Comparison of the Flow around Circular and Rectangular Emergent Cylinders with Subcritical and Supercritical Conditions

Abstract

There are multiple initiatives aimed at strengthening coastal communities against tsunami disaster risks, such as growing vegetation belts, construction of embankments, moats, and different hybrid alternatives. To find a solution for strengthening the coastal buildings themselves, we firstly reviewed the flow phenomena around a single emergent (circular and rectangular) cylinder (case C1), which was considered as a piloti-type column under different Froude conditions, and evaluated the formation of surface bow-waves, hydraulic jump detachment, and wall-jet-like bow-waves. Secondly, the flow characteristics were investigated under the same Froude conditions with side-by-side two-cylinder (case C2) and four-cylinder (case C4) arrays in an open channel. Surface bow-wave length ($L_{Bw}$) increased by 7–12% over the rectangular cylinders (RCs) compared to the circular cylinders (CCs) with a subcritical flow. For the supercritical flow with a 1/200 bed slope, hydraulic jump detachment was observed in relation to the Froude number. The observed length of the hydraulic jump detachment ($L_{jump}$) varied between 3.1–8.5% and 4.2–12.9% for the CCs and RCs in the supercritical flow with a 1/200 bed slope. In addition, the wall-jet-like bow-wave height ($h_{jet}$) over the CCs was increased by 37% and 29% compared to the RCs with a supercritical flow and zero bed slope (orifice-type flow). For case C4, a hydraulic jump was observed for the supercritical flow over the horizontal channel bed. Finally, empirical equations were defined concerning the geometrical shape and arrangement based on the experiment data for the single and side-by-side configurations of the cylinders to validate the height of the wall-jet-like bow-wave as the most critical flow property.

1. Introduction

The Indian Ocean Tsunami (IOT) in 2004, Sulawesi Tsunami Indonesia (STI) in 2018, and Great East Japan Tsunami (GEJT) in 2011 destroyed numerous coastal structures. These were the most catastrophic failures ever recorded in human history [1,2,3]. After the GEJT, Japan’s Ministry of Land, Infrastructure, Transport, and Tourism (MLIT) divided the tsunami impact into two groups, referencing the recurrence interval: level 1 (recurrence interval of less than a hundred years) and level 2 (recurrence interval of several hundred to a thousand years) tsunamis [2]. Developing a mitigation strategy to safeguard coastal communities from level 2 tsunamis, which have the capability to overtop coastal embankments, has thus become essential. After the 2011 GEJT, the mitigation strategy and the proposed methodologies switched from a single defense system to a compound and/or hybrid mitigation system that integrates artificial and natural defenses against the hydrodynamic forces of the tsunami waves [3]. Numerous researchers have integrated different settings and numerical approaches into physical investigations. These include a combination of a coastal forest and moat [4]; double embankment systems [5]; a combination of an embankment, moat, and forest [6,7]; a mangrove forest [8]; a canal and dune [9]; a moat and/or canal behind the coastal embankment [10,11,12]; a multi-defensive line strategy [13]; vertical double-layer vegetation behind a coastal embankment [14]; and vegetation with different geometric conditions in the crest of a coastal embankment for energy reduction [15]. Moreover, coastal vegetation has been recognized as the most convenient natural approach that can be used to reduce the energy of tsunami waves [16,17,18]. The vegetation bio-shield can mitigate the hydrodynamic force of level 1 and level 2 tsunamis through a single mitigation approach. However, the only problem is the mitigation effect, as a single approach would be less beneficial during a disastrous situation [17]. Therefore, employing a hybrid defense system (HDS) or an alternative design for the coastal building structures must be investigated [1,2,3].

Cuomo et al. [19] examined the flow behavior around a building structure during a flood event. They highlighted the need for modern risk assessment techniques for research addressing critical infrastructures’ vulnerability. According to the computational and experimental assessments, the interaction of a tsunami traveling inland with the confined environment leads to a difference in flow depths and velocities, indicating the location affected from the Froude number [20,21,22,23]. Thus, an early study by Manawasekara et al. [23] highlighted the importance of an opening in the building frame for minimizing flood-induced loads, while Chock et al. [24] showed the need for a safe design against tsunami wave hydrodynamic loadings. Furthermore, the use of an internal break-away wall has been shown to be helpful in protecting against the collapse of buildings from increased loading during tsunamis and coastal flood catastrophes [25].

Consequently, during the hydrodynamic phase, when a break-away wall and internal partitions have eliminated the impact forces, the influences of the building design and its structural strength become critical [26]. Investigations in the aftermath of the 2018 STI also showed that elevated buildings, such as vertical parking garages and commercial buildings with multiple stories with porous construction, saved many lives [27]. Consequently, it would be advisable to elevate building structures by adding piloti-type columns up to a particular height, and the minimum height should be equal to the height of a single floor. Previous studies concentrated on the loading process over the building structures and the impact forces produced by tsunami-like surges. However, evaluating the flow behavior around these piloti-type columns in piloti-type building structures under different Froude conditions is compulsory to produce safer structures regarding hydrodynamic loading and the scouring process [23,24,28]. Yamamoto et al. [29] evaluated the flood damage throughout Japan and suggested that a new approach to develop a resilient environment in coastal and riverine regions by using piloti-type buildings would be the best solution for the adaptation measures. Dang-Vu et al. [30] also evaluated the influence of introducing additional shear walls, which can be constructed on the first-floor level, as a strengthening approach for piloti-type buildings subjected to earthquake loadings.

The hydrodynamic forces acting on an impermeable structural system in an open-channel flow are well-characterized under various flow conditions [27,31,32,33,34,35]. Mignot and Riviere [32] and Riviere et al. [35] focused on the bow waves’ key characteristics and the features of the flow behavior around a blunt rectangular obstacle in a supercritical flow. Mignot and Riviere [32] and Riviere et al. [35] also evaluated the formation of the detachment hydraulic jump and wall-jet-like bow-wave over the emergent obstacle’s front face while changing the flow depth, obstacle geometry, and channel slope. However, there is an unknown factor that is yet to be confirmed when the approach flow is a kind of tsunami inundation current with subcritical and supercritical characteristics through the coastal structures, especially with regard to piloti-type buildings, as Yamamoto et al. [29] and Samarasekara et al. [36] have explained.

Therefore, this study aimed to characterize the flow phenomena over single circular and rectangular emergent cylinders, assuming a piloti-type column, in a steady-state flow under different Froude number and channel conditions to address the subcritical and supercritical flow behaviors of an approaching tsunami flow current along the shoreline. Additionally, emergent circular cylindrical arrays were tested as piloti-type column groups, assuming they were connected to a coastal building under the same Froude conditions. The primary purpose of evaluating the flow fields over the piloti-type single cylinder or cylindrical arrays was to define the maximum possible spacing between the ground surface and the first ceiling level of the piloti-type building in order to propose more beneficial and safer designs for resilient structures as a resilience approach in the future.

2. Materials and Methods

2.1. Experimental Facility

Figure 1 shows the experimental setup arranged inside an open channel over a smoothly coated plywood bed, which was used to avoid frictional losses. The adjustable gate fixed upstream near the inlet regulated the flow with the different experimental characteristics to achieve supercritical flow conditions. The channel width ($B$) was 0.5 m, and the cylinders were selected with the parameters $D\le 60$ mm and $W\le 30$ mm to avoid lateral confinement effects from the side walls.

Table 1. Water depths, geometric characteristics, and Froude numbers considered in the experiment.

Description	Value	$\mathbf{Froude} \mathbf{Number} \left(\mathit{F}\mathit{r}\right)$
Diameter of the circular cylinder $D$ (mm)	32, 47, 60
Width of the rectangular cylinder $W$ (mm)	30, 20
Water depth $h_o$ (mm) (subcritical)	30.7, 37.0, 45.5, 49.1, 55.7	0.59, 0.63, 0.65, 0.67, 0.70
Water depth $h_o$ (mm) (supercritical with a 1/200 bed slope)	20.5, 26.7, 30.7, 38.1, 44.2	1.27, 1.37, 1.40, 1.41, 1.45
Water depth $h_o$ (mm) (supercritical with zero bed slope)	53.2, 45.5, 38.0, 30.7, 23.5	1.45, 1.52, 1.70, 1.74, 1.99

summarizes the preliminary characteristics for the experiments in this study.

The definitions of the notations shown in Figure 1a are as follows: $L$ is the working length of the channel (=12.5 m); $L_u$ is the distance to the emergent cylindrical obstacle from the vertical gate (=2.8 m); $L_B$ is the length of the fixed bed installed inside the channel (=6.5 m); $H_B$ is the height of the bed fixed inside the channel (=0.1 m); $Q_{in}$ is the inlet flow discharge; the upstream water depth ($h_o$) was measured at a distance equal to $10D$ (or $10W$) from the upstream face of the emergent cylinder; the mean velocity of the flow was obtained with $U(=Q_{in}/B{h}_o)$; $H$ is the depth of the channel (=0.4 m); $D$ is the diameter of the circular cylinder; and $W$ is the width of the rectangular cylinder. Changing the inlet discharge, channel slope ($S_o$), and cylinder dimensions ($D$ or $W$) made it possible to independently modify the approach flow conditions $h_o/D$ (or $h_o/W$) and the Froude number ($Fr$). The details for each experimental case considered in the experiments are presented in

Table 2. Experimental cases considered.

Case	Case Name
Case C1-CC	Single emergent circular cylinder
Case C1-RC	Single emergent rectangular cylinder
Case C2-1	Side-by-side two-cylinder array (center-to-center spacing $S_Y$ was 200 mm)
Case C2-2	Side-by-side two-cylinder array (center-to-center spacing $S_Y$ was 100 mm)
Case C4-1	Side-by-side four-cylinder array (spacings of $S_X$$\mathrm{and} S_Y$ were 100 mm and 100 mm)
Case C4-2	Side-by-side four-cylinder array (spacings of $S_X$$\mathrm{and} S_Y$ were 200 mm and 200 mm)

.

As illustrated in Figure 1b,c, the same procedure was followed with the side-by-side emergent cylinder array placed inside the experimental flume. Select side-by-side emergent cylinder arrays were produced using 32 mm diameter circular cylinders, which represented the piloti-type column pairs. The notations shown inside the cylinder arrays (case C2 and case C4) in Figure 1b,c indicate the points where water depths were observed at the defined locations along the center line $C_0$, $C_1$, and $C_2$.

2.2. Instrumentation and Flow Observations

In the experiment facility, an electromagnetic flow meter (EFM) attached to the closed circuit was used to measure the channel’s inlet flow rate. Flow was controlled using a computer-driven software package called HYDRA (v2.02), which connected to the circulating pump of the experimental flume. Water depth upstream of the emergent cylinder was measured using a point gauge, as shown in Figure 1, at a distance of $10D$ (or 10$W$) for both the circular and rectangular emergent cylinders. The uncertainty values regarding the discharge measurement ($∆Q_{in}$) of the flow meter in the channel and the water depth measurement upstream ($∆h_o$) were 0.005 L/min and 0.01 mm, respectively. The pointwise x-directional velocity in each case was recorded using an electromagnetic flow current meter (EFCM), which had an uncertainty ($∆U$) of about 0.01 cm/s. Three high-resolution cameras were used to capture the flow variation inside the experimental flume, as shown in Figure 1a. Finally, an image analysis technique was used to analyze the images to calculate the average water surface elevation over the emergent cylinders in each case under both the subcritical and supercritical flow conditions. To calculate the average water depth value, images were captured over a period of 30 s by the cameras from the respective direction such that 30 pictures were used in each case (Figure 1a).

2.3. Dimensional Analysis

The experiment cases tabulated in Table 2 represent piloti-type columns with different geometries and arrangements under subcritical and supercritical flow conditions over horizontal and sloping beds with steady-state flow (Figure 1). The following dimensional variables were used to define the flow: upstream water depth ($h_o$), upstream velocity ($U$), emergent cylinders’ dimensions (diameter $D$ or width $W$), properties of water (dynamic viscosity $\mu$, density $\rho$, and the surface tension with air $\sigma$), bed and obstacle roughness ($k_s$), and gravitational acceleration ($g$). The obstruction length was ignored for the emergent rectangular cylinders due to its lower contribution to the flow structures, whether subcritical or supercritical [31,32,37]. This research focused on flow features, such as surface bow-waves, hydraulic jump detachment, and wall-jet-like bow-waves. Regarding the flow circumstances, these flow characteristics were dimensionally defined to develop an empirical relationship based on multiple linear regression analysis [32,35]. Equation (1) was used to characterize the primary parameters selected for the dimensional analysis of the cylindrical column.

$$(\mathrm{flow} \mathrm{form}, \mathrm{wall}-\mathrm{jet})=f(h_o, h_F, D, U, \rho , \mu , g, \sigma , k_s)$$

(1)

Vaschy–Buckingham’s pi theorem was used to develop the empirical relationship, taking into account the fundamentals of the flow behavior (Figure 1 and Table 1). The most important flow type observed in the entire experiment series was the wall-jet-like bow-wave along the front face of the cylinder [32,35] under the supercritical flow conditions. Yamamoto et al. [29] and Dang-Vu et al. [30] have highlighted the importance of determining the level of elevation and its influence on the seismic characteristics when evaluating the stability of a building structure. Hence, this paper describes an empirical relationship by considering only this wall-jet-like bow-wave flow condition.

2.3.1. Single Emergent Cylinder

For the supercritical flow, Equation (2) describes the identified non-dimensional relationship for case C1 concerning the wall-jet characteristics in the present experiment. The height of the wall-jet ($h_{jet}$) was made dimensionless relative to the kinetic height, and it finally became $h_{jet}^{*}=h_{jet}/\left(U^2/2g\right)$. Furthermore, the formation of the wall-jet-like bow-wave ($h_{jet}$) was identical to the supercritical flow with horizontal and sloping beds. The only difference was that the wall-jet height over the front surface in the sloping channel bed was lower than the supercritical flow with a horizontal channel bed [38,39,40].

$$h_{jet}^{*}=f\left(Fr,Re,\frac{h_o}{D}\right)$$

(2)

2.3.2. Side-by-Side Two-Sided and Four-Sided Emergent Cylinder Arrays

When a subcritical or supercritical flow passes through an emergent cylinder array (cases C2 and C4), the primary parameter is the water depth in front of the cylinder array. Referring to Equation (3) for the cylinder array, the non-dimensional wall-jet height ($h_{jet}^{*}$) can be explained by the spacing ($S_Y$) in the transverse direction (Figure 1). The Froude number ($Fr$), upstream water depth ($h_o$), and water depth at the center of the Y-Y axis at the first column pair ($h_C$) were used to define the wall-jet height $\left(h_{jet}\right)$. The $h_C$ value was denoted as $C_o$ for case C2 and $C_1$ for case C4.

$$h_{jet}^{*}=f\left(Fr,Re,\frac{h_o}{S_Y},\frac{h_C}{S_Y}\right)$$

(3)

3. Results and Discussion

3.1. Subcritical Flow through an Emergent Cylinder

In the subcritical flow, the Reynolds number ($Re$) varied in the range of 8900–23,600. A surface bow-wave was generated at the front surface of the cylinder when the subcritical flow passed through case C1. For the flows where $Fr$ < 1, referring to Figure 2a, only the surface bow-wave was apparent in the subcritical regime, and the flow deflection originated upstream from case C1, as shown in Figure 2b. When the $h_o/D$ or $h_o/W$ ratio became smaller, a larger surface bow-wave in terms of flow depth ($h_o$) and geometry could be observed. Figure 2a shows both the CC and RC. The distance from the front face of the cylinder to the surface bow-wave toe ($L_{bw}$) was visible with the streamlined separation. The $L_{Bw}$ value increased with increasing water depth and the cylinder geometry ($D$ or $W$). The non-dimensional surface bow-wave length $L_{Bw}^{*}$ (=$L_{Bw}/D$ or $L_{Bw}/W$) is tabulated in

Table 3. Non-dimensional surface bow-wave length.

$\mathit{F}\mathit{r}$	${\mathit{L}}_{\mathit{B}\mathit{w}}^{\mathit{*}}$$(={\mathit{L}}_{\mathit{B}\mathit{w}}/\mathit{D}$$\mathbf{or} {\mathit{L}}_{\mathit{B}\mathit{w}}/\mathit{W})$
	32 mm dia.	47 mm dia.	60 mm dia.	30 × 40 mm	20 × 20 mm
0.59	0.56	0.42	0.37	0.91	1.03
0.63	0.57	0.49	0.42	1.07	1.18
0.65	0.70	0.55	0.50	1.25	1.30
0.67	0.79	0.58	0.52	1.37	1.43
0.70	0.87	0.69	0.61	1.54	1.53

for each geometry considered in the present experiment. When the subcritical flow passed through cases C2 and C4, similar behavior was observed over the front surface, such as the formation of a surface bow-wave. Figure 2c shows the surface bow-wave formation for case C2. As shown in Figure 2c,d, the parabolic pathway of this surface bow-wave merged at the center line along the flow direction when the flow passed through cases C2 and C4.

As shown in Figure 2b, when changing the geometry of the cylinder ($D$ or $W$), the length of the surface bow-wave ($L_{Bw}$) remains proportional to the $h_o/D$ (or $h_o/W$) [32,35,41]. When the diameter or width of the emergent cylinder increases, the surface bow-wave’s length ($L_{Bw}$) increases due to the blockage ratio.

The importance of determining the interaction in the formation of the bow-wave is, as McDonald [42] has explained, due to the fact that the bow wave can be characterized as steady, unsteady, or quasi-steady. In the subcritical flow condition, due to the breaking dynamics of the incoming flow imposed by the surface tension, when the obstacle becomes larger, a larger bow-wave is observed [41,42,43]. Hence, reducing the interaction in the bow-wave would be beneficial for lowering the scouring and stability near the structure in terms of the Froude conditions of the incoming flow [29,30,32,36,41,42,43,44].

3.2. Supercritical Flow through Emergent Cylinders

For the supercritical flow, the primary concern was the formation of the wall-jet-like bow-wave. When evaluating the wall jets’ behavior, there was a scale effect with respect to the field-measured data as the entire experimental setup considered the field-measured data after the post-tsunami survey to finalize the experiment scale [1,2,27,45,46,47]. Furthermore, based on the overall objective of the experiment, deciding on a minimum elevation for the building structure was a secondary solution following the defensive measures on the coastline in the case of a tsunami flow approaching [1,2,3,16,29,30,48]. Hence, the laboratory-produced wall-jet-like bow-waves seemed to be extremely similar to those seen in nature and represented in this study. However, when the upstream water depth ($h_o$) or the geometry of the cylinder ($D$ or $W$) decreased significantly, scale effects and the capillary effects became apparent. Therefore, the scale effect in the experimental setup was checked with the supercritical flow, both with the channel bed kept horizontal and with a slope, to confirm that pressure affected the wall-jet height and its characteristics, as discussed by Riviere et al. [35]. Furthermore, in order to prevent any lateral-confinement effects from the side walls of the channel, flows with high $h_o/D$ or $h_o/W$ were created in the channel by employing small-size CCs and RCs, as tabulated in Table 1 [35,44,49]. Moreover, extended droplets from the wall jet’s tip were seen in the vicinity of the front face of the emergent cylinder in the present experiment due to the capillary effects. These minor capillary and wall-jet changes did not affect the recorded time-averaged water depth in front of the emergent cylinders [32,35]. Furthermore, the capillary effect does not affect the flow transition from a surface bow-wave or detached hydraulic jump to a wall jet, even in laboratory-scale experiments [32,35,37,41,42,43].

3.2.1. Supercritical Flow over the Sloping Channel Bed

The bed slope of the channel was set to 1/200 to achieve the supercritical flow condition, replicating the tsunami flow over sloping ground [27,32,35,36,48]. Under the sloping bed condition, the Reynold’s number ranged between 10,700 and 35,700. Each physical model (case C1, case C2, and case C4) was kept inside the experimental flume to identify the flow structures and characteristics. As shown in Figure 3, when $Fr$ > 1 and the channel had a 1/200 bed slope, wall jets were demonstrated over the front face of the cylindrical columns in case C1 with increasing upstream water depth ($h_o$). In case C1, for the high $h_o/D$ or $h_o/W$ ratios corresponding to smaller cylinder size, the wall jet could be evacuated as a parallel jet where the flow was blocked by the CC or RC columns with a relatively small detached hydraulic jump [31,32,35,41,43]. However, for the low $h_o/D$ or $h_o/W$ ratios, which corresponded to larger cylinder dimensions in relation to the upstream water depth ($h_o$), wall jets occurred with a more significant detached hydraulic jump due to the slight increase in water depth near the upstream face resulting from the bluntness of the emergent cylinder.

As shown in Figure 4a,b for case C1 over a 1/200 sloping bed, with high $h_o/D$ and $h_o/W$ ratios, wall-jet-like bow-waves ($h_{jet}$) with a minor detachment length occurred (Figure 3). Furthermore, with low $h_o/D$ ratios (with 60 mm dia. CCs), small wall jets with a more significant detached hydraulic jump were observed [41,43]. The formation of the hydraulic jump detachment and the variations in its length ($L_{jump}$) for case C1 relative to the geometry ($D$ or $W$) for the 1/200 sloping bed are tabulated in

Table 4. Non-dimensional hydraulic jump detachment length ($L_{jump}^{*}$) and non-dimensional wall-jet bow-wave height ($h_{jet}^{*}$) relative to the geometry for case C1 for the supercritical flow when the channel bed slope equaled 1/200.

Non-Dimensional Property	Upstream Water Depth $\left({\mathit{h}}_{\mathit{o}}\right)$—1/200 Sloping Channel Bed (mm)					Cylinder Dimension
	20.5	26.7	30.7	38.1	44.2
$L_{jump}/D$	0.45	0.53	0.68	0.65	0.74	32 mm dia.
	0.31	0.37	0.45	0.52	0.58	47 mm dia.
	0.32	0.33	0.39	0.48	0.59	60 mm dia.
$L_{jump}/W$	0.69	0.81	0.93	1.08	1.11	30 × 40 mm
	0.83	0.92	1.07	1.20	1.39	20 × 20 mm
$h_{jet}/D$	0.64	0.93	1.33	1.29	1.47	32 mm dia.
	0.39	0.62	0.73	0.91	0.96	47 mm dia.
	0.33	0.49	0.60	0.73	0.81	60 mm dia.
$h_{jet}/W$	0.63	0.97	1.15	1.36	1.57	30 × 40 mm
	1.01	1.52	1.62	2.12	2.45	20 × 20 mm

. As shown in Table 4, the non-dimensional hydraulic jump detachment length $L_{jump}/D$ (or $L_{jump}/W$) relative to the geometry demonstrated higher values for the RCs than the CCs. From a general perspective, when $Fr$ > 1 for supercritical flows, the wall-jet ($h_{jet}$) height is the most critical parameter for the channel horizontal and 1/200 sloping bed cases as uplift forces can be created over the ceiling level of a piloti-type building [29,30,35].

Table 4 summarizes the non-dimensional wall-jet height $h_{jet}/D$ (or $h_{jet}/W$) relative to the geometry for the RCs and CCs. For cases C2 and C4, the hydraulic jump detachment resembled a surface bow-wave due to the increased water depth at the front. Figure 4c,d show the side views of the flow profile for cases C2 and C4, respectively. In cases C2 and C4, a relatively low wall-jet-like bow-wave height was observed, and the surface bow-wave was not similar to that in case C1 over the 1/200 sloping channel bed, as shown in Figure 4. Furthermore, the height of the wall jet ($h_{jet}$) slightly increased with an increase in the $S_Y$ spacing, as shown in

Table 5. Non-dimensional values for surface bow-wave length ($L_{Bw}^{*}$) and wall-jet height ($h_{jet}^{*}$) relative to the spacing of $S_Y$.

Non-Dimensional Property	Upstream Water Depth $\left({\mathit{h}}_{\mathit{o}}\right)$—1/200 Sloping Channel Bed (mm)					Case	${\mathit{S}}_{\mathit{Y}}$$\left(\mathbf{mm}\right)$
	20.5	26.7	30.7	38.1	44.2
$L_{Bw}^{*}$	0.17	0.19	0.22	0.24	0.26	C2-1	100
$h_{jet}^{*}$	0.21	0.33	0.37	0.48	0.58
$L_{Bw}^{*}$	0.07	0.09	0.10	0.12	0.13	C2-2	200
$h_{jet}^{*}$	0.13	0.17	0.19	0.26	0.32
$L_{Bw}^{*}$	0.17	0.18	0.20	0.23	0.26	C4-1	100
$h_{jet}^{*}$	0.19	0.33	0.37	0.50	0.61
$L_{Bw}^{*}$	0.08	0.10	0.12	0.13	0.14	C4-2	200
$h_{jet}^{*}$	0.11	0.18	0.21	0.27	0.33

.

Table 5 shows the average changes in the non-dimensional surface bow-wave length $L_{Bw}^{*}$ (=$L_{Bw}/S_Y$) and the non-dimensional wall-jet bow-wave height $h_{jet}^{*}$ (=$h_{jet}/S_Y$) calculated relative to the $S_Y$ spacing. Arnaud et al. [50] explained that, when a tsunami flows through a cylinder array (porous structure), due to the refraction and diffraction effects resulting from the adjacent cylinders in the array, the wall-jet-like bow-waves demonstrate minor changes, as seen in the results obtained for case C1.

3.2.2. Supercritical Flow over the Horizontal Channel Bed

The supercritical flow over the horizontal bed was achieved using a gate fixed at the channel’s inlet, as shown in Figure 1. The flow reached a supercritical condition after exiting from the gate, following a vena contracta at the gate exit, without any transition to the front face of the cylinder in each case. The water depth ($h_0$) and Froude number ($Fr$) used under this flow condition were selected in order to maintain the flow transition represented by the formation of a hydraulic jump downstream in all cases in the present experiment and to achieve the described fluid conditions [31,32,34,35]. This flow replicated the flow immediately after a tsunami crosses a shoreline and flows through the frontline buildings following the mitigation measures, attacking the frontal structures on its flow path. In this flow condition, the Reynolds number ($Re$) varied in the range of 20,400–44,800.

Figure 5 shows the results related to flow forms when $Fr$ > 1 and when the channel slope was kept horizontal with flow controlled by a sluice gate, as shown in Figure 1. Due to the supercritical condition and the kinetic energy of the flow, for both the high and low $h_o/D$ and $h_o/W$ ratios, wall jets were observed over the front face for case C1. When the flow reached the obstacle front, it was converted to a potential head from a kinetic head when $Q_{out}>Q_{in}$, as it was blocked by the CC or RC, as shown in Figure 6a [35,37,38]. Figure 6 shows that, when the supercritical flow reached the cylinder front in case C1, case C2, and case C4, its kinetic energy was converted to a wall-jet-like bow-wave and dissipated as it skirted the emergent cylinder through the formation of a lateral jet [32,35]. The height of the wall-jet bow-wave was several times that of the upstream water depth ($h_o$). Therefore, the non-dimensional wall-jet height $h_{jet}^{*}$ ($=h_{jet}/h_o$) was evaluated relative to the upstream water depth ($h_o$).

Table 6. Non-dimensional wall-jet height $h_{jet}^{*}$ ($=h_{jet}/h_o$) relative to the upstream water depth ($h_o$) for the supercritical flow over the horizontal channel bed for case C1.

Non-Dimensional Property	Upstream Water Depth $\left({\mathit{h}}_{\mathit{o}}\right)$—1/200 Sloping Channel Bed (mm)					Cylinder Dimension
	23.5	30.7	38.0	45.5	53.2
$h_{jet}^{*}$ $(=h_{jet}/h_o)$	2.92	2.19	1.88	1.48	1.29	32 mm dia.
	3.23	2.39	2.16	1.72	1.46	47 mm dia.
	3.48	2.66	2.28	1.79	1.50	60 mm dia.
	2.94	2.16	1.91	1.47	1.26	30 × 40 mm
	2.43	1.82	1.61	1.24	1.06	20 × 20 mm

shows the non-dimensional wall-jet height ($h_{jet}/h_o$) relative to the upstream water depth ($h_o$) for the supercritical flow over the horizontal channel bed.

Furthermore, we observed a slight, abrupt water depth increase over the RC compared to the CC at the toe of the wall jet, as shown in Figure 6a. For the CCs, when increasing the cylinder diameter, the contact area for the flow also increased, but, due to its shape, no abrupt increase in water depth occurred at the front except for the wall jet. As shown in Figure 6a, the supercritical flow in case C1 was skirted with a smaller aerated region over the CCs compared to the RCs due to its geometry [35]. A similar flow phenomenon occurred at the front face for the supercritical flow in case C2, as shown in Figure 6b. With the supercritical flow in case C4, a small bow-wave was formed over the cylinders’ front face in the second row, as shown in Figure 6c. Figure 6d shows the skirting of the wall jet around the circular cylindrical column surface and the dissipation from both sides as lateral jets in case C4.

3.3. Variation in the Depth of the Flow through the Side-by-Side Cylinder Array

Concerning the results of the present experiment, significant differences were observed in the flow characteristics inside the cylinder array, including in case C2 and case C4, for both the subcritical and supercritical flows. Among the Froude characteristics selected in the present experiment, supercritical flow with a horizontal channel bed was given extra attention due to the behavior of the flow in case C4. The flow depth in case C4 changed along the center line for the supercritical flow over the horizontal channel bed. In case C4, water depth drastically changed at the points $C_o$, $C_1$, and $C_2$ in relation to the spacing in the y-direction ($S_Y$) and x-direction ($S_X$), as shown in Figure 1c. The supercritical flow formed a hydraulic jump inside the cylinder array. This hydraulic jump formation would create an additional upthrust over the bottom floor of piloti-type structures during a supercritical overland tsunami flow [28,29]. Figure 7 shows the variation in the non-dimensional water depth relative to the upstream water depth measured at each point considered in case C4 (see Figure 1c).

The percentage increment in the water depth for the supercritical flow over the horizontal channel bed in case C4-1 at point $C_1$ ranged between 28.8 and 34.5%, while at point $C_o$, it ranged between 36.4 and 45.7%, and at point $C_2$, it ranged between 30.5 and 37.5%. Further, for case C4-2 with supercritical flow over the horizontal channel bed, the percentage increments in the water depth at points $C_1$, $C_o$, and $C_2$ were 27.7–30.5%, 32.4–42.0%, and 26.7–33.0%, respectively. This scenario showed the opposite results as for the supercritical flow over a 1/200 sloping channel bed. Due to this phenomenon over the sloping bed, the flow became hypercritical downstream following the cylinder arrays in cases C2 and C4 [39,40,51,52].

3.4. Development of the Conceptual Models and Explanation of the Flow Transition

Out of the three flow characteristics considered in the present experiment, the most critical flow type in all the cases was the formation of the wall-jet-like bow-wave over the front face of the emergent cylinders. Furthermore, in an elevated building, the wall-jet-like bow-wave produces an additional uplifting pressure force over the superstructure due to the water depth increase inside the cylinder array, as explained in Section 3.3 for the supercritical flow [28]. This increase in water depth inside the cylinder array (case C4) further increases the risk affecting the stability of the building structure [23,24,27,28,35,45,46,53,54]. Hence, developing an empirical equation to predict this wall-jet height would be necessary for sustainable structural design in the future.

3.4.1. Single Emergent Cylinders in the Supercritical Flow

A similar phenomenon as with the supercritical flow over the horizontal bed was observed for the supercritical flow with a 1/200 bed slope, but the height of the wall jet was lower than in the horizontal bed case due to the flow transition in front of the cylinder. In both the supercritical flows considered, the wall jet dissipated by skirting the cylinder surface and was converted to a lateral jet after reaching its maximum possible height. Figure 8a represents the time-average relative wall-jet height $h_{jet}^{*}\left[=h_{jet}/\left(U^2/2g\right)\right]$ as a function of the upstream Froude number observed over the emergent cylinders in case C1 for the supercritical flow with both horizontal and 1/200 sloping beds. As shown in Figure 8a, most of the values of the $h_{jet}^{*} \mathrm{w}\mathrm{e}\mathrm{r}\mathrm{e} > 1$. This was because the surface velocity over the front face of the cylinder was more significant than the mean upstream velocity ($U$). Utilizing Graf and Altinakar’s [52] explanation, the upper bound of the experimental values for the non-dimensional wall-jet height ($h_{jet}^{*}$) was equal to 2.071 ≈ (1.15/0.8)² in the present experiment, as shown in Figure 8a by the dashed line. In Figure 8a, open symbols refer to the supercritical flow over a 1/200 bed slope, and closed symbols refer to the supercritical flow with a horizontal bed in case C1. Equation (4) was developed to predict the dimensionless wall-jet height ($h_{jet}^{*}$) based on the present experimental results for the supercritical flow with a horizontal bed in case C1. Also, for the rectangular cylinders, term $D$ in Equation (4) is replaced as term $W$. Equation (4) was subsequently validated for the supercritical flow data observed in the 1/200 bed-slope case.

$$h_{jet}^{*}=1.72{\left(Fr\right)}^{-0.38}\ast {\left(\frac{h_o}{D}\right)}^{-0.33}$$

(4)

According to the t-test results, the p-values of the selected dimensionless parameters $Fr$ and ($h_o/D$) in Equation (4) were lower than 0.05, which meant they satisfied the null hypothesis. The values for the experimental dimensionless wall-jet height versus those for the predicted wall-jet height obtained with Equation (4) are presented in Figure 8b. They indicated that the selected parameters in Equation (4) explained all the response data around the mean. For the circular cylinders, the average percentage errors in the predictions of the non-dimensional wall-jet height using Equation (4) were −0.25% and −6.79% for the supercritical flow over a horizontal bed and over a 1/200 sloping bed, respectively. Each case’s correlation coefficient is also shown in Figure 8b.

Moreover, Equation (4) was tested to predict the wall-jet height over the emergent rectangular cylinders, and the correlation plots for the predicted versus experimental values for each case, along with the correlation coefficients, are shown in Figure 8c for both the horizontal and 1/200 sloping bed supercritical flow conditions in case C1. The applicability of Equation (4) for the prediction of the dimensionless wall-jet height for supercritical flow over horizontal and sloping beds was evaluated in terms of the percentage of error. The average error percentages were 0.27% and −7.51% for the supercritical flow over a horizontal bed and the supercritical flow over a 1/200 sloping bed, respectively.

Additionally, the empirical equation developed by Riviere et al. [35], adopted and as Equation (5) was tested with modifications in the present experiment for a circular and rectangular emergent cylinders in case C1. In Equation (5), $h$ is upstream water depth, $F$ is Froude number, and $R$ is obstacle width. To apply the present experiment, the term $R$ in Equation (5) is replaced by $D$ for the circular cylinders and replaced by $W$ for the rectangular cylinders in case C1, respcetively.

$$h_{jet}^{*}=1.7{\left(\frac{h}{R}\right)}^{-0.24}{F}^{[-0.24\left(h/R\right)-0.17]}$$

(5)

Referencing Equation (5), the correlation plot for the experimental and predicted dimensionless wall-jet height is shown in Figure 8d. The goodness of fit ($R^2$) value for the calculated dimensionless wall-jet height was 0.977 for the supercritical flow when the channel bed was kept horizontal and 0.954 for the supercritical flow when the channel bed sloped by 1/200. Furthermore, the average percentage errors in the prediction of the dimensionless wall-jet height using Equation (5) for the circular and rectangular emergent cylinders in case C1 were 3.24% and −3.76%, respectively.

3.4.2. Side-by-Side Emergent Cylinder Array in the Supercritical Flow

Similar to the explanation in Section 3.2 and Section 3.3, when water flowed through the cylinder arrays in cases C2 and C4, the formation of the wall jet depended on the parameters given in Equation (6). The parameters are also shown in Equation (4) for the formation of the wall-jet-like bow-wave for the supercritical flow in cases C2 and C4. The water depth between the cylinders ($h_C$) along the centerline was considered in relation to the spacing between the cylinders ($S_Y$), as shown in Figure 1b,c. Equation (6) was formulated using the dataset for the wall jet with a supercritical flow passing through the cylinder array on the horizontal channel bed. Then, Equation (6) was validated with the 1/200 sloping channel bed dataset to predict the dimensionless wall-jet height.

$$h_{jet}^{*}=3.89\ast {\left(Fr\right)}^{-2.15}\ast {\left(\frac{h_o}{D}\right)}^{-0.85}{\left(\frac{h_C}{S_Y}\right)}^{0.05}$$

(6)

Figure 9a,b show the correlation plots for the predicted wall-jet height compared to the experimental results for the supercritical flow with horizontal and 1/200 sloping channel beds for both cases C2 and C4. For each case, the goodness of the decision coefficient ($R^2$) is shown on the graph. According to the plotted graphs shown in Figure 9 for cases C2 and C4, the non-dimensional wall-jet height over the sloping bed was slightly underestimated. The average percentage errors for case C2 in the results predicted by Equation (6) were 2.1% and −7.8% for the horizontal and 1/200 sloping beds, respectively. Furthermore, the average percentage errors for case C4 in the results predicted by Equation (6) were 3.4% and −4.5% for the horizontal and 1/200 sloping beds, respectively.

4. Summary

This study primarily examined the characteristics of subcritical flow, supercritical flow with a horizontal bed, and supercritical flow with a sloping bed around single circular and rectangular emergent cylinders in an open channel. Furthermore, side-by-side circular cylinder arrays (Figure 1) with varying spacings, as shown in Table 1, were tested to evaluate the flow characteristics with the same Froude number conditions. Three types of flow were observed during the experiment: surface bow-waves, detached hydraulic jumps, and wall-jet-like bow-waves [32,35,41,43,44]. The surface bow-wave appeared over the front face of the cylinders due to the subcritical flow condition with the single (case C1) and side-by-side emergent cylinder arrays (cases C2 and C4). This was the reason for the streamlined curvature of the flow, which flowed around the cylinders (Figure 2), and the length of the surface bow-wave ($L_{Bw}$) changed in relation to the geometry of the emergent cylinder. The development of the surface bow-wave length ($L_{Bw}$) could be altered by changing the $h_o/D$ (or $h_o/W$) ratio in relation to the Froude condition in the subcritical flow with a single emergent cylinder or the $h_C/S_X$ ratio for side-by-side cylinder arrays.

Among these three flow types, wall-jet-like bow-waves were given extra attention in terms of the Froude number and the channel conditions. The wall-jet-like bow-wave was observed in the supercritical flow when the channel bed was kept horizontal and when it was sloped. The supercritical flow with the horizontal bed reached the emergent cylinder without any transition, and on the sloping bed, due to the bluntness of the cylinder, there was a water depth increase upstream resulting from the detached hydraulic jump in addition to the wall jet observed [28,32,41,43]. A wall-jet-like bow-wave was observed for the supercritical flow when it passed through the side-by-side cylinder arrays [35,38,41,51,55]. However, the shape of the emergent cylinder may be changed to correct the existing findings quantitatively. Placing more streamlined obstacles in contact with a supercritical flow, such as bridge piers or piloti-type columns with circular or rectangular cross-sections, may enable flow evacuation with the presence of a wall-jet-like bow-wave, as previously explained.

Following the post-tsunami surveys, the experimental setup employed field-measured data to determine the minimum elevation for building structures above ground level that would maintain a free surface beneath them as a secondary solution following defensive measures on the coastline in the case of a tsunami approach [3,56]. Hence, the laboratory-produced wall-jet-like bow-waves seemed to be extremely similar to those seen in nature and represented in Figure 4 and Figure 6. However, when the upstream water depth ($h_o$) or cylinder geometry parameters ($D$ or $W$) decreased significantly, scale and capillary effects became apparent. The scale effect in the experimental setup was checked to confirm that the pressure affected the wall-jet height and its characteristics, as discussed by Riviere et al. [35]. Due to the capillary effects, extended droplets from the wall jet’s tip were seen in the vicinity of such obstacles. These minor capillary and wall-jet changes did not affect the recorded average water depth in front of the cylinders. Such capillary effects are insignificant at the actual scale relevant to hydraulic engineering applications. They do not affect the form of the transition from a wall jet to a surface bow-wave or a detached hydraulic jump, even in laboratory-scale experiments [32,35,44].

As seen in the sloping roadways during urban floods [26,53] or the overland flow following a tsunami catastrophe as a tsunami run-up, different obstacles may respond in different ways to a supercritical flow [32,35]. Inland flow eventually reaches a supercritical regime after a tsunami and interacts with concurrently emergent man-made (buildings, automobiles, etc.) and natural (hills, etc.) obstructions [45,46,54,56]. The supercritical inflow must, therefore, skirt these obstacles on all sides since they are impermeable obstructions mounted on the base/bed and emerging from the free surface. This demonstrates that the primary contradiction may be explained by one aspect of the open-channel flow: its vertical confinement between the free surface and the bottom. In subcritical and supercritical flows, the flow may evacuate the constrained region between the bed and the free surface through a wall jet or a surface bow-wave/detached hydraulic jump [28,29,32,35]. Furthermore, without the need for a horizontal streamline curve, a supercritical flow workaround may occur beyond the free stream. However, sometimes, particularly when the cylinder/obstacle diameter ($D$) or width ($W$) are much greater than the upstream water depth ($h_o$), a wall jet may not always be able to clear the whole volume of the water blocked by the obstacle [28]. This phenomenon replaces the wall-jet bow-wave with a hydraulic jump detachment. However, to form a hydraulic jump detachment, the cross-stream width of the obstruction should be larger than the upstream water depth ($h_o$) in the supercritical flow [35]. For the supercritical flows, by altering the $h_o/D$ (or $h_o/W$) ratio, the Froude characteristics of the formation of the detached hydraulic jump and the wall-jet-like bow-wave can be adjusted.

However, the current findings may be quantitatively adjusted by modifying the geometry of the emerging cylindrical obstacles, including the single and side-by-side arrangements. The transition’s threshold ratio $h_o/D$ or $h_o/W$ is reduced with the Froude number, confirming the theoretical hypothesis that the change is triggered by mass conservation [32,34,35]. Concentrating on the wall-jet-like bow-wave, the empirical equations presented in this article show that mass conservation also impacts the height of the wall jet above the free surface flow, which was determined using $Fr$ and $h_o/D$ or $h_o/W$. Lastly, the wall jet showed height oscillations, and these oscillations were likely due to the reversed spillage over the upstream face of the obstacle and related to the rising and lowering water cycle. From a general perspective, the determination of the wall-jet height and its prediction using the empirical equations defined in this article will be beneficial for the identification of relationships between wall jets and the upstream water depth ($h_o$) in the construction of piloti-type buildings as a resilience approach against tsunami destructiveness in the future.

5. Conclusions

In the present study, we conducted a model-scale experiment to investigate the flow characteristics over emergent single and side-by-side cylinder arrays in an open channel with different Froude numbers and channel slopes. The Froude numbers lay within the subcritical to supercritical zones and different cylinder sizes and water depth ratios ($h_o/D$ or $h_o/W$) were used, including a side-by-side cylinder arrangement with different spacings in the x and y directions ($S_X$ and $S_Y$). The following conclusions were derived after completing a series of experiments in the present study:

• The height of the wall-jet ($h_{jet}$) was altered with the changes in the $h_o/D$ (or $h_o/W$) ratio in case C1 and with the changes in the $h_C/S_X$ ratio in the side-by-side cylinder arrays (cases C2 and C4) for both the supercritical conditions considered;
• The scale effects did not influence the wall-jet height ($h_{jet}$) in the laboratory-scale experiment, and the extended droplets observed at the wall jets’ tips due to the capillary effect also did not affect the time-averaged wall-jet height;
• For the supercritical flows, when the $h_o/D$ (or $h_o/W$) ratio was > 1, a wall-jet-like bow-wave could develop in front of the obstacle. Furthermore, when the $h_o/D$ (or $h_o/W$) ratio was <1, a distinct detached hydraulic jump could occur in front of the obstacle with different geometrical characteristics;
• For the supercritical flow, the flow depth could increase within the side-by-side square cylinder array (case C4) in relation to the x and y directional spacing in the cylinder array and the sloping characteristics of the experimental flume;
• The derived equations were used to predict the non-dimensional wall-jet height with the supercritical flow over the single and side-by-side cylinder arrays, which can be helpful in designing piloti-type building structures in urban coastal environments as a resilience measure against future tsunami risks.

In the present experiment, due to the lack of facilities, the $S_X$ and $S_Y$ spacings were limited to two cases for the side-by-side cylinder arrays to avoid sidewall effects from the experimental flume on the flow structure. Hence, additional research is needed with variations in the $S_X$ and $S_Y$ spacings to evaluate the flow phenomena inside the cylinder arrays further. Moreover, flow following a coastal embankment due to tsunami overtopping creates downstream subcritical and supercritical flow conditions. In addition, emergent single and side-by-side cylinder arrays can be tested with a flow following the overtopped coastal embankment.