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Oscillatory Forces on Square Cylinder and Cubes

Christensen, E. D.; Nielsen, F.; Borup, M.

Published in:Coastal Structures 2019

Link to article, DOI:10.18451/978-3-939230-64-9_085

Publication date:2019

Document VersionPublisher's PDF, also known as Version of record

Link back to DTU Orbit

Citation (APA):Christensen, E. D., Nielsen, F., & Borup, M. (2019). Oscillatory Forces on Square Cylinder and Cubes. In N.Goseberg, & T. Schlurmann (Eds.), Coastal Structures 2019 (pp. 851-861). Bundesanstalt für Wasserbau.https://doi.org/10.18451/978-3-939230-64-9_085

Conference Paper, Published Version

Christensen, Erik; Nielsen, Frederik; Borup, MikkelOscillatory Forces on Square Cylinder and Cubes

Verfügbar unter/Available at: https://hdl.handle.net/20.500.11970/106701

Vorgeschlagene Zitierweise/Suggested citation:Christensen, Erik; Nielsen, Frederik; Borup, Mikkel (2019): Oscillatory Forces on SquareCylinder and Cubes. In: Goseberg, Nils; Schlurmann, Torsten (Hg.): Coastal Structures2019. Karlsruhe: Bundesanstalt für Wasserbau. S. 851-861.https://doi.org/10.18451/978-3-939230-64-9_085.

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Abstract: Large scale porous structures are a very common part of coastal and offshore structures such as breakwaters for coastal or harbors protection, and scour protection layers for offshore wind turbine foundations. Much research has been carried out on for instance breakwaters from a statistical point of view. This has led to design formulas. In this paper, we present an approach where the forces on large porous structures were measured directly. The goal was to investigate the variability of the wave forces on different structural elements depending on the spatial position in the protection layer. The study indicated that the elements or stones on the edge of a protection layer experienced the largest force away from the protection layer under the influence to an oscillatory outer flow (i.e. a simplification of wave dynamics). The KC-number ranged from 2 to 12. The results indicated a potentially spreading mechanism and explained instability of the outer positioned stones.

Keywords: Oscillatory forces, forces on cubes, porous structures, stone protection layer

1 Introduction

Large porous structures are used in for instance stone covers for pipelines, scour protection, armor- and filter layers in rubble mound breakwaters. They present an area where currents, waves and turbulence often have a large influence on the structural design. The development of the coastal and offshore areas are expected to be intensified, where the last couple of decades offshore wind development will be supplemented by a more general implementation of marine infrastructure in the near future. The development includes energy facilities, e.g. offshore wind farms, exploitation of wave energy, and development and implementation of marine aquaculture. Studies of porous structures are generic in nature as porous structures often are an important part of common marine structures. “New” protection measures such as artificial reefs that apart from acting as coastal protection also support recreational activities such as surfing, fishing, and diving activities will increase the use of porous structures. Porous structures can also be seen in other structures such as fish cages (Chen and Christensen, 2018), floating breakwaters (Christensen et al., 2018) and in studies of porous bed as in Corvaro et al. (2010) and Nielsen et al. (2013).

The objective of the experimental study, we present here, was to achieve a fundamental understanding of forces on individual elements/stones in for instance a protection layer. The final research goal is detailed understanding of forces and flow over and inside for instance scour protection layers under the influence of waves.

The design of scour protection layers have previously been linked to the enhanced bed shear stresses around mono-piles, where the inception of motion as found for sediment transport was used, see for instance Sumer and Fredsøe (2006a). De Vos et al. (2012, 2011) developed empirical design methodologies with a static as well as a dynamic approach. Their approaches resemble the methodology that has been utilized for developing design formulas for rubble mound breakwaters, such as Van der Meer (1987), where the damage of the structure was studied based on derived non-dimensional parameters.

Oscillatory Forces on Square Cylinder and Cubes

E. D. Christensen, F. T. Nielsen & M. Borup

Technical University of Denmark, Kgs. Lyngby, Denmark

Coastal Structures 2019 - Nils Goseberg, Torsten Schlurmann (eds) - © 2019 Bundesanstalt für Wasserbau

ISBN 978-3-939230-64-9 (Online) - DOI: 10.18451/978-3-939230-64-9_085

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The studies mentioned above did not study the internal flow mechanism between armor stones in detail. However, some studies have investigated the internal flow mechanisms of protection layers. In Jensen et al. (2014) pressure gradients were investigated along the sloping front of a breakwater for regular waves, and in Jensen et al. (2017) simulated flow in stone cover-layers and estimated lift forces on individual stones in a toe protection. Earlier, Grace and Evans (1979) conducted an experiment that studied instability of small concrete cubes under the influence of oscillatory flow. Dixen et al. (2012) studied suction of sand from a stone/armour layer on a sand bed under the influence of waves. Hald (1998) measured forces on armour stones in a rubble mound breakwater.

In this study, we measured forces in the in-line and transverse direction of the flow. Force coefficients were estimated in the same way as in classical works like for instance Sarpkaya (1976), and Bearman et al. (1984). An initial verification of the experimental set-up was made by comparison to Bearman et al. (1984) for square cylinders. No literature was found on measurements of force coefficients on cubes attached to a wall. Li (1989) studied the forces on free hanging cubes and spheres under the influence of oscillatory flow, which have been utilized for qualitative comparisons. More details on the experimental set-up, methodologies, and results can be found in Nielsen and Borup (2019).

The remaining part of the paper includes an introduction to test conditions and methodologies used to for instance estimate force coefficients. This is followed by a section on the experimental set-up and applied equipment. The results section describes validation tests and the focus of the study, the forces on cubes. Finally, summary and conclusions are given.

2 Test conditions and methodologies

Two different test scenarios were established. The first scenario consisted of one or four square cylinders. These tests were carried out as a supplement to the test scenarios presented in Bjørke et al. (2018) and as a validation of the present used experimental procedures. The second set-up investigated the forces on a collection of cubes close to a wall under the influence of an ambient oscillating flow. Therefore, the second set-up was an attempt to study the hydrodynamic forces, eventually induced by waves, on a collection of elements (stones) on a seabed.

Due to mutually exclusive scaling laws, dynamic similarity between physical model experiments and full scale is only attainable at full scale. The physical model experiments were scaled using Froude scaling. Froude scaling together with geometrical similarity allowed reproduction of the Keulegan-Carpenter number, KC. In order to reduce the influence of viscous effects, and therefore the Reynolds number, the structural elements were square cylinders or cubes. This reduced the uncertainty of location of separation points as these were believed to be determined by the sharp edged. The KC-number varied from 2-12 indicating the inertia force contributed significantly to the total force.

The test model was mounted to a carriage, and the carriage oscillated the entire model set-up in a filled wave flume with otherwise calm water. The position of the model was measured with a distance laser and the forces were measured with single dimension force transducers. This gave time-series of position, speed, acceleration, and forces. The analyzed force time-series was also used to estimate force coefficients for a Morison-force model.

For a body in oscillatory flow, the Morison formulation for the total in-line force reads: 𝐹𝐹𝑖𝑖𝑖𝑖 = ½𝜌𝜌𝐶𝐶𝐷𝐷𝐴𝐴𝑈𝑈2 + 𝜌𝜌𝐶𝐶𝑀𝑀𝑉𝑉𝑉𝑉𝑉𝑉�̇�𝑈 (1)

The first term on the right-hand side is the drag force and the second term is the hydrodynamic mass force (inertia force). 𝜌𝜌 is the water density, A is the cross-sectional area, Vol is the volume of the body, U is the undisturbed velocity, �̇�𝑈 the undisturbed acceleration, CD is the drag coefficient and CM is the inertia coefficient including the Froude Krylov contribution, i.e. effect of the outer pressure gradient. The Froude-Krylov force was not included in the measurements and is not accounted for in the force times series. The effect of a potential a Froude-Krylov force on the hydrodynamic mass coefficient, CM, was accounted for by; 𝐶𝐶𝑀𝑀 = 𝐶𝐶𝑚𝑚 + 1. Here 𝐶𝐶𝑚𝑚is the hydrodynamic mass coefficient directly determined from the measurements.

The drag coefficient, CD, and the inertia coefficient, CM, were determined for individual square cylinders and cubes. The method of least squares fit determined the force coefficients, see for instance Sumer and Fredsøe (2006b).

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For all conducted tests the period of the oscillations was, T = 3.0 s. To achieve different KC-numbers the maximum stroke, and therefore the maximum oscillating velocity changed from test to test. The flow parameters are given in Tab. 1. The actual parameters might have deviated slightly from the tabulated values. However, in the analyses the actual maximum speeds and accelerations were used.

Tab. 1. Flow parameters used in the experiments with the two set-ups

𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑉𝑉𝑃𝑃,𝑇𝑇 = 3.0 𝑠𝑠 𝑈𝑈𝑚𝑚 [𝑚𝑚/𝑠𝑠] 0.027 0.053 0.080 0.107 0.133 0.160 𝐾𝐾𝐶𝐶 [−] 2 4 6 8 10 12 𝑅𝑅𝑃𝑃 [−] 1067 2133 3200 4267 5333 6400

2.1 Test set-up 1

Forces on square cylinders were measured in set-up 1. The width of the cylinder was 0.040 m and the length of the test cylinder was 0.40 m. The test cylinder connected to a dummy cylinder, which was fasten to a pvc-plate. Four different configurations were tested; 1 cylinder, 2×1 cylinders, 1×2 cylinders, and 2×2 cylinders. Fig. 1 illustrates the set-up in a 2×2 configuration. Only results from a single cylinder and the 2×2 cylinder configuration are presented in this paper.

Fig. 1. Set-up of 2×2 square cylinders. The dimensionless distance between the square cylinders was d/D = 1.0.

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2.2 Test set-up 2

Fig. 2. Experimental set-up with cubes in a 6×6 configuration.

The second set-up was configured with a number of cubes. The side length of the cubes were 0.040 m. A part from initial testing of a single cube four configurations were tested; 3×3 cubes, 3×6 cubes, 6×3 cubes and 6×6 cubes. We only present test results from 3×3 and 6×6 configurations. Fig. 2 illustrates a 6×6 configuration of cubes installed in the flume, and Fig. 3 shows a plane view of 3×3 and 6×6 configurations. Note that the forces were only measured on the 9 cubes to the upper left corner due to symmetry in the configurations.

Fig. 3. Position of the cubes on the horizontal PC-plate in a 3×3 (left) and 6×6 (right) configurations.

3 Experimental set-up

Fig. 4. Definition sketch of the carriage mounted on a railing system.

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The experiments were conducted in DTU-MEKs hydraulic laboratory. The wave flume had a length of 25 m, a width 60 cm, and a total depth of 80 cm. Two motors, one of each side of the flume, of the type (ROBOT-ZONE, 165 RPM, 12V) moved the carriage on top of a wave flume along a railing system. In the experiments, the carriage established an oscillating motion of the physical model in an otherwise calm water as illustrated in Fig 4.

3.1 Force Measurements

Two different load cells were used in the experiment. Set-up 1 used the TAL220 load cell that has a capacity up to 10 kg, while set-up 2 used the Phidgets RB-Phi-117 that had a capacity of 0.78 kg. All force cells were calibrated with a calibration rig before conducting the measurements.

3.2 Position measurements

A distance laser, SICK DT50-2, attached to the stationary frame above the flume measured the position of the carriage, as illustrated in Fig. 4. For verification purposes, the laser was seconded by an accelerometer (SparkFun ADXL335) attached on top of the carriage following the oscillation generated by the two motors.

4 Results

4.1 Validation tests of experimental set up

Two tests were conducted to validate the experimental set-up and procedures. The first test case was a square cylinder, while the second test included four square cylinders with the individual distance of half and one cylinder width. Similar test results for the single square cylinder are available in Bearman et al. (1984), and for the second test case in Bjørke et al. (2018).

4.1.1 One Square cylinder

Fig. 5 shows an example of the measured inline force for the square cylinder for KC = 12. The signal was filtered with a cut-off frequency of 5 Hz using the Hamming filter. As it appears from the figure the major characteristics of the inline force were still captured with this cut-off frequency. Spectral analyses showed a clear peak at 50 Hz, which was assumed to be the electric power grids mains frequency. Other sources for electrical noise were not clearly identified.

Fig. 5. Example of measured raw and filtered inline force on a square cylinder, for KC = 12.

To show the general trend in the force time-series the phase (or ensemble) averaged forces were found, and Fig. 6 shows the results for KC equal to 6 and 12. The oscillating force appeared similar to each half period suggesting that the experimental set-up behaved equally in both directions in terms of speed, accelerations, and forces.

Fig. 7 shows comparisons of force coefficients found based on the least-squared method the force coefficients and results presented in Bearman et al. (1984), experimental results, and in Bjørke et al. (2018), numerical results using the computational fluid dynamics based on OpenFOAM. In general, a

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very good agreement was achieved. Even for small KC-numbers, where the drag contribution to the total force is known to be small, the agreement was good. It was only for KC smaller than 2 that differences appeared.

Fig. 6. Example of the measured phase-averaged inline force for: Left: 𝐾𝐾𝐶𝐶 = 6.0, Right: 𝐾𝐾𝐶𝐶 = 12.0.

Fig. 7. Comparison of force coefficients in the Morison Equation for a square cylinder. The inertia coefficient

4.2 Four square cylinders

d/D = 0.5

d/D = 1.0

Fig. 8. Phase-averaged forces on four square cylinders, Red lines: Left 1 and left 2, Blue lines: Right 3 and right 4. Black line: Reference force for single square cylinder. The relative velocity is on the right axis as a reference.

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The forces on four square cylinders were measured with the same procedures as for the single square cylinder. The relative velocity was positive from left to right in the set-up (opposite of the carriage motion). Fig. 8 shows phase-averaged forces for the four cylinders for two different spacing, d/D = 0.5 and 1.0. Lee positioned cylinders resulted in the largest forces. For instance, the flow orientated from left to right at around 𝜔𝜔𝜔𝜔 = 40 ° gave the largest force on the cylinders to the right, i.e. on the lee side of the other two cylinders (blue lines in Fig. 8). This phenomenon was also observed and reported in Bjørke et al. (2018).

4.3 Tests with cubes attached to a wall in oscillatory flow

To validate the set-up with cubes comparisons were made to the experiments reported in Li (1989), even though that his experiments related to a free hanging cube in oscillatory flow. Our cube was attached to a wall, and therefore the comparison should be seen as indicative and mainly qualitatively.

We found the same tendencies in our experiments with a cube attached to a wall as reported in Li (1989). The cube attached to a wall gave larger drag coefficient as well as larger inertia coefficient compared to a free hanging cube. The force coefficients were in general lower than the force coefficients found for a square cylinder. This was expected as the flow is three-dimensional and therefore easier can adapt to a geometry of a cube compared to a cylinder. The larger force coefficients for an object attached or close to a wall is a well-known phenomenon. This has earlier been reported for a circular cylinder in Yamamoto et al. (1974).

Fig. 9. Comparison of force coefficients on a free cube, (Li, 1989), and on a cube attached to a wall, our experiments

4.3.1 3×3 cubes

KC = 6.0 KC = 12.0

Fig. 10. Phase-averaged inline forces on cubes in the 3×3 configuration. The forces were normalized with the maximum inline force on a single cube. Blue lines (-, --, -.) are legends for cube 1,2, and 3. Red lines (-, --, -.) are legends for cube 4,5, and 6. Green lines (-, --, -.) are legends for cube 7,8, and 9.

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Tests with 3×3 cubes followed the tests with one single cube. Fig. 10 shows phase-averaged forces on the nine cubes normalized with the maximum force on a single cube. In both cases, KC = 6.0 and KC = 12.0, the maximum forces on the cubes were for all cubes reduced compared to a single cube. Cubes on the lee side of the configuration experienced typically the largest maximum force during each half cycle of the oscillatory flow. In all cases, the middle three cubes experienced the weakest force. It could be argued that the reduction of the maximum forces for all cubes is related to three-dimensional effects. Here the cube configuration diverts the flow to such an extent that the forces reduce. Compared to the case with four square cylinders we see that a two-dimensional structure cannot take advantage of the same effect, as the maximum force on the cylinders in the 2×2 cylinder configuration increased compared to a single cylinder.

Fig. 11 shows the total horizontal force vector for a 3×3 configuration at different phases of the oscillation. We observed that when corner-cubes experienced the maximum force, away from the group of cubes, the force orientated towards the centerline of the configuration.

Fig. 12 shows phase-averaged forces from tests with 6×6 cubes. Again, we used the maximum force on a single cube to normalize forces on the cubes. As in 3×3 configuration, the forces were for all cubes reduced compared to the single cube. The forces on cubes situated inside the 6×6 configuration experienced approximately the same force, while the three cubes on the boundary experienced the largest force in each half cycle of the oscillation. The force was largest in the half cycle where it pointed away from the cubes.

Fig. 11. The resulting force on each cube over a full oscillating period for the 3×3 configuration for KC = 12.

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4.3.2 6×6 cubes

KC = 6.0 KC = 12.0

Fig. 12. Phase-averaged inline forces on cubes in the 6×6 configuration. The forces were normalized with the maximum inline force on a single cube. Blue lines (-, --, -.) are legends for cube 1,2, and 3. Red lines (-, --, -.) are legends for cube 4,5, and 6. Green lines (-, --, -.) are legends for cube 7,8, and 9.

Fig. 13 shows that the cube situated in the corner of the 6×6 configuration had its largest force directed towards the centerline of the configuration. For all the other cubes the transverse force was in general smaller than the transverse forces on the corner-cubes in the 3×3 configuration.

Fig. 14 illustrates the shadow effect, - or armoring effect -, of a group of cubes as function of KC. Increasing KC-number enhanced the armoring effect. For small KC-number the flow motion is expected to be local and with only small vortices and no vortex shedding, which could explain the lower effect for low KC-numbers. For both configurations, the reduction of the force appeared to reach a plateau around KC = 6-8, and then the force continues to reduce for increasing KC. For circular cylinders vortex-shedding starts for KC > 7, and therefore a change of the flow regime could explain the plateau. Note, that if a Froude-Krylov force was added the effect of the armoring effect could be smaller.

Fig. 13. The resulting force on each cube over a full oscillating period for the 6×6 configuration for KC = 12.

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Fig. 14. Reduction of the maximum force of the three middle cubes in the 3×3 configuration, left, and three cubes to the right in the 6×6 configuration.

5 Summary and conclusions

We studied the forces on square cylinders and cubes in an oscillatory flow. The experimental set-up with simple force cells in large groups have been validated.

The results with a single square cylinder corresponds to results earlier reported in the literature in for instance Bearman et al. (1984) and Bjørke et al. (2018). In the 2×2 cylinder configuration the lee side cylinder experienced the largest force away from the configuration. These forces were in general larger than the maximum force on a single square cylinder.

The forces on a single cube were mainly qualitative verified against literature, but the measured forces were believed to be well represented by the experiments. The maximum force on the cubes was smaller for groups of cubes compared to the single cube. It was argued that this could be a three-dimensional effect, as it was not observed for the square cylinders. In the 3×3 configuration the largest force was found on cubes on the lee side of the configuration in each half cycle, however the trend did not appear as strong as in the cases with cylinders.

In the 6×6 configuration, the forces on the second and third row of cubes were approximately the same while the exposed cubes experienced the largest forces, and typically, when they were situated on the lee side of the configuration.

The effect of reduced forces increased with increasing KC-number. The effect was largest for the 6×6 configuration, where the forces on the inner cubes were reduced up to 25% at KC = 12. The reduction was only 18 % for the inner row of cubes in the 3×3 configuration. If a Froude-Krylov force was added the armoring effect is expected to be smaller.

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