PROCEDIMIENTO DE COORDINACIÓN DE AISLAMIENTO [ 2 ]

This chapter discusses the development of the 1:70 and 1:17 scale mooring line models. In section one the reasoning behind the selection of a prototype experimental mooring system is presented. This is followed by a brief introduction to relevant scaling analysis procedures in section two. Section three discusses the application of scaling analysis in the design of an experimental model. Prior to the conclusions of section five, section four discusses the methods for modelling mooring line elasticity.

As discussed in section one, it is important to have a good physical understanding of the response of systems to ensure accurate representation by mathematical modelling. In particular, the mathematical modelling of non-linear systems generally requires the application of numerous assumptions to ensure, firstly that a solution is possible and, secondly, from an engineering design perspective, that this solution is achieved in an efficient manner. Physical validation of these engineering models is normally not possible at prototype scale because of the large sizes involved and therefore the use of smaller scale models is required.

5.1 Selection of an appropriate prototype model

The prototype mooring system selected to be modelled experimentally is that defined in the comparative mooring line damping study of Brown and Mavrakos (1997). Reference to the study of Brown and Mavrakos is abbreviated as the ISSC study. The ISSC study required the modelling of two mooring systems; a shallow water continuous chain line system in 82.5m water depth, and a moderately deep water chain/wire mix system, in 500m.

Reasons for basing the test programme parameters upon the mooring systems considered in the ISSC study include:

• The large number of contributors to the ISSC case study represents a significant proportion of the mooring analysis and design methodologies that are presently in existence. Hence the results presented are indicative of state of the art mooring evaluation techniques.

• The requirement of, relatively simple, sinusoidal horizontal excitation of the line suspension point lends itself well to experimental analyses as well as data interpretation.

• The existence of previous experimental data helps give confidence to results. Wichers & Huijsmans (1990) experimental results, on which the ISSC study is partly based, provide an excellent starting point on which to compare and develop the present experimental model.

The discrepancies that were highlighted within the ISSC study however require that an additional independent experimental verification be carried out to improve confidence in the numerical analysis codes. In particular the discrepancies were that although dynamic tension amplitudes exhibit reasonable agreement across all participants' results, the hydrodynamic damping predictions are seen to vary by up to 60% of the mean time domain results.

5.2 Scale models

Appropriately scaled models are capable of providing a complete picture of the true physical behaviour and expected phenomena of the desired prototype. The ability of the scale model to effectively mirror a prototype is however dependent upon the scaling methods applied to the relevant prototype particulars. In this section a general discussion of scaling procedures is presented.

There are four fundamental dimensions in terms of which all other physical qualities may be expressed. They are mass, length, time and temperature. It is the appropriate scaling of each of these prototype dimensional qualities that determines a scale model’s ability to replicate the necessary modes of the prototype.

The type of scaling that may be applied to each of the individual or collective groups of fundamental dimensions are categorised within four key scaling methodologies: • Geometric

• Kinematic • Static • Dynamic

In geometry the corresponding points of two formations (that are not necessarily geometrically similar) are known as homologous points. Within geometric scaling the

prototype geometric (length) properties are reduced by a constant value, hence the homologous parts of a prototype and its model are composed of homologous points. When in motion, if these homologous points lie along a homologous path at proportional time there is said to be kinematic scaling.

Providing that the material density is constant between prototype and a geometrically scaled model, static similarity will be achieved. However the use of dissimilar material density properties between prototype and model may cause static similarity to be achieved without correct geometric scaling.

Assuming that the prototype and its scale model have an equivalent proportional distribution of mass, and are exposed to proportional total forces, dynamic similarity is attained. In achieving dynamic similarity, the model and prototype will automatically have achieved kinematic similarity. However, this does not necessarily mean that geometric similarity is achieved.

Table 5.1 lists the scaling parameters discussed above according to the type and grouping of dimensional quality. It is noted that the temperature dimensional quality has not been included within the scaling analysis due to the assumed achievement of temperature similarity between prototype and model environment. In the scaling of model mooring lines it is imperative that dynamic similarity is achieved.

5.3 Static & dynamic models

This section discusses the static and dynamic model development. Initially an inelastic model line is assumed with elastic considerations discussed later in section 5.4.

5.3.1 Concept of the static mooring line model

Webster (1995) assumes that the tension at the suspension point of a catenary mooring line undergoing sinusoidal oscillations is a function of at least 15 variables, including dynamic and static components. From this assumption for the total line tension, six variables can be extracted that are responsible for the static catenary tension, , these being

Arranging the variables of equation 5.1 into non-dimensional groupings enables the definition of a static mooring line according to the following parameters:

Lg Length of suspended mooring line to water depth ratio,

H known as the ‘scope’ of the line.

Ratio of static pre-tension to weight of submerged vertical line. Together with the mooring line scope, this parameter wH

governs the static geometry, or profile, of the line.

D Diameter of the mooring line to water depth. In typical

H mooring depths this is small.

It is noted that the static line tension, F^, comprises a horizontal and vertical component represented by and F^g respectively. Each of these components are derived from the solution of the catenary equations presented in section 3.1.

Consideration of the line scope reveals that the water depth of a typical test facility governs the scaling of the line length. The geometric scale factor for mooring lines is hence governed by the ratio of prototype to model test facility water depth. Application of this ratio as the geometric scale factor provides the required model dimensions, as presented in table 5.2.

During the oscillation of the mooring line suspension point, caused for example by the environmental excitation on the attached vessel, the correct design of the catenary mooring line ensures that at the extreme amplitude of oscillation a large percentage of the line will remain on the seafloor. As an inelastic mooring line cannot stretch there is no requirement to model the entire length of grounded line provided that at the anchor a length of line always remains on the seabed to ensure zero vertical loading on the anchor. Section 2.3 provides a further discussion of this.

It is noted that geometric scaling using the water depth parameter infers that the modelling of a deep water mooring line in a relatively shallow facility will require a long, and very thin line. This will result in a model scale line of extremely low mass. This difficulty precludes the experimental static modelling of the 500m water depth mooring system of Brown and Mavrakos (1997). The decision not to model the deeper water situation is also based upon dynamic concerns addressed in section 5.3.4.

The water depths of the two test facilities used in the experimental programme determine the appropriate model mooring line characteristics. From table the tests for the 1:70 model require a line with submerged weight per unit length of less than IN . The measurement of such small static loads and subsequent variation of these loads with the excursion of the point of suspension is extremely difficult and is likely to compound experimental errors. Similar concerns are discussed by Moxnes & Larsen (1998). In their paper a comparison of identical FPSO model tests at ‘traditional’ scale (1:55) and ‘ultra-small’ scale (1:170) is made.

In order to circumvent the possible introduction of such measurement errors the experimental investigation at 1:70 model scale achieves static similarity without geometric similarity. Tests performed using the 1:17 model are however carried out using a geometrically scaled mooring line. Indeed the ability to perform tests using equivalent models of differently scaled parameters is a useful validation of the methodology.

To select the appropriate model chain to be used in the 1:70 model scale tests several types of commercially available chain were analysed, the dimensions of which are presented in table 5.3. The highlighted row of table 5.3 indicates the dimensions of the chain selected for the 1:70 experimental tests. It is the non­ geometric scaling of chain diameter that causes the 1:70 scale model line to deviate from the true geometric scaling. This also results in a compromise of the scaled model line weight. The required, and actual, 1:70 model line dimensions are presented in table 5.4 with the geometrically scaled 1:17 model presented for comparison. Given that the static catenary profile is dependent on the pre-tension angle, the effective distortion of the geometric scaling does not prevent static similarity. The implications of this scaling methodology on dynamic similitude are discussed in later sections.

5.3.3 Concept of the dynamic mooring line model

Dynamic similarity between a prototype and its scaled model requires that the proportional distribution of mass and environmental forces are equivalent. Providing that the model mooring achieves static similitude then the distribution of mass may be considered satisfactory.

The exposure to equivalent environmental forces requires a consideration of the fluid medium that surrounds the model. In general, experimental scale model tests that are required to determine the environmental loading on a system are performed in an

environment which is identical to that of the prototype. For example, the water in which the model chain line is immersed has identical properties to that of the prototype system. The inability to scale the surrounding medium requires careful consideration because of the influence that the medium will have on any environmentally induced loads. In the case of a model immersed in water, consideration must be given to the effects of viscosity. The fluid loads on a mooring line that are dependent upon the viscous nature of water as a result of the line motion through water may be crudely broken down into the following criteria:

• Hydrodynamic loading which is considered in terms of drag and added mass forces.

• Vortex induced vibration (VIV).

5.3.3.1 Drag and added mass coefficients

The dominance of drag or inertial hydrodynamic force components depends upon the growth and separation of the fluid boundary layer on the mooring line. A measure of the boundary layer thickness and nature of the flow on the body is provided by the Reynolds (Re) number, the ratio of inertial to viscous shearing forces in the fluid. The drag coefficient is dependent upon the boundary layer characteristics and therefore depends upon the Re number. Although theoretical calculation of drag forces is possible, assuming inviscid fluid properties, present analytical methods are unable to determine viscous drag loads and thus empirical methods are required for their determination.

Numerous experimental studies have been performed using smooth cylinders for visualisation of fluid flow phenomena and the empirical derivation of associated drag coefficient variation for specific body characteristics and fluid flow parameters. A principal text for comprehensive discussion of the physics of these phenomena is Sarpkaya and Isaacson (1981). Figure 5.1 presents empirical drag coefficients of a smooth cylinder for increasing Re number Schlichting (1968). The transition from laminar to turbulent flow, the critical flow regime, occurs over a small range of Re numbers. For a smooth cylinder this type of flow leads to a sharp reduction in drag force and corresponds to the movement around the cylinder of the flow separation point.

The effect of Re number on drag coefficient in figure 5.1 assumes that the fluid flow around the body has uniform velocity, however the action of an impulsive fluid load

on smooth cylinders can induce an increase in drag coefficient by up to approximately 30% (Sarpkaya(1966)). Also included in figure 5.1 are drag coefficients for model scale chain specimens derived from experimental investigations (Huse (1992)), Over the range 5x10^<Re<1x10^ the upper end of the subcritical range, the chain drag coefficients are seen to be approximately double those of the equivalent smooth cylinder, a trend that is observed to continue with increasing Re number. It is interesting to note that small diameter chains, over the critical range 50<Re<200, exhibit similar drag coefficients to those of the larger dimension chain specimens.

As the fluid accelerations require forces that are exerted on the body in the form of a pressure distribution, the relative acceleration of the local fluid acts as if the body is of greater mass than that which it actually is. This added mass phenomenon, quantified in terms of the added mass coefficient, depends on the geometry of the body and the condition of the fluid surrounding the body, which in turn is dependent upon both the Re number and, a second non-dimensional number, the Keulegan Carpenter (KG) number. For harmonic flow the KG number indicates the relative amplitude of the body, or fluid, oscillation to the diameter of the body. This ratio is of particular importance in the determination of the dominance of the inertial or drag loading of the model line.

Typically the inertia loading of a body is defined in coefficient form (Gm) and is generally represented by the added mass coefficient plus the Froude-Krylov force that occurs if the body is stationary in an accelerating fluid. The Froud-Krylov force is here defined as the force required to accelerate the displaced fluid volume with the acceleration of the surrounding fluid.

For a body in a given fluid the loading on the body is dependent upon the sum of drag and inertia loads. Which of these loads will dominate the hydrodynamic loading of a body depends in a complex manner upon body dimension, body (or fluid) oscillation parameters and the properties of the fluid in which the body is immersed. For example, if the body, or fluid, undergo harmonic oscillation of insufficient frequency and amplitude to enable separation of the boundary layer from the body the drag forces are small and inertial loads will dominate. If the Re number for high frequency oscillations is sufficient to induce separation, and the amplitude of oscillation large enough to ensure approximate uni-directional motion, the boundary layer separates from the body shedding vortices into the free stream. In this situation drag forces dominate the hydrodynamic loading, and an empirical drag coefficient may be determined for the body. If however the motion does not effectively

approximate to uni-directional flow, although drag forces dominate, a complicated fluid field is induced where the wake from flow separation during one oscillation cycle may be incident upon the body for subsequent cycles. In such cases the drag forces acting on the body are specific to body and oscillation characteristics. The development of this type of complex fluid field is possibly responsible for the cross­ coupling of high and low frequency line tensions, as described in section 3.2.1.

The relationship between oscillation parameters and body dimension is provided by the KC number, which is defined by

TTJ

K c = i ^ , (5.2)

where T , U ^and D represent the oscillation period, relative velocity of the body in the fluid and body diameter respectively.

Only a limited quantity of work has been performed on the development of drag and added mass coefficients for chains. Brown et al (1995) comment on mooring line damping and the particular relevance that the correct determination of drag coefficient has upon mooring calculations. Tests performed by Huse (1992) involved single frequency oscillations of a wide variety chain diameters, of model and prototype scale, and limited oscillation parameters. Brown et al (1997) describes a development of these experimental tests involving a large model chain specimen of approximately 1/3rd prototype scale oscillating at single and combined wave and drift frequencies for a range of KC numbers. Their results indicate that in harmonic flow tests the measured drag coefficient is 20% greater than that of Huse. The influence of bi-harmonic flow causes an increase in drag coefficient of 33% relative to that of harmonic flow.

S.3.3.2 Vortex induced vibrations (VIV)

The production of musical tones, Aeolian tones, from wires exposed to the wind has been known of since ancient times. In 1878 Strouhal discovered that the Aeolian tones generated were proportional to the wind speed divided by the wire diameter. A result of this observation was the development of the Strouhal number for vortex shedding.

As the fluid field develops around the body the asymmetric shedding of vortices from opposing sides of the body results in an oscillating side force, usually normal to the flow direction, inducing a body vibration. If the fluid flow parameters are such that the

shedding frequency approaches the natural frequency of the body, the shedding and natural frequency eventually become synchronized. This synchronization leads to a possible increase of the shed vortex strength which results in the following:

• A change of the vortex shedding frequency from the stationary cylinder shedding frequency to the cylinder vibration frequency causing the vortex shedding to 'lock­ in'. The broad band of Strouhal numbers that exist over the transition Re range, where the flow about a body changes from laminar to turbulent nature, are a direct result of the complicated flow pattern that exists during the flow transition. This flow pattern is significantly modified by the out of plane oscillation of the body and hence creates this broad band response.

• A dramatic increase in the body drag coefficient.

• Possible catastrophic failure. Cases of lock-in at the resonant frequency of structures are well documented and of particular concern for ice-laden power cables. Luongo and Piccardo (1995) present a simplified analytical model with numerical verification for an investigation into the response of iced suspended cables experiencing self-excited aeroelastic oscillations, in the vertical plane, due to wind.

Although VIV may induce catastrophic structural failures, the VIV induced motions are themselves self-limiting. For a smooth cylinder, as the amplitude of oscillation