Wave forces on submarine pipelines near a plane boundary

Forces induced by regular waves on submarine pipelines resting on as well as near a plane boundary and aligned parallel to wave fronts of the oncoming waves are investigated experimentally. The inline hydrodynamic coefficients of drag and inertia are evaluated through the use of Morison equation and the least squares method. The transverse force is analysed in terms of maximum transverse force and transverse root mean square (r.m.s.) coefficients. The resulting inline and transverse hydrodynamic coefficients are correlated with the period parameter or Keulegan-Carpenter number and relative clearance of the pipeline from the plane boundary. The effect of depth parameter on these coefficients and the correlation between maximum transverse force and transverse r.m.s. coefficients are also reported.     

An experimental investigation of hydrodynamic coefficients for a vertical truncated rectangular cylinder due to regular and random waves

The research into hydrodynamic loading on ocean structures has concentrated mostly on circular cross-section members and relatively limited work has been carried out on wave loading on other cross-sections such as rectangular sections. These find applications in many offshore structures as columns and pontoons in semi-submersibles and tension-leg platforms. The present investigation demonstrates the behaviour of rectangular cylinders subject to wave loading and also supplies the hydrodynamic coefficients for the design of these sections.This paper presents the results of wave forces acting on a surface piercing truncated rectangular cylinder set vertically in a towing tank. The experiments are carried out in a water depth of 2.2 m with regular and random waves for low Keulegan–Carpenter number up to 6. The rectangular cylinder is of 2 m length, 0.2 m breadth and 0.4 m width with a submergence depth of 1.45 m from still water level. Based on Morison equation, the relationship between inertia and drag coefficients are evaluated and are presented as a function of KC number for various values of frequency parameter β, for two aspect ratios of cylinders, equals to 1/2 and 2/1. Drag and inertia coefficients obtained through regular wave tests are used for the random wave analysis to compute the in-line force spectrum.The results of the experiments show the drag and inertia coefficients are strongly affected by the variation in the aspect ratios of the cylinder. The drag coefficients decreases and inertia coefficients increases with increase in Keulegan–Carpenter number up to the range of KC number tested. The random wave results show a good correlation between measured and computed force spectrums. The transverse forces in both regular and random waves are found to be small compared to in-line forces.     

Forces on a smooth submarine pipeline in random waves — A comparative study

Forces on a smooth submarine pipeline, fixed horizontally near a plane boundary, have been investigated under random wave conditions. The submarine pipeline was subjected to Pierson-Moskowitz spectrum (P-M spectrum) at various energy levels. The water particle kinematics were computed based on the linear random wave model and the Morison equation was chosen as the wave force predictor model. The inline hydrodynamic coefficients of drag and inertia were evaluated using two different methods, one in the frequency domain and the other in the time domain. Five mathematical formulations were considered for the analysis of transverse wave forces and these were compared in terms of the correlation coefficient. The transverse force was also analyzed in terms of the transverse root mean square (rms) coefficient. The inline hydrodynamic coefficients of drag and inertia and the transverse rms coefficient were correlated with the Keulegan-Carpenter number or period parameter, the relative clearance of the pipeline from the bed and the depth parameter. Finally, the results of the random wave tests were compared with those of regular waves under similar pipeline conditions.     

Hydrodynamic coefficients for inclined cylinders

The wave induced dynamic pressures around a circular cylinder of diameter 0.2 m due to regular waves were measured in a wave flume in a water depth of 1 m and in a wave basin in a water depth of 3 m. The experimental investigations were carried out with the cylinder inclined along and against the direction of wave propagation. The least-squares technique was employed to evaluate the coefficients of drag (C_D) and inertia (C_M) from the sectional force time histories obtained by integrating the measured circumferential pressure distribution. The variation of drag and inertia coefficients are presented as a function of Keulegan-Carpenter number (KC) for different inclinations of the cylinder. The comparison between the measured and the theoretical force derived from the evaluated hydrodynamic coefficients is found to be good.     

流向Morison 型波浪力的高阶矩分析

通过理论研究定量地说明流向Morison波浪力,即拖曳力和惯性力的高阶统计矩随采样次数增加的规律.主要应用二阶Stokes波理论,推导了流向Morison波浪力的前四阶累积量.计算了作用于实际海底管线上的流向Morison波浪力的偏斜度和峰度.结果表明,随着采样次数的增加,拖曳力和惯性力的偏斜度和峰度驱于收敛.文中给出的方法为后续理论工作奠定了基础.     

Wave forces acting on a submerged horizontal circular cylinder in oblique waves and on a vertical cylinder in deep waves

This paper presents a method of estimating wave forces acting on a submerged horizontal circular cylinder fixed in oblique waves.The experiments show that drag and inertia coefficients in beam sea are available for calculating the wave forces in oblique waves.Wave forces exerted on a vertical circular cylinder in deep waves are also investigated.The experimental results show that wave forces acting on the vertical cylinder coincide approximately with hydrodynamic forces acting on a submerged circular cylinder in an oscillating fluid.     

Experimental study of hydrodynamic forces and wave forces exerted on a truss structure

The drag and added mass coefficients of a truss leg of an ocean platform are obtained by using the forced-oscillation technique in a still water. Higher order forces and lift forces are also measured.The drag and inertia coefficients of the truss leg model are obtained by measuring the wave forces acting on it in regular deep waves. The moment lever of the wave force is compared with theoretical results.     

In-line force on a cylinder translating in oscillatory flow

Experiments were conducted with smooth and sand-roughened cylinders moving with constant velocity in a sinusoidally oscillating flow to determine the drag and inertia coefficients and to examine the effect of wake biasing on the modified Morison equation. The various flow parameters such as the relative cylinder velocity. Reynolds number, and the Keulegan-Carpenter number were varied systematically and the in-line force measured simultaneously. The principal results, equally valid for both smooth and rough cylinders, are as follows: the drag coefficient decreases with increasing relative current for a given Reynolds number and Keulegan-Carpenter number; the effect of wake biasing on the drag and inertia coefficients is most pronounced in the drag-inertia dominated regime; and the two-term Morison equation with force coefficients obtained under no-current conditions is not applicable to the prediction of wave and current induced loads on circular cylinders.     

Wake II model for hydrodynamic forces on marine pipelines including waves and currents

The Wake II model for the determination of the hydrodynamic forces on marine pipelines is extended to include currents and waves. There are two main differences between the Wake II and the traditional model. First, in the Wake II model the velocity is modified to include the pipe's encounter with the wake flow when the velocity reverses. Second, the model uses time dependent drag and lift coefficients. The flow field is assumed to be the linear superposition of regular waves and uniform current and is treated as wave only but in two different phases. The model requires eight empirical parameters that are obtained from comparisons with field data for various Keulegan–Carpenter numbers and current to wave ratios. The effective velocity and the force predictions are compared with field data from Exxon Production Research Company and with the conventional model. The model gives satisfactory results and predicts lift forces that in shape, magnitude and phase relative to the velocity are in very close agreement with measured forces. For the horizontal forces the results are very accurate. A substantial improvement is obtained over the predictions with the conventional model. This work is applicable to the design of submarine pipelines laying on the sea bottom in water depths where waves or waves and currents contribute to the hydrodynamic forces.     

Improved assessments of wave-induced fatigue for free spanning pipelines

For subsea pipeline projects, the costs related to seabed correction and free span intervention are often considerable. Development of reliable methods for fatigue analyses of pipelines in free spans contributes to minimize costs without compromising pipeline integrity. Assessment of wave-induced fatigue damage on multi-span pipelines is investigated, and improved analysis methods are suggested in this paper. A time-domain (TD) algorithm is developed, which accounts for non-linear hydrodynamic loading and dynamic interaction between adjacent spans. The proposed TD approach is employed to evaluate linearized frequency-domain (FD) solutions from recognized design standards and to study the dynamic response of multi-span pipelines to direct wave loading. Differences between multi- and single-span analyses are described for the first time, and the common assumption that the main fatigue damage contribution comes from the fundamental mode is demonstrated not to hold for multi-spans. An improved FD solution capable of predicting multi-mode response is derived and demonstrated to give accurate fatigue life estimates for multi-span pipelines.     

半潜式平台内孤立波载荷计算方法及其试验验证

针对半潜式平台的立柱群和沉箱群,设计了两套独立的载荷测量系统,利用大型重力式密度分层水槽,在不同来波方向下对孤立波中半潜式平台载荷进行了系列模型试验。研究表明,对平台立柱部分,其内孤立波载荷可以用Morison公式进行计算,基于试验结果建立了Morison公式中其拖曳力系数以及惯性力系数的经验公式;对于半潜式平台的沉箱部分,当来波方向与其中纵剖面不平行时,其水平内孤立波载荷同样可以使用Morison公式进行计算,并建立了Morison公式中其拖曳力系数以及惯性力系数的经验公式;当来波方向与半潜式平台中纵剖面平行时,沉箱群的水平内孤立波载荷可以采用Froude-Krylov公式进行计算;同时,在不同来波方向下沉箱群的垂向载荷同样可以采用Froude-Krylov公式进行计算。     

WAVE-CURRENT FORCE COEFFICIENTS FOR ISOLATED PILE

Based on the linear wave theory and model experiment results, the wave-current force coefficients for isolated pile are investigated, using Morison's formula to calculate the wave-current force. A formula is'presented for determining the relative coefficients of the drag, in which the feature value of the wave-current field is. proposed and used as an important parameter. According to the maximum wave-current force measured in the experiment, the coefficients of the drag force and inertia force are determined by statistical method of two-variable regression so that both of them are fitted in optimum.     

Parameter identification of a compliant nonlinear SDOF system in random ocean waves by reverse MISO method

The determination of the drag and inertia coefficients, which enter into the wave force model given by Morison's equation, is particularly uncertain and difficult when a linear spectral model is used for ocean waves, and the structure is compliant and has nonlinear dynamic response. In this paper, a nonlinear System Identification method, called Reverse Multiple Inputs–Single Output (R–MISO) is applied to identify the hydrodynamic coefficients as well as the nonlinear stiffness parameter for a compliant single-degree-of-freedom system. Four different types of problems have been identified for use in various situations and the R–MISO has been applied to all of them. One of the problems requires iterative solution strategy to identify the parameters. The method has been found to be efficient in predicting the parameters with reasonable accuracy and has the potential for use in the laboratory experiments on compliant nonlinear offshore systems.     

A Prediction Method of Internal Solitary Wave Loads on the Semi- Submersible Platform

An investigation into the prediction method for internal solitary waves (ISWs) loads on the columns and caissons of the semi-submersible platform found on three kinds of internal solitary wave theories and the modified Morison Equation is described. The characteristics of loads exerted on the semi-submersible platform model caused by the ISWs have been observed experimentally, and the inertial and drag coefficients in Morison Equation are determined by analyzing the forces of experiments. From the results, it is of interest to find that Reynolds number, KC number and layer thickness ratio have a considerable influence on the coefficients. The direction of incoming waves, however, is almost devoid of effects on the coefficients. The drag coefficient of columns varies as an exponential function of Reynolds number, and inertia coefficient of columns is a power function related to KC number. Meanwhile, the drag coefficient of caissons is approximately constant in terms of regression analysis of experimental data. The results from different experimental conditions reveal that the inertia coefficient of caissons appears to be exponential correlated with upper layer depths.

     

Allowable span length of submarine pipeline in shallow water

Because of the complex geological conditions of the seabed, submarine pipelines buried beneath the ocean floor become suspended over the seabed under the long-term scour of waves eroding the surrounding sediment. Further, most oil fields were built in offshore areas while the country was developing. This gives the waves seen in shallow water obvious nonlinear features, and the abnormal characteristics of these waves must be considered when calculating their hydrodynamic forces. Particularly under such conditions, these suspended spans of submarine pipelines are prone to damage caused by the action of the external environment load. Such damages and eventual failures may result not only in great property losses but also pollution of the marine environment. The span length of these areas is a key predictive factor in pipeline damages. Therefore, determining the allowable span length for these submarine pipelines will allow future projects to avoid or prevent damage from excessive suspended span lengths. Expressions of the hydrodynamic loads placed on suspended spans of pipeline were developed in this work based on the first-order approximate cnoidal wave theory and Morison equation. The formula for the allowable free span length was derived for the common forms of free spanning submarine pipeline based on the point where maximum bending stresses remain less than the material’s allowable stress. Finally, the allowable free span length of real-world pipelines was calculated for a subsea pipeline project in Bohai Bay. This research shows that, with consideration for the complicated marine environment, existing suspended spans are within allowable length limitations. However, continuing to limit the length of these submarine pipeline spans in the Nanpu oil field will require ongoing attention.     

New model to determine forces at on-bottom slender pipelines

The present paper proposes a numerical model to determine horizontal and vertical components of the hydrodynamic forces on a slender submarine pipeline lying at the sea bed and exposed to non-linear waves plus a current. The new model is an extension of the Wake II type model, originally proposed for sinusoidal waves (Soedigdo et al., 1999) and for combined sinusoidal waves and currents (Sabag et al., 2000), to the case of periodic or random waves, even with a superimposed current. The Wake II type model takes into account the wake effects on the kinematic field and the time variation of drag and lift hydrodynamic coefficients. The proposed extension is based on an evolutional analysis carried out for each half period of the free stream horizontal velocity at the pipeline. An analytical expression of the wake velocity is developed starting from the Navier–Stokes and the boundary layer equations. The time variation of the drag and lift hydrodynamic coefficients is obtained using a Gaussian integration of the start-up function. A reduced scale laboratory investigation in a large wave flume has been conducted in order to calibrate the empirical parameters involved in the proposed model. Different wave and current conditions have been considered and measurements of free stream horizontal velocities and dynamic pressures on a bottom-mounted pipeline have been conducted. The comparison between experimental and numerical hydrodynamic forces shows the accuracy of the new model in evaluating the time variation of peaks and phase shifts of the horizontal and vertical wave and current induced forces.     

A coupled model of submerged vegetation under oscillatory flow using Navier–Stokes equations

This work presents a new model for wave and submerged vegetation which couples the flow motion with the plant deformation. The IH-2VOF model is extended to solve the Reynolds Average Navier–Stokes equations including the presence of a vegetation field by means of a drag force. Turbulence is modeled using a k–ε equation which takes into account the effect of vegetation by an approximation of dispersive fluxes using the drag force produce by the plant. The plant motion is solved accounting for inertia, damping, restoring, gravitational, Froude–Krylov and hydrodynamic mass forces. The resulting model is validated with small and large-scale experiments with a high degree of accuracy for both no swaying and swaying plants. Two new formulations of the drag coefficient are provided extending the range of applicability of existing formulae to lower Reynolds number.     

Wave force coefficients for horizontally submerged rectangular cylinders

The results of wave force measurements carried out on a section of horizontally submerged rectangular cylinders, which are used as pontoons in many offshore structures, are reported in this paper. Two rectangular cylinders with aspect (depth–breadth) ratios equal to $\frac{1}{2}$ and $\frac{3}{4}$ and a square section (aspect ratio=1.0) cylinder are chosen for this study. Experiments are carried out in a wave tank at a water depth of 2.2 m at low Keulegan–Carpenter (KC) numbers to measure the horizontal and vertical wave forces acting on a 100 mm section, located at mid-length of the cylinders. For each cylinder, tests are carried out for two relative depths of submergence of 2.68 and 4.68. Measured wave forces in regular and irregular waves are then used to derive drag (C_D) and inertia coefficients (C_M). The analysis show that at very low KC numbers the inertia coefficients for all cylinders approached the potential flow values for both horizontal and vertical forces. The drag coefficients at low KC numbers exhibited large values and they decreased sharply with increase in KC number. For the square cylinder, where relatively a large KC number is obtained compared to other cylinders, inertia coefficients reached minimum values in the range of KC of about 3–4 and increased thereafter. In this range, C_M values are about 50% or so, smaller than the same at KC close to zero. The results of the experiments reveal that aspect ratio has large influence on hydrodynamic coefficients.     

Drag and inertia coefficients for a circular cylinder in steady plus low-amplitude oscillatory flows

Computer simulations of steady plus low-amplitude oscillatory flow about a circular cylinder are reported at a fixed Reynolds number of 150 based on the steady component. The conventional Keleugan–Carpenter number based on the oscillatory component is fixed at π/5. The oscillation frequency is varied so as to study a wide spectrum of flows where inertial forces dominate at one end and viscous drag forces at the other as a function of the modified Keleugan–Carpenter number. The hydrodynamic force on the cylinder in-line with the flow direction is represented by Morison's equation and an extended version with three terms. The drag and inertia coefficients in Morison's equation are determined by least-squares fits to data directly computed from integration of skin friction and pressure distributions around the periphery of the cylinder. The root-mean-square value of the residue of reconstructed minus directly-computed forces varies between 2 and 41% depending on the flow parameters. Comparable results can be obtained with a semi-theoretical approach using inviscid inertia and quasi-steady viscous drag terms. Physical explanations for the variation of the force coefficients are provided and implications for pertinent flow–structure interactions are discussed.     

Hydrodynamic Forces and Instability of Submarine Pipelines Under Waves and Currents： A Review

Stability design of submarine pipelines is a very important procedure in submarine pipeline engineering design. The calculation of hydrodynamic forces caused by waves and currents acting on marine pipelines is an essential step in pipeline design for stability. The hydrodynamic forces-induced instabilities of submarine pipelines should be regarded as a wave/ current-pipeline-seabed interaction problem. This paper presents a review on hydrodynamic forces and stability research of submarine pipelines under waves and currents. The representative progress including the improved design method and guideline has been made for the marine pipelines engineering design through experimental investigations, numerical simulations and analytical models. Finally, further studies on this issue are suggested.