This section investigates the influence of changes in cross-section and corner ratio on wave evolution and the resulting nonlinear wave amplification around the given cylinder, subjected to both short and long waves. First, the physical mechanism of wave amplification around the cylinder is described based on the spatial contours of wave evolution. Following this, the associated harmonic patterns around each cross-section are analyzed about these findings.
3.1. Wave Interaction with a Cylinder with Different Cross-Sections Subject to Short Waves
Figure 4,
Figure 5,
Figure 6 and
Figure 7 illustrate the normalized spatial contour of wave evolution around each cylinder subjected to a short incident wave, viewed from both the top and front perspectives. As shown in
Figure 4(I(a)), Type-1 wave scattering for
is associated with a concentric and symmetric wave field. During the interaction of a single cycle of the incident wave with the cylinder, there are two instances of Type-1 wave scattering, occurring at the front and back stagnation points, as seen in
Figure 4(I(a)) and
Figure 6(I(f)).
The incident wave traveling downstream bends around the cylinder and is divided into two lateral edge waves on either side, as depicted in
Figure 4(I(b)). At the back-corner point, these lateral edge waves move downstream and reach the back stagnation point of the cylinder, resulting in high-frequency and fully developed Type-2 wave scattering, as shown in
Figure 5(I(c,d)). When the wave trough reaches the front stagnation point, the lateral edge waves circulate upstream, as illustrated in
Figure 6(I(e,f)). For short incident waves, these lateral edge waves barely reach the front-corner point of the cylinder, as seen in
Figure 7(I(g)). After the passage of the wave zero up-crossing at the front point, the waves move downstream, as shown in
Figure 7(I(h)).
For other cross-sections subjected to short incident waves, there are two instances of Type-1 wave scattering at the front and back stagnation points, as shown in
Figure 4(a(II–V)) and
Figure 6(f(II–V)). Additionally, there is one instance of Type-2 wave scattering at the back of the cylinder, as seen in
Figure 5(d(II–V)). The transition from a circular to a square corner ratio is highlighted by an increase in the maximum run-up height at the front of the cylinder, as shown in
Figure 4(a(II–V)). This change in corner ratio from circular to sharp square results in increased wave–structure interactions, leading to a systematic decrease in the energy and momentum of the lateral edge waves. As the corner ratio decreases gradually,
to
, the edge wave energy decreases, and the corner becomes more pronounced. Consequently, there is a stronger interaction with the back-corner point, which is particularly evident at
, as shown in
Figure 6(f(IV)).
Following the change in the corner ratio from
to
, the energy of the lateral edge waves reduces and they slow down as they circulate the cylinder. This results in a wave–wave interaction at the shoulder and consequently weaker interaction; see
Figure 7(g(V)). There is an occurrence of slightly developed wave scattering Type-2 at the front of the given cylinder for
; see
Figure 6(IV(e,f),V(e,f)). For the corner ratio of
, where the corner is visible, the interaction of the negative horizontal velocity with the front corner forms a new set of lateral edge waves, see
Figure 6(e(IV,V),(f(IV,V)) and
Figure 7(g(IV,V)). Similar to the
for the other cross-sections, the lateral edge waves traveling upstream can barely reach the front-corner point; see
Figure 7g.
The overall view of the harmonic values of second, third, and fourth of the total wave elevation at front-corner and also front stagnation points, see
Figure 8c–e, interestingly, indicates the occurrence of slightly developed wave scattering Type-2 at the front of the given cylinder for
; see
Figure 6(IV(e,f),V(e,f)). Comparing the occurrence of wave scattering Type-2 for
at the front and back of the given cylinder, there is less dynamic pressure and rise of water level and accordingly less steep waves; see
Figure 5d. This is related to the weak wave–wave interaction during the wave phase,
.
Concerning the information from the analysis of time history and spatial contour of wave evolution, harmonic analysis can provide a better and more quantitative understanding of the nonlinear nature of wave amplification and high-frequency wave field around each given cylinder. Here, the mean value and harmonics of the total wave elevation at given wave probes up to the fifth are analyzed. In continuation, in this study, it is shown to what extent the corner ratio contributes to the nonlinear wave field around the given cylinders.
The comparison of the mean value and harmonics of the total wave elevation around the given cross-section in terms of wave probe location,
, normalized by incident-minded
, is presented in
Figure 8. In addition, the harmonic contours of the diffracted wave for all the given five cross-sections in the short wave are presented in
Figure 9,
Figure 10 and
Figure 11. In a general view of
Figure 8, the systematic variation in corner ratio, there is a systematic change in harmonic pattern and values for cross-sections in the range,
, for harmonics rather than second. In addition, a different pattern is observed for
, or harmonics rather than first, particularly for second harmonic which is related to the sharp corner effect.
The first harmonic,
, pattern of
includes two peaks at front and back stagnation points with the values of
, respectively, and one depression at the back-corner point with the values of
; see
Figure 8b,
Figure 9a, and
Figure 10a. The front peak value is much larger than the back peak. These two peaks are more related to the run-up/-down with the arrival of incident wave crest/trough during the occurrence of wave scattering Type-1 and Type-2 at the front and back stagnation points, respectively, see
Figure 4(I(a)),
Figure 5(I(c)), and
Figure 6(I(f)). As mentioned before, the lateral edge waves traveling around the cylinder experience large energy and momentum reduction caused by both wave–wave and wave–structure interaction. Therefore, there is a significant reduction in first harmonic value from the front to the back-corner point which results in a depression value. Changing the cross-section in the range of,
, a similar harmonic pattern with a smooth change in the value is also observed. As expected, with the reduction in corner ratio, there is a systematic increase in first harmonics at the front stagnation point from
for
to the
for
; see
Figure 8b. It was mentioned before that a reduction in corner ratio increases the cylinder circumference and number of stagnation points. It finally, results in larger wave amplification and accordingly systematic rise of 1st harmonics in front of the given cylinder.
Increasing the cylinder circumference from,
, to the,
, enhances the wave–cylinder interaction, and causes more lateral edge wave damping. Consequently, this affects the wave scattering Type-2 and suppresses the rise of the water level at the back of the cylinder, see
Figure 5c,d and
Figure 6e. Furthermore, there is more reduction in 1st harmonics at the back-corner and also the back stagnation point which is more apparent for,
, see
Figure 8b. As seen in
Figure 8c, the second harmonic,
, pattern of
consists of three peaks at the front, shoulder, and back with the values of
, respectively. In addition, there are two depression points at the front corner and back corner, with values of,
. In
Figure 9b and
Figure 10b, the second harmonics show a spreading pattern like a bird’s footprint behind the given cylinder.
The large wave amplification and consequently, steep wave run-up during the occurrence of wave scattering Type-1 mainly contribute to the front peak of the second harmonic at the front of the cylinder, see
Figure 4(I(a)),
Figure 9b, and
Figure 10b. The back peak value firstly and mainly is related to the steep and high-frequency wave field caused by wave scattering Type-2; see
Figure 5(I(d)). Secondly, it is due to the reduced wave run-up caused by wave scattering Type-1 at the back of the given cylinder, secondary crest; see
Figure 6(f(I)). Considering the short incident wave interacting with the cylinder,
, there is a strong interaction between edge waves and negative horizontal velocity related to the arrival of a wave trough. This results in the large peak value at the shoulder, see
Figure 6(f(I)) and
Figure 8c.
Following the passage of zero-down crossing and trough at the back-corner and front-corner, respectively, see
Figure 4(I(a)) and
Figure 6(I(e)), there are two depression points, see
Figure 8c. They are related to the weak interaction of edge waves traveling upstream with the opposite incident wave. The energy of edge waves at the back corner initiated from the back is larger. On the other hand, there is a considerable suppression of edge wave energy at the shoulder which leads to weak edge wave interaction with the front corner. Thus, the value of the 2nd harmonic at the back-corner is larger compared to the front-corner value.
For the variation in the cross-section in the range of,
, slight and moderate systematic reduction for the second harmonic is observed at the front and back stagnation points, respectively; see
Figure 8c. The reduction in corner ratio and accordingly increase in the cross-section circumference leads to more wave–structure interaction during wave scattering Type-1 and more lateral edge wave energy suppression during wave scattering Type-2. Thus, there is a weaker edge wave collision which results in a less steep wave run-up at the front and back stagnation points, respectively. Changing the corner ratio, the harmonic pattern changes, and a considerable variation in second harmonic at back-corner, shoulder, and front-corner points are observed as well; see
Figure 9b and
Figure 10b. As seen in
Figure 6e, the interaction of edge waves with opposite incident waves occurs at the arrival of zero-down crossing at the back-corner point. For the cross-section in the range of,
, there is a systematic increase of 2nd harmonics at the back-corner point.
It was explained before that the energy of lateral edge waves circulating the cylinder in the upstream direction decreases with the reduction in corner ratio. In contrast, as the geometrical presence of the corner becomes visible, there is strong wave–structure interaction at the back-corner point which is highlighted for
; see
Figure 10(b(IV)). For,
, although the corner is sharp, regarding the back stagnation point second harmonic value, there is even more edge wave energy suppression, see
Figure 10(b(V)).
This leads to weaker wave–structure interaction at the back-corner point compared to
. Wave–structure interaction at the back-corner point wave–wave interaction with the opposite incident wave and then wave–structure interaction with the given cylinder circumference slows down the progressive lateral edge waves. So, as seen in
Figure 6f, the lateral edge waves interact with the opposite incident wave at different wave phases between zero-down crossing to the trough. As a result, there is weaker wave–wave interaction with the corner ratio reduction, which is apparent as a depression point for both
at the shoulder; see
Figure 8c.
Energy reduction due to continuous wave–wave and wave–structure interaction, consequently, slows down lateral edge waves. It is observed for all the given cross-sections that the lateral edge waves cannot reach the front-corner point subject to short incident waves, see
Figure 6f,
Figure 7g,
Figure 9b, and
Figure 10b. However, interestingly, there is another set of lateral edge waves for
where the geometrical presence of the corner is visible. It is caused by the interaction of negative horizontal velocity with the passage of incident wave zero-down crossing and then trough at the front-corner point, see
Figure 6(e(IV,V),f(IV,V)). Hence, a notable and systematic rise of the second harmonic is observed at the front-corner point for these cross-sections, see
Figure 10b.
The mean value,
, pattern of,
, includes two peaks at front and back stagnation points with the positive values of
, respectively. There are also two depressions at the shoulder and back-corner point with the negative values of,
, see
Figure 8a. Steep and high wave run-up during wave scattering Type-1 shifts up the mean value significantly and results in a peak at the front stagnation point, see
Figure 4(I(a)) and
Figure 11(I). Similarly, steep wave run-up and rise of water level due to wave scattering Type-2 and then wave scattering Type-1 shifts up the mean value which is observed as a peak at the back stagnation point, see
Figure 5d and
Figure 6f. The interaction of edge waves traveling upstream with the opposite incident wave at about the arrival of the zero-down crossing phase shifts down the mean value, see
Figure 8a. This leads to the negative value of depression at the back-corner point, see
Figure 6(I(e)). Similarly, as the edge waves interact with the opposite incident wave at about the arrival of the trough phase at the shoulder point, it also shifts down the mean value, see
Figure 6(I(f)). As a result, there is a larger depression with a negative value at the shoulder compared to the back-corner point. As the corner ratio decreases, a slight and systematic increment of peak mean value at the front stagnation point is observed due to a smooth increase in wave run-up height, see
Figure 11(II–V). On the opposite, a drop in wave run-up height at the back stagnation point leads to a moderate systematic reduction in peak mean value. For,
, the corner effect becomes strong. Hence, the wave–structure interaction in addition to the interaction of edge waves with the opposite incident wave at different phases between zero-down crossing to the trough shifts down the mean value. Therefore, a considerable negative value of depression is observed at the back-corner point, see
Figure 8a. The interaction of edge waves with the opposite incident wave at the shoulder point occurs at phases between zero-down crossing to the trough for the corner ratio in the range of,
; see
Figure 6f. Consequently, the mean value shifts up, systematically, from negative for
to the positive value for
at the shoulder point. For
, the change in mean value follows the systematic variation at all the wave probes, except, shoulder and back-corner points. As seen in
Figure 6e,f, although the corner is sharp, the weak wave–structure interaction at both back-corner and shoulder points shifts up the mean value significantly compared to the other cross-sections.
For higher harmonics rather than second, excluding,
, there is a systematic change for all cross-sections at all the given wave probe locations; see
Figure 8d–f,
Figure 9c,d, and
Figure 10c,d. In
Figure 9c,d and
Figure 10c,d, the third harmonics indicate diffraction waves with a stronger pattern in a much wider region behind the given cylinder and the pattern is similar to that of the second harmonics. Following the second to the fifth harmonics, the harmonic values at the back stagnation point for,
, compared to the other cross-section suggests the occurrence of considerably weak wave scattering Type-2. It is related to the significant suppression of lateral edge wave energy interacting with the cross-section circumference at the back of the given cylinder, see
Figure 5d.
It was mentioned before that, as the corner effects become strong,
, another set of edge waves is induced at the front-corner point. This finally results in a slightly developed wave scattering Type-2 front stagnation point, see
Figure 6e,f and
Figure 7g. As these new sets of edge waves move to the front stagnation point, interact with the given cross-section circumference. This interaction suppresses their energy and leads to a small rise of water and accordingly, a small steep wave run-up at the front stagnation point, see
Figure 6(f(IV,V)) and
Figure 7(g(IV,V)). It is worth noting that, this second wave run-up at front of the cylinder for,
, does not contribute to the 2nd harmonics but a notable contribution to the 3rd and 4th harmonics, see
Figure 9c,d and
Figure 10c,d. The overall analysis of higher harmonics rather than first harmonic for all cross-sections reveals that the wave amplification and free-surface nonlinearly are more observed at the backward part of the given cylinder; see
Figure 9c,d and
Figure 10c,d. Besides, as the geometrical presence of corners becomes visible, corners also contribute significantly to the nonlinear wave field around the given cylinder which is evident in mean value and third and particularly in second harmonics; see
Figure 9b–d,
Figure 10b–d and
Figure 11(II–V).
3.2. Wave Interaction with a Cylinder with Different Cross-Sections Subject to Long Waves
Here, for the long-wave cases,
Figure 12,
Figure 13,
Figure 14 and
Figure 15 show the normalized spatial contour of wave evolution around each given cylinder with the top and front view. With the change in the incident wavelength from T = 7 s to T = 15 s, a smaller wave–structure interaction and accordingly reduced dynamic pressure and wave amplification at both front and back stagnation points are observed. Besides, the wave run-up at the back point is less steep and the height is smaller compared to the short-wave case. The wavelength variation is highlighted with the occurrence of fully developed wave scattering Type-2 at the front of the given cylinder, where the lateral edge wave can reach the front of the cylinder before the arrival of zero-up crossing, see
Figure 14e. In general, the formation of two-wave scattering Type-2 is observed during the interaction of one wave cycle for,
, at the back and then the front; see
Figure 12(I(a)) and
Figure 14(I(e)). Unlike the short-wave case, there is a shift back for the phase of edge wave interaction with the back-corner, shoulder, and front-corner points.
Change in the cross-section from circular to square, , increases the wave–structure interaction. The corner effect is observed as the main contribution to the further lateral edge wave energy and momentum damping. For the cross-sections in the range of, , the lateral edge waves experience energy reduction mostly due to the interaction with the circumference. The lateral edge waves passing the front-corner point collide and form wave scattering Type-2 at the front of the given cylinders. For square cross-sections which are identified with sharp corners,, the evolution of the free surface is somewhat different.
At the front of the cylinder with sharp corners, there are lateral edge waves that are from the remaining of the previous incident wave cycle. They get damped more than the other cross-sections and therefore, cannot propagate downstream; see
Figure 13(c(II–V)). However, interestingly, a set of lateral edge waves is observed due to the interaction of positive horizontal velocity with the sharp back-corners after the passage of incident wave zero-up crossing. They collide at the back stagnation point, and initiate the formation of wave scattering Type-2 slightly sooner than the other cross-sections; see
Figure 13(V(c,d)). In the case of the square cross-section, the lateral edge waves (damped) at the back cannot prorogate upstream and contribute to the formation of wave scattering Type-2 at the front.
Here, the harmonic analysis of total wave evolution at given wave probes can provide useful information on the wavelength effect on the nonlinear wave field around each cylinder. The comparison of normalized mean value and harmonics of the total wave elevation around the given cross-section subject to the long incident wave is presented in
Figure 16. In addition, the harmonic contours of the diffracted wave for all the given five cross-sections in the long wave are presented. Overall, following the systematic variation in corner ratio in the range of
, a systematic but with a slight change in harmonic pattern and value is observed compared to short wave cases. The harmonics are somewhat different for square cross-sections,
, which is related to the sharp corner effect. In general, the first harmonics are small at given wave probes with an average of
. This is related to the smaller wave diffraction and wave amplification by increasing wavelength from T = 7 s to T = 15 s. The wavelength effect on the harmonic contours of the diffracted wave for a circular cylinder is presented in
Appendix A.
The first harmonic,
, pattern of,
, similar to the short-wave case, includes two peaks at front and back stagnation points with the values of,
, respectively. In addition, there is one depression at the back-corner point with the values of,
, see
Figure 16b. As mentioned before, the front and back peaks are caused by the run-up/-down during wave scattering Type-1 and Type-2 at the front and back stagnation points, respectively. Moreover, energy and momentum reduction from the front- to the back-corner point leads to the mentioned depression in first harmonic. Systematic reduction in corner ratio increases the dynamic pressure and accordingly, wave amplification which is observed as a slight and systematic rise in first harmonic at the front stagnation point; see
Figure 12a. On the contrary, with a change in cross-section from circular to square, the lateral edge waves experience more energy damping due to more wave–structure interaction. Therefore, there is a weaker lateral edge wave collision that causes less steep and smaller height wave run-up and leads to a slight and systematic drop in first harmonic at shoulder, back-corner, and back stagnation points, see
Figure 16b.
Figure 16c, indicates the second harmonic,
, the pattern, which is completely different from the short wave case, see
Figure 9b and
Figure 10b. The harmonic pattern of
consists of two peaks at front and back stagnation points with the values of
, respectively, and one depression point at the back-corner with the value of
. The back peak value is mainly related to the steep and high-frequency wave field caused by wave scattering Type-2. This value is considerably smaller than the short-wave case because of weak lateral edge wave collision due to the smooth rate of change in velocity components. Furthermore, the contribution of wave scattering Type-1 is small due to the weak interaction of the negative horizontal velocity of the wave trough with the back stagnation point. There is a considerable suppression of edge wave energy at the back of the cylinder which results in the depression value of the second harmonic. Firstly, it is the result of the weak wave–wave interaction of edge waves traveling upstream with the opposite incident wave. Secondly, it is related to the weak wave interaction with the back-corner point before the arrival of the zero-down crossing. It was mentioned before, unlike the short-wave case, the phase in which the edge waves interact with the back-corner, shoulder, and front-corner points shift back. After the passage of the zero-down crossing at the shoulder point, a weak interaction of lateral edge waves with negative velocity components of the incident wave is observed. This weak wave–wave interaction somewhat enhances lateral edge wave energy and causes to some extent stronger wave–structure interaction compared to the back-corner point. Hence, there is an increase in the second harmonic value from the back corner to the shoulder point, see
Figure 16c,
Figure A2b and
Figure A3b. Similarly, the interaction of lateral edge waves passing the front-corner point with the negative horizontal velocity of the incident wave before the arrival of the wave trough considerably increases their energy.
It is observed that a continuous rise in the second harmonic value, initiated from the shoulder point, indicates that the peak value at the front is significantly higher than the other points around the cylinder, see
Figure A2b and
Figure A3b. This reveals the important contribution of the occurrence of wave scattering Type-2 at the front, see
Figure 14e, besides the small contribution by less steep wave run-up during the occurrence of wave scattering Type-1, compared to the short-wave case; see
Figure 8c and
Figure 16c. Unlike the short-wave case, there is no considerable variation in second harmonic at corners, and the corner effect is observed as more further damping of edge wave energy. Therefore, change in cross-section from circular to square,
. As seen in
Figure 16c, there is a systematic and small reduction in second harmonics from the back to shoulder points. Afterward, the slight augmentation of lateral edge wave energy leads to a slight rise in second harmonics from the shoulder to the front point. For,
, the harmonic pattern is similar and the change in value is systematic (except the front point) but moderately evident.
It was shown that there is more edge wave energy reduction for
compared to the other cross-sections, and the edge waves get damped before being able to travel around the cylinder; see
Figure A3b(V). Therefore, there is no record of lateral edge waves from the back to the shoulder point, and consequently, a notable reduction in second harmonics is observed. The interaction of the induced lateral edge waves by the front-corner point is weaker than one for the other cross-sections at the front to form wave scattering Type-2 and it is apparent as a considerable reduction of second harmonics at the front for,
.
The mean value,
, pattern of
includes two peaks at front and back stagnation points with the positive values of
, respectively; see
Figure A3(IV). There is also one depression at the shoulder point with the negative value of
; see
Figure 16a. The occurrence of two wave run-ups during wave scattering Type-1 and Type-2 shifts up the mean value and results in a notable peak value at front stagnation. The back peak which is smaller than the front peak is mainly related to the shift-up in the mean value caused by wave run-up and the rise of water level during wave scattering Type-2; see
Figure A4. The negative value of depression is the outcome of lateral edge waves’ weak interaction with the shoulder point following the passage of incident wave zero-down crossing that shifts down the mean value.
The interaction of lateral edge waves with the back corner at the wave phase before the arrival of zero-down crossing results in a shift-down in the mean value (negative). In contrast, the interaction of lateral edge waves with the negative horizontal velocity at the front corner causes a considerable rise in mean value (positive) which is also larger than the value at the back corner. As the corner ratio decreases in the range of,
, a slight rise in the mean value is observed due to the smooth increase in wave run-up at the front, see
Figure A4(II–V), whereas a smooth decrease in wave run-up at the back results in a slight drop of the mean value; see
Figure A4(IV). In continuation, a weaker interaction of lateral edge waves at the back-corner and then shoulder leads to a moderate rise in mean value at these points which is evident fore,
; see
Figure A4(V).
The mean value at the front corner indicates a moderate drop, which is related to the contribution of the corner effect in lateral edge wave enhancement at this point. For,
, the harmonic pattern is similar but unlike the other cross-sections, the value does not follow a systematic change. The second wave run-up height and accordingly the mean value is smaller at the front stagnation point, comporting to the other cross-sections. As seen in
Figure 13(V(c)), there is no record of the interaction of lateral edge waves at the shoulder. Furthermore, there is a considerably weak interaction of lateral edge waves at the back corner and also a weak interaction of induced lateral edge waves at the front corner. In general, these interactions shift up significantly the mean value at these points compared to the other cross-sections.
For higher harmonics rather than second, the harmonic patterns are similar and there is mostly systematic slight change for all the cross-sections at all the given wave probe locations. The overall analysis of higher harmonics rather than first harmonic, reveals that in all the cross-sections subject to the long wave, the wave amplification and free-surface nonlinearly are more observed at the forward part of the given cylinder, see
Figure A2c,d and
Figure A3c,d. This is mainly related to the interaction of negative velocity components of the incident wave with the forward part of the given cylinder and the occurrence of wave scattering Type-2 at the front. Besides, as the geometrical presence of the corner becomes visible, unlike short-wave cases, corners weakly contribute to the nonlinear wave field around the given cylinder.