Second-order forced waves in an eccentrically-rotating circular cylinder

Second-order forced waves in an eccentrically-rotatingcircular cylinder

A.N. Williamsa, *, J.H. Vazquezb

aDepartment of Civil & Environmental Engineering, University of Houston, Houston, TX 77204-4791, USAbDepartment of Maritime Systems Engineering, Texas A&M University, Galveston, TX 77553-1675, USA

Received 22 May 1995; accepted 19 March 1998

Abstract

The free-surface motion induced by the constant rotation of a ¯uid-®lled circular cylinder about avertical axis o�set from its geometrical center is investigated theoretically. Assuming potential ¯ow, andusing a perturbation parameter de®ned in terms of the axis o�set, the theoretical analysis includes termsup to second-order in this parameter. The velocity potentials at ®rst- and second-order are expressed aseigenfunction expansions involving unknown coe�cients which are subsequently determined through theboundary conditions. Numerical results are presented that illustrate the in¯uence of the various problemparameters on the free-surface elevation and the induced velocity ®eld in the ¯uid at both ®rst- andsecond-order. # 1998 Elsevier Science Ltd. All rights reserved.

1. Introduction

Since cylindrical structures containing ¯uid appear in a wide range of di�erent engineeringapplications, the dynamics of ¯uid-®lled cylinders have been the subject of many diversestudies. Although a wide range of literature exists on applications where viscosity hasimportant in¯uence on the dynamics of the entrained ¯uid, the following discussion isrestricted to situations where the dynamics of the ¯uid may reasonably be predicted byconsidering it to be inviscid.Several investigators have studied the e�ects of base excitation on the dynamic response of

¯uid-®lled cylindrical containers. The e�ects induced by the lateral motion of ¯uid storagetanks has received considerable attention from the seismic community for many years.Although the earliest studies considered the tank containing the ¯uid as rigid [1], later

International Journal of Engineering Science 37 (1999) 395±406

0020-7225/98/$19.00 # 1998 Elsevier Science Ltd. All rights reserved.PII: S0020-7225(98)00072-X

PERGAMON

* Corresponding author.

investigations included tank ¯exibility e�ects and developed semi-analytical methods based oneigenfunction expansion techniques for the prediction of ¯uid±tank interactions [2±4].A nonlinear analysis of waves propagating around the free surface of water contained in a

circular basin of ®nite uniform depth has been presented by Bryant [5]. He found that resonantwaves may be generated not only by external forcing near the basin's natural frequency, butalso from the internal forcing of some wave components by others. Standing radial cross-wavesin an annular wave tank were investigated by Becker and Miles [6]. They found that thesecond-order Stokes-wave expansion for deep-water, standing gravity waves may becomesingular for circular containers.Despite the large amount of published work which exists on the forced ¯uid motions inside a

circular cylinder, the authors do not know of any analysis which addresses the topic of thepresent paper, namely the free-surface motions induced by the constant rotation of a ¯uid-®lledcylindrical structure about a vertical axis o�set from its geometrical center.

2. Theoretical development

The geometry of the problem is shown in Fig. 1. A circular cylinder of radius R, ®lled with¯uid to a depth h, is subjected to a constant angular velocity o about a vertical axis o�set by Efrom that through its geometrical center. Cylindrical coordinates (r, y, z) are de®ned with thez-axis pointing vertically upwards from the origin O at the still water level. If the geometricalcenter of the cylindrical cross-section at time t is at (xc, yc) then the equation describing thecylinder surface is (x ÿ xc)

2+( y ÿ yc)2=R 2. If the time origin is de®ned such that (xc,

yc) = (ÿE sin ot, E cos ot), then with (x, y) = (r cos y, r sin y) the instantaneous cylindersurface is

r � F�y; t� � E sin�yÿ ot� � �R2 ÿ E2 cos2�yÿ ot��1=2: �1�For E<<R this equation may be written as

r � R� E sin�yÿ ot� ÿ �E2=2R�cos2�yÿ ot� �O�E3� �2�The ¯uid is assumed inviscid and incompressible and its motion irrotational. Therefore, itsmotion may be described in terms of a velocity potential f(r, y, z; t) where the ¯uid velocityvector q = Hf. This velocity potential is required to satisfy Laplace's equation in the region of¯ow, i.e.

r2f � 0 r � R; 0 � y � 2p; ÿh � z � Z�r; y; t�; �3�and is subject to boundary conditions on the cylinder bottom, cylinder sides and ¯uid free-surface, namely

fz � 0 on z � ÿh; �4�

fr � Ft � 1

r2Fyfy on r � F�y; t�; �5�

A.N. Williams, J.H. Vazquez / International Journal of Engineering Science 37 (1999) 395±406396

Zt � Zrfr �1

r2Zyfy ÿ fz � 0 on z � Z�r; y; t�; �6a�

gZ� ft �1

2�rf�2 � 0 on z � Z�r; y; t�; �6b�

in which z = Z(r, y; t) denotes the instantaneous free-surface, g is the acceleration due togravity, and subscripts denote partial derivatives.The velocity potential and free-surface elevations are now expressed as perturbation series in

terms of the eccentricity parameter E, namely

f�r; y; z; t� � Ef�1��r; y; z; t� � E2f�2��r; y; z; t� � . . . ; �7a�

Z�r; y; t� � EZ�1��r; y; t� � E2Z�2��r; y; t� � . . . ; �7b�Substituting the above series forms into the governing equation and boundary conditions,expanding the free-surface and cylinder wall conditions about z = 0 and r = R, respectively,and gathering terms of like order in E, replaces the above nonlinear problem by a series oflinear problems, one at each order of E.

Fig. 1. De®nition sketch.

A.N. Williams, J.H. Vazquez / International Journal of Engineering Science 37 (1999) 395±406 397

The ®rst-order, O[E], problem is

r2f�1� � 0 r � R; 0 � y � 2p; ÿh � z � 0; �8�

f�1�z � 0 on z � ÿh; �9�

f�1�r � ÿo cos�yÿ ot� on r � R; �10�

Z�1�t ÿ f�1�z � 0 on z � 0; �11a�

gZ�1� � f�1�t � 0 on z � 0: �11b�A suitable form for f (1) which satis®es Eqs. (8)±(11) is given by

f�1��r; y; z; t� � A0J1�k0r�cosh k0�z� h� �X1n�1

AnI1�knr�cos kn�z� h�( )

cos�yÿ ot�; �12�

in which J1 and I1 are the Bessel function and modi®ed Bessel function of the ®rst kind,respectively. The wavenumber k 0 satis®es the linear dispersion relation o 2=gk0 tanh k0h, andthe k n n = 1, 2, . . . , are the positive real roots of o 2+gk n tan k nh = 0. The coe�cients An,n = 0, 1, 2, . . . , are determined from the cylinder wall boundary condition, Eq. (10), utilizingthe orthogonality properties of the vertical eigenfunctions.The second-order, O[E 2], problem is

r2f�2� � 0 r � R; 0 � y � 2p; ÿh � z � 0; �13�

f�2�z � 0 on z � ÿh; �14�

f�2�r � ÿo2R

sin 2�yÿ ot� ÿ sin�yÿ ot�f�1�rr �1

R2cos�yÿ ot�f�1�y on r � R; �15�

Z�2�t ÿ f�2�z � ÿf�1�r Z�1�r ÿ1

r2f�1�y Z�1�y � Z�1�f�1�zz on z � 0 �16a�

gZ�2� � f�2�t � ÿZ�1�f�1�zt ÿ1

2�rf�1��2 on z � 0; �16b�

with a combined free-surface boundary condition given by

f�2�tt � gf�2�z �1

gf�1�t �f�1�tt � gf�1�z �z ÿ ��rf�1��2�t on z � 0: �17�

The solution to the second-order problem is obtained by decomposing f (2) into twocomponents. Each component is required to satisfy only one inhomogeneous boundarycondition, either on the free-surface or the cylinder wall, and is required to satisfy a


homogeneous condition on the other boundary. Therefore, f 2=G (2)+c (2) such that

G�2�tt � gG�2�z � 0 on z � 0; �18a�

C�2�tt � gC�2�z �1

gf�1�t �f�1�tt � gf�1�z �z ÿ ��rf�1��2�t on z � 0; �18b�

while

G�2�r � ÿo2R

sin 2�yÿ ot� ÿ sin�yÿ ot�f�1�rr �1

R2cos�yÿ ot�f�1�y on r � R; �19a�

C�2�r � 0 on � R: �19b�A suitable form for G (2)ÿ is given by

G�2��r; y; z; t� � B0J2�m0r�coshm0�z� h� �X1n�1

BnI2�mnr�cos mn�z� h�( )

sin 2�yÿ ot�; �20�

in which J2 and I2 are the Bessel function and modi®ed Bessel function of the ®rst kind,respectively. The wavenumber m 0 satis®es a modi®ed (free-wave) dispersion relation 4o 2=gm0tanh m0h, and the m n n = 1, 2, . . . , are the positive real roots of 4o 2+gm n tan k nh = 0. Thecoe�cients Bn, n = 0, 1, 2, . . . , are determined from the cylinder wall boundary condition,Eq. (19)a), utilizing the orthogonality properties of the vertical eigenfunctions in a mannersimilar to that used in the solution to the ®rst-order problem.A suitable form for c (2) is given by

C�2��r; y; z; t� �X1n�1

CnJ2�lnr�cosh ln�z� h�sin 2�yÿ ot�; �21�

where, in order to satisfy the homogeneous cylinder wall boundary condition, Eq. (19)b), thewavenumbers l n n = 1, 2, . . . , are de®ned as the roots of J

02(l nR) = 0. The coe�cients Cn

are obtained from Eq. (18)b) by substituting the ®rst-order potential and its derivatives intothe right-hand side and utilizing the orthogonality properties of the Bessel functions over theinterval 0 E r E R.Once the velocity potentials at ®rst- and second-order have been determined, various

quantities of engineering interest may be found. The surface elevation in the cylinder may beobtained from the dynamic free-surface boundary conditions, Eq. (11b) and (16b), while thevarious velocity components may be obtained by direct di�erentiation of the appropriatepotentials.

3. Numerical results and discussion

A computer program has been developed to implement the above theory. Numerical testinghas determined that, for the range of cylinder dimensions and excitation frequencies of interest,


the in®nite series for the velocity potentials at ®rst- and second-order may be truncated after 50terms. Increasing the number of terms beyond this value did not change the computed resultsby more than 1±2%, indicating that convergence had essentially been achieved.Fig. 2 shows the maximum values of the dimensionless ®rst-order and second-order

instantaneous free-surface elevations in the cylinder, namely EZ (1)(r, y; 0)/R and E 2Z (2)(r, y; 0)/R, as a function of koR for the case h/R = 2, E/R = 0.025. It can be seen that the ¯uidexhibits resonant responses at both ®rst- and second-order throughout the frequency range of

Fig. 2. Variation of maximum ®rst-order (Ð) and second-order (± ±) instantaneous free-surface elevation in cylinderat t = 0 for h/R = 2, E/R = 0.025. Bottom ®gure shows same information as top ®gure but with an expandedvertical scale.


interest. The two di�erent scales in the ®gure allow detailed inspection of the ¯uid response atnon-resonant frequencies. The ®rst-order resonances occur at wavenumbers n n n = 1, 2, 3,. . . , related to the zeros of the ®rst derivative of the Bessel function of the ®rst kind of orderone, i.e. at wavenumbers n n which satisfy J

01(n nR) = 0. It can be seen that resonant

wavenumbers n1, and n2, occur within the frequency range of interest. The resonant behaviorobserved in the second-order free-surface elevation is derived from two sources. Firstly, thesecond-order surface elevation exhibits a resonant response at wavenumbers corresponding tothose at which the ®rst-order solution becomes resonant, i.e. at wavenumbers n n n = 1, 2,these will be termed `®rst-order' resonances. However, the second-order solution also exhibits aresonant response at the wavenumbers l n n = 1, 2, 3, . . . , de®ned as the roots ofJ02(l nR) = 0. These are true `second-order' resonances, and it is noted that the ®rst-order

solution is ®nite at these locations. In the present case, the ®rst seven wavenumbers l n appearwithin the frequency range of interest.Fig. 3 shows the ®rst-order, second-order and total (®rst- plus second-order) instantaneous

free-surface elevations in the cylinder at t = 0 at koR = 2, with h/R = 2, and E/R = 0.025.Both surface and contour plots are shown. It should be noted that the second-order free-surface elevation consists of both oscillatory and steady terms. At this frequency, the amplitudeof the oscillatory component of second-order free-surface elevation is found to be roughlyequal to that of the ®rst-order elevation (107%), while the mean second-order elevation isapproximately 7% of the ®rst-order amplitude. The total free-surface elevation has a maximumvalue which is a 103% increase over the ®rst-order (i.e. the total is approximately twice the®rst-order estimate) and a minimum value which is an increase of 48% over the ®rst-orderquantity. Therefore, it can be seen that the e�ect of the second-order terms is to amplify boththe crests and the troughs in the surface pro®le. Fig. 4 presents the instantaneous (t = 0) totalhorizontal and vertical ¯uid particle velocity components at z = 0 for the same data used togenerate Fig. 3. The horizontal velocity is shown as a vector plot in which the lengths of thearrows are proportional to the magnitude of the velocity. The variation of the vertical velocitycomponent in the cylinder is presented as a contour plot. Examination of the vector plot inFig. 4 indicates that, although symmetric about the x-axis, the ¯ow pattern is not circular. Itcan be seen that there are three pairs of circulation cells in the ¯ow which are approximatelyelliptical in shape. The locations of these cells are indicated by the dashed lines in the ®gure.At the instant shown, the ¯uid particles are migrating from the second quadrant to the ®rst,while travelling around the circulation cells. Finally, it is noted that there are three stagnationpoints on the x-axis, at x = ÿ 0.2R, 0.4R and x = 0.6R, approximately. The locations of thestagnation points are denoted by the closed circles in the ®gure. The vertical velocity is notedto be antisymmetric about the x-axis. Although not shown explicitly herein, it has been foundthat at this frequency the vertical velocity is dominated by the second-order component.Fig. 5 shows the ®rst-order, second-order and total (®rst- plus second-order) instantaneous

free-surface elevations in the cylinder at t = 0 at koR = 4, with h/R = 2, and E/R = 0.025.Again, both surface and contour plots are shown. At this higher frequency, the amplitude ofthe oscillatory component of second-order free-surface elevation is found to be approximately75% of the ®rst-order elevation, while the mean second-order elevation is approximately 2% ofthe ®rst-order amplitude. The total free-surface elevation has a maximum value which is a 50%increase over the ®rst-order estimate and a minimum value which is an increase of 18% over


Fig. 3. First-order, second-order and total (®rst- plus second-order) instantaneous free-surface pro®les forkoR = 2.0, h/R = 2.0, and E/R = 0.025.


the ®rst-order quantity. Fig. 6 presents the instantaneous (t = 0) total horizontal and vertical¯uid particle velocity components at z = 0 for the same data used to generate Fig. 5. At thishigher frequency, it can be seen that the induced ¯uid velocity ®eld is more complex than thatfor koR = 2 and, due to the shorter wavelengths, a clear ¯ow pattern cannot be easilydiscerned from the ®gure. It can be seen, however, that from y = 458 to 1808 the horizontalvelocity is almost tangential to the cylinder wall at this instant. Also, a pair of `kidney-shaped'circulation cells are observed in the ¯ow. Again, the approximate locations of these cells areindicated by the dashed lines. Near the cylinder wall, rapid changes in the direction of thehorizontal velocity are observed. Finally, at this frequency, a stagnation point occurs atx = ÿ 0.2R, approximately. Again, this location is marked by a closed circle in the ®gure. Asfar as the vertical velocity is concerned, the anti-symmetry about the x-axis is noted. Also, thevertical velocity gradient in the cylinder is greater at this frequency than in the previous casebecause of the smaller wavelengths involved.

Fig. 4. Total (®rst- plus second-order) instantaneous horizontal and vertical liquid particle velocities at t = 0, at

z = 0, for koR = 2.0, h/R = 2.0, and E/R = 0.025.


Fig. 5. First-order, second-order and total (®rst- plus second-order) instantaneous free-surface pro®les at t = 0, forkoR = 4.0, h/R = 2.0, and E/R = 0.025.


To place the results shown in Figs. 3±6 into context, for a cylinder of radius R = 1 m,depth h = 2 m, having an o�set E = 2.5 cm, the range of surface elevations and vertical andhorizontal velocities will now be examined. For a dimensionless frequency koR = 2, thecorresponding tangential velocity on the cylinder wall would be vt=4.43 m/s. The range of thefree-surface elevation is ÿ47 cm±47 cm at ®rst-order, ÿ45 cm±61 cm for the second-order, andÿ70 cm±1 m for the combined ®rst- plus second-order. The associated range for the horizontalvelocity is 0.3 m/s±3.75 m/s, while the vertical velocity range is ÿ4.6 m/s±4.6 m/s. Beforediscussing the ranges for the second case presented, it should be noted that koR = 2 is nearone of the resonant frequencies associated with this system (see Fig. 2). The second case, atkoR = 4, corresponds to a tangential velocity vt=6.26 m/s to ®rst-order. While this case hasalmost 50% higher rotational frequency than the ®rst case, it exhibits much lower ranges forthe free-surface elevation and vertical and horizontal velocities since this frequency is not closeto any of the resonant frequencies of the system. The free-surface range is ÿ9 cm±9 cm at the®rst order, ÿ7 cm±7 cm. at second-order, and ÿ11 cm±14 cm for the combined ®rst- plus

Fig. 6. Total (®rst- plus second-order) instantaneous horizontal and vertical liquid particle velocities at t = 0, atz = 0, for koR = 4.0, h/R = 2.0, and E/R = 0.025.


second-order. Finally, the horizontal velocity values range from 0±1.22 m/s while the verticalvelocity ranges from ÿ86 cm/s±86 cm/s. As intuitively expected, for those oscillationfrequencies away from resonance, the e�ect of the 2.5 cm. o�set on the ¯uid motion inside the1 m-radius cylinder is relatively small.

4. Conclusions

A theoretical solution has been developed to describe the free-surface motion induced by theconstant rotation of a ¯uid-®lled cylindrical structure about a vertical axis o�set from thatthrough its geometrical center. The ¯uid is assumed inviscid and the motion irrotational. Thetheoretical development is based on potential ¯ow theory. A perturbation parameter has beende®ned in terms of the axis o�set and the subsequent analysis included terms up to second-order in this parameter. The velocity potentials at ®rst- and second order were expressed aseigenfunction expansions and the unknown potential coe�cients were determined throughapplication of the cylinder wall and free-surface boundary conditions. The solution at second-order consists of two terms: one satis®es a homogeneous free-surface boundary condition but anon-homogeneous boundary condition on the cylinder wall; while the other satis®es ahomogeneous cylinder wall boundary condition and a non-homogenous free-surface boundarycondition at this order. These terms give rise to the body-forced and free-surface (locked)waves, respectively. Numerical results have been presented that illustrate the in¯uence of thevarious problem parameters on the free-surface elevation and the induced velocity ®eld in the¯uid at both ®rst- and second-order. It is found that the ¯uid motion inside the cylinder ischaracterized by a series of `®rst-order' and `second-order' resonant responses occurring at thezeros of the derivatives of the Bessel function of the ®rst kind of order one and two,respectively. A series of laboratory experiments are planned by the authors to con®rm thephenomena predicted by the theoretical model presented herein, these results will be reportedin due course.

References

[1] G.W. Housner, Dynamic pressure on accelerated ¯uid container, Bulletin Seismological Society of America 47(1957) 15±35.

[2] M.A. Haroun, G.W. Housner, Dynamic characteristics of liquid storage tanks, Journal of Engineering

Mechanics Division, ASCE 108 (1982) 783±800.[3] M.A. Haroun, G.W. Housner, Complications in free vibration analysis of tanks, Journal of Engineering

Mechanics Division, ASCE 108 (1982) 801±818.[4] M.A. Haroun, Vibration studies and tests of liquid storage tanks, Journal of Earthquake Engineering and

Structural Dynamics 11 (1983) 179±206.[5] P.J. Bryant, Nonlinear progressive free waves in a circular basin, Journal of Fluid Mechanics 205 (1989) 453±

467.

[6] J.M. Becker, J.W. Miles, Standing radial cross-waves, Journal of Fluid Mechanics 222 (1991) 471±499.[7] J.W. Miles, Resonantly forced, nonlinear gravity waves in a shallow rectangular tank, Wave Motion 7 (1985)

291±297.


Documents

Second-order forced waves in an eccentrically-rotating circular cylinder