing a series of successive calculations, an error is obtained not
greater than 1-2 per cent in the area of the spectral maximum (0.9 :::; a :::; 1.5,
where a= o-f!Jmax)· Moreover, the calculation error is not greater than 3-5
per cent in the frequency ranges (0.8 :::; a :::; 0.9) and (1.5 < a :::; 2.5) and not
greater than 5-10 per cent in the ranges (0. 7 :::; a :::; 0.8) and (2.5 < a :::; 3.5).
In addition, the assessment of the numerical accuracy of the conservation of
the collision integral is performed. The expression Gni(k) (4.1) is integrated
over a wave vector k. The error of the estimate of conserving the non-linear
energy transfer is not higher than 1 per cent.
Then the calculations of the integral ( 4.1) are analysed, comparing with
the results obtained by other investigators. As a criterion, the results of Hasselmann and Hasselmann (1981) are used. They present different functions
for the non-linear energy transfer which are calculated for the JONSWAP
spectrum at different determining parameters.
The calculation results for the one-dimensional function Gni(IJ), obtained
by Hasselmann & Hasselmann ( 1981) for the peakness parameter 1 = 7 and
angular energy distribution rv cos 2 (;3), are shown in Fig. 4.4a. This figure
also shows calculations by the algorithm proposed in this monograph. As it
is seen, the coincidence of results is quite good, taking into account that the
frequencies used in the calculation are somewhat different.
Although the agreement between the calculation results for peakness
1 = 3.3 is good enough, there are some differences at frequencies IJ > 1.21Jmax·
Comparing the calculation results, it can be concluded that the calculations in the present study are of a more stable (smooth) character. For
peakness 1 = 1.0 this difference is already significant (see Fig. 4.4b). The
proposed algorithm gives a noticeably smoother curve indicating greater
stability of the result obtained. This conclusion becomes even more pronounced, if the calculations of a two-dimensional function Gnl ( IJ, ;3) are
compared.
4 Physical Mechanisms Forming the Wave Spectrum in Deep Water
Tests of the collision integral calculation algorithm and assessment
of the calculation accuracy. Before further calculations the experiments
are made to test the aforementioned integration algorithm, using the quadrature formulas ( 4.9) described above. For this purpose the calculations of the
non-linear transfer integral for the JONSWAP spectrum are made using a different number of integrand points. Thus, the parameter n is assumed to be
equal to 2, 3, 4, 5, 7 and 8, respectively. The results of these calculations
are presented in Fig. 4.3. They show that with increasing n, the calculation
results converge quite rapidly to its accurate value. The numerical values become practically indiscernible for n > 5. For n = 4 the relative differences in
the energy range are not more than 15 per cent (of the corresponding value
at n = 8) sufficient for most practical calculations. The integration error is
estimated as 1-2 per cent with n = 7.
The accuracy for the typical JONSWAP spectrum is estimated by repeated calculations with double the number of grid points until the difference of the calculation results for two successive cases reaches the prescribed
value. By executing a series of successive calculations, an error is obtained not
greater than 1-2 per cent in the area of the spectral maximum (0.9 :::; a :::; 1.5,
where a= o-f!Jmax)· Moreover, the calculation error is not greater than 3-5
per cent in the frequency ranges (0.8 :::; a :::; 0.9) and (1.5 < a :::; 2.5) and not
greater than 5-10 per cent in the ranges (0. 7 :::; a :::; 0.8) and (2.5 < a :::; 3.5).
In addition, the assessment of the numerical accuracy of the conservation of
the collision integral is performed. The expression Gni(k) (4.1) is integrated
over a wave vector k. The error of the estimate of conserving the non-linear
energy transfer is not higher than 1 per cent.
Then the calculations of the integral ( 4.1) are analysed, comparing with
the results obtained by other investigators. As a criterion, the results of Hasselmann and Hasselmann (1981) are used. They present different functions
for the non-linear energy transfer which are calculated for the JONSWAP
spectrum at different determining parameters.
The calculation results for the one-dimensional function Gni(IJ), obtained
by Hasselmann & Hasselmann ( 1981) for the peakness parameter 1 = 7 and
angular energy distribution rv cos 2 (;3), are shown in Fig. 4.4a. This figure
also shows calculations by the algorithm proposed in this monograph. As it
is seen, the coincidence of results is quite good, taking into account that the
frequencies used in the calculation are somewhat different.
Although the agreement between the calculation results for peakness
1 = 3.3 is good enough, there are some differences at frequencies IJ > 1.21Jmax·
Comparing the calculation results, it can be concluded that the calculations in the present study are of a more stable (smooth) character. For
peakness 1 = 1.0 this difference is already significant (see Fig. 4.4b). The
proposed algorithm gives a noticeably smoother curve indicating greater
stability of the result obtained. This conclusion becomes even more pronounced, if the calculations of a two-dimensional function Gnl ( IJ, ;3) are
compared.