numerical analysis of stability of a stationary or rotating circular cylindrical shell containing...

International Journal of Mechanical Sciences 68 (2013) 258–269

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International Journal of Mechanical Sciences

0020-74

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n Corr

E-m

mvp@ic

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Numerical analysis of stability of a stationary or rotating circular cylindricalshell containing axially flowing and rotating fluid

S.A. Bochkarev n, V.P. Matveenko

Institute of Continuous Media Mechanics RAS, Acad. Korolev Str. 1, Perm 614013, Russia

a r t i c l e i n f o

Article history:

Received 4 September 2012

Received in revised form

15 January 2013

Accepted 18 January 2013Available online 29 January 2013

Keywords:

Classical shell theory

Cylindrical shell

Potential fluid

Combined flow

Finite-element method

Stability

03/$ - see front matter & 2013 Elsevier Ltd. A

x.doi.org/10.1016/j.ijmecsci.2013.01.024

esponding author. Tel.: þ7 3422378308.

ail addresses: [email protected] (S.A. Bochk

mm.ru (V.P. Matveenko).

a b s t r a c t

The stability of stationary or rotating cylindrical shells interacting with a rotating internal fluid flow is

studied. The paper presents the results of the finite element solutions for shells having different linear

dimensions and subjected to various boundary conditions. It has been found that the form of stability

loss in the stationary and rotating shells under the action of the fluid flow, having both the axial and

circumferential components, depends on the type of the boundary conditions specified at their ends. It

has been shown that for different variants of boundary conditions and different geometrical dimensions

rotation of the fluid in a stationary shell or co-rotation of the shell and the fluid may increase or

decrease the critical velocity of the axial fluid flow.

& 2013 Elsevier Ltd. All rights reserved.

1. Introduction

Under field conditions, shells of revolution as an integral partof some technical applications may interact with an axial androtational flow of a fluid or gas. Taken alone, the axial flow of thefluid or its rotation exerts a destabilizing effect on the elasticstructure. Their combined action would thus be expected to causemore noticeable changes in the dynamic behavior of the structureand will require a comprehensive analysis. The background of theproblem, including a review of relevant analytical studies andsome examples of practical applications, in which a flowing androtating fluid is an essential part of engineering systems, isoutlined in [1].

Thus far, there have been a few studies dealing with thedynamic behavior of elastic stationary or rotating shells subjectedto a rotational gas flow [2–7]. Natural vibrations and stability ofrotating cylindrical shells conveying liquid or gas rotating withthe same angular velocity are considered in [2,3]. In [2] theinvestigation is carried out using the linear Donnell theory ofshells and Euler’s equations of motion for non-viscous incom-pressible liquid. It has been found that rotation of the shell andthe fluid leads to a decrease in the critical flow velocity asopposed to the case where their rotation is neglected. In [3], the

ll rights reserved.

arev),

aerodynamic pressure is defined in the framework of the linear-ized potential theory. The results of this study show that the flowof gas exerts inessential effect on the natural vibrations of therotating shell conveying co-rotating gas.

The analysis of a stationary shell subject to a rotating fluid inthe presence or absence of the axial velocity component has beenmade in [4,5]. In [4], based on the linear Sanders theory of shellsthe authors analyzed the propagation of harmonic waves in thin-walled circular cylindrical shells made of orthotropic and iso-tropic materials. The equations of fluid motion are written in theform adopted in [2]. It has been shown that for the examinedconfigurations the impact of the fluid flow on the natural vibra-tion frequencies of shells with a fluid is much stronger than thatof a mere rotation. A similar approach has been used in [5]. Here,the results of the analysis indicate that rotation of the fluid causesan excessive reduction in the critical fluid velocity compared tothe case when the fluid rotation is neglected.

The analysis of the dynamic behavior of cylindrical shells ofrevolution, interacting with an annular flow of compressible gas,which has both the axial and tangential velocity components, iscarried out in [6,7]. Here the analytical methods are employed toexamine the behavior of infinitely long cylindrical shells placed ina circular channel with a rigid external wall. In [6], the numericalresults are presented for a particular case of the shell in anannular gas flow, the velocity of which has a tangential compo-nent only. It has been shown that the coalescence of the forwardand backward propagating waves may cause the loss of stabilityin the form of the traveling wave flutter. In [7], the numerical

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z

x

v

uw

r

h

s

U f

s

Fig. 1. Computational scheme.

S.A. Bochkarev, V.P. Matveenko / International Journal of Mechanical Sciences 68 (2013) 258–269 259

calculations for the flow having both velocity components weremade without taking into account the mutual influence of theaxial and tangential components on the system stability. Notethat the authors of [6] offered a general problem formulation,which admits rotation of the inner shell. Nevertheless, they do notinvestigate the influence of the shell rotation on the dynamicbehavior of the system. The results of some experimental studiesdealing with the analysis of the cylindrical shell stability underthe action of an annular swirl flow are presented in [8,9].

The focus of recent paper [10] is the finite element analysis ofthe natural vibrations and stability of stationary and rotatingcircular cylindrical shells of finite length subjected to the internalflow of non-viscous compressible fluid. Here, it has been foundthat regardless of the type of the boundary conditions the loss ofstability in stationary shells occurs in the form of a flutter,manifesting itself as a coalescence of forward and backwardpropagating waves, whereas in the case of co-rotating shells theallowance made for the initial circumferential tension caused bythe centrifugal forces guards against the loss of stability of theshell and the fluid rotating with the same angular velocity.

In paper [11] the authors have cast doubt on the validity of theresults obtained in [2], which were later brought under closeexamination in paper [1]. It was found that in the framework ofthe analytical solution proposed in [2], it would be mathemati-cally impossible to determine the occurrence of instability in thesystem, the behavior of which is simulated taking into accountthe destabilizing effect of both the axial flow and the rotating flowinside the rotating shell. Based on these considerations theauthors try to find plausible physical explanation of the arisingmathematical complications suggesting that the loss of stabilityin such cases is impossible even though it seems to be physicallyreasonable. The analysis of this situation is being continued inrecently published paper [12]. Here, it was shown that at certainvalues of the angular velocities of the rotating shell and co-rotating fluid and the velocity of the axial flow there may bezones, in which the fluid is in the stagnant state. However, in thiswork no attempts have been made to provide an explanation ofhow this phenomenon affects the stability of the shell conveyingfluid. In [1] the authors also report the failure of the attempt toreproduce the results of work [5].

Natural vibrations of hollow rotating shells were investigatedmore extensively [13–22], for which purpose various numerical–analytical and numerical methods have been used including thefinite element method. These studies have a dual purpose: todemonstrate the possibilities of the applied methods and toinvestigate the dependence of vibrations of rotating single- andmulti-layer shells on such factors as geometrical dimensions,various boundary conditions, centrifugal and Coriolis forcesinclusive of initial circumferential stresses caused by rotation.Some of the aspects of the above investigations were examined indetail in monograph [23].

The effect of the internal fluid (gas) flow with an axial velocitycomponent on the stability of the cylindrical shell has beenstudied more comprehensively. The background of these investi-gations and soundly reasoned analytical and numerical researchdone in this field are also discussed in [1]. Here, we will outlineonly some of the numerous numerical studies [24–34], which arebased on the finite element method.

In [24], open shells of revolution are analyzed using a hybridfinite element, in which the exact functions of displacement aredefined directly by the equations of the Sanders shell theory.Here, as in the analytical studies, an exact expression for pressureof the flowing incompressible fluid written in terms of the Besselfunction has been derived in the framework of the potentialtheory. A similar approach for computation of the hydrodynamicpressure was used in works [25–27]. In work [25], the emphasis

was placed on studying the orthotropic preloaded cylindricalshells, whereas the other two works are concerned with theanalysis of the conical shells. Compressibility of the flowing fluidis taken into account in work [28], which offers the finite elementalgorithm, in which a preloaded cylindrical shell of revolution isdescribed in the framework, a three-dimensional theory ofelasticity. The hydrodynamic pressure is determined from Euler’sequation meeting specially stated dynamic boundary conditions,which take into account the fluid flow. In [29–32], the authorsalso use the potential theory for description of the fluid flow.Here, the finite-element algorithms are proposed for studyingmulti-layer, viscoelastic composite cylindrical and conical shellsconveying a compressible fluid. The equation for the perturbationvelocity potential is derived by applying the Bubnov–Galerkinmethod. The numerical investigations made in [29] yield therelationship between the fluid velocity and the number of the lowfrequency harmonic independently of the geometrical parametersof the shell and boundary conditions. In [32], it was shown thatfor simply supported or clamped–clamped shells, an increase ofthe cone opening angle may lead to a change in the mode ofstability loss from that of divergence to a flutter. Differentvariants of the hybrid technique based on the boundary integralequation method for calculating the effect of the shell–fluidinteraction and the finite element method for determining thedynamic properties of elastic structures are discussed in [33,34]for the case of incompressible fluid.

Papers [2–7] mentioned above are generally concerned withinfinitely long shells, so the influence of the boundary conditionsfor stationary and rotating shells conveying fluid with the axialand circumferential components of velocity is still poorly under-stood. In this paper, the numerical solution of the examinedproblem is found based on the potential theory, which is used todescribe a compressible fluid flow. On the one hand, practicalapplication of this theory is seldom the case, yet on the otherhand it essentially simplifies the problem solution allowingimplementation of intensive numerical experiments, which willpave the way toward further research, based on more accuratemodels for description of the fluid behavior.

From the above discussion it is clear that this study has beenundertaken to clarify two very important issues: to verify thepossibility of the stability loss in the systems consisting of astationary or rotating shell of a finite length subjected to axiallyflowing and rotating fluid and to estimate the influence of variousboundary conditions specified at the ends of the shells on thecharacter of its dynamic behavior.

2. Constitutive relations

Let us consider an elastic cylindrical shell of length L, radius R

and thickness h (Fig. 1). The shell is either stationary or rotatesabout its longitudinal axis with an angular velocity Os. The shell

S.A. Bochkarev, V.P. Matveenko / International Journal of Mechanical Sciences 68 (2013) 258–269260

conveys a perfectly compressible fluid, which moves with theaxial velocity U and simultaneously rotates with the angularvelocity Of , which in the general case is equal to an angularvelocity of the rotating shell. It is necessary to find such acombination of the axial fluid velocity and the angular velocityof the shell or/and fluid, at which the shell loses its stability.

To describe the motion of the rotating fluid in the volume Vf

we introduce in our consideration the perturbation velocitypotential f, which in the cylindrical coordinate system ðr,y,xÞ inthe case of small perturbations is described by the wave equation[35]:

r2f�

1

c2

@

@tþU

@

@x

� �2

f¼2Of

cM@2f@x@y

þ1

c

@2f@y@t

� �

þO2

f

c2

@2f@y2�r

@f@r

� �, ð1Þ

where c is the velocity of sound in a liquid, M¼U/c is the Machnumber. The pressure exerted by a fluid Pf on the elastic structureis defined by the linearized Bernoulli formula:

Pf ¼�rf

@f@tþU

@f@sþOf

@f@y

� �on the surface Ss ¼ Sf \ Ss: ð2Þ

Here rf is the specific density of the fluid; s is the meridionalcoordinate of the shell; Sf and Ss are the surfaces, which bound thefluid and flow volumes, respectively. The shell–fluid interface Ssmust satisfy the impermeability condition:

@f@r¼@w

@tþU

@w

@sþOf

@w

@y, ð3Þ

where w is the normal component of the shell displacementvector. The perturbation velocity potential at the shell entranceand exit must meet the following boundary conditions:

x¼ 0 : f¼ 0, x¼ L : @f=@x¼ 0: ð4Þ

Application of the Bubnov–Galerkin method to the partialdifferential equation for the perturbation velocity potential (1)satisfying boundary conditions (3) and (4) yields the system ofequations [32]:

Xmf

l ¼ 1

ZVf

@Fl

@r

@Fk

@rþ

1

r2

@Fl

@y@Fk

@yþ½1�M2

�@Fl

@x

@Fk

@x

�"

þ2Of M

c

@2Fl

@x@yFkþ

O2f

c2

@2Fl

@y2Fk�r

@Fl

@rFk

� �!dV

#fal

þXmf

l ¼ 1

ZVf

2U

c2

@Fl

@xFkþ

2Of

c2

@Fl

@yFk

� �dV _falþ

ZVf

FlFk

c2dV €fal

" #

�Xms

p ¼ 1

ZSs

NpFk dS _wapþ

ZSs

U@Np

@sFkþ

"Of

@Np

@yFk

!dSwap

#¼ 0,

k¼ 1,mf : ð5Þ

Here, mf and ms are the numbers of the finite elements used todiscretize the fluid Vf and shell Vs domains; fal and wap are thenodal values of the fluid perturbation velocity potential and shelldisplacements; F and N are the shape functions for the perturba-tion velocity potential and normal component of the displace-ment vector of the shell.

In the classical theory of shells based on the Kirchoff–Lovehypothesis the components of the deformation vector in thecurvilinear coordinates ðs,y,zÞ can be written as [36]

e11 ¼ E11þzk11, e22 ¼ E22þzk22, e12 ¼ E12þzk12,

where

E11 ¼ e1þ1=2y21, k11 ¼ k1 ð132Þ,

E12 ¼ e12þy1y2, k12 ¼ 2t ð6Þ

and

e1 ¼@u

@s, e2 ¼

1

R

@v

@yþw

� �, e12 ¼

@v

@sþ

1

R

@u

@y, y1 ¼�

@w

@s,

y2 ¼1

Rv�

@w

@y

� �, k1 ¼�

@2w

@s2 1, k2 ¼

1

R2

@v

@y�@2w

@y2

� �,

t¼ 1

R

@v

@s�@2w

@s@y

� �,

where u and v are the meridional and circumferential compo-nents of the displacement vector, yi are the angles of rotation ofthe non-deformable normal.

The components of the shell deformations (6) written in thematrix form are

e¼ enþ1=2Ee,

where e¼ fE11,E22,E12,k11,k22,k12tgT, en ¼ fe1,e2,e12,k1,k2,2tgT isthe linear part of the strain, e¼ f0,0,0,0,y1,y2,0,0,0gT, E is thematrix of linear factors, which has non-zero components E15 ¼ y1,E16 ¼ y2, E35 ¼ y2, E36 ¼ y1.

The physical relations between the vector of the generalizedforces and moments T and the strain vector en are represented inthe matrix form:

T ¼ fT11,T22,T12,M11,M22,M12gT ¼Den:

For isotropic materials the non-zero components of the stiff-ness matrix D are defined in terms of the elasticity modulus E,Poisson’s ratio n and shear modulus G:

nD11 ¼D12 ¼D21 ¼ nD22 ¼ nEh=ð1�n2Þ, D33h2=12¼D66 ¼ Gh3=12,

nD44 ¼D45 ¼D54 ¼ nD55 ¼ nEh2=12=ð1�n2Þ:

The strain energy of the shell U supplemented with theadditional terms, taking into account the preload force, isexpressed by

2U ¼

ZSs

eTnDen dSþ

ZSs

eTr0e dS: ð7Þ

The matrix of the initial forces and moments r0, the elements ofwhich are determined from the condition ETDe0

n¼ r0e, has the

non-zero values s055 ¼ T11, s0

66 ¼ T22.The kinetic energy T of the shell, rotating with the angular

velocity Os is given by [21]

2T ¼

ZVs

rsð _u2þð _v�ROs�OswÞ

2þð _wþvOsÞ

2Þ dV , ð8Þ

where rs is the specific density of the shell material.The work W done by the hydrodynamic forces is expressed by

W ¼

ZSs

Pf w dS: ð9Þ

3. Numerical implementation

For numerical implementation of the proposed model we usedthe semi-analytical version of the finite element method based onthe Fourier series expansion in terms of the circumferentialcoordinate y:

u¼X1j ¼ 0

u_

j cos jyþX1j ¼ 0

u^

j sin jy,

v¼X1j ¼ 0

v_

j sin jy�X1j ¼ 0

v^

j cos jy,

w¼X1j ¼ 0

w_

j cos jyþX1j ¼ 0

w^

j sin jy,


fa ¼X1j ¼ 0

f_

j cos jyþX1j ¼ 0

f^

j sin jy: ð10Þ

Here, j is the number of the harmonic.Expressing in Eq. (10) the symmetric ðu

_

j,v_

j,w_

j,f_

jÞ and antisym-metric ðu

^

j,v^

j,w^

j,f^

jÞ unknowns in terms of the nodal displace-ments, we obtain the following expressions for the shell and fluidfinite elements:

U ¼ fu,v,wgT ¼Nde ¼ ½N_

N^

�fd_

ed^

egT, ð11Þ

/a ¼ F/e ¼ ½F_

F^

�ff_

ef^

egT: ð12Þ

Here, N and F are the matrices of the shape function, de and /e arethe vectors of the nodal unknowns.

For the shell we used the finite element in the form of atruncated cone with a linear polynomial approximation of themeridional and circumferential components and cubic polyno-mial approximation of the normal component. For the fluid weused a triangle finite element with a linear approximation of theperturbation velocity component.

Using Eq. (11) we obtain

en ¼ Bde, e¼Gde, ð13Þ

where B and G are the matrices relating the strains en and e to thenodal displacements.

The equation of motion is derived based on Lagrange’s equation:

d

dt

@L

@ _dT

!�@L

@dT¼Q : ð14Þ

Here, d and _d are the generalized displacements and velocities,L ¼ T�U is the Lagrange function, Q ¼ @W=@dT are the generalizedforces. Using Eqs. (7)–(9) and taking into account Eqs. (11)–(13) weobtain from Eq. (14) the system of equations, which in the matrixform can be written as

ðKsþKgs�Kc

sÞdþMs€dþCo

s_dþrf ðCsf

_/aþAsf /aþAosf /aÞ ¼ 0: ð15Þ

Here: Ks ¼P

ms

RSs

BTDB dS is the stiffness matrix; Ms ¼P

ms

RSsr0

NTN dS is the mass matrix; Kgs ¼

Pms

RSs

GTr0G dS is the matrix of

the geometrical stiffness; Kcs ¼

Pms

RSsr0O

2s NTX1N dS is the matrix

of centrifugal force effect; Cos ¼

Pms

RSs

2r0OsNTX2N dS is the

matrix of Coriolis force effect; Csf ¼P

ms

RSs

NTF dS; Asf ¼

Pms

RSs

U

NT@F=@s dS; Ao

sf ¼P

ms

RSsOf N

T@F=@y dS; r0 ¼

Rhrs dz; O1 23 ¼�1,

O1 32 ¼ 1; O2 22 ¼O2 33 ¼ 1; matrices and vectors with a super-script ‘‘0’’ are determined by solving the axisymmetric static

problem Ksd¼ F , where f ¼ f0 0 r0RO2s g

T.

Using Eqs. (11) and (12) Eq. (5) can be written in a matrix formas

ðKf þKof þAc

f þAcof Þ/aþMf

€/a�ðCcf þCo

f Þ_/a

�Cfs _wa�ðAfsþAofs Þwa ¼ 0, ð16Þ

where

Kf ¼Xmf

ZVf

@FT

@r

@F

@rþ

1

r2

@FT

@y@F

@yþ@FT

@x

@F

@x

!dV , Mf ¼

Xmf

ZVf

FTF

c2dV ,

Kof ¼

Xmf

ZVf

O2f

c2

@2FT

@y2F�r

@FT

@rF

" #dV , Co

f ¼�Xmf

ZVf

2Of

c2

@FT

@yF dV ,

Acf ¼�

Xmf

ZVf

M2 @FT

@s

@F

@sdV , Ao

f ¼Xmf

ZVf

2OfM

c

@2FT

@x@yF dS,

Ccf ¼

Xmf

ZVf

2U

c2

@FT

@xF dV , Cfs ¼�

Xms

ZSs

FTN dS,

Aofs ¼

Xms

ZSs

Of@N

T

@yF dS, Afs ¼�

Xms

ZSs

UFT @N

@sdS:

Thus, the analysis of rotating shells conveying a rotating fluidreduces to a simultaneous solution of two systems of Eqs. (15) and(16). A combined system of equations takes the following form:

ðKþAÞfd /agTþMf €d €/ag

TþCf _d _/agT ¼ 0, ð17Þ

where

K¼KsþKg

s�Kcs 0

0 �rf ðKf þKof Þ

" #, M¼

Ms 0

0 �rf Mf

" #,

C¼ rf

Cos Csf

Cfs Ccf þCo

f

" #, A¼ rf

0 Asf þAosf

AfsþAofs Ac

f þAof

24

35:

Expressing the perturbed motion of the shell and the fluid by

d¼ q expðinltÞ, /a ¼/ expðinltÞ,

where q and / are the functions of the coordinates, in ¼ffiffiffiffiffiffiffi�1p

, andl¼ l1þ inl2 is the characteristic coefficient, we finally arrive at

ðK�l2Mþ inlCþAÞfq /gT ¼ 0: ð18Þ

The problem solution reduces to a computation and analysis ofthe eigenvalues l of the system (18). The complex eigenvalues arecalculated based on the Muller method [37]. The computationalefficiency of the algorithm can be increased by renumbering thedegree of freedom based on the Cathill–McKee algorithm [38].

4. Numerical results

The numerical simulations have been carried out for a cylindricalshell (E¼200 GPa, n¼ 0:29, rs ¼ 7812 kg=m3, R¼1 m, h¼0.01 m),conveying axially flowing and rotating fluid ðrf ¼ 103 kg=m3Þ. Forthe selected parameters the critical velocity, at which the systemloses stability will be essentially lower than the sound velocity in aliquid. On the one hand, for the examined configurations such asimplified version of the problem formulation has insignificanteffect on the critical velocities leading to a loss of stability [32],but on the other hand, it simplifies to some extent the analysis ofthe calculated eigenvalues. All the results of numerical investigationpresented below were obtained for 40 shell finite elements and1000 fluid finite elements. Such a size of the finite element meshprovides a comfortable computation speed for solving a spectralproblem with an appropriate accuracy [32].

In the numerical examples discussed below we will considercylindrical shells simply supported (v¼w¼0; SS) or rigidlyclamped (u¼ v¼w¼ @w=@s¼ 0; CC) at both ends and cantilev-ered shells (CF).

4.1. Testing of the numerical algorithm

To test the algorithm proposed for the problem of a hollowrotating shell we compared the obtained results with the resultspresented in [18]. To this end, we considered a cylindrical shellsubjected to various combinations of boundary conditions and thefollowing system parameters: L/R¼5, R/h¼500, j¼1. The numberof half waves in the meridional direction m is equal to one. Fig. 2ashows the variation of the dimensionless frequency o¼ RlDðD¼ ½rsð1�n2Þ=E�0:5Þ with the angular velocity of shell rotationOs ðrev=sÞ.

The results obtained in the framework of the developed algo-rithm agree well with the results from [18] (denoted in Fig. 2a bycircles) almost for a full range of the examined angular velocitieswith only a few exceptions. Under the boundary conditions CC one

0 15 30 45

0

0.8

1.6

2.4

0 1.9 3.8 5.70

5

10

15

0

0.5

1

1.5

m = 1

m = 2

m = 3

DU

m = 8

m = 6

m = 7

SFU

1

21 2

U

-0.4

Fig. 3. Real and imaginary parts (a) and loci (b) of dimensionless eigenvalues o versus dimensionless axial velocity of the fluid U : (a) shell simply supported at both ends;

(b) cantilevered shell.

0 60 120 180 240

s

0

1

2

3

4

0 3 6 9 120

1

2

3

4

m= 1

m= 2

m= 3

m = 4

SSCC

CF

D

Fig. 2. Dimensionless natural frequencies o (a) and real parts of eigenvalues X (b) versus angular velocity of the shell Os (rev/s) and dimensionless axial velocity of the

fluid flow L: solid and dashed lines—the results of computation; circles—Ref. [18]; dash-dotted lines—Ref. [39].


of the modes reaches zero at the angular velocity which is 9.3%higher than the velocity found in [18].

In the next example we consider a steel cylindrical clamped–clamped shell conveying an incompressible fluid ðrf ¼ 103 kg=m3Þ.The computations were made for the following parameters: L/R¼25.9, h/R¼0.0227, rf =rs ¼ 0:1282, j¼2, n¼ 0:3. Fig. 2b showsvariation of the first four dimensionless eigenvalues X¼ ReðRl=U0Þ � 10 with the dimensionless fluid velocity L¼U=U0 � 102

ðU0 ¼ ½1=D� ¼ 5387Þ.In Fig. 2b, solid lines denote the results obtained in the present

work and dash-dotted lines—the results of numerical–analyticalinvestigations [39]. A rather good agreement between our resultsand the results of study [39], in which the authors used the firstfour terms in the Galerkin series expansion, is observed only forthe first two eigenvalues. We have found that the loss of stabilityin the form of divergence occurs for the first mode (m¼1) atLD ¼ 6:586, whereas in [39] this happens at the velocity of 6.4.

4.2. Shell conveying fluid: modes of stability loss

Our prime interest here is with the analysis of limiting cases,which will allow us to estimate the influence of both the axial and

circumferential components of the fluid velocity on the dynamicbehavior of the stationary and rotating shells. The results ofnumerical computations are presented for L/R¼4 and j¼4. It iswell known [1] that the axial fluid flow, as well as its circumfer-ential rotation, exerts a destabilizing effect on the elastic body.However, in this case, the loss of stability is governed by differentmechanisms and may be of static nature (by divergence) and ofdynamic nature (by flutter), which have two forms.

In the case when the fluid flow has only axial velocity compo-nent, the form of the stability loss depends on the type of theboundary conditions [1]. Here we present the results of computa-tion, which lend support to this statement. The shells clamped orsimply supported at both ends lose stability in the form ofdivergence. This can be illustrated by Fig. 3a, in which the real o1

and imaginary o2 parts of the first three dimensionless eigenvalues(o¼ RlD� 102, o1 ¼ ReðoÞ, o2 ¼ ImðoÞ) are plotted as a functionof the dimensionless axial fluid velocity U ¼UD� 102. Here and inwhat follows, dash-dotted lines correspond to the imaginary part ofeigenvalues. When the fluid velocity UD approaches the value of4.183, the real part of the first mode vanishes and there appears acouple of imaginary parts, equal in magnitude but opposite in sign,which corresponds to the loss of stability in the form of divergence.


Shells clamped at the end where the flow enters the shell andfree at the other end lose stability in the form of a single modeflutter. As an illustration, we refer to Fig. 3b, which shows the lociof three dimensionless eigenvalues in a complex plane, whichwere obtained at different values of the dimensionless axial fluidvelocity. A distinguishing feature of this configuration is thatbeginning with the velocity U 47 the variations in the imaginaryparts of the eigenvalues are essentially non-monotonic. AtUSF ¼ 9:048 the imaginary part of the seventh mode becomesnegative, which corresponds to the loss of stability in the form ofa single-mode flutter.

In the case of angular rotation of the shell or fluid the naturalfrequency splits into two values, which corresponds to theappearance of the forward and backward propagating waves.Fig. 4 shows the real and imaginary parts of the first twodimensionless eigenvalues o as a function of the dimensionlessangular velocity of the fluid Of ¼ ROfD� 102 for shells undervarious boundary conditions. An increase in the rotational velo-city of the fluid causes an increase in the eigenvalues correspond-ing to the forward propagating waves (solid line) and a decreasein the eigenvalues corresponding to the backward propagatingwaves (dashed line). At a certain value of the rotational velocitythe real part of the backward wave of the first mode becomes zeroand begins to grow with a further increase of the rotationalvelocity. The real parts of both waves of the first mode coalesce at

0.60 1.2 1.80

3

6

9

0

1

2

3

m = 1

m = 2

CF 21

f

Fig. 4. Real o1 and imaginary o2 parts of dimensionless eigenvalues against the dimen

(b) cantilevered shell.

0 0.6 1.2 1.80

4

8

12

m = 1

m = 2

1

Fig. 5. Real parts of dimensionless eigenvalues o against the dimensionless rotation

supported at both ends; (b) cantilevered shell.

the rotational velocity OF . This results in the appearance of theimaginary parts equal in magnitude but opposite in sign, whichcorresponds to the loss of stability in the form of a two-mode(coupled-mode) flutter. In this case, the loss of stability of shellsunder different boundary conditions follows the same scenario.

In Fig. 5 the real parts of the first two dimensionless eigenva-lues o are plotted against the dimensionless rotational velocity Ofor the variant of numerical simulation, in which the shell and thefluid inside it rotates with the same angular velocity O ¼O f ¼Os.In this case, the loss of stability does not occur. As shown in [10],in the shell containing a co-rotating fluid a stabilizing effect isexerted by the initial circumferential tension caused by thecentrifugal forces.

In our simulations we used such angular velocities of the shell,at which the value of the normal deviation due to the action of thecentrifugal force lies within the limit of the shell thickness orslightly exceeds it.

Under a combined action of the axial flow and angular rotationof the fluid inside the stationary shell or the rotating shell, whoserotational velocity is equal to the angular velocity of the fluid, theloss of stability of the shell occurs in the form specified by theboundary conditions.

Fig. 6 shows the plots of dimensionless eigenvalues of astationary or a rotating shell simply supported at both ends. Here,as before, dashed lines denote the real parts of the eigenvalues,

0 0.6 1.2 1.80

3

6

9

0

1

2

3

m = 1

m = 2

CF

1 2

f

sionless angular velocity of the fluid O f : (a) shell simply supported at both ends;

0 0.6 1.2 1.80

4

8

12

m = 1

m = 2

1

al speed of the shell containing a co-rotating fluid O ¼O f ¼Os: (a) shell simply

0 1.4 2.8 4.20

3.5

7

10.5

0

0.5

1

1.5

0 1.9 3.8 5.70

5

10

15

0

0.7

1.4

2.1

m = 2

m = 1

m = 1

m = 2

m = 2

m = 1

m = 1

m =2

U

1 2

U

1 2

CFU

CFU

Fig. 6. Real o1 and imaginary o2 parts of the dimensionless eigenvalues versus the dimensionless axial fluid velocity U for shells simply supported at both ends:

(a) Os ¼ 0, O f ¼ 0:946; (b) Os ¼O f ¼ 1:89.

0 4 8 12 16

0

1.5

3

4.5

0 3 6 9 12

0

1.5

3

4.5

m = 3

m = 2

m = 1

m = 1

m = 2

m = 3m = 3

m = 2

m = 1

m = 1

m = 2

m = 3

-0.75-0.75

1

2

1

2

SFU SFU

Fig. 7. Loci of dimensionless eigenvalues o plotted against the dimensionless fluid velocity U for a cantilevered shell: (a) Os ¼ 0, O f ¼ 0:946; (b) Os ¼O f ¼ 1:89.

Table 1

The values of dimensionless critical axial velocity U and angular velocities O f of the fluid rotation for shells with various boundary conditions and different numbers of the

harmonic j for L/R¼4.

Variants of boundary

conditions

Velocity Number of the harmonic j

1 2 3 4 5 6 7

SS U 9.2007 6.9441 4.2919 4.1832 4.6801 5.1399 5.5567

O f28.0534 6.2379 2.3388 1.5853 1.7161 2.0654 2.4731

CC U 9.2972 9.1124 6.7735 5.4637 5.3771 5.4731 5.7311

O f33.1576 9.4823 3.9895 2.3079 1.9739 2.1569 2.5087

CF U 10.2486 10.2688 10.3151 9.0481 8.5113 8.5588 8.8472

O f11.7165 2.3923 1.1575 1.2775 1.6317 2.0309 2.4517


corresponding to backward waves, solid lines stand for forwardwaves and dash-dotted line stand for imaginary parts of theeigenvalues. Compared to the variant presented in Fig. 4a rotationof the fluid leads to a change in the form of stability loss, which inthe present case occurs in the form of a coupled-mode flutter.A similar scenario is observed in the case of clamped–clampedshells. It should be noted that in the case when the shell and thefluid rotate simultaneously, variation of the imaginary parts occursnon-monotonically and has a small region where they decrease.

For cantilevered shells the angular rotation of the fluid in astationary or rotating shell has no effect on the form of stabilityloss. This statement can be illustrated by Fig. 7, where eigenvalue

loci plotted versus the axial fluid flow. Here the loss of stabilityoccurs due to the appearance of a negative imaginary part of thethird-mode eigenvalues at the dimensionless axial velocity of thefluid flow USF ¼ 6:07 in the case of a stationary shell and thedimensionless axial velocity USF ¼ 6:04 in the case when the shelland the fluid rotate simultaneously.

4.3. Stability analysis of a stationary shell containing axially flowing

and rotating fluid

The values of the dimensionless critical axial velocities ofthe fluid flow U and the dimensionless angular velocity of the


fluid Of in relation to the number of the harmonic j are given inTable 1 for different variants of boundary conditions. The analysisof these results shows that compared to the critical axial velo-cities the difference in the critical angular velocities of the fluid inshells subjected to different boundary conditions is more pro-nounced for lower harmonics and disappears almost completelyat higher harmonics. What is also interesting, the cantileveredshells containing a rotating fluid are less stable than in the case of

Table 2

The values of dimensionless critical axial velocities U and angular velocities of the

fluid rotation O f for shells with various boundary conditions and linear dimen-

sions for j¼4.

Variants of

boundary

conditions

Velocity Ratio of the shell length to its radius L/R

1 2 4 6 8 10

SS U 5.7892 4.7854 4.1832 3.5235 3.5808 3.2133

O f3.7625 2.4303 1.5853 1.3025 1.0054 0.9032

CC U 6.9004 6.0992 5.3771 5.0154 4.5713 4.3623

O f4.4898 2.9698 1.9739 1.5532 1.3417 1.1516

CF U 9.9978 9.1364 8.5113 8.2253 7.9346 7.8835

O f2.7203 1.8147 1.1575 0.9032 0.7618 0.5894

Table 3

The values of the dimensionless critical axial velocities U at different numbers of th

dimensionless angular velocities of the fluid rotation O f for L/R¼4.

O fVariants of boundary

conditions

Number of the harmonic j

1 2

0 SS 9.2007 6.9441

CF 10.2485 10.2688

0.1891 SS 9.2007 6.9409

CF 10.2449 10.2506

0.3783 SS 9.2007 6.9310

CF 10.2436 10.2535

0.5674 SS 9.2007 6.9146

CF 10.2425 10.2563

0.7566 SS 9.2005 6.8917

CF 10.2412 10.2591

0.9457 SS 9.2003 6.8619

CF 10.2399 10.2332

1.1348 SS 9.2001 6.8255

1.3239 SS 9.1999 6.7823

1.5131 SS 9.1996 6.7321

Table 4

The values of the dimensionless critical angular velocity of the fluid O f at different num

different dimensionless axial velocities of the fluid flow U for L/R¼4.

U Variants

of boundary conditions


1 2

0 SS 28.0534 6.2379

CF 11.7093 2.3923

0.0019 CF 10.8465 2.1843

0.9457 SS 27.9952 6.1785

1.8914 SS 27.8205 5.9967

CF 10.8596 2.2152

2.8371 SS 27.5258 5.6817

3.7828 SS 27.1063 5.2123

CF 10.9178 2.3379

5.6742 CF 11.0153 2.6715

7.5656 CF 11.1531 4.1439

their interaction with an axial fluid flow when they are lesssubject to instability.

Table 2 shows the values of the dimensionless critical axialvelocity U and dimensionless angular velocity of the fluid rotationOf at different ratios of the shell length to its radius L/R. Theresults displayed in the table suggest that for the examinedvariants of boundary conditions an increase in the shell lengthcauses a decrease in the critical velocities.

In simulation of the combined effect of both velocity compo-nents the stability boundary was defined through assigning afixed value to one of the velocities and searching for a criticalvalue of the other velocity.

Table 3 presents the values of the dimensionless criticalvelocities of the axial fluid flow U in relation to different numbersof the harmonic j obtained for shells under various boundaryconditions at different values of the dimensionless angularvelocity of the fluid rotation Of . The data listed in the tableclearly demonstrate that for shells conveying fluid with the axialvelocity component the inclusion of the fluid rotation intoconsideration has a destabilizing effect, on the system no matterwhat the type of the boundary conditions is used, and withincreasing angular velocity the stability boundary decreases.

The values of the dimensionless critical angular velocity of thefluid Of in relation to the number of the harmonic j, obtained atdifferent values of the dimensionless axial flow velocity U under

e harmonic j for shells under various boundary conditions, obtained at different

3 4 5 6 7

4.2919 4.1832 4.6801 5.1399 5.5567

10.3151 9.0481 8.5113 8.5588 8.8472

4.2772 4.1492 4.6638 5.1274 5.5467

10.3219 8.9263 8.1151 8.1215 8.4447

4.2324 4.0459 4.6146 5.0901 5.5155

10.3004 8.3201 7.5622 7.6995 8.0693

4.1571 3.8707 4.5307 5.0249 5.4622

10.2924 7.7965 7.1841 7.2775 7.6535

4.04987 3.6181 4.4077 4.9231 5.3827

10.2841 7.2753 6.3180 6.8255 7.2089

3.9084 3.2767 4.2320 4.7591 5.2683

10.0431 6.1522 5.8336 6.0109 6.7222

3.7293 2.8229 3.8655 4.5030 5.1043

3.5082 2.1999 3.2027 4.1593 4.8851

3.2368 1.1819 2.3139 3.7051 4.6044

bers of the harmonic j for shells under various boundary conditions, obtained at

3 4 5 6 7

2.3388 1.5853 1.7161 2.0654 2.4731

1.1575 1.2775 1.6317 2.0309 2.4517

1.0303 1.1064 1.3762 1.6685 1.9644

2.2782 1.5389 1.6828 2.0274 2.4232

2.0892 1.3952 1.5794 1.9193 2.2924

1.0399 1.0981 1.3643 1.6578 1.9525

1.7339 1.1302 1.4094 1.7469 2.1106

1.0826 0.6405 1.1623 1.4843 1.8503

1.1872 1.1658 1.1932 1.3821 1.6745

1.6673 1.2597 1.0874 1.0909 1.2573

2.0880 0.6394 0.3743 0.4837 0.6049


various boundary conditions are listed in Table 4. For shells simplysupported at both ends conveying a rotating fluid an allowancemade for the axial component of the fluid exerts a destabilizingeffect and with increasing axial fluid velocity the stability boundarydecreases. For cantilevered shells such unambiguous dependence is

0 1.3 2.6 3.9 5.2 6.5 7.8 9.1

U

0

0.5

1

1.5

2

2.5

f

stable

flutter

CF

CC

SS

Fig. 8. Stability diagrams for shells under various boundary conditions subjected

to a combined action of the dimensionless axial flow U and dimensionless

rotational flow O f of the fluid: j¼4, L/R¼4.

0 0.5 1 1.5 20

0.25

0.5

0.75

1

stable

flutter

CF

SSCC

Fig. 9. The total critical flow velocity x versus the ratio of the angular to axial

velocity of the flow z for shells under various boundary conditions under the

combined action of the axial flow and rotational flow of the fluid: j¼4, L/R¼4.

0 2 4 6

U

0

1.3

2.6

3.9

stable

flutterL /R = 2

4

610

Fig. 10. Stability diagram at j¼4 for shells simply supported at both ends (a) and cant

action of the dimensionless axial flow U and dimensionless rotational flow O f .

not observed. It should be noted that taking account of even aminimal axial velocity U ¼ 0:0019 leads to the appearance of newcritical values of the angular velocity for all numbers of theharmonics considered in this study. Moreover, for lower harmonicsat certain values of the axial fluid velocity one can observe anincrease of the stability boundary.

A detailed analysis of stability carried out for different num-bers of the harmonic, for example for j¼4, allows us to concludethat in the presence of the axial and angular components of thefluid velocity the character of the stability boundary stronglydepends on the type of the boundary conditions and lineardimensions of the system.

For shells simply supported or clamped at both ends thestability boundary does not depend on the exhaustive search ofthe values of the angular and axial velocities (Fig. 8). Forcantilevered shells the variant of the exhaustive search, in whichthe axial velocity is fixed and a search is made for the value of theangular velocity, at which the system loses stability, is shown inFig. 8 by a solid line. Another variant, in which the angularvelocity has a fixed value and a search is made for the axialvelocity leading to the loss of stability is denoted by a dashed line.A possible reason for the difference in the critical velocities is thatfor shells under such boundary conditions the existence of theaxial flow velocity may lead to an increase in the critical angularvelocity of the fluid flow. These results suggest that for cantilev-ered shells the axial fluid flow within some regimes exerts astabilizing effect, whereas for shells under other boundary con-ditions it has only a destabilizing effect.

The effect of a dramatic variation in the dimensionless criticalangular velocity Of for cantilevered shells due to the appearanceof the axial velocity component is readily illustrated by thefollowing results of computation. In the absence of the axial fluidflow the critical angular velocity is equal to 1.277, and due to theappearance of even a minimal axial flow it reduces to 1.106.

To gain a better understanding of the effect of one velocitycomponent on the other, we refer to Fig. 9 where the results ofcomputations are represented in the form of the dependence ofthe total critical flow velocity x¼ ðU2

þR2O2Þ1=2=ðU9O ¼ 0Þ on the

ratio of the angular velocity to the axial velocity of the fluidz¼ RO=U. The use of the dimensionless quantities x and z wasfirst proposed in [5]. Note that the quantity z is inverselyproportional to the Rossby number Ro¼U=ðROÞ, which is one ofthe main units of measurement used in the dynamics of rotatingfluids [1]. From the results presented in Fig. 9 for cantilevered

0 4 8 12

U

0

1

2

3

L/R = 2

4

610stable

flutter

ilevered shells (b) having different linear dimensions L/R subjected to a combined


shells we can draw a conclusion that for the examined variants ofthe boundary conditions an increase in the angular velocity of thefluid is crucial for reducing the total critical flow velocity.

Fig. 10 shows the stability diagrams for shells with differentlinear dimensions L/R and various boundary conditions, which aresubjected to a combined action of the axial and rotational fluidflow. The results presented in the diagrams allow us to estimatethe influence of the geometrical dimensions of shells on thestability boundary. In particular, for cantilevered shells the extentof the stabilizing effect of the axial flow velocity essentiallydepends on the linear dimensions of the system—the smallerthe dimensions, the higher the stability boundary. With increas-ing L/R the stabilizing effect vanishes. It is quite possible that forthe examined variants of the boundary conditions such a behaviorof cantilevered shells under a combined action of both velocitycomponents is attributed to the hydrodynamic damping, whichalways exists in the system even at minimal velocities of theaxial flow.

4.4. Stability analysis of a rotating shell containing axially flowing

and co-rotating fluid

The results of numerical simulation show that in the casewhen the shell and the fluid rotate simultaneously with the sameangular velocity the loss of stability does not occur (Fig. 5).

Table 5

The values of the dimensionless critical axial velocities of the fluid U at different num

rotation O ¼Os ¼O f for shells under various boundary conditions for L/R¼4.

O Variants of boundary

conditions


1 2

0 SS 9.2007 6.9441

CF 10.2485 10.2688

0.9457 SS 9.2266 6.9464

CF 10.2393 10.2332

1.8914 SS 9.2249 6.9532

CF 10.2317 10.2190

2.8371 SS 9.2223 6.9645

CF 10.2232 10.2058

3.7828 SS 9.1926 6.9802

CF 10.2141 10.1945

4.7285 SS 9.1879 7.0003

CF 10.2043 10.1854

0 3 6 90

3.5

7

10.5

SSIII

SSI

SSIVSSII

flutter

U

Fig. 11. Stability diagram for a shell simply supported at both ends (a) and cantilev

dimensionless angular rotation of the shell and fluid O: j¼4, L/R¼4.

However the appearance of the axial flow velocity may causethe loss of stability in the form of a single or two-mode flutterdepending on the type of the boundary conditions. Table 5 liststhe values of the critical values of the dimensionless axial flow U

for different numbers of the harmonic j and dimensionlessangular velocities of the shell and fluid rotation O. The computa-tions were made for L/R¼4 and two variants of the boundaryconditions. From these results it follows that for cantileveredshells an increase in the velocity of simultaneous rotation of theshell and the fluid leads to a decrease in the critical velocity of theaxial fluid flow, whereas for shells simply supported at both endsthis dependence is of inverse character.

A more comprehensive analysis can be made based on the datashown in Fig. 11, in which we present the stability boundaryobtained for shells simply supported at both ends and cantilev-ered shells under a combined action of the dimensionless axialflow U and simultaneous dimensionless angular rotation of theshell and the fluid O. Here we consider different scenarios, inwhich the initial circumferential tension and fluid rotation iseither neglected or taken into account. Identification of thesevariant using Roman numerals was made according to Table 6. Inthe case of simultaneous rotation of the shell and the fluid andwith account for the initial circumferential tension an increase inthe angular velocity of the system including shells simply sup-ported at both ends leads to an increase in the critical value of the

bers of the harmonic j and dimensionless angular velocities of the shell and fluid

3 4 5 6 7

4.2919 4.1832 4.6801 5.1399 5.5567

10.3151 9.0481 8.5113 8.5588 8.8472

4.3319 4.2649 4.6952 5.1444 5.5594

10.2798 7.4595 6.7905 6.9193 7.2768

4.4457 4.4607 4.7442 5.1595 5.5679

9.8194 6.0358 5.7118 5.6795 6.1711

4.6180 4.6856 4.8348 5.1919 5.5844

8.6573 4.6173 4.4368 4.7472 5.0719

4.8289 4.8938 4.9655 5.3211 5.6165

7.8465 3.7680 4.2715 4.0050 4.8444

5.0589 5.0799 5.1777 5.5972 5.6857

5.8267 3.4489 3.4639 3.5776 4.0682

0 3 6 90

4

8

12

CFIII

CFII

CFIV

CFI

U

ered shell (b) under a combined action of the dimensionless axial flow U and

Table 6Variants of computation of the stability boundary in the system under the

combined action of the axial fluid flow and simultaneous angular rotation of the

shell and the fluid.

Variant number I II III IV

Fluid rotation Of a0 Of a0 Of ¼ 0 Of ¼ 0

Initial circumferential tension Kgs a0 Kg

s ¼ 0 Kgs a0 Kg

s ¼ 0


axial flow velocity (curve SSI), and for cantilevered shells—to adecrease in the critical axial velocity (curve CFI). The results forshells simply supported at both ends do not qualitatively agreewith the data reported in [2] for infinitely long shells. This can beattributed to the fact that in [2] as well as in a number of otherworks the authors neglect the initial circumferential tensioncaused by the centrifugal forces. This statement is supported bythe results of numerical simulations, which do not take intoaccount the initial circumferential tension (curves SSII and CFII).In this case as with work [2], an increase in the angular velocity ofthe shell and fluid rotation leads to an abrupt decrease in the axialvelocity of the flow no matter what the type of the boundaryconditions.

Fig. 11 presents the results of numerical simulations in theabsence of fluid rotation. In this case, for both types of theboundary conditions rotation of the shell alone has a stabilizingeffect and hence rotation of the fluid has a destabilizing effectboth for a stationary and a rotating shell (Fig. 4).

The analysis of the dependence of the total critical flowvelocity x on the quantity inversely proportional to the Rossbynumber allows us to conclude that the influence of the angularvelocity of the shell and fluid rotation on the total critical velocityof the fluid flow is most pronounced in cantilevered shells.

Numerical calculations for shells having other linear dimen-sions, in particular for L/R¼10, show that an increase in the shelllength does not lead to qualitative changes in the results.

5. Conclusion

In this study, we have analyzed the stability of stationary orrotating elastic circular cylindrical shells interacting with a fluidflow having both the axial and circumferential components ofvelocity. A mathematical formulation of the problem and itsnumerical implementation based on the finite element algorithmhave been considered. The proposed numerical algorithm hasbeen used to investigate the stability of shells with differentlinear dimensions subjected to various boundary conditions.

It has been shown that for a stationary shell containing arotating fluid and for a rotating shell with a co-rotating fluid theaxial fluid flow leads to the loss of stability, the form of whichdepends on the type of the boundary conditions prescribed at theends of the shell.

In the case of stationary shells, the fluid flow having the axialand circumferential velocity components has an essential effecton the stability boundary, which manifests itself in a decrease ofcritical velocities, at which the loss of stability occurs. It should benoted, however, that for short cantilevered shells the axial fluidflow may lead to an increase in the critical angular velocities ofthe fluid rotation.

In the case of rotating shells conveying a co-rotating fluid thecharacter of the dynamic behavior is specified by the type of theboundary conditions, and, accordingly, an increase of the angularvelocity may cause a growth or decrease of the critical axialvelocities leading to the onset of instability. It has been found thatin the context of a simplified formulation, which ignores the

initial circumferential tension caused by the centrifugal forces, forshells under various boundary conditions an increase in theangular velocity leads to an abrupt decrease in the criticalvelocities of the axial flow.

The results obtained in the present study qualitatively agreewith the results of works [2,5] for infinitely long shells and settledoubts expressed by the authors of [1,11,12] as to a possibility ofstability loss in shells conveying a rotating fluid.

Acknowledgments

The work was supported by the Russian Foundation for theFundamental Research (Grant 09-01-00520).

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numerical analysis of stability of a stationary or rotating circular cylindrical shell containing...

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