Water wave theories and Wave loads

Stochastic Analysis of Offshore Steel Structures - An analytical Appraisal. Springer. 2013. Halil Karadeniz, Vedat Togan

Chapter3 Water wave theories and Wave loads

Introduction

海洋是人类已使用的地球重要部分，他们占了大约70％地球，含有丰富的成分和未开发的能源。大洋用于运输、粮食资源、军事用途等。
很多注意力都集中在发现上海底碳氢化合物的开采和利用[3]可再生能源[4]。海洋环境非常复杂和混乱，有时会很平静，主要是受到大气现象的干扰，偶尔在构造区域被地球地震运动激发而导致在生活和经济方面的灾难后果[1]。海洋表面的扰动是不规则的水波，主要由风产生。风产生风暴、地震、月球和月球的引力等来源孙，物理海浪可分为以下几种类型：源力、波形运动、水深、波长、波高度和周期等。月亮和太阳的引力吸引力产生海浪。==风从最小到最大的顺序是毛细管波或波纹、超重力波、重力波和次重力波。==风暴、滑坡、地震和爆炸可能会产生海啸，它是比风力产生的更长波和周期，以及更大的灾难性波浪。除了这些波浪，有时在海域还会发生很大的波浪。这些被称为怪波（freak waves）[5]。尽管其原因尚不清楚，已知的物理机制暗示了对怪异的波浪现象可能是波浪-电流相互作用、几何和衍射聚焦，由于色散和调制不稳定性引起的聚焦，碰撞和大气作用[5]。

海洋中最常见的波浪环境是由于风在不同阶段产生波浪毛细管、海浪和涌浪。风产生的波的大小取决于风速、持续时间和获取时间（风在吹过的水面长度比上水深）。在完全平静的海面上，风几乎没有抓地力，因此在水表面上掠过，水移动产生涡流和小波纹形成毛细波。由于这些涟漪，风更好地抓住了水面，它变得更粗糙。逐渐地，它发展出短波的不规则波传播在不同的方向。这种波浪状况称为海浪，它取决于风的持续时间和获取时间，海浪发展成一个完全发达的海面，达到最大大小，速度和超出抓取时间的周期，即风将最大的能量传递给了海浪，此时大海浪已经完全发展。

当海浪移出风力输入能量区域时，称为作为浪涌。它停止增长并获得通常的能量总大小以及光滑圆润的外形。由于它失去了能量输入，因此在传播过程中，波开始衰减。海浪的表面起伏不规则，可以由许多振幅不同周期、长度和方向的规则谐波组成，它们总共构成一个随机的表面高程。这些波形的条件定义了随机海浪谱[6，7]，分析方法本质上是用来定量的。海浪也可以根据相对水深d/L分为深水波、浅水波和非常浅水波，其中d是静止水深，L是波长[8]。从理论上讲，规则波可以是由满足在水面和海底多个边界条件的波动方程确定。根据简化边界条件的程序，水波理论分为两种，线性和非线性（有限振幅），请参阅[8-21]。由于海浪是随机的，因此随机描述实际上是在实践中使用的[6，7]，它构成了基本海洋结构随机分析的输入[22，23]。除其他外波浪是最重要的现象，对海洋中的海上结构施加载荷必须将结构设计为可以长期承受的随机力。由于海浪在海底结构上产生主要的动态载荷短期和长期都会造成疲劳损伤，在本章中，将介绍水波及其随机表示。然后，进行波浪载荷的随机分析计算程序基于波荷载公式的水-结构相互作用，着重强调附加质量和流体动力阻尼。

Introduction to Wave Theories

Water surface waves are obtained from inviscid, incompressible and irrotational flow under certain boundary conditions by using the principles of hydrodynamics [24]. To obtain wave equations the continuity of water is used. The continuity in fluid dynamics states that the fluid mass is conserved, i.e., in a steady state process, the rates of mass entering and leaving a system are equal which leads to the continuity equation [10], in general,
∂ρ∂t+∇⋅(ρV)=0\frac{\partial\rho}{\partial t}+\nabla\cdot (\rho V) =0∂t∂ρ+∇⋅(ρV)=0
where ρ\rhoρ is the mass density and VVV is the velocity vector of the water, ∇\nabla∇ is the divergence operator defined as
∇=∂∂xi+∂∂yj+∂∂zk\nabla=\frac{\partial}{\partial x}i+\frac{\partial}{\partial y}j+\frac{\partial}{\partial z}k∇=∂x∂i+∂y∂j+∂z∂k

In addition to the translational motion, one other important definition in fluid motion is the rotation of fluid particles. The rotation vector is defined as
Ω=12∇⊗V\Omega=\frac{1}{2}\nabla \otimes VΩ=21∇⊗V
where ⊗\otimes⊗ denotes a vector product. The components in xxx, yyy, and zzz directions are
Ωx=12(∂w∂y−∂v∂z)\Omega_x=\frac{1}{2}\left(\frac{\partial w}{\partial y}- \frac{\partial v}{\partial z}\right)Ωx=21(∂y∂w−∂z∂v)
Ωy=12(∂u∂z−∂w∂x)\Omega_y=\frac{1}{2}\left(\frac{\partial u}{\partial z}- \frac{\partial w}{\partial x}\right)Ωy=21(∂z∂u−∂x∂w)
Ωz=12(∂v∂x−∂u∂y)\Omega_z=\frac{1}{2}\left(\frac{\partial v}{\partial x}- \frac{\partial u}{\partial y}\right)Ωz=21(∂x∂v−∂y∂u)

The irrotationality of the flow states that the rotation components are all zero, i.e., Ωx=0\Omega_x=0Ωx=0, Ωy=0\Omega_y=0Ωy=0 and Ωz=0\Omega_z=0Ωz=0, and the flow undergoes only the translational motion. The continuity and irrotationality conditions of the flow, given by Eqs. (3.1c) and (3.2b), are the basic equations of water waves that satisfy a number of boundary conditions at the bottom and on the free surface of the sea:

At the bottom, the velocity components normal to the bottom surface will be zero and, for a body in the wave, the normal velocity of the water is equal to the normal velocity of the body. If the body is fixed, then the normal water velocity will be zero [14]. These are stated as, if the z axis is in vertical direction,
For inviscid and irrotational unsteady flows, the Bernoulli equation can be obtained from the spatial integration of the Navier–Stokes equation [10] as written in vector notations by

Symbol	Designation
SWL	Still water level
η\etaη	Free surface elevation
HHH	Wave height
LLL	wavelength
η^\hat \etaη^	Wave amplitude
ddd	Water depth
CCC	Wave celerity
CgC_gCg	Wave group velocity
uuu, www	Water particle velocities in horizontal and vertical directions
TTT	Wave period
$x $	Wave frequency (rad/s)
mmm	Wave number
α\alphaα	Wave steepness

Stokes Wave Theory

Other Wave Theory

As presented in Sect. 3.2.1 the Stokes wave theories are most suitable for deep and intermediate water depth. Even higher order terms in the Stokes theory for steeper waves produce unrealistic results. For shallower water, a finite-amplitude wave theory is required. Cnoidal wave theory [9, 36–38] and, in very shallow water, solitary wave theory [9, 39, 40], are the analytical wave theories most commonly used for shallow water. Solutions in the cnoidal wave theory are obtained in terms of elliptical integrals of the first kind. The solitary wave theory is a special case of the cnoidal wave theory at one limit, and at the other limit it is identical with the linear wave theory. As the relative depth decreases the cnoidal wave becomes the solitary wave, which has a crest that is completely above the still water level and has no trough. The cnoidal wave theory covers a large class of long waves with finite amplitudes. It is presented in terms of two parameters, k2 and Ur, where k2 depends on the water depth, the wave length and height, which is one of the independent variables in the elliptical function. The parameter Ur is the Ursell parameter defined as Ur ¼ L2H=d3 which depends on the wave steepness and the relative water depth. This parameter defines the range of application of the wave theories. In general, the cnoidal wave theory is applicable for Ur[25, the Stokes theory is applicable for Ur\10 and both theories are applicable for Ur ¼ 10 25: The limiting case of (k2 = 0) results in the small amplitude wave theory while (k2 = 1) results in the solitary wave theory. The forms (water elevation) of different waves are shown in Fig. 3.2. Since the nonlinear wave theories are not the main topic covered in this book they are not further presented herein. The interested readers should consult related references [8, 9, 14, 16, 27, 32–41]. As the linear (Airy) wave theory is used for the stochastic analysis of offshore structures further in this book, it is presented below in detail.

Linear (Airy) Wave Theory

The linear wave theory (also known as the Airy wave theory) is the simplest and most useful theory among other wave theories. It assumes small wave steepness and small relative water depth (H/d), which allows the free surface boundary conditions to be linearized and satisfied at the mean water level (still water level, SWL). It is equivalent to the first-order Stokes wave theory. The linearized differential equations are stated from Eqs. (3.5a) and (3.5b) as written by,
∇2Φ=0→{Bottom boundary condition. …:∂Φ∂z∣z=−d=0Free surface boundary condition :→(∂2Φ∂t2+g∂Φ∂z)z=0=0\nabla^{2} \Phi=0 \rightarrow\left\{\begin{array}{l} \text { Bottom boundary condition. } \ldots:\left.\frac{\partial \Phi}{\partial z}\right|_{z=-d}=0 \\ \text { Free surface boundary condition }: \rightarrow\left(\frac{\partial^{2} \Phi}{\partial t^{2}}+g \frac{\partial \Phi}{\partial z}\right)_{z=0}=0 \end{array}\right. ∇2Φ=0→{ Bottom boundary condition. …:∂z∂Φ∣∣z=−d=0 Free surface boundary condition :→(∂t2∂2Φ+g∂z∂Φ)z=0=0

The horizontal and vertical water particle velocities are calculated from,

Water particle velocities : →{u=∂Φ∂x=−iη^ωcosh⁡m(z+d)sinh⁡mdei(ωt−mx)w=∂Φ∂z=η^ωsinh⁡m(z+d)sinh⁡mdei(ωt−mx)\begin{array}{l} \text { Water particle velocities : } \rightarrow\left\{\begin{array}{l} u=\frac{\partial \Phi}{\partial x}=-i \hat{\eta} \omega \frac{\cosh m(z+d)}{\sinh m d} \mathrm{e}^{i(\omega t-m x)} \\ w=\frac{\partial \Phi}{\partial z}=\hat{\eta} \omega \frac{\sinh m(z+d)}{\sinh m d} \mathrm{e}^{i(\omega t-m x)} \end{array}\right. \end{array} Water particle velocities : →{u=∂x∂Φ=−iη^ωsinhmdcoshm(z+d)ei(ωt−mx)w=∂z∂Φ=η^ωsinhmdsinhm(z+d)ei(ωt−mx)
The corresponding accelerations for small amplitude waves are calculated from the time derivatives of these velocities as written
Water particle accelerations: →{u˙=∂u∂t=η^ω2cosh⁡m(z+d)sinh⁡mdei(ωt−mx)w˙=∂w∂t=iη^ω2sinh⁡m(z+d)sinh⁡mdei(ωt−mx)\begin{array}{l} \text { Water particle accelerations: } \rightarrow\left\{\begin{array}{l} \dot{u}=\frac{\partial u}{\partial t}=\hat{\eta} \omega^{2} \frac{\cosh m(z+d)}{\sinh m d} \mathrm{e}^{i(\omega t-m x)} \\ \dot{w}=\frac{\partial w}{\partial t}=i \hat{\eta} \omega^{2} \frac{\sinh m(z+d)}{\sinh m d} \mathrm{e}^{i(\omega t-m x)} \end{array}\right. \end{array} Water particle accelerations: →{u˙=∂t∂u=η^ω2sinhmdcoshm(z+d)ei(ωt−mx)w˙=∂t∂w=iη^ω2sinhmdsinhm(z+d)ei(ωt−mx)
The water particle displacements are calculated from the time integration of
velocities as written by,
Water particle displacements: →{ξx=∫udt=−η^cosh⁡m(z+d)sinh⁡mdei(ωt−mx)ξz=∫wdt=−iη^sinh⁡m(z+d)sinh⁡mdei(ωt−mx)\begin{aligned} \text { Water particle displacements: } \rightarrow &\left\{\begin{array}{l} \xi_{x}=\int u \, \mathrm{d} t=-\hat{\eta} \frac{\cosh m(z+d)}{\sinh m d} \mathrm{e}^{i(\omega t-m x)} \\ \xi_{z}=\int w \, \mathrm{d} t=-i \hat{\eta} \frac{\sinh m(z+d)}{\sinh m d} \mathrm{e}^{i(\omega t-m x)} \end{array}\right. \end{aligned} Water particle displacements: →{ξx=∫udt=−η^sinhmdcoshm(z+d)ei(ωt−mx)ξz=∫wdt=−iη^sinhmdsinhm(z+d)ei(ωt−mx)
The real parts of these displacements form an elliptical orbit around a fixed point (x0, z0) as shown in Fig. 3.4. The equation of the orbit is
Water particle orbit: →((ξx0−x0)2cosh⁡2m(z0+d)+(ξz0−z0)2sinh⁡2m(z0+d))=η^2sinh⁡2md\text { Water particle orbit: } \rightarrow\left(\frac{\left(\xi_{x_{0}}-x_{0}\right)^{2}}{\cosh ^{2} m\left(z_{0}+d\right)}+\frac{\left(\xi_{z_{0}}-z_{0}\right)^{2}}{\sinh ^{2} m\left(z_{0}+d\right)}\right)=\frac{\hat{\eta}^{2}}{\sinh ^{2} m d} Water particle orbit: →(cosh2m(z0+d)(ξx0−x0)2+sinh2m(z0+d)(ξz0−z0)2)=sinh2mdη^2
With these velocity and acceleration information the wave forces on structural members can be calculated using the Morison’s equation [35] as it will be explained later in this chapter. However, since the deep water condition is used further in this book, the velocity potential function, velocity and acceleration components, and the orbit equation for deep water condition are presented below.

Formulation for Deep Water Condition

Deep water approximations: →{tanh⁡md≈1sinh⁡m(z+d)≈emzsinh⁡mdcosh⁡m(z+d)≈emzsinh⁡md\text { Deep water approximations: } \rightarrow\left\{\begin{array}{l} \tanh m d \approx 1 \\ \sinh m(z+d) \approx \mathrm{e}^{m z} \sinh m d \\ \cosh m(z+d) \approx \mathrm{e}^{m z} \sinh m d \end{array}\right. Deep water approximations: →⎩⎨⎧tanhmd≈1sinhm(z+d)≈emzsinhmdcoshm(z+d)≈emzsinhmd
Velocity potential function :→Φ=η^gωemzei(ωt−mx)\text { Velocity potential function }: \rightarrow \quad \Phi=\hat{\eta} \frac{g}{\omega} \mathrm{e}^{m z} \mathrm{e}^{i(\omega t-m x)} Velocity potential function :→Φ=η^ωgemzei(ωt−mx)
Water elevation ∴→η=−iη^ei(ω−r−m)x\therefore \rightarrow \eta=-i \hat{\eta} e^{i(\omega-r-m) x}∴→η=−iη^ei(ω−r−m)x
Dispersion relationship :→ω2=mg: \rightarrow \omega^{2}=m g:→ω2=mg
Water particle velocities... :→{u=η^−iωemzei(ωt−mx)w=η^ωemzei(ωt−mx)Water particle accelerations :→{u˙=η^ω2emzei(ωt−mx)w˙=iη^ω2emzei(ωt−mx)\begin{aligned} &\text { Water particle velocities... } \quad: \rightarrow\left\{\begin{array}{l} u=\hat{\eta}-i \omega \mathrm{e}^{m z} \mathrm{e}^{i(\omega t-m x)} \\ w=\hat{\eta} \omega \mathrm{e}^{m z} \mathrm{e}^{i(\omega t-m x)} \end{array}\right.\\ &\text { Water particle accelerations }: \rightarrow\left\{\begin{array}{l} \dot{u}=\hat{\eta} \omega^{2} \mathrm{e}^{m z} \mathrm{e}^{i(\omega t-m x)} \\ \dot{w}=i \hat{\eta} \omega^{2} \mathrm{e}^{m z} \mathrm{e}^{i(\omega t-m x)} \end{array}\right. \end{aligned} Water particle velocities... :→{u=η^−iωemzei(ωt−mx)w=η^ωemzei(ωt−mx) Water particle accelerations :→{u˙=η^ω2emzei(ωt−mx)w˙=iη^ω2emzei(ωt−mx)
Water particle displacements : →{ξx=−η^emzei(ωt−mx)ξz=−iη^emzei(ωt−mx)\text { Water particle displacements : } \rightarrow\left\{\begin{array}{l} \xi_{x}=-\hat{\eta} \mathrm{e}^{m z} \mathrm{e}^{i(\omega t-m x)} \\ \xi_{z}=-i \hat{\eta} \mathrm{e}^{m z} \mathrm{e}^{i(\omega t-m x)} \end{array}\right. Water particle displacements : →{ξx=−η^emzei(ωt−mx)ξz=−iη^emzei(ωt−mx)
Water particle orbit : →(ξx0−x0)2+(ξz0−z0)2=η^2e2mz0\text { Water particle orbit : } \rightarrow\left(\xi_{x_{0}}-x_{0}\right)^{2}+\left(\xi_{z_{0}}-z_{0}\right)^{2}=\hat{\eta}^{2} \mathrm{e}^{2 m z_{0}} Water particle orbit : →(ξx0−x0)2+(ξz0−z0)2=η^2e2mz0

Stochastic Description of Ocean Waves and Short-Term Sea States

Sea waves have irregular profiles changing randomly in time and space. They cannot be determined in a deterministic way like those presented in above sections. Sea waves are random in nature and mostly short crested. During a relatively short period, they are composed of infinite number of independent regular waves with different amplitudes, frequencies, traveling directions, and phases that are all random. Since we assume that these individual regular waves are determined from the linear wave theory, the superposition rule is applied to form a random wave group traveling along an arbitrary direction with individual waves scattered around it. Since there is infinite number of waves in a random wave group with different mplitudes, in the light of central limit theorem, the random surface displacement (surface elevation) becomes a Gaussian random process [7, 42–46] with zero mean value. A random wave profile obtained from the superposition of a number of individual independent regular waves is shown in Fig. 3.5. Over a time interval of about 3 h at some locations on the sea surface, the surface elevations g are measured relative to the still water level and, in this way, a number of records are obtained. Under the same sea condition, over another time intervals with about 3 h long at different locations, the measurements of the surface elevations are repeated and different records are obtained. From the comparison of all records of the surface elevation it can be seen that the surface elevation g would have a similar appearance and its general properties would not change. Thus, a single record of the surface elevation represents its random characteristics. This observation reveals that, in a short term, the water surface elevation g is a stationary ergodic process. Similarly, the water particle velocities and accelerations become also stationary ergodic processes. In a short term, random waves are usually described in terms of sea states. A unidirectional sea state is a stationary ergodic process described by the parameters, the significant wave height Hs and mean zerocrossings period Tz, and also by a spectral function SggðxÞ of the surface elevation g, which is defined between ð0x1Þ as known to be the sea spectrum, or wave energy spectral density function [14, 22].

Transfer Functions of a Random Wave in Deep Water

Statistics and Spectral Functions of the Water Elevation in the Short Term

The Pierson–Moskowitz Sea Spectrum

PM sea spectrum : →Sηη(ω)=Aω5exp⁡(−Bω4)\text { PM sea spectrum : } \rightarrow S_{\eta \eta}(\omega)=\frac{A}{\omega^{5}} \exp \left(-\frac{B}{\omega^{4}}\right) PM sea spectrum : →Sηη(ω)=ω5Aexp(−ω4B)
A=αηg2and B=5ωp4/4A=\alpha_{\eta} g^{2} \quad \text { and } \quad B=5 \omega_{p}^{4} / 4 A=αηg2 and B=5ωp4/4
Peak wave frequency ( Hsrepresents severity): →ωp=(165αηg2Hs2)1/4\text { Peak wave frequency ( } H_{s} \text { represents severity): } \rightarrow \quad \omega_{p}=\left(\frac{16}{5} \frac{\alpha_{\eta} g^{2}}{H_{s}^{2}}\right)^{1 / 4} Peak wave frequency ( Hs represents severity): →ωp=(516Hs2αηg2)1/4
Hs&Tzrepresent severity: {A=αηg2=4π3Hs2Tz4ωp=2πTz(45π)1/4→peak wave frequency \begin{aligned} H_{s} \& T_{z} \text { represent severity: } &\left\{\begin{aligned} A &=\alpha_{\eta} g^{2}=\frac{4 \pi^{3} H_{s}^{2}}{T_{z}^{4}} \\ \omega_{p} &=\frac{2 \pi}{T_{z}}\left(\frac{4}{5 \pi}\right)^{1 / 4} & \rightarrow\text { peak wave frequency } \end{aligned} \right. \end{aligned} Hs&Tz represent severity: ⎩⎪⎪⎪⎨⎪⎪⎪⎧Aωp=αηg2=Tz44π3Hs2=Tz2π(5π4)1/4→ peak wave frequency

The JONSWAP Sea Spectrum

JONSWAP sea spectrum : →Sηη(ω)=Aω5exp⁡(−Bω4)γg(ω)\text { JONSWAP sea spectrum : } \rightarrow S_{\eta \eta}(\omega)=\frac{A}{\omega^{5}} \exp \left(-\frac{B}{\omega^{4}}\right) \gamma^{g(\omega)} JONSWAP sea spectrum : →Sηη(ω)=ω5Aexp(−ω4B)γg(ω)
g(ω)=exp⁡[−12(ω−ωpωpσ)2]g(\omega)=\exp \left[-\frac{1}{2}\left(\frac{\omega-\omega_{p}}{\omega_{p} \sigma}\right)^{2}\right] g(ω)=exp[−21(ωpσω−ωp)2]
σ={0.07if ω≤ωp0.09if ω>ωp\sigma=\left\{\begin{array}{cl} 0.07 & \text { if } \omega \leq \omega_{p} \\ 0.09 & \text { if } \omega>\omega_{p} \end{array}\right. σ={0.070.09 if ω≤ωp if ω>ωp
Peak wave frequency (Hsrepresents severity): →ωp=(fp(γ)αηg2Hs2)1/4\text { Peak wave frequency }\left(H_{s} \text { represents severity): } \rightarrow \quad \omega_{p}=\left(\frac{f_{p}(\gamma) \alpha_{\eta} g^{2}}{H_{s}^{2}}\right)^{1 / 4}\right. Peak wave frequency (Hs represents severity): →ωp=(Hs2fp(γ)αηg2)1/4
fp(γ)=3.19714(1−0.286ln⁡γ)f_{p}(\gamma)=\frac{3.19714}{(1-0.286 \ln \gamma)} fp(γ)=(1−0.286lnγ)3.19714

Station	A(m)	B(m)	C	Location
India	0.80	2.70	1.22	Atlantic Ocean
Juliett	0.90	2.70	1.24	Atlantic Ocean
Sevenstones	0.6	1.67	1.21	Atlantic Ocean
Morecambe Bay	0.0	0.78	1.05	Irish Sea
Mersey bay	0.00	0.69	1.01	Irish Sea
Varne	0.00	1.05	1.30	North Sea
Smith’s Knoll	0.08	0.89	1.28	North Sea

Directional Wave Spectrum

Wave-Current Interaction

Probabilistic Description of Sea States in the Long Term

Morison’s Equation and Wave Forces on Structural Members

The Morison’s equation is defined as a distributed wave force per unit length of the pile and is normal (perpendicular) to the pile. It is written as,
Morison’s equation : →f=CD∣u∣u+CMu˙\text { Morison's equation : } \rightarrow f=C_{D}|u| u+C_{M} \dot{u} Morison’s equation : →f=CD∣u∣u+CMu˙
in which the first term is the drag force contribution and the second one is the inertia force contribution, uuu and u˙\dot uu˙ are respectively the velocity and acceleration components of a water particle which are normal to the pile, ∣⋅∣|\cdot|∣⋅∣ denotes an absolute value, CDC_DCD and CMC_MCM are respectively drag and inertia force constants defined as,
Drag and inertia force constants: →{CD=Dρwcd/2CM=πD2ρwcm/4\text { Drag and inertia force constants: } \rightarrow\left\{\begin{array}{l} C_{D}=D \rho_{w} c_{d} / 2 \\ C_{M}=\pi D^{2} \rho_{w} c_{m} / 4 \end{array}\right. Drag and inertia force constants: →{CD=Dρwcd/2CM=πD2ρwcm/4
in which DDD is the diameter of the pile, ρw\rho_wρw is the density of water, cdc_dcd and cmc_mcm are respectively the drag and inertia force coefficients of the Morison’s equation. These coefficients are functions of the Reynolds number (ReR_eRe), Keulegan–Carpenter number (KcK_cKc) and the roughness of the cylinder [14, 22, 23, 51, 91, 92]. API recommends the following drag and inertia values for circular cylinders [93, 94] for large Keulegan–Carpenter number (Kc>30)(K_c \gt 30)(Kc>30):

for smooth cylinders: cd=0.65c_d = 0.65cd=0.65 and cm=1.6c_m = 1.6cm=1.6
for rough cylinders: cd=1.05c_d = 1.05cd=1.05 and cm=1.2c_m = 1.2cm=1.2

For Kc>30_Kc \gt 30Kc>30 these values are modified by a wake amplification factor. The DNV rules [51] accept the same values for the smooth and rough cylinders, except the inertia force coefficient cmc_mcm. The DNV rules define higher cm values than the API recommended practices. For Kc<3Kc \lt 3Kc<3, cmc_mcm can be assumed to be independent of KcK_cKc and equal to the theoretical value of (cm=2.0)(c_m =2.0)(cm=2.0) for both smooth and rough cylinders. In the case of Kc>3K_c > 3Kc>3, it is calculated from [51] as,
cm=max⁡{1.6,[2−0.044(Kc−3)]→smooth cylinder 1.2,[2−0.044(Kc−3)]→rough cylinder c_{m}=\max \left\{\begin{array}{l} 1.6,\left[2-0.044\left(K_{c}-3\right)\right] \rightarrow \text { smooth cylinder } \\ 1.2,\left[2-0.044\left(K_{c}-3\right)\right] \rightarrow \text { rough cylinder } \end{array}\right. cm=max{1.6,[2−0.044(Kc−3)]→ smooth cylinder 1.2,[2−0.044(Kc−3)]→ rough cylinder
For low Keulegan–Carpenter numbers, a detailed discussion on the drag and inertia force coefficients is presented in [95]. It is reported [96] that, for (Kc<10)(K_c \lt 10)(Kc<10), the inertia force becomes dominant, for (10<Kc<20)(10 \lt K_c \lt 20)(10<Kc<20), both inertia and drag force components are significant and for (Kc>20)(K_c \gt 20)(Kc>20), the drag force becomes dominant. The diameter of the cylinder, DDD, is also an influential factor in the wave force regime. The Morison’s equation is appropriate for slender members. However, for large members, it can also be applied with modification of the inertia force coefficient cmc_mcm. This is done by using wave diffraction theory to calculate dynamic pressure, which is explained in the following section.

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