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

  Breaking wave load of a slender cylinder in a cylinder group  II

Contents

  Contents  II

  Nomenclature  III

  Figures  V

Tables VII   

1.  Introduction  1

2.  State of knowledge  4

2.1.  Problem statement and procedure of analysing the state of knowledge  4

2.2.  Flow around a cylinder  4

2.2.1.  Steady flow  5

2.2.2.  Oscillatory flow  8

2.3.  Forces on a single cylinder  8

2.3.1.  Drag force  9

2.3.2.  Mass force  10

2.3.3.  Drag and mass coefficient  13

2.4.  Breaking waves  14

2.5.  Single cylinder in breaking waves  18

2.6.  Cylinder group  20

2.6.1.  Tandem arrangement  21

2.6.2.  Side by side arrangement  24

2.7.  Experimental investigations with cylinder groups  27

2.8.  Summary of existing results  35

3.  Experimental investigation  37

3.1.  Experimental set-up  37

3.2.  Measuring instruments  39

3.3.  Cylinder group configurations  40

3.4.  Testing programme  41

4.  Evaluation of the experimental results  48

4.1.  Wave kinematics  48

4.1.1.  Wave height  48

4.1.2.  Wave celerity  53

4.1.3.  Wave length  55

4.2.  Total force on single cylinder  56

4.3.  Total force on cylinder groups  60

4.3.1.  Tandem arrangement  60

4.3.2.  Side by side arrangement  62

5.  Summary and concluding remarks  68

6.  Acknowledgements  71

7.  References  72

Breaking wave load of a slender cylinder in a cylinder group

Nomenclature

c wave celerity C cylinder group coefficient c p pressure coefficient drag coefficient C ^D mass coefficient C ^M d water depth water depth at the breaking location d ^b critical water depth d ^cr e gap between cylinder and wall D diameter of the cylinder u δ

water particle acceleration

δ t f force per unit length drag force per unit length f d

inertia force per unit length f ^m total force per unit length f ^total F force drag force F d

total force on one cylinder within a cylinder group F ^group horizontal force component F ^H F m inertia force F p pressure force total force on a single cylinder F ^single F τ skin friction force H wave height wave height in deep water H 0

wave height at the breaking location H ^b wave height at the cylinder H ^c generated wave height H ^gen KC Keulegan-Carpenter number l*w*d wave flume dimension L wave length L 0 wave length in deep water

Breaking wave load of a slender cylinder in a cylinder group

LC loading case virtual mass M ^V M moment

M group moment on the top of the measuring cylinder within a cylinder group M single moment on a single cylinder added mass m ^a mass of the displaced water m ⁰ p pressure R radius of the cylinder Re Reynolds number s wave steepness S separation point centre spacing S ^C gap S ^G SWL sea water level T wave period generated wave period T ^gen t time t d duration of the force of one single wave temporal centre gravity t ^s u horizontal particle velocity in the wave v velocity of the mass of water displaced water volume V ^D WG wave gauge x,y coordinates η water level elevation maximum elevation of the free water surface η ^b density of water ρ ^w ν kinematic viscosity area of the impact λ*η ^b λ‘ vertical asymmetry factor of a breaking wave µ horizontal asymmetry factor of a breaking wave front steepness of a breaking wave ε X,B

  Breaking wave load of a slender cylinder in a cylinder group  V

  Figures  

  Fig. 3 1 Problem statement, objective and proceeding  2

  Fig. 3 2 Definition sketch (modified from Sumer and FredsØe, 1997 )  5

  Fig. 3 3 Flow patterns around a circular cylinder (Sumer and FredsØe, 1997 )  7

  Fig. 3 4 Drag force components (modified from Oumeraci, 2005 )  9

  Fig. 3 5 Pressure coefficient c p at a single cylinder in a) subcritical flow regime and  

  b) supercritical flow regime (Sumer and FredsØe, 1997 )  10

  Fig. 3 6 Displaced water volume (modified from Oumeraci, 2005 )  12

  Fig. 3 7 Pressure distribution for a single cylinder in (a) stationary ideal and  

  (b) instationary subcritical flow (Oumeraci, 2005 )  12

  Fig. 3 8 Drag coefficient as a function of the Reynolds number (Oumeraci, 2005 )  13

  Fig. 3 9 Types of breaking wave (Dean and Dalrymple, 1984 )  15

  Fig. 3 10 Wave of limiting steepness in deep water (CERC, 1977 )  16

  Fig. 3 11 Parameter of breaking waves (Wiencke, 2001 )  17

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

  (Oumeraci, 2005 )  18

  Fig. 3 13 Definition sketch of the impact (Wiencke, 2001 )  19

  Fig. 3 14 Definition sketch of two cylinders in a tandem arrangement  

  (elevation and top view)  21

  Fig. 3 15 Pressure coefficient distribution around two cylinders in a tandem  

  arrangement, (Zdravkovich, 1977 )  22

  Fig. 3 16 Velocity distribution around cylinders in a side by side arrangement at  

  a) S G 0 2 D and b) S G 2 0 D (Hori, 1959 )  25

  Fig. 3 17 Flow around a) free cylinder and b) a near-wall cylinder  

  (Sumer and FredsØe, 1997 )  26

  Fig. 3 18 Configurations of cylinder groups (Apelt and Piorewicz, 1986 )  28

  Fig. 3 19 Relation between F group /F single and wave steepness in a tandem  

  arrangement with a) three cylinders and b) two cylinders  

  (modified from Apelt and Piorewicz, 1986 )  29

  Fig. 3 20 Relation of force ratio and wave steepness in a side by side arrangement  

  (modified from Apelt and Piorewicz, 1986 )  31

  Fig. 3 21 Configurations of cylinder groups (Chakrabarti, 1982 )  32

  Fig. 3 22 Mean curves for C M and C D in a side by side arrangement with a) three  

  cylinder and b) five cylinder (Chakrabarti, 1982 )  33

  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 )  34

  Fig. 3 1 Cross-section and perspective of the model set-up with the instrumented  

  cylinder (Sparboom et al , 2005 )  38

  Breaking wave load of a slender cylinder in a cylinder group  VI

  Fig. 3 2 Longitudinal section (Sparboom et al , 2005 )  39

  Fig. 3 3 Configurations of cylinder groups (Sparboom et al , 2005 )  41

  Fig. 3 4 Loading cases for the selected cylinder group configurations  46

  Fig. 3 5 Wave loading cases and forces on vertical cylinder in experiments in  

  Wiencke (modified from Oumeraci, 2003 )  47

  Fig. 4 1 Water level elevation η near the breaking point  49

  Fig. 4 2 Water level elevation η above SWL  50

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

  case  51

  Fig. 4 4 Characteristical points of the wave crest (Wiencke, 2001 )  53

  Fig. 4 5 Celerity of characteristic points of the wave crest  55

  Fig. 4 6 Moment M of the single cylinder  56

  Fig. 4 7 Measured moments M as a function of the time t  58

  Fig. 4 8 Maximum values of measured moments at the single cylinder  

  (configuration no 1 )  59

  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 S G D  61

  Fig. 4 10 Coefficient C 2 in dependence of the different loading cases  62

  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 S G D  63

  Fig. 4 12 Coefficient C 7 in dependence of the different loading cases  64

  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 S G 3 D  65

  Fig. 4 14 Coefficient C 12 in dependence of the different loading cases  65

  Fig. 4 15 Relation between force ratio and the size of the gap at constant  

  wave height in side by side arrangements  66

  Breaking wave load of a slender cylinder in a cylinder group  VII

  Tables  

  Tab. 3 1 Drag coefficient as a function of the Keulegan-Carpenter number  

  (Barltrop et al , 1990 )  13

  Tab. 3 2 Mass coefficient as a function of the Reynolds number (CERC, 1977 )  14

  Tab. 3 3 Mass coefficient as a function of the Keulegan-Carpenter number  

  (Barltrop et al , 1990 )  14

  Tab. 3 4 Overview over past experimental investigations  36

  Tab. 3 1 Channels for data acquisition  40

  Tab. 3 2 Test programme for breaking waves (Sparboom, 2005 )  42

  Tab. 3 3 Definition of the loading cases and description of the characteristic  

  appearance of the waves (modified from Wiencke, 2001 )  44

  Tab. 4 1 Mean values of the breaker height h b and the wave height at cylinder h c  52

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

  of the loading case  54

  Tab. 4 3 Overview of the results of the coefficient C of the three cylinder groups  67

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

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.

Breaking wave load of a slender cylinder in a cylinder group

group = (1.1) C * F F single

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.

Fig. 3-1 Problem statement, objective and proceeding

Breaking wave load of a slender cylinder in a cylinder group

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.

Breaking wave load of a slender cylinder in a cylinder group

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.

Breaking wave load of a slender cylinder in a cylinder group

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:

max ⋅ D u

Re

u max = maximum horizontal water particle velocity [m/s], D= diameter of the cylinder [m], ν = 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).

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).

Breaking wave load of a slender cylinder in a cylinder group

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*10 ⁵ , the transitional flow regime where 3*10 ⁵ < Re <3.5*10 ⁵ and the supercritical flow regime where 3.5*10 ⁵ < Re < 5*10 ⁵ . 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*10 ⁶ . When the Reynolds number finally increases over a value of 4.5*10 ⁶ 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.

Breaking wave load of a slender cylinder in a cylinder group

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.

Breaking wave load of a slender cylinder in a cylinder group

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

max ⋅

KC

u max = 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 F total acting on a single slender cylinder is due to a drag force F D . In an oscillatory flow an additional component, namely the mass force F M , 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 f total in terms of forces per unit length of the cylinder:

Breaking wave load of a slender cylinder in a cylinder group

= f

total

D= cylinder diameter [m], w ρ = water density [t/m³], u= horizontal water particle velocity [m/s], C M = mass coefficient [-], C D = 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 C D = 1.75 for breaking waves (CERC, 1977). The two components and the force coefficients C M and C D are going to be investigated in detail in the following paragraphs.

2.3.1. Drag force

The drag force F D due to the friction force F τ and the pressure force F P 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.

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

Breaking wave load of a slender cylinder in a cylinder group

In Fig. 3-5 the distribution of the pressure force F P is shown in form of a pressure coefficient c p = 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).

Fig. 3-5 Pressure coefficient c p 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 F M is caused by the acceleration of the fluid in the immediate surroundings of the structure. The accelerated mass M V is called virtual mass, which is composed of the displaced water mass m o and the so called added mass m a .