Impingement of propeller jet on a vertical quay wall

Ocean Engineering 183 (2019) 73–86

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Impingement of propeller jet on a vertical quay wall Maoxing Wei a, b, Yee-Meng Chiew b, * a b

Ocean College, Zhejiang University, Zhoushan, 316021, China School of Civil and Environmental Engineering, Nanyang Technological University, Singapore, 639798

A R T I C L E I N F O

A B S T R A C T

Keywords: Propeller jet Impinging jet Turbulent flow field PIV POD

This study experimentally investigates the mean and turbulent flow fields of a propeller jet impinging on a vertical quay wall. Four impingement distances, namely, wall clearances (¼ longitudinal distance between the propeller and vertical wall), were used to examine their influence on the flow behaviors. In order to investigate the characteristics of the three-dimensional impinging jet, both the streamwise (jet central plane) and transverse (impingement plane) flow fields were measured for each impingement distance using the particle image velocimetry (PIV) technique. Based on the wall clearance, three regions are identified, namely free, impingement and wall jet regions. Additionally, the results show that the evolution of the impinging jet is governed by two mechanisms: jet diffusion and wall obstruction. Their relative importance under varying wall clearances was interpreted in terms of the mean flow patterns and energy dissipation. Moreover, the proper orthogonal decomposition (POD) method is employed to identify the dominant flow structures associated with these two mechanisms in terms of the POD modes and their corresponding dominant frequency, which in turn provides a quantitative means to single out the individual flow mechanism.

1. Introduction Ever since the Age of Discovery, marine transport has been under­ going continuous development to satisfy the ever-changing demands of the world trade. Permanent International Association of Navigation Congresses (PIANC, 2015) reported that during the last decade, the shipping industry development was mainly portrayed by an increase in vessel capacity, which inevitably is accompanied by the increase in engine power. As the most widely used device for ships’ propulsions (Wang et al., 2018), propellers produce a discharge of water with high kinetic energy which is referred to as the propeller jet (Wei et al., 2017). Due to its complex flow characteristics, the swirling propeller jet has attracted extensive attention of many researchers. A careful review of published literature reveals that most past studies are focused on free expanding propeller jet, i.e., an unconfined jet in the absence of boundary effects induced by seabed or quay structures (Albertson et al., 1950; Berger et al., 1981; Blaauw and Kaa, 1978; Felli et al., 2006, 2011; Felli and Falchi, 2018; Hamill and Kee, 2016; Hamill et al., 2015; Hamill, 1987; Hsieh et al., 2013; Lam et al., 2010; Verhey, 1983). Moreover, Lam et al. (2011) provided a thorough review of the empirical equations used to predict the velocity distribution within an unconfined propeller jet. On the other hand, the flow behaviors of a confined

propeller jet, which is more common in navigation channel and harbor basin, i.e., in close proximity to the seabed or quay walls, has not received the same kind of attention as its unconfined counterpart. Thus far, limited studies on this issue are those by Johnston et al. (2013) and Wei et al. (2017), who examined plane bed boundary effects on the evolution of the propeller jet. During vessel berthing and deberthing, the jet from the rotating propeller blades directly impinges onto the quay structures, either in the form of a vertical (closed quay) or slope wall (open quay), producing an impinging swirling jet. Consequently, a quay structure near a ship pro­ peller often undergoes scouring, which is a growing concern as it may result in structural instability or even failure. PIANC (2015) documented that ship propeller-induced jet flow is the main cause of local scour that forms around quay structures in harbor basins. de Gijt and Broeken (2005) stated that the propeller-induced scour hole might reach several meters in depth if the bed material is sand or non-cohesive soils with low undrained shear strength. Hamill et al. (1999) experimentally investi­ gated the propeller-induced scour hole around a vertical quay wall and found that due to wall confinement, the maximum scour depth was significantly increased compared to that of an unconfined propeller scour, especially for small wall clearances. Wei and Chiew (2017, 2018) studied propeller scour around an open quay with a slope and reported

* Corresponding author. E-mail address: [email protected] (Y.-M. Chiew). https://doi.org/10.1016/j.oceaneng.2019.04.071 Received 29 November 2018; Received in revised form 18 March 2019; Accepted 22 April 2019 Available online 16 May 2019 0029-8018/© 2019 Published by Elsevier Ltd.

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Ocean Engineering 183 (2019) 73–86

that the maximum scour depth can be either increased or decreased depending on the varying toe clearance (¼ longitudinal distance be­ tween propeller face and slope toe). It may, therefore, be inferred from these studies that both the quay type and clearance have a close rela­ tionship with the impinging propeller jet and resulting scour depth. However, the underlying scouring mechanisms still have not been fully understood due to the lack of a detailed measurement of the impinging flow field. Although impinging jets have been widely studied due to its extensive industrial applications such as cooling, heating and drying surfaces (Nozaki et al., 2003), most of them were focused on turbulent heat transfer. Felli et al. (2010) conducted an experimental study on swirling jet impingement by using a ducted propeller, which is often used in the jetting industry. However, for ships, especially those equipped with large propellers, it is the non-ducted propeller that is commonly used to gain higher speeds. For a non-ducted propeller jet, its slipstream is without the confinement of a shroud and likely will feature different characteristics from a ducted one. To the authors’ best knowledge, studies on a ship propeller jet impingement upon a quay structure and its contributions to the associated scouring action have not been previously addressed. Since the impinging jet flow is the actual driving force to the potential scouring action, this study is focused on examining the three-dimensional flow behaviors of an impinging swirling jet generated by a non-ducted ship propeller with a simple configuration of a vertical quay wall on a plane bed. The mean and turbulent characteristics of the flow fields in both the streamwise and transverse planes are presented and discussed in terms of the observed flow patterns, based on which a brief discussion of its implications on the related scouring mechanism is given. Furthermore, the proper orthogonal decomposition (POD) method is applied to extract the dominant flow structure in the streamwise plane. With a full picture of the development of the impinging propeller jet, the results of this study are expected to provide an improved insight into the associated scouring mechanisms in future studies and to facilitate related numerical modelings.

(height ¼ water level) � 60 cm (width ¼ flume width) � 2cm (thick­ ness). A photo of the experimental model setup is shown in Fig. 1. With the primary objective of examining the wall-impingement effect on the development of a propeller jet, the tests were performed at a constant propeller rotational speed of n ¼ 545 rpm (revolution per minute) with four wall clearances (i.e., Xw ¼ Dp, 2Dp, 3Dp, and 4Dp). Each test was carried out under the bollard pull condition in still water with a depth of 0.6 m, and the propeller axis was located at 0.3 m depth. Hsieh et al. (2013) investigated a free expanding propeller jet in the same flume and suggested that the boundary effects of such a set-up (free surface, bot­ tom, and sidewalls) have negligible influence on the development of the jet. The glass bottom and sidewalls of the flume enable optical obser­ vations through the use of the particle image velocimetry (PIV) tech­ nique. The PIV system comprises a 5 W air-cooling laser with a wavelength of 532 nm as the light source and a high-speed camera. The high-speed camera used (Phantom Miro M-120 with a Nikkor 60 mm f/ 2.8 prime lens) has a maximum resolution of 1920 � 1200 px2, 12-bit depth and over 1200 maximum fps (frame per second) at 1152 � 1152 px2. Aluminum particles with d50 of 10 μm and specific density of 2.7 were used as seeding particles, whose settling velocity is estimated to be 92.6 μm/s using Stoke’s law and is negligible compared with the pro­ peller jet velocity. The beam emitted from the laser source passed through the optics, resulting in a laser light fan of 1.5 mm thickness. To avoid the undesirable effect due to possible free surface fluctuation, the laser sheet was cast upward through the glass bottom of the flume. A Right-Handed Coordinate System (o-xyz) is adopted in this paper, in which the origin is located at the center of the propeller disk. The xaxis is streamwise-oriented along the bottom centerline; the y-axis along the spanwise-oriented towards the starboard and the z-axis along the upward vertical. Accordingly, the velocity components (u-w) and (v-w) correspond to the (x-z) and (y-z) planes, respectively. To obtain the three-dimensional characteristics of an impinging swirling jet, both the streamwise central plane (y ¼ 0) and the transverse impingement plane (x ¼ Xw) were measured using the planar PIV. For the streamwise measurement, the laser sheet was set to align with the propeller rotation axis as shown in Fig. 2(a); for the transverse measurement, the laser sheet was set to be immediately upstream of the vertical quay wall and perpendicular to the propeller axis, and the field of view (FOV) was captured through a 45� mirror located downstream of the transparent vertical wall as shown in Fig. 2(b). The streamwise and transverse planes were measured at 300 fps and 1000 fps, respectively, and over 18,000 and 25,000 frames were recorded for each case in these two planes. Accordingly, the total measuring time for the streamwise and transverse planes are over 60 s and 25 s, respectively, which are long enough for capturing sufficient cycles (2725 and 1135) of the periodic flow gener­ ated by the 5-bladed propeller rotating at 545 rpm. For image postprocessing, the Laplacian filter function is used to reduce light reflec­ tion/scattering by virtue of its ability to filter out low-frequency signals and enhancing high-frequency signals. Consequently, time-series vector calculations were performed in Davis 8.4.0, in which a multi-pass iter­ ation with window sizes from 64 � 64 px2 to 32 � 32 px2 was adopted in the cross-correlation analysis. The size of the measured field of view and corresponding spatial resolution are tabulated together with the test conditions in Table 2. The efflux velocity (Uo) is defined as the maximum velocity taken from the mean velocity distribution across the initial efflux plane (Ryan, 2002). It must be stated that since the propeller blades periodically pass through the efflux plane, it is difficult to extract any meaningful data there. As a result, the efflux velocity values given in

2. Experimental setup The tests in this study were conducted in a recirculating flume with dimensions of 11 m length, 0.6 m width and 0.7 m depth in the Hy­ draulic Modeling Laboratory, Nanyang Technological University. The propeller model used is a five-bladed Wageningen B-Series propeller with an overall diameter (Dp) of 7.5 cm and a hub diameter (Dh) of 1.0 cm. The geometrical properties of the propeller are given in Table 1. Although nearly all commercial vessels have a central rudder behind the propeller, which can split the flow into an upper and lower jet (Roba­ kiewicz, 1966; Fuehrer et al., 1981, etc.), this study, for the sake of simplicity as per many recently published articles in this area (Hamill and Kee, 2016; Wei and Chiew, 2017, 2018; etc.), has not included a rudder. Its inclusion clearly would have provided a more realistic behavior of the flow field of the propeller wash and should be included in future researches. Consequently, in this study, the final design of the propeller rig simply consists of a single screw propeller model. The propeller rig was mounted on an appropriately designed movable car­ riage that spans transversely across the flume and could be moved along the longitudinal rail located on the sidewalls of the flume. In such a way, the propeller was able to operate at different clearances from the model quay wall, namely the wall clearance Xw, which is defined as the lon­ gitudinal distance between the propeller face and vertical wall. The quay model was built using an acrylic plate with a dimension of 60 cm Table 1 Propeller characteristics. Propeller diameter, Dp (cm)

Hub diameter, Dh (cm)

Blade number, N

Pitch ratio, P0

Blade area ratio, β

Thrust coefficient, KT

Toque coefficient, KQ

Advance ratio, J

7.5

1.0

5

1.04

0.62

0.46

0.06

0 (bollard pull)

74

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Ocean Engineering 183 (2019) 73–86

Fig. 1. Photograph of the experimental model setup.

Table 2 are taken as the maximum axial velocity in the vicinity of the actual efflux plane. Following Blaauw and Kaa (1978), the Reynolds U D

impingement region possesses a relative high-pressure gradient that deflects the flow direction to be parallel to the wall in the resulting wall jet region (Beltaos, 1976). Similarly, as a propeller jet impinges onto a vertical wall, it is expected that two mechanisms would govern the evolution of the jet, i.e., jet diffusion (dominant in the free jet region) and wall obstruction (dominant in the impingement region). Due to the presence of the hub at the center of the propeller disk, the free expanding (i.e., unconfined) propeller jet is characterized by a double-peak (off-center) profile in the zone of flow establishment (ZFE) and a single peak at the center in the zone of established flow (ZEF). At the small wall clearance of Xw ¼ Dp, Fig. 3(a) exhibits a noticeable “spread-out” feature of the double-peak profile (hereinafter referred to as the upper and lower jet streams) at the immediate near wake region downstream of the propeller face. It may be inferred that the influence of the wall obstruction has extended to the propeller face, leaving no space for the free jet region to form. Following the classification of Beltaos (1976), the impingement region under the current context could be further divided into a pair of spread-out jet streams and a triangle-like recirculating zone (denoted by the red dot line in the lower row of Fig. 3) between the spread-out streams, in which two recirculating eddies with opposite directions form on both sides of the centerline of propeller axis (z/Dp ¼ 0), as is clearly illustrated by the streamlines in Fig. 3(a). It should be noted that these two eddies are not perfectly symmetrical about the centerline, which may be attributed to the instability of the hub vortex. This will be further discussed in a later section by means of the transverse flow results. When the two spread-out jet streams impinge and attach onto the vertical wall at z/Dp � � 0.7, they resemble two wall jets, one moving upward away from, and the other downward towards, the bed [see Fig. 3(a)]. Similarly, as the wall clearance increases to Xw ¼ 2Dp, Fig. 3(b) shows that wall obstruction still dominates the jet evolution since the upper and lower jet streams immediately spread out from the outset and no sign of the free jet region is present. Due to the larger wall clearance, Fig. 3(b) reveals a larger (triangular) recirculating zone that pushes the wall jet region farther upward/downward toward z/Dp ¼ ~ � 1. When the wall clearance is increased to Xw ¼ 3Dp, Fig. 3(c) shows that jet diffusion appears to dominate wall obstruction because the double-stream configuration (red contour) is able to take on a free jet pattern, as those presented in Hsieh et al. (2013) and Wei et al. (2017). Moreover, unlike the development observed in Figs. 3(a) and (b), Fig. 3(c) shows that the size of the

nD L

numbers of the jet flow (Reflow ¼ 0ν p ) and propeller (Reprop ¼ νp m ) also are calculated and included in Table 2, in which ν is kinematic viscosity of water and Lm is the characteristic length dependent on the propeller � � �� 1 geometry defined as Lm ¼ βDp π 2N 1 DDhp . 3. Time-averaged mean flow patterns 3.1. Streamwise flow field Fig. 3 shows the comparison of the time-averaged mean streamwise velocity fields at the streamwise central plane (y ¼ 0) under varying wall clearances. For better understanding, the results are presented in two ways: (1) the upper row presents the contour map of the in-plane mean velocity with the superimposed streamlines; (2) the lower row shows the vector plot, where the magnitude and direction of the velocity vector are qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ðu2 þ w2 Þ and arctanðw=uÞ, respectively. For easy comparison be­

tween different sub-plots, the same scaling of color bar and vector length are used for plots of contour and vectors, respectively. In addition, the loci of the local maximum velocity in the upper and lower jet streams are also depicted in the vector plot by a red solid line, representing the trajectory of each stream. The recirculating region associated with the reverse flow (u < 0) is enveloped by a red dash line (i.e., the contour line of u ¼ 0). Furthermore, since the horizontal and vertical axes have the same scale, the figure is undistorted and the true shape of the jet is preserved; the spread-angle of the jet is thereby illustrated by two blue dash lines, which will be discussed later. The contour map in Fig. 3 shows that the wall effect on the flow structure is clearly discernible as the propeller jet is placed near a ver­ tical wall and impinges onto it at a right angle. Consequently, the measured efflux velocity and the associated Reflow are significantly reduced in the near clearance cases of Xw ¼ Dp and 2Dp, although Reprop was kept constant at the same rational speed (see Table 2). Beltaos and Rajaratnam (1974) studied the impingement of a turbulent jet from a nozzle onto a solid surface and identified three regions, namely free jet, impingement and wall jet regions. The free jet region exists at some distance from the wall where wall perturbation is negligible, and the 75

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Fig. 2. Experimental setup of PIV measurement: (a) streamwise measurement configuration and (b) transverse measurement configuration. Table 2 Summary of experimental conditions. Test no.

Dp (cm)

Xw/Dp

n (rpm)

Reprop

Uo (m/s)

Reflow

Measuring plane

Measuring location

Field of view (cm2)

Resolution (cm)

1 2 3 4 5 6 7 8

7.5 7.5 7.5 7.5 7.5 7.5 7.5 7.5

1 1 2 2 3 3 4 4

545 545 545 545 545 545 545 545

1.15 � 104 1.15 � 104 1.15 � 104 1.15 � 104 1.15 � 104 1.15 � 104 1.15 � 104 1.15 � 104

0.47 0.47 0.52 0.52 0.59 0.59 0.60 0.60

3.53 � 104 3.53 � 104 3.90 � 104 3.90 � 104 4.43 � 104 4.43 � 104 4.50 � 104 4.50 � 104

Streamwise (x-z) Transverse (y-z) Streamwise (x-z) Transverse (y-z) Streamwise (x-z) Transverse (y-z) Streamwise (x-z) Transverse (y-z)

y ¼ 0, central plane x ¼ Xw, impingement y ¼ 0, central plane x ¼ Xw, impingement y ¼ 0, central plane x ¼ Xw, impingement y ¼ 0, central plane x ¼ Xw, impingement

10.5 � 22 17 � 17.5 18 � 22 17 � 17.5 25.5 � 22 17 � 17.5 33 � 22 17 � 17.5

0.29 0.24 0.29 0.24 0.29 0.24 0.29 0.24

recirculating zone decreases with increasing wall clearance. This possibly is attributed to the fact that most jet energy has been dissipated due to jet diffusion before it impinges onto the wall, thus resulting in a less intense recirculating flow. This is further confirmed in Fig. 3(d) where the recirculating eddies have evidently diminished and replaced by a moderate flow deflection as inferred by the streamlines in Fig. 3(d). Additionally, at Xw ¼ 4Dp, the double-stream configuration shows no obvious spread-out feature until the jet approaches the near-wall region around x/Dp ¼ ~3. In order to quantify the development of the spread-out feature, the

plane plane plane plane

trajectories of the upper and lower jet streams are more explicitly illustrated in the lower row of Fig. 3 by a pair of curves (red line) rep­ resenting the loci of the local maximum velocity. By doing so, one can identify the angle between the spread-out potions of the loci curves. In general, Figs. 3(e) and (f) show that after the initial spread-out at the outset, the two streams undergo a further deflection, causing them to be parallel to the wall when the jet approaches the near-wall region (within ~0.5Dp upstream of the wall). Figs. 3(g) and (h), on the other hand, show that the loci curves exhibit a quasi-parallel (to each other) feature in the near wake (i.e., the free jet region) before spreading out in the 76

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Fig. 3. Comparison of streamwise time-averaged velocity fields with different wall clearances: (a,e) Xw ¼ Dp; (b,f) Xw ¼ 2Dp; (c,g) Xw ¼ 3Dp; (d,h) Xw ¼ 4Dp.

near-wall region (within ~1Dp upstream of the wall). It may, therefore, be concluded that in all the wall clearance cases, a common wallenforced spread-out is present in the near-wall region, which is directly related to the wall obstruction. The spread-out angle is also shown by drawing the tangential lines to the deflected curves at their inflection points. The so obtained wall-enforced spread-angle is found to be reduced from 156.0� at Xw ¼ Dp to 98.5� at Xw ¼ 4Dp, which quan­ titatively signifies a decreasing wall obstruction mechanism. In summary, based on the results of the streamwise measurement, three regions with distinctive flow behaviors can be identified under varying wall clearances, namely free jet, impingement (consists of spread-out streams and recirculating eddies) and wall jet regions. It must be stated that a quantified demarcation between each region, which likely is dependent on the wall clearances as well as the propeller type and rotational speed etc., is beyond the scope of this study.

although Fig. 4(b) is a 90� -offset of Fig. 4(a). Outside the central swirling region, the data reveal that the velocity distribution resembles that of the circular jet without swirl. As a result, the streamlines form a ring that divides the flow field into two distinctive portions, an “attracting-spiral” pattern (hub vortex) inside the ring and a “repelling-spiral” pattern (wall jet) outside the ring [See Fig. 4(c)]. In particular, the attracting-spiral streamlines are consistent with the counterclockwise swirling rotation and their “sink” overlaps with the lowest velocity deduced from the background contour. These two patterns are maintained until Xw ¼ 3Dp [see Figs. 4(c), (f), and 4(i)]. It should be stated that neither the streamline ring is perfectly circular [as compared to that of an uncon­ fined jet shown in Wei (2018)] nor the sink point located at the sym­ center. This is possibly attributed to the instability of the hub vortex as it impinges onto the wall as suggested by Felli et al. (2010), who reported that the intensity of the hub vortex spiral decreases with increasing wall clearance, which is also confirmed by the results of this study in that the two “interlocking spirals” progressively diminish, as shown in Figs. 4(d), (e), (g), and (h). When Xw ¼ 4Dp, Figs. 4(j) and (k) show no sign of the swirling spiral and the streamlines in Fig. 4(l) exhibit a “source flow” pattern which reflects a wall jet flow prevailing along the wall, as was also observed in the streamwise velocity results (see Fig. 3). Moreover, the right column of Fig. 4 also presents the impinging flow field from the viewpoint of the in-plane resultant velocity, in which the central low-velocity circle that represents the slow-moving stagnation region is in reasonable agreement with the projection of the recirculating zone that is observed in the streamwise flow field.

3.2. Transverse flow field The transverse mean velocity fields at the impingement plane under different wall clearance cases are depicted in Fig. 4, in which the left and central columns represent the mean lateral (v) and vertical (w) velocity components, respectively; the right column represents the in-plane qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi resultant velocity [ ðv2 þ w2 Þ]. Additionally, the vectors and stream­

lines are also superimposed in the contour maps of the individual and resultant velocity plots, respectively, in the figure. It is not hard to imagine that a horizontal circular jet without swirl will spread out radially in the form of concentric circles when it im­ pinges onto a vertical wall, and the decomposed velocity components will be axial symmetric about the centerline. However, this is not the case for a swirling impinging jet. It is interesting to note that as the swirling jet impinges on the vertical wall with a small clearance, the contour maps of the individual velocity components exhibit two “interlocking spirals” in the central core region as shown in Figs. 4(a) and (b). In this study, a left-hand propeller was used, which means that it rotates counterclockwise to provide the forward thrust. As such, inside the central swirling region, the data in Fig. 4(a) expectedly shows that the lateral velocity (v) is mainly positive and negative in the third (lower left) and first (upper right) quadrants, respectively. The same “inter­ locking spirals” feature is present in the vertical velocity component

3.3. Depth velocity profiles In order to further examine impingement effects on the individual velocity components, the vertical profiles of u, v and w at selected lon­ gitudinal locations (x ¼ Dp, 2Dp, 3Dp, and 4Dp) are extracted from Figs. 3 and 4 and shown in Fig. 5. It should be noted that the v-profiles are only available at the impingement plane of x ¼ Xw since the transverse flow fields were only measured there. For Xw ¼ Dp, Fig. 5(a) reveals that when the swirling jet impinges onto the wall, both the v- and w-profiles exhibit the characteristic of an undulating type S-curve. Specifically, for the v-profile, the data exhibits a zero-crossing point that dividing the profile into a negative upper and a positive lower half, which signifies a counterclockwise swirling that 77

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Ocean Engineering 183 (2019) 73–86

Fig. 4. Comparison of transverse time-averaged velocity components in impingement plane with different wall clearances: (a–c) Xw ¼ Dp; (d–f) Xw ¼ 2Dp; (g–i) Xw ¼ 3Dp; (j–l) Xw ¼ 4Dp.

78

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Fig. 5. Comparison of time-averaged velocity component profiles (u, v, w) at selected downstream locations with different wall clearances: (a) Xw ¼ Dp; (b) Xw ¼ 2Dp; (c) Xw ¼ 3Dp; (d) Xw ¼ 4Dp.

complies with the propeller rotation. One may therefore infer that the zero-crossing point of v-profile represents the center of the hub vortex whose location persists around z/Dp ¼ 0. On the other hand, the undu­ lating S-shape curve associated with the w-profile comprises three zerocrossing points with two turning points. The three points (S1, S2, and S3) along the line of w ¼ 0 form at z/Dp � � 0.6 and 0, in which S1/S3 are zero-up-crossing and S2 is zero-down-crossing. The “zero-up-crossing” is

defined as the point where the sign of w changes from negative (downward flow) to positive (upward flow) with increasing z/Dp, and vice-versa. Therefore, the former and latter points signify “diverging” and “converging” flow motions, respectively. Specifically, looking at the upper-half profile for example, as the upper jet stream impinges onto the wall, it leads to a flow divergence thereby producing an upward flow and a downward flow at the upper zero-up-crossing point (S1). This 79

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downward flow meets with the other upward flow that is affected by the lower jet stream at the middle zero-down-crossing point (S2), conse­ quently creating two recirculating eddies with opposite directions [see Fig. 3(a)]. It may therefore be surmised that the two zero-up-crossing points (S1 and S3) pinpoint the locations where the spread-out jet streams impinge onto the wall thereby causing the formation of recir­ culation eddies at the impingement plane. For Xw ¼ 2Dp, Fig. 5(b) shows that at x ¼ Dp, the u-profile indicates a symmetrical curve about the propeller axis (i.e., z/Dp ¼ 0) with a pair of positive double-peaks, representing the upper and lower jet streams and a negative central zone that signifies the reverse recirculating flow. Similar to that in Fig. 5(a), the w-profile at x ¼ Dp also features three zero-crossing points. However, the direction of the flow reverses in that the zero-down-crossing points form on the upper and lower sides, and the zero-up-crossing in the middle, of the vertical axis. Beyond the upper and lower zero-down-crossing points, the negative and positive vertical velocities reflect entrainment of the ambient stationary fluid into the jet streams; conversely, the w-profile between them present a centrosym­ metric form with a positive peak on the upper half and a negative peak on the lower half, whose locations coincide with the double-peak of the u-profile, indicating the spread-out trend of the jet streams. Upon impingement, Fig. 5(b) shows that at x ¼ 2Dp, the w-profile reverts to a curve that is similar to that of Fig. 5(a), while the v-profile shows an offcenter zero-crossing point, indicating that the swirling center has drifted due to the instability of the hub vortex. A comparison of Figs. 5(c) and (d) at x ¼ Dp reveals that both the uand w-profiles show reasonable agreement between these two wall clearance cases (Xw ¼ 3Dp and 4Dp), indicating negligible interference from the wall and thus the free jet region persists there. In particular, the u-profile shows the iconic double-peak feature as that of a free propeller jet. At the same time, it is worth noting that the w-profile features two zero-down-crossing points in the upper and lower sides and one zero-upcrossing point in the middle, similar to that in Fig. 5(b). As discussed earlier, the zero-up-crossing point (S2) reflects a “diverging” flow mo­ tion with upward w (positive) on its upper and downward w (negative) on its lower sides. This profile could be a reflection of the self-expansion of the jet in the absence of wall effects. As the jet moves to x ¼ 2Dp, Fig. 5 (c) shows that the u-profile exhibits a wider double-peak and a very small axial velocity at the center when compared to those of Fig. 5(d), which are respectively due to the spread-out effect and the intrusion of the recircuiting zone induced by wall obstruction. As for the w-profile, Fig. 5(c) shows that it has a similar shape as that observed in the free jet region at x ¼ Dp. However, it should be noted, based on its u-profile, that the jet has already entered the impingement region where the wall obstruction would have dominated jet development. Therefore, the diverging flow here could more likely be attributed to the influence of the wall rather than self-expansion. Farther downstream, Fig. 5(c) shows that when the jet impinges onto the wall at x ¼ 3Dp, the w-profile is found to revert to the S-curve resembling those at x ¼ Xw in Figs. 5(a) and (b), albeit with reduced magnitude due to energy dissipation during the jet diffusion phase. Meanwhile, the v-profile shows that the lateral velocity has been significantly decreased compared to its counterparts in Figs. 5(a) and (b), whereas the marginal native/positive value in the upper/lower halves still demonstrate a trace of the swirling rotation. For Xw ¼ 4Dp, Fig. 5(d) shows that the w-profile essentially loses all the undulating pattern at x ¼ 2Dp, indicating the cessation of free jet expansion. Afterwards, the three zero-crossing points, which signifies the spread-out motion of the jet streams, reappear at x ¼ 3Dp. One may therefore deduce that the jet undergoes a further expansion, which, however, is unlikely to be attributed to jet diffusion but rather wall ef­ fects. At this juncture, the wall obstruction also exerts a flattening effect on the double-peak feature of the u-profile, eventually causing it to attach to the u ¼ 0 line at x ¼ Xw. In addition, upon impingement at x ¼ Xw ¼ 4Dp, the w-profile shows an overlapped segment with w ¼ 0 (within z/Dp ¼ �~0.42), instead of three zero-crossing points as shown in Figs. 5(a)-5(c). This is because the magnitude of the jet streams has

been significantly reduced before they impinge onto the wall. The absence of a vertical flow here explains why a recirculating flow region is not formed between the deflected upward/downward wall jets [see Fig. 3(d)]. Furthermore, one may infer that if the jet was given more space to expand, the two jet streams would have eventually merged into a single peak before x ¼ 4Dp. This is consistent with the suggestion of Stewart (1992), who stated that an unconfined propeller jet would enter the zone of established flow at x/Dp ¼ 3.25. 4. Momentum exchange and energy dissipation 4.1. Reynolds shear stress and vorticity field in streamwise plane Fig. 6 shows the distributions of non-dimensional vorticity

(¼ωy Dp =Uo ) and Reynolds shear stress (¼ u’ w’ =U2o ) in the upper and lower rows, respectively; and both of them are superimposed with the mean velocity vectors. The out-of-plane vorticity is calculated by ωy ¼

∂w ∂x

∂u and U (¼ 0.60 m/s) is taken as the efflux velocity of the case of o ∂z Xw ¼ 4Dp (see Table 2), which is assumed to be free of the wall effect. As a propeller jet enters an open (quiescent) water body, a shear layer (referred to as the outer shear layer hereafter) forms at the interface between the high-velocity jet and the surrounding stagnant fluid. In the case of an impinging jet, due to the presence of the recirculating zone that forms between the spread-out jet streams discussed earlier, a sec­ ondary shear layer (referred to as the inner shear layer hereafter) de­ velops at the interface of the spread-out peaks and the inner reverse flow in the recirculating zone. Fig. 6 clearly shows the formation of intense vorticities both along the outer and inner shear layers. Comparing the vorticity contour and the superimposed vectors, it can be observed that the interface (white color) between the positive (red color) and negative (blue color) vorticities coincides with the loci of the peak velocity vec­ tor. Due to the dominant effect of wall obstruction when the wall clearance is small, Figs. 6(a) and (b) show that the intense vorticity occurs immediately in the near-wake region of the propeller since the recirculating zone has extended there [see Figs. 3(e) and (f)]. However, Fig. 6(c) shows that the vorticity along the inner shear layer only ap­ pears to be pronounced beyond x/Dp ¼ 1 as the double streams start to spread out there, which signifies the beginning of the impingement re­ gion. As the wall clearance further increases, Fig. 6(d) shows that both the inner and outer vorticity tend to fade out due to increasing jet diffusion and decreasing wall obstruction effects. A comparison of the distributions of Reynolds shear stress and vorticity fields reveals that the regions of intense Reynolds stress are at the same location as those with pronounced vortices; this reinforces the fact that the magnitude of the Reynolds shear stress is highly correlated with the presence of the vortices. Moreover, the superimposed vectors show that the positive Reynolds shear stress generally overlaps with a positive horizontal ve­ locity gradient (∂u/∂z) which in turn denotes a clockwise rotation of the vortex; in other words, a negative vorticity value, and vice-versa. However, a closer examination reveals that around the near-wall re­ gion where the jet streams impinge onto the wall [circled by red dash lines in Figs. 6(e), (f), and (g)], the sign of Reynolds shear stress seems to be opposite to the velocity gradient that can be deduced from the superimposed vector plot. This possibly is attributed to the fact that it is the higher vertical velocity gradient (∂w/∂x) that dominates the shear layer there as its horizontal counterpart has been dampened by the vertical wall. A similar phenomenon was also observed in the near wake region of a circular cylinder (Hsieh, 2008).

4.2. AKE and TKE distributions in streamwise and transverse planes The streamwise distributions of non-dimensional average kinetic energy (AKE) and turbulent kinetic energy (TKE) superimposed with the mean vector plot are presented in Fig. 7, where AKE ¼ 12 ðu2 þ w2 Þ=U2o

and TKE ¼ 34 ðu’2 þ w’2 Þ=U2o . It must be stated that only the in-plane 80

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Fig. 6. Comparison of streamwise time-averaged vorticity fields and Reynolds shear stress with different wall clearances: (a,e) Xw ¼ Dp; (b,f) Xw ¼ 2Dp; (c,g) Xw ¼ 3Dp; (d,h) Xw ¼ 4Dp.

velocity components are used here due to the two-dimensional nature of the planar PIV measurement. As a result, the TKE calculation is based on

stagnant surrounding fluid. A closer observation of the AKE contour and the superimposed velocity vectors reveals that high AKE region mainly concentrates in a narrow band along the loci of peak velocity vectors. Conversely, the lower row of Fig. 7 shows that the high TKE region is prominent in a wider band along the outer and inner shear layers as identified in Fig. 4. Specifically, at Xw ¼ Dp and Xw ¼ 2Dp, where the wall obstruction is dominant, Figs. 7(e) and (f) show that the higher TKE zone basically resides in the inner shear layer due to the higher velocity gradient between the jet stream and the inner recirculating flow. As the

the assumption of isotropic turbulence where v’2 ¼ 12 ðu’2 þ w’2 Þ (Chandrsuda and Bradshaw, 1981; Sciacchitano and Wieneke, 2016). As its mathematical expression implies, the AKE distribution shown in Fig. 7 is inherently consistent with that of mean velocity distribution in Fig. 3, whereas the square operator further distinguishes the jet streams, which contains most of the average kinetic energy, from the relatively

Fig. 7. Comparison of distributions of average kinetic energy (AKE) and turbulent kinetic energy (TKE) in streamwise plane with different wall clearances: (a,e) Xw ¼ Dp; (b,f) Xw ¼ 2Dp; (c,g) Xw ¼ 3Dp; (d,h) Xw ¼ 4Dp. 81

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relative importance of jet diffusion and wall obstruction changes at Xw ¼ 3Dp, Fig. 7(g) reveals that the higher TKE zone migrates to the outer shear layer with the spreading jet streams, meanwhile a pair of secondary TKE regions (circled by the red dash curves) associated with the inner shear layer departs from its outer counterpart to be attached to the shrinking recirculating zone close to the wall. At the same time, a low-turbulent core can be observed immediately downstream of the propeller hub, which again resembles a free propeller jet thus confirm­ ing the presence of the free jet region there. As the wall clearance further increases to Xw ¼ 4Dp, Fig. 7(h) shows that the secondary TKE region has diminished due to the absence of the recirculating flow and the major TKE is found to be concentrated along the outer shear layer in the free jet region. Furthermore, according to the sign of Reynolds shear stress and the vector gradient shown in Fig. 4, one can deduce that the TKE pro­

increases, Figs. 8(b)-8(d) depict a larger stagnation region while the AKE magnitude in the surrounding wall jet region has been significantly reduced. In general, one may surmise that the intensity of AKE within the wall jet region is inversely proportional to the dominance of jet diffusion. This finding explains the results of Hamill et al. (1999), in which they measured the local scour depth around a vertical quay wall in a closed quay and reported that the maximum scour depth mono­ tonically decreases with increasing wall clearances. However, in the case of an open quay, Wei and Chiew (2017) stated that the maximum scour depth exhibits a sine wave trend with increasing toe clearances. This possibly is due to the fact that jet impingement on a slope surface is a more complex phenomenon and needs further investigations. The lower row of Fig. 8, on the other hand, depicts the overall TKE in the transverse plane; it shows a consistent correlation with its counter­ part in the streamwise plane. In other words, both are highly dependent on the wall obstruction, whose dominance decreases with increasing wall clearances. A noticeable feature of Fig. 8(e) is that the distribution of high TKE exhibits a “circular dot”, in which the quasi-circular ring represents the shear layer that divides the transverse flow into two parts [see Fig. 5(c)]. The dot in the center overlaps with the core of the hub vortex as it clearly can be observed from the superimposed vectors. As the wall clearance increases, the ring-like TKE distribution progressively diminishes while the vortex core remains until Xw ¼ 2Dp. At the same time, it can be observed that, in general, the stagnation region appears to contain a higher TKE magnitude than its surrounding wall jet region. This indicates that the main source of TKE is the instability of the hub vortex, while the wall jet possesses the major AKE.

duction term is positive ( ρu’ w’ du dz > 0) along the shear layer, con­ firming an energy transfer from the mean (AKE) to the turbulent field (TKE) (Bailly and Comte-Bellot, 2015). It must be stated that a rigorous interpretation of turbulent energy flux should be based on a complete calculation of TKE budget terms which require concurrent measurement data of all three fluctuating components (u’, v’, w’). Therefore, it should be noted that the present discussion only considers the in-plane com­ ponents and further efforts are needed in future studies. Similarly, Fig. 8 depicts the distributions of non-dimensional AKE and TKE at the impingement plane (i.e., x ¼ Xw), in which AKE ¼ 12 ðv2 þ

w2 Þ=U2o and TKE ¼ 34 ðv’2 þ w’2 Þ=U2o . Unlike that observed in the streamwise plane in Figs. 7 and 8 reveals that the overall AKE magnitude in the transverse plane shows a decreasing trend with increasing wall clearance. Specifically, due to the confined jet diffusion at a small wall clearance at Xw ¼ Dp, Fig. 8(a) shows that upon impingement significant AKE has been converted to the wall jet region (in red), which could contribute significantly to the formation of a local scour hole if an erodible bed is present. Meanwhile, the quasi-circular region with low AKE in the center represents a stagnation region. As the wall clearance

5. Proper orthogonal decomposition (POD) analysis As discussed in the preceding sections, the flow field of the impinging swirling jet is governed by the jet diffusion and wall obstruction mechanisms under varying wall clearances. Meyer et al. (2007)

Fig. 8. Comparison of distributions of average kinetic energy (AKE) and turbulent kinetic energy (TKE) in transverse plane with different wall clearances: (a,e) Xw ¼ Dp; (b,f) Xw ¼ 2Dp; (c,g) Xw ¼ 3Dp; (d,h) Xw ¼ 4Dp. 82

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suggested that using PIV snapshots are insufficient to fully identify the dominant dynamic flow structures since it contains random fluctuations and thus may be a superposition of several structures. For this reason, in order to detect and distinguish the flow structures in complex flows, the proper orthogonal decomposition (POD) method often is applied to PIV data due to its ability to extract the dominant flow structure from the less coherent fluctuations (Yang et al., 2016). A detailed introduction of this method can be found in Sirovich (1987), Meyer et al. (2007), Yang et al. (2016), etc. In this study, the snapshot POD analysis proposed by Siro­ vich (1987) is applied in the streamwise flow field based on 1000 snapshots. In the following discussion, the terms of POD modes and coefficients are used. In brief, POD modes are the eigenfunctions of the velocity auto-correlation operator and form an orthogonal set (Feng

et al., 2011); and POD coefficients are determined by projecting the fluctuating part of the velocity filed onto the POD modes (Meyer et al., 2007). Therefore, the coefficient indicates a correlation between the corresponding POD spatial mode and the instantaneous flow field (Yang et al., 2016). Meyer et al. (2007) reported that the first few modes capture the most energetic and hence largest structures of the flow in terms of en­ ergy considerations. Fig. 9 shows the equivalent vorticity fields of the first four modes, in which the annotated percentage value represents the energy fraction of the associated POD mode compared to the total ki­ netic energy of velocity fluctuations. It should be noted that the super­ imposed vectors are only meant for revealing the equivalent vorticity pattern and their length does not have physical meaning until they are

Fig. 9. Comparison of equivalent flow structures of the first four modes under different wall clearances. First row: Mode 1; second row: Mode 2; third row: Mode 3; fourth row: Mode 4. 83

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combined with their corresponding POD coefficients used in the reconstruction of snapshots (Meyer et al., 2007). Corresponding to Fig. 9, the power spectra of the POD coefficients for each mode are shown in Fig. 10. In addition, the energy fraction and cumulative energy percentage also are included in the last row in Fig. 10. The former represents the percentage of the total kinetic energy of velocity fluctu­ ations associated with each POD mode, and the latter clearly shows that the first few modes represent significantly more energy than the rest. The following discussion will be mainly focused on the first four modes. Starting with the right column of Fig. 9, it can be seen that at a large wall clearance of Xw ¼ 4Dp, the vorticity fields of the first four POD modes reveal almost the same structure of a series of isolated tip vortices in a coherent pattern with alternating sign in the near wake region of the propeller within 0 < x/Dp < 2. Farther downstream, random flow vari­ ations with less coherence are observed in the near-wall region (2 < x/ Dp < 4), which likely is due to the random nature of the turbulent wake and the effect of wall perturbation. Meanwhile, all the first four modes at Xw ¼ 4Dp represent a comparable energy fraction of ~5%, indicating that they provide an equivalent contribution in constructing the flow field. It may, therefore, be inferred that these four modes together can be interpreted as different phases of the same phenomenon. In addition, it is interesting to note that the abovementioned coherent tip vortices pattern resembles those found in a free propeller jet (Di Felice et al., 2004); one, therefore, may infer that Modes 1–4 at Xw ¼ 4Dp describe a dominant structure that is associated with the jet diffusion mechanism. This structure is then found to diminish in the fifth mode and beyond (not shown in the figure), which seem to represent the random variation of the flow structure associated with the wall obstruction mechanism. Referring to the power spectrum plots, the right column (Xw ¼ 4Dp) in

Fig. 10 reveals a dominant pulse frequency of 9 Hz in all the first four modes. It is interesting to note that this frequency is exactly the same as the propeller rotating frequency (545 rpm �9 Hz), corroborating the inference that jet diffusion dominates the flow field at Xw ¼ 4Dp. Consequently, it may be inferred that the coherent vortices pattern (shown in Fig. 9), which corresponds to the dominant frequency of 9 Hz, is closely related to the flow structure associated with the free jet diffusion. Based on this hypothesis, one may surmise that at small wall clearance cases where the jet diffusion mechanism is less dominant, the coherent vortices pattern that signifying free jet diffusion would appear in the higher order modes, which is associated with lower relative energy. As the wall clearance decreases to Xw ¼ 3Dp, the third column of Fig. 9 shows that the “coherent vortices pattern”, which signifies free jet diffusion, only can be observed in Mode 3 [Fig. 9(k)] and becomes conspicuous in Mode 4 [Fig. 9(o)]. Accordingly, Modes 3 and 4 are in reasonable agreement with a secondary peak frequency of 9 Hz that stands out in their corresponding power spectrum plots in Fig. 10, suggesting that they are associated with the free jet diffusion mecha­ nism. On the contrary, the first two modes shown in Figs. 9(c) and (g) exhibit a relative random variation in the vorticity field, indicating a strong effect of wall obstruction on the unstable flow structure; not surprisingly, their corresponding power spectrum plot shows a single peak frequency close to zero, indicating that the periodic characteristic of the swirling flow has been totally pinned down in the dominant flow structure. This may suggest that the first two modes primarily describe the flow structure associated with the wall obstruction mechanism. Similarly, when the wall clearance is reduced to Xw ¼ 2Dp, a comparison between the second column of Figs. 9 and 10 reveals that the first two

Fig. 10. Power spectrum of first four POD coefficients (first four rows) and energy distribution of the POD modes (last row). 84

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modes correspond to a single peak frequency close to zero without showing any sign of the “coherent vortices pattern”. In fact, they are hardly recognizable even in Modes 3 and 4 although their corresponding power spectrum plots do exhibit a secondary peak frequency of 9.3 Hz, which is reasonably close to that of the free jet. Turning to the first column of Fig. 9 where the wall clearance of Xw (¼ Dp) is small, it is interesting to note that the frequency of 9 Hz associated with the free jet diffusion is shown as a secondary peak in all the first four modes, while the dominant frequency is found to be 3.9 Hz (Modes 1–3) and 5.1 Hz (Mode 4), which may be considered as the response frequency of the wall-impingement. A possible explanation of this dual-peak phenomenon is that within a small distance of x/Dp � 1, both the jet diffusion and wall obstruction influences are significant, thus their impacts are concurrently present in each mode, creating a strong coupling. However, since the free jet frequency holds the lower peak in all the modes, its corresponding flow structure, i.e., coherent vortices pattern, is still hardly identifiable in the POD spatial modes, which is similar as that of Modes 3 and 4 at Xw ¼ 2Dp.

The POD method was employed to identify the coherent flow structures in the streamwise plane. The data related to the first four POD modes of all the wall clearance cases reveal that the relative dominance between the jet diffusion and wall obstruction mechanisms may be quantitatively interpreted by the POD analysis in terms of the decom­ posed flow structures (POD mode) and the peak frequency obtained from the power spectrum of their corresponding POD coefficients. The dominant flow structure associated with jet diffusion is identified by a series of coherent tip vortices regularly distributed in the near wake region of the propeller, whose corresponding peak frequency is consis­ tent with that of the propeller rotation; while the wall obstruction ap­ pears as randomly distributed small-scale vortices without a dominant periodic characteristic originated from the swirling of the propeller rotation. Acknowledgments The authors acknowledge the financial support provided by Nanyang Technological University for this research.

6. Conclusions

Appendix A. Supplementary data

This paper presents an experimental study of a swirling propeller jet impinging onto a vertical quay wall using PIV techniques. Under varying wall clearances, the jet development was found to be governed by two mechanisms, namely jet diffusion and wall obstruction. As a result of the trade-off interaction between them, three flow regions can be identified in the streamwise flow field, namely free jet, impingement and wall jet regions, in which the impingement region consists of two flow patterns (spread-out streams and recirculating eddies), which serve as a transi­ tion between the free and wall jet regions. For the small clearance cases (such as Xw ¼ Dp and 2Dp in this study), the results show that the jet immediately enters the impingement region as the upper and lower streams spread out at the outset to create a recirculating zone between them. As the wall clearance increases, the free jet has more space to develop and tend to squeeze the recirculating eddies which eventually diminish at Xw ¼ 4Dp. The decreasing dominance of the wall obstruction is parametrized by the spread-angle identified from the trajectories of the spread-out jet streams. Results of the mean velocity and turbulence measurements at the impingement plane show that the recirculating zone of the streamwise flow field corresponds to a central swirling core dominated by the hub vortex. Inside the swirling core, the contour of the individual velocity component shows two “interlocking spirals”, while the corresponding streamlines exhibit a pattern of “attracting-spiral”. On the other hand, outside the core, the streamlines present a pattern of “repelling-spiral” that represents the wall jet region. The impingement effect on all the three velocity components is further examined with the help of their depth profiles at selected locations. Particularly, the vertical velocity wprofiles vividly illustrate the diverging motion of the upper and lower jet streams, which elucidates the underlying mechanism as either selfexpanding (jet diffusion) or forced-spreading (wall obstruction). A point-by-point comparison of the Reynolds shear stress and vorticity distributions in the streamwise plane highlights the correlation between them. The major momentum exchange switches from the inner shear layer (between jet streams and recirculating eddies) to the outer shear layer (between jet streams with surrounding water body) with increasing wall clearance, as the former and latter are respectively dominated by the wall obstruction and jet diffusion. As for the energy dissipation, the streamwise AKE was found to increase with increasing wall clearance while its transverse counterpart exhibits an opposite trend; which implies a reducing scouring capacity of the impinging jet at the base of the wall. Moreover, the streamwise TKE distribution reveals that turbulence production mainly takes place around the shear layer identified from the Reynolds stress and vorticity fields, whereas the transverse TKE is primarily associated with the instability of the hub vortex.

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