Extreme wave loads on submerged water intakes in shallow water*

CORNETT Andrew, HECIMOVICH Mark, NISTOR Ioan

1. Ocean, Coastal and River Engineering, National Research Council, Ottawa, Canada

2. Department of Civil Engineering, University of Ottawa, Ottawa, Canada, E-mail: andrew.cornett@nrc-cnrc.gc.ca

Extreme wave loads on submerged water intakes in shallow water*

CORNETT Andrew1,2, HECIMOVICH Mark2, NISTOR Ioan2

1. Ocean, Coastal and River Engineering, National Research Council, Ottawa, Canada

2. Department of Civil Engineering, University of Ottawa, Ottawa, Canada, E-mail: andrew.cornett@nrc-cnrc.gc.ca

(Received October 9, 2014, Revised November 3, 2014)

This paper provides new guidance concerning the hydrodynamic loads on submerged intake structures located in shallow water under breaking and non-breaking waves. Results from a series of experiments conducted in a large wave flume at 1:15 scale to study the hydrodynamic forces exerted on a generic intake structure located on a sloping seabed in shallow water below breaking and non-breaking irregular waves are presented. Based on analysis of the experimental data, empirical relationships are developed to describe the peak loads in terms of characteristic wave parameters such as significant wave height and peak wave period. The distribution of the peak loads across different parts of the intake structure is also described. Drag and inertia force coefficients for the horizontal forcing on the intake structure and for the main structural sub-components are derived and presented. It is shown that the well-known Morison equation, with appropriate drag and inertia force coefficients, can provide reasonable estimates of the moderate horizontal loads, but the peak loads are less well predicted.

wave force, water intake, drag force, inertia force, coastal engineering

Introduction

Submerged water intake structures are commonly used to draw-in water for cooling and to support various processes at industrial facilities such as desalination plants, power generating stations, smelters and refineries. A typical submerged water intake structure features a vertical intake pipe protruding several meters above the seabed, set below a solid horizontal plate known as a velocity cap (see Fig.1). The velocity cap, which is typically supported above the mouth of intake pipe on three or more columns, serves to promote horizontal flow into the inlet, prevent vortex formation and help protect local fish species[1]. In many coastal areas, submerged water intakes are located in depths less than 10 m and are exposed to forcing from breaking waves during storms. Estimating wave-induced forces and moments on these structures for design purposes is challenging, particularly when they are located in the surf zone. This is due to the non-linearity of the orbital velocities under large breaking and nonbreaking waves in shallow water, and the complexity of the interaction between the unsteady flow and the intake structure[2]. Large waves breaking in the surf zone can exert induce substantial horizontal and vertical forces on submerged intake structures–loads that are difficult to predict with acceptable certainty[3-5].

Fig.1 Typical submerged intake structures with solid velocity caps (source: www.google.com)

There has been considerable research over many decades concerning the hydrodynamic forces on objects, and particularly cylinders, in oscillatory flows. However, relatively few studies have focused on intake structures, and even fewer have considered openintake structures with velocity caps exposed to highly nonlinear oscillatory flows. Mogridge and Jamieson[6]used experimental methods to investigate the loads on a submerged capped intake structure due to breaking and non-breaking waves, and found linear correlations between the wave heights and the induced horizontal forces. Nakota and Houser[7]used physical hydrodynamic modelling to investigate the loads on a submerged velocity cap due to non-breaking regular waves. More recently, Pita and Sierra[8]described several different types of intake structure, defined the principal design parameters, and described their recent experience with design, manufacture and installation of such structures. Raju et al.[9]used a sophisticated numerical model (OpenFoam) to estimate wave forces on a caisson-type water intake structure located on the seabed, and compared their predictions with results from laboratory experiments conducted with regular waves.

In 1950, Morison et al.[10]proposed the Morison equation (also known as the MOJS equation after the original authors), in which the hydrodynamic force on a submerged cylinder due to oscillatory flows is expressed as a summation of drag and inertia forces. Over the years, many researchers have conducted experiments to determine the most appropriate force and inertia coefficients for various situations (smooth and rough cylinders, low, medium and high Reynolds number, single and multiple cylinders, etc.). For example, Zdravkovich[11]investigated the effects that two cylinders spaced at varying distances had on the recorded forces and presented drag coefficient data dependant on the relative spacing. Sarpkaya[12]tested a circular array of tubes surrounding a central pipe in oscillatory flow and found that force coefficients vary with the Keulegan-Carpenter (KC) number, which represents the ratio between the amplitude of the water particle orbits and the cylinder diameter. Sarpkaya[12]found that the inertia coefficient increased with increasing KC, while the drag coefficient decreased. Burrows et al.[13]published force coefficient data for a submerged cylinder in random waves, and used the Morison equation to predict the wave-induced forces on the cylinder. Sparboom et al.[14]studied cylinder groups in breaking and non-breaking wave conditions and found that the wave force for all cylinder group configurations (for cylinders in close proximity) increased with increasing wave height and wave period. Despite this previous research, the relationship between the wave conditions and the resulting hydrodynamic loads on realistic intake structures remains unclear, and no simple methods are available to predict wave loading on realistic intake structures.

This paper describes a research study in which physical scale model experiments were conducted to determine the hydrodynamic loads exerted by breaking and non-breaking irregular waves on a typical submerged intake structure comprised of a circular intake pipe protruding vertically above the seabed and located below a solid horizontal velocity cap supported by four circular columns. The design of the intake structure is shown in Fig.2. At prototype scale, the circular intake pipe has an external diameter of 2.1 m and protrudes 2.3 m above the seabed. The four support columns each have an external diameter of 0.52 m and a length of 3.1 m, while the velocity cap has a diameter of 5.2 m and a thickness of 0.2 m. The projected area, A, of the intake pipe in the vertical plane is 4.85 m2, while the projected area of the four circular columns is 6.48 m2. The projected area of the velocity cap in the vertical plane is 1.0 m2, while the projected area of the velocity cap in the horizontal plane is 21.2 m2.

The experiments were conducted in a 2.0 m wide by 97 m long by 2.9 m deep wave flume, at a geometric scale of 1:15. Scaling laws derived from Froude scaling principles have been used to estimate prototype quantities from the data measured in physical model. The wave-induced loads on the entire structure and on the different parts of the intake (intake pipe, columns, velocity cap) have been determined for a range of water depths in irregular and regular wave conditions.

1. Physical model experiments

1.1 Scaling considerations

The intake structure shown in Fig.2 was modeled at 1:15 scale according to scaling laws derived from similarity of the Froude number ()Fr in the model and prototype. This implies that all model lengths were reduced by a factor of 15, all velocities and durations were reduced by a factor of 3.87, all forces were reduced by a factor of 3 375, and all moments were reduced by a factor of 50 625. Since wave mo-tion and wave-structure interactions are mainly governed by the balance of gravitational and inertial forces acting on water particles, similitude of the Froude number, together with geometric similitude, ensures that the model provides a realistic simulation of these processes. All values presented in this paper have been converted to prototype scale following standard Froude scaling laws.

Fig.2 Design of the typical submerged intake structure considered in this study (m)

Ideally, the Reynolds number ()Re, which measures the ratio of viscous forces to inertial forces, would also be the same in the model and prototype situations. However, this ideal situation could not be achieved, and in these experiments the Reynolds number was distorted by a factor of 58, which implies that the Reynolds number in the model was 58 times smaller than in the prototype situation. Hence, the viscous forces in the model (responsible for the formation of eddies and the generation of drag forces) were smaller than in the prototype situation. However, since roughturbulent flow conditions were maintained in the model, as in nature, the scale effects due to this distortion of the Reynolds number are expected to remain relatively small.

1.2 Experimental setup

All experiments were conducted in a 97 m long, by 2 m wide, by 2.9 m deep wave flume located in the Ocean, Coastal and River Engineering Laboratory (OCRE) of the National Research Council of Canada (NRC), located in Ottawa, Canada. The NRC-OCRE flume features a powerful wave machine that can generate waves up to 1.2 m in height, and is equipped with a sophisticated active wave absorption system for suppressing wave energy reflected from test structures.

Fig.3 Experimental setup

A scale model of a typical sloping foreshore was constructed within the flume for this study in order to produce realistic depth-limited breaking wave conditions typical of coastal areas with sloping foreshores. The rigid bathymetry consisted of a short section with a steep slope (1:6) at low elevation, followed by a longer section with a milder 1:23 slope extending up to the water line. A gently sloping (1:10 grade) porous gravel beach with very low wave reflectance was installed behind the test site. This gravel beach was able to absorb virtually all of the incident wave energy. The experimental setup is shown in Fig.3.

Fig.4 Model intake structure

A 1:15 scale model of the prototype intake structure was designed and fabricated using a combination of aluminum, timber and PVC materials (see Fig.4). The model intake structure was designed such that hydrodynamic loading on different parts of the structure could be measured using a single force sensor. The force sensor, a water-proof six-axis load cell manufactured by AMTI Corp., was installed in a small pit cast into the sloping bathymetry below the model intake. The model intake was installed where the local bottom elevation was -9 m at full scale (0.6 m at model scale). In the first configuration, the forces and moments acting on all parts of the model intake were transferred to the load cell and measured. In the second configuration, only the forces acting on the central intake pipe were measured. In the third configuration, only the forcing on the central intake pipe and the four columns was transferred to the load cell and measured. The solid velocity cap and support columns always remained in place, but were sometimes disconnected from the load cell. Force and moment data were collected for a broad range of regular and irregular wave conditions, for all three model configurations, and for three different water levels. The flow of water into the intake was not modelled in these experiments.

Capacitance wave gauges were used to measure vertical fluctuations of the free surface ()η at various locations in the flume. An array of five gauges1(-ηη5) was placed in the constant-depth offshore zone, near the wave machine. Three wave gauges were placed near the model structure: gauge6η was located 0.5 m up-wave from the model intake, gauge7η was located beside the intake, and gauge8η was located 0.5 m down-wave from the structure. The wave-induced kinematics beside the model intake were measured using a high-resolution 3-D acoustic velocity sensor (ADV) manufactured by Nortek Inc.. This instrument measured the three orthogonal components of velocity (,,)u v w along the x, y and z directions, respectively, where u is defined positive in the direction of wave propagation and w is positive upwards. A PC-based data acquisition system was used to acquire data from these instruments during testing.

The velocity and force instruments were calibrated before and after testing to ensure that the measured data was reliable. The capacitance wave gauges were re-calibrated several times during the study to ensure that the wave measurements were reliable. The dynamic characteristics of the model intake structure and the force measurement system were investigated using a series of impulse response tests. The model intake structure was very stiff in all three configurations, such that the natural frequency of free vibrations (in air) was approximately 13 Hz at full scale (~47 Hz at model scale), therefore much higher than the frequency of the wave forcing. Thus, any spurious inertial forces that might arise due to vibration of the model could be easily separated from the hydrodynamic loading using low-pass filtering methods. Since the model had low mass, the magnitude of the spurious inertial forcing was also relatively small compared to the hydrodynamic forcing. These results confirmed that the force measurement system was able to reliably measure hydrodynamic forcing at frequencies below ~10 Hz full scale.

1.3 Testing program

The local water depth at the intake has a significant influence on the wave properties and the degree of wave breaking around the structure. In this study, the model intake structure was exposed to a series of regular and irregular waves at three different water levels, -2 m, 0 m and ±2 m. At the location of the structure, the water depths tested were =h7.5 m, 9.5 m, and 11.5 m at full scale, or 0.5 m, 0.6 m and 0.7 m at model scale. Nine different irregular sea states, with peak periods, pT, of 8.0 s, 11.0 s and 14.0 s and significant wave heights,sH, of 3.0 m, 4.5 m and 6.0 m were generated for each water level. These incident wave conditions were defined in the offshore region near the wave generator, and due to the effects of shoaling and wave breaking, the wave conditions at the intake were often significantly different. This range of water levels, significant wave heights, and peak periods was used to produce a broad range of wave conditions at the model intake, including depthlimited conditions with extensive wave breaking (see Fig.5). All wave conditions were generated, measured and tuned to ensure good agreement with target conditions, prior to testing with the model intake structure. The test duration for irregular waves was 2 h at prototype scale (~31 min at model scale). This ensured a minimum of at least 500 unique waves per irregular wave test. Three tests with regular waves were also completed for each water depth, corresponding to wave heights of 3.0 m, 4.5 m, and 6.0 m, all with 11 s period. The entire set of test conditions was repeated for each structure configuration.

Fig.5 Wave breaking at the model intake structure

1.4 Data analysis

The wave, velocity and force data was analyzed using standard statistical, time-domain (zero-crossing) and frequency domain (spectral) methods. A righthanded Cartesian coordinate system (,,)x y z was adopted for reporting forces and moments recorded on the intake structure, in which x was defined positive in the direction of wave propagation (along the flume), y was positive across the flume, and z was positive upwards. The origin of the reference system was takenat the seabed at the center of the intake pipe. Since the force sensor was re-zeroed before the start of each test when fully submerged, only the hydrodynamic forces due to the wave action were recorded; the static forces due to gravity and buoyancy were excluded. The force sensor provided measurements of the wave forces acting on the model in the x, y and z directions (Fx, Fy,Fz), and moments about these three axes (Mx,My,Mz), referenced to the centroid of the instrument (below the seabed). The overturning moments at the seabed were calculated from the moments recorded at the instrument centroid. A digital low-pass filter was employed to remove any possible spurious inertial forcing due to vibration of the model intake structure. Several derived time histories were computed from the force sensor outputs, including the magnitude of the force in the horizontal plane (Fh), the magnitude of the total force ()F and the total overturning moment (Mo). A peak detection algorithm was used to detect independent peaks (positive and negative) in the measured and derived time histories, and several different statistics were calculated to characterize the extreme values. These included the minimum and maximum values (Fmin,Fmax) and the positive and negative peak values associated with low occurrence probability, such as the 95-percentile 95()F, 98-percentile (F98) and 99-percentile (F99) values. In this paper, the 95F statistic, defined as the 95-percentile value of the cumulative distribution formed from the set of independent positive force peaks (one peak per wave), has been adopted to characterize the extreme forces produced by irregular waves. While the95F value is always smaller than the maximum value, it provides a more statistically stable and reliable measure of the peak forcing due to irregular waves, and is useful for identifying sensitivities and dependencies to variables such as wave height, wave period, water depth, and so on.

2. Results and discussion

As the incident waves propagated over the sloping bathymetry towards the model intake structure, they underwent changes due to the effects of shoaling and subsequent depth-limited breaking in the surf zone. The waves generated orbital velocities in the water around the intake structure, and these flows were responsible for exerting hydrodynamic forces on the intake. In this study the ratio of the significant wave height (Hs) at the intake to the local water depth ()h varied from 0.24 to 0.75. Very little wave breaking occurred for Hs/h＜0.35, whereas the degree of wave breaking was extensive for Hs/h＞0.6. Wave shoaling was responsible for increasing the wave heights at the intake, while wave breaking worked to attenuate wave heights and modify the distribution of wave heights, reducing the size of the maximum waves passing over the intake. As is shown in Fig.5, in many tests, the largest waves broke directly above the intake, generating highly complex and turbulent 3-D flows around the intake structure. The forces generated by these highly turbulent flows were also complex.

Fig.6 Comparison of wave spectra at offshore and inshore locations

2.1 Wave shoaling and surf-zone transformation

In this subsection we discuss briefly the changes in wave properties that occurred as the waves propagated over the sloping bathymetry from deeper water to the intake. The offshore and inshore (at the location of the intake) wave spectra for two test conditions are compared in Fig.6. For these particular tests, the depth at the intake is 9.5 m at full scale, while the offshore depth is 25.1 m. The area under each spectrum indicates the total amount of energy, while the spectral shape indicates the distribution of wave energy with frequency. For these two cases, the wave energy at the intake is less than at the offshore location, and there is a significant re-distribution of energy towards both lower and higher frequencies. The loss of energy is primarily due to the effects of depth-limited wave breaking, while the re-distribution of energy indicates that the nonlinearity of the waves has increased in shallow water as expected through the growth in both low-frequency and high-frequency harmonics. The low frequency harmonics are sometimes referred to as infra-gravity waves, and are responsible for the setdown (temporary lowering of the mean free surface) below wave groups (groups of large waves) and the corresponding set-up between wave groups. The high frequency harmonics, which are generally bound to the primary waves, are responsible for the asymmetric wave profiles and height increases characteristic of waves in shallow water. These high-frequency harmonics cause the wave crests to become higher and sho-rter, and at the same time cause the wave troughs to become shallower and longer (see Fig.7).

Fig.7 Example of wave conditions measured above the intake (offshore: Tp=14s , Hs=6 m )

Fig.8 Wave height distributions for four representative test conditions

Not only are the spectral shape and the shape of the wave profiles modified in shallow water (near the intake) compared to conditions in deeper water offshore, the distribution of wave heights is also affected by the reduction in water depth. In deeper water outside the surf zone, wave heights are generally distributed according to the Rayleigh distribution. However, in the surf zone, because depth-limited wave breaking has a proportionally greater impact on the larger waves, the larger waves are attenuated more than the moderate waves, and the wave height distribution narrows. This can be seen in Fig.8, where the distribution of wave heights observed above the intake and in the offshore location are compared for four different representative test conditions. The test conditions with Hs=3m are not much affected by wave breaking, hence the wave height distribution at the intake retains the Rayleigh form for these cases. However, the tests with Hs=6 m waves (offshore) are strongly affected by depth-limited breaking, and the wave height distributions are seen to be modified accordingly. Use of the Rayleigh distribution can lead to over-estimation of maximum wave heights in conditions in the surf zone where depth-limited breaking occurs.

Fig.9 Effects of shoaling and wave breaking on extreme wave height (=9.5 mhat intake)

Shoaling generally increases wave heights in shallower water depths, whereas depth-limited breaking dissipates wave energy and reduces wave heights. The combined effect of these two opposing mechanisms can be seen in Fig.9, which compares the1/100H statistic (average height of the highest 1% of waves) computed from the wave records measured at the intake and offshore locations. The data shows that for less energetic conditions (Hs=3m), the largest waves grow higher as they reach the intake, whereas the opposite holds true for the more energetic conditions (Hs=6 m). Shoaling works to increase the wave heights in both cases, but the effects of depth-limited breaking override the effects of shoaling for the larger Hs=6 m waves, but not for the smaller Hs= 3 m waves. The Hs=4.5 m wave conditions represent an intermediary stage, where the effects of shoaling and wave breaking are more or less in balance. Wave breaking dominates for the shorter-period conditions (Tp=8s), whereas shoaling dominates for the longer waves (Tp=11 s and 14 s).

2.2 Character of wave kinematics and forcing

Our focus now shifts to the general character of the orbital velocities near the intake and the hydrodynamic loading on the intake structure due to the nonlinear shallow-water waves. Figure 10(a) shows an example of the free surface time history recorded abovethe intake, for the case of offshore waves with Tp= 14 s and Hs=6 m . Figure 10(b) shows the corresponding horizontal and vertical velocities recorded beside the model intake, while Fig.10(c) shows the corresponding horizontal and vertical loading for the whole intake structure. The current meter was located ~0.5 m away from the model intake, where the kinematics were only marginally affected by the presence of the intake. The horizontal and vertical velocities are both oscillatory, reflecting the elliptical orbital motion due to the waves. The vertical velocities are significantly smaller than the horizontal velocities, as expected in such shallow-water conditions. There is marked asymmetry in the vertical velocities, with positive velocities (upwards) exceeding negative velocities (downwards). In many tests, high-frequency fluctuations were observed in the velocity signals, due to the strong turbulent fluctuations and aeration caused by local wave breaking (see Fig.5).

Fig.10 Typical measured time histories at the intake location for incident waves with Tp=14s , Hs=6 m

A reasonable estimate of the peak horizontal velocities at the intake can be obtained when the peak wave height above the intake is known. The observed relationship between peak horizontal velocity, quantified by the 99 percentile value of the velocity (Ux,99%), and the peak wave height, quantified by the H1/100statistics, is plotted in Fig.11. As is shown in this figure, the simple expression

with a=0.37 and b=1.34 provides a good fit to the experimental data (r2=0.89) and can be used for preliminary analysis.

Fig.11 Observed relationship between horizontal velocity and wave height

Fig.12 Distributions of horizontal velocity and horizontal force for four representative test conditions

As was expected, the horizontal and vertical forces on the intake are both oscillatory, reflecting the oscillatory nature of the kinematics. The vertical andhorizontal forces are of similar magnitude, even though the vertical velocities and accelerations are considerably smaller than the horizontal velocities and accelerations. The relatively large vertical forces are due to the shape of the intake structure, and the velocity cap in particular, which has a large projected area in the horizontal plane, but a much smaller projected area in the vertical plane. The positive vertical forces (uplift) exceed the negative vertical forces, which is consistent with the asymmetry in the vertical kinematics noted above.

The distributions of horizontal velocity (measured beside the model intake) and horizontal force for four representative test conditions are compared in Fig.12. For these cases, the depth at the intake was h=9.5 m and the forcing on the entire intake structure was measured. These results demonstrate that the orbital velocities increase with increasing wave height and increasing wave period as expected. These results also show that the horizontal forcing on the intake structure increases with increasing wave height, however, the influence of wave period is less clear. This lack of clarity is likely due to the fact that the forcing is linked to both the velocity and acceleration of the flow, and the fact that the velocities tend to increase with increasing wave period, whereas the accelerations tend to diminish with increasing wave period.

Figure 10 shows a typical example of the wave and force data measured during a test with irregular waves in which the loading on the entire intake structure was recorded. The peak forces in the direction of wave propagation tend to occur ahead of the wave crests, when the free surface is rising quickly. This phasing suggests that a portion of the horizontal load is due to drag forces, which are maximized when the horizontal velocities are maximized below the wave crests, while a portion is due to inertial forces, which are proportional to the horizontal acceleration of the fluid past the intake. This suggests that the horizontal forcing could potentially be estimated using the Morison equation, which assumes that the hydrodynamic forcing ()F t due to an oscillatory flow can be approximated as the summation of a drag force proportional to the frontal area and u u, and an inertia force proportional to the submerged volume and du/dt,

The application of the Morison equation model and its skill at predicting loading for this type of structure is discussed further in Subsections 2.6 and 2.7. Before considering the Morison equation, the simple empirical relationships that have been developed for predicting peak loads in terms of characteristic wave parameters such as significant wave height and peak period are considered in Subsection 2.3 and 2.4. The distribution of the peak wave load across different parts of the intake structure is considered in Subsection 2.5.

2.3 Peak horizontal force

The peak horizontal loads for a given water level and wave condition were consistently largest for model configuration 1 (loading on the entire structure), slightly smaller for model configuration 3 (loading on the intake pipe and columns), and much smaller for configuration 2 (loading on the intake pipe only). As was expected, peak vertical loads, both upwards and downwards, were much larger for configuration 1 (velocity cap included) than for either configurations 2 or 3 (velocity cap excluded).

Fig.13 Variation of peak horizontal force with local Hm0/h for all three model configurations

Fig.14 Influence of peak wave period on the variation of peak horizontal force with local Hm0/h for model configuration 1 (whole structure)

Figure 13 shows the variation in peak horizontal force, Fx,95, versus local Hm0/h. Results for all three model configurations are plotted together in this figure. Despite the relatively large scatter, there is a clear trend of increasing peak horizontal force with increasing significant wave height. As can be seen in Fig.14, which shows results for the entire intake structure (configuration 1), much of the scatter is due to variation in peak wave period. This figure demonstrates that waves with longer periods tend to exert largerpeak horizontal forces, while waves with shorter periods exert smaller peak forces. Part of this result can be attributed to the fact that for depth-limited conditions, individual waves with longer periods and longer wavelengths tend to grow higher before breaking, compared to comparable waves with shorter periods. Similar results were obtained for the other model configurations.

Fig.15 Variation of peak horizontal force with localmax/Hh for all three model configurations

Fig.16 Variation of peak uplift force with local Hm0/h

In these experiments, the maximum wave height, Hmax, recorded at the structure ranged from 0.5 up to 1.2 times the local water depth. As is shown in Fig.15, the scatter seen in Fig.13 is reduced significantly when the peak horizontal forces for each model configuration are plotted againstmax/Hh, instead of Hm0/h. This demonstrates that the peak horizontal loads (Fx,95) are more closely related to the maximum wave height than to the significant wave height. The trends seen in Fig.15 can be approximated using a simple power relationship of the form

For the entire intake structure, a=126 kN , b=1.75 and r2=0.84. For the central intake pipe only, a= 53kN, b=1.56 and r2=0.83. While for the central intake pipe and the four circular columns, a= 101kN, b=1.82 and r2=0.80.

2.4 Peak vertical force (uplift)

The vertical load on the intake structure is dominated by the forcing attracted by the horizontal velocity cap, and very small vertical loads were recorded for model configurations 2 and 3 when the velocity cap was disconnected from the load cell. Figure 16 shows the variation in the peak uplift force (Fz,95) with local Hm0/h for model configuration 1 (loading on the entire intake structure). The peak vertical uplift loads, which are considerably larger than the peak horizontal loads examined above, increase with increasing significant wave height and also increase with decreasing water depth. The observed variation of peak uplift force with Hm0/h can be well described by a simple exponential equation of the form

where =8.57a, =5.25b and2=0.91 r.

Fig.17 Influence of local water depth on peak uplift force

Fig.18 Influence of peak wave period on peak uplift force

The results plotted in Figs.17 and 18 illustrate theminor influences of water depth and wave period on peak uplift loads. Peak uplift forces were largest at the lowest water level, when the velocity cap was located close to the free surface. The results also indicate that for depth limited conditions, waves with longer periods tend to exert larger peak uplift forces than do waves with similar wave heights but shorter periods. However, the influences of water depth and wave period are secondary compared to the wave height.

2.5 Loading on intake structure components

As mentioned previously, the model intake structure was mounted to the force sensor in three different ways that allowed the forces on different parts of the intake structure to be measured independently. In configuration 1, the forces and moments on the entire structure were measured. In configuration 2, only loads on the central pipe were measured, while in configuration 3, the loads on the central pipe and the four support columns were measured. The velocity cap always remained in place, but was disconnected from the force sensor in configurations 2 and 3. Using this information, loading on the velocity cap due to each wave condition can be estimated by differencing results for configurations 1 and 3. Similarly, loading on the four support columns can be estimated by differencing results for configurations 2 and 3. (Loading on the central intake pipe was measured directly in configuration 2.).

Ideally, these differencing operations could be evaluated in the time-domain at every time step, producing time histories of the forcing on each structure component for each wave condition. However, in practice this approach often produced large errors due to small differences in the timing of the maxima in each record and small variations in the wave breaking processes in each test. Instead, this differencing operation has been performed using statistical and characteristic parameters that describe the distribution of peak forces for each model configuration. For example, the F95value for the four circular columns has been estimated by subtracting the95F value for configuration 2 from the95F value for configuration 3. This differencing operation was performed to estimate a full set of extreme values (including 98F, 99F and maxF) for each component of force and moment.

One drawback of this approach is that it can produce unrealistic estimates when the numbers being differenced are both very small and close to zero. Hence, it should only be used in cases where substantial loads were measured. It is also important to recognize that peak loads due to different waves can be subtracted from each other using this approach. Despite these caveats, this statistical approach is able to provide reasonable and useful estimates of extreme wave loads for structural components for which loads could not be measured directly.

Fig.19 Peak horizontal force on intake pipe, support columns and velocity cap, =9.5 mh

Typical results from this analysis are presented in Fig.19. This figure shows, for nine different test conditions with =9.5 mh, the portion of the peak horizontal force attracted by the central intake pipe, the four circular support columns and the horizontal velocity cap. The important influences of significant wave height and peak wave period on the peak horizontal loads can also be seen in this figure. Similar results were obtained for tests conducted with other water depths. On average, the central intake pipe, with a projected area of 4.85 m2, attracts 44% of the peak horizontal load, while the four circular columns, with a combined projected area of 6.48 m2, attract 37% of the peak horizontal load. The velocity cap, with a frontal area of 1.0 m2in (in the vertical plane) attracts 19% of the peak horizontal load on average. These results suggest that the peak horizontal loads are not simply proportional to the frontal area of each structural component. Much of the horizontal loading on the velocity cap is likely due to inertial forces which are proportional to the fluid acceleration and the submerged volume instead of the projected frontal area.

2.6 Morison equation force coefficients

If one neglects the velocity cap, the intake structure is essentially a collection of five cylinders, the central intake pipe surrounded by the four support columns. It is possible that the well-known Morison equation (2) may be used to provide a useful prediction of the loading due to waves, at least the forcing in the direction of wave propagation. To test this hypothesis, the flow velocity and force data from tests with regular waves was used to determine drag and inertia force coefficients for the whole intake structure and for each main sub-component (central pipe, velocity cap and support columns). Velocity and acceleration time histories recorded beside the structure were used to fit the Morison equation (2) to the time history of the horizontal forcing induced by regular waves.Flow acceleration was obtained by taking the first derivative of the measured velocity.

Fig.20 Comparison of measured and predicted horizontal forcing in regular waves (H=3m,T=11s,h=7.5 m )

Inertia and drag force coefficients (CM,CD) were determined by fitting the Morison equation to force, velocity and acceleration time histories due to regular waves. In a comparative analysis of various methods, Wolfram and Naghipour[15]found that for estimating force coefficients, the weighted least squares method provided the best fit to experimental force data. Following this method, the drag and inertia force coefficients are calculated as follows:

whereoF is the observed force measured during tests with regular waves. Drag and inertia force terms without coefficients are represented by Df and If, respectively. The weighted least squares method was used to determine best fit drag and inertia force coefficients for each test with regular waves. Time histories with data for at least ten wave cycles were used in every case. Figure 20 shows an example of the agreement between the Morison equation model and the forcing measured on the full structure, for the case of a regular waves with H=3m, T=11s and h=7.5 m water depth. The measured force signal features short periods of positive forcing, peaking at ~80 kN, alternating with longer periods of negative forcing, each containing two negative peaks, both smaller than -30 kN. The non-symmetric form of this forcing, with larger positive peaks and smaller negative peaks, can be attributed to the nonlinearity of the shallow water waves, which feature higher crests and shallower troughs. In this case, the Morison equation provides a reasonable prediction of the positive forcing, but significantly over-estimates the negative forcing. In reality, drag and inertia coefficients vary over each wave cycle, so using one set of coefficients for the entire wave cycle cannot be expected to provide a perfect fit.

The Keulegan-Carpenter number, KC, represents the ratio of the water particle orbital amplitude to the cylinder diameter, and can be written as

whereMU is the maximum horizontal orbital velocity during the wave cycle, T is the wave period, and D is the cylinder diameter. It provides an indication of the relative importance of the inertia and drag force terms, with inertia forces dominating for small KC numbers, and drag forces dominating for large KC numbers[16]. The force coefficients determined in this study for the central pipe alone and the set of four support columns are shown in Fig.21 as a function of KC. Drag forces are dominant for the slender columns (posts), whereas both drag and inertia terms are important for the central intake pipe.

Fig.21 Inertia and drag force coefficients derived for the central pipe and the set of 4 support columns (posts)

Fig.22 MC for various structure sub-components (configuretion 1 (full structure), pipe (central intake pipe), Configuration 3 (central pipe + 4 support columns), posts (4 support columns)

Figure 21 reveals a roughly linear increase in the magnitude of the inertia coefficients (CM) with increasing KC number and slightly decreasing values for the drag coefficients (CD), for both the pipe and the set of four posts. These force coefficient values and their variation with the KC number are similar to those obtained experimentally by Sarpkaya[12]for a similar arrangement of tubes placed around a central pipe.

Fig.23 DC for various structure sub-components

Force coefficients for both the full structure (configuration 1) and the pipe plus 4 posts (configuration 3) were also determined. In this case, due to the small range in the KC number, the force coefficients are plotted in Fig.22 and Fig.23 against the local wave height normalized by the local water depth (/)H h.

2.7 Force prediction using the morison equation

The force coefficients determined by fitting the Morison equation to experimental data from tests with regular waves have been used, together with the velocity signal measured beside the structure, to model and predict the forcing on the intake structure due to irregular waves. The forcing on the intake structure has been estimated using two methods. Method 1: by summing the forcing estimated separately for each of the three structural sub-components (central intake pipe, set of four support columns, and the velocity cap), and Method 2: by estimating the forcing acting on the entire structure directly using a single pair of force coefficients. The force coefficients used to predict the forcing for each irregular wave test condition were selected from the data presented above in Subsection 2.6, which was derived from testing with regular waves. The force coefficients were selected on the basis of the similarity between the regular and irregular wave test conditions.

The velocity data recorded during each irregular wave test was used together with the Morison equation (1) to predict the hydrodynamic forcing due to irregular waves for: (a) the total structure, (b) the central pipe alone, and (c) the central pipe plus the set of four support columns. Figure 24 shows a typical example of the horizontal forcing on the intake structure measured during testing with irregular waves. It is observed that the positive force peaks are generally larger than the negative peaks, reflecting the non-linearity of the shallow water waves. Figure 25 shows an example of the reasonable agreement obtained between the measured horizontal force and the horizontal force predicted using the Morison equation, for a sub-set of the record shown in Fig.24. The main features of the horizontal forcing on the structure were successfully reproduced using the Morison equation (2) combined with measured kinematics and the new force coefficients, however, details such as the magnitude of the largest negative and positive peak forces were generally less well predicted.

Fig.24 Time-history of horizontal force due to irregular waves (Hm0=3m ,Tp=8s ,h=9.5 m)

Fig.25 Agreement between measured (solid line) and predicted (dotted line) horizontal force due to irregular waves (full structure, Hm0=3m ,Tp=8s ,h=9.5 m)

An assessment of the ability of the Morison equation model to predict peak forces for each irregular wave test revealed that the Morison equation model was able to predict average or RMS forces reasonably well, but was unable to predict peak (95th percentile and higher) forces with acceptable accuracy. For each structural configuration, there was at least one case where positive peak forces (95th percentile) were under-estimated by as much as 40%. This suggests that the Morison equation model is an unreliable predictor of the extreme forces due to the nonlinearity of irregular waves in shallow water. Part of this less than ideal performance is likely due to the fact that the force coefficients were selected based on the character of the significant wave conditions in each sea state, and not on the character of the extreme waves, which were generally more than 1.5 times larger than the significant waves.

3. Summary and conclusions

A series of scale model experiments have been conducted at 1:15 scale in a large wave flume to investigate the hydrodynamic loading exerted on a typical submerged intake structure located in the surf zone due to breaking and non-breaking irregular waves. The model intake structure consisted of a circular intake pipe protruding above the seabed surrounded by four circular columns supporting a circular velocity cap located above the intake mouth. The experiments reported in this paper were conducted at 1:15 scale following Froude scaling principles, as gravitational and inertial forces were dominant compared to viscous forces. The Reynolds numbers at model scale were 58 times smaller than the corresponding prototype values, indicating that the flows would be more turbulent at full scale. Rance[17]and Sarpkaya[12]conducted experiments on force coefficients for single cylinders in isolation as well as multiple cylinders in close proximity, and concluded that force coefficients for multiple cylinders placed in proximity are only very weakly dependent on the Reynolds number. This suggests that scale effects are small for these types of structures, and that the results of these large scale model experiments will be applicable in prototype situations. However, future investigations focused on quantifying scale effects for these types of structures are recommended.

The model intake structure was located on a sloping foreshore and was exposed to a range of regular and irregular wave conditions at three water levels, while the wave properties, kinematics, and 6-axis forces on the intake structure were all measured continuously. The measurements have been analyzed to investigate the relationship between the wave conditions and the forcing for different parts of the intake and for the whole structure.

A fairly broad range of shallow-water wave conditions has been investigated in this study, the ratio of significant wave height to water depth (Hm0/h) at the intake structure varied between 0.24 and 0.75, while the maximum wave height at the intake ranged from 0.5 up to 1.2 times the local water depth. The shape of the wave spectra, the asymmetry of the wave profiles, and distribution of wave heights were all modified at the intake relative to their form in deeper water. Wave shoaling was responsible for increasing the wave heights in shallower water at the intake, while depth-limited wave breaking worked to dissipate wave energy, attenuate wave heights and modify the distribution of wave heights, reducing the size of the maximum waves passing over the intake.

The waves generated orbital velocities in the water column around the intake which in turn exerted oscillatory forces on the intake structure. The peak positive horizontal forces (in the direction of wave propagation) were generally larger than the peak negative forces, reflecting the nonlinearity of the shallow water wave conditions. This asymmetry in forcing generally increased for larger waves and shallower water depths. The peak positive horizontal forces on the intake structure were found to increase with increasing wave height and with increasing wave period. This result can be partly attributed to the fact that for depth-limited conditions, waves with longer periods (and longer wavelengths) tend to grow higher before breaking compared to comparable waves with shorter periods. For the conditions considered in this study, the relationship between the peak horizontal force on the intake (Fx,95) and the dimensionless maximum wave height (Hmax/h) can be approximated by a simple power relationship of the form Fx,95=a( Hmax/h)b, where the values of the coefficients a and b are given in Subsection 2.3. This simple empirical formula can be used to obtain preliminary estimates of the peak horizontal wave loads on intake structures similar to the one considered in this study.

Wave loading in the vertical direction, which is dominated by the forcing attracted by the horizontal velocity cap, was found to be considerably larger than the forcing in the horizontal direction. The peak uplift loads were found to increase with increasing wave height and with decreasing water depth. The peak vertical loads increased as the free surface approached the velocity cap. For the conditions considered in this study, the relationship between the peak uplift force (Fz,95) and the dimensionless significant wave height (Hm0/h) can be well described by a simple exponential relationship of the form Fz,95=aexp[b( Hm0/h)], where the values of the coefficients a and b are given in Subsection 2.4.

In this study, experiments were conducted in which the wave-induced hydrodynamic loading on the entire intake structure was measured, and these experiments were repeated with different parts of the intake disconnected from the force sensor. Useful estimates of extreme wave loads for the intake components that were disconnected have been obtained by differencing extreme force statistics derived from the repeated experiments. This statistical approach is able to provide reasonable and useful estimates of extreme wave loads for components for which loads could not be measured directly. Results from this analysis indicate that, on average, the central intake pipe attracts 44% of the peak horizontal load, the four circular columns attract 37% of the peak horizontal load, and the velocity cap attracts 19% of the peak horizontal load. These results suggest that the peak horizontal loads are not simply proportional to the frontal area of each structural component. For the central intake pipe andthe velocity cap, much of the forcing is likely due to inertia forces which are proportional the fluid acceleration and the submerged volume instead of the projected frontal area.

Using the weighted least squares method, drag and inertia force coefficients were derived for the intake structure and for the main structural sub-components, from data recorded in experiments with regular waves. The force coefficients determined in this study are in good agreement with findings from other studies conducted with similar flow conditions and structure geometries. The force coefficients varied with KC number and with normalized wave height (/)H h as expected. However, the Morison equation model, combined with measured kinematics and constant force coefficients, generally failed to adequately predict the highly asymmetric character of the regular wave forcing, which mirrored the nonlinearity of the shallow water waves.

The force coefficients derived from measurements in tests with regular waves were used to predict the forcing on the intake structure due to irregular waves. The general character and timing of the horizontal forcing was successfully predicted, as was the magnitude of the forcing due to smaller amplitude waves. However, the magnitude of the peak positive forces was generally under-estimated, while the magnitude of the peak negative forces was generally over-estimated. This suggests that the Morison equation model may be an unreliable predictor of the extreme forces exerted on submerged water intakes by nonlinear irregular waves in shallow water.

The study described in this paper provides some new insights into the hydrodynamic loads on submerged intake structures in shallow water under breaking and non-breaking waves. Scale model tests at large scale are recommended to optimize designs to suit local conditions and to determine extreme wave loads for design of important structures.

Acknowledgements

The authors wish to thank the NRC’s Ocean, Coastal and River Engineering Laboratory (NRCOCRE) for supporting this research and contributing access to their facilities, equipment and technical support. Financial support from the Natural Science and Engineering Research Council (NSERC) of Canada is also acknowledged.

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10.1016/S1001-6058(15)60454-1

* Biography: CORNETT Andrew (1961-), Male, Ph. D.,

Research Program Leader, Marine Infrastructure, Energy and Water Resources