Breaking wave load of a vertical slender cylinder within a cylinder group

Contents

Nomenclature

Figures

Tables

1. Introduction

2. State of knowledge
2.1. Problem statement and procedure of analysing the state of knowledge
2.2. Flow around a cylinder
2.2.1. Steady flow
2.2.2. Oscillatory flow
2.3. Forces on a single cylinder
2.3.1. Drag force
2.3.2. Mass force
2.3.3. Drag and mass coefficient
2.4. Breaking waves
2.5. Single cylinder in breaking waves
2.6. Cylinder group
2.6.1. Tandem arrangement
2.6.2. Side by side arrangement
2.7. Experimental investigations with cylinder groups
2.8. Summary of existing results

3. Experimental investigation
3.1. Experimental set-up
3.2. Measuring instruments
3.3. Cylinder group configurations
3.4. Testing programme

4. Evaluation of the experimental results
4.1. Wave kinematics
4.1.1. Wave height
4.1.2. Wave celerity
4.1.3. Wave length
4.2. Total force on single cylinder
4.3. Total force on cylinder groups
4.3.1. Tandem arrangement
4.3.2. Side by side arrangement

5. Summary and concluding remarks

6. Acknowledgements

7. References

Nomenclature

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Figures

Fig. 3-1 Problem statement, objective and proceeding

Fig. 3-2 Definition sketch (modified from Sumer and FredsØe, 1997)

Fig. 3-3 Flow patterns around a circular cylinder (Sumer and FredsØe, 1997)

Fig. 3-4 Drag force components (modified from Oumeraci, 2005)

Fig. 3-5 Pressure coefficient cp at a single cylinder in a) subcritical flow regime and b) supercritical flow regime (Sumer and FredsØe, 1997)

Fig. 3-6 Displaced water volume (modified from Oumeraci, 2005)

Fig. 3-7 Pressure distribution for a single cylinder in (a) stationary ideal and (b) instationary subcritical flow (Oumeraci, 2005)

Fig. 3-8 Drag coefficient as a function of the Reynolds number (Oumeraci, 2005)

Fig. 3-9 Types of breaking wave (Dean and Dalrymple, 1984)

Fig. 3-10 Wave of limiting steepness in deep water (CERC, 1977)

Fig. 3-11 Parameter of breaking waves (Wiencke, 2001)

Fig. 3-12 (a) Quasi-static and (b) dynamic component of a breaking wave (Oumeraci, 2005)

Fig. 3-13 Definition sketch of the impact (Wiencke, 2001)

Fig. 3-14 Definition sketch of two cylinders in a tandem arrangement (elevation and top view)

Fig. 3-15 Pressure coefficient distribution around two cylinders in a tandem arrangement, (Zdravkovich, 1977)

Fig. 3-16 Velocity distribution around cylinders in a side by side arrangement at a) SG = 0.2D and b) SG = 2.0D (Hori, 1959)

Fig. 3-17 Flow around a) free cylinder and b) a near-wall cylinder (Sumer and FredsØe, 1997)

Fig. 3-18 Configurations of cylinder groups (Apelt and Piorewicz, 1986)

Fig. 3-19 Relation between Fgroup/Fsingle and wave steepness in a tandem arrangement with a) three cylinders and b) two cylinders (modified from Apelt and Piorewicz, 1986)

Fig. 3-20 Relation of force ratio and wave steepness in a side by side arrangement (modified from Apelt and Piorewicz, 1986)

Fig. 3-21 Configurations of cylinder groups (Chakrabarti, 1982)

Fig. 3-22 Mean curves for CM and CD in a side by side arrangement with a) three cylinder and b) five cylinder (Chakrabarti, 1982)

Fig. 3-23 Relation of a) mass force and b) drag force and the spacing in a tandem arrangement in regular waves (Smith and Haritos, 1997)

Fig. 3-1 Cross-section and perspective of the model set-up with the instrumented cylinder (Sparboom et al., 2005)

Fig. 3-2 Longitudinal section (Sparboom et al., 2005)

Fig. 3-3 Configurations of cylinder groups (Sparboom et al., 2005)

Fig. 3-4 Loading cases for the selected cylinder group configurations

Fig. 3-5 Wave loading cases and forces on vertical cylinder in experiments in Wiencke (modified from Oumeraci, 2003).

Fig. 4-1 Water level elevation η near the breaking point

Fig. 4-2 Water level elevation η above SWL

Fig. 4-3 Wave height at the different wave gauges in dependence on the loading case

Fig. 4-4 Characteristical points of the wave crest (Wiencke, 2001)

Fig. 4-5 Celerity of characteristic points of the wave crest

Fig. 4-6 Moment M of the single cylinder

Fig. 4-7 Measured moments M as a function of the time t

Fig. 4-8 Maximum values of measured moments at the single cylinder (configuration no.1)

Fig. 4-9 Maximum values of measured moments at the single cylinder (configuration no.1) and the cylinder group in a tandem arrangement with a gap SG=D

Fig. 4-10 Coefficient C2 in dependence of the different loading cases

Fig. 4-11 Maximum values of measured moments at the single cylinder (configuration no.1) and the cylinder group in a side by side arrangement with a gap SG=D

Fig. 4-12 Coefficient C7 in dependence of the different loading cases

Fig. 4-13 Maximum values of measured moments at the single cylinder (configuration no.1) and the cylinder group in a side by side arrangement with a gap SG=3D

Fig. 4-14 Coefficient C12 in dependence of the different loading cases

Fig. 4-15 Relation between force ratio and the size of the gap at constant wave height in side by side arrangements

Tables

Tab. 3-1 Drag coefficient as a function of the Keulegan-Carpenter number (Barltrop et al., 1990)

Tab. 3-2 Mass coefficient as a function of the Reynolds number (CERC, 1977)

Tab. 3-3 Mass coefficient as a function of the Keulegan-Carpenter number (Barltrop et al., 1990)

Tab. 3-4 Overview over past experimental investigations

Tab. 3-1 Channels for data acquisition

Tab. 3-2 Test programme for breaking waves (Sparboom, 2005)

Tab. 3-3 Definition of the loading cases and description of the characteristic appearance of the waves (modified from Wiencke, 2001)

Tab. 4-1 Mean values of the breaker height hb and the wave height at cylinder hc

Tab. 4-2 Minimum and maximum values of the wave celerity in dependence of the loading case

Tab. 4-3 Overview of the results of the coefficient C of the three cylinder groups

1. Introduction

Slender cylinders are widely used as a structural element in offshore structures. Oil platforms, jetties and piers are often supported by group of cylinders, which are arranged closely spaced. The Morison equation (Morison et al., 1950) constitutes a simple tool to calculate the wave force on one single cylinder. To what extent the cylinders, which are arranged in a group, affect each other is extensively unclear. As these group interference effects are not considered in the Morison equation there is a lack of a generally accepted formula to calculate the individual forces on cylinders within cylinder groups.

In this student research project the special loading case of breaking waves acting on cylinder groups is examined. Breaking waves developed from wave superposition during a storm may cause great impact loads also in deep water. The investigation of breaking waves leads to the upper bound of possible loads on offshore structures. A closer analysis of the so called impact force and the validation of former assumptions of considering it is not part of this paper. The main focus lies on the interactions between cylinders arranged in groups when a single breaking wave impinges the group or a part of it.

These interactions are investigated based on large-scale experiments, which have been performed in summer 2004 in the Large Wave Flume (GWK) at the Coastal Research Centre (FZK) in Hanover. Fifteen configurations of cylinder groups have been examined, including one configuration with one single cylinder and fourteen configurations of groups up to three cylinders arranged in row or transversely. The single cylinder and one cylinder in each cylinder group are equipped with strain gauges on the top, which measure the bending moments during the tests. These measuring cylinders, in the single arrangement and in the group arrangements, have the same position in the wave flume. Therefore the comparison of the measured bending moments of the single cylinder with those of the cylinder in the group provides information about the influence of the adjacent elements in a cylinder group. The results of the single cylinder test can be taken as a reference for the results of the cylinder group’s tests.

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A coefficient C is going to be developed which describes the increase or decrease of the wave load on the measuring cylinder in the group compared to the single cylinder. The coefficient C will be analysed as a function of the breaking wave conditions, the configuration and the spacing between the cylinders. In this paper only a selection of the fifteen performed tests is going to be analysed. The following flow chart shows the problem statement, the objective and the proceeding of this paper.

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Fig. 3-1 Problem statement, objective and proceeding

As required in the task, the state of knowledge is first described in chapter 2. Therein the required report of previous experiments is given. The recent experiments in Hanover are reported in chapter 3. Therein the experimental set-up, the measurement techniques and the testing programme are briefly described. The fourth chapter deals with the evaluation of these experiments. Finally, in chapter 5, the main results are briefly summarised. A required comparison with the published results of previous experiments is also given in chapter 5. The same chapter includes furthermore a prospect with continuative aspects for further research.

2. State of knowledge

2.1. Problem statement and procedure of analysing the state of knowledge

The Morison equation gives a simple engineering tool to determine the wave force acting on slender cylinders. Slender means, that the structure with a certain diameter D is sufficiently small compared to the wave length L (D/L < 0.05). In this case diffraction effects are negligible. The equation is applicable for vertical, circular, slender cylinder with a smooth surface without neighbouring cylinders. In an arrangement of closely spaced cylinders the influence of the neighbouring cylinders on the surrounded fluid field is likely to be expected. As the Morison equation is only applicable for single cylinder, the need of a modification of the given equation in order to consider this interaction is obvious and proposed in Morison et al. (1950).

In this chapter the basic principles of single cylinders in a fluid flow are first described. Furthermore the wave kinematics of breaking waves are studied especially the breaking point as an important parameter in the following examination. Afterwards it is made use of the investigation of single cylinder subjected to breaking waves for further interpretation of cylinder groups. Afterwards the previous knowledge about cylinder groups is presented. Previous publications describing experimental investigations with different configurations of cylinders in breaking and non-breaking waves are briefly reviewed and analysed furthermore. Tab. 3-4 summarises the main parameters of the wave and model set-up of these past investigations. A conclusion closes this chapter.

2.2. Flow around a cylinder

In this paragraph the description of the flow around a cylinder is separated into two different flow types, namely the steady flow and the oscillating flow. In a steady flow the fluid is characterised by a smooth uniform movement with a locally constant velocity. The oscillating flow is characterised by acceleration of the water particles.

2.2.1. Steady flow

A steady flow is described by the Reynolds number Re and undergoes tremendous changes as the Reynolds number increases from zero. The Reynolds number is formulated as:

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_umax= maximum horizontal water particle velocity [m/s], D= diameter of the cylinder [m],
[illustration not visible in this excerpt]= kinematic viscosity [m²/s] (1.0*10-6 m²/s at 20°C)

While considering a smooth circular cylinder in a steady current, two different flow regions are established, namely the wake and the boundary layer. A definition sketch is given in Fig. 3-2. The wake region extends over a wide distance, which is comparable with the diameter D, while the boundary layer extends over a comparatively small thickness (Sumer and FredsØe, 1997).

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Fig. 3-2 Definition sketch (modified from Sumer and FredsØe, 1997)

In Fig. 3-2 the separation point locates the separation of the boundary layer from the surface of the structure. For small values of Reynolds number no separation occurs. The separation first appears when the Reynolds number becomes 5. At that value the appearance of a fixed pair of vortices at the rear part of the cylinder is found. The next range is characterised by a laminar vortex street (40 < Re < 200). The wake becomes partly turbulent where 200 < Re < 300. At Reynolds numbers greater than 300 the following three ranges could be described: the subcritical flow regime where 300 < Re < 3*105, the transitional flow regime where 3*105< Re <3.5*105 and the supercritical flow regime where 3.5*105 < Re < 5*105. The subcritical range is characterised by a completely turbulent wake but the boundary layer over the cylinder surface remains laminar. The next Reynolds number regime, the so called transitional flow regime, shows a turbulent boundary layer separation at one side of the cylinder. The boundary layer separation becomes turbulent at both sides of the cylinder in the supercritical flow regime, but the boundary layer is not completely turbulent. The transition to turbulence is located somewhere between the stagnation point and the separation point. The boundary layer becomes turbulent at one side when the Reynolds number reaches the value of about 1.5*106. When the Reynolds number finally increases over a value of 4.5*106 the boundary layer becomes turbulent over the whole cylinder surface. This flow regime is called transcritical flow regime (Sumer and FredsØe, 1997). A detailed classification with a brief description of each flow regime is given in Fig. 3-3, which shows examples of flow patterns and occurring vortex shedding at all Reynolds numbers.

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Fig. 3-3 Flow patterns around a circular cylinder (Sumer and FredsØe, 1997)

As described above, vortex shedding occurs when the Reynolds number reaches values greater than 40. For these values the boundary layer is separated by the surface of the cylinder and a shear layer is formed. A significant amount of vorticity contained in the boundary layer is fed into the shear layer and causes it to roll up into a vortex. Likewise a vortex is formed at the other side of the cylinder, which rotates in the opposite direction (Sumer and FredsØe, 1997).

2.2.2. Oscillatory flow

In case of a smooth circular cylinder exposed to an oscillatory flow, an additional parameter is to be taken into consideration. The so called Keulegan-Carpenter number KC is defined by

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_umax= maximum horizontal water particle velocity [m/s], T= wave period [s], D= cylinder diameter [m]

Small Keulegan-Carpenter numbers testify that the orbital motion of the water is small relative to the width of the cylinder. At a Keulegan-Carpenter number smaller than 1.1 a flow separation behind the cylinder does not occur. On the other hand, a large Keulegan-Carpenter number indicates a quite large motion of the water particles relative to the total width of the cylinder, resulting in separation and probably vortex shedding. For very large Keulegan-Carpenter numbers (KC → ∞) each half period of a wave motion can be assumed as a steady flow. The different flow regimes vary as a function of the Reynolds number (for more details compare Sumer and FredsØe, 1997).

2.3. Forces on a single cylinder

In section 2.2 the flow around the cylinder was described. This flow will exert resultant forces on the cylinder. These forces might be classified into an in-line force, which affects the cylinder in the in-line direction, and a lift force acting on the cylinder in the transverse direction. In this paper only the in-line force is investigated. In case of a steady flow, the total in-line force Ftotal acting on a single slender cylinder is due to a drag force FD. In an oscillatory flow an additional component, namely the mass force FM, is to be considered as a result of the acceleration of the water. The wave force is given by the Morison equation (Formula 2.3). The equation considers the two components of the total force ftotal in terms of forces per unit length of the cylinder:

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D= cylinder diameter [m], [illustration not visible in this excerpt]= water density [t/m³], u= horizontal water particle velocity [m/s], CM = mass coefficient [-], CD= drag coefficient [-].

Generally, the larger load of breaking waves compared to non-breaking waves is considered by multiplying the drag coefficient with a factor of 2.5. The outcome of this is a drag coefficient of CD = 1.75 for breaking waves (CERC, 1977). The two components and the force coefficients CM and CD are going to be investigated in detail in the following paragraphs.

2.3.1. Drag force

The drag force FD due to the friction force Fτ and the pressure force FP and is related to the water particle velocity u. In Formula 2.1 the drag force is determined with the velocity-squared term in form of u*|u| to ensure that the drag force has always the equal direction of the velocity.

The contribution of the friction drag to the total drag force is less than 2-3%, so that the friction drag can be omitted in most of the cases (Sumer and FredsØe, 1997). Fig. 3-4 shows a definition sketch with the two components.

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Fig. 3-4 Drag force components (modified from Oumeraci, 2005)

In Fig. 3-5 the distribution of the pressure force FP is shown in form of a pressure coefficient cp= p/ (ρ*U²/2) for different values of the Reynolds number. Therein S denotes the separation points of the flow from the surface of the cylinder. The stagnation point of the flow causes a pressure maximum at the front of the cylinder. Furthermore, the pressure distribution shows a constant negative pressure behind the separation point. This indicates that the wake region is calm compared to the outer-flow region. A conspicuous movement of the separation point to the rear part of the cylinder occurs when the flow regime changes from subcritical to supercritical. Simultaneously the negative pressure at the rear part is reduced, which leads to a reduction in the drag (Sumer and FredsØe, 1997).

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Fig. 3-5 Pressure coefficient cp at a single cylinder in a) subcritical flow regime and b) supercritical flow regime (Sumer and FredsØe, 1997)

2.3.2. Mass force

The so called mass force FM is caused by the acceleration of the fluid in the immediate surroundings of the structure. The accelerated mass MV is called virtual mass, which is composed of the displaced water mass mo and the so called added mass ma.

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MV = virtual mass, m_o = mass of the displaced water, m_a = added mass

The displaced water volume is shown in Fig. 3-6. The displaced water mass mo is independent from the shape of the structure and to determine by the formula given in Fig. 3-6.

In order to explain the force Fma as a result of the acceleration of the added mass, the comparison with an ideal stationary flow is added. In an ideal stationary flow the water particle velocity is constant so that no water particle acceleration occurs. A single cylinder in an ideal stationary flow with the pressure distribution on its surface is shown in Fig. 3-7 (a). The forces acting on a solid body surrounded by water have to be in balance.

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In case of an ideal flow without water particle acceleration, from Formula (2.4) results the added mass force Fma to be zero. According to Formula (2.6) the pressure force on the front and on the back has the same amount, which results from the flow not to separate from the cylinder.

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Fig. 3-6 Displaced water volume (modified from Oumeraci, 2005)

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Fig. 3-7 Pressure distribution for a single cylinder in (a) stationary ideal and (b) instationary subcritical flow (Oumeraci, 2005)

In case of an unsteady flow the water particle acceleration causes the force Fma according to Formula (2.4). The amount of this force corresponds to the sum of the front and back force with the opposite sign:

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2.3.3. Drag and mass coefficient

In formula 2.1 two dimensionless force coefficients, the mass coefficient CM and the drag coefficient CD, are included. In general the mass coefficient CM and drag coefficient CD are determined experimentally.

The drag coefficient CD is influenced by the structural shape and the flow characteristics and additionally by surface conditions. Fig. 3-8 shows the drag coefficient CD as a function of the Reynolds number. The given values are recommended by CERC (1977).

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Fig. 3-8 Drag coefficient as a function of the Reynolds number (Oumeraci, 2005)

The dependence of the drag coefficient on the Keulegan-Carpenter number is given with the following values described in Barltrop et al. (1990):

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Tab. 3-1 Drag coefficient as a function of the Keulegan-Carpenter number (Barltrop et al., 1990)

The mass coefficient is a function of structural shape, flow characteristics and the arrangement of structural elements relative to each other (Gibson and Wang, 1977). The mass coefficient CM indicates which added mass proportional to the displaced water mass has to be accelerated.

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In case of a circular cylinder in a laminar flow the added mass has the same value as the displaced water mass ma = mo. The CERC (1977) recommended the values given in Tab. 3-2 for the mass coefficient CM for the different ranges of Reynolds numbers mentioned above:

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Tab. 3-2 Mass coefficient as a function of the Reynolds number (CERC, 1977)

CERC (1977) does not state the mass coefficient to depend on the KC number. In Barltrop et al. (1990) the following values are recommended for the mass coefficient CM in dependence of the Keulegan-Carpenter number:

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Tab. 3-3 Mass coefficient as a function of the Keulegan-Carpenter number (Barltrop et al., 1990)

2.4. Breaking waves

Offshore structures are subjected to current, wind, earthquake, ice, ship-collision and in particular to swell, including non-breaking and breaking waves. Latter are classified as spilling, plunging and surging breakers, depending on their appearance (Fig. 3-9). The steepness of the wave front is the most critical factor for the type a breaking wave as which it will be formed.

In shallow water a wave is breaking because of a change of wave parameters. The bottom decelerates the wave which steeps the wave at the same time. By reaching a limiting steepness the wave begins to break and in doing so dissipates a part of its energy. In deep water breaking waves may occur when the critical wave steepness is exceeded due to wave-wave-interaction.

Spilling breakers occur at flat bottom slopes. As a wave moves toward the beach the wave steepness increases gradually and the peak of the crest gently slips down the face of the wave. The water at the crest of a wave may create foam as it spills over. Plunging breakers occur on steeper slopes. This type of breaker is characterised by a large quantity of water at the crest curling out of the wave. In doing so, a temporarily tube of water on the wave face is formed before the water plunges down the face of the wave against the trough with a violent splashing. Large amounts of air are trapped within the tube. A loud explosive sound arises when the trapped air is released. The wave energy is released on a small area. After the initial impingement, there is a tendency for repeated actions by smaller vortices formed after the first impact (Lin und Hwung, 1992).

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a) Spilling breaker

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b) Plunging breaker

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c) Surging breaker

Fig. 3-9 Types of breaking wave (Dean and Dalrymple, 1984)

Surging breakers occur at a steep bottom slope. The breaker tongue moves with a small velocity, so that it does not unscrew from the wave front. Surging breakers only occur with a steep bottom and a high reflection rate, whereas plunging and spilling breakers can appear in flat and deep water.

The dissipated energy of plunging breakers is limited on a temporal and local area. In case of a spilling breaker the energy is released more slowly. Surging breakers release a minor amount of energy. Thus the force acting on a cylinder subjected to plunging breakers describes the upper bound (Wiencke, 2001). During the breaking process air bubbles are entrained between the breaker front and the structure. If a plunging wave impinges on a cylinder the air can escape to both sides in contrary to e.g. a vertical wall. Air is first trapped when the breaker tongue impinges the water in front of the wave, which describes an already broken wave. Thus, when observing a cylinder, trapped air is not to be considered. For the investigation of cylinder groups the water spray which arises from the impinging of the breaker front on the cylinder is of note. The water, deflected by each cylinder may impinge the neighbouring cylinders in form of spray and in that way may heighten the force on it. The experiments reported in this paper were performed with plunging breakers in deep water. Thus a further description of plunging breakers is needed by defining parameters of a breaking wave approaching the breaking point. The breaking point is defined as the point where foam first appears on the wave crest, where the front face of the wave first becomes vertical or where the wave crest first begins to curl over the face of the wave (CERC, 1977).

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Fig. 3-10 Wave of limiting steepness in deep water (CERC, 1977)

In deep water the limiting steepness mentioned above is given by:

s= H₀/L₀ = 0,142 ≈ 1/7 [-] (2.9)

H₀= wave height in deep water [m], L₀= wave length in deep water

This limiting steepness occurs when the crest angle is 120°, as shown in Fig. 3-10. Then the water particle velocity u at the wave crest equals the wave celerity c (CERC, 1977). Myrhaug and Kjeldsen (1986) stated that the wave steepness s does not represent a unique definition of breaking waves as several asymmetric waves can exist within this total steepness, but with a different crest front steepness. Assuming that in general breaking waves do not appear with a symmetric shape Myrhaug and Kjeldsen (1986) defined parameters for considering the asymmetry of a wave (Fig. 3-11). They introduced the vertical asymmetry factor λ’, the horizontal asymmetry factor μ and the crest front steepness εX,B. The crest front steepness εX,B describes the mean crest front inclination related to the sea water level. The steepness is not constant above the sea water level. Especially the formation of the breaker tongue (compare Fig. 3-9 b)) causes a changing shape of the breaker front during the process of wave breaking (Wiencke, 2001).

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Fig. 3-11 Parameter of breaking waves (Wiencke, 2001)

While analysing the force exerted by breaking waves acting on a cylinder this development of the breaking wave is to be considered. Thus the distance between the breaking point and the cylinder represents an important parameter, as stated more precisely in the following paragraph.

2.5. Single cylinder in breaking waves

The force exerted by breaking waves acting on a cylinder can be subdivided into a quasi-static and a dynamic component. The quasi static component is equivalent to the force due to non-breaking waves. It varies with the water level elevation. The maximum horizontal force FH of non-breaking waves has values up to the 2.5-fold compared to the force due to water without wave motion. The duration of the force of one single wave td corresponds to half of the wave period (Oumeraci, 2005).

The maximum force exerted by breaking waves is dynamic in nature and reaches greater values than produced by non-breaking waves. This so called impact force is produced by the steep wave front and large horizontal acceleration at the front of the breaker (Morison et al., 1950). The maximum horizontal force FHmax reaches values of about the 2.5 to 15-fold in comparison to water without wave motion. Characteristic for the impact force is the short duration td with 0.1% to 1% of the wave period (Oumeraci, 2005). The two diagrams in Fig. 3-12 illustrate these coherences.

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Fig. 3-12 (a) Quasi-static and (b) dynamic component of a breaking wave (Oumeraci, 2005)

The duration of the impact is defined as the time during which the cylinder enters the wave, more precisely the time between completely uncovered and completely submerged (Easson and Greated, 1985). Afterwards the submerged cylinder experiences forces correspond to a non-breaking wave during the passage of the rest of the wave crest.

A theoretical description of the impact force is given in Wiencke (2001). There the area of the impact (λ*ηb) is the portion of the cylinder that is struck by the breaker front. The wave front is assumed to be vertical and hits the cylinder simultaneously along the height of the impact area. Fig. 3-13 shows a sketch of a breaking wave and its parameters while impinging a single cylinder.

Wiencke (2001) examined the impact force on a single cylinder due to breaking waves based on large-scale experiments in the Large Wave Flume (GWK) in Hanover and concluded the impact force to depend on the position of the breaking point related to the cylinder. The highest force occurs for the case of a wave breaking just in front of the cylinder (compare Fig. 3-5, loading case 3).

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Fig. 3-13 Definition sketch of the impact (Wiencke, 2001)

A wave breaking further from the cylinder causes a twofold impact. First the breaker tongue impinges the cylinder and afterwards the wave front itself. The impact force then has smaller value compared to the maximum forces exerted by a breaking wave just in front of the cylinder (Wiencke, 2001). Griffiths et al. (1992) also determined the greatest forces exerted on a single slender cylinder at the breaking point (defined as the point where the wave first becomes vertical), because of the impact acting on the structure over the full height of the wave front.

Wiencke considered the so called pile-up effect, which arises from considering the physical principle of the deformation of the free water surface. This effect describes an accumulation of the displaced water at the edge of the cylinder, causing a prior immersion of the cylinder. When ignoring this effect the duration of the impact is wrongly lengthened because the pressure transducers are earlier submerged by the pile-up effect and therefore the maximum forces are reduced.

2.6. Cylinder group

The assumption that a cylinder arranged in a group behaves in an identical manner like a single one is only justified if the cylinders are sufficiently apart. Otherwise the interference effects between closed spaced cylinders drastically changes the surrounded flow and produces unexpected forces, pressure distributions and intensifies or suppresses vortex shedding (Zdravkovich, 1977). A cylinder group is defined as a conglomeration of at least two cylinders. Related to the wave direction the cylinders can be arranged staggered or in row. Latter may be aligned as a tandem arrangement, where the cylinders are arranged one behind the other at any longitudinal spacing, or as a side-by-side arrangement, where the cylinders face the flow at any transverse spacing. In this paper only cylinders in row are taken into consideration. A tandem arrangement of two cylinders with its parameters and designation used in the following is shown exemplarily in Fig. 3-14. The figure shows the cylinder group subjected to horizontal wave forces due to breaking waves, which are investigated in this paper. The spacing between the cylinders was differently defined in former investigations. Therefore a clear unique definition is required. The spacing SC describes the centre spacing of the cylinders. The gap between the cylinders is designated with SG.

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Fig. 3-14 Definition sketch of two cylinders in a tandem arrangement (elevation and top view)

In the following two paragraphs the changes in the flow pattern for tandem and side by side arrangements are described by analysing the pressure distribution, the velocity profiles and vortex shedding. All adduced experiments were performed with a cylinder group in an air flow. Subsequent a general estimation for excepted results of these experiments in Hanover are going to be made.

2.6.1. Tandem arrangement

First of all, the change in the pressure distribution of closely spaced cylinders in a tandem arrangement is investigated. Hori (1959) carried out experiments in a wind tunnel with two cylinders with D = 5 mm and a Reynolds number of 8.103 at three different centre spacing. Fig. 3-15 shows the pressure distribution around the upstream and downstream cylinder arranged in a tandem arrangement.

The pressure distribution of the upstream cylinder is similar to the pressure distribution of the single cylinder, shown in Fig. 3-5 a). The presence of the downstream cylinder only affected the pressure distribution of the upstream cylinder on its rear part. With an increasing of the centre spacing SC from 1.2D to 3D the drag force at the rear part was reduced about 20% (Zdravkovich, 1977).

The downstream cylinder has a low negative pressure at its front, which has the corresponding value of the pressure of the upstream cylinder. This fact is an indication, that the flow in the gap is almost stagnant. The negative pressure in front of the downstream cylinder exceeds the negative pressure at its rear part. Hence, the downstream cylinder experiences a thrust force (Hori, 1959). On both sides of the downstream cylinder two symmetrical maximums are recognizable. This indicates the existence of so called reattachment points, which are the result of the reattachment of the flow separated by the upstream cylinder. With an increasing in the centre spacing SC the reattachment points appear further to the front of the cylinder (Zdravkovich 1977).

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Fig. 3-15 Pressure coefficient distribution around two cylinders in a tandem arrangement, (Zdravkovich, 1977)

In case of a centre spacing of 1.2D no maximum appears at the sides of the downstream cylinder. This indicates that the separated flow unites behind the downstream cylinder. Thus the downstream cylinder is positioned in the total shelter of the upstream cylinder.

Zdravkovich and Stanhope (1972) examined a tandem arrangement at higher Reynolds numbers (1*105 – subcritical range). There the centre spacing SC varies from 1.5D to 7D. For centre spacing up to 3.5D they obtain the same results as found at lower Reynolds numbers in Hori (1959). For tandem arrangements with a centre spacing SC wider than 3.5D the pressure distributions of the different arrangements are similar among themselves but no reattachment points were noticeable. The only difference was found in a reduction of the stagnation pressure of the downstream cylinder with the decrease in the centre spacing. In an arrangement with a centre spacing of 5D the presence of the upstream cylinder causes a reduction of the stagnation pressure coefficient from cp = 1 for the single cylinder to cp = -0.2 for the downstream cylinder in the tandem arrangement. (Zdravkovich, 1977)

Investigations of the velocity profiles in the gap between the two cylinders (at 0.25SG, 0.5SG and 0.75SG) and behind the downstream cylinder (at 0.25SG) were published in Zdravkovich and Stanhope (1972). For all spacing up to 3.5D the velocity was similar and showed low velocity in the gap. The magnitude of the velocity in the wake behind the downstream cylinder was considerably greater than in the gap. The flow pattern in the gap experienced a sudden change when the centre spacing SC increases beyond 3.5D. Beyond a centre spacing of 3.5D the velocity profiles in the gap and the wake became similar and show a nearly full developed flow approaching the downstream cylinder.

As obtained from the pressure distribution, the flow in the gaps at a centre spacing of up to 3D is stagnant. This is verified by the results of the velocity profiles. The described reduction of the stagnation point pressure at the downstream cylinder with the decreasing in size of the gap indicates a shelter effect of the upstream cylinder.

A single cylinder in a flow causes vortex shedding at its rear part depending on the Reynolds number. Fig. 3-3 shows the flow patterns of a single cylinder at different Reynolds numbers. If a second cylinder is positioned close behind another, it is likely to be expected that the vortex shedding is influenced of its presence. In experiments with a tandem arrangement of two cylinders vortex shedding occurred behind the upstream cylinder for a centre spacing SC greater than 4D. Contrary to the upstream cylinder, vortex shedding was detected in the whole range of spacing behind the downstream cylinder (Zdravkovich, 1977).

Concluding the paragraph, the experiments revealed a sheltering effect in a tandem arrangement for the downstream cylinder, which depends on the size of the gap between the cylinders. The flow in the gap is nearly stagnant. Vortex shedding is often suppressed by the presence of the downstream cylinder.

2.6.2. Side by side arrangement

Hori (1959) also examined the pressure distribution on the surface of cylinders which where positioned in a side by side arrangement. Although the geometrical arrangement of the two cylinders is symmetrical in regard to the flow, the pressure distribution differs from each other. This indicates the existence of an interaction between cylinders in a side by side arrangement.

In the range of centre spacing SC from 1.1D to 2.2D the flow around two cylinders in a side by side arrangement is of a bistable nature, which means that the cylinders experiences two bistable forces. The pressure at the rear part of a cylinder changes from one steady value to another or simply fluctuated between the two extremes (Zdravkovich, 1977).

Another distinctive feature is the change of the position of the stagnation point. The stagnation point of both cylinders approaches the other cylinder with a decreasing in the size of the gap SG from 2D to 0.2D. Furthermore the stagnation region broadens out (Hori, 1959).

The velocity profiles in the wake of two cylinders in a side by side arrangement are investigated in Hori (1959). At a distance of 0.9D and 6D from the transverse axis of the two cylinders the velocity was measured. In Fig. 3-16 (a) and (b) the results are shown as a ratio of the measured velocity U to the main stream velocity U∞. At a spacing of SG = 0.2D the velocity in the gap of the cylinders does not reach the height of the velocity on the outer margins of the cylinders. The velocity becomes equal in the gap and at the outer margins at a spacing of SG = 2.0D.

In case of a spacing SG = 0.2D the curve shows two valleys at x/d = 0.9 (position right behind the cylinder), which indicates a weak flow regime at the rear part of the cylinders. Farther downstream at x/d = 6 these two ranges of low velocities are coalesced and form one single valley. At a spacing SG = 2.0D these two valleys occur also at x/d = 0.9, but are not combined at x/d = 6.

Hori (1959) found this to correlate with the observed vortex shedding, as the frequency at a spacing SG = 0 correspond to that of a single cylinder of 2D, whereas at a spacing SG = 2D the frequency is almost equal to a single cylinder with a diameter 1D.

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Fig. 3-16 Velocity distribution around cylinders in a side by side arrangement at a) SG = 0.2D and b) SG = 2.0D (Hori, 1959)

In the paper of Zdravkovich (1977) another experiment is presented. At a centre spacing SC from 3D to 2.5D the vortices of the two cylinders in a side by side arrangement are symmetric related to the axis of the gap. At a spacing SC of 2D the gap flow is deflected upwards. The vortex shedding becomes weak at a centre spacing SC of 1.5D, while the gap flow is biased to one side and wide narrow wakes are formed behind the cylinders. The same trend continuous at SC = 1.25D with a biased flow in the gap changing over either downwards or upwards (Zdravkovich, 1977).

Sumer and FredsØe (1997) investigated the changes in the flow around a cylinder, which is placed near a wall. As this scenario resembles the side by side arrangement the results are presented at this point. A conformance could not be assumed but the rough basics of the physical effects might help for further interpretation. Fig. 3-17 shows a sketch of a cylinder in a free flow and arranged near-wall. The separation points are marked with S.

The authors found out, that the stagnation point moves from the centre of the cylinder towards the wall. While the stagnation point is located at about Φ = 0° it moves to the angular position of up to Φ = -40°according to the gap ratio e/D, where e is the gap between the cylinder and the wall and D is the diameter of the cylinder. Similar results were found in (Hori, 1959) for two cylinders in a side by side arrangement like mentioned above. Furthermore, the angle position of the separation point changes. The separation point at the free-stream side of the cylinder is stagnant while the separation point on the wall-side moves downstream, as shown in Fig. 3-17 b)

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Fig. 3-17 Flow around a) free cylinder and b) a near-wall cylinder (Sumer and FredsØe, 1997)

The vortex shedding was found to be suppressed for gap ratio values smaller than e/D= 0.3. This suppression of vortex shedding is linked with the asymmetry in the development of the vortices on the two sides of the cylinders. The vortex on the free-stream side of the cylinder grows larger and stronger than the vortex on the wall-side. Therefore the interaction of the two vortices is largely inhibited, which results in a partial or complete suppression of the regular vortex shedding. Finally, the suction on the free-stream side of the cylinder is larger than on the wall-side. This effect disappears with a restored symmetric pressure distribution at a gap ratio of e/D = 1 (Sumer and FredsØe, 1997).

2.7. Experimental investigations with cylinder groups

In the last chapters the theoretical description of breaking waves and slender cylinders in a flow, single positioned or arranged in a group, was reported. In the following the past performed experiments are reported and analysed to give additional practical results. According to the author’s knowledge, only one examination with cylinder groups in breaking waves has been published. Therefore publications with cylinder groups in non-breaking waves are added.

Apelt and Piorewicz (1986) investigated groups of slender cylinders subjected to breaking waves. In a wave flume with the dimensions 26 m x 0.9 m x 0.6 m and 15 m x 3 m x 0.5 m they investigated wave breaking occurring on the bottom slope of 1:15. The authors found the critical water depth dcr, where the wave force has the maximum value at the cylinder, to be always shoreward from the breaking point and given by:

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The experiments were carried out with waves with a wave steepness H_o/L_o of 0.01, 0.02, 0.03 and 0.05. Apart from a single cylinder, two tandem arrangements of three and two cylinders were considered as well as one side-by-side arrangement of three cylinders. In Fig. 3-18 the three different arrangements of cylinders are shown.

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Fig. 3-18 Configurations of cylinder groups (Apelt and Piorewicz, 1986)

The investigated cylinders have diameters of D = 10.2 cm and 15.3 cm and the gap between the cylinders SG ranged from 0.5D to 5.0D. It should be noted that the width of the spacing in the configuration with a tandem arrangement was limited by the wave flume facilities so that in these configurations only spacing up to 4D were investigated. The instru­mented cylinder was placed where the maximum breaking wave force was experienced with the single cylinder at the critical water depth dcr (compare Formula 2.10). The authors differentiate between the effect of plunging breakers (Ho/Lo ≤ 0.03) and spilling breakers (H_o/L_o > 0.03).

They found out, that in case of the configuration with a tandem arrangement of three cylinders the measuring cylinder in the centre was less loaded than a single cylinder in equal conditions. They determined a dependence on the wave steepness. At small wave steepness of H_o/L_o<0.01 a shelter effect by the presence of the upstream cylinder up to 35% compared to a single cylinder was detected.

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Fig. 3-19 Relation between Fgroup/Fsingle and wave steepness in a tandem arrangement with
a) three cylinders and b) two cylinders (modified from Apelt and Piorewicz, 1986)

This high sheltering effect occurs only in plunging breakers. The sheltering effect decreases with an increase of the wave steepness. So in spilling breakers a sheltering effect of maximal 20% can be expected. They found only a little effect of the size of the gap for this arrangement. So they concluded the force on the centre cylinder approximately as:

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for all gaps SG ≤ 5D and wave steepness H₀/L₀ ≤ 0.05. The authors used the linear character of the curve in Fig. 3-19, which has a slope of y/x = 4.5 and a y-axis intercept of 0.65 for the curve elongated to the y-axis (compare Fig. 3-19 a)).

The great sheltering effect for the centre cylinder occurring in plunging breakers compared to spilling breakers indicates that the impact of the breaker front plays a role for the magnitude of sheltering effect, as an impact does not occur in spilling breakers. The power of the impact separates the wave so that the downstream cylinder experiences only an alleviated, sideways impact of the breaker front. As the size of the gap over the range tested has only a little effect on the wave force of the centre cylinder, this separation seem to be stable over a length of 5D.

In case of two cylinders in a tandem arrangement, the wave force of the instrumented upstream cylinder depends on the wave steepness. Compared to a single cylinder the upstream cylinder, with a second cylinder arranged in behind, experienced smaller forces in plunging breakers (up to 13%) and slightly larger forces in spilling breakers (up to 13%). In this arrangement the spacing has also little influence. Over the range covered by the experiments, the authors described the force on the upstream cylinder approximately by:

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