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Physical modelling and similitude of marine structures
Ocean Engineering 26 (1999) 111–123
Physical modelling and similitude of marine structures Dracos Vassalos The Ship Stability Research Centre, Department of Ship and Marine Technology, University of Strathclyde, Strathclyde, UK Received 19 May 1997; received in revised form 1 July 1997; accepted 2 September 1997
Abstract This review paper is aimed at providing information on and at explaining the appropriate use of models in the design of marine structures. Emphasis is placed on the derivation of scaling laws and on limitations of accuracy. Similitude, dimensional analysis and the use of governing equations in model experiment scaling are explained before addressing the modelling of some common ocean engineering tests. 1998 Elsevier Science Ltd. All rights reserved. Keywords: Physical modelling; Similitude; Marine structures
1. Introduction 1.1. General Systems involving the natural phenomena of the oceans, the atmosphere, geographical features, and so on, are notoriously difficult to investigate in detail. Thus for the purpose of analysis and from the point of view of economics, we are compelled to make use of models of the full-scale (often called the ‘prototype’) system. It is possible with laboratory models to ensure some control over the many variables of influence so that those of particular interest can be isolated for special study. Other reasons for using models include: 앫 To study phenomena defying analytical solution, or those for which no experimental results or empirical methods are available 0029-8018/99/$—see front matter 1998 Elsevier Science Ltd. All rights reserved. PII: S 0 0 2 9 - 8 0 1 8 ( 9 7 ) 1 0 0 0 4 - X
112
앫 앫 앫 앫 앫 앫 앫
D. Vassalos / Ocean Engineering 26 (1999) 111–123
To improve conceptual and physical understanding of a problem To verify (or disprove) theoretical results Data acquisition Calibration of a theoretical (numerical) model To select alternative designs To forecast full-scale behaviour To identify governing parameters.
In problems where analytical and/or empirical methods will yield a reliable solution the use of models is not justified, as being more expensive. In the absence of these, however, it must be understood that the actual cost of model testing will usually be only a small fraction of 1% of the overall capital sums involved and it may well be a great deal less than the cost of corrective action which may be necessary as a result of failure to carry out tests. In the majority of cases that naval architects have to deal with, and especially those involving fluid phenomena, it is very seldom that a problem can be tackled by theoretical means alone. Here the art of engineering must be practised with experience, judgement, ingenuity and patience if useful results are to be obtained and correctly interpreted, and prototype performance predicted from these. 1.2. Types of experiments Experiments can be broadly classified into the following two categories (Couch et al., 1984): 1. Steady State: Typically, resistance and propulsion experiments where the model is tested in a steady condition at forward speed. The data collected is usually time-averaged to render the results time independent. 2. Unsteady: Typically, seakeeping/ocean engineering experiments where, as a result of the transient loads produced by the ocean environment, such experiments are characterised by their dynamic non-steady nature. Data collected is now time dependent with results shown either as time domain statistics, e.g., rms, significant values, averages, etc., or as frequency domain functions, e.g. RAO’s (Response Amplitude Operators) and response spectra. A further distinction can be made between seakeeping and ocean engineering experiments: 앫 Seakeeping usually implies a dynamic process involving ship like vessels generally with forward speed. 앫 Ocean Engineering usually refers to the dynamic behaviour of stationary or nearly stationary structures (floating or fixed). Finally, in the context of either seakeeping or ocean engineering, the following types of experiment can be considered: 앫 Rigid Body Response—motions, velocities, accelerations, relative motions, pressures. The results will be RAO’s and/or motion statistics.
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앫 Hydrodynamic Forces—Added mass and damping components (forces excitation at various frequencies), wave excitation component (restrained model). 앫 Flexural Body Response—Experiments investigating stresses or structural deflections of vessels and offshore structures. The emphasis in the paper will be on ocean engineering type experiments.
2. Brief historical background Historical records show that experiments with models of ships have been performed since the time of Leonardo da Vinci. However, it was not until W.E. Froude (1874), that model test predictions were established as a valuable engineering tool— in his case for the resistance of a full-scale ship from model resistance experiments in a towing tank. Interest in the seaworthiness of ships is as old as estimates of resistance and propulsion, but the accurate extrapolation of model test results to fullscale was not possible until the principles of spectral and transfer functions were established, as recorded in the seminal paper of St Dennis and Pierson (1953). Over the years, model tests have demonstrated their usefulness in solving problems relating to both preliminary design and retrofitting. The obvious benefits of a successful model test programme can be an optimum hydrodynamic and/or structural design which will enhance the economic performance of the marine structure.
3. Model experiments scaling Even though suitable measurements are generally easier to achieve in testing models that at full-scale, scaling problems can never by completely overcome, and it must be admitted that model tests in the controlled artificial environment of the laboratory always lack something of the uncertain harsh reality of the real world experienced by marine structures. In view of the difficulties in properly scaling an experiment, if a novel one is being planned, it is often prudent first to construct a small crude model for observational purposes only, in order to detect possible sources of problem, and to assist in planning the more detailed model tests. The basic principles of scaling laws and the main approaches to planning model experiments will be considered, under three headings. 3.1. Physical similarity The concept of physical similarity implies the need to ensure that, in planning the experiment, a certain similarity should be maintained between the model and the prototype. In general, two systems are said to be physically similar (in respect of certain specified physical quantities) when the ratio of corresponding magnitudes of these quantities is the same between the systems. This proposition must be satisfied not only by the contents of the system but also by the system boundary, inputs and
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outputs. In other words, it involves not only the physical model itself, but also the test facility, the environment and the model response as shown in Fig. 1. The most important problems associated with marine structures involve geometric, kinematic and dynamic similarities. 3.1.1. Geometric similarity (similarity of shape) If the specified physical quantities are lengths, the similarity between the systems is known as ‘geometric similarity’. Where boundaries of solid bodies or patterns of motion are considered, the characteristic property of geometric similarity is that the ratio of any length in one system to the corresponding length in the other system is the same throughout. This ratio is usually called the scale factor; i.e. adopting the suffix ‘p’ to denote prototype and ‘m’ to denote model.
(scale factor) =
冉冊
Lp . Lm
Corollaries of geometric similarity are: Areap = 2Aream Volumep = 3Volumem→massp = 3massm, (if p = m) Ip(mk)2 = 5Im, kp = km. A serious departure from geometric similarity may arise from the failure to scale the roughness of the solid boundaries. Accurate representation of surface roughness entails scaling not only the heights of individual protuberances but also their distribution over the surface, which is usually determined by the manufacturing process. To overcome this deficiency several turbulence stimulators have been devised. 3.1.2. Kinematic similarity (similarity of motion) When the flow pattern in a model system is geometrically similar to that in the prototype system then these two systems are said to possess kinematic similarity. Since, however, solid boundaries themselves consist of streamlines, geometric similarity of models is a prerequisite. Similarity of motion also implies geometric similarity and similarity of time intervals. Thus, as the corresponding lengths and corre-
Fig. 1.
The physical similarity domain.
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sponding time intervals in two systems are in a fixed ratio, so the corresponding velocities (and accelerations) must also be in a fixed ratio of magnitude at corresponding times. An example of kinematic similarity is found in a planetarium where parts of the Universe are reproduced to a given length scale factor, and planetary motions are copied in a fixed ratio of time intervals.
3.1.3. Dynamic similarity (similarity of forces) Dynamic similarity exists when the ratio of corresponding forces is constant. The forces that may be relevant include: inertial⇒(mass × acceleration) gravitational⇒(mass × g) viscous⇒(shear stress × area) elastic⇒(modulus of elasticity × area, or bulk modulus × area, or c2 × area − isentropic flow) 앫 pressure⇒(⌬P × area) 앫 capillary.
앫 앫 앫 앫
The fluid inertia force is common to all fluid dynamics problems and consequently any other relevant force is conveniently introduced as a ratio to this force. To form these ratios, the individual forces are first expressed in terms of a number of relevant physical parameters, i.e. l, length v, velocity g, gravitational acceleration k, bulk modulus E, modulus of elasticity ␥, surface tension. Using these parameters, certain important ratios can be formed as shown in Table 1. If one is seeking to preserve dynamic similarity between the model and prototype systems, and hence also geometric and kinematic similarity, it is necessary to preserve the relevant magnitudes of all the relevant forces as the physical parameters change magnitude. If, for example the two forces considered to be relevant are the inertial and viscous fluid forces, then dynamic similarity entails the constancy of Rn between the model and prototype. If, however, inertial, viscous and gravitational forces are all important, this would require a simultaneous scaling of both Fn and Rn, i.e. keeping their ratio constant. However, Rn/Fn = (g1/2l3/2)/ and keeping this ratio constant for a substantial change in ‘l’ would be impossible. In cases such as this, partial dynamic similarity would normally be adopted by scaling properly what is assumed to be the important quantities, whilst taking steps to ensure that the improperly scaled quantities are of little importance, otherwise correction factors may be used.
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Table 1 Common dimensionless numbers Description
Expression
Name
Inertia Force Gravity Force
V √gl
Fn
Inertia Force Viscous Force
vl
Rn
Inertia Force Elastic Force
v2 v2 v2 or K E c2
冉冊 冉 冊
Inertia Force Surface Tension Force
v
l ␥
Inertia Force Pressure Force
v
⌬P
冉冊
Ca(Ma)2
Wb
1/2
Eu
1/2
3.2. Dimensional analysis A common approach to the planning of model experiments lies in carrying out a dimensional analysis of the situation at hand. This enables the relevant factors influencing a particular problem to be assembled in a manner which is suggestive of its underlying structure and indicative of requirements that the model should satisfy. Consider, for example, the heave motion of a ship and its model (Lloyd, 1988). Without any detailed knowledge of the physical processes involved it might be surmised that heave amplitude will be a function of wave amplitude and frequency, the speed and heading, and the size, shape and inertias of the hull. In addition, fluid properties such as density, gravitational acceleration and viscosity should be relevant. On the basis of the above, and following the standard dimensional analysis methodology, it follows zo = f1兵o,,,␣,L,[H],[I],,g,其
(1)
where, f = some function of, H = hull shape matrix and [I] = inertias matrix Expression (1) could further be written in the form
再
zo o = f2 , o L
冪g, √gL ,, L
v
冎
[H] [I] vL , . , L L5
(2)
What Eq. (2) is implying is that the non-dimensional heave amplitude will be the same at both model and full scale provided that all the non-dimensional parameters on the RHS of the equation have the same numerical values at model and full scale. This requirement dictates the conditions required for the model experiment and it will be educational to explore this a little further.
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3.3. Non-dimensional wave amplitude
冉冊 冉冊 o L
=
p
( o )p o → = . L m ( o )m
3.4. Non-dimensional wave frequency
p
冪 g = 冪 g → Lp
Lm
m
m
= p√.
From the relationships
=
2 2g 2 ;T= ;k= →m = p/; Tm = Tp/√; km = kp. 2 g
3.5. Non-dimensional speed Identical Fn’s yield Vm = Vp/√. On the other hand, identical Rn’s require Vm = Vp . Clearly, unless = 1 the above requirements cannot both be simultaneously satisfied, a conclusion already arrived at. It may be observed, however, that maintaining Rn constant is not a practical proposition (a prototype speed of 30 knots and a of 30 would require a model test speed of 900 knots!). Fortunately, viscous forces do not play a prominent part in rigid body motions. The requirement of keeping Rn constant may therefore be waived provided that the right precautions are taken, i.e. adopting as large a model size as possible, and using turbulence stimulators if this is judged appropriate. Finally, assuming that geometric similarity is satisfied and linearity prevails, Eq. (2) may be written as
冦
冧 冦
冧
冪g,Fn, or f 冪g,F , .
zo =f o 3
L
4
L
e
n
(3)
Extending the above argument to irregular waves, this becomes: RMSz = f5
冦
H1/3 ,T0 L
冧
冪L,Fn, . g
Summarising the above, the experiment scaling laws are shown in Table 2. 3.6. Governing equations Model experiment scaling may also be derived by an examination of the governing equations of a problem when these are known. These equations must hold for both
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Table 2 Experiment scaling laws (Froude scaling) Prototype quantity
Model quantity
Mass, mp Length, Lp Time, Tp Velocity, Vp Acceleration, ␣p Angle, p · Angular velocity, p Angular acceleration, ¨ p Pressure/stress, Pp Frequency, fp Force, Fp Moment, Mp
mp × ⫺3 Lp × ⫺1 Tp × ⫺1/2 Vp × ⫺1/2 ␣p × 1 · p × 1 p × 1/2 ¨ p × Pp × ⫺1 fp × 1/2 Fp × ⫺3 Mp × ⫺4
the model and the prototype and can be used to develop appropriate scaling laws. Consider, for example, the classic single D.O.F. spring–mass system, where the equation of motion is given by: mx¨(t) + cx˙(t) + kx(t) = F(t).
(4)
Related parameters of interest include the natural frequency n and the damping ratio ␥ defined as:
n =
冉冊 k m
1 2
c . 2√km
;␥=
(5)
Eq. (5) must be also held for both the model and the prototype. Applying Eq. (4) to the model and introducing appropriate scaling factors, e.g. m = mp/mm, etc., yields −1 2 −1 −1 −1 −1 ¨ p + (−1 ˙ p + (−1 m x t mpx c x t)cpx k x )kpxp = F FP.
(6)
Multiplying (6) by kx and comparing the corresponding terms between (4) and (6), yields
2t k tk kx = = = 1. m c F
(7)
Applying also (5) to model and prototype ( is constant),
omegan =
冉冊 k m
1/2
; c = (km)1/2.
Solving, for example, (7) and (8) in terms of x, m and t
k =
1 m m mx ; = ; = 2 ;omegan = . 2t c t F t t
(8)
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Adopting Froude scaling, x = , m = 3, t = 1/2 →k = 2; c = 2.5; F = 3. The above is known as ‘group theory’ approach.
4. Model scaling laws for ocean engineering tests 4.1. Non-dimensional parameters Nearly all ocean engineering structures, whether they be fixed platforms or floating vessels employ circular tubes for part of their structure. Although steady current situations are of interest for all such structures, the prime design loading arises from wave action as a result of the varying velocity and acceleration due to water particles motion. It is well known that the drag coefficient of long smooth cylinders subjected to steady flow exhibit a large change at Rn ⬇ 5 × 105. In waves, the drag coefficient and added mass coefficient for a circular cylinder appear to vary with both Rn and the Keulegan–Carpenter Number (NKC = UmT/D; period parameter). The NKC describes the viscous scale effects acting on circular cylinders in sinusoidal flow fields. For small values of the parameter, the values of added mass and drag coefficients vary considerably. At higher values (increasing period of the flow oscillation), the coefficients approach constant values, dependent only on the instantaneous Rn. The majority of offshore structures will experience their severest loading when Rn is in the super-critical range and only a few items, e.g. drill pipes, mooring ropes, etc., will experience maximum loading while in the sub-critical or critical ranges. As the diameter of the member increases, the relative significance of the velocity and acceleration induced forces changes, with the inertia forces beginning to predominate until the stage is reached where there is substantial reflection of the waves with diffraction also occurring as shown in Fig. 2. Another viscous phenomenon associated with relatively low flow velocities is the rate at which long, bluff bodies such as cables or risers shed vortices. An important parameter for these types of flow is the Strouhal number (S = fD/U; vortex shedding frequency parameter). If the frequency at which the vortices are shed matches one of the natural frequencies of the structure, serious dynamic loading can occur. The subject of time dependent flows acting on slender members, such as tubular structural members, is of high interest to the offshore industry. If viscous forces dominate and Reynolds scaling law, is applied: vm = vp; v˙m = v˙p3; Fm = Fp. The viscous forces, if scaled properly, are the same at model and full scale while inertia forces are smaller by a factor 3. 4.2. Modelling of wave forces The general problem concerning the forces acting on a fixed body of characteristic dimension D, when subjected to a regular wave train of length L, height H in water
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Fig. 2.
Regions of validity–force prediction methods for a fixed pile.
depth d is of fundamental importance. Any time-invariant force F, such as the maximum in-line force, may be expressed in functional form as: F = f(,,D,H,L,d,g)
(9)
from which it may be derived that F = f(d/L,H/L,D/L,Rn). gHD2
(10)
In model tests the constancy of the dimensionless parameters d/L (wave–depth parameter) and H/L (wave steepness) between the model and the prototype ensures that the wave train is suitably represented. A selection of alternative parameters in the literature include: H/d, kH, H/gT2, HL2/d3 (Ursell parameter), kd, d/L, d/gT2 and T√d/g. Any four independent parameters will suffice. Depending on the size of D/L the following cases arise:
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4.3. D/L > 0.2-wave forces in the diffraction regime When D/L is sufficiently large, wave diffraction or scattering is important. Furthermore, the amplitude of water particle oscillation relative to D is necessarily small and the flow separation effects may be neglected. Therefore Rn may be omitted and the flow may be treated on the basis of potential theory. In addition if H/L is small, F may be considered to be proportional to H and thus H/L may also be omitted. Thus, in the usual linear diffraction problem it follows: F = f(d/L,D/L) or f(D/d,D/L). gHD2
(11)
In deep water, F = f(D/L). gHD2
(12)
4.4. D/L ⬍ 0.2-wave forces in the flow separation regime If D/L is small, the body will no longer scatter the incident wave field and D/L has no direct physical significance. Under such conditions, however, NKC assumes importance. Eq. (10) now becomes F = f(d/l,H/L,NKC,Rn). gHD2
(13)
This case can further be subdivided into two problems depending on the degree of importance of Rn. 4.4.1. Rn effects unimportant Provided changes in Rn are taken to be admissible, Eq. (13) represents a situation that can be reproduced in the laboratory, i.e. the lack of ability of maintaining high Rn in model tests is assumed to be of no major importance. In this case, under certain conditions, a further simplification may be possible. For example, the 2-D sinusoidal flow normal to the axis of a vertical cylinder can be characterised by Um (maximum horizontal wave particle velocity) and T. Then the time-invariant sectional force F⬘ may be written as F⬘ = f(NKC,Rn); NKC = UmT/D; Rn = UmD/. 1/2DU2m
(14)
Correspondingly, the instantaneous time-dependent force Ft⬘ would be written as Ft⬘ = f(NKC,Rn,t/T). 1/2DU2m
(15)
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D. Vassalos / Ocean Engineering 26 (1999) 111–123
4.4.2. Rn effects important This case presents the most significant source of difficulties in the modelling of offshore structures. Since it is not possible to model a free-surface flow so as to maintain both Fn and Rn constant, several approaches attempting to by-pass this difficulty have been devised. It should be stated that when no flow separation occurs or if it is located so as not to influence the overall loads on the structure, the Rn effect may be neglected. Some of the approaches worth mentioning are: 앫 restricting the simulation to 2-D flow by using a piston or a U-shaped tube (Sarpkaya and Isaacson, 1981) 앫 body oscillating in otherwise still water, e.g. Strathclyde ULOC (Wolfram, 1991) 앫 using turbulence stimulators. 4.5. Modelling of elastic structures When taking account of the dynamic response of an elastic member, a number of additional parameters are needed to characterise its behaviour. These include its density s, modulus of elasticity E, and damping properties, conveniently characterised by the damping ratio which is partially structural and partially hydrodynamic ␥sh. The Cauchy number also becomes yet another dimensionless parameter to consider. In wave force problems (Fn constant), Eq. (13) would be extended to F = f(d/L,H/L,NKC,Rn,s/D,U2/E,␥sh). gHD2
(16)
Ideally all three additional parameters should be held constant between the model and the prototype, but in practice this may not be possible. On the basis of Froude scaling, structural deflections will scale by and the same applies to both moduli of elasticity E and rigidity G. Angular rotations remain unchanged. In many problems the most important effect of E lies in describing the structure’s fundamental natural frequency, fn, and it is then convenient to adopt the alternative U/fnL (reduced velocity) in place of U2/E. The required frequency may be obtained by a model which is itself rigid but which is elastically mounted, rather than any flexibility in the model itself. When a fully elastic model of the entire structure is required some flexibility is achieved by attempting to scale EI correctly rather than E or I in isolation. Alternatively, segmented models connected by variable springs or special materials may be used.
5. Concluding remarks Based on the information presented in the foregoing two key remarks are noteworthy: 앫 There is no doubting the usefulness and necessity of physical models in the design of marine structures.
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앫 Before any testing is undertaken and more importantly before any results from model tests are to be used, the limitations of the particular test must be clearly understood and appreciated.
References Couch, R.B. et al., 1984. The use of model basins in the design of ships and marine structures. Trans. SNAME. Froude, W., 1874. On experiments with H.M.S. Greyhound. Trans. INA. Lloyd, A.R.J.M., 1988. Seakeeping: Ship Behaviour in Rough Weather. Horwood, London. Sarpkaya, T., Isaacson, M., 1981. Mechanics of Wave Forces on Offshore Structures. Van Nostrand Reinhold, New York. St Dennis, M., Pierson, W.J., 1953. On the motions of ships in confused seas. Trans. SNAME. Wolfram, J., 1991. A novel underwater hydrodynamic experiment facility and first results with smooth and marine growth covered circular cylinders. Trans. RINA.
Physical modelling and similitude of marine structures Dracos Vassalos The Ship Stability Research Centre, Department of Ship and Marine Technology, University of Strathclyde, Strathclyde, UK Received 19 May 1997; received in revised form 1 July 1997; accepted 2 September 1997
Abstract This review paper is aimed at providing information on and at explaining the appropriate use of models in the design of marine structures. Emphasis is placed on the derivation of scaling laws and on limitations of accuracy. Similitude, dimensional analysis and the use of governing equations in model experiment scaling are explained before addressing the modelling of some common ocean engineering tests. 1998 Elsevier Science Ltd. All rights reserved. Keywords: Physical modelling; Similitude; Marine structures
1. Introduction 1.1. General Systems involving the natural phenomena of the oceans, the atmosphere, geographical features, and so on, are notoriously difficult to investigate in detail. Thus for the purpose of analysis and from the point of view of economics, we are compelled to make use of models of the full-scale (often called the ‘prototype’) system. It is possible with laboratory models to ensure some control over the many variables of influence so that those of particular interest can be isolated for special study. Other reasons for using models include: 앫 To study phenomena defying analytical solution, or those for which no experimental results or empirical methods are available 0029-8018/99/$—see front matter 1998 Elsevier Science Ltd. All rights reserved. PII: S 0 0 2 9 - 8 0 1 8 ( 9 7 ) 1 0 0 0 4 - X
112
앫 앫 앫 앫 앫 앫 앫
D. Vassalos / Ocean Engineering 26 (1999) 111–123
To improve conceptual and physical understanding of a problem To verify (or disprove) theoretical results Data acquisition Calibration of a theoretical (numerical) model To select alternative designs To forecast full-scale behaviour To identify governing parameters.
In problems where analytical and/or empirical methods will yield a reliable solution the use of models is not justified, as being more expensive. In the absence of these, however, it must be understood that the actual cost of model testing will usually be only a small fraction of 1% of the overall capital sums involved and it may well be a great deal less than the cost of corrective action which may be necessary as a result of failure to carry out tests. In the majority of cases that naval architects have to deal with, and especially those involving fluid phenomena, it is very seldom that a problem can be tackled by theoretical means alone. Here the art of engineering must be practised with experience, judgement, ingenuity and patience if useful results are to be obtained and correctly interpreted, and prototype performance predicted from these. 1.2. Types of experiments Experiments can be broadly classified into the following two categories (Couch et al., 1984): 1. Steady State: Typically, resistance and propulsion experiments where the model is tested in a steady condition at forward speed. The data collected is usually time-averaged to render the results time independent. 2. Unsteady: Typically, seakeeping/ocean engineering experiments where, as a result of the transient loads produced by the ocean environment, such experiments are characterised by their dynamic non-steady nature. Data collected is now time dependent with results shown either as time domain statistics, e.g., rms, significant values, averages, etc., or as frequency domain functions, e.g. RAO’s (Response Amplitude Operators) and response spectra. A further distinction can be made between seakeeping and ocean engineering experiments: 앫 Seakeeping usually implies a dynamic process involving ship like vessels generally with forward speed. 앫 Ocean Engineering usually refers to the dynamic behaviour of stationary or nearly stationary structures (floating or fixed). Finally, in the context of either seakeeping or ocean engineering, the following types of experiment can be considered: 앫 Rigid Body Response—motions, velocities, accelerations, relative motions, pressures. The results will be RAO’s and/or motion statistics.
D. Vassalos / Ocean Engineering 26 (1999) 111–123
113
앫 Hydrodynamic Forces—Added mass and damping components (forces excitation at various frequencies), wave excitation component (restrained model). 앫 Flexural Body Response—Experiments investigating stresses or structural deflections of vessels and offshore structures. The emphasis in the paper will be on ocean engineering type experiments.
2. Brief historical background Historical records show that experiments with models of ships have been performed since the time of Leonardo da Vinci. However, it was not until W.E. Froude (1874), that model test predictions were established as a valuable engineering tool— in his case for the resistance of a full-scale ship from model resistance experiments in a towing tank. Interest in the seaworthiness of ships is as old as estimates of resistance and propulsion, but the accurate extrapolation of model test results to fullscale was not possible until the principles of spectral and transfer functions were established, as recorded in the seminal paper of St Dennis and Pierson (1953). Over the years, model tests have demonstrated their usefulness in solving problems relating to both preliminary design and retrofitting. The obvious benefits of a successful model test programme can be an optimum hydrodynamic and/or structural design which will enhance the economic performance of the marine structure.
3. Model experiments scaling Even though suitable measurements are generally easier to achieve in testing models that at full-scale, scaling problems can never by completely overcome, and it must be admitted that model tests in the controlled artificial environment of the laboratory always lack something of the uncertain harsh reality of the real world experienced by marine structures. In view of the difficulties in properly scaling an experiment, if a novel one is being planned, it is often prudent first to construct a small crude model for observational purposes only, in order to detect possible sources of problem, and to assist in planning the more detailed model tests. The basic principles of scaling laws and the main approaches to planning model experiments will be considered, under three headings. 3.1. Physical similarity The concept of physical similarity implies the need to ensure that, in planning the experiment, a certain similarity should be maintained between the model and the prototype. In general, two systems are said to be physically similar (in respect of certain specified physical quantities) when the ratio of corresponding magnitudes of these quantities is the same between the systems. This proposition must be satisfied not only by the contents of the system but also by the system boundary, inputs and
114
D. Vassalos / Ocean Engineering 26 (1999) 111–123
outputs. In other words, it involves not only the physical model itself, but also the test facility, the environment and the model response as shown in Fig. 1. The most important problems associated with marine structures involve geometric, kinematic and dynamic similarities. 3.1.1. Geometric similarity (similarity of shape) If the specified physical quantities are lengths, the similarity between the systems is known as ‘geometric similarity’. Where boundaries of solid bodies or patterns of motion are considered, the characteristic property of geometric similarity is that the ratio of any length in one system to the corresponding length in the other system is the same throughout. This ratio is usually called the scale factor; i.e. adopting the suffix ‘p’ to denote prototype and ‘m’ to denote model.
(scale factor) =
冉冊
Lp . Lm
Corollaries of geometric similarity are: Areap = 2Aream Volumep = 3Volumem→massp = 3massm, (if p = m) Ip(mk)2 = 5Im, kp = km. A serious departure from geometric similarity may arise from the failure to scale the roughness of the solid boundaries. Accurate representation of surface roughness entails scaling not only the heights of individual protuberances but also their distribution over the surface, which is usually determined by the manufacturing process. To overcome this deficiency several turbulence stimulators have been devised. 3.1.2. Kinematic similarity (similarity of motion) When the flow pattern in a model system is geometrically similar to that in the prototype system then these two systems are said to possess kinematic similarity. Since, however, solid boundaries themselves consist of streamlines, geometric similarity of models is a prerequisite. Similarity of motion also implies geometric similarity and similarity of time intervals. Thus, as the corresponding lengths and corre-
Fig. 1.
The physical similarity domain.
D. Vassalos / Ocean Engineering 26 (1999) 111–123
115
sponding time intervals in two systems are in a fixed ratio, so the corresponding velocities (and accelerations) must also be in a fixed ratio of magnitude at corresponding times. An example of kinematic similarity is found in a planetarium where parts of the Universe are reproduced to a given length scale factor, and planetary motions are copied in a fixed ratio of time intervals.
3.1.3. Dynamic similarity (similarity of forces) Dynamic similarity exists when the ratio of corresponding forces is constant. The forces that may be relevant include: inertial⇒(mass × acceleration) gravitational⇒(mass × g) viscous⇒(shear stress × area) elastic⇒(modulus of elasticity × area, or bulk modulus × area, or c2 × area − isentropic flow) 앫 pressure⇒(⌬P × area) 앫 capillary.
앫 앫 앫 앫
The fluid inertia force is common to all fluid dynamics problems and consequently any other relevant force is conveniently introduced as a ratio to this force. To form these ratios, the individual forces are first expressed in terms of a number of relevant physical parameters, i.e. l, length v, velocity g, gravitational acceleration k, bulk modulus E, modulus of elasticity ␥, surface tension. Using these parameters, certain important ratios can be formed as shown in Table 1. If one is seeking to preserve dynamic similarity between the model and prototype systems, and hence also geometric and kinematic similarity, it is necessary to preserve the relevant magnitudes of all the relevant forces as the physical parameters change magnitude. If, for example the two forces considered to be relevant are the inertial and viscous fluid forces, then dynamic similarity entails the constancy of Rn between the model and prototype. If, however, inertial, viscous and gravitational forces are all important, this would require a simultaneous scaling of both Fn and Rn, i.e. keeping their ratio constant. However, Rn/Fn = (g1/2l3/2)/ and keeping this ratio constant for a substantial change in ‘l’ would be impossible. In cases such as this, partial dynamic similarity would normally be adopted by scaling properly what is assumed to be the important quantities, whilst taking steps to ensure that the improperly scaled quantities are of little importance, otherwise correction factors may be used.
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Table 1 Common dimensionless numbers Description
Expression
Name
Inertia Force Gravity Force
V √gl
Fn
Inertia Force Viscous Force
vl
Rn
Inertia Force Elastic Force
v2 v2 v2 or K E c2
冉冊 冉 冊
Inertia Force Surface Tension Force
v
l ␥
Inertia Force Pressure Force
v
⌬P
冉冊
Ca(Ma)2
Wb
1/2
Eu
1/2
3.2. Dimensional analysis A common approach to the planning of model experiments lies in carrying out a dimensional analysis of the situation at hand. This enables the relevant factors influencing a particular problem to be assembled in a manner which is suggestive of its underlying structure and indicative of requirements that the model should satisfy. Consider, for example, the heave motion of a ship and its model (Lloyd, 1988). Without any detailed knowledge of the physical processes involved it might be surmised that heave amplitude will be a function of wave amplitude and frequency, the speed and heading, and the size, shape and inertias of the hull. In addition, fluid properties such as density, gravitational acceleration and viscosity should be relevant. On the basis of the above, and following the standard dimensional analysis methodology, it follows zo = f1兵o,,,␣,L,[H],[I],,g,其
(1)
where, f = some function of, H = hull shape matrix and [I] = inertias matrix Expression (1) could further be written in the form
再
zo o = f2 , o L
冪g, √gL ,, L
v
冎
[H] [I] vL , . , L L5
(2)
What Eq. (2) is implying is that the non-dimensional heave amplitude will be the same at both model and full scale provided that all the non-dimensional parameters on the RHS of the equation have the same numerical values at model and full scale. This requirement dictates the conditions required for the model experiment and it will be educational to explore this a little further.
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3.3. Non-dimensional wave amplitude
冉冊 冉冊 o L
=
p
( o )p o → = . L m ( o )m
3.4. Non-dimensional wave frequency
p
冪 g = 冪 g → Lp
Lm
m
m
= p√.
From the relationships
=
2 2g 2 ;T= ;k= →m = p/; Tm = Tp/√; km = kp. 2 g
3.5. Non-dimensional speed Identical Fn’s yield Vm = Vp/√. On the other hand, identical Rn’s require Vm = Vp . Clearly, unless = 1 the above requirements cannot both be simultaneously satisfied, a conclusion already arrived at. It may be observed, however, that maintaining Rn constant is not a practical proposition (a prototype speed of 30 knots and a of 30 would require a model test speed of 900 knots!). Fortunately, viscous forces do not play a prominent part in rigid body motions. The requirement of keeping Rn constant may therefore be waived provided that the right precautions are taken, i.e. adopting as large a model size as possible, and using turbulence stimulators if this is judged appropriate. Finally, assuming that geometric similarity is satisfied and linearity prevails, Eq. (2) may be written as
冦
冧 冦
冧
冪g,Fn, or f 冪g,F , .
zo =f o 3
L
4
L
e
n
(3)
Extending the above argument to irregular waves, this becomes: RMSz = f5
冦
H1/3 ,T0 L
冧
冪L,Fn, . g
Summarising the above, the experiment scaling laws are shown in Table 2. 3.6. Governing equations Model experiment scaling may also be derived by an examination of the governing equations of a problem when these are known. These equations must hold for both
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Table 2 Experiment scaling laws (Froude scaling) Prototype quantity
Model quantity
Mass, mp Length, Lp Time, Tp Velocity, Vp Acceleration, ␣p Angle, p · Angular velocity, p Angular acceleration, ¨ p Pressure/stress, Pp Frequency, fp Force, Fp Moment, Mp
mp × ⫺3 Lp × ⫺1 Tp × ⫺1/2 Vp × ⫺1/2 ␣p × 1 · p × 1 p × 1/2 ¨ p × Pp × ⫺1 fp × 1/2 Fp × ⫺3 Mp × ⫺4
the model and the prototype and can be used to develop appropriate scaling laws. Consider, for example, the classic single D.O.F. spring–mass system, where the equation of motion is given by: mx¨(t) + cx˙(t) + kx(t) = F(t).
(4)
Related parameters of interest include the natural frequency n and the damping ratio ␥ defined as:
n =
冉冊 k m
1 2
c . 2√km
;␥=
(5)
Eq. (5) must be also held for both the model and the prototype. Applying Eq. (4) to the model and introducing appropriate scaling factors, e.g. m = mp/mm, etc., yields −1 2 −1 −1 −1 −1 ¨ p + (−1 ˙ p + (−1 m x t mpx c x t)cpx k x )kpxp = F FP.
(6)
Multiplying (6) by kx and comparing the corresponding terms between (4) and (6), yields
2t k tk kx = = = 1. m c F
(7)
Applying also (5) to model and prototype ( is constant),
omegan =
冉冊 k m
1/2
; c = (km)1/2.
Solving, for example, (7) and (8) in terms of x, m and t
k =
1 m m mx ; = ; = 2 ;omegan = . 2t c t F t t
(8)
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Adopting Froude scaling, x = , m = 3, t = 1/2 →k = 2; c = 2.5; F = 3. The above is known as ‘group theory’ approach.
4. Model scaling laws for ocean engineering tests 4.1. Non-dimensional parameters Nearly all ocean engineering structures, whether they be fixed platforms or floating vessels employ circular tubes for part of their structure. Although steady current situations are of interest for all such structures, the prime design loading arises from wave action as a result of the varying velocity and acceleration due to water particles motion. It is well known that the drag coefficient of long smooth cylinders subjected to steady flow exhibit a large change at Rn ⬇ 5 × 105. In waves, the drag coefficient and added mass coefficient for a circular cylinder appear to vary with both Rn and the Keulegan–Carpenter Number (NKC = UmT/D; period parameter). The NKC describes the viscous scale effects acting on circular cylinders in sinusoidal flow fields. For small values of the parameter, the values of added mass and drag coefficients vary considerably. At higher values (increasing period of the flow oscillation), the coefficients approach constant values, dependent only on the instantaneous Rn. The majority of offshore structures will experience their severest loading when Rn is in the super-critical range and only a few items, e.g. drill pipes, mooring ropes, etc., will experience maximum loading while in the sub-critical or critical ranges. As the diameter of the member increases, the relative significance of the velocity and acceleration induced forces changes, with the inertia forces beginning to predominate until the stage is reached where there is substantial reflection of the waves with diffraction also occurring as shown in Fig. 2. Another viscous phenomenon associated with relatively low flow velocities is the rate at which long, bluff bodies such as cables or risers shed vortices. An important parameter for these types of flow is the Strouhal number (S = fD/U; vortex shedding frequency parameter). If the frequency at which the vortices are shed matches one of the natural frequencies of the structure, serious dynamic loading can occur. The subject of time dependent flows acting on slender members, such as tubular structural members, is of high interest to the offshore industry. If viscous forces dominate and Reynolds scaling law, is applied: vm = vp; v˙m = v˙p3; Fm = Fp. The viscous forces, if scaled properly, are the same at model and full scale while inertia forces are smaller by a factor 3. 4.2. Modelling of wave forces The general problem concerning the forces acting on a fixed body of characteristic dimension D, when subjected to a regular wave train of length L, height H in water
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Fig. 2.
Regions of validity–force prediction methods for a fixed pile.
depth d is of fundamental importance. Any time-invariant force F, such as the maximum in-line force, may be expressed in functional form as: F = f(,,D,H,L,d,g)
(9)
from which it may be derived that F = f(d/L,H/L,D/L,Rn). gHD2
(10)
In model tests the constancy of the dimensionless parameters d/L (wave–depth parameter) and H/L (wave steepness) between the model and the prototype ensures that the wave train is suitably represented. A selection of alternative parameters in the literature include: H/d, kH, H/gT2, HL2/d3 (Ursell parameter), kd, d/L, d/gT2 and T√d/g. Any four independent parameters will suffice. Depending on the size of D/L the following cases arise:
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4.3. D/L > 0.2-wave forces in the diffraction regime When D/L is sufficiently large, wave diffraction or scattering is important. Furthermore, the amplitude of water particle oscillation relative to D is necessarily small and the flow separation effects may be neglected. Therefore Rn may be omitted and the flow may be treated on the basis of potential theory. In addition if H/L is small, F may be considered to be proportional to H and thus H/L may also be omitted. Thus, in the usual linear diffraction problem it follows: F = f(d/L,D/L) or f(D/d,D/L). gHD2
(11)
In deep water, F = f(D/L). gHD2
(12)
4.4. D/L ⬍ 0.2-wave forces in the flow separation regime If D/L is small, the body will no longer scatter the incident wave field and D/L has no direct physical significance. Under such conditions, however, NKC assumes importance. Eq. (10) now becomes F = f(d/l,H/L,NKC,Rn). gHD2
(13)
This case can further be subdivided into two problems depending on the degree of importance of Rn. 4.4.1. Rn effects unimportant Provided changes in Rn are taken to be admissible, Eq. (13) represents a situation that can be reproduced in the laboratory, i.e. the lack of ability of maintaining high Rn in model tests is assumed to be of no major importance. In this case, under certain conditions, a further simplification may be possible. For example, the 2-D sinusoidal flow normal to the axis of a vertical cylinder can be characterised by Um (maximum horizontal wave particle velocity) and T. Then the time-invariant sectional force F⬘ may be written as F⬘ = f(NKC,Rn); NKC = UmT/D; Rn = UmD/. 1/2DU2m
(14)
Correspondingly, the instantaneous time-dependent force Ft⬘ would be written as Ft⬘ = f(NKC,Rn,t/T). 1/2DU2m
(15)
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4.4.2. Rn effects important This case presents the most significant source of difficulties in the modelling of offshore structures. Since it is not possible to model a free-surface flow so as to maintain both Fn and Rn constant, several approaches attempting to by-pass this difficulty have been devised. It should be stated that when no flow separation occurs or if it is located so as not to influence the overall loads on the structure, the Rn effect may be neglected. Some of the approaches worth mentioning are: 앫 restricting the simulation to 2-D flow by using a piston or a U-shaped tube (Sarpkaya and Isaacson, 1981) 앫 body oscillating in otherwise still water, e.g. Strathclyde ULOC (Wolfram, 1991) 앫 using turbulence stimulators. 4.5. Modelling of elastic structures When taking account of the dynamic response of an elastic member, a number of additional parameters are needed to characterise its behaviour. These include its density s, modulus of elasticity E, and damping properties, conveniently characterised by the damping ratio which is partially structural and partially hydrodynamic ␥sh. The Cauchy number also becomes yet another dimensionless parameter to consider. In wave force problems (Fn constant), Eq. (13) would be extended to F = f(d/L,H/L,NKC,Rn,s/D,U2/E,␥sh). gHD2
(16)
Ideally all three additional parameters should be held constant between the model and the prototype, but in practice this may not be possible. On the basis of Froude scaling, structural deflections will scale by and the same applies to both moduli of elasticity E and rigidity G. Angular rotations remain unchanged. In many problems the most important effect of E lies in describing the structure’s fundamental natural frequency, fn, and it is then convenient to adopt the alternative U/fnL (reduced velocity) in place of U2/E. The required frequency may be obtained by a model which is itself rigid but which is elastically mounted, rather than any flexibility in the model itself. When a fully elastic model of the entire structure is required some flexibility is achieved by attempting to scale EI correctly rather than E or I in isolation. Alternatively, segmented models connected by variable springs or special materials may be used.
5. Concluding remarks Based on the information presented in the foregoing two key remarks are noteworthy: 앫 There is no doubting the usefulness and necessity of physical models in the design of marine structures.
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앫 Before any testing is undertaken and more importantly before any results from model tests are to be used, the limitations of the particular test must be clearly understood and appreciated.
References Couch, R.B. et al., 1984. The use of model basins in the design of ships and marine structures. Trans. SNAME. Froude, W., 1874. On experiments with H.M.S. Greyhound. Trans. INA. Lloyd, A.R.J.M., 1988. Seakeeping: Ship Behaviour in Rough Weather. Horwood, London. Sarpkaya, T., Isaacson, M., 1981. Mechanics of Wave Forces on Offshore Structures. Van Nostrand Reinhold, New York. St Dennis, M., Pierson, W.J., 1953. On the motions of ships in confused seas. Trans. SNAME. Wolfram, J., 1991. A novel underwater hydrodynamic experiment facility and first results with smooth and marine growth covered circular cylinders. Trans. RINA.