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Simulation of tension controlled cable deployment
Simulation of tension controlled cable deployment JOHN
W. LEONARD
Department of Civil Engineering, Oregon State University, Corvallis, OR
S T E P H E N R. KARNOSKI Naval Civil Enghleering Laboratory, Port Htteneme, CA
The theory and a numerical algorithm were developed for the deployment o f cable from a ship traveling with constant speed and direction in a flowing fluid. Attention was focused on steadystate conditions, i.e., horizontal, steady horizontal flow varying in magnitude and direction with depth but uniform in plan, and on tension controlled payout o f the cable.
1. I N T R O D U C T I O N When cable is deployed from a shipboard canister or reel with a pre-set peel-point tension, the rate o f cable deployment is dependent on the balance o f external forces (gravitational, fluid dynamic, and frictional) over the length of the cable in the water column with the tension at the ship. When the payout rate is less than the ship speed, the cable will tend to be dragged along the b o t t o m . When it is desired to place more cable between two points on the bottom than the straight-line distance between those two points, the payout rate must be greater than the ship speed. This generates slack cable on the b o t t o m which is desirable because of smaller tensions in the installed cable, no dragging, less chance of snags and smaller suspensions over irregular bottoms. Simplified models o f slack deployment can be generated 35'37 based on balance o f fluid forces normal to the cable by the normal c o m p o n e n t of the wet weight of the cable. These models require the assumption of straight configurations in the water column and the assumption o f uniform fluid flow (no shear or rotation). More refined models which allow spatially variable fluid flows can be developed based on either numerical
integration along the cable or on solution o f matrix equations for discretized equations of motion. See Table 1 for a listing o f references on various methods of numerical analysis o f cables without payout. For deployment of extremely long cables with low levels o f tension the latter a p p r o a c h (e.g., finite element models) will require high precision operations on large sets o f equations. The former a p p r o a c h (e.g. direct integration) may require numerous integration steps but will not require storage of large arrays of variables and will easily a c c o m m o d a t e highly variable flow fields. The governing differential equations and boundary conditions for the description of the nonlinear boundary value problem of cable deployment in a 3-D setting are developed in the following secton using the works by Leonard lz, t7 and by Chiou 4 as bases for the derivation with payout added. A solution algorithm based on iterations for payout velocity and on direct integration of the cable equations along the curved cable length is then presented and illustrative examples are presented for deployment in both uniform and sheared nonplanar current profiles.
Table 1. References on Numerical Methods Lumped-Parameter
Finite Element
Imaginary Reactions
Direct Integration
Seide128 Thresher and Nath 3~ Liu ~s Nuckolls and Dominguezz3 Leonard and Nath js Sanders25 Ablow and SchechterI
Leonard and Reeker~6 Leonard II FellipaS Webster3~ Ma, Leonard and Chu 2~ Leonard and Nath~5 Lo and Leonard'9 Celmer, Stevens, and Coe 2
Skop and O'Hara 29 Dominguez and Smith7 Nath 2z Watson and Kuneman3z Rosenthaland Skop24
Gay ~~ Schram and Reyle27 Wang3 Choo and Casarellas Chang and Pilkeya DeZoysa6 Wingham and Kreshaven36 Leonard ~2-,4
Accepted September 1989. Discussion closes July 1990.
34
Applied Ocean Research, 1990, Vot. 12, No. 1
9 1990 Computational Mechanics Publications
Simulation o f tension controlled cable deployment: J. IV. Leonard and S. R. Karnoski 2. D E R I V A T I O N OF EQUATIONS Shown in Fig. 1 is a definition sketch o f the deployment from a ship of a light-weight cable in a sheared nonplanar fluid flow field. A Cartesian coordinate system is attached to the ship moving at constant speed over a horizontal ocean bottom. The origin of the coordinate system is taken at the location o f the point 'B' where the cable contacts the bottom. An arc length curvilinear coordinate system is used to describe the cable behavior at an arbitrary point along the cable. In the first part of this section are described the assumptions made to mathematically model the deployment process. Secondly, the geometrical constraints and material behavior are developed from some of those assumptions. Equilibrium equations are then generated for a differential length of cable subject to body forces and fluid drag. Finally, boundary conditions at the touchdown point 'B' and at the ship 'C' are developed.
where ~z is the unit vector in the xz direction and /'p is the unit tangent to the cable at ' P ' with positive direction in the direction of increasing arc length coordinates from the bottom point 'B'. 7. In order to maintain steady form the touchdown point 'B' moves with constant velocity and direction matching that of the ship, i.e. ~'B = Pc. 8. If the payout speed v < [ ~'c[, the cable is dragged along the bottom at p o i n t ' B ' and the tangent to the cable at the bottom is t8 = ~2, i.e. the cable has zero slope at the touchdown point. 9. If the payout speed v>>.lVcl, slack cable is deposited on the bottom at point 'B' and the tangent to the cable at the bottom is t8 = Ols~l + 028~Z
(2)
where 0iB = direction cosines of t'B such that 028 = Vclv
(3a)
018 = 11 - (Vc]v)21 1/2
(3b)
i.e. [Tp at 'B' is the vertical free fall velocity.
Kinematic assttmptions 1. The cable in steady state spans between point 'B' on the bottom and point 'C' at the ship traveling at steady speed Vc in a straight line in" the xz direction. 2. Bottom is horizontal. 3. At an arbitrary point ' P ' on the cable the particle (point on the cable) velocity vector is ~'v and its magnitude is invariant along the cable in order to maintain a steady form for the cable. 4. The water velocity vector at point 'P' is denoted by r',,. and is assumed to be uniform in the xz-x3 plane with no component in the x~ direction. 5. The water velocity vector may vary with xl, i.e. sheared current. 6. At point 'C' the cable deploys from the ship with speed v relative to the moving ship. Thus, the particle velocity at any point along the cable is ~'v = Vce2 -- We
(1)
Dynanffc equilibriunt assunlptions 10. Cable is subject to uniaxial stress with no flexure, shear or torsion. 11. Cable is either elastic or inextensible with stressstrain relation o = Eoe
(4)
where a = uniaxial stress, Eo = elastic modulus (or co if inextensible) e = elastic strain = 0 , 2 - 1)]2 (5) X = elongation ratio = dS]dSo (6) dS = stretched differential length dSo = unstretched differential length 12. Mass is conserved and the density of material does not change upon stretching. Thus the stretched wet weight o f a differential length can be related to the unstretched dry weight of a differential length by w dS = Wo dSo(Sg- l)]Sg = Wo d S ( S g - 1)]>,S~
V
!
;
,
\
I
9
'I
'/_
Deployed Cable
D~
!
,/ v
$ i
iXt
VF
I
I I
I
,
_ Vel0city Profile
Figure 1. Definition sketch o f cable deployment
(7)
where w = unit stretched wet weight Wo = unit unstretched dry weight Sg = specific gravity of cable Also, the mass m per unit stretched length is m = pA = poAo]X = WolkG
(8)
where p, O0 are cable densities and A, Ao crosssectional areas o f the stretched and unstretched lengths, respectively, and G = gravitational constant. 13. If deployment at the ship is passively controlled, the cable tension T at the particle detaching from the shipboard canister or reel is constant and is given by T = To + Kv z
(9)
where To = peel point tension for the canister or reel and K = c o n s t a n t o f proportionality for unspooling from the cable pack.
Applied Ocean Research, 1990, Vol. 12, No. 1
35
Simulation o f tension controlled cable deployment: J. IV. Leonard and S. R. Karnoski 14. Fluid drag forces on the slender cable are taken from the relative velocity form o f the Morison equation.2~'26 (Derivation in a subsequent section.) 15. The independence principle for normal and tangential drag is invoked, i.e. the tangent and normal drag forces are separately calculated from the tangential and normal components of the relative fluid velocity. 16. The drag coefficients Corand C~x in the Morison equation are Reynolds' number dependent and could vary with relative roughness of the cable. In this work, they are taken as constant values. 17. Added mass effects are included from the inertial term of the Morison equation 2~'26 with steady particle acceleration equal to the normal acceleration due to curvature changes in the cable.
Geometrical and material behavior The position vector to an arbitrary point ' P ' on the cable is /~ = X~&~
For an elastic (or inextensible) material
c = olEo = TIEoAo
where E 0 = m o d u l u s o f elasticity and A o = c r o s s sectional area of unstretched strength member in the cable. If the cable is inextensible, E is taken as infinite. Thus, the elongation ratio is X = ,,11+ 2TIEoAo
Buoyant force vector ,~b = - (w dS)~l By assumption 3 of Section 2.1.2
where Xi = coordinates o f point ' P ' and ~i = Cartesian base vectors. Repeated indices imply summation over the range 1,2,3. The unit tangent vector t'at ' P ' is
-fib = -- [W0 d S ( S g - l)/kSg] 5ii~i
-
dXi - dS
~i
(12)
then, the governing equations for positions Xi are
(14)
and in terms of components in the X~ direction T = T~i
(15)
Replacing i" in equation (14) by equation (12), we can obtain relationships between 0i and T~ as
O, = 751T
where hii = Kronecker delta = 1 if i = j , or else = 0 if i~j.
Fhdd drag force vector By the independence principle for drag on a slender inclined cylinder using the relative velocity form of the Morison equation 2~'z6 (Assumptions 14 and 15) Fa = 89pjD dS{ Co.v[ VRNI PR~' + rcCor[ VRTI VRTI
dXi = 0i (13) dS We will define the tension vector T b o t h in terms of the magnitude T in the direction of t" T = 7"[
(24) where 0 f = fluid density, D = d i a m e t e r of cable, CD.V= normal drag coefficient, C o r = t a n g e n t i a l drag coefficient, ~:RN normal component of relative velocity vector, and ~'nr= tangential component o f relative velocity vector. The t e r m s ] PRN[ and I VRr[ denote the magnitudes of ~'rRN and VRr, respectively. =
%
%.
(16)
and, therefore, the governing equations for positions X~ can be written as dX~ =--7'" dS T
(17)
which are nonlinear first-0rder equations for X~ in terms o f tension components Ti with T = [ TjTil 1/2
(18)
The strain c is related to the elongation ratio X by X = ,/1 + 2c
(19)
where
X = dSldSo 36
(23)
(11)
where dS is the stretched differential length. If we define direction cosines Oi o f / " by
[ = Oi~i
(22)
Equilibrittm At a general point on a cable a stretched differential length can be isolated as a free body as shown in Fig. 2. The external forces acting on that free body are the buoyant weight, and drag and added inertia forces due to relative motion through the fluid. These external forces are balanced by the variation in end point tensions over the differential length and the inertial force due to the cable acceleration.
(10)
[ = d/~ dS
(21)
(20)
Applied Ocean Research, 1990, Vol. 12, No. 1
Figure 2. Free body diagram
Shmdation o f tension controlled cable deployment: J. IV. Leonard attd S. R. Karnoski The relative velocity vector is
have
~ = P,,.- f'l, = V,,.i~i- Vc~2 + v[
(25)
where V,,.~are the Cartesian components of V,,.. Recalling equation (12), we find
= [ v,,~oj- Vc~:oA 7+ v7 = [V, f l y - Vc6jzOj+ v]OiYi = VRri6i
(26a) (26b)
where Vnr~= Cartesian components of PRr. The normal velocity vector iS~'R N = ~'tR -
~'tR T
VRri)ei + V[ = ( V.,y - VcSi2) (&j - OiOj)6i = VRxi6i
fit= +X-ing
mdSOp=Pb+Pd+P1+
I PRrl = I VRTiVRril I/Z
" (28b)
Therefore, the drag force is
-T
(34)
XG
dS[~ -~- + r~
+ XsgGw~vt-:MX[ff.--~+Ti
--~j (35)
(29a) dS
where
~'RN[ VRNi Fori = ~,p:V~rC~rlP~rl gl~r~ FDNi = 89
(31)
where hp,v denotes the normal component of cable acceleration and t~nN is the relative normal acceleration. The relative inertial force is derived for acceleration normal to the cable according to Assumption 8 as CI~IIV(IRN
MXV / ~ - ~ - +
T~"
~i
XSg
6il
__~
The term dt[dS is the normal fi to the cable, and therefore the particle acceleration is normal to the cable. Evaluating di'[dS by equations (12) and (16), we have ei
=
(29c)
(30)
-]-
I+
- ~,o:DI CDu] PRu[ VRNi + CorrC l Vgrl VRril
d(-v[) d~ v
= _ v 2 di" dS
XG T
(29b)
Added fluid inertia free vector The added fluid inertia force is that due to relative normal acceleration of the cable through the water using a relative acceleration form of the inertial contribution to the Morison equation. The acceleration of the particle 'P' under steady state conditions is given by
= +p.tVdS
T + ~--~ d
Combining terms and eliminating dS, we have
ff-a = ( Fo,vi~i + Fori6i) dS
f t = ply dS
(31)
VRN. The (28a)
ae,'~, = -- ( I R N ~-. -- V I~l~
ei
Therefore, by equations (8), (23), (29) and (33), we can write the components of equation (34) as
(27a) (27b)
[ PRUl = { VRxiVRNil l/z
dF',o d~'pdS ~ ; = dt - d--S d t -
M , v [ ' ~ ' ~ - + T/
Equations o f motion We set the sum of forces on the differential segment shown in Fig. 2 equal to the inertial force on the segment
: ( V w i -- Vc~i2 -
where VRxi=Cartesian _components of magnitudes of PRN and Vnr are
dS
vZ(l+
C~IN~
(36)
Equation (36) is a nonlinear first-order equation for Ti in terms of relative velocities VR,~7and Vnri (which involve Oi = 75[T) given by equations (26) and (27). Note that if decoupled equations for dl)]dS are sought, the final term on the right-hand-side of equation (36) will have to be approximated in that the formal derivative 1
dTj
gives coupling terms for dT~]dS. In the numerical solution procedures presented in section 3, the term d[dS ( l / T ) will be evaluated approximately by a backward difference.
Boundary conditions Equations (17) and (36) constitute six first-order differential equations for the six dependent variables of location X~ and tension components T~, i = 1, 2, 3. The independent variable is the arc length coordinate S. Thus, six boundary conditions at the two ends need to be specified. There are also two auxiliary variables - the payout velocity v and total length of cable Su suspended in the water column. By assumption 3, v is taken as invariant with S.
(32)
V = v o l u m e per unit stretched length of cable and CAIN = added mass coefficient for normal acceleration. Writing the fluid inertial force in terms of the mass nt of the cable and using equation (8) of Assumption 12 we
Bottom boundary conditions The origin of the Cartesian coordinate, .hi, and the arc length coordinate, S, are taken at the touchdown point 'B' of the cable on the ocean bottom. Thus, at Applied Ocean Research, 1990, VoL 12, No. 1
37
Simulation o f tension controlled cable deployment: J. IK Leonard and S. R. Karnoski S = 0 we have three boundary conditions. X~(0);
i = 1, 2, 3
(38)
Also, the touchdown point by Assumption 7 moves with constant velocity Vc. Therefore, by Assumption 9 if payout velocity v is passively controlled, the direction cosines at S = 0 are given by equations (3) and can be expressed in terms of tension components as Tl (0) = T[ 1 - (Vc/v) 2 ] 1/2
(39a)
7"2(0) = TVc[v
(39b)
7"3(0) = 0
(39c)
T*= T]woDe
(43d)
v* = v/Vc
(430
V,*i= V~,~/Vc
(43f)
Nondimensional scaling numbers are defined below:
fD = 89
= ratio o f drag forces to cable weight (44a)
em= woDe/EoAo = ratio o f cable weight to stiffness (44b)
(
r/= ~
1+--~]
Therefore, we have two boundary conditions
= ratio of inertial force to cable weight (44c)
Tt(O)/Tz(O) = [ (v/Vc) 2 - I I 1/2
(40a)
X = ,11 + 2cmT*
(44d)
7"3(0)
(40b)
K* = KV~/woDe
(440
T~ = To/woDe
(44f)
S,* = S,,/De
(44g)
=0
The magnitude Tis undetermined. It should be a small quantity dependent on static frictional resistance of a short segment of cable lying on the bottom near the touchdown point. Zajac 37 stated that T(0) is identically zero when slack cable is being deposited on the bottom. He used that assumption to justify a kink in the cable at the bottom and hence a straight-line solution under uniform current conditions.
Then the governing equations become:
Location (i = 1, 2, 3)
dX*
T*
(45a)
dS* - T*
Canister boundary conditions By assumptions 1 and 2 the location of the ship is at a constant depth De above the ocean bottom. Therefore we have a boundary condition XI(S,,) = De (at unspecified length S,).
(41)
Equilibriunl (i = 1, 2, 3)
dT:
~"
d-~
(45b)
(42)
If, instead, active control of payout is assumed, equation (42) is replaced by specifying v at the ship. In these sections we have specified seven boundary conditions, equations (38), (40), (41) and (42) for the six field variables X~, Ti, i = 1, 2, 3 and the two auxiliary variables v, S,. An eighth condition is obtained if we assume the tension magnitude T(0) at S = 0 to be a small parameter. If, instead, active control o f payout is assumed, equation (42) is replaced by specifying v at the ship and Assumption 8 is used to replace equation (40b) by the condition T~ (0)= 0.
Summary o f nondintensionalized equations It is necessary to nondimensionalize the variable in order to scale the variables and equations for purposes o f numerical solution. Geometric variables are nondimensionalized by De = depth of water from ship to ocean bottom. Velocities are nondimensionalized by Vc= ship speed. Tensions and forces are nondimensionalized by w0D~ = dry unit weight times water depth. Let
38
I(s - l)
-- fD[ CDx[ ~'~NI V~xi + CDrrr [ ~'~T[ V~ri]
At the ship, the payout is passively controlled. Therefore, by Assumption 13 we have the equilibrium boundary condition
T(S,) = To + kv 2 (at unspecified length S,,)
1
dS* = [1 + ,Tv*2/XT *] ( - - ~
X* = XdDe
(43a)
S* = S/De
(43 b)
T* = Ti/woDe
(43c)
Applied Ocean Research, 1990, Vol. I2, No. 1
Boundary conditions At S * = 0 (bottom) X* = 0;
i = I, 2, 3
(46a)
T* --, 0 T*tT* It z = [ v*2 - 1] 1/z
(46b)
T~' = 0
(46d)
(46c)
At S* = S,* (ship) X~' = 1
(460
T* = To*+ K * v *z
(46f)
Equations (45) constitute a boundary value p~'oblem with six first-order equations in the six dependent values X*, T* and two auxiliary variables v*, &,*subject of the eight boundary conditions given by equation (46). In the next section solution algorithms are described for that boundary-value problem. 3. SOLUTION A L G O R I T H M S Equations (45) constitute a sixth-order two-point boundary value problem subject to the boundary conditions given by equations (46). Because these are six firstorder equations, they can be cast into the form of an
Simulation o f tension controlled cable deployment: J. IV. Leonard attd S. R. Karnoski initial value problem which can be integrated from the origin at the bottom along the arc length of the cable to the termination point at the ship. When the initial value problem is integrated from an assumed value o f payout velocity, the combined solution to the initial value problem is then forced to meet the boundary conditions at the terminal point by iterating on the assumed payout velocity.
tor P,§ dp
to
=f@)
(49)
at integration step (n + I) in terms o f the solution vector y,, at the previous step n is given by )',,+t =.9, + ~(/~0 + 2/~, + 2K2 +/~3) + 0(AS 5) (50) where
Iterative sohttion If in equations (45) and boundary condition equation (46f) we assume the payout velocity to be known, equations (45) become an initial value problem which can be integrated numerically by the procedure described subsequently in section 3.2. The boundary condition equation (46e) can be satisfied by terminating the numerical integration when the calculated X~' = 1. This yields the value of the auxiliary variable S, (suspended length) for the assumed payout velocity. The boundary condition equation (46f) is then used to calculate the error in payout tension at the ship, i.e., we calculate a residual R*(v*) = T* - (To + K*v *z)
(47)
We see that the residual given by equation (47) is a continuous function of payout velocity in that T* at the ship is a function of v* terms in the equilibrium equations, equations (45b) and (26). Thus, a binary search technique of iterations can be adopted to calculate v*. A first estimate at v is taken as a weighted average o f the ship speed Ve and fluid velocities. Successive values o f v* are assumed, equation (45) are integrated and R*(v*) calculated. Where R*(v*) changes sign, this denotes that a root to equation (47) has been bracketed for which R(v*) = 0. Successive halving of that interval will then converge on the value of v* for which R(v*) <. allowable error. A final integration of the equations then yields the locations and tension components at every integration point along the cable for that value o f v*.
Initial value problem solution The six first-order equations given by equations (45) are integrated numerically as an initial-value problem subject to the six initial values o f Xl*, T* given at s = 0 (bottom) by equations (46a to 46d). To decouple the derivatives of Ti in equation (2.5.3b), the term involving d]dS*(1]T*) is approximated by a backward difference, i.e., d clS*
(~___.) ( - ~ --
+)/AS*
A'-%=ASf(Sn,yn)
(514)
Kt =Asf(s.+ 89 zxs, P. + 89 kz= ~sf(s. + l As, p,, + 89
(51b)
A~ = A s f ( s , + AS, P,, +/~a)
(51d)
(51c)
The error in equation (50) is o f the order AS 5. Use o f equations (50) and (51) to solve equations (45) requires evaluation of the right hand sides o f equation (45) four times during each step in the integration from S, to Sn+ l,
4. E X A M P L E S Given in this section are three illustrative examples of the theory and solution methods described in the previous sections. In each example a different parameter was varied to demonstrate its effect on cable deployment. Certain properties were common to all examples. Numerical values of these properties are given in Table 2. Inextensible cable was assumed. The properties which were different in the three examples are given in Table 3 along with indications o f the parameters varied in each of the examples. The cable sizes, weights and peel point tensions for the three examples span a regime of realistic properties for passively deployed cables.
Example 1. Effect of ship velocity In the first example the ship velocity was varied from 0.5 ft/sec to 3.0 ft/sec in the absence of current. This leads to a two-dimensional problem in a vertical plane.
Table 2. Common propertiesfor exaJ~Iples Error tolerance = 0.01~0 Gravitational constant = 32.174 ft/secz Seawater density = 1.99 slugs/ft3 Water depth = 5000 ft Unspooling viscosity = 0.0 Added mass coefficient = 2.0 Normal drag coefficient= 2.0
(48,
-
where Tp*= value of T* at the prior integration step and AS* = size o f integration step. Spatial variations in the fluid velocity can be handled easily. Values of V,i(Xt) at station points through the depth are specified. Then as equations (45) are integrated, values of V,i(S) at each integration point are interpolated based on calculated values of X~. The equations can be integrated numerically by the fourth-order Runge-Kutta method.34 The solution vec-
Table 3. Differing propertiesfor e.ramples Property
Example 1 Example2
Example3
Diameter Dry weight Peel Point Force
2.0 in. 0.5 in. 1.50 lbs]ft 0.20lbs]ft 400 lbs 20 lbs
0.l in. 0.01 lbs/ft 2 Ibs
Ship speed Varied Tang. Drag. Coeff. 0.017 Current None
3 ft]sec 3 ft]sec Varied 0.017 Side (0.75 ft/sec) Varied (sheared)
Applied Ocean Research, 1990, VoL 12, No. 1 39
Simulation o f tension controlled cable deployment: J. IV. Leonard and S. R. Karnoski The effect of a faster ship velocity on the cable geometry is a cable which is more inclined from the vertical and therefore one with a greater length of cable in the water column. Profile views of the cable at differing ship speeds are shown in Fig. 3. We see that Zajac's 37 assumption of straight cable geometry is justified for the case of uniform current. Only a slight curvature adjacent to the bottom is present, as seen in Fig. 4.
Example 2. Effect o f tangential drag coefficient In the second example the tangential drag coefficient was varied from 0.010 to 0.042 in the presence of a side current of 0.75 ft/sec. This leads to a three-dimensional problem, but still one with uniform current. The variation of response with tangential drag coefficient is of interest because of present uncertainties regarding formulas and numerical values for that coefficient for small cables with large inclinations in low Reynolds number flow conditions. The tangential drag coefficient does not change the geometry of the cable. Figures 5 and 6 are the profile and plan views, respectively, of the cable in the water column for all values of tangential drag coefficient considered. However, the coefficient does affect the payout rate of the cable. The payout rate influences the deployed slack. One measure of deployed slack is the ratio of particle velocity at the ship (v-Vc) to the ship C~SLE D~AM[TER - ' 2 . 0 INCHES
/
S
r = o.s
fcl|
1.0 file
l.S
4
/
/
/
speed (Vc) expressed as a percentage o70 Slack = ( v / V c - 1) • 100
(52)
Shown in Fig. 7 is the variation of percent slack with tangential drag coefficient. The percent slack varies from about 86~ to 180% with most variations for smaller drag coefficients and less variation for larger drag coefficients. Thus, the payout rate of the cable, and hence the deployed slack, are sensitive to the assumed magnitude of the tangential component of hydrodynamic drag. PROFILE VIEW
g z~ o~
3
o~
P
,
,
,
i
2
i
,
4
i
,
5
,
h
8
,
,
10
12
LON,.CITUDI~L EAY (FEET)
Figure 5. Vertical profile f o r side current attd f o r all CDT
/ 0
J
-0 2
~
e
.~
l|
|. '
9S
,
~|
-0.4 -0,6
-o~ -
-1.2
~ ..... 2
w 5
1
-2.2 -2.4 -2.6 -
0
0
4
8
12
,
.
i
16
20
24
(Thousands)
28 -32
LONGITUDLP~L LAY (FEET)
i
i
2
4
~
.
L
6
i
.
8
'l
4
i
10
12
(Thousands) LONC~TU~:NJ,L LAY (FEET)
Figure 3. Vertical profiles f o r varying canister velocities
Figure 6. Plan view f o r side current attd f o r all COT C~BLIE D ~ E T I ~ - 2.0 I,ICHE$
~o
"i
/
1B
-
/
j
j-q|
=o 180
IS
lEO
14
140
1.1
~"
130
12
.~
120
It
'~
ItO 1C0 90 8O
~
>o
10
9 8 7 6
EO
5
~O
4
.t~
x
20~
2
7~
I
10
0
0
4
8
12
16
Z'O
24
2:8
I.Df~CdlU'Dtt,U~ LAY (FEI~)
Figure4. Near-bottom profiles f o r varying canister velocities 40
Applied Ocean Research, 1990, Vol. 12, No. 1
0
,
,
h
,
0.01
0 02
0 03
0.04
TANOENT~L CnAO CCEFFIC:ENT
Figure 7. Effect o f tangent drag coefficient on deployed slack
Simulation .of tension controlled cable deployment: J. IV. Leonard and S. R. Karnoski Example 3. Effect o f sheared current In the third example a sheared nonplanar current profile was considered. At the ship the current was assumed to be in the direction of vessel motion and to have the same speed (net zero velocity at ship). The current magnitude was assumed to decrease linearly with depth and the direction to also change linearly with depth. At the bottom the current is directed toward the side and the magnitude was varied from 0.5 ft/sec to 3.0 ft/sec. The resulting cable configurations for each case are shown in Figs 8 and 9 for the profile and plan views, PRCFILE~EW 5
~_r
c~~ , O= =.~ ~
2
i
, 2
,
, 9
,
= , , ~ 8 (Tnouso~Os) L,,CNGITU.~:NaLLAY (FEET)
=
, 10
=
, 12
* 14
Figure 81 Vertical view for sheared currenls PL~ ~[W
0 -I -2 -3 -4 -5
~,g
-7
"3g
-s
~
f
t
l
respectively. A fully three-dimensional response is obtained when such sheared nonplanar current profiles are considered. Also, this example shows that tensions in the cable can exceed the peel-point tensions when threedimensional currents are encountered. Shown in Fig. 10 are the cable tensions at each point along the cable for the differing magnitudes of bottom side current. 5. CONCLUSION The theory and numerical algorithm were developed to predict the configuration and tension components during the passively controlled payout of a cable from a moving ship for steady-state deployment conditions. From those assumptions it was possible to derive six first-order differential equations for the three location coordinates and three tension components at an arbitrary point along the cable. Two auxiliary variables were defined, the length of cable suspended in the water column and the payout velocity. A combined iterative and direct integration method was described to solve those differential equations subject to boundary conditions at the bottom and at the ship. Runge-Kutta fourth-order integration was used to propagate solutions along the cable from the bottom to the ship and a binary search technique was used to iterate for the payout velocity. Examples were generated to demonstrate the capability of the computer program developed for modeling steady-state deployment of cable. Uniform 2-D and 3-D current distributions were considered and solved readily. The solutions for payout velocity and slack deposited on the bottom were seen to be somewhat sensitive to the value assumed for the tangential drag coefficient. Sheared and twisted curents 3-D were also generated and solved. The computer program was seen to be able to handle general depth-varying velocity profiles.
till ['* I
-10
ACKNOWLEDGEMENT -12
This work was supported in part by the USN Office of Naval Research under the University Research Initiative (URI) Contract No. N00014-86-K-0687 and in part by the Naval Civil Engineering Laboratory.
-:3 -14 -13 iotta
";16 2
9
C~rr*ac
$F*.4 - 3.0 I t / i
I0
6
12
LCNC:Tb~;~ALL~Y (FKET)
Figure 9. Plan views for sheared currents
REFERENCES
10
8
r
~rl.nt 3.0 It/,
e.d -
7
~
4 3
2 loft I
0
,--i
,
0
2
i
i 9
.
.
. 6
.
. B
.
i 10
(7~o~sondl) ~.~C L[NCTM (r~ET)
~ 12
i
, 14
,
, 16
,
, I~
Figure 10. Effect of sheared currents on cable tension
1 Ablow, C. M. and Schechter, S. Numerical simulation of undersea cable dynamics, Ocean Engineerblg, 1983, 10(6), 443-57 2 Celmer, T. N., Stephens, T. C. and Coe, J. M. Numerical simulation of towed cables, Ocean Enghteering, 1983, 10(2), ! 19-32 3 Chang, P. Y. and Pilkey, W. D. Static and dynamic analysis of mooring lines, Journal o f Hydronauties, 1973, 7, 29-34. 4 Chiou, R.-B. Nonlinear static analysis of long submerged cable segments by spatial integration. M. S. Report, Civil Eng. D e p t . , Oregon State University, July, 1987 5 Choo, Y. 1. and Casarella, M. J. Configuration of a towline attached to a vehicle moving in a circular path, Journal o f ltydronautics, 1972, 6, 51-7 6 DeZoysa, A. P. K. Steady-state analysis of undersea cables, Ocean Engineering, 1978, 5, 209-23 7 Domfnguez, R. F. and Smith, C. E. Dynamic analysis of cable systems. Journal o f the Struetttral Division, ASCE, 1972, 98, 1817-1934 8 Fellipa, C. A. Finite element analysis of three-dimensional cable structures, Procs., Int. Conf. on Computational Methods in Nonlinear Mechanics, 1974, 31 i-24
Applied Ocean Research, 1990, Vol. 12, No. 1 41
S i n n d a t i o n o f tension controlled cable d e p l o y m e n t : J. IV. L e o n a r d a n d S. R. K a r n o s k i 9
10 ii 12 13 14 15 16 17 18
19 20 21 22
42
Ferriss, D. H. Design calculations associated with the laying of submarine communication cables, National Physical Laboratory Report, Contract NPL-DMA-A-76, Teddington, Middlesex, UK, March, 1984 Gay, S. M. New engineering techniques for application to deep water mooring, ASME paper n 66-Pet-31, 1966 Leonard, J. W. Curved finite element approximations to nonlinear cables. Procs., 4th Offshore Technology Conference, 1972, I, 225-36 Leonard, J. W. Newton-Raphson iterative method applied to circularly towed cable]body systems, Engineering Structures, 1979, 1, 73-80 Leonard, J. W. Convergence acceleration method for ocean cables, Preprint No. 80-131, ASCE Spring Meeting, Portland, 1980 Leonard, J. W. Minicomputer simulation of tethers for trawling and towing operations, Procs., OSDS '82, Oregon State University, Corvallis, 1982, 489-508 Leonard, J. W. and Nath, J. H. Comparison of finite element and lumped parameter methods for oceanic cables, Engineering Structures, 1981, 3, 153-167 Leonard, J. W. and Recker, W. W. Nonlinear dynamics of cables with low initial tensions, Journal of the Engineering Mechanics Division, ASCE, 1972, 98, 293-311 Leonard, J. W. Tension Structures, McGraw Hill Book Co., NY, 1988 Liu, F. C. Establishment of initial configurations in lumped parameter simulations of underwater cable dynamics. Technical Memo M-44-77-9, Naval Civil Engineering Laboratory, Port Hueneme, CA, 1977 Lo, A. and Leonard, J. W. Dynamic analysis of underwater cables. Journal of the Engineering Mechanics Division, 1982, ASCE, 108, 605-21 Ma, D., Leonard, J. W. and Chu, K.-H. Slack-elasto-plastic dynamics of cable systems, Journal of the Engineering Mechanics Division, ASCE, 1979, 105, 207-22 Morison, J. R., O'Brien, M. P., Johnson, J. W. and Schaaf, S. A. The forces exerted by surface waves on piles. Petroleum Transactions, AIME, 1950, 189, 149-54 Nath, J. H. Tether element geometry from steady loads, Journal of the IVaterway, Port, coastal and Ocean Division, ASCE, 1979, 105, 193-97
A p p l i e d Ocean Research, 1990, Iiol. 12, N o . 1
23 24 25 26 27 28 29 30 31 32
33 34 35 36 37
Nuckolls, C. E, and Dominguez, R. F. Large displacement mooring dynamics. Procs., 9th Offshore Technology Conference, 1977, 3, 19-30 Rosenthal, F. and Skop, R. A. Guyed towers under arbitrary loads, Journal of the Structural Division, ASCE, 1980, 106, 679-92 Sanders, J. V. A three-dimensional dynamic analysis of a towed system, Ocean Engineering, 1982, 9(5), 483-99 Sarpkaya, T. and lsaacson, M. St. Q. Mechanics of wave forces on structures, Van Nostrand Reinhold Co., NY, 1981 Schram, J. W. and Reyle, S. P. A three-dimensional analysis of towed systems, Journal of Hydronautics, 1968, 4, 203-20 Seidel, L. H. On the dynamics of a mooring line, Look Lab, 1971, University of Hawaii, Honolulu, HI, 2(1) Skop, R. A. and O'Hara, G. J. The method of imaginary reactions: a new technique for analyzing structural cable systems, Marine Tech. Soc. Journal, 1970, 4, 21-30 Thresher, R. W. and Nath, J. H. Anchor-last deployment simulations by lumped masses, Journal of the IVaterways, Harbors and Coastal Enghteering Division, ASCE, 1975, 101,419-34 Wang, H. T. Effect of nonplanar current profiles on the configuration of moored cable systems. Procs., 4th Offshore" Technology Conference, 1972, 1,215-24 Watson, T. V. and Kuneman, J. E. Determination of the static equilibrium configuration of externally redundant submerged cable arrays, Procs., 6th Offshore Technology Conference, OTC 2323, 1975 Webser, R. L. Nonlinear static and dynamic response of underwater cable structures using the finite element method. Procs., 7th Offshore Technology Conference, OTC 2322, 1975 Weeg, G. P. and Reed, G. B. Introd,tction to Nunterical Analysis, Blaisdell Publ. Co., Waltham, MA., 1966 Western Instruments Corp. Model of the Newtonian deployment of cables in water, NCEL Contract Report N00123-86-D-0295, WIC, Ventura, CA, June, 1988 Wingham, P. J. and Kreshaven, N. R. Predicting the equilibrium depth of a body towed by a faired cable, Ocean Engineering, 1978, 5, 15-35. Zajac, E. E. Dynamics and kinematics of the laying and recovery of submarine cable The Bell Systent Technical Journal, 1975, 1129-1207
W. LEONARD
Department of Civil Engineering, Oregon State University, Corvallis, OR
S T E P H E N R. KARNOSKI Naval Civil Enghleering Laboratory, Port Htteneme, CA
The theory and a numerical algorithm were developed for the deployment o f cable from a ship traveling with constant speed and direction in a flowing fluid. Attention was focused on steadystate conditions, i.e., horizontal, steady horizontal flow varying in magnitude and direction with depth but uniform in plan, and on tension controlled payout o f the cable.
1. I N T R O D U C T I O N When cable is deployed from a shipboard canister or reel with a pre-set peel-point tension, the rate o f cable deployment is dependent on the balance o f external forces (gravitational, fluid dynamic, and frictional) over the length of the cable in the water column with the tension at the ship. When the payout rate is less than the ship speed, the cable will tend to be dragged along the b o t t o m . When it is desired to place more cable between two points on the bottom than the straight-line distance between those two points, the payout rate must be greater than the ship speed. This generates slack cable on the b o t t o m which is desirable because of smaller tensions in the installed cable, no dragging, less chance of snags and smaller suspensions over irregular bottoms. Simplified models o f slack deployment can be generated 35'37 based on balance o f fluid forces normal to the cable by the normal c o m p o n e n t of the wet weight of the cable. These models require the assumption of straight configurations in the water column and the assumption o f uniform fluid flow (no shear or rotation). More refined models which allow spatially variable fluid flows can be developed based on either numerical
integration along the cable or on solution o f matrix equations for discretized equations of motion. See Table 1 for a listing o f references on various methods of numerical analysis o f cables without payout. For deployment of extremely long cables with low levels o f tension the latter a p p r o a c h (e.g., finite element models) will require high precision operations on large sets o f equations. The former a p p r o a c h (e.g. direct integration) may require numerous integration steps but will not require storage of large arrays of variables and will easily a c c o m m o d a t e highly variable flow fields. The governing differential equations and boundary conditions for the description of the nonlinear boundary value problem of cable deployment in a 3-D setting are developed in the following secton using the works by Leonard lz, t7 and by Chiou 4 as bases for the derivation with payout added. A solution algorithm based on iterations for payout velocity and on direct integration of the cable equations along the curved cable length is then presented and illustrative examples are presented for deployment in both uniform and sheared nonplanar current profiles.
Table 1. References on Numerical Methods Lumped-Parameter
Finite Element
Imaginary Reactions
Direct Integration
Seide128 Thresher and Nath 3~ Liu ~s Nuckolls and Dominguezz3 Leonard and Nath js Sanders25 Ablow and SchechterI
Leonard and Reeker~6 Leonard II FellipaS Webster3~ Ma, Leonard and Chu 2~ Leonard and Nath~5 Lo and Leonard'9 Celmer, Stevens, and Coe 2
Skop and O'Hara 29 Dominguez and Smith7 Nath 2z Watson and Kuneman3z Rosenthaland Skop24
Gay ~~ Schram and Reyle27 Wang3 Choo and Casarellas Chang and Pilkeya DeZoysa6 Wingham and Kreshaven36 Leonard ~2-,4
Accepted September 1989. Discussion closes July 1990.
34
Applied Ocean Research, 1990, Vot. 12, No. 1
9 1990 Computational Mechanics Publications
Simulation o f tension controlled cable deployment: J. IV. Leonard and S. R. Karnoski 2. D E R I V A T I O N OF EQUATIONS Shown in Fig. 1 is a definition sketch o f the deployment from a ship of a light-weight cable in a sheared nonplanar fluid flow field. A Cartesian coordinate system is attached to the ship moving at constant speed over a horizontal ocean bottom. The origin of the coordinate system is taken at the location o f the point 'B' where the cable contacts the bottom. An arc length curvilinear coordinate system is used to describe the cable behavior at an arbitrary point along the cable. In the first part of this section are described the assumptions made to mathematically model the deployment process. Secondly, the geometrical constraints and material behavior are developed from some of those assumptions. Equilibrium equations are then generated for a differential length of cable subject to body forces and fluid drag. Finally, boundary conditions at the touchdown point 'B' and at the ship 'C' are developed.
where ~z is the unit vector in the xz direction and /'p is the unit tangent to the cable at ' P ' with positive direction in the direction of increasing arc length coordinates from the bottom point 'B'. 7. In order to maintain steady form the touchdown point 'B' moves with constant velocity and direction matching that of the ship, i.e. ~'B = Pc. 8. If the payout speed v < [ ~'c[, the cable is dragged along the bottom at p o i n t ' B ' and the tangent to the cable at the bottom is t8 = ~2, i.e. the cable has zero slope at the touchdown point. 9. If the payout speed v>>.lVcl, slack cable is deposited on the bottom at point 'B' and the tangent to the cable at the bottom is t8 = Ols~l + 028~Z
(2)
where 0iB = direction cosines of t'B such that 028 = Vclv
(3a)
018 = 11 - (Vc]v)21 1/2
(3b)
i.e. [Tp at 'B' is the vertical free fall velocity.
Kinematic assttmptions 1. The cable in steady state spans between point 'B' on the bottom and point 'C' at the ship traveling at steady speed Vc in a straight line in" the xz direction. 2. Bottom is horizontal. 3. At an arbitrary point ' P ' on the cable the particle (point on the cable) velocity vector is ~'v and its magnitude is invariant along the cable in order to maintain a steady form for the cable. 4. The water velocity vector at point 'P' is denoted by r',,. and is assumed to be uniform in the xz-x3 plane with no component in the x~ direction. 5. The water velocity vector may vary with xl, i.e. sheared current. 6. At point 'C' the cable deploys from the ship with speed v relative to the moving ship. Thus, the particle velocity at any point along the cable is ~'v = Vce2 -- We
(1)
Dynanffc equilibriunt assunlptions 10. Cable is subject to uniaxial stress with no flexure, shear or torsion. 11. Cable is either elastic or inextensible with stressstrain relation o = Eoe
(4)
where a = uniaxial stress, Eo = elastic modulus (or co if inextensible) e = elastic strain = 0 , 2 - 1)]2 (5) X = elongation ratio = dS]dSo (6) dS = stretched differential length dSo = unstretched differential length 12. Mass is conserved and the density of material does not change upon stretching. Thus the stretched wet weight o f a differential length can be related to the unstretched dry weight of a differential length by w dS = Wo dSo(Sg- l)]Sg = Wo d S ( S g - 1)]>,S~
V
!
;
,
\
I
9
'I
'/_
Deployed Cable
D~
!
,/ v
$ i
iXt
VF
I
I I
I
,
_ Vel0city Profile
Figure 1. Definition sketch o f cable deployment
(7)
where w = unit stretched wet weight Wo = unit unstretched dry weight Sg = specific gravity of cable Also, the mass m per unit stretched length is m = pA = poAo]X = WolkG
(8)
where p, O0 are cable densities and A, Ao crosssectional areas o f the stretched and unstretched lengths, respectively, and G = gravitational constant. 13. If deployment at the ship is passively controlled, the cable tension T at the particle detaching from the shipboard canister or reel is constant and is given by T = To + Kv z
(9)
where To = peel point tension for the canister or reel and K = c o n s t a n t o f proportionality for unspooling from the cable pack.
Applied Ocean Research, 1990, Vol. 12, No. 1
35
Simulation o f tension controlled cable deployment: J. IV. Leonard and S. R. Karnoski 14. Fluid drag forces on the slender cable are taken from the relative velocity form o f the Morison equation.2~'26 (Derivation in a subsequent section.) 15. The independence principle for normal and tangential drag is invoked, i.e. the tangent and normal drag forces are separately calculated from the tangential and normal components of the relative fluid velocity. 16. The drag coefficients Corand C~x in the Morison equation are Reynolds' number dependent and could vary with relative roughness of the cable. In this work, they are taken as constant values. 17. Added mass effects are included from the inertial term of the Morison equation 2~'26 with steady particle acceleration equal to the normal acceleration due to curvature changes in the cable.
Geometrical and material behavior The position vector to an arbitrary point ' P ' on the cable is /~ = X~&~
For an elastic (or inextensible) material
c = olEo = TIEoAo
where E 0 = m o d u l u s o f elasticity and A o = c r o s s sectional area of unstretched strength member in the cable. If the cable is inextensible, E is taken as infinite. Thus, the elongation ratio is X = ,,11+ 2TIEoAo
Buoyant force vector ,~b = - (w dS)~l By assumption 3 of Section 2.1.2
where Xi = coordinates o f point ' P ' and ~i = Cartesian base vectors. Repeated indices imply summation over the range 1,2,3. The unit tangent vector t'at ' P ' is
-fib = -- [W0 d S ( S g - l)/kSg] 5ii~i
-
dXi - dS
~i
(12)
then, the governing equations for positions Xi are
(14)
and in terms of components in the X~ direction T = T~i
(15)
Replacing i" in equation (14) by equation (12), we can obtain relationships between 0i and T~ as
O, = 751T
where hii = Kronecker delta = 1 if i = j , or else = 0 if i~j.
Fhdd drag force vector By the independence principle for drag on a slender inclined cylinder using the relative velocity form of the Morison equation 2~'z6 (Assumptions 14 and 15) Fa = 89pjD dS{ Co.v[ VRNI PR~' + rcCor[ VRTI VRTI
dXi = 0i (13) dS We will define the tension vector T b o t h in terms of the magnitude T in the direction of t" T = 7"[
(24) where 0 f = fluid density, D = d i a m e t e r of cable, CD.V= normal drag coefficient, C o r = t a n g e n t i a l drag coefficient, ~:RN normal component of relative velocity vector, and ~'nr= tangential component o f relative velocity vector. The t e r m s ] PRN[ and I VRr[ denote the magnitudes of ~'rRN and VRr, respectively. =
%
%.
(16)
and, therefore, the governing equations for positions X~ can be written as dX~ =--7'" dS T
(17)
which are nonlinear first-0rder equations for X~ in terms o f tension components Ti with T = [ TjTil 1/2
(18)
The strain c is related to the elongation ratio X by X = ,/1 + 2c
(19)
where
X = dSldSo 36
(23)
(11)
where dS is the stretched differential length. If we define direction cosines Oi o f / " by
[ = Oi~i
(22)
Equilibrittm At a general point on a cable a stretched differential length can be isolated as a free body as shown in Fig. 2. The external forces acting on that free body are the buoyant weight, and drag and added inertia forces due to relative motion through the fluid. These external forces are balanced by the variation in end point tensions over the differential length and the inertial force due to the cable acceleration.
(10)
[ = d/~ dS
(21)
(20)
Applied Ocean Research, 1990, Vol. 12, No. 1
Figure 2. Free body diagram
Shmdation o f tension controlled cable deployment: J. IV. Leonard attd S. R. Karnoski The relative velocity vector is
have
~ = P,,.- f'l, = V,,.i~i- Vc~2 + v[
(25)
where V,,.~are the Cartesian components of V,,.. Recalling equation (12), we find
= [ v,,~oj- Vc~:oA 7+ v7 = [V, f l y - Vc6jzOj+ v]OiYi = VRri6i
(26a) (26b)
where Vnr~= Cartesian components of PRr. The normal velocity vector iS~'R N = ~'tR -
~'tR T
VRri)ei + V[ = ( V.,y - VcSi2) (&j - OiOj)6i = VRxi6i
fit= +X-ing
mdSOp=Pb+Pd+P1+
I PRrl = I VRTiVRril I/Z
" (28b)
Therefore, the drag force is
-T
(34)
XG
dS[~ -~- + r~
+ XsgGw~vt-:MX[ff.--~+Ti
--~j (35)
(29a) dS
where
~'RN[ VRNi Fori = ~,p:V~rC~rlP~rl gl~r~ FDNi = 89
(31)
where hp,v denotes the normal component of cable acceleration and t~nN is the relative normal acceleration. The relative inertial force is derived for acceleration normal to the cable according to Assumption 8 as CI~IIV(IRN
MXV / ~ - ~ - +
T~"
~i
XSg
6il
__~
The term dt[dS is the normal fi to the cable, and therefore the particle acceleration is normal to the cable. Evaluating di'[dS by equations (12) and (16), we have ei
=
(29c)
(30)
-]-
I+
- ~,o:DI CDu] PRu[ VRNi + CorrC l Vgrl VRril
d(-v[) d~ v
= _ v 2 di" dS
XG T
(29b)
Added fluid inertia free vector The added fluid inertia force is that due to relative normal acceleration of the cable through the water using a relative acceleration form of the inertial contribution to the Morison equation. The acceleration of the particle 'P' under steady state conditions is given by
= +p.tVdS
T + ~--~ d
Combining terms and eliminating dS, we have
ff-a = ( Fo,vi~i + Fori6i) dS
f t = ply dS
(31)
VRN. The (28a)
ae,'~, = -- ( I R N ~-. -- V I~l~
ei
Therefore, by equations (8), (23), (29) and (33), we can write the components of equation (34) as
(27a) (27b)
[ PRUl = { VRxiVRNil l/z
dF',o d~'pdS ~ ; = dt - d--S d t -
M , v [ ' ~ ' ~ - + T/
Equations o f motion We set the sum of forces on the differential segment shown in Fig. 2 equal to the inertial force on the segment
: ( V w i -- Vc~i2 -
where VRxi=Cartesian _components of magnitudes of PRN and Vnr are
dS
vZ(l+
C~IN~
(36)
Equation (36) is a nonlinear first-order equation for Ti in terms of relative velocities VR,~7and Vnri (which involve Oi = 75[T) given by equations (26) and (27). Note that if decoupled equations for dl)]dS are sought, the final term on the right-hand-side of equation (36) will have to be approximated in that the formal derivative 1
dTj
gives coupling terms for dT~]dS. In the numerical solution procedures presented in section 3, the term d[dS ( l / T ) will be evaluated approximately by a backward difference.
Boundary conditions Equations (17) and (36) constitute six first-order differential equations for the six dependent variables of location X~ and tension components T~, i = 1, 2, 3. The independent variable is the arc length coordinate S. Thus, six boundary conditions at the two ends need to be specified. There are also two auxiliary variables - the payout velocity v and total length of cable Su suspended in the water column. By assumption 3, v is taken as invariant with S.
(32)
V = v o l u m e per unit stretched length of cable and CAIN = added mass coefficient for normal acceleration. Writing the fluid inertial force in terms of the mass nt of the cable and using equation (8) of Assumption 12 we
Bottom boundary conditions The origin of the Cartesian coordinate, .hi, and the arc length coordinate, S, are taken at the touchdown point 'B' of the cable on the ocean bottom. Thus, at Applied Ocean Research, 1990, VoL 12, No. 1
37
Simulation o f tension controlled cable deployment: J. IK Leonard and S. R. Karnoski S = 0 we have three boundary conditions. X~(0);
i = 1, 2, 3
(38)
Also, the touchdown point by Assumption 7 moves with constant velocity Vc. Therefore, by Assumption 9 if payout velocity v is passively controlled, the direction cosines at S = 0 are given by equations (3) and can be expressed in terms of tension components as Tl (0) = T[ 1 - (Vc/v) 2 ] 1/2
(39a)
7"2(0) = TVc[v
(39b)
7"3(0) = 0
(39c)
T*= T]woDe
(43d)
v* = v/Vc
(430
V,*i= V~,~/Vc
(43f)
Nondimensional scaling numbers are defined below:
fD = 89
= ratio o f drag forces to cable weight (44a)
em= woDe/EoAo = ratio o f cable weight to stiffness (44b)
(
r/= ~
1+--~]
Therefore, we have two boundary conditions
= ratio of inertial force to cable weight (44c)
Tt(O)/Tz(O) = [ (v/Vc) 2 - I I 1/2
(40a)
X = ,11 + 2cmT*
(44d)
7"3(0)
(40b)
K* = KV~/woDe
(440
T~ = To/woDe
(44f)
S,* = S,,/De
(44g)
=0
The magnitude Tis undetermined. It should be a small quantity dependent on static frictional resistance of a short segment of cable lying on the bottom near the touchdown point. Zajac 37 stated that T(0) is identically zero when slack cable is being deposited on the bottom. He used that assumption to justify a kink in the cable at the bottom and hence a straight-line solution under uniform current conditions.
Then the governing equations become:
Location (i = 1, 2, 3)
dX*
T*
(45a)
dS* - T*
Canister boundary conditions By assumptions 1 and 2 the location of the ship is at a constant depth De above the ocean bottom. Therefore we have a boundary condition XI(S,,) = De (at unspecified length S,).
(41)
Equilibriunl (i = 1, 2, 3)
dT:
~"
d-~
(45b)
(42)
If, instead, active control of payout is assumed, equation (42) is replaced by specifying v at the ship. In these sections we have specified seven boundary conditions, equations (38), (40), (41) and (42) for the six field variables X~, Ti, i = 1, 2, 3 and the two auxiliary variables v, S,. An eighth condition is obtained if we assume the tension magnitude T(0) at S = 0 to be a small parameter. If, instead, active control o f payout is assumed, equation (42) is replaced by specifying v at the ship and Assumption 8 is used to replace equation (40b) by the condition T~ (0)= 0.
Summary o f nondintensionalized equations It is necessary to nondimensionalize the variable in order to scale the variables and equations for purposes o f numerical solution. Geometric variables are nondimensionalized by De = depth of water from ship to ocean bottom. Velocities are nondimensionalized by Vc= ship speed. Tensions and forces are nondimensionalized by w0D~ = dry unit weight times water depth. Let
38
I(s - l)
-- fD[ CDx[ ~'~NI V~xi + CDrrr [ ~'~T[ V~ri]
At the ship, the payout is passively controlled. Therefore, by Assumption 13 we have the equilibrium boundary condition
T(S,) = To + kv 2 (at unspecified length S,,)
1
dS* = [1 + ,Tv*2/XT *] ( - - ~
X* = XdDe
(43a)
S* = S/De
(43 b)
T* = Ti/woDe
(43c)
Applied Ocean Research, 1990, Vol. I2, No. 1
Boundary conditions At S * = 0 (bottom) X* = 0;
i = I, 2, 3
(46a)
T* --, 0 T*tT* It z = [ v*2 - 1] 1/z
(46b)
T~' = 0
(46d)
(46c)
At S* = S,* (ship) X~' = 1
(460
T* = To*+ K * v *z
(46f)
Equations (45) constitute a boundary value p~'oblem with six first-order equations in the six dependent values X*, T* and two auxiliary variables v*, &,*subject of the eight boundary conditions given by equation (46). In the next section solution algorithms are described for that boundary-value problem. 3. SOLUTION A L G O R I T H M S Equations (45) constitute a sixth-order two-point boundary value problem subject to the boundary conditions given by equations (46). Because these are six firstorder equations, they can be cast into the form of an
Simulation o f tension controlled cable deployment: J. IV. Leonard attd S. R. Karnoski initial value problem which can be integrated from the origin at the bottom along the arc length of the cable to the termination point at the ship. When the initial value problem is integrated from an assumed value o f payout velocity, the combined solution to the initial value problem is then forced to meet the boundary conditions at the terminal point by iterating on the assumed payout velocity.
tor P,§ dp
to
=f@)
(49)
at integration step (n + I) in terms o f the solution vector y,, at the previous step n is given by )',,+t =.9, + ~(/~0 + 2/~, + 2K2 +/~3) + 0(AS 5) (50) where
Iterative sohttion If in equations (45) and boundary condition equation (46f) we assume the payout velocity to be known, equations (45) become an initial value problem which can be integrated numerically by the procedure described subsequently in section 3.2. The boundary condition equation (46e) can be satisfied by terminating the numerical integration when the calculated X~' = 1. This yields the value of the auxiliary variable S, (suspended length) for the assumed payout velocity. The boundary condition equation (46f) is then used to calculate the error in payout tension at the ship, i.e., we calculate a residual R*(v*) = T* - (To + K*v *z)
(47)
We see that the residual given by equation (47) is a continuous function of payout velocity in that T* at the ship is a function of v* terms in the equilibrium equations, equations (45b) and (26). Thus, a binary search technique of iterations can be adopted to calculate v*. A first estimate at v is taken as a weighted average o f the ship speed Ve and fluid velocities. Successive values o f v* are assumed, equation (45) are integrated and R*(v*) calculated. Where R*(v*) changes sign, this denotes that a root to equation (47) has been bracketed for which R(v*) = 0. Successive halving of that interval will then converge on the value of v* for which R(v*) <. allowable error. A final integration of the equations then yields the locations and tension components at every integration point along the cable for that value o f v*.
Initial value problem solution The six first-order equations given by equations (45) are integrated numerically as an initial-value problem subject to the six initial values o f Xl*, T* given at s = 0 (bottom) by equations (46a to 46d). To decouple the derivatives of Ti in equation (2.5.3b), the term involving d]dS*(1]T*) is approximated by a backward difference, i.e., d clS*
(~___.) ( - ~ --
+)/AS*
A'-%=ASf(Sn,yn)
(514)
Kt =Asf(s.+ 89 zxs, P. + 89 kz= ~sf(s. + l As, p,, + 89
(51b)
A~ = A s f ( s , + AS, P,, +/~a)
(51d)
(51c)
The error in equation (50) is o f the order AS 5. Use o f equations (50) and (51) to solve equations (45) requires evaluation of the right hand sides o f equation (45) four times during each step in the integration from S, to Sn+ l,
4. E X A M P L E S Given in this section are three illustrative examples of the theory and solution methods described in the previous sections. In each example a different parameter was varied to demonstrate its effect on cable deployment. Certain properties were common to all examples. Numerical values of these properties are given in Table 2. Inextensible cable was assumed. The properties which were different in the three examples are given in Table 3 along with indications o f the parameters varied in each of the examples. The cable sizes, weights and peel point tensions for the three examples span a regime of realistic properties for passively deployed cables.
Example 1. Effect of ship velocity In the first example the ship velocity was varied from 0.5 ft/sec to 3.0 ft/sec in the absence of current. This leads to a two-dimensional problem in a vertical plane.
Table 2. Common propertiesfor exaJ~Iples Error tolerance = 0.01~0 Gravitational constant = 32.174 ft/secz Seawater density = 1.99 slugs/ft3 Water depth = 5000 ft Unspooling viscosity = 0.0 Added mass coefficient = 2.0 Normal drag coefficient= 2.0
(48,
-
where Tp*= value of T* at the prior integration step and AS* = size o f integration step. Spatial variations in the fluid velocity can be handled easily. Values of V,i(Xt) at station points through the depth are specified. Then as equations (45) are integrated, values of V,i(S) at each integration point are interpolated based on calculated values of X~. The equations can be integrated numerically by the fourth-order Runge-Kutta method.34 The solution vec-
Table 3. Differing propertiesfor e.ramples Property
Example 1 Example2
Example3
Diameter Dry weight Peel Point Force
2.0 in. 0.5 in. 1.50 lbs]ft 0.20lbs]ft 400 lbs 20 lbs
0.l in. 0.01 lbs/ft 2 Ibs
Ship speed Varied Tang. Drag. Coeff. 0.017 Current None
3 ft]sec 3 ft]sec Varied 0.017 Side (0.75 ft/sec) Varied (sheared)
Applied Ocean Research, 1990, VoL 12, No. 1 39
Simulation o f tension controlled cable deployment: J. IV. Leonard and S. R. Karnoski The effect of a faster ship velocity on the cable geometry is a cable which is more inclined from the vertical and therefore one with a greater length of cable in the water column. Profile views of the cable at differing ship speeds are shown in Fig. 3. We see that Zajac's 37 assumption of straight cable geometry is justified for the case of uniform current. Only a slight curvature adjacent to the bottom is present, as seen in Fig. 4.
Example 2. Effect o f tangential drag coefficient In the second example the tangential drag coefficient was varied from 0.010 to 0.042 in the presence of a side current of 0.75 ft/sec. This leads to a three-dimensional problem, but still one with uniform current. The variation of response with tangential drag coefficient is of interest because of present uncertainties regarding formulas and numerical values for that coefficient for small cables with large inclinations in low Reynolds number flow conditions. The tangential drag coefficient does not change the geometry of the cable. Figures 5 and 6 are the profile and plan views, respectively, of the cable in the water column for all values of tangential drag coefficient considered. However, the coefficient does affect the payout rate of the cable. The payout rate influences the deployed slack. One measure of deployed slack is the ratio of particle velocity at the ship (v-Vc) to the ship C~SLE D~AM[TER - ' 2 . 0 INCHES
/
S
r = o.s
fcl|
1.0 file
l.S
4
/
/
/
speed (Vc) expressed as a percentage o70 Slack = ( v / V c - 1) • 100
(52)
Shown in Fig. 7 is the variation of percent slack with tangential drag coefficient. The percent slack varies from about 86~ to 180% with most variations for smaller drag coefficients and less variation for larger drag coefficients. Thus, the payout rate of the cable, and hence the deployed slack, are sensitive to the assumed magnitude of the tangential component of hydrodynamic drag. PROFILE VIEW
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Figure 5. Vertical profile f o r side current attd f o r all CDT
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Figure 3. Vertical profiles f o r varying canister velocities
Figure 6. Plan view f o r side current attd f o r all COT C~BLIE D ~ E T I ~ - 2.0 I,ICHE$
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Figure4. Near-bottom profiles f o r varying canister velocities 40
Applied Ocean Research, 1990, Vol. 12, No. 1
0
,
,
h
,
0.01
0 02
0 03
0.04
TANOENT~L CnAO CCEFFIC:ENT
Figure 7. Effect o f tangent drag coefficient on deployed slack
Simulation .of tension controlled cable deployment: J. IV. Leonard and S. R. Karnoski Example 3. Effect o f sheared current In the third example a sheared nonplanar current profile was considered. At the ship the current was assumed to be in the direction of vessel motion and to have the same speed (net zero velocity at ship). The current magnitude was assumed to decrease linearly with depth and the direction to also change linearly with depth. At the bottom the current is directed toward the side and the magnitude was varied from 0.5 ft/sec to 3.0 ft/sec. The resulting cable configurations for each case are shown in Figs 8 and 9 for the profile and plan views, PRCFILE~EW 5
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c~~ , O= =.~ ~
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,
, 9
,
= , , ~ 8 (Tnouso~Os) L,,CNGITU.~:NaLLAY (FEET)
=
, 10
=
, 12
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Figure 81 Vertical view for sheared currenls PL~ ~[W
0 -I -2 -3 -4 -5
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-7
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~
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respectively. A fully three-dimensional response is obtained when such sheared nonplanar current profiles are considered. Also, this example shows that tensions in the cable can exceed the peel-point tensions when threedimensional currents are encountered. Shown in Fig. 10 are the cable tensions at each point along the cable for the differing magnitudes of bottom side current. 5. CONCLUSION The theory and numerical algorithm were developed to predict the configuration and tension components during the passively controlled payout of a cable from a moving ship for steady-state deployment conditions. From those assumptions it was possible to derive six first-order differential equations for the three location coordinates and three tension components at an arbitrary point along the cable. Two auxiliary variables were defined, the length of cable suspended in the water column and the payout velocity. A combined iterative and direct integration method was described to solve those differential equations subject to boundary conditions at the bottom and at the ship. Runge-Kutta fourth-order integration was used to propagate solutions along the cable from the bottom to the ship and a binary search technique was used to iterate for the payout velocity. Examples were generated to demonstrate the capability of the computer program developed for modeling steady-state deployment of cable. Uniform 2-D and 3-D current distributions were considered and solved readily. The solutions for payout velocity and slack deposited on the bottom were seen to be somewhat sensitive to the value assumed for the tangential drag coefficient. Sheared and twisted curents 3-D were also generated and solved. The computer program was seen to be able to handle general depth-varying velocity profiles.
till ['* I
-10
ACKNOWLEDGEMENT -12
This work was supported in part by the USN Office of Naval Research under the University Research Initiative (URI) Contract No. N00014-86-K-0687 and in part by the Naval Civil Engineering Laboratory.
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REFERENCES
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,--i
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2
i
i 9
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. 6
.
. B
.
i 10
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~ 12
i
, 14
,
, 16
,
, I~
Figure 10. Effect of sheared currents on cable tension
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Applied Ocean Research, 1990, Vol. 12, No. 1 41
S i n n d a t i o n o f tension controlled cable d e p l o y m e n t : J. IV. L e o n a r d a n d S. R. K a r n o s k i 9
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42
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