#### 5.3. Euler Equations

As a result of the computing procedures previously outlined (

Section 3.2), results related to the Euler-equations-derived flow fields are presented.

It is to be preliminarly noted that the following figures incorporate three types of information, namely: (i) they depict the flow fields in terms of flow structures as extracted with the swirling-strength (or also

${\lambda}_{ci}$) criterion [

62], and represented at given threshold values (

${{\lambda}_{ci}|}_{th}$, see Alfonsi and Primavera [

64]), at different instants; (ii) “lines” eventually detectable in the visualizations actually represent borders between structures; (iii) the external surfaces of the flow structures are colored with the spanwise component of the vorticity (

${\omega}_{y}$), so that the sign and the intensity of the local-particle rotation can be inferred from the visualizations (more intense

${\omega}_{y}$ coloring denotes stretching of vortex lines, less intense

${\omega}_{y}$ coloring denotes shrinking of vortex lines, reddish areas denote positive

${\omega}_{y}$, bluish areas denote negative

${\omega}_{y}$, greyish areas denote zero

${\omega}_{y}$).

Figure 14a,b show top views of the computing domain at

$t={t}_{{F}_{max}}^{Euler}$ (the instant at which the Euler-derived wave field exerts the maximum force on the cylinder) and

$t={t}_{{R}_{max}}^{Euler}$ (the instant at which the Euler-derived wave field exhibits the maximum runup on the upstream external surface of the cylinder), respectively. The fields are rather regular and substantially symmetric. The flow field exhibits a number of tubular vortical structures concentrated at the free surface, where, from the distribution of these structures, one obtains the perception of the free-surface configuration in terms of wave crests, troughs, accumulation of fluid mass upstream from the cylinder, and progressive development of the wave-diffraction phenomenon (see groups of lines

a to

d in

Figure 14). In particular, at

$t={t}_{{F}_{max}}^{Euler}$ there exist a remarkable accumulation of fluid mass—mainly irrotational—upstream from the cylinder, while at

$t={t}_{{R}_{max}}^{Euler}$ both positive and negative vorticity contaminates the latter mass of fluid.

Overall, the nature of the above-mentioned free-surface tubular structures is that of being largely irrotational, exception made for the

previously-generated vorticity contained in some of them that propagates (conservatively) across the field, according to the progression of the wave-cylinder interaction process. From

Figure 14 it can be also noticed that the areas characterized by positive and/or negative, previously-generated

${\omega}_{y}$, are well defined and confined, showing that no viscous diffusion of vorticity occurs, as expected (in the absence of viscosity, no vorticity diffusion occurs).

**Figure 14.**
Flow structures as obtained from the solution of the primitive-variable Euler equations. Top view of free-surface profile: (**a**) $t={t}_{{F}_{\mathrm{max}}}^{Euler}$; (**b**) $t={t}_{{R}_{\mathrm{max}}}^{Euler}$.

**Figure 14.**
Flow structures as obtained from the solution of the primitive-variable Euler equations. Top view of free-surface profile: (**a**) $t={t}_{{F}_{\mathrm{max}}}^{Euler}$; (**b**) $t={t}_{{R}_{\mathrm{max}}}^{Euler}$.

Figure 14 further shows that the wave crest- and trough-structures are substantially irrotational, while positive and/or negative vorticity is present in the ascending and/or descending portions of the wave, approximately at half-way between a crest and a trough. Moreover, it can be noted that, at

$t={t}_{{F}_{max}}^{Euler}$, the flow structures encompassed in the group of lines

a (located in a descending portion of the wave, upstream from a trough) are characterized by strong positive vorticity, the structures encompassed by the group of lines

b (right after a trough) are characterized by less intense positive vorticity, those encompassed by the group of lines

c (downstream from a crest) are characterized by strong negative vorticity far away from the cylinder, and by strong positive vorticity nearby the cylinder (caused by the pressure field in the proximity of the cylindrical body), while the structures encompassed by the group of lines

d are characterized by remarkably-less intense negative vorticity.

In going from $t={t}_{{F}_{max}}^{Euler}$ to $t={t}_{{R}_{max}}^{Euler}$ one can also notice that the structures encompassed in the group of lines a exhibit an increase of their positive ${\omega}_{y}$-field, due to the turning and stretching of vortex lines, as caused by the progression of the wave-to-cylinder approaching process, while the structures encompassed by the group of lines b also increase their ${\omega}_{y}$ field (of either sign), due to the turning and stretching of vortex lines nearby the cylinder. Moreover, the structures encompassed by the group of lines c increase their negative ${\omega}_{y}$-field, due to the turning of vortex lines right downstream from the cylinder, while those encompassed by the group of lines d are subjected to remarkable turning and stretching between the wave trough and wave crest, downstream from the cylinder.

In

Figure 15,

Figure 16 and

Figure 17, upstream-, side-, and downstream-close-up views of the structure field are shown in the vicinity of the cylinder external wall, at

$t={t}_{{F}_{max}}^{Euler}$ and

$t={t}_{{R}_{max}}^{Euler}$, respectively. These figures show that other types of structures develop underwater, all of them of inviscid type. Arrow 1 indicates a structure completely encompassing the cylindrical body, that persists in going from

$t={t}_{{F}_{max}}^{Euler}$ to

$t={t}_{{R}_{max}}^{Euler}$. Upstream from the cylinder, a rather complex structure is visible (structure 2/3), that only slightly changes its shape in going from

$t={t}_{{F}_{max}}^{Euler}$ to

$t={t}_{{R}_{max}}^{Euler}$ (note that, in the subsequent Navier-Stokes-field representations, structures 2 and 3 will appear as distinct).

Additional structures are present. In

Figure 15 and

Figure 16, arrow 4 denotes an upstream structure that represents the underwater counterpart of the accumulation of fluid mass upstream from the cylinder, that verifies when the oncoming wave approaches the cylindrical body. In

Figure 16 and

Figure 17, arrow 5 denotes a downstream structure that represents the underwater remnant of the wave-cylinder interaction process. In

Figure 15,

Figure 16 and

Figure 17, a right-side structure and a left-side structure are also visible, mirroring the way trough which the structure field propagates from upstream to downstream from the cylinder, in the absence of viscous forces.

**Figure 15.**
Close-up at the cylinder surface of inviscid-flow structures as obtained from the solution of the Euler equations. Upstream view: (**a**) $t={t}_{{F}_{\mathrm{max}}}^{Euler}$; (**b**) $t={t}_{{R}_{\mathrm{max}}}^{Euler}$.

**Figure 15.**
Close-up at the cylinder surface of inviscid-flow structures as obtained from the solution of the Euler equations. Upstream view: (**a**) $t={t}_{{F}_{\mathrm{max}}}^{Euler}$; (**b**) $t={t}_{{R}_{\mathrm{max}}}^{Euler}$.

**Figure 16.**
Close-up at the cylinder surface of inviscid-flow structures as obtained from the solution of the Euler equations. Side view: (**a**) $t={t}_{{F}_{\mathrm{max}}}^{Euler}$; (**b**) $t={t}_{{R}_{\mathrm{max}}}^{Euler}$.

**Figure 16.**
Close-up at the cylinder surface of inviscid-flow structures as obtained from the solution of the Euler equations. Side view: (**a**) $t={t}_{{F}_{\mathrm{max}}}^{Euler}$; (**b**) $t={t}_{{R}_{\mathrm{max}}}^{Euler}$.

**Figure 17.**
Close-up at the cylinder surface of inviscid-flow structures as obtained from the solution of the Euler equations. Downstream view: (**a**) $t={t}_{{F}_{\mathrm{max}}}^{Euler}$; (**b**) $t={t}_{{R}_{\mathrm{max}}}^{Euler}$.

**Figure 17.**
Close-up at the cylinder surface of inviscid-flow structures as obtained from the solution of the Euler equations. Downstream view: (**a**) $t={t}_{{F}_{\mathrm{max}}}^{Euler}$; (**b**) $t={t}_{{R}_{\mathrm{max}}}^{Euler}$.

#### 5.4. Navier Stokes Equations

As a result of the computing procedures previously outlined (

Section 3.3), results related to the Navier-Stokes-equations-derived flow fields, are presented (as for the kind of information incorporated in the following figures, one can refer to the previous section).

Figure 18a,b show top views of the computing domain at

$t={t}_{{F}_{\mathrm{max}}}^{NS}$ (the instant at which the Navier-Stokes-derived wave field exerts the maximum force on the cylinder), and

$t={t}_{{R}_{\mathrm{max}}}^{NS}$ (the instant at which the Navier-Stokes-derived wave field exhibits the maximum runup on the external surface of the cylindrical body), respectively.

The fields are rather regular and substantially symmetric, and the structure field exhibits a number of tubular and—in this case—mainly flattened vortical structures, concentrated at the free surface. Again, from the distribution of these structures, one obtains the perception of the free-surface configuration in terms of wave crests and troughs, and also of the accumulation of fluid mass upstream from the cylinder, and the progressive development of the wave-diffraction phenomenon related to the wave-cylinder interaction process (lines

a to

q in

Figure 18). The nature of the free-surface flattened (viscous) tubular structures is that of being predominantly irrotational far away from the cylinder, though becoming rotational when they actually interact with the cylindrical body. Moreover, the flattened structures tend to stay irrotational in their free-surface portion, while becoming rotational immediately under the free surface. This is due to the presence of a negligible shear-stress field at the free surface, and to a non-negligible shear-stress-field under the free surface.

**Figure 18.**
Flow structures as obtained from the solution of the primitive-variable Navier-Stokes equations. Top view of free-surface profile: (**a**) $t={t}_{{F}_{\mathrm{max}}}^{NS}$; (**b**) $t={t}_{{R}_{\mathrm{max}}}^{NS}$.

**Figure 18.**
Flow structures as obtained from the solution of the primitive-variable Navier-Stokes equations. Top view of free-surface profile: (**a**) $t={t}_{{F}_{\mathrm{max}}}^{NS}$; (**b**) $t={t}_{{R}_{\mathrm{max}}}^{NS}$.

The structure fields shown in

Figure 18 also exhibit a number of strong and not flattened tubular structures immediately under the free surface, mainly located upstream from the cylinder, that represent the underwater consequence of the manifestation of the wave troughs at the free surface in their viscous interaction with the cylindrical body. An inspection to

Figure 18 also shows that, through the field, zones characterized by more and/or less intense coloring (either reddish or bluish) are present in the field, and also that, typically at the free surface, a more intense

${\omega}_{y}$ coloring (either reddish or bluish) mainly verifies in the vicinity of the wave troughs. This latter phenomenon is due to the presence of the above-mentioned strongly-rotational underwater tubular structures, whose vorticity field breaks up the predominantly-irrotational character of the structures at the free-surface.

Moreover, from

Figure 18a, at

$t={t}_{{F}_{\mathrm{max}}}^{NS}$, one can notice that the presence of the cylindrical body induces an inversion in the sign of

${\omega}_{y}$ in the trough immediately upstream from the cylinder (from reddish to bluish in front of the cylinder), being the latter bluish zone progressively pushed towards the boundaries of the domain at

$t={t}_{{R}_{\mathrm{max}}}^{NS}$ (

Figure 18b). What happens there is that the presence of the cylindrical body in front of the ascending portion of the wave determines an inversion of the fluid-particle orbital paths, with the consequence that the shear-stress field induces a strong negative

${\omega}_{y}$-field in the zone right in front of the cylinder.

In

Figure 19,

Figure 20 and

Figure 21, upstream-, side- and downstream-close-up views of the structure field are shown in the vicinity of the cylinder external wall, at

$t={t}_{{F}_{\mathrm{max}}}^{NS}$, and

$t={t}_{{R}_{\mathrm{max}}}^{NS}$, respectively. From these figures it can be noticed that other types of viscous-flow structures develop under the free surface. The establishment of the latter structures actually represents the more relevant difference between the Navier-Stokes- (and/or eventually Euler-) flow-fields, and the velocity-potential-derived flow fields, so unveiling physical phenomena that are impossible to detect by only analyzing the potential-derived fields.

Figure 19,

Figure 20 and

Figure 21 show first—as also previously mentioned—the presence of two main underwater tubular, not flattened structures upstream from the cylinder and under the wave trough, characterized by high positive vorticity. In going from

$t={t}_{{F}_{\mathrm{max}}}^{NS}$ to

$t={t}_{{R}_{\mathrm{max}}}^{NS}$ these structures grow in dimensions, and the one that is nearest to the cylinder develops a more extended, massive sub-structure, right in front of the cylinder itself. These structures in the whole represent the underwater consequence of the manifestation of the trough at the free surface, where the aforementioned more extended sub-structure represents the underwater counterpart of the wave-to-cylinder approaching process.

**Figure 19.**
Close-up at the cylinder surface of viscous-flow structures as obtained from the solution of the Navier-Stokes equations. Upstream view: (**a**) $t={t}_{{F}_{\mathrm{max}}}^{NS}$; (**b**) $t={t}_{{R}_{\mathrm{max}}}^{NS}$.

**Figure 19.**
Close-up at the cylinder surface of viscous-flow structures as obtained from the solution of the Navier-Stokes equations. Upstream view: (**a**) $t={t}_{{F}_{\mathrm{max}}}^{NS}$; (**b**) $t={t}_{{R}_{\mathrm{max}}}^{NS}$.

**Figure 20.**
Close-up at the cylinder surface of viscous-flow structures as obtained from the solution of the Navier-Stokes equations. Side view: (**a**) $t={t}_{{F}_{\mathrm{max}}}^{NS}$; (**b**) $t={t}_{{R}_{\mathrm{max}}}^{NS}$.

**Figure 20.**
Close-up at the cylinder surface of viscous-flow structures as obtained from the solution of the Navier-Stokes equations. Side view: (**a**) $t={t}_{{F}_{\mathrm{max}}}^{NS}$; (**b**) $t={t}_{{R}_{\mathrm{max}}}^{NS}$.

**Figure 21.**
Close-up at the cylinder surface of viscous-flow structures as obtained from the solution of the Navier-Stokes equations. Downstream view: (**a**) $t={t}_{{F}_{\mathrm{max}}}^{NS}$; (**b**) $t={t}_{{R}_{\mathrm{max}}}^{NS}$.

**Figure 21.**
Close-up at the cylinder surface of viscous-flow structures as obtained from the solution of the Navier-Stokes equations. Downstream view: (**a**) $t={t}_{{F}_{\mathrm{max}}}^{NS}$; (**b**) $t={t}_{{R}_{\mathrm{max}}}^{NS}$.

Additional and differently-shaped flow structures are present in the underwater environment nearby the cylinder at both

$t={t}_{{F}_{\mathrm{max}}}^{NS}$ and

$t={t}_{{R}_{\mathrm{max}}}^{NS}$. In

Figure 19,

Figure 20 and

Figure 21, arrow 1 indicates a structure that completely encompasses the cylindrical body. At

$t={t}_{{F}_{\mathrm{max}}}^{NS}$, structure 1 is characterized by strong negative vorticity on the cylinder downstream side, by strong positive vorticity laterally, and by an additional-structure superposition phenomenon (structures 2 and 3) on the upstream side. Structure 2 is characterized by high values of positive vorticity, while structure 3 is characterized by less intense

${\omega}_{y}$ (

Figure 19a). Moreover, through

Figure 19,

Figure 20 and

Figure 21, arrows 4 and 5 indicate flow structures that develop respectively upstream (structure 4) and downstream (structure 5) from the cylinder, immediately under the free-surface level. Structure 4 appears to be the underwater counterpart of the phenomenon of accumulation of fluid mass that verifies upstream from the cylinder when the wave approaches the cylindrical body. Structure 5 appears as being the downstream underwater remnant of the cylinder-wave interaction process.

Overall, both structure field and vorticity field strongly change in going from

$t={t}_{{F}_{\mathrm{max}}}^{NS}$ to

$t={t}_{{R}_{\mathrm{max}}}^{NS}$. While at

$t={t}_{{F}_{\mathrm{max}}}^{NS}$ the underwater field is populated by structures 1, 2, 3, 4, 5, at

$t={t}_{{R}_{\mathrm{max}}}^{NS}$, structures 2 and 3 have disappeared, structure 1 is characterized by strong positive

${\omega}_{y}$ on the cylinder upstream side (exception made for a small central area,

Figure 19b), structure 4 has separated in two portions and changed the sign of

${\omega}_{y}$ (

Figure 19b), while on the cylinder downstream side, structure 5 still persists, though smaller than previously (

Figure 21b).

Additional information on the phenomenon at hand can be derived from specific structure-field representations.

Figure 22 and

Figure 23 show close-ups of the structure field (block 3 of the computational domain,

Figure 9), at

$t={t}_{{F}_{\mathrm{max}}}^{NS}$ and

$t={t}_{{R}_{\mathrm{max}}}^{NS}$, respectively.

The left portion of each of these figures represents a bottom close-up view of the structure field, as represented with the same criterion previously used, such that the external surfaces of the flow structures are colored with the values assumed by the spanwise component of the vorticity (

${\omega}_{y}$, reddish areas denote positive

${\omega}_{y}$, bluish areas denote negative

${\omega}_{y}$, greyish areas denote zero

${\omega}_{y}$, more intense coloring corresponds to high

${\omega}_{y}$-values, less intense coloring corresponds to lower

${\omega}_{y}$-values). The right portions of

Figure 22 and

Figure 23 also represents the structure field at

$t={t}_{{F}_{\mathrm{max}}}^{NS}$ and

$t={t}_{{R}_{\mathrm{max}}}^{NS}$, where now the external surfaces of the structures are colored with the values assumed by the pressure, in a darker-to-lighter color scale (bluish areas denotes low values of the pressure, in the vicinity of the free surface, rosy areas denote higher values of the pressure, more deep under the free surface).

**Figure 22.**
Close-up views of structures as obtained from the solution of the Navier-Stokes equations. Bottom view of computing domain at $t={t}_{{F}_{\mathrm{max}}}^{NS}$. The external surfaces of structures are colored with: (**a**) wave-field spanwise vorticity; (**b**) wave-field pressure values.

**Figure 22.**
Close-up views of structures as obtained from the solution of the Navier-Stokes equations. Bottom view of computing domain at $t={t}_{{F}_{\mathrm{max}}}^{NS}$. The external surfaces of structures are colored with: (**a**) wave-field spanwise vorticity; (**b**) wave-field pressure values.

**Figure 23.**
Close-up views of structures as obtained from the solution of the Navier-Stokes equations. Bottom view of computing domain at $t={t}_{{R}_{\mathrm{max}}}^{NS}$. The external surfaces of structures are colored with: (**a**) wave-field spanwise vorticity; (**b**) wave-field pressure values.

**Figure 23.**
Close-up views of structures as obtained from the solution of the Navier-Stokes equations. Bottom view of computing domain at $t={t}_{{R}_{\mathrm{max}}}^{NS}$. The external surfaces of structures are colored with: (**a**) wave-field spanwise vorticity; (**b**) wave-field pressure values.

The following can be observed.

In

Figure 22a it is possible first to observe some phenomena already depicted in

Figure 18a, namely the existence—upstream from the cylinder—of a spanwise tubular (not flattened) structure, as part of a more complex wave-trough flow structure, that includes the vorticity-inversion zone at center, in front of the cylinder. It also to be noted how the intensity of the vorticity field under the free-surface level is higher upstream from cylinder than downstream).

Figure 22b further shows that the peripheral portion of the above-mentioned tubular structure lies more in depth (lighter-blue areas in

Figure 22b near the boundaries of the computing domain) than its central part.

Figure 22 also shows the presence of structure 4 (see also at

Figure 19 and

Figure 20). This structure, at

$t={t}_{{F}_{\mathrm{max}}}^{NS}$, is characterized by negative vorticity (

Figure 22a), and by a remarkable depth just in front of the cylinder (

Figure 22b). Other structures are detectable, especially in

Figure 22a, namely those already characterized by lines

k,

l,

m,

n,

o,

p,

q in

Figure 18a. This is the structural character of the flow field in the vicinity of the cylinder, at the time at which the wave exerts the maximum force on the obstacle.

Figure 23 shows the flow-field representation at the time at which the wave exhibits the maximum runup on the external surface of the cylindrical body. Also here (see also at

Figure 18b) a complex tubular structure can be recognized upstream form the cylinder in

Figure 23a, that exhibits—in its central part—a massive sub-structure just in front of the cylinder. Note how this sub-structure is characterized by less intense vorticity with respect to other portions of the same structure. From

Figure 23b can be also observed that the new sub-structure, and its adjacent portions, extend remarkably in deep under the free surface, in a difference with respect to what shown in

Figure 22b (at the previous instant), where the central portion of the structure upstream from the cylinder lied, in practice, at the free-surface level.

Figure 23 also shows the presence of structure 4 (see also at

Figure 19 and

Figure 20) as characterized—this time—by positive vorticity (

Figure 23a, the structure has broken-up in two parts), and, again, by a remarkable depth under the free-surface, in front of the cylinder (

Figure 23b).

Further, as a result of the computing procedures previously outlined (

Section 4.2), results in terms of proper orthogonal flow modes are presented. Top views (the free-surface level) of a reconstructed velocity field, based on the three most energetic eignefunctions of the decomposition, are shown in

Figure 24,

Figure 25,

Figure 26,

Figure 27 and

Figure 28, in terms of constant wave-flow

x-velocity, through a time interval equal to a wave period, in a comparison with the structure field (same computing subdomain).

Figure 24a (

$T=0.000$,

T is the wave period) shows that, at the beginning of the wave period (the oncoming wave is only slightly altered by the presence of the cylinder), the free surface is mainly irrotational. On the upstream side of the cylinder, the diffraction tracks are clearly detectable in terms of borders among structures, where track

a denotes the ascending portion of the wave, track

b denotes the wave crest, and track

e denotes the still-water level (note that, due to the particular nature of information incorporated in these figures, the track lettering is now different from that adopted in

Figure 18). Only very small zones of positive (reddish) spanwise vorticity are present, immediatly upstream from the cylinder, at the cylinder wall. Overall, the field is regular and symmetric.

Figure 24b shows the flow field in terms of KL modes. It can be first noted that there is no direct correspondence between the field visualized in

Figure 24a and that shown in

Figure 24b, meaning that the energy content of the flow distributes through the field with no direct correspondence with the topological-structure distribution

. The only similarity between the two visualizations is mirrored by the position of track

a, also indicated in

Figure 24b.

**Figure 24.**
Flow-field representation at $T=0.000$ (T is wave period), top view (free-surface level): (**a**) flow-structures (colored with spanwise vorticity); (**b**) surfaces of constant x-velocity from reconstruction based on the three most energetic KL modes.

**Figure 24.**
Flow-field representation at $T=0.000$ (T is wave period), top view (free-surface level): (**a**) flow-structures (colored with spanwise vorticity); (**b**) surfaces of constant x-velocity from reconstruction based on the three most energetic KL modes.

**Figure 25.**
Flow-field representation at $T=0.144$ (T is wave period), top view (free-surface level): (**a**) flow-structures (colored with spanwise vorticity); (**b**) surfaces of constant x-velocity from reconstruction based on the three most energetic KL modes.

**Figure 25.**
Flow-field representation at $T=0.144$ (T is wave period), top view (free-surface level): (**a**) flow-structures (colored with spanwise vorticity); (**b**) surfaces of constant x-velocity from reconstruction based on the three most energetic KL modes.

Further upstream from track

a, in

Figure 24b, other tracks are present (oblique arrows) that actually represent borders between flow structures with high kinetic-energy content. A further inspection to

Figure 24b shows that the field upstream from the cylinder can be seen as divided in two zones, namely a first zone right in front of the obstacle, characterized by remarkably-small flow scales, and a second zone characterized by larger scales. A flow-mode border is also detectable in

Figure 24b, between these two zones of high-kinetic-energy scale-organization. In

Figure 25a,b the fields at

$T=0.144$ are represented.

Figure 25a shows the evolution of the diffraction tracks with respect to the previous instant. On the upstream side of the cylinder, track

d denotes the descending portion of the wave, track

e is now interrupted by track

f, and track

g now denotes the presence in the field of the already mentioned rather complex wave trough. This latter

g-structure actually mirrors the previously-outlined vorticity-inversion phenomenon in front of the cylinder (this section). Also in this case, the only detectable similarity between the visualization of

Figure 25a and that of

Figure 25b is the position of track

d, also indicated in

Figure 25b. Again, further upstream from track

d, in

Figure 25b, other tracks are present (oblique arrows), representing borders between flow structures with high kinetic-energy content, and a further inspection to

Figure 25b shows that the field upstream from the cylinder is also divided in two zones, i.e., a first zone characterized by small flow scales, and a second zone characterized by larger flow scales. A flow-mode border also appears in

Figure 25b, between these two zones of high-kinetic-energy scale-organization. In

Figure 26a,b the fields at

$T=0.500$ are represented.

Figure 26a shows again the evolution of the diffraction tracks with respect to the previous instant. On the upstream side of the cylinder, tracks

f and

g have in practice collapsed into a unique structure (now approaching the cylinder), tracks

h and

i denote the ascending portion of the wave (toward the crest), while track

k indicates the presence of a new crest in the domain. Also in this case, the only detectable similarity between the visualization of

Figure 26a and that of

Figure 26b, is the position of track

f, and, again, further upstream from track

f, in

Figure 26b, other tracks are present (oblique arrows), representing borders between structures with high kinetic-energy content. A further inspection to

Figure 26b shows that the field upstream from the cylinder is again divided in two zones, a first small-scale zone, and a second large-scale zone, and a flow-mode border is again detectable in

Figure 26b. It can be noted that the presence of the aforementioned discontinuity between smaller and larger high-kinetic-energy flow scales (and mirrored by the presence of a distinct flow-mode border) lasts from

$T=0.000$ up to

$T=0.500$ i.e., in practice, along the whole wave-to-cylinder approaching phase. In

Figure 27a,b the fields at

$T=0.750$ are represented. In

Figure 27a, on the cylinder upstream side, track

i clearly mirrors the interaction between wave and cylinder, while the wave-crest track

k is approaching the cylinder, assuming, at the same time, a complex configuration in front of the cylinder. In this case, the only detectable similarity between the fields represented respectively in

Figure 27a,b, is track

i, while, in

Figure 27b, other tracks are present (oblique arrows), always representing borders between flow structures with high kinetic-energy content. Differently from the previous instants, it clearly appears from

Figure 27b, that no more discontinuities between smaller and larger high-kinetic-energy flow scales are present, and the field of high-energy scales continuously evolves from the center to the periphery of the domain, in forming an elongated pattern. In

Figure 28a,b the fields at

$T=1.000$ are represented. In

Figure 28a, on the cylinder upstream side, the crest-track

k is now approaching the cylinder, while tracks

l and

m denote the descending portion of the wave. The configuration of the high-energy scales of

Figure 28b is similar to that of the previous instant, where the similarity between the two fields can be now recognized in the position of track

k.

Overall, the flow fields (actually the iso-u-fields) that have been reconstructed by using the three most energetic eigenfunctions of the KL decomposition, further show that the borders of the flow structures are actually borders among high-energy flow structures of tubular nature, the latter appearing—with respect to their energy content—the most significant type of viscous-flow structures that have been found in the wave-cases considered in the present work.

**Figure 26.**
Flow-field representation at $T=0.500$ (T is wave period), top view (free-surface level): (**a**) flow-structures (colored with spanwise vorticity); (**b**) surfaces of constant x-velocity from reconstruction based on the three most energetic KL modes.

**Figure 26.**
Flow-field representation at $T=0.500$ (T is wave period), top view (free-surface level): (**a**) flow-structures (colored with spanwise vorticity); (**b**) surfaces of constant x-velocity from reconstruction based on the three most energetic KL modes.

**Figure 27.**
Flow-field representation at $T=0.750$ (T is wave period), top view (free-surface level): (**a**) flow-structures (colored with spanwise vorticity); (**b**) surfaces of constant x-velocity from reconstruction based on the three most energetic KL modes.

**Figure 27.**
Flow-field representation at $T=0.750$ (T is wave period), top view (free-surface level): (**a**) flow-structures (colored with spanwise vorticity); (**b**) surfaces of constant x-velocity from reconstruction based on the three most energetic KL modes.

**Figure 28.**
Flow-field representation at $T=1.000$ (T is wave period), top view (free-surface level): (**a**) flow-structures (colored with spanwise vorticity); (**b**) surfaces of constant x-velocity from reconstruction based on the three most energetic KL modes.

**Figure 28.**
Flow-field representation at $T=1.000$ (T is wave period), top view (free-surface level): (**a**) flow-structures (colored with spanwise vorticity); (**b**) surfaces of constant x-velocity from reconstruction based on the three most energetic KL modes.