On the practical evaluation of hydrodynamic coefficients of sem-submersible platforms, based on systematic model tests

ON THE PRACTICAL EVALUATION OF HYDRODYNAMIC COEFFICIENTS OF SEMISUBMERSIBLE PLATFORMS, BASED ON SYSTEMATIC MODEL TESTS

R. KISHEV A. IVANOV2and L. POMERANETZ2

1Seakeeping Laboratory, BSHC, 9003, Varna (Bulgaria) 2Seakeeping Laboratory, KSRI, 196158, Leningrad (USSR)

SUMMARY

The paper summarizes basic results from the experimental investigation of the hydrody-namic coefficients in motion equations for prediction of catamaran-type semisubmersible platform's behaviour in waves. By the method of forced oscillations, the effect of hull geometric characteristics, fluid viscosity and oscillation frequency on added masses and damping coefficients has been established for a series of semisubmersible drilling platforms (SSDP) models. Particular attention has been paid to the investigation of semisub's hydro-dynamic characteristics at horizontal motion in the low frequency range. The data obtained have been systematized by regression analysis for use in computations of the dynamic and ki-nematic characteristics of free-floating and anchored offshore structures in waves.

1. INTRODUCTION

In the development of calculation methods for complex evaluation of floating structured behaviour in real operating conditions, parti-cular importance is attashed to the hydrodyna-mic coefficients in the equations of motion. The existing practical methods for computati-on of added mass and damping coefficients of

semisubmersible offshore structures are deve-loped using a number of empirical assumptions. Many of these methods do not take into account

in particular, the hydrodynamic flow interac-tion between construcinterac-tion elements located ne-ar each other, the influence of fluid viscosity and frequency of oscillations on the hydrody-namic characteristics of motion. The most re-liable account of the above-mentioned factors can be taken by special model tests.

The present paper includes general results of a systematic experimental investigation of added mass and damping coefficients, which are necessary for prediction of SSDP behaviour in waves.

fluß-MiIUm 2, 2

CO Duft

WI.: 016. ¡bNâ-. 015a181836

2. MODEL SERIES

In order to determine the hydrodynamic mo-tion characteristics of the floating structu-'res, a systematic series of semisubmersible catamaran-type models (Fig. 1) was developed.

The influence of model hull form on added mass and damping coefficients was evaluated using the following dimensionless parameters:

2a/B - horizontal clearance/pontoon breadth ratio;

B/H - pontoon breadth/pontoon height ratio;

D/B - vertical columns diameter/pontoon breadth ratio;

1/H - draught/pontoon height ratio. As can be seen in Fig. 2 and corresponding Table 1, the choosen parameters varied in the ranges most common for semisubmersible's design practice.

Q

-tÍIIN

Fig. 2 Parametric grid

Table i Geometrical parameters of SSDP modl series

The fourth parameter T/H for each of the seven models varied in a sufficiently broad range (0.8 - 3.0), covering different regimes of platform operation.

3. TESTING METHOD

The forced oscillation method was used to determine the added mass and damping coeffici-ents of SSDP models. With the aid of oscilla-tion devices(PMM at mooscilla-tions in the horizontal plane and vertical motion oscillator in the vertical plane), the models were subjected to harmonic oscillations:

X.=X.sin(t) , i=1,2. . .6

('1) the amplitude x. and frequency w of which' were known.

The frequency of model oscillations in the' horizontal plane varied in the range ( 0.25 -1.25 )1/s, and in the vertical plane C 1.0

-8.0 )1/s.

A system of inductive gages was used to measure the exciting forces and moments, and the model motions were registered by a poten-tiometric pick-ups.

The BSHC automated system PMMAS [2] was used for acquisition, registration and proces-sing of the experimental data.

The hydrodynamic coefficients were determi-ned in a coordinate system, whoze horizontal

plane coincided with the still water surface, and the mutually perpendicular vertical planes coincided with the midship section and longi-tudinal centerplane, respectively.

4. LOW FREQUENCY HORIZONTAL MOTION The gross dimensions of the modern SSDP compared to wave parameters bring the hydro-dynamic forces exciting platform motion in the low frequency region, where the speeds of flow round the elements of construction are small and the percentage of viscous forces compo-nents is elatively high. In this area, the wave damping predicted by the potential theo-ry on which the engineering methods for moti-on calculatimoti-on are based, does not practical-ly exist, and this usualpractical-ly leads to incorrect results particularly at predicting forces in horizontal plane, refering to the positioning problems.

In real conditions, the presence of techno-logical roughness lead to nonsymmetric vortex separation from the cylindrical construction elements, as well as to forming of separation flow around the sharp pontoon edges. At this, the vortex separation is principally nonperi-odical due to the preceeding motion generated vorticity. The picture will be stronger expre sed in the area of joints. In these conditi-ons, the damping strongly depends on the fre-quency as well as on motion amplitude, as il-lustrated in Fig. 3.

T

Fig. 3 Non-dimensional sway damping

Model No 2a/B B/H 0/B 1 4.7 2.5 .67 2 3.3 2.0 .67 3 3.3 3.0 .67 4 3.3 2.5 .67 5 3.3 2.5 .40 6 2.0 2.5 .67 7 3.3 2.5 .80

bn thé bàsis

of'systematic

test results analysis it was concluded, that within the frames of the studied frequency range, the damping force in the horizontal plane can be approximated using the formula:

,i=1,2,6 (2)

where the coefficients B. and B. can be

considered linear frequency11functis. As should have been expected, the inertia forces are tô a smaller extent subjected to the influence of the flow specifics in the low f requency area. The added mass coefficients proved independent of the motion amplitude and in the studied frequency range-linearly depen-dent on the frequency, as shown in Fig. 4.

The systematical change of parameters of the geometric series SSDP models enabled the establishing of relation between the hydrody-namic coefficients and the main dimension ra-tios on the basis of regression analysis. The,

results obtained are. illustrated in Appendix 2. They can be used for computer calculation pror gram's composition as well as for simulating the SSDP behaviour in waves on computer.

The experimental.results served as a criteL non for the acceptability of the theoretical prediction of hydrodynamic coefficiênts. The calculation of added mass coefficient of ponH toons was made on the basis of the strip the-! ory, applying the Kochin-Frank method (1] while the vertical column's influence was ta_L ken into account after the weil-known.formula

A__(T_H)p1rD2/4 ₍₃₎

the applicability of which is verified by the control tests with single column [3].

Further, the total values of the SSDP motiL on coefficients were calculated as follows:

A..- nf ¿.dx

ii

Li

# nzA. (4) .12. 15 lo 05 D IB 0.8 0.67 0.4 005 0.10 0.15 0.20 WND

Fig. 4 Non-dimensional sway added mass

where:

n - number of pontoons; m -

number of

columns;

Ai'.- added mass of pontoon cross-section

22.

contour.

When comparing the càlculation results af-, ter formula (4) with experiment, it can be se-en that the deviations are directly related to the relative submergence of pontoons. Obvi-ously, the test results are under the mutual influence between the vértical column and the respective pontoon section, which is not con-j sidered by (4). Therefore, we recommend new calculation formula of the type:

A..'n(f al.dKJ adX)

(5)

9.1 9.2

where:

£2=qDm

9.1=L-2.2.

added mass of the cross-section con-tour of the pontoon with the adjacent column;

q.. -

coefficient accounting the mutual in-fluence between cross-sections with and without vertical column.

It is found [3], that this coefficient tends toward unity at T/H1 and with increase of the relative draft decreases exponentially.

Fig. 5 gives the exemplary dependence of the value q on the column diameter.

The dependence (5) is applicable for the cäses lacking mutual influence of pontoons, 2a/B=3.O being accepted as a boundary. In the case of 2a/B3.0 the strip coefficients of added mass a.. should be calculated using the method of ingrai equations, modified for the twin (niultihull) cylinders.

0/B

1.0

Fig. 5 Coefficient of column-pontoon

- interaction-in sway 0.4 0,2

.L

H 26 3.0

5. VERTICAL MOTION

The vertical motion tests (i=3,4,5) were implemented in wide frequency range and near-ly all regimes were accompanied with conside-rable, and at high speeds of motion - inten-sive wave generation. No steady tendency for the dependence of hydrodynamic coefficients on the motion amplitudes was established, since the deviations were within the frames of experimental errors.

As it is shown in Fig. 6 and 7, the pon-toón draft considerably influences the motion coefficients in vertical plane. At motions on the free surface, the coincidence of theore-tical [3] and experimental results is comple-te but for the low frequency area ,where the measured value is much lower. At deepening of pontoons near below the free surface and es-pecially in regimes of motion, at which edges of pontoon decks intersect the fluid surface, intensive wave generation leads to strong disturbances in exciting as well as restoring forces, which determines the obvious nonline-arity of reaction.

With the deepening of pontoons further downward, the picture obviously improves and

M33 4,0

3,0

2.0

1.0

the dependence on the frequency of oscilla-tion becomes nearly straight line. In the area of high frequencies (WND 1), steady disagreement of theoretical and experimental results is detected.

The appropriate mapping of the families of the complex surfaces shown in Fig. 6 and 7 by means of linear regression analysis, using directly geometrical parameters as indepen -dent variables, proved to be impossible. Ac-counting for the character of the surface cross-cuts and the tendency to smooth with deepening the pontoons, we offer slightly different form of analysis, characterized by inserting of parameters, nonlinear in respect of the relative draft and frequency, in the linear regression procedure, thus separating the influence of these factors from the in-fluence of the hull geometry. In such formu-lation, equation (Al) takes the form:

M N

B D a

,Jki(w)

(6)

A

_ii

(B

_ii

)=

_k

where for the cases of vertical motions for instance is given:

0,2

0,4

0,6

1,0 2,0 _{T/ H} 3,0

Fig. 6 Non-dimensional heave added mass as a function of frequency and relative draft

C 0.5'

'/4.,

N

\

!À%

\

ü2 0/. pic1.c2.c3 B B C1=a1+a2j a3( 1.# a42j

C3= 1.

+ a7(1)2 T/H 3= 1O.exp(-5.w) 54_ lOexp(-2.2T/H)

The sample tables of a coefficients are given in Appendix 2, and in Fig. 8 the calcu-lation after formula (6) is compared with the original experimental values, thus showing the satisfactory approximation of results. The complete tables of regression coeffici-ents for the remaining hydrodynamic motion characteristics are given in [8].

For the purpose of unifying the calculati-on approach in the cases when standard strip method programs for calculation of motion co-efficients are used, the applicability of

0.6

formulae type (5) is investigated, so that correspondence with .experiment would be best. Here the correction coefficient qjj (i=3,4,5) appeared nearly constant,varying in the fra-mes 0.6-0.7 and in areas of average and high frequencies it is applicable to damping coef-ficients as well.

6. INFLUENCE 0F GEOMETRIC PARAMETERS ON THE HYDRODYNAMIC COEFFICIENTS

The dependence of added masses and damping on the SSDP hull geometry is presented on se-ries exemplary Figures 9 to 18. Detailed ana-lysis of the material obtained during the se-rial tests [5],[6], shows tendencies of the influence of each investigated parameter as

fol lows:

6.1 Diameter of vertical column

In accordance with the physics of phenome-na, the increase of column diameter leads to smooth increase of the non-dimensional added mass at horizontal motion (Ajj , i=1,2,6) which is more strongly felt at the end of the investigated range. On the contrary, the cor-responding damping coefficients decrease with the increase of the relative diameter. In the case of high frequency oscillations in verti-cal plane it was found that the increase of

to

2.0

T/H

Fig. 7 Non-dimensional heave damping as a function of frquency and relative draft

0.8 0fr o 0.4

2A/B=3.3 OIBO.67

B/H=2.5.

A22 1.0 2.4 H 0.06 0.12 0.18

- UND

0? o

--0 05 1.0 1.5 UNO

Fig. 8 Matching of experimental points by polynomial regression

column diameter leads to decrease in the non-dimensional coefficients of added masses and damping

(a1),i=3,4,5.

_{This could be} ex-plained by the decrease of the pontoon decks area and changes in the wave generation regi-me in the fluid round the columns. In most of the cases the change is nearly linear. As a result of the prevailing influence of the ho-rizontal motion at the formation of the trans-verse stream flow, with the increase of D/B the cross-couplin coefficients

4

and

B4

increase, while A24and

B5

decrease. 6.2 Horizontal clearance

Here the results should be discussed for each direction of motion separately, as the studied parameter influences the hydrodynamic coefficients in different ways.

The coefficients of surge added masses and damping for

2a/B

3.0

are practically the same. Below this boundary, the decrease of clearance is more markly felt and, for exam-ple, leads to increase of coefficients

A1

and

Bj

with 20-25 % in average, at

_2a/B=2.0.

In sway, the narrowing of the clearance, on the contrary, leads to decrease of coeff i-cients

A2

and

B2 ,

which is particularly felt at pontoon situated near free surface. Here again, in principle, the value of the horizontal clearance starts influencing the coefficients below

2a/B = 3.0.

The hydrodynamic coefficients of heaving motion increase with shortening the clearance, this effect being more strongly expressed on the free surface and for the cases of 2a/B_<3.

This can be also stated for the case of pitch motion, which is explained by the mutual

in-fluence of both motions.

As for the rolling motion, the sharp incre-ase of the added masses and damping with in-crease of clearance is quite expected and

explicable. This change follows parabolic de-pendence and is only slightly dependent on draft.

The yaw coefficients also considerably de-pend on clearance, increasing with the incre-ase of

2a/B

value. It is interesting to noti-ce that on the surfanoti-ce this change is linear, and with submerging the pontoon the change follows parabolic dependence. Such effect is observed for the remaining horizontal motions as well (i

= 1,2).

The change of the surge-pitch and sway-roll cross-coupling coefficients follows the above-mentioned pattern, respectively.

The coefficients

A5

and

B5

decrease

whi-le

A4

and

B4

increase with increasing the horizontal clearance, at which in the prevai-ling cases their dependence on 2a/B is linear. This dependence can be related to the fact, that the cross-coupling terms appear as

sta-tical moments of inertia.

6.3 Pontoon board height

With the increase in pontoon height, i.e., with the increase of the oscillating body's

front projection area, the dimensionless coef-ficient A

, i=1,2,6

increases as well. In the rest of the cases

(Bli , i=1,2,6

and

A(B) i=3,4,5)

the hydrodynamic coeffici-ents decrease with the increase of H. Accor-dingly, the dimensionless coefficients A1*5

A24 and B24 increase and the coefficient

B5

decreases, which indicates the predominating effect of the columns at coupled surge-pitch motions. The abovementioned dependence is va-lid for all relative submergences investiga-ted. In almost all of the cases, and particu-larly at great pontoon draughts, the coeff i-cients' dependence on the _B/H ratio is nearly linear.

/'6

£ 012

AV

AV__

= w 0.20

12

0,4 0.6 0,8 0/B 0.2 0.8 0.4

Fig. 11 Sway-to-roll cross-coupled damping dependence on column diameter and frequency

T/H: 2,4

1.0 1.2

A..

A

ii

_{- M}

i =

1,2,3

A..

A..

_ji

i = 4,5,6

i = 1,2,3

0.4 i =

4,5,6

MLV 0.2

Fig. 9 Surge added mass dependence Fig. 10 Heave added mass dependence

On column diameter on column diameter

and frequency and frequency

0.4 0.6 0.8

D/B

0.4 0.6 0.8

0.1 0.2 0.4 1.0 o

.-24

0.8 2 0.2 o 1.0

8

v.2

ÍA

'-w

AM

_/W

0.6

:2.4

5.0

-

2a/B

0.8 1.0 12 Fig. 12 Sway added mass dependence Fig. 13 Roll added mass dependence

on horizontal clearance on horizontal clearance

and frequency and frequency

Fig. 14 Heave damping dependence Fig. 15 Sway-to-roll cross-coupled

on horizontal clearance added mass dependence

and frequency on horizontal clearance

and frequency

lß 2.0 3.0 5n

-'--2aIB

Fig. 16 Yaw added mass dependence on cross section extension and frequency

20

Lb

4

w-.

Fig. 18 Surge-to-pitch cross-coupled added mass dependence

on cross section extension and frequency

1.2

2.0 2.5 3.0

B/H

Fig. 17 Pitch added mass dependence on cross section extension and frequency

7. CONCLUSIONS AND RECOMMENDATIONS

7.1 On the basis of systematic series tests of SSDP models, diagrams are plotted and ana-lytically described for the evaluation of the hydrodynamic coefficients of motion, depen -ding on the main geometrical parameters. This enables the computerizing of calculations of motion of the semisubmersible type structures under consideration, as well as the solving of a number of design and service problems related to transportation in waves, settle -ment and positioning of the platform.

7.2 By comparing the calculated and experi -mental values of the added masses and damping, the conditions for applicability of standard strip methods in the generalized design scheme of SSDP behaviour evaluation in real ser -vice conditions are established. The application of the regression dependences recommen -ded above, however, leads to a considerable shortening in the calculation time at satis -factory accuracy.

7.3 It is advisable to continue the investi-gation of hydrodynamic coefficients into and out of the range of geometrical parameters discussed in the present paper, as well as in the higher frequencies range for the case of horizontal motions, where the viscous and

wa-2.0 2.5 3.0

B/H

2.0 2.5 3.0

vecomponents are of nearly the samorder.

ThUs the prediction accuracy for both added masses and damping coefficients can be impro-ved on the basis of the specifying and expan-ding of the reference diagrams.

7.4 The viscous components of hydrodynamic forces, appearing in the case of low frequen cy horizontal motions are, most probably, sub-jected to a significant scale effect. This problem must be given further and thorough consideration, in order to apply properly the experimental results in full-scale calcula -i tions.

ACKNOWLEDGEMENTS

L The work has been carried out at the Bul-garian Ship Hydrodynamic Center. Thanks are due to all collaborators of the BSHC Seake -eping Department, who hawe taken part in the preparation and carrying out of the cycles of tests. The help of Mr. Z. Ziatev in the adop-tion and practical applicaadop-tion of the prog.-, rams for experimental data regression analy-sis is highly appreciated.

REFERENCES

Baranova I.,Voitkunskayá A. ,Kishev R., Rachnianin N.,Shebalov A. - Evaluation of Hydrodynamic Caracteristics of Motion of an Arbitrary Contour by the Kochin-Frank Method - J. Problems in Shipbuilding, Vol. 39, Leningrad, 1984

Instruction Manual: PMMAS - Automated Data Aquizition System - BSHC, Varna, 1982 Kishev R. - Comparative Analysis of Calculation and Experimental Results of Hydro -dynamic Motion Coefficients Investigation on Semisub's Elements Models - BSHC, yama 1983

Kishev R. Spasov S. - Viscous effects on Semisub's Low Frequency Horizontal Motion Damping - Joint Bulgarian-Russian Seminar on Behaviour of Ships and Marine Structu-res in Waves, Leningrad, 1984

Kishev R. ,Spasov S. - Analysis of Experi-mentally Obtained Results for the Hydrodynamic Motion Coefficients of Semisubmer -sible Model Series - BSHC, Varna, 1984 6,. Spasov S. ,Kishev R. - Experimental

Evalua-tion of Cross-coupling Terms in EquaEvalua-tions of Motion of Semisubmersible Platforms -Joint Bulgarian-Russian Seminar on Behaviour of Ships and Marine Structures in Wa

-ves, Leningrad, 1984

Ziatev Z. - An Algoritm and Computer Prog. ram for Multiple Linear Regression Analy sis - 1O-th SMSSH, BSHC, Varna, 1981 Kishev R. - Evaluation of Added Masses and Damping Coefficients of Semisubmersible

Platforms on the Basis of Systematic Model Tests Results - Final Report., BSHC, Varna

1985

lo

APPENDIX. 1

A METHOD. OF EXPERIMENTAL DATA REGRESSION ANALYSIS

Systematic diagrams of hydrodynamic added mass and damping coefficients have been plot-ted for the needs of practical application of results in offshore structures' design. The analytical description of these diagrams is realized on the basis of the multiple linear regression analysis. The calculation method

ised is described in details in [7].

From the graphics enclosed it is evident, that the hydrodynamic coefficients' dependen-' ce on the main geometrical parameters can be presented in the form of low order polynoms ,(JL3). More complicated is the problem of. describing the complex 3-D dependence of ver-tical motions' coefficients on draught and frequency. The preliminary data analysis has shown, however, that by introducing composite nonlinear type parameters (e.g. containing 'exponents or logarithms), this dependence can be also reduced to the polynomial one.

The wanted model can be represented as

L

p.R.

(Al)

where R

-

known functions of the indepen-:dent parameters

x

of the type

R. =

_i

₁

i1.9?i2.

₂

N

The p coefficients' calculation algorithm 'envisages the seeking,of an optimum compositi 'on of the sum (Al) on the basis of the crite-rion for maximum decrease in the residual sum of the squares of the deviations for each successive step. Fisher's criterion is used as significance criterion.

APPENDIX 2

COEFFICIENTS IN THE REGRESSION DEPENDENCES FOR CALCULATION OF THE DIMENSIONLESS ADDED MASSES AND DAMPING COEFFICIENTS (Sample tables. A detailed description cai be found in [8]..)

Regression coefficients for calculation of sway damping term

B2

cy=3.7%

I

(1±0)

jO

-Regression c6efficiénts and powers.

id

formula (6) for pitch added mass calculátion

'i:

Regression coefficients for calculation of sway damping terni

B2

pi

0 2.0652 1.. 1 0.3735 (B/H)(2a/B)(T/H) 2 0.0038 (D/B)2(B/H)(2a/B)2(T/H)3 3 0.1189 (D/B)(2a/B)2(T/H)2 4 -0.1014 (2a/B)2(T/H) 5 -0.0066 ('2a/B)(T/H) 6 -0.0564 (D/B)2(B/H)(2a/B)2(T/H)2 7 0.7752 (D/B)(T/H). 8 -0.8332 (B/H) 9 -0.1788 (2a/B)(T/H)2 10 -0.0444 (D/B)(B/H)(2a/B)(T/H)2

a=4.5%

j coefficients aL , L = 1-7 powers ki k i a1 a2 a4 a5 a7

1234

-0.048 0.071 -0.012 -0.208 025 15.843 -13.641

0000

2 -12. 370 11.311 -2.029 -0.262 .031 -0.917 0.573

3 32

1 3 -4. 113 -5.543 0.895 0.189 .022 0.397

-0.793 3022

4 2.954 8.854 -1 .259 -0.184 .022 0.065 -0.47 1

3122

5 -0.003 -0.027 0.004 -0.143 .016 5.703 -3.724 1 3 0 1 6 8.858 -7.085 1 .604 -0.4 12 .052 4.934 -7.4 14

2022

7 0.531 -0.364 0.070 -0.310 .041 -0.732 0.264

3,2

0 1 8 -0.00 1 0.00 1 -0.00 1 -0.211 .025 -23.360

17.500 0400

9 68.190 -47.732 8.095 -0.629 .092 -6. 166

7.120 0201

10 0.112 0.246 -0.079 -0.126 .012 -2.012 1 .471

1022

11 1 .642 -1 .498 0.273 -0.190 .021 0.542

-0.213 2321

12 0.013 0.004 -0.00 1 -0.333 .043 -1.277

0.503 0221

13 6.586 -5.624 0.979 -0.157 .015 1.782

-1 .744 3311

14 -0.006 0.006 -0.001 -0.2 14 .024 -0.229

-0.533 2420

15 -0.109 -0.124 0.040 -0.211 .025 -1.700 i . 158 0 0 1 2 16 -3.538 3.150 -0.508 -0.228 .026 1:. 1 14

-1 .940 3010

17 0.220 -0.198 0.032 -0.308 .037 -1.331 0.68 1

3320

18 -13.071 9.141 -2.060 -0.354 .043 0.046

-1.168 2122

pi R1 O 0.1946 1 -0.1970 T/H 2 -0.0183 (DIB)(T/H)3 -3 0.0005 B/H )C1/H )3 4 0.0243 5 0.0416 (D/B )

2)3

(T/H)2 * 6 0.0055 (B/H)(2a/B) 7 -0.0029 (D/B)(B/H)(2a/B)(T/H)