.I. Fluid Me&. (1988), O O ~ . 187, pp. 507532
`Printed in Great Britain
`
`507
`
`The dynamics of towed flexible cylinders
`Part 1. Neutrally buoyant elements
`
`By A. P. D O W L I N G
`Department of Engineering, University of Cambridge, Trumpington Street,
`Cambridge C H 2 lPZ, UK
`
`(Received 20 March 1987)
`
`The transverse vibrations of a thin, flexible cylinder under nominally constant
`towing conditions are investigated. The cylinder is neutrally buoyant, of radius uA
`with a free end and very small bending stiffness. As the cylinder is towed with
`velocity U , the tangential drag causes the tension in the cylinder to increase from zero
`a t its free end to a maximum a t the towing point. Transverse vibrations of the
`cylinder are opposed by a normal viscous drag force. Both the normal and tangential
`viscous forces can be described conveniently in terms of drag coefficients C, and
`C,. The ratio C,/C,
`has a crucial effcct on the motion of the cylinder. The form of
`thc transverse displacement is found to be greatly influenced by the existence of a
`critical point a t which the effect of tension in the cylinder is cancelled by a fluid
`loading term. Matched asymptotic expansions are used to extend the solution across
`this critical point to apply the downstream boundary condition. Displacements well
`upstream of the critical point have a simple form, while nearer to the critical point
`the solution depends on whether the normal drag coefficient C, is greater or less than
`one-half C,.
`The typical acoustic streamer geometry considered is found to be stable to
`transverse displacements a t all towing speeds. Forced perturbations of frequency w
`are investigated. At low frequencies they propagate effectively along the cylinder
`with speed U . At higher frequencies they are attenuated.
`The effect of a rope drogue of length I,,
`radius aR, is investigated. Provided
`is very small, the drogue has the same effect as a small increase in the
`ol,a,/Ua,
`length of the cylinder. However a t higher frequencies and for small values of the ratio
`C,/C,
`attaching a drogue may be disadvantageous because it reduces the
`attenuation of high-frequency disturbances as they propagate down the cylinder.
`
`1. Introduction
`Towed instrumentation packages in the form of long flexible cylinders are used
`extensively to detect and analyse acoustic signals in the ocean. A typical geometry
`is illustrated in figure 1 . It consists of a heavier-than-water cable attached a t one end
`to a ship and a t the other to a neutrally buoyant slender cylinder containing a sonar
`array. This cylinder is sometimes referred to as an acoustic ‘ streamer ’ or ‘ array ’.
`There may possibly be a rope at the downstream end of the cylinder acting as a
`drogue. If such an arrangement is to give good resolution of the acoustic signals it
`detects, the instantaneous shape of the acoustic streamer must be known. When the
`ship maintains a constant velocity, the cylinder is straight and horizontal. However,
`changes in the ship’s path will make i t deform. In these two papers we analyse linear
`departures from the ideal case due, for example, to changes in ship speed or heading.
`
`17
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`

`508
`
`A . P. Dowling
`
`-
`
`Heavier-than-water cable
`
`Rope drogue 3 Neutrally' buoyant
`
`cylinder containing
`sonar array
`FIGURE 1 . Typical geometry for a ship towing an array
`
`The aim of this work is to provide a simple means by which the shape of the towed
`system can be predicted either from the ship's path or from an accelerometer a t the
`leading edge of the cylinder. Part 1 deals with the displacements of the neutrally
`buoyant elements, while Part 2 investigates the propagation of disturbances along
`the negatively buoyant cable. The results of Part 1 provide the downstream
`boundary conditions for the cable in Part 2.
`Computer programs have been developed to calculate the three-dimensional path
`of a towed system as a ship manoeuvres (see for example Ivers & Mudie 1973, 1975;
`Huston & Kamman 1981 ; Sanders 1982; Ablow & Schechter 1983). In general these
`packages require considerable computing resources and, if they are to run in real
`time, certain simplifying assumptions must be made. Since we are investigating
`small departures from constant velocity, we adopt a different approach, and linearize
`the transverse equations of motion. Pa'idoussis (1966, 1968) derived a linearized form
`of the transverse momentum equation for neutrally buoyant flexible cylinders with
`an axial flow. A term has been omitted from these early versions and Pa'idoussis
`(1973) gives the correct form of the equation of motion. Disturbances of frequency
`o satisfy a linear fourth-order differential equation. The coefficient of the fourth
`derivative depends on the bending stiffness of the cylinder. Perturbations of
`cylinders whose response depends on their bending stiffness have been extensively
`studied in the literature (see for example Hawthorne 1961 ; Pa'idoussis 1966, 1968,
`1973 ; Lee 1978 ; Prokhorovich, Prokhorovich & Smirnov 1982).
`However, acoustic streamers are very long in comparison with their radius a*, and,
`for motions with wavelengths comparable with the cylinder length, the restoring
`force due to bending stiffness is exceedingly small. It is therefore appropriate to
`recognize this and neglect the effect of the bending stiffness over most of the cylinder.
`This approximation has been made by Ortloff & Ives (1969), Kennedy (1980),
`Kennedy & Strahan (1981) and Lee & Kennedy (1985). The differential equation
`then reduces to second order, the coeficient of the highest derivative being
`T ( x ) -ponai U 2 , where T ( x ) is the tension in the cylinder and varies along its length.
`U is the mean flow velocity and pa the density of the surrounding fluid; pa nai U 2
`arises due to the effect of fluid loading.
`In $ 2 4 we consider a thin, neutrally buoyant, flexible cylinder with its
`downstream end unrestrained. T ( x ) then vanishes a t this free end, and increases
`along the cylinder due to tangential drag. It is well known that the transverse
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`

`Dynamics of towed Jlexible cylinders. Part 1
`
`509
`
`displacements of a tensioned string in vacuo satisfy a hyperbolic differential
`equation. When fluid loading is included, the equation remains hyperbolic over most
`of its length, but is elliptic near the free end, having a regular singular point a t x,,
`a critical position a t which T(x,) = po 7cai U 2 . Ortloff & Ives and Kennedy & Strahan
`base their work on Pa’idoussis’ early erroneous equation of motion, and find one of
`the solutions of their linear, second-order equation to be unbounded a t x,. They
`therefore reject this solution and the downstream boundary condition and describe
`the response of the cylinder in terms of the other (finite) solution. However, when the
`correct form of Pa’idoussis’ equation is used, and for reasonable values of the drag
`coefficients, both solutions are finite at the critical position, although one solution
`has a branch point there. Hence, before the downstream boundary condition can be
`applied, further investigation is needed to see how this solution behaves as x crosses
`x,. In the region of x, the response is controlled by the bending stiffness of the
`cylinder. We therefore use the fourth-order equation in the region of x,, and the
`method of matched asymptotic expansions to join these ‘inner’ bending solutions to
`the ‘outer’ tension-dominated response. In this way, the general solution for the
`vibration of a fluid-loaded cylinder, in the limit of small bending stiffness, can be
`found. Application of the free-end boundary condition then leads to an analytical
`expression for transverse displacements of the towed cylinder at frequency w . The
`displacements are found to have a simple form well upstream of x,. Nearer to x, the
`expression is more complicated and depends on whether the normal drag coefficient
`C, is greater or less than half the tangential drag coefficient C,.
`In 93 we investigate the stability of a neutrally buoyant towed cylinder by seeing
`whether there are any free modes that grow in time. A practical streamer geometry
`is found to be stable a t all towing speeds.
`Since the towed cylinder is stable, it is appropriate to determine its response to
`forcing at its upstream end. A t low frequencies for which wl,/U
`is small the
`disturbances propagate along the cylinder, virtually unchanged in amplitude and
`with a phase speed U , while for higher frequencies the disturbances decay in
`amplitude along the streamer. The first form of motion is often described as ‘worm-
`in-a-hole’ because all points of the cylinder take the same track. This motion is
`compatible with observations of low-frequency oscillations of towed arrays.
`So far we have assumed the end of the cylinder to be free. However in many
`practical situations it is attached to a rope drogue. In 95 we investigate the effect of
`a rope drogue. We use the work in the earlier sections to describe the transverse
`motions of the rope which has a free end, and apply continuity of displacement and
`force a t the junction of the drogue and cylinder. Ifwl,a,/Ua,
`is very small, the main
`effect of the drogue is found to be the same as an increase in length of the cylinder
`by an amount lRaR/aA, where a, and I , are the radius and length of the rope drogue
`respectively. At higher frequencies a drogue can have an adverse effect if the ratio
`C,/C,
`is small. Then attaching a drogue reduces the attenuation of high-frequency
`transverse disturbances as they propagate down the cylinder.
`
`2. The transverse motion of a neutrally buoyant flexible cylinder
`Consider an acoustic streamer consisting of a long flexible cylinder of length I,,
`radius a,, towed in the negative x-direction a t a constant speed U . If the cylinder is
`neutrally buoyant its mean position is horizontal. We will investigate linear
`departures from this arrangement and choose a frame of reference in which the
`distant fluid has a velocity (U,O,O), with the origin a t the mean position of the
`
`17-2
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`510
`
`A . P. Dowling
`
`FIGURE 2. Linear perturbations of the towed cylinder, viewed in a frame of reference in which
`the distant fluid has velocity (a, 0 , O ) .
`
`FIGURE 3. Forces acting on a small length, Sx, of the neutrally buoyant cylinder.
`
`upstream end of the cylinder as shown in figure 2. It follows from the neutral
`buoyancy of the cylinder, and the linearity of the disturbances, that perturbations
`in the (y = 0)- and ( z = 0)-planes satisfy identical uncoupled equations. It is
`thzrefore sufficient just to investigate the motion in one plane, (2 = 0) say.
`The equation of motion of the cylinder may be derived by considering the balance
`of forces on a small length as shown in figure 3. Let T(x) be the variable tension in
`the cylinder, FN and FT the viscous forces acting on the cylinder per unit length in
`the local normal and tangential directions respectively, and FA the inviscid force due
`to the acceleration of the virtual mass of the cylinder. Resolving in the x-direction
`gives, to zeroth order in the perturbations,
`aT
`FT+- = 0.
`ax
`The transverse momentum balance gives, to first order in the disturbances,
`
`a4y
`ay
`-FA-FN+FT--B--,
`ax
`ax4
`where m is the mass of the cylinder per unit length and B is its bending stiffness. This
`is Pa'idoussis' equation. The term FT aylax was omitted in Pa'idoussis' early work (see
`for example Pai'doussis 1966, 1968). This error was later corrected (Pai'doussis 1973
`and Pa'idoussis & Yu 1976) but unfortunately the earlier erroneous form has been
`adopted in much of the towed-array literature (Ortloff & Ives 1969; Kennedy 1980;
`Kennedy & Strahan 1981). As Pai'doussis (1973) points out, omitting the term
`FT aylax is equivalent to taking the tangential viscous force to act in the x-direction
`rather than in the instantaneous tangential direction.
`FA is the force required to accelerate the neighbouring fluid as the cylinder
`deforms. Provided the flow does not separate, the expression derived by Lighthill
`
`where po is the fluid density.
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`

`

`Dynamics of towed Jlexible cylinders. Part 1
`
`51 1
`
`The viscous forces acting on a long, thin flexible cylinder are discussed by Taylor
`(1952). He proposes the form
`FN = $po V 2 {2aA C, sin2 i + 2naA CN sin i},
`F, = po V 2 na, C, cos i,
`where V is the magnitude of the relative velocity between the cylinder and the
`distant flow, and i is the angle between this relative velocity and the local tangent.
`For linear perturbations of a neutrally buoyant element, V is equal to U and i is
`small,
`sini x i x l3+%, cosi x I.
`ax
`ri at
`These expressions therefore reduce to
`
`(2.46)
`
`( 2 . 4 ~ )
`
`FT = po nu* u2 c,.
`With this form for the tangential drag the longitudinal momentum equation (2.1)
`may be integrated immediately to give
`
`(2.56)
`
`( 2 . 5 ~ )
`
`T ( x ) = T(1,) +ponaA U 2 CT(ZA-x),
`(2.6)
`T ( l A ) is the tension a t the downstream end of the array, and vanishes if the end is
`free. Then T(O), the tension at the upstream end, is directly proportional to the drag
`coefficient C,. Hence G, may be inferred from measurements of T ( 0 ) . Data from
`large-scale experiments suggest C, = 0.0025 (Andrew private communication 1984).
`Ni & Hansen (1978) obtained similar values of C, for a range of Reynolds numbers
`in their rig experiments. There is less evidence about the appropriate value of
`G,. Taylor discusses in some detail how the value of C, would vary in the range
`0 < C, < C, depending on the type of roughness on the cylinder. We will therefore
`investigate the effect of varying C, within this range.
`When the expressions for FA and FN in (2.3) and (2.5) are substituted into the
`transverse momentum equation (2.2) they lead to
`porn- a2y = ( T ( x ) - p o n a ~ U 2 ) - - p o n a ~ -+2U-
`a2y
`at2
`a x 2
`
`axat
`a2y
`
`(Z
`
`The coefficient of the second derivative of y vanishes a t a position x,, where T(x,) is
`equal to the fluid-loading term po nu; U 2 . Using (2.6) to rewrite T(x) shows that
`
`where
`
`T(x)-p0naiU2 = p o ~ ~ A U 2 C T ( ~ c - ~ ) ,
`T ( z A ) -5
`PO naA u2CT
`
`x, = 1,+
`
`cT ‘
`
`x, lies on the cylinder if
`
`1, 2 xc 2 0,
`
`i.e. if
`
`(2.8)
`
`(2.10)
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`

`

`512
`
`A . P. Dowling
`
`Since we are considering linear disturbances we may investigate each Fourier
`component separately. For modes with time dependence eiwt, equation (2.7) reduces
`to
`
`This may be cast into non-dimensional form by scaling lengths on L (typically
`of order I*). When a new variable X = x / L is introduced, and with y(x,t) =
`Re (Y(X) eiWt ), the transverse momentum equation becomes
`
`d4 Y
`dY
`d2Y
`d x
`e 3 ~ - ( X c - X ) ~ + b - + i 1 2 b Y
`
`= 0,
`
`(2.12)
`
`e = (B/L3po ~a~ U2C,)%, is a small parameter because bending forces in the cylinder
`are very much less than the tension forces.
`
`(2.13)
`
`Q is the non-dimensional frequency, wL/U, and X , is the non-dimensional critical
`position, x,/L.
`Away from regions with intense gradients the contribution from the fourth-order
`derivative in (2.12) is negligible because e is small, and the equation reduces to
`
`= 0,
`
`(2.14)
`
`d2Y d Y
`(X,-X)--b--iiabY
`d x 2 dx
`a second-order ordinary differential equation with a regular singular point at
`X = X,. This only differs from the equation investigated by Ortloff & Ives (1969)
`and Kennedy & Strahan (1981) (derived from Pa'idoussis' erroneous version) in that in
`their equation the coefficient of the first derivative is, in this notation, - 1 - b rather
`than -b. We will see later that this apparently small difference has considerable
`consequences.
`The occurrence of a singularity a t X , is not an artefact of the linearization. Ablow
`& Schechter (1983) investigate finite-difference solutions of general cylinder motion.
`They find that the matrix to be inverted is singular a t the end of the cylinder where
`the tension vanishes. Since they have omitted a virtual mass term, which is the
`generalization of our linearized expression pa nai U 2 a2y/i3x2 to arbitrary motion,
`their singularity is entirely equivalent to our singularity at X,. Ablow & Scheehter
`get around this difficulty in an ad hoc way by applying the downstream boundary
`condition at' a point P, a short distance from the free end, and assuming the cylinder
`to be straight between P and the free end. However for the linear disturbances
`considered here the effect of the singularity on the form of the solution can be
`investigated analytically.
`Two solutions of (2.14) may be obtained by the standard method (Ince 1956) of
`trying a series solution of the form
`Y(X) = (X,-X)" c an(Xc-X)n.
`
`0
`
`n=O
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`

`

`Dynamics of towed jlexible cylinders. Part 1
`The indicia1 equation shows CT to be equal either to 0 or t o 1 - b. The general solution
`of (2.14) is then found to be
`O0 (iQb(X, -X))%
`Y(X) = P x
`n ! ( n + b - l ) !
`P and Q are arbitrary constants. Y ( X ) is a linear combination of two independent
`series solutions. The first solution is analytic while the second has a branch point a t
`X,. Both series converge for all X. Equation (2.13) shows that
`
`+ Q ( X , -X)’-b z
`
`O0 (iQb(X, -X))%
`n-O n ! ( n + l - b ) ! ’
`
`(2.15)
`
`513
`
`and so for 0 < C, < C, both solutions are finite a t the critical position X,.
`In a typical problem, boundary conditions for the transverse motion are given at
`the two ends of the cylinder X = 0 and X = lA/L. If the inequalities expressed in
`(2.10) are satisfied these two points are on either side of the critical position X,. Let
`us now for definiteness consider a cylinder with a free downstream end at which
`T(1,) vanishes. Then the first inequality in (2.10) is automatically satisfied. Practical
`acoustic streamers are sufficiently long to ensure that C, 1, > aA, hence also meeting
`the second inequality. The critical point X, therefore lies somewhere between the two
`ends where the boundary conditions are specified. Before these boundary conditions
`can be applied, we need to determine how the solution expressed in (2.15) varies as
`X passes through X,. There is no difficulty with the first series solution because it is
`continuous a t X,, but the second has a branch point there and the relevant cut must
`be found. Since the second series solution has large gradients in the vicinity of X,,
`bending forces become important and the full fourth-order equation (2.12) is needed
`to determine the form of the deflections. The method of matched asymptotic
`expansions may be used to match these ‘inner’ bending solutions to the ‘outer’
`tension-dominated disturbances described in (2.15).
`It is worth noting that Ortloff & Ives (1969) and Kennedy & Strahan (1981) did
`not have these difficulties. The second independent solution to their equation is
`infinite a t the critical point. They therefore reject it and the downstream boundary
`condition, and use the other bounded solution to describe motion in X < X,. But we
`have seen that when the correct form of Pai’doussis’ equation is used, both
`independent solutions are finite at X, for reasonable values of the normal drag
`coefficient. The behaviour of the solutions as X varies across X , must therefore be
`investigated in detail so that the downstream boundary condition may be applied.
`The solutions of (2.12) are to be determined for small values of E and 0 < Re (b).
`Let us shift the origin by introducing a new variable r = X,-X. Then with
`Y(X) = $(r,e), $ satisfies
`
`d$
`d2$
`d4$
`E3--r--b-++Qb#
`dr2
`dr4
`dr
`For small e, it is appropriate to seek a series solution in powers of the small parameter
`e3 which appears in (2.16) ;
`$(r, e) = $,(r) + e3&(r) + . . . .
`
`.
`
`= 0.
`
`(2.16)
`
`The equation for #o(r) is
`
`r T + b A - i Q b # o d2$, d$
`
`
`dr,
`dr
`
`= 0.
`
`(2.17)
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`

`

`C
`
`(iS2br)n
`n ! ( n + 1 - 6 ) !
`
`(2.18)
`
`The general solution of this second-order equation ha,s been given in (2.15) and is
`(iObr)"
`#o(r) = P c n=O n ! ( n + b - l ) !
`Gradients of #o(r) become large near r = 0. Away from this region $o(r) dcscribes
`deflections of the towed cylinder in which the response is determined by its mass,
`tension and fluid loading.
`In the region of r = 0, the surface response is controlled by bending stiffness, and
`# ( r , E ) varies rapidly. For this region then let us strctch the spatial coordinate and
`introduce R = r/t: with #(r, e ) = @(R, E ) . An inner expansion for the displacements
`can be obtained by expanding @ in ascending powers of E
`@(R, E ) = @,(R) + e@,(R) + . . ,
`
`514
`
`A . P. Dowling
`
`a;
`
`O0 satisfies
`
`(2.19)
`
`The four independent solutions of this linear fourth-order equation are determined
`in integral form in the Appendix where, in particular, their asymptotic forms for
`large IRI are evaluated. It is found that for large positive R
`
`A , B, C and D are arbitrary constants multiplying the four independent solutions
`and are to be determined by matching to the outer solution. When this inner solution
`is rewritten in terms of the outer variable r it becomes
`
`Hence for r p E we have
`(iQbr)"
`(iQ6r)n
`+ (Q, + &3 ) y l - b c n=o n ! (n+ 1 - 6) !
`#o(r) = p c 12=o n ! (n+b- i ) !
`with Q2 = -Betixb (1 -b) ! (6-2) ! cbP1 and Q3 = -Ce-iixb (1 -6) ! (6-2) ! &'.
`VC'hen r / e is large and negative with 1 9 Irl 9 E , the asymptotic form for O,,(R)
`evaluated in the Appendix (see equation (A 26)) shows that
`
`(2.23)
`
`O0
`
`- r)ib-?! 8-Eb
`ac.4 2
`+
`( 1 - b ) ! (6-2)!
`
`( Q2 e-is + Q3 eis ),
`
`(2.24)
`
`r 9 - 1 , with B = n ( & ~ ) + $ ( - r / e ) ~ .
`for - E +
`The first term on the right-hand side of (2.24) is just the series solution, describing
`the balance between tension and inertial forces, that is valid everywhere. The second
`and third terms describe motions downstream of the critical position with large
`gradients whose form is influenced by the bending stiffness of the cylinder.
`
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`

`

`Dynamics of
`
`towed flexible cylinders. Part 1
`
`525
`
`Returning to the spatial variable X
`O0 (iQb(X, -X))n
`Y(X) = P c
`%=,, n ! ( n + b - l ) !
`
`and
`
`+ nu
`
`with
`
`(2.253)
`
`(X -x,
`ei-3
`(Q2e-is+Q3eio) for 1 $- X-X, + e,
`(1 - b ) ! (b - 2) !
`4 ) + $( (X -x,)/E)t
`0 = .($-a
`To summarize then, perturbations of the cylinder upstream of the critical point are
`described by two linearly independent series solutions. The displacement typically
`varies over a non-dimensional lengthscale of order unity and the bending stiffness of
`the cylinder, characterized by the small non-dimensional parameter E , is unimportant
`in this range. The first series solution is always slowly varying and so is valid for all
`X . The second has a branch cut at the critical position, X,, and large gradients near
`it. This second solution is extended across X = X, in the Appendix by using the full
`equation of cylinder motion including bending stiffness and is displayed in (2.25 6).
`Not only is the appropriate branch cut determined by equation ( 2 . 2 5 b ) , but it also
`shows the solution downstream of X,
`to contain wavelike disturbances with
`wavelengths of the order of e. Bending stiffness therefore has an important influence
`
`on the cylinder motion throughout this region. If either Q2 or Q3 is non-zero, the
`displacements of the cylinder vary over a long lengthscale well upstream of X , , but
`have very small wavelengths throughout the tail region downstream of X,. The
`coefficients P, Q2 and Q3 may be determined from the boundary conditions a t the two
`ends of the array, X = 0 and X = LA = 1JL.
`The two boundary conditions appropriate for a free end of a flexible cylinder have
`been derived by Hawthorne (1961). One condition is that there be no bending
`moment ;
`d2 Y
`- = 0
`d x 2
`The second condition arises from the transverse momentum balance for the tapered
`portion of the cylinder at its end. If this portion is very short its momentum is
`negligible, and the forces acting on it must cancel. The only significant forces on a
`short end a t which the tension vanishes are due to bending resistance and the virtual
`mass of the fluid near the end. This has been evaluated by Hawthorne (1961) and
`PaTdoussis (1966) and takes the form
`
`a t X = L , .
`
`(2.26)
`
`where f is a non-dimensional parameter less than unity introduced to account for
`departures of the flow from two-dimensionality. In view of the uncertainty in the
`value off, it is reassuring that our results will be found not to depend on the details
`of this boundary condition. For motions with time dependence eiwt (2.27) becomes
`
`(2.27)
`
`(2.28)
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`

`

`516
`
`A . P. Dowling
`
`We now substitute for Y(X) from equation (2.25b) into the two boundary conditions
`(2.26) and (2.28). I n the limit of small bending stiffness, this gives
`
`(eiOE-inb - e-illE+ixb
`
`2i( 1
`z;E&+f&fb
`(b- 2) ! ( 1 -b) !
`
`)-
`
`flQ2E2 eieE
`( b + l ) ! '
`
`(2.29)
`
`(2.30)
`Q 3 = - Q 2 e-i28~ 1
`and
`E is the distance between the critical point and the free end. It is apparent from (2.9)
`that
`E = L,-X, = a,/LC,.
`(2.31)
`In deriving (2.29) the product SZE has been assumed to be small in comparison with
`unity. Acoustic streamers are sufficiently long in comparison with their radius for
`this to be a good approximation a t the low frequencies of interest here. 8, is defined
`8, = ~ ( i b - 2) + g ( E / ~ ) f .
`(2.32)
`by
`The solution of (2.29) and (2.30) has quite a different form according to whether
`Re@) is greater or less than one-half. For Re(b) greater than a half these two
`boundary conditions reduce to the very simple statement that
`
`Qz w Q3 - p&!.
`
`(2.33)
`I n the limit of small bending stiffness, E tends to zero and Q2 and Q3 are negligible in
`comparison with P. Then the motion upstream of the critical position is given by
`for E < X, - X, Re (b) > $.
`
`Y(X) = P z: (iQb(Xc
`
`%-,,
`
`n ! ( n + b - l ) !
`
`(2.34)
`
`(2.35)
`
`, (2.36)
`
`When Re@) is less than one-half, (2.29) and (2.30) become
`
`and the cylinder displacement is described by
`
`sin 8,
`b( -b) !
`(iSZb(X, -X))n
`n-,, n ! (n + b - 1) ! + sin (8, - r b ) (b + 1 ) !
`Q2El+b (X, -X)l-b
`(iQb(X, -X))']
`x c
`n-O n ! ( n + l - b ) !
`
`for G 4 X,-X and Re(b) < t.
`We have determined a general expression for the propagation of deflections of
`frequency w along a towed cylinder with a free end in the limit of small bending
`st'iffness. The method of solution involves extending the upstream expression for
`transverse displacements across a critical point, a t which the restoring force due to
`tension is cancelled by a fluid-loading effect, so that the downstream boundary
`condition may be applied. The general solution upstream of the critical point is given
`in (2.34) and (2.36), and is seen to have a different form according to whether Re(b)
`is greater or less than one-half. These can be combined into a statement that
`sin 8,
`+H(0.5-Re(b))
`sin (8, - zb)
`SZ2E1+b (X, - X ) l - b C
`
`n-,, n ! ( n + b - l ) !
`
`X-
`
`b(-b)!
`(b+ l ) !
`
`(iSZb(X, -"))I
`
`n=,, n ! ( n + l - b ) !
`
`, (2.37)
`
`PGS v. WESTERNGECO (IPR2014-00688)
`WESTERNGECO Exhibit 2045, pg. 10
`
`

`

`Dynamics of towed JEexible cylinders. Part 1
`
`517
`
`for tz < X,-X
`and SZE small. An inspection of this expression shows that, when
`lengths are non-dimensionalized on the cylinder length l,,
`the displacement at
`position X = x/1, depends on the non-dimensional frequency 52 = ol,/U,
`the ratio
`and the value of the parameter aA/lACT. I n a particular example the constant
`C,/C,
`P will be determined by the upstream boundary.
`Well upstream of the critical position this form for Y ( X ) simplifies to
`a, (iSZb(X, - X ) ) n
`Y(X) = P c
`n=O n ! ( n + b - I ) ! *
`
`(2.38)
`
`For Re (b) greater than a half, this solution is valid throughout the region X , - X % E .
`But when Re (6) is less than one-half, it only holds in X,-X % E. Then, once X,-X
`is comparable in magnitude to E , the distance from the critical point to the free end,
`the second series in (2.37) is important and leads to large gradients of the
`displacement Y ( X ) . For real frequencies, Re (b) is equal to C,/C,. Hence, whenever
`the normal drag coefficient is less than half the tangential drag coefficient, the
`displacement of the cylinder has large gradients upstream of the critical position. It
`is interesting to note that (2.38) describes the motion that would be obtained by
`applying a boundary condition i 0 Y +dY/dX = 0 a t the critical point, i.e. a
`condition that the cylinder has no normal velocity there. The displacement
`downstream of the critical position X , has a more complicated form, and the full
`expression defined by (2.25b) and the boundary conditions (2.29) and (2.30) must be
`used. The slope of the cylinder is large, but there is little practical interest in the
`solution in this region.
`Ortloff & Ives (1969) expressed their solution to a similar equation in terms of
`Bessel functions of complex argument and order. Y ( X ) can be rewritten in a similar
`form. The series expansion for Jb-l shows that
`(iQb(X,-X))"
`r, npo n ! ( n + b - I ) !
`where a = (-4iSZb)i, the sign of the square root being chosen so that Re(a) is
`positive. Hence, (2.38) is equivalent to
`
`= ( ~ a ( X c - x ) ~ ) l - b J b - l (a(X, - X ) $ .
`
`(2.39)
`
`( a ( X , - X ) i ) ,
`Y ( X ) = P(;a(X,-X)i)l-bJ,-,
`(2.40)
`for 0.5 < Re (b) and X,-X 9 6 or for 0 < Re (b) < 0.5 and X, - X 9 E. This form for
`Y ( X ) will be found to be convenient when investigating high-frequency disturbances.
`
`3. The stability of a towed flexible cylinder
`The general solution for the deflections of a towed cylinder derived in $ 2 may be
`used to investigate its stability by seeing whether free modes for a fixed upstream end
`grow or decay in time.
`It follows from (2.38) that a condition of no transverse displacement a t X = 0 is
`equivalent to
`(i52bX,)n
`c ,=,n!(n+b-l)!
`The complex non-dimensional eigenfrequencies 52 may be determined from this
`equation. If one of the roots has negative imaginary 52, a free mode grows in time and
`the system is unstable. If, however, all the roots of (3.1) have positive imaginary 52
`
`= 0.
`
`PGS v. WESTERNGECO (IPR2014-00688)
`WESTERNGECO Exhibit 2045, pg. 11
`
`

`

`518
`
`A . P. Dowling
`
`the system is stable. It is interesting to note that (3.1) shows that the non-
`dimensional eigenfrequencies 52 are determined only by the values of the parameters
`and are independent of the free-stream velocity U . It therefore
`C,/C,
`and a,/l,C,
`implies that if the towed cylinder is unstable a t any flow speed, it is unstable for all
`speeds !
`We introduce a function f(52) defined by
`
`(iQbXJn
`f(52) = ( b - l ) ! 2
`nso n ! (n+b- l ) ! '
`
`OcI
`
`(3.2)
`
`f(i2) is analytic in the lower half Q-plane and its number of zeros, N , within a closed
`contour r in this region is given by Cauchy's theorem as
`
`(3.3)
`
`The derivative f ' ( Q ) may be evaluated by differentiating (3.2) term by term. The
`integral (3.3) was evaluated for C,/C, = 0.25 and 0.75, a,/l,C, = 0.033 and a
`contour r consisting of the real 52-axis from 52 = 30 to - 30 closed by a semicircle in
`the lower half 52-plane. In both cases N was found to be zero, showing that f(s2) has
`no zeros in Im(52) < 0 with 1521 E < 1. Hence we conclude that for this typical
`acoustic streamer geometry the towed cylinder is stable for all towing speeds.
`
`4. The forced vibration of a towed flexible cylinder
`Let us now determine how disturbances produced by vibration of the upstream
`end of a towed neutrally buoyant cylinder propagate along it. Without loss of
`generality we will take the amplitude of the transverse vibrations of the upstream
`end to be unity, and write
`
`y(0, t ) = coswt,
`Since y(x, t ) = Re( Y(X) eiot) the boundary condition for Y ( X ) is
`
`(4.1)
`
`Y ( 0 ) = 1.
`(4.2)
`The coefficient P in the solution (2.38) can be determined from the boundary
`condition (4.2) to give a form for the displacements of the cylinder:
`
`(iQbX,)"
`O0 (iQb(X,-X))"/
`Y(X) = c
`O0
`n=O n ! ( n + b - 1) ! m-O m ! (m+b-- l