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Aspects of the hydrodynamic loading on towed arrays and flexible risers
Aspects of t.he hydrodynamic loading on towed arrays and flexible risers R. C. T. Rainey WS Atkins Engineering Sciences Ltd, Woodcote Grove, Ashley Road, Epsom, Surrey KT18 5BW, UK
The hydrodynamic loads on towed arrays and flexible risers can be divided into 'inertial' loads, due the basic need for the water to move aside as the array or riser passes, and 'drag' loads reflecting the size of the wake left behind. The latter must inevitably be calculated empirically from experimental data, because of the great inherent complexity of the turbulent processes in wakes. The former, however, can be calculated analytically, and this has been done for many years on towed arrays, although not on flexible risers. It is first shown under which circumstances the analytical formula used on towed arrays gives a worthwhile improvement over the Morison-type semi-empirical inertial forces used in the analysis of flexible risers. The improvements occur when the array or riser is tightly curved, which is of course a practically important situation, because it can lead to mechanical damage. It is next shown, however, that in this tightly-curved case the analytical formula itself breaks down, because an assumption used in its derivation is violated. The question is therefore posed of how the analytical formula might be generalized, and a number of alternative approaches to that problem are considered.
Keywords: hydrodynamic loads, flexible cylinders, towed arrays, flexible risers This paper is concerned with the calculation of the hydrodynamic loading on towed arrays (of the types used for geophysical prospecting at sea or in anti-submarine naval operations) and flexible risers, which both pose the problem of a long flexible cylinder moving under water. The central concept in the analysis of hydrodynamic phenomena of this kind is the classical division of the flow field into two components (see, for example, Reference 1, equation 2.4.13; the third component, which describes the effects of compressibility, can be ignored because its effect on hydrodynamic loads is negligible): (1) the potentialflow, i.e., the essential movement of the water necessary to allow the passage of the flexible cylinder with the creation of the minimum possible fluid kinetic energy, and in particular without the creation of any wake (2) the vorticity-induced flow, which is the flow field remaining when the potential-flow velocity at each point is subtracted from the total velocity. It consists This paper was originally presented at a meeting on 'Flexible risers' held 9 January 1989 at University College London, UK.
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E n g . S t r u c t . 1 9 8 9 , Vol. 1 1, O c t o b e r
essentially of the water velocity associated with the wake and the boundary layer next to the cylinder surface The importance of this twofold breakdown of the flow field is that the total hydrodynamic load is the sum of the hydrodynamic loads felt in the two flows considered separately. This classical result appears at first sight surprising, since the total hydrodynamic pressure is not the sum of the hydrodynamic pressures in the two flows considered separately. It is, however, a necessary consequence of the total momentum of the water (suitably defined) being the sum of momenta in the two separate flows, which is to be expected since momentum is directly proportional to velocity. The potential flow is responsible for the 'inertial' part of the hydrodynamic loading, which is the consequence of the acceleration of the water as it nears the cylinder and its deceleration after the cylinder has passed. The vorticity-induced flow is responsible for the 'drag' part of the hydrodynamic loading, which is the consequence of the kinetic energy which the cylinder leaves behind in its wake. 0141-0296/89/040248-06/$03.00 © 1989 Butterworth & Co (Publishers) Ltd
Hydrodynamic loading on towed arrays and flexible risers: R.C.T. Rainey The determination of the vorticity-induced flow is formidably difficult, since it is dominated by the problem of the separation of the boundary layer and the turbulent processes within it, both topics of enormous inherent complexity. In practice there is no alternative to an essentially empirical approach, using drag coefficients to characterize the drag loads in a suitable non-dimensional manner. It is therefore sometimes argued that there is no point in taking any great care over the inertial hydrodynamic loads, and that they should also be treated empirically. This approach has historically been followed in the empirical calculation of hydrodynamic loads on offshore structures and flexible risers by 'Morison's equation' (i.e., with empirical inertia coefficients as well as empirical drag coefficients; see, for example, Reference 2, equation 5.43, summarized in Figure 1), and indeed is justifiable whenever the hydrodynamic load is dominated by drag, as it commonly is. It was pointed out by Lighthill 3'4, however, that the inertial load deserves more attention in those cases where drag does not dominate, and that it is commonly not given such attention through ignorance of the fundamental theoretical result on load summation cited above. Lighthill's call has recently been answered by the derivation by the author of a general-purpose exact formula for the inertial loads on the cylindrical members of fixed or floating offshore structures 5. It is the purpose of this paper to review the calculation procedures available for the similar exact calculation of inertial hydrodynamic loads on towed arrays and flexible risers, which are both non-rigid, and thus present a more difficult case than an offshore structure. At the same time, this paper aims to explore the circumstances under which such improvements in the calculation of inertial loads are practically significant, which is also a more difficult question to decide than on an offshore structure.
Normal
.•
L~
~"
and thus is exactly equivalent to a Morison inertia term with a coefficient of pA. His second term - 2 p A U ( ~ 2 y /
Normal water acceleration, F N
~
Inertial load = aA N + bF N I Drag load = cV 2 + dV 2
J
a , b , c , d are empirical coefficients
Conventional 'Morison-type' empirical formulation of hydrodynamic loads on a flexible riser
Figure I
8xdt), however, has no equivalent in the Morison formulation, and his third - pA U2(O2y/8x 2) is of the form - p A U 2{curvature} This is exactly equivalent to the mechanical effect of an axial tension T, which produces a side force Ta2y/dx 2. Interestingly, though, it has the opposite sign---evidently Lighthilrs hydrodynamic force tends to buckle the array, whereas an axial tension tends to straighten it. To explore the conditions under which Lighthill's second and third terms (i.e., those beyond the Morison formulation) influence the behaviour of a towed array, we can write the overall equation of motion of the array in Figure 2 as (1 + r)pA
- pA {lateral acceleration}
VA
Normal riser acceleration, AN
Towed array case: Lighthill solution for straight towing For the case of a towed array under normal operating conditions (i.e., being towed straight and level) and thus undergoing only small deflections from a straight line, the problem of the inertial hydrodynamic loads was completely solved by Lighthill 6 (see Figure 2) in his studies of the swimming motion of slender fishes. The application to towed arrays was immediat.ely noticed by Hawthorne 7, and there has followed a series of papers of increasing mathematical complexity8-12 (these papers by no means all agree, incidentally) which follow the implications on array behaviour of the Lighthill inertial loads when combined with various formulations of the hydrodynamic drag. For our purposes all these subsequent analyses are incidental: we are concerned in this paper only with the hydrodynamic forces at the starting point of those analyses, and in particular the link between Lighthill's inertial loads and those in the Morison-type empirical formulation used for flexible risers (Figure 1). We see that his first term, --pA(d2y/dt2), is of the form
Axial / relative _ ~'velocity,
relative, VN velocity
t32y+ 2pA U ~d2y
-
(T-
pAU 2) ff~x2 ~2y = 0
where r is the array density (relative to water). This equation includes the full Lighthill inertial loads, but includes the drag loads only via the tension T(the tension in a straight towed array comes of course from longitudinal drag on the tail streamer and on the array itself). It can be written in the form t32Y
",
692Y
,'~2 t~2Y
at~+z~ff[--P
~x2=0
where ct = U/(1 + r) and f12 = T/(1 + r)pA - U2/(1 + r) = C 2 - U2/(1 q- r) We see at once that without Lighthill's second and third terms, or if the towing speed U of the array is zero, this equation reverts to the standard wave equation for a stretched string 13 632y ~t 2
t~2y Co2 ~ x 2 = 0
Eng. Struct. 1989, Vol. 11, October
249
Hydrodynamic loading on towed arrays and flexible risers. R.C. 1-. Rainey Lighthill 6 analysed small deflections of a slender fish holding station in a steady current:
0
I\\\\\
/ / / ~ -
Load = F (per unit length) -25 o
C u r r e t n t velocity = U
-~/
~
x
-so g f-
F = -pA (~t + U~-~)2y
V
Increasing speed
-75
32Y + 2U 32Y
= -pA
( "~'~ 3t
+ U2 -32Y )
3x~t
ax 2
a
where p = water density, A = cross-sectional area
Figure 2
I
-100
I
I
Analytical formulation of inertial loads
A
0 -
~
l
"~
25
~"--
//
which allows the propagation of transverse waves f ( x - Cot) with velocity Co. Otherwise, waves f ( x - ct) with velocity c can be supported if c2f " - 2 a c f " - fl2f,,= 0
I
-
c
That is, if
2
c)
c 2 - 2~c - ~2 = 0 5O
and thus if
b
l
l
I
l
Time
c = + ~//~ + ~
+=
Figure 3
Predicted time histories of changes in (a) tension and (b) depth of a towed array (150 degrees turn)
= + , / c ~ - u 2 / ( I + r) + U'/(I + r? + U/(I + r)
= _ Co(1 + (U/Co)2[1/(1 + r) - 1/(1 + r) 2] +...) +
Ul(1 + r)
+ Co + U/(1 + r) if U << C o
Thus waves can be supported if the wave velocity is Co + U/(1 + r) downstream or
Co - U/(1 + r) upstream
In view of the obvious central importance of these lateral-wave speeds to any analysis of the dynamics of the array, we conclude that Lighthill's extra hydrodynamic loads (those omitted in the Morison formulation) are important if the longitudinal water velocity along the array becomes comparable with the propagation speed o f lateral waves along the array (i.e., the propagation speed as calculated ignoring Lighthill's extra loads). To judge when this occurs in practice, we note that the definition of the lateral-wave speed COabove implies that it equals the water velocity required to give a drag of T on a body of cross-section 2(1 + r)A (and unity drag coefficient). Thus a typical 5 cm radius neutrally-buoyant towed array at a modest tension of 1 tonne has a lateral-wave speed of 25 m/s. This is 10 times the typical operational towing speed of 2.5 m/s; we conclude that the effects of Lighthill's extra terms will only be noticeable when the tensions are much lower. For example, if the
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Eng. Struct. 1989, Vol. 11, October
tension drops to 0.01 tonne, then the lateral-wave speed becomes 2.5 m/s also, and downstream-travelling waves will (see above) propagate at three times the speed of upstream-travelling waves, which is a marked contrast with the single wave speed of the Morison formulation. The circumstance under which such low tensions occur is well known: it is during tight turning manoeuvres of the towing vessel (see Figure 3 taken from Reference 14, which describes the WS Atkins computer program TOWS3D, and compares its predictions with the results of recent Royal Navy towed array trials), when the array is forced into a tightly-curved configuration, and can buckle and break.
Flexible riser case: inapplicability of the Lighthill solution The problem of a tightly curved towed array neatly introduces the topic of flexible risers, which are typically deployed in configurations which can become tightly curved (again with risk of damage) under the action of extreme waves. Figure 4 shows the extreme motions of a flexible riser configuration, as predicted by the WS Atkins computer program AQWAFLEX 15. It appears that the details of the inertial hydrodynamic loads (described above) will be just as important to the central design problem of tight riser curvature in extreme waves as they are to the problem of tight curvature in towed arrays, and for just the same reason. We are therefore presented with the important question of whether the Lighthill analysis holds in the case of a
Hydrodynamic loading on towed arrays and flexible risers: R.C.T. Rainey -0.00
-0.10 Minimum
D
-0.20
--
-0.30
m
i
%
-0.40
--
X r-
-0.50
-0.60
- -
- -
P r o d u c t i o n
-0.70
-0.80 --x~x--
Control b u n d l e Export I offloading
-0.90
I
- I .00 - I .2
- I .0
,
I -0.8
I
I -0.6
I
I
i
,
-0.4
-0.2
I
i
0
0.2
X - d i s t a n c e x 102(m}
Figure 4
C o m p u t e d extreme positions of flexible risers in survival w a v e s
tightly-curved flexible cylinder. The answer is that it does not. This can be seen by considering the case of a flexible riser formed into a rigid circular hoop and held obliquely in a uniform current, as shown in Figure 5. Referring to Figure 2, we see that the first two of Lighthill's hydrodynamic load terms will be zero, because the riser is not moving. Only the last one will be non-zero, and it will produce a force proportional to the curvature of the hoop and the local water velocity along that part of the riser. However, since the curvature of the riser is everywhere directed towards its centre, the inertial hydrodynamic force will be also, and will thus give zero turning moment about the hoop diameter perpendicular to the current, shown in Figure 5. And this is incorrect, because it is known that the inertial hydrodynamic loads on such a hoop will reduce to a 'Munk moment' about that hoop diameter, and this moment will be non-zero because the added mass of the hoop as a whole is different in the 'broadside' and 'edgewise' directions (see, for example, Reference 16, equation 338). The problem is evidently that the Lighthill analysis assumes that the angle of the riser to the flow is small, and in this case it is not. It is therefore natural to ask if Lighthill's original analysis 6 can be extended to cover this case. At a superficial level Lighthill's argument is a simple momentum one (which, incidentally, resembles the derivation of the lateral forces resulting from internal flow in the riser--here there is no analogous complexity
about the angle of the riser to the flow, and the force in our fixed hoop example will be a simple centrifugal one): at any point the riser is moving laterally relative to the water at a velocity
so that the lateral momentum of the water per unit length must be (given that the added mass per unit length is pA)
and thus the rate of change of this momentum, which must be equal and opposite to the lateral force, is
pA ~ + U ~ x
y
It is tempting to extend this thinking to the case in Figure 5, where the relative velocity of the water changes around the hoop in a more complicated way. This, however, would be to miss the essential subtlety of LighthiU's analysis. He stresses that momentum arguments of this type (which actually go back to Munk's derivation of the 'negative centrifugal force' on a turning airship, see Reference 17, Figure 3) are suggestive only,
Eng. Struct. 1989, Vol. 11, October
251
Hydrodynamic loading on towed arrays and flexible risers." R.C.T. Rainey
Current y
Figure 5
Riser formed into a rigid circular hoop and fixed obliquely in a steady current (i.e., current not aligned with axis of hoop)
and goes on to prove his result by direct integration of surface pressures. The reason for this caution is the classical difficulty that when a body moves in an infinite expanse of water the momentum of the water associated with that movement is not defined, because the water velocity does not decay sufficiently rapidly with distance from the body to make the momentum volume integral convergC. This famous mathematical difficulty is based on a very real physical phenomenon - - that if the expanse of water is not infinite, but bounded by distant rigid walls, then when the body accelerates the walls as a whole will feel an acceleration reaction which does not tend to zero as we consider ever more-distant wails. (We can see that this must be true by considering the case of the distant walls, and thus the water as a whole, oscillating rigidly back and forth. By moving cyclically in a suitable relative phase, the immersed body would be able to extract useful work from the inertial forces from the moving water - - the walls must therefore be able to 'feel that this is going on', to enable them to put in an equivalent amount of energy to that taken out by the body). Thus when we try to circumvent the problem of momentum definition by imagining the water to have distant boundaries (finitely distant boundaries will of course always make the momentum volume integral well defined), our momentum arguments are greatly complicated by the necessity for considering the reaction forces felt on these distant boundaries. In view of this fundamental problem it is perhaps surprising that a naive momentum argument nevertheless gives the right answer in the case of Figure 2. Lest this be thought typical, a case where it gives the wrong answer is that of a neutrally-buoyant straight cylinder moving broadside out into the air from an expanse of still water. The popular momentum argument 2 would tell us that the cylinder velocity must double, since the effective mass of the cylinder has halved (its added mass in this case equalling its own mass) - - this, however, is to create energy from nowhere 5 because twice the velocity and half the mass gives twice the kinetic energy. On the other hand, the approach of direct integration of surface pressures is formidably arduous algebraically - - Lighthill 6 required 20 long equations as against three short ones for the momentum argument (see also Reference 5, where it is shown that this is inevitably so, because the leading pressure terms cancel out by 'D' Alembert's paradox'). And the more arduous the
252
Eng. Struct. 1989, Vol. 11, October
algebra, the lower the probability that the final answer is correct. Indeed, authors striving to obtain correct answers in this field (rather than merely impress the naive reader!) invariably lay great stress on cross-checks with other, shorter, arguments is Be all these difficulties as they may, the logic of this paper is that Lighthill's refinements to the calculation of inertial hydrodynamic loads, already widely applied in the study of towed arrays, appear to matter most in the very case (i.e., the highly-curved one) where that method is inapplicable. If this impasse is to be broken, Lighthill's result has got to be generalized, and there appear in view of the above to be only three realistic options for doing it: (1) Lighthill's combination of a heuristic momentum argument backed by a rigorous, but long, direct integration of surface pressures (2) a rigorous momentum argument. This might be constructed, for example, by the device of two parallel confining walls, described elsewhere by Lighthill 3 (3) a rigorous energy argument. This method was pioneered by Taylor 18 and is the basis for the general rigid-body result of Rainey 5 Conclusions
Although the hydrodynamic drag on towed arrays and flexible risers has inevitably to be calculated empirically from experimental data, there is no reason why the hydrodynamic inertial forces should not be calculated analytically from first principles. Indeed, such an analytical result (significantly more sophisticated than the Morison-type empirical formulation) has been applied for many years to towed arrays. Unfortunately, it appears that the case when this extra sophistication makes a practical difference is that of the tightly-curved towed array or flexible riser (important because it is typically tight curvature which causes mechanical damage), and in this case the analytical result is inapplicable. Deriving a more general, universally applicable, version of the analytical result appears feasible. An encouraging recent development is the derivation of a related general-purpose analytical description of the hydrodynamic inertial forces on offshore structures. Acknowledgements
The author acknowledges the improvements to this paper suggested by contributors to the discussion at the Seminar on 9 January. Specifically, Dr. R. M. Carson drew the author's attention to the works cited above by Hawthorne, Paidoussis, Ortloff and lves, and Lee; and Dr. E. Kodaissi drew the author's attention to the effects of internal flow mentioned above. References
1 Batchelor, G. K. An Introduction to Fluid Dynamics, 1st edn, Cambridge University Press, 1967 2 Sarpkaya, T. and Isaacson, M.de St.Q. Mechanics o f Wave Forces on Offshore Structures, Van Nostrand Reinhold, New York, 1981 3 Lighthill, Sir James 'Waves and hydrodynamic loading' Proc. 2nd Int. Conf. on the Behaviour o f Offshore Structures, 1, BHRA Fluid Engineering, Cranfield, 1979, 1~10
Hydrodynamic loading on towed arrays and flexible risers: R.C.T. Rainey 4 5 6 7 8 9 10 11 12
Lighthill, Sir James 'Fundamentals concerning wave loading on offshore structures', J. Fluid Mech. 1986, 173, 667~81 Rainey, R. C. T. 'A new equation for calculating wave loads on offshore structures', J. Fluid Mech. 1989, 204, 295-324 Lighthill, Sir James 'Note on the swimming of slender fish', J. Fluid Mech. 1960, 9, 305-317 Hawthorne, W. R. 'The early development of the Dracone flexible barge', Proc. Inst. Mech. Engrs 1961, 175, 65-83 Paidoussis, M. P. 'Dynamics of flexible slender cylinders in axial flow', J. Fluid Mech. 1966, 26, 717-751 Paidoussis, M. P. 'Stability of towed, totally submerged flexible cylinders', J. Fluid Mech. 1968, 34, 273-297 Ortloff, C. R. and lves, J. 'On the dynamic motion of a thin flexible cylinder in a viscous stream', J. Fluid Mech. 1969, 38, 713-720 Lee, T.S.'StabilityoftheOrtloff-lvesequation',J. FluidMech. 1981, 110, 293-295 Dowling, A. P. 'The dynamics of towed flexible cylinders, parts 1 &
13 14
15 16 17 18
2', J. Fluid Mech. 1988, 187, 507 571 Kreysig, E. Advanced Engineering Mathematics, 3rd Edn, Wiley, New York, 1972 Andrew, R. N., Bull, P. W. and Smith, S. L. 'Hydrodynamic modelling of towed array sonars', Proc. Int. Syrup. on AntiSubmarine Warfare, 1, Royal Institution of Naval Architects, London, 1987 Smith, S. L. and Lyons, R. A. User Manual for the Computer Program AQWAFLEX, WS Atkins Engineering Sciences Ltd, Epsom, Surrey, 1986 Lighthill, Sir James An Informal Introduction to Theoretical Fluid Mechanic's, Clarendon Press, Oxford, 1986 Munk, M. M. Aerodynamics of Airships, div.Q in Aerodynamic Theory, VI, (Ed. W. F. Durand) Julius Springer, Berlin, 1936 Taylor, Sir Geoffrey 'The forces on a body placed in a curved or converging stream of fluid', Proc. Roy Soc. Lond. 1928, AI20, 260-283
Eng. Struct. 1989, Vol. 11, October
253
The hydrodynamic loads on towed arrays and flexible risers can be divided into 'inertial' loads, due the basic need for the water to move aside as the array or riser passes, and 'drag' loads reflecting the size of the wake left behind. The latter must inevitably be calculated empirically from experimental data, because of the great inherent complexity of the turbulent processes in wakes. The former, however, can be calculated analytically, and this has been done for many years on towed arrays, although not on flexible risers. It is first shown under which circumstances the analytical formula used on towed arrays gives a worthwhile improvement over the Morison-type semi-empirical inertial forces used in the analysis of flexible risers. The improvements occur when the array or riser is tightly curved, which is of course a practically important situation, because it can lead to mechanical damage. It is next shown, however, that in this tightly-curved case the analytical formula itself breaks down, because an assumption used in its derivation is violated. The question is therefore posed of how the analytical formula might be generalized, and a number of alternative approaches to that problem are considered.
Keywords: hydrodynamic loads, flexible cylinders, towed arrays, flexible risers This paper is concerned with the calculation of the hydrodynamic loading on towed arrays (of the types used for geophysical prospecting at sea or in anti-submarine naval operations) and flexible risers, which both pose the problem of a long flexible cylinder moving under water. The central concept in the analysis of hydrodynamic phenomena of this kind is the classical division of the flow field into two components (see, for example, Reference 1, equation 2.4.13; the third component, which describes the effects of compressibility, can be ignored because its effect on hydrodynamic loads is negligible): (1) the potentialflow, i.e., the essential movement of the water necessary to allow the passage of the flexible cylinder with the creation of the minimum possible fluid kinetic energy, and in particular without the creation of any wake (2) the vorticity-induced flow, which is the flow field remaining when the potential-flow velocity at each point is subtracted from the total velocity. It consists This paper was originally presented at a meeting on 'Flexible risers' held 9 January 1989 at University College London, UK.
241t
E n g . S t r u c t . 1 9 8 9 , Vol. 1 1, O c t o b e r
essentially of the water velocity associated with the wake and the boundary layer next to the cylinder surface The importance of this twofold breakdown of the flow field is that the total hydrodynamic load is the sum of the hydrodynamic loads felt in the two flows considered separately. This classical result appears at first sight surprising, since the total hydrodynamic pressure is not the sum of the hydrodynamic pressures in the two flows considered separately. It is, however, a necessary consequence of the total momentum of the water (suitably defined) being the sum of momenta in the two separate flows, which is to be expected since momentum is directly proportional to velocity. The potential flow is responsible for the 'inertial' part of the hydrodynamic loading, which is the consequence of the acceleration of the water as it nears the cylinder and its deceleration after the cylinder has passed. The vorticity-induced flow is responsible for the 'drag' part of the hydrodynamic loading, which is the consequence of the kinetic energy which the cylinder leaves behind in its wake. 0141-0296/89/040248-06/$03.00 © 1989 Butterworth & Co (Publishers) Ltd
Hydrodynamic loading on towed arrays and flexible risers: R.C.T. Rainey The determination of the vorticity-induced flow is formidably difficult, since it is dominated by the problem of the separation of the boundary layer and the turbulent processes within it, both topics of enormous inherent complexity. In practice there is no alternative to an essentially empirical approach, using drag coefficients to characterize the drag loads in a suitable non-dimensional manner. It is therefore sometimes argued that there is no point in taking any great care over the inertial hydrodynamic loads, and that they should also be treated empirically. This approach has historically been followed in the empirical calculation of hydrodynamic loads on offshore structures and flexible risers by 'Morison's equation' (i.e., with empirical inertia coefficients as well as empirical drag coefficients; see, for example, Reference 2, equation 5.43, summarized in Figure 1), and indeed is justifiable whenever the hydrodynamic load is dominated by drag, as it commonly is. It was pointed out by Lighthill 3'4, however, that the inertial load deserves more attention in those cases where drag does not dominate, and that it is commonly not given such attention through ignorance of the fundamental theoretical result on load summation cited above. Lighthill's call has recently been answered by the derivation by the author of a general-purpose exact formula for the inertial loads on the cylindrical members of fixed or floating offshore structures 5. It is the purpose of this paper to review the calculation procedures available for the similar exact calculation of inertial hydrodynamic loads on towed arrays and flexible risers, which are both non-rigid, and thus present a more difficult case than an offshore structure. At the same time, this paper aims to explore the circumstances under which such improvements in the calculation of inertial loads are practically significant, which is also a more difficult question to decide than on an offshore structure.
Normal
.•
L~
~"
and thus is exactly equivalent to a Morison inertia term with a coefficient of pA. His second term - 2 p A U ( ~ 2 y /
Normal water acceleration, F N
~
Inertial load = aA N + bF N I Drag load = cV 2 + dV 2
J
a , b , c , d are empirical coefficients
Conventional 'Morison-type' empirical formulation of hydrodynamic loads on a flexible riser
Figure I
8xdt), however, has no equivalent in the Morison formulation, and his third - pA U2(O2y/8x 2) is of the form - p A U 2{curvature} This is exactly equivalent to the mechanical effect of an axial tension T, which produces a side force Ta2y/dx 2. Interestingly, though, it has the opposite sign---evidently Lighthilrs hydrodynamic force tends to buckle the array, whereas an axial tension tends to straighten it. To explore the conditions under which Lighthill's second and third terms (i.e., those beyond the Morison formulation) influence the behaviour of a towed array, we can write the overall equation of motion of the array in Figure 2 as (1 + r)pA
- pA {lateral acceleration}
VA
Normal riser acceleration, AN
Towed array case: Lighthill solution for straight towing For the case of a towed array under normal operating conditions (i.e., being towed straight and level) and thus undergoing only small deflections from a straight line, the problem of the inertial hydrodynamic loads was completely solved by Lighthill 6 (see Figure 2) in his studies of the swimming motion of slender fishes. The application to towed arrays was immediat.ely noticed by Hawthorne 7, and there has followed a series of papers of increasing mathematical complexity8-12 (these papers by no means all agree, incidentally) which follow the implications on array behaviour of the Lighthill inertial loads when combined with various formulations of the hydrodynamic drag. For our purposes all these subsequent analyses are incidental: we are concerned in this paper only with the hydrodynamic forces at the starting point of those analyses, and in particular the link between Lighthill's inertial loads and those in the Morison-type empirical formulation used for flexible risers (Figure 1). We see that his first term, --pA(d2y/dt2), is of the form
Axial / relative _ ~'velocity,
relative, VN velocity
t32y+ 2pA U ~d2y
-
(T-
pAU 2) ff~x2 ~2y = 0
where r is the array density (relative to water). This equation includes the full Lighthill inertial loads, but includes the drag loads only via the tension T(the tension in a straight towed array comes of course from longitudinal drag on the tail streamer and on the array itself). It can be written in the form t32Y
",
692Y
,'~2 t~2Y
at~+z~ff[--P
~x2=0
where ct = U/(1 + r) and f12 = T/(1 + r)pA - U2/(1 + r) = C 2 - U2/(1 q- r) We see at once that without Lighthill's second and third terms, or if the towing speed U of the array is zero, this equation reverts to the standard wave equation for a stretched string 13 632y ~t 2
t~2y Co2 ~ x 2 = 0
Eng. Struct. 1989, Vol. 11, October
249
Hydrodynamic loading on towed arrays and flexible risers. R.C. 1-. Rainey Lighthill 6 analysed small deflections of a slender fish holding station in a steady current:
0
I\\\\\
/ / / ~ -
Load = F (per unit length) -25 o
C u r r e t n t velocity = U
-~/
~
x
-so g f-
F = -pA (~t + U~-~)2y
V
Increasing speed
-75
32Y + 2U 32Y
= -pA
( "~'~ 3t
+ U2 -32Y )
3x~t
ax 2
a
where p = water density, A = cross-sectional area
Figure 2
I
-100
I
I
Analytical formulation of inertial loads
A
0 -
~
l
"~
25
~"--
//
which allows the propagation of transverse waves f ( x - Cot) with velocity Co. Otherwise, waves f ( x - ct) with velocity c can be supported if c2f " - 2 a c f " - fl2f,,= 0
I
-
c
That is, if
2
c)
c 2 - 2~c - ~2 = 0 5O
and thus if
b
l
l
I
l
Time
c = + ~//~ + ~
+=
Figure 3
Predicted time histories of changes in (a) tension and (b) depth of a towed array (150 degrees turn)
= + , / c ~ - u 2 / ( I + r) + U'/(I + r? + U/(I + r)
= _ Co(1 + (U/Co)2[1/(1 + r) - 1/(1 + r) 2] +...) +
Ul(1 + r)
+ Co + U/(1 + r) if U << C o
Thus waves can be supported if the wave velocity is Co + U/(1 + r) downstream or
Co - U/(1 + r) upstream
In view of the obvious central importance of these lateral-wave speeds to any analysis of the dynamics of the array, we conclude that Lighthill's extra hydrodynamic loads (those omitted in the Morison formulation) are important if the longitudinal water velocity along the array becomes comparable with the propagation speed o f lateral waves along the array (i.e., the propagation speed as calculated ignoring Lighthill's extra loads). To judge when this occurs in practice, we note that the definition of the lateral-wave speed COabove implies that it equals the water velocity required to give a drag of T on a body of cross-section 2(1 + r)A (and unity drag coefficient). Thus a typical 5 cm radius neutrally-buoyant towed array at a modest tension of 1 tonne has a lateral-wave speed of 25 m/s. This is 10 times the typical operational towing speed of 2.5 m/s; we conclude that the effects of Lighthill's extra terms will only be noticeable when the tensions are much lower. For example, if the
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tension drops to 0.01 tonne, then the lateral-wave speed becomes 2.5 m/s also, and downstream-travelling waves will (see above) propagate at three times the speed of upstream-travelling waves, which is a marked contrast with the single wave speed of the Morison formulation. The circumstance under which such low tensions occur is well known: it is during tight turning manoeuvres of the towing vessel (see Figure 3 taken from Reference 14, which describes the WS Atkins computer program TOWS3D, and compares its predictions with the results of recent Royal Navy towed array trials), when the array is forced into a tightly-curved configuration, and can buckle and break.
Flexible riser case: inapplicability of the Lighthill solution The problem of a tightly curved towed array neatly introduces the topic of flexible risers, which are typically deployed in configurations which can become tightly curved (again with risk of damage) under the action of extreme waves. Figure 4 shows the extreme motions of a flexible riser configuration, as predicted by the WS Atkins computer program AQWAFLEX 15. It appears that the details of the inertial hydrodynamic loads (described above) will be just as important to the central design problem of tight riser curvature in extreme waves as they are to the problem of tight curvature in towed arrays, and for just the same reason. We are therefore presented with the important question of whether the Lighthill analysis holds in the case of a
Hydrodynamic loading on towed arrays and flexible risers: R.C.T. Rainey -0.00
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tightly-curved flexible cylinder. The answer is that it does not. This can be seen by considering the case of a flexible riser formed into a rigid circular hoop and held obliquely in a uniform current, as shown in Figure 5. Referring to Figure 2, we see that the first two of Lighthill's hydrodynamic load terms will be zero, because the riser is not moving. Only the last one will be non-zero, and it will produce a force proportional to the curvature of the hoop and the local water velocity along that part of the riser. However, since the curvature of the riser is everywhere directed towards its centre, the inertial hydrodynamic force will be also, and will thus give zero turning moment about the hoop diameter perpendicular to the current, shown in Figure 5. And this is incorrect, because it is known that the inertial hydrodynamic loads on such a hoop will reduce to a 'Munk moment' about that hoop diameter, and this moment will be non-zero because the added mass of the hoop as a whole is different in the 'broadside' and 'edgewise' directions (see, for example, Reference 16, equation 338). The problem is evidently that the Lighthill analysis assumes that the angle of the riser to the flow is small, and in this case it is not. It is therefore natural to ask if Lighthill's original analysis 6 can be extended to cover this case. At a superficial level Lighthill's argument is a simple momentum one (which, incidentally, resembles the derivation of the lateral forces resulting from internal flow in the riser--here there is no analogous complexity
about the angle of the riser to the flow, and the force in our fixed hoop example will be a simple centrifugal one): at any point the riser is moving laterally relative to the water at a velocity
so that the lateral momentum of the water per unit length must be (given that the added mass per unit length is pA)
and thus the rate of change of this momentum, which must be equal and opposite to the lateral force, is
pA ~ + U ~ x
y
It is tempting to extend this thinking to the case in Figure 5, where the relative velocity of the water changes around the hoop in a more complicated way. This, however, would be to miss the essential subtlety of LighthiU's analysis. He stresses that momentum arguments of this type (which actually go back to Munk's derivation of the 'negative centrifugal force' on a turning airship, see Reference 17, Figure 3) are suggestive only,
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Hydrodynamic loading on towed arrays and flexible risers." R.C.T. Rainey
Current y
Figure 5
Riser formed into a rigid circular hoop and fixed obliquely in a steady current (i.e., current not aligned with axis of hoop)
and goes on to prove his result by direct integration of surface pressures. The reason for this caution is the classical difficulty that when a body moves in an infinite expanse of water the momentum of the water associated with that movement is not defined, because the water velocity does not decay sufficiently rapidly with distance from the body to make the momentum volume integral convergC. This famous mathematical difficulty is based on a very real physical phenomenon - - that if the expanse of water is not infinite, but bounded by distant rigid walls, then when the body accelerates the walls as a whole will feel an acceleration reaction which does not tend to zero as we consider ever more-distant wails. (We can see that this must be true by considering the case of the distant walls, and thus the water as a whole, oscillating rigidly back and forth. By moving cyclically in a suitable relative phase, the immersed body would be able to extract useful work from the inertial forces from the moving water - - the walls must therefore be able to 'feel that this is going on', to enable them to put in an equivalent amount of energy to that taken out by the body). Thus when we try to circumvent the problem of momentum definition by imagining the water to have distant boundaries (finitely distant boundaries will of course always make the momentum volume integral well defined), our momentum arguments are greatly complicated by the necessity for considering the reaction forces felt on these distant boundaries. In view of this fundamental problem it is perhaps surprising that a naive momentum argument nevertheless gives the right answer in the case of Figure 2. Lest this be thought typical, a case where it gives the wrong answer is that of a neutrally-buoyant straight cylinder moving broadside out into the air from an expanse of still water. The popular momentum argument 2 would tell us that the cylinder velocity must double, since the effective mass of the cylinder has halved (its added mass in this case equalling its own mass) - - this, however, is to create energy from nowhere 5 because twice the velocity and half the mass gives twice the kinetic energy. On the other hand, the approach of direct integration of surface pressures is formidably arduous algebraically - - Lighthill 6 required 20 long equations as against three short ones for the momentum argument (see also Reference 5, where it is shown that this is inevitably so, because the leading pressure terms cancel out by 'D' Alembert's paradox'). And the more arduous the
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algebra, the lower the probability that the final answer is correct. Indeed, authors striving to obtain correct answers in this field (rather than merely impress the naive reader!) invariably lay great stress on cross-checks with other, shorter, arguments is Be all these difficulties as they may, the logic of this paper is that Lighthill's refinements to the calculation of inertial hydrodynamic loads, already widely applied in the study of towed arrays, appear to matter most in the very case (i.e., the highly-curved one) where that method is inapplicable. If this impasse is to be broken, Lighthill's result has got to be generalized, and there appear in view of the above to be only three realistic options for doing it: (1) Lighthill's combination of a heuristic momentum argument backed by a rigorous, but long, direct integration of surface pressures (2) a rigorous momentum argument. This might be constructed, for example, by the device of two parallel confining walls, described elsewhere by Lighthill 3 (3) a rigorous energy argument. This method was pioneered by Taylor 18 and is the basis for the general rigid-body result of Rainey 5 Conclusions
Although the hydrodynamic drag on towed arrays and flexible risers has inevitably to be calculated empirically from experimental data, there is no reason why the hydrodynamic inertial forces should not be calculated analytically from first principles. Indeed, such an analytical result (significantly more sophisticated than the Morison-type empirical formulation) has been applied for many years to towed arrays. Unfortunately, it appears that the case when this extra sophistication makes a practical difference is that of the tightly-curved towed array or flexible riser (important because it is typically tight curvature which causes mechanical damage), and in this case the analytical result is inapplicable. Deriving a more general, universally applicable, version of the analytical result appears feasible. An encouraging recent development is the derivation of a related general-purpose analytical description of the hydrodynamic inertial forces on offshore structures. Acknowledgements
The author acknowledges the improvements to this paper suggested by contributors to the discussion at the Seminar on 9 January. Specifically, Dr. R. M. Carson drew the author's attention to the works cited above by Hawthorne, Paidoussis, Ortloff and lves, and Lee; and Dr. E. Kodaissi drew the author's attention to the effects of internal flow mentioned above. References
1 Batchelor, G. K. An Introduction to Fluid Dynamics, 1st edn, Cambridge University Press, 1967 2 Sarpkaya, T. and Isaacson, M.de St.Q. Mechanics o f Wave Forces on Offshore Structures, Van Nostrand Reinhold, New York, 1981 3 Lighthill, Sir James 'Waves and hydrodynamic loading' Proc. 2nd Int. Conf. on the Behaviour o f Offshore Structures, 1, BHRA Fluid Engineering, Cranfield, 1979, 1~10
Hydrodynamic loading on towed arrays and flexible risers: R.C.T. Rainey 4 5 6 7 8 9 10 11 12
Lighthill, Sir James 'Fundamentals concerning wave loading on offshore structures', J. Fluid Mech. 1986, 173, 667~81 Rainey, R. C. T. 'A new equation for calculating wave loads on offshore structures', J. Fluid Mech. 1989, 204, 295-324 Lighthill, Sir James 'Note on the swimming of slender fish', J. Fluid Mech. 1960, 9, 305-317 Hawthorne, W. R. 'The early development of the Dracone flexible barge', Proc. Inst. Mech. Engrs 1961, 175, 65-83 Paidoussis, M. P. 'Dynamics of flexible slender cylinders in axial flow', J. Fluid Mech. 1966, 26, 717-751 Paidoussis, M. P. 'Stability of towed, totally submerged flexible cylinders', J. Fluid Mech. 1968, 34, 273-297 Ortloff, C. R. and lves, J. 'On the dynamic motion of a thin flexible cylinder in a viscous stream', J. Fluid Mech. 1969, 38, 713-720 Lee, T.S.'StabilityoftheOrtloff-lvesequation',J. FluidMech. 1981, 110, 293-295 Dowling, A. P. 'The dynamics of towed flexible cylinders, parts 1 &
13 14
15 16 17 18
2', J. Fluid Mech. 1988, 187, 507 571 Kreysig, E. Advanced Engineering Mathematics, 3rd Edn, Wiley, New York, 1972 Andrew, R. N., Bull, P. W. and Smith, S. L. 'Hydrodynamic modelling of towed array sonars', Proc. Int. Syrup. on AntiSubmarine Warfare, 1, Royal Institution of Naval Architects, London, 1987 Smith, S. L. and Lyons, R. A. User Manual for the Computer Program AQWAFLEX, WS Atkins Engineering Sciences Ltd, Epsom, Surrey, 1986 Lighthill, Sir James An Informal Introduction to Theoretical Fluid Mechanic's, Clarendon Press, Oxford, 1986 Munk, M. M. Aerodynamics of Airships, div.Q in Aerodynamic Theory, VI, (Ed. W. F. Durand) Julius Springer, Berlin, 1936 Taylor, Sir Geoffrey 'The forces on a body placed in a curved or converging stream of fluid', Proc. Roy Soc. Lond. 1928, AI20, 260-283
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