262

EXPERIMENT ll5

263

264

266

266

267

268

269

270

27!

272

273 z O I a LQV:

I I +

274 OA

O IX

92

~1

t

+ i I + + I I

275

276

277

278

EXPERIMENT 116

279

280

281

282

283

284

285

286

287

288

289 C QK U'.I � 'M

C C

z z

V. C/3Q

~ ~

+ +

290

291 z zC 0

& CfJ

Mt 0

Pt M

K J!OC

~ ~

Cf.' C/.'t

O Q

I + + +

292

E iY P E R I H E i"1 T 1 1 7

293

294

295

296

297

298

299

300

301

302

303

304

305

4. REFERENCES

1. Bishop, R. E. D. and Hassan, A. Y., 1964, "The Li ft andDrag Forces on a Circular Cylinder in a Flowing Fluid",Proceedin s of the Ro al Societ, Series A, Vol. 277,pp. 32-50.

2. Dahlquist, G. and Bjorck, A,, 1974, Numerical Methods,Prentice Hall, New Jersey.

3. Dean, R. B., Milligan, R. W, and Wootton, L. R,, 1977,An Ex erimental Stud of Flow Induced Vibrations, AtkinsResearch and Development, Epsom, Surrey, England, ReportDecember !977/l.

4. Internationa't Mathematical and Statistical Library IMSL!,Reference Manual, 1 981, Edition 8, IMSL, Inc .

5. King, R., 1977, "A Review of Vortex Shedding Research andIts Application", Ocean En ineerin , Vol. 4, pp. 141-171,London: Pergamon Press.

6. Mercier, J. A., 1973, Lar e Am litude Oscillation of aCircular C linder in a Low S eed Stream, Ph.D. Thesis,Stevens Institute of echno ogy, Department of MechanicalEngineering, Ann Arbor, MI: University MicrofilmsOrder No. UM74-884.

7. Moeller, M. J. and Leehey, P,, 1982, "Measurements ofFluctuating Forces on an Oscillating Cylinder in a CrossFlow", Proceedin s of the Third International Conferenceon the Behavior of Offshore Structures, August 1982,Vol 2, pp. 681-689. New York: Hemisphere Publishing Co.

8. Patri kalakis, N. M., 1983, Theoretical and Ex erimentalProcedures for the Prediction of the 0 namic Behavior of

Marine Risers, Ph.D. Thesis, MIT, Department of OceanEngineering.

9. Sarpkaya, T., 1977a, Transverse Oscillations of a CircularC linder in Uniform Flow, Naval Postgraduate School,Report No. NPS-69SL 071, Monterey, CA.

10. Sarpkaya, T., 1977b, "In-Line and Transverse Forces onCylinders in Oscillatory Flow at High Reynolds Numbers",Journal of Shi Research, Vol. 21, No. 4, pp. 200-216.

306

11. Sarpkaya, T., 1979, "Vortex Induced Oscillations",J 1 f A 1' d M h ' , Vol, 46, pp. 241-258,

12. Staubli, T., 1983, "Calculation of the Vibration of anElastically Mounted Cylinder Using Experimental Datafrom Forced Oscillation", J 1 f Fl d EASNE Transactions, June 19 p

13. Toebes, G. H., 1969, "The Unsteady Flow and Wake Near anOscillating Cylinder", Journal of Basic En ineerinASME Transactions, September 969, pp. 93-505 andDecember, 1969, pp. 859-862.

14. Vandiver, J. K., 1983, "Drag Coefficients of LongFlexible Cylinders", Proceedings, 15th Off hConference, Houston, TX, Yo1. 1, P p

15. Veri ey, R. L. P. and Moe, G., 1979, The Forces on aC linder Oscillatin in a Current, Norwegian Institute ofTechnology, River and Harbour Laboratory, SINTEF ReportNo. STF60A79061.

307

APPENDIX A

Figure A-1. Rigid Cylinder Results

1.0

0.5

A X 0 ~ 0 0 2 0 6 1 0 1.4 1.8 2 0f /f

e s

Range of Synchronism of Vortex Formationwith Forced Oscillations Orthogonal to

a lJni form Stream, adapted from Mercier

lg73!.

309

Figure A 2 Rrgsd Cylinder Results

2. ~

= Am litude of Inertia Force Per UnitInertia Coefficient c> = mp i u e

2 Function of U Parametri call y v ithLength/oA ~ x+1 as a unc0

O f Sinusoid Oscillations Orthogonal to aRespect to x/O, or inu

Uni form Stream, Mercier �973!.

Nomen cl a ture:

p Density of the Fluid

A = mO /42

0 Cyl i nder Di arne terC i rcul a r Frequency o f Osc i 1 1 a ti on

x Amplitude of Oscillation

c = c + 1M m

310

Figure A-3: Rigid Cylinder Results

Drag Coefficient c = Amplitude of Drag Forced

Per Unit I.ength Parallel to the Oscillation/

0.5pD~ x as a Function of' U* Parametrically2 2

with Respect to x/0, for Sinusoid Oscillations

Orthogonal to a Uniform Stream, Mercier �973!.

311

Figure A-4: Rigid Cylinder Results

x/D

1,501.25

cD

0.20 0.30 5 f 0/U for x/D ~ 1.500 05' 00.20 0,30 5 for x/D 1,250.10

05 for x/D ~ 0.750

5 for x/00

0.10 0.20 0.30 5 for x/D ~ 1.030 0.10 0.20 0.30

0.50 0 0.10 0.20 0.30

0 ' 0.10 020 0 30S for x/D x 0,250

Average Drag Coefficient cD = Average Drag Per UnitLength/0.5pDV as a Function of S = 1/U* Para-2

c 0

metrically with Respect to x/D, for Sinusoid

Oscillations Orthogonal to a Uniform Stream, Mercier� 973! .

312

Fi gure A-5: Ri gid Cyl inder Resul ts

Cp Average Drag Coefficient, Lift Coefficient andStrouhal Number for a Fixed Rigid Smooth Circular

Cylinder in a Uniform Stream as a Function of

Reynolds Number, Hishop and Hassan �964!.

313

APPENDIX 8

314

PREDICTION OF THE RESPONSE OF A SPRING MOUNTED RIGIDCYLINDER IN A UNIFORM STREAM USING RIGID CYLINDER EXPERIMENTS

Let us consider a rigid cylinder of mass M, diameter 0 , and

length L, mounted on elastic springs of stiffness K, and dashpots

of coefficient c, and permitted to respond orthogonally to a

uniform stream. The basic assumption in the subsequent analysis

is that the response motion x t! is monochromatic, i.e.:

x t! = Asin <at!

with unknown amplitude, A, and circular frequency, ~=2-f. Under

this assumption, estimates of the overall hydrodynamic forceX

orthogonal to the stream F t!, can be made from corresponding

forced sinusoid motion rigid cylinder experiments using equation �!.

The displacement, x t!, obeys the following equation:

X ~Mx"" + cx" + Kx = F t!

tt

Awhere subscript t denotes derivative with respect to time.

Introducing equations �! and B.l ! in equation B.Z!, we find

the following two simultaneous nonlinear equations to determine

V*=2>'lt /~D and non-zero values of a=A/D* ' c e e'

316

A comparison of the results of our procedure with a spring

mounted rigid cylinder experiment is shown in Figure B-l. The

experimental data shown in Figure B-l are derived from Dean et al.

�977!. Information about our theoretical prediction is given in

Table B-1. The maximum calculated amplitude underpredicts the

maximum measured amplitude by less than 165. Possible explanation

for this difference is that in our prediction, Reynolds number and

aspect ratio were not scaled correctly. Another possible explana-

tion is that the values of c and cd from available rigid cylinder

experiments are not very accurate. For example, rigid cylinder

results from Mercier �973! and Sarpkaya �977a! show large dis-

crepancies. As noted in Sarpkaya �977a!, lack of resolution led

Nercier �973! to occasionally fair his experimental data in a

misleading way. Mercier's �973! data 'have been digitized and used

in our work because they extend to larger values of "a", which are

of particular interest for large amplitude forced oscillations of

flexible cylinders, Patrikalakis �983!. From Figure B-l we also

see that for U* > 5.75 approximately, we have been unable to find

a non-zero solution for "a" from equations B.3! and 8.4!. The most

probable explanation for this is that our assumption for the response,

see equation B.l!, is no longer valid.

An analysis similar to the one presented in this Appendix has

been published concurrently by Staubli �983! and corroborates our

findings.

E

Hz!

1.69

U*Un

Crn

3.75 4.78 0.23 3.46

4.75 5.29 1.93 0.63 1.95

5.00 5.31 2.02 0.66 1.51

5.25 S.32 2.12 0.69 1.11

5.50 5.34 2.22 0.71 0.77

4.00 5.02 1.72 0.29 3.25

4.25 5.07 1.80 0.37 2.67

4.50 5.15 1.88 0.45 2.21

-1.87

-1.46

-1.11

-0.86

-0.60

-D.SS

-0.50

-0.47

Table B-l: Information about the Prediction of the

Response of the Spring Mounted Cylinder

o f Fi gure B-1 .

318

F~gu~e 8-1: Comoarisons of Spring Mounted Rigid CylinderResponse with Theoretical Predictions

1,0

o,e

2.0

1.5

1.0 .04,0 6,0 7.05.0

Un8.03.0

Plots of the Measured Vortex "Shedding" Frequency, f, the Measured

and Calculated Response Frequency, f , and the Measured and Calculated

Non-Dimensional Response Half Amplitude, Y/D, for a Smooth Spring

Mounted Rigid Cylinder Oscillating Orthogonally to a Uniform Water

Stream. Measured Data are Derived from Dean, et al, �977 !. Model

Characteristics: f =2.15 Hz, D=25.4 mm, m=2.93, ~=13, 6=0.147,

1C' =0.91, Re=2680 ta 10370.

319

= 25M /oD L2

K

= V D/vc

Re

max mi nEnvelopes of Strouhal Number for a Fixed Rigid

Cylinder in a Uniform Stream, Derived from

Figure 2 of Chapter IV of Patrikalakis �983!.

's is t' he Frequency of Lift on a Fixed Rigid

Cylinder in a Uniform Stream, Corresponding

to a Strouhal Number, St=f D/V, equal to 0.2.s c'

is the Vortex "Shedding" Frequency Measured in

the Wake of the Spring Mounted Cylinder.

is the Frequency of Primary Lift Motion of the

Spring Mounted Rigid Cylinder.

NOMENCt ATURE FOR ADDITIONAL SYMBOI S SHOWN IN FIGURE B-1