LABORATORY STUDY OF DRAG AND INERTIA FORCES ON A BRANCHED CORAL COLONY OF ACROPORA PALMATA

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1 Proceedings of the 6 th International Conference on the Application of Physical Modelling in Coastal and Port Engineering and Science (Coastlab16) Ottawa, Canada, May 10-13, 2016 Copyright : Creative Commons CC BY-NC-ND 4.0 LABORATORY STUDY OF DRAG AND INERTIA FORCES ON A BRANCHED CORAL COLONY OF ACROPORA PALMATA JUAN DAVID OSORIO-CANO 1, ANDRES F. OSORIO 1, HOCINE OUMERACI 2 1 Research Group in Oceanography and Coastal engineering, OCEANICOS, Department of Geosciences and Environment, Universidad Nacional de Colombia at Medellín, Colombia, jdosori0@unal.edu.co, afosorioar@unal.edu.co ABSTRACT 2 Technische Universität Braunschweig, Leichweiß-Institute for Hydraulic Engineering and Water Resources,Germany, h.oumeraci@tu-braunschweig.de This study aims at improving the understanding of near-bed hydrodynamics in a coral reef by examining the drag and inertia coefficients of the species Acropora palmata. This coral species was selected for parameterization due to its capacity to survive in wave/current exposed areas with moderate to high energy environments, where its branches and the global structure of its colonies play an important role for wave damping. Small scale experiments were carried out for two different flow regimes: steady and oscillatory flow. For this purpose, a current flume as well as a wave flume were used. In-line forces, flow velocities and water surface elevations over 3-D models of A. palmata were measured for two main configurations: a single coral and a group of corals. The results show that under steady flow conditions, the drag coefficient (C D) can be well represented as a function of the Reynolds number by means of a power law equation, C D = are b + c (determination coefficient, R 2 >0.98). Also, under oscillatory flow conditions, for the studied rigid coral with flat branches, it is shown that the inertia force (F M) dominates over the drag component (F D), explaining up to 83% of the total force exerted on the coral structure. KEWORDS: Force coefficients, Wave energy, Drag force, Inertia force, Coral reefs. 1 INTRODUCTION Coral reefs not only provide food and shelter for at least one quarter of all ocean species, contributing to preserve biodiversity hotspots and sites for tourism activities, fishing and recreation (Burke et al. 2011), but also, they work as barriers that protect much of the world's coastlines against sea waves. The physical processes involved in dissipating wave energy for these reef systems are generally strongly influenced by steep slopes, heterogeneous bathymetry and large bottom roughness, which involve complex wave transformation processes, including refraction, shoaling, reflection and wave breaking accompanied by an enhancement of frictional energy dissipation (Monismith 2007; Young 1989; Lugo-Fernández, H. Roberts & Suhayda 1998). On the other hand, as waves move from deep waters over a steep reef-slope towards a flat reef-plateau, these wave transformation processes become highly non-linear (Jensen 2002). Wave transformation and wave attenuation have been studied primarily in the steep transition zone to the outer reef flat in laboratory models of fringing reefs (e.g., Gourlay 1994; Gourlay 1996; Gourlay & Colleter 2005; Massel & Gourlay 2000) or over natural fringing and barrier reefs (e.g., Hardy & Young 1996; Lowe et al. 2005; Young 1989; Lugo-Fernández et al. 1998). In terms of wave attenuation due to bottom friction, the most important aspects to be considered are related not only to the geometry or geomorphology of the reef profile, but also to the roughness and porosity of the reef surface. These aspects are highly correlated with the density of coral colonies and species distribution along the reef and have to be considered in order to understand their influence on wave attenuation. For example on reef flats with rough bottoms in shallow waters, frictional dissipation has been considered as a primary component of the total wave energy dissipation (Falter et al. 2004; Lowe et al. 2007; Lowe, Falter, et al. 2005; Nelson 1996). 1

2 Understanding the mechanics of the wave boundary layer is important to accurately model and predict near-shore currents that, in turn, affect coastal erosion, pollutant dispersal, and the health of coastal ecosystems. The wave flow over these boundaries has typically been approached considering homogenous bottom roughness. However, Pawlak & Maccready (2002) examined how oscillatory flows react to different roughness scales and showed that both the length scales and distribution of roughness elements introduce mechanisms that can enhance energy dissipation near the boundary. These results suggest that for an accurate determination of bed friction processes, a detailed characterization of bed roughness is needed along with effective parameterization schemes. Despite all the contributions to this field of research, parameterization of bed roughness is still not an exact science. Theory suggests that roughness length scales and distribution both play an important role in the mechanisms of wave energy dissipation (Nunes & Pawlak 2008). The understanding of the physical processes that occur over submerged reefs is still a research topic, particularly in near-shore environments with shallower water depths and larger bottom roughness. In addition, very few studies have been carried out, both from the theoretical and numerical perspectives, to investigate the role of corals in mitigating storm waves or even tsunamis. Therefore, this study is aimed at contributing to improve the understanding of near-bed dynamics over coral reefs by examining the force coefficients. Specifically, the effect of the flow regime (oscillatory or steady flow conditions) on the drag and inertia coefficients of a branched coral species will be studied. Towards this aim, the paper is structured as follows: Section 2 describes the materials and methods for steady and oscillatory regimes. Section 3 contains the results for the drag and inertia coefficients. Finally, the discussion and conclusions are drawn in Section 4. 2 MATERIALS AND METHODS 2.1 Selection of coral species for parameterization The coral species Acropora palmata, called elkhorn coral (Figure 1a), is selected for parameterization due to its geometrical characteristics and its ability to survive in moderate to high energy environments (reef crest or fore reef terrace) associated with the incident waves and currents. Its branches and the global structure of its coral colony play an important role on the wave energy dissipation due to bottom friction. However, there is still a gap of knowledge regarding the contribution to the wave damping caused by its roughness. This is mostly due to measurement difficulties and the high costs of field campaigns. There are several ways to obtain hydrodynamic information of Acropora palmata. On the one hand, the coral could be extracted from its natural habitat and the sample could then be tested under laboratory conditions. Nevertheless, this branched coral can reach lengths of up to 2m by 2m (in a plan view) which makes the extraction not only nearly impossible but also not feasible due to environmental restrictions. As an alternative, researchers could use coral skeletons. But then again, even if these were available, experiments would be expensive given that a large flume would be necessary to perform the laboratory tests at a real scale. For these reasons, in this study we use a 3-D model of the Acropora palmata. This model is created based on the general shape of this branched coral, as observed during a field campaign at San Andrés and Providencia islands, Colombian Caribbean Sea (Figure 1(a)). The morphological characteristics of the coral are scaled to generate a digital model, which is designed using a free and open-source 3-D graphic software called Blender. Then, the 3-D physical model is obtained using a 3-D printer. For this study two different configurations of corals are printed and tested, (i) a single coral colony of A. Palmata (scale 1:8) and (ii) a group of six coral colonies (scale 1:12) (Figure 1(b)). Figure 1. (a) Acropora palmata colonies at San Andres Island (Colombian Caribbean Sea) and (b) 3-D model of A. palmata elaborated with ABS plastic material using a 3D printer machine. 2

3 For this study, the structure of this coral is considered completely stiff. Accordingly, the 3-D model is printed using two rigid materials: ABS (Acrylonitrile Butadiene Styrene) and PLA (Poly Lactic Acid) plastic. ABS is made from oilbased resources with a high melting point, while PLA is a biodegradable type of plastic, manufactured from plant-based resources such as cornstarch or sugar cane. However, ABS is chosen since prototypes printed with this material are stronger than those made of PLA. 2.2 Experimental set-up for steady flow conditions Small-scale experiments are performed in a horizontal current flume (8 m long, 0.3 m wide and 0.6 m height) at the Leichweiß-Institute for Hydraulic Engineering and Water Resources (LWI) in Braunschweig, Germany. The water surface elevation is measured using four wave gauges (WG), as shown in figure 2. In-line forces are obtained with a force transducer mounted on the bottom of the coral model, while flow velocities are obtained with an electromagnetic current meter (ECM) Experiments are carried out using five different water depths (0.24 m, 0.20 m, 0.16 m, 0.13 m and 0.07 m) in combination with six flow velocities (0.05 m/s, 0.1 m/s, 0.2 m/s, 0.3 m/s, 0.45 m/s and 0.55 m/s). This leads to different Reynolds numbers, which have been used by several authors to evaluate the behaviour of the drag coefficient on marine organisms under different scenarios of flow velocities and water depths (Samuel & Monismith 2013; Paul & Amos 2011). Figure 2. Model set-up and measuring techniques for the experiments performed in the current flume 2.3 Experimental set-up for oscillatory flow conditions (regular waves) For the case of regular waves, experiments are carried out in the smaller wave flume ( Berliner Rinne ) of the Leichtweiß- Institute für Wasserbau (LWI), TU Braunschweig, Germany. This wave flume is made out of a smooth material (Plexiglas) and has a width of 0.30 m. It also has length of m, with two different sections: Section 1, with a height of 0.53 m in the first 3.42 m and Section 2, with a height of 0.38 m in the next 15.9 m. In addition, this wave flume has a piston type wave paddle capable of generating regular sinusoidal waves. Different scenarios are tested considering four different water depths (0.24m, 0.20m, 0.16m, 0.13m) in combination with three wave heights (0.075m, 0.05m and 0.03m) and three wave periods (2.0s, 1.5s, 1.0s). A total number of 60 tests are performed: 30 tests for the single coral model and 30 tests for the group of corals model. All experiments correspond to an intermediate water depth (1/20 < d/l < L/2). Figure 3. Model set-up and measuring techniques for the experiments performed in the Berliner Rinne wave flume. 3

4 3 RESULTS 3.1 Drag forces and drag coefficients under steady flow Figure 4 shows the comparison of the square mean flow velocity (u 2 ) and mean total forces (F) obtained for the different water levels, for the single coral model and for the group of corals. For the latter, the in-line force corresponds to the total force exerted on the whole structure (group of 6 corals). Figure 4. Comparison of square flow velocities (u 2 ) vs hydraulic total force (F) for different water levels, considering (a) single coral model and (b) group of 6 corals. Results show that a linear relationship between the flow velocity (u 2 ) and the total force (F) can be considered for both for a single coral (Fig. 4a) and for the group of corals (Fig. 4b). Also, for the single coral experiments, forces increase as the water level increases. However, in general, for the case of the group of corals the forces behave differently from those on the single for the different water depths tested, given that in the former case the maximum forces are obtained for the minimum water level. These results can be explained by considering that for the group of corals the area exposed to the flow is higher compared to that of a single coral, which leads to an increase in the total force over the whole structure. In order to estimate the drag coefficient (C D ), the non-dimensional equation (Eq. 1) was used. In this formulation, F D is the total drag force under steady state conditions, ρ is the density of the fluid, C D is the drag coefficient, u is the free-stream velocity, and A p is the frontal or projected area of the coral structure normal to the flow direction (Figure 5). C D = 2F D ρu 2 A p (1) Figure 5. Projected area Ap perpendicular to the flow direction for each coral model. 4

5 The Reynolds number (Re=ρuL/μ) will be used to assess the behaviour of the drag coefficient under different scenarios. Hence, for its estimation, it is necessary to properly define the characteristic length scale (L). Several authors have proposed different definitions for L according to the geometry and the configuration of the structure inside the water column. Samuel and Monismith (2013) presented an approach to evaluate the drag coefficient C D as function of the Reynolds number using real corals skeletons of Pocillopora verrucosa, Stylophora pistillata and Porites compressa. Their results were related to simple geometrical shapes like solid and hollow spheres. Thus, they defined the characteristic length scale as the diameter of a sphere with the same projected area as the projected area (perpendicular to flow direction) of the coral colony, that is L = 4A p /π. Other authors have represented this parameter as the diameter or length of a vertical cylinder, equivalent in terms of the hydraulic behaviour to the coral colony (Lowe et al., 2005b). In our case, none of these formulations for the characteristic length scale are appropriate to represent our branched coral species. Moreover, given the morphology (size and shape) of Acropora palmata it is not possible to simplify this coral species using a single geometrical shape. For this reason, we use an alternative equation for L based on the volume of the coral structure (V s) and the projected area (A p), as, L = V s /A p. Figure 6(a) and (b) show the drag coefficients (C D ) obtained using Eq. 1 vs. the Reynolds numbers (Re), for different flow velocities and water depths. In both cases (the single coral and the group of corals), C D appears to decrease as the Reynolds number increases. Furthermore, it can be seen that the drag coefficient decays rapidly for small values of Re (Re<1000) and is highly dependent on the water depth, while for larger numbers of Re, C D reaches a constant value (approximately C D = 1.2 at Re > ). In addition, as expected, the magnitude of C D (for each flow velocity) is lower for the case of the group of corals compared to the single coral case. Figure 6. Comparison of drag coefficients (CD) vs. Reynolds numbers (Re) under unidirectional flow for (a) single coral and (b) the group of corals at different water depths. To quantify the behaviour of the drag coefficient as a function of the Reynolds number, a power law is fitted to the data, as C D = are b + c. This is achieved using a nonlinear least squares regression algorithm, with bounds set at a 95% confidence level. Results of this fitting process and the goodness of fit for different water levels are presented in Tables 1 (single coral) and 2 (group of corals). In all cases results are statistically significant with values of the coefficient of determination R 2 >0.98. Table 1. Power law model for the drag coefficient (CD) and goodness-of-fit for a single Coral Single Coral Water level Power law, CD = are b + c SSE R 2 RMSE WL Re WL Re WL Re WL Re WL Re

6 Table 2. Power law model for drag coefficient (CD) and goodness-of-fit for the group of corals Group of Corals Water level Power law, CD = are b + c SSE R 2 RMSE WL Re WL Re WL Re WL Re WL Re Hydraulic resistance of coral colonies under oscillatory flow There are three types of hydrodynamic forces that act upon a coral colony as a wave passes over it: inertial, lift and drag. This study focuses mainly on the drag forces induced by horizontal water velocity generated by passing waves and inertial forces induced by water acceleration. Velocity-induced lift force, which acts vertically on a colony, is typically an order of magnitude less than the drag force (Madin et al., 2006) and is not considered in this study. Furthermore, the drag force is expected to be dominant for slender coral bodies, i.e. branched corals, whereas the inertia force is expected to be dominant for large coral bodies, i.e. massive coral species. Lift forces for branched corals are expected to be small given their geometric shape (Baldock et al., 2014). The drag (C D ) and inertia (C M ) coefficients are highly influenced by the geometrical characteristics of the coral, its structural integrity and the flow dynamics characterized by the Reynolds number (Re) and Keulegan-Carpenter (KC) number (Keulegan & Carpenter 1958). This KC number is defined as KC=uT/L, where u is the maximum horizontal flow velocity, T is the wave period and L is the characteristic length scale defined for this study as L=V s/a p. The determination of C D and C M so far has been derived indirectly from the measured velocities and surface elevations of laboratory experiments (or field measurements) using the Morison equation (Morison et al., 1950), as F T = F D + F M = 1 2 ρc DA p u u + ρc M V s u t, (2) where: F T : Total force [N], C D : drag coefficient [-], C M : inertia coefficient [-], F D : drag force [N], F M : inertia force [N], A p : frontal area of the object [m 2 ], ρ: water density [kg/m 3 ], V s : volume of the object [m 3 ], u: flow velocity [m/s] and u t : flow acceleration [m/s 2 ]. Equation (2) consists of two terms, the drag force (F D ) as a function of flow velocity and the inertia force (F M ) as a function of flow acceleration. For the interaction of waves and corals, both forces are significant, depending on the coral species, water level and hydrodynamic conditions. Figure 7 presents an example of the series of in-line forces on the tested group of corals, orbital velocities and the free surface elevation, for a particular case of regular waves (H=0.03 m, T=1 s, WL=0.13 m). The series of flow acceleration (dotted red line in Figure 7) is obtained by calculating the local derivative of the velocity ( u ), considering a time step of t=0.01s. t Figure 7. Comparison of the in-line force FT, horizontal flow velocity u and flow acceleration ( u ) obtained for the tested group of corals (H=0.03m T=1.0 s, WL=0.13m) t 6

7 Figure 7 shows that the free surface (η) is in phase with the horizontal orbital velocity u and that there is a phase difference of 90 between the flow velocity u and acceleration ( u ). That is, acceleration is maximum when the velocity is zero, and acceleration is zero when flow velocity is maximum. On the other hand, it should be pointed out that the maximum in-line force values do not occur at the same time flow velocity is maximum. t There are several methodologies for the determination of the drag and inertia coefficients under oscillatory flow conditions (Recio & Oumeraci 2007). For practical purposes, the least squares method (Recio & Oumeraci 2007; Journée & Massie 2001), is implemented to account for force coefficients C D and C M from the experiments conducted in the small wave flume Berliner Rinne, as described below Least squares method This method assumes that the predicted total force F P can be estimated as follows, where: F P = λ D u u + λ M u t, (3) λ D = 1 2 ρc DA p (4) λ M = ρc M V s (5) By defining ε 2 = [F P F T ] 2 as the quadratic total error between the predicted and the measured forces over the total length of measurements, then ε 2 u = [λ D u u + λ F M t T] 2. (6) Since the least squares method requires that the error ε 2 should be minimal, then following system of equations is obtained: λ D u 4 + λ M u u u t = u u F T λ D u u u t + λ M ( u t ) 2 = u t F T ε 2 λ D = 0, and ε2 λ M = 0, and thus the This system can be solved simultaneously to get λ D and λ M. Then these values can be replaced in Eqs. (4) and (5) to obtain force coefficients C D and C M. (7) Figures 8(a) and (b) show the drag and inertia coefficients as a function of the Reynolds number, Re, for all water levels, while Figure 8(c) and (d) show these coefficients as a function of Keulegan-Carpenter number, KC. The colors are associated with each water level as follows: water level=0.24m (WL24, black), water level=0.20m (WL20, red), water level=0.16m (WL16, blue), water level=0.13m (WL13, magenta). For each water level, two different periods are plotted: T=2.0 s and T=1.0 s, differentiated by either circles or stars respectively. The results for T=1.5s had to be discarded from the analysis in Figure 8 since they did not show a coherent behaviour in comparison with the other tests. From Figure 8a &b, it is hardly possible to find any relationship between the force coefficients and the Reynolds number, as was found above for steady flow conditions. Regarding the coefficients as a function of KC number, the values obtained for the inertia coefficient C M (Figure 8d) range from 6.5 for the lowest wave period (T=1s) to 5-9 for the highest period (T=2.0 s). On other hand, C D values (Figure 8c) vary from 0.5 to 4 for the lowest period, while for the highest period, drag coefficient tends to be around 3 and 4. The transition point between low and high periods occurs around KC=14. 7

8 Figure 8. Drag coefficients (unfilled markers) and inertia coefficients (filled markers) as a function of the Reynolds number (a and b) and the Keulegan-Carpenter number (c and d), for each water level and wave periods T=2.0 s (circles) and T=1.0 s (stars). From the estimated drag coefficients using the least squares method, the drag force (F D) and inertia force (F M) components are obtained using the Morrison equation (Eq. 2). Figure 9a shows an example of the comparison between the predicted total force (F P) and its components (F D and F M) with the total force (F T) measured by the force sensor mounted under the structure of the group of corals. These results correspond to the wave conditions presented in Figure 7 (H = 0.03 m, T=1.0 s, WL = 0.13m). Figure 9. (a) Comparison of the total measured in-line force (FT) vs. the predicted force (FP) and its components (FD and FM) and (b) Relationship between FT and FM, for the group of corals (H=0.03m T=1 s, WL=0.13m). As expected, F D is in phase with the free surface elevation (η) and the flow velocity. Hence, there is also a phase difference of 90 between both force components F D and F M. This means that F D is maximum when orbital velocity is maximum, i.e. when flow acceleration is minimum (du/dt ~ 0) and, F M is maximum when acceleration is maximum, i.e. when orbital velocity is minimum (u ~ 0). Moreover, the results in Figure 9a suggest that for this particular case (H = 0.03 m, T = 1.0s, WL = 0.13m), the magnitude of the inertial force (F M) dominates over the drag component (F D). Figure 9b shows a comparison between the total measured force (F T) and the predicted inertial force (F M) for the same case of regular waves shown in Figure 9a. These results indicate that the total force for the group of corals might be described essentially (to 83%) by the inertial component (F M). Similar results were obtained for other tests with different water depths, wave heights and wave periods, although they are not presented here for reasons of brevity. 8

9 4 DISCUSSION AND CONCLUDING REMARKS Under steady flow conditions ( u = 0), where the forces of inertia are zero, there is a clear linear relationship between t the square of the velocities (u 2 ) and the forces for the studied branched coral models (Figure 4). However, in nature, this condition is only valid where the effect of the waves over structures can be considered negligible, that is, in environments were currents dominate (e.g. tidal currents). In this case, the study of the drag coefficient as a function of the water depth becomes important, since it has been demonstrated that it varies considerably with varying water levels, specifically for low Reynolds numbers (Re <1000). On the other hand, for larger Re numbers, a constant value of the drag coefficient, C D, (approximately C D = 1.2 at Re > ), was obtained. The results also showed that C D can be well represented as a function of Re by means of a power law equation, C D = are b + c (determination coefficient, R 2 >0.98). A similar behavior was found by Samuel & Monismith (2013) who studied drag forces over other branched coral species (Pocillopora verrucosa, Stylophora pistillata and Porites compressa), for which values of C D varied between 2.5 and 0.6 for <Re< , respectively. In addition, our results have shown that the effect of the arrangement of the coral colonies (single coral or group of corals) on the drag coefficient becomes relevant. This is because the projected area perpendicular to the flow (A p), is a variable that depends not only on the amount of corals but also on the distribution and separation of each colony. For this reason, we consider that the definition of the characteristic length scale (L=V s/a p), based on the ratio between the total volume of the structure V s and projected area A p, is appropriate. Under oscillatory flow conditions, previous studies have reported values of drag coefficients that deviate from the ones found in this study, which range from 1.3 to 3.7 (600<Re<1700 at a water level of 0.24 m). For example, Rosman and Hench (2011) presented a summary of C D values for different coral species, including branched corals like A. palmata, for which C D appears to be around (Lugo-Fernández et al., 1998). In addition, Storlazzi et al. (2005), reported values of C D around 0.85 for some other branched corals like Montipora capitata, Porites compressa, Porites lobata, Pocillapora meandrina based on the range of values of determined for corals and rough cylinders by Denny (1988, 1994) and Gerhart et al. (1993). However, they do not report the corresponding range of the Re number, nor the projected area or the distribution of their studied coral colony. For this reason, we believe that the deviations in C D values could possibly be due to differences in the experiments such as the number of the studied coral colonies, flow conditions, water depths or the shape of the studied species. As for the inertia coefficient, there are no reported values in the literature for the species A. palmata under field conditions or laboratory experiments. Bodies with shapes that deflect the path of the water that moves around them tend to have higher values of C M than bodies that do not deflect flows as much (Kaandorp & Kübler 2001). This behavior is reflected in the results obtained for the studied species, since A. Palmata has rigid and flat branches in each colony (perpendicular to the flow) that change the flow patterns between the coral and the surrounding area. This generates high values of C M associated to inertia forces (F M), which in this study were found to explain to a large degree the total force exerted over the structure, compared to the drag component (F D). In this way, the importance of this study lies not only in the evaluation of the drag coefficient but also in the study of the inertia coefficient, which becomes relevant under oscillatory flow conditions and which has not been sufficiently studied for branched corals such as the Acropora palmata. Although the number of trials are still not enough to allow us to propose new formulations or to conclude on the relationship between the force coefficients (C D and C M ) and the dimensionless numbers (Re, KC) under oscillatory flow conditions, they are a starting point towards the characterization of these coefficients in branched corals, in which the drag coefficient, but mostly the inertia coefficient should be taken into consideration. Furthermore, these results will be used to systematically validate a free and open source Computational Fluid Dynamics (CFD) model like OpenFOAM, due its capabilities to reproduce hydrodynamic processes around complex and heterogeneous bottom shapes. The validated model will then be applied to perform a parameterization study in order to extend the conditions tested in the physical wave flume. The results might serve as basis to conclude on the dependence of those coefficients with respect to wave characteristics (i.e. wave height and period), water depth, and the number and distribution of colonies of A. palmata or even other species of interest. ACKNOWLEDGEMENTS We would like to thank LWI for providing access to the laboratory and the institution s facilities. The work by J. D. Osorio-Cano is supported by COLCIENCIAS. Laboratory experiments were partly funded by CEMarin. Travel support to Braunschweig, Germany was provided by the program Enlazamundos-Alcaldía de Medellín. 9

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