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High-temperature ductility of large-grained TiC
Journal of Fluids and Structures (1988) 2, 245-261
ON THE DYNAMICS OF C U R V E D PIPES TRANSPORTING FLUID PART II: EXTENSIBLE THEORY A. K.
MISRA,M. P. PAIOOUSSISAND K. S. VAN
Department of Mechanical Engineering, McGill University, Montreal, Quebec, Canada H3A 2K6 (Received 16 June 1987 and in revised form 20 January 1988) The paper treats the effect of internal flow on the statics, dynamics and stability of extensible curved pipes. The cases of pipes clamped at both ends and pipes clamped at the upstream end but free to slide axially at the downstream end are considered. For pipes clamped at both ends, the stressed shape is symmetrical for inviscid flow but asymmetrical for viscous flow; there is also a significant difference between the steady-state combined tension and pressure force calculated for the two flows. The stability behaviour of a clamped-clamped curved pipe, however, does not depend on whether viscosity is taken into account or not; the natural frequencies do not change much with increasing flow velocity and the system thus never loses stability. This conclusion is consistent with that obtained in the first part of this work using a modified inextensible theory and with the predictions of other extensible theories. There is, however, a substantial change in the stability behaviour if the downstream end of the pipe is allowed to slide axially, even though it is still clamped as far as transverse displacement is concerned. The pipe then loses stability by flutter at sufficiently high flow velocities. 1. I N T R O D U C T I O N IN THE FIRSTPART of this work [1], the centreline of the pipe has been assumed to be inextensible, i.e., the centreline strain in the pipe is taken to be zero. Two variants of the inextensible theory were considered: in the first case, the steady-state forces induced by the pressure and centrifugal forces were neglected, while in the second, they were taken into account. There was significant difference in the results obtained from the two theories, at least for pipes supported at both ends. The pipes lose stability by buckling (divergence) at a sufficiently high flow velocity according to the first theory; however, when the steady-state forces are taken into account, in the second version of the inextensible theory, the pipes remain stable for all flow velocities. The objective of this paper is to present an analysis of the problem without assuming that the centreline of the pipe is inextensible. This analysis is called here the extensible theory. The results obtained from this theory are compared with the results yielded by the two inextensible theories, leading to the final recommendations as to the most appropriate theory for the problem at hand. This paper also presents results for a combination of boundary conditions not considered so far by any previous investigation, i.e., when one end is completely clamped, while the other end is clamped in terms of lateral constraint but free to slide axially. This configuration is of practical importance, since in many situations, in order to allow for thermal stress relief, it is not desirable to use totally clamped supports at the two ends of the pipe, using instead a sliding support. 0889-9746/88/030245 + 17 $03.00
(~) 1988 Academic Press Lmalted
246
n . K . MISRA E T A L .
2. G O V E R N I N G E Q U A T I O N S O F M O T I O N The system under consideration, as in Part I, consists of a curved pipe of length L, having a uniform cross-sectional area A,, mass per unit length Mr, flexural rigidity E1 and shear rigidity GJ (Figure 1). The pipe is initially in a plane having an arbitrary centreline shape. It conveys a stream of fluid of mass Mi per unit length. The flow is assumed to be a plug flow with constant velocity U. Strictly speaking, for an extensible pipe, the area of cross-section changes during the oscillations of the pipe. Hence, the flow velocity varies slightly with time. This variation however, is likely to have a small effect on the dynamics of the pipe and is neglected in this paper. The equations governing the motion of the combined pipe-fluid system, valid for both inextensible and extensible pipes, have been derived in the first part of this work [1]. In nondimensionalized form they are In~° + on;') + [ n ( n ~ + o n , ) ] ' + o n
"
iv
-
]/O/Xo + U- : ( n "i -Jr
073' -3I- 8 )
+ [2/31ati(0; + 003) + k01 + (1 +/3.)#, 1 = 0,
(1)
tokyo + C,2n'~] + [2/3~atiO; + zxO2 + (1 +/3a)#2] = 0,
(2)
.(,72 - ow") - A 0 ( V / ' + One) + ( n n ~ ) ' -
- n ' + o(n~' + on;) + o n ( n ; + ons) + rO~o+ a:o(n; + on,)] - [/Suzti(//; - 8//1) + zX//, + (1 +/~.)#,1 = 0, amed)centrehne
j~
The stramed centrehne
2"0
g
"0
t
x
g/ Figure 1. Kinematics of the system; definition of coordinate systems and coordinates used.
(3)
CURVED PIPES TRANSPORTING FLUID. PART II
247
[ o ( o ~ - ~ ) - A(W" + 0o~)l + [o~bl = o,
(4)
where th = u / L ,
~h=v/L,
"~- U L ( M f / E I ) 1/2, A=GJ/EI,
~13=w/L,
~a = M a / ( M f Ai"Mr),
¢r=IJ(Mr+M,)L
y = (Mr + My - A o p p ) g L 3 / E I , l i = (Aipi - AoPe - Qz)L2/E1,
2,
= t [ E I / ( M t + MI)L4] ~/2,
~=s/L,
~a m_l~la/(Mf ..~ Mt),
M=A,L2/L
= Md(M~ +
O=L/Ro,
M,), (5)
lip = ( A t p i -- A o p ¢ ) L 2 / E I , A = c L 2 / [ E I ( M f + Mr)] 1/2 and
F~ = A e / c .
In the above, u and v are transverse deformations of the pipe in and normal to the plane of the pipe, respectively; w is the longitudinal displacement, ~p is the twist of the tube; s is the distance of a point on the pipe measured along the centreline; M, and c are the added mass per unit length and the coefficient of viscous damping due to the surrounding fluid, associated with transverse motion; /14a and g play similar roles in longitudinal motion; p; and Pe are the pressure in the internal and external fluids, respectively; Qz is the axial force; 0Lxo, etc. are the direction cosines of the gravity acceleration vector. Prime and dot denote differentiation with respect to ~ and 3, respectively. Unlike the inextensible case, the longitudinal strain e = ( O w / a s ) - (u/Ro) --/:O. Thus, u and w, or equivalently ~/a and t/3, are not directly related to each other in the present case. Furthermore, the axial force Qz is given by e z = E A t e = E A t 3ss
;
(6)
hence, using equation (5), the dimensionless pressure-tension combined force 11 may be expressed as II = lip - M(t/; - 001 ).
(7)
Once rh and t/3 are known, YI can be determined from equation (7). It may be noted that this procedure can be used only for the extensible case, since the last term vanishes if the pipe is inextensible.
2.1. EQUATIONS GOVERNING THE STATIC EQUILIBRIUM OF THE PIPE As in the inextensible case, each of the displacements and twist ~/1, t/2, ~/3, and ~p can be imagined to consist of a steady or static part and a perturbation about the static part, i.e., ~1 ~__n l q_ ?]~ '
7j2_. ~ _l_ ?]~,
~j3 = ?]; ..1_/j~ '
~]) = /po q_ ~/),,
(8)
the superscripts ° and * denoting the steady and perturbed parts, respectively. The equations governing the steady part are obtained from equations (1)-(4) by deleting the time-dependent terms and are given by (t/~'v + 00~') + [II°0/~ ' + 0~7~)]' + 0II ° - yO~xo+ aZ(~lT ' + 0~/~' + 0) = 0,
(9)
0l~ iv - 0~p°") - A0(~0 °" + 0~/~") + (II%/~')' - 7¢~y0+ t/zr/~" = 0,
(10)
- n °' + o(,71" + o,7~") + o n ° ( n l ' + 0 7 ; ) + ~,O~zo+ a2o(,11 ' + o n ; ) = o,
(11)
0 ( 0 ~ , ° - r/~") - A O p ° " + 0r/~") = 0.
(12)
248
A.K. MISRA E T
AL,
It may be noted that equations (9) and (11) governing the static in-plane displacements are decoupled from the equations of out-of-plane static equilibrium (10) and (12). Furthermore, if the gravity effect is negligible (i.e., y = 0), or if the pipe initially lies in a vertical plane (i.e., 0% = 0), then the static out-of-plane deformations r/~ and ~po vanish. All the cases considered in this work satisfy this requirement and hence, ~/~ and ~p° are always zero. Using equation (7), the in-plane static equilibrium equations (9) and (11) can be rewritten after linearization as (,7~~ + 0,~") + [n~(,7~' + 0,7~)1' - s~0(,7~' - 0,7~) + a 2 ( ~ " + 0 ~ ' ) + 0 ( n ~ + ~2) _ ro~ ° = 0, -~(~;"-
(13)
0,7~') - 0(,7~" + 0,7~") - onp(,7?' + 0,7;) - 0242(,77' + 0,7;) + n ; - rO~zo= o.
(14) Once the in-plane static displacements have been evaluated from equations (13) and (14), the steady combined force II ° can be determined from equation (7). In the derivation of equations (13) and (14), it has been assumed that the static displacements are small. This assumption is validated later by examining the results. 2.2. GOVERNING EQUATIONS OF MOTION ABOUT THE EQUILIBRIUM POSITION
Substituting equation (8) into the equations of motion (1)-(4), subtracting the static equilibrium equations (9)-(12), using equation (7) and neglecting the second and higher order pertrubation terms, one obtains
- 0 s O ( n ; ' - 0 ~ t ) + a z ( n t " + o n ; ' ) + 2fl~/2~i(0 ~, + o 0 ; ) + A 0 t + (1 + ~ a ) # t = 0, ('7~ ' ~ -
0~,*") - 0 A ( ~ , * " + 0'7~") + [ n ° , 7 ~ ' ] ' - ~ [ , ? ~ , ( , ~ ,
(15)
- 0,7~)1'
+t32t/f" + 2fll/2~i1~ ' + A//~ + (1 + fl,)i)~' = O,
(16)
o ( , 7 t ' + 0 ~ " ) + o n o ( o ~ , + 0,7~) - 0~(,7~' + o ~ ) ( n ~ ' - o,7t)
-/~//~' - (1 + fia)#~ = O, 0(0~p* - t/~") - A(tp*" + 0r/~'") + o~)* = O,
(17)
(18)
where equations (15) and (17) govern in-plane motion, while equations (16) and (18) correspond to out-of-plane motion. Similarly to the case of the static equilibrium equations, the in-plane and out-of-plane perturbation equations may be solved separately; however, the out-of-plane perturbations depend on the in-plane static displacements through FI°. It may be noted that the out-of-plane perturbation equations (16) and (18) in the present extensible case are identical to those for an inextensible pipe [1] provided that t/~ = 0 and that the steady state pressure-tension effects are taken into account. As has been mentioned earlier in Section 2.1, the former condition is satisfied if the pipe initially lies in a vertical plane or if the gravity effects are negligible. Since all the cases considered in the present work satisfy these conditions, out-of-plane motion need not
CURVED PIPES TRANSPORTINGFLUID. PAR']?II
249
be analysed again,t and from here on efforts are directed towards the analysis of in-plane motion only. 3. FINITE E L E M E N T ANALYSIS OF IN-PLANE M O T I O N Solutions of the equations governing the static equilibrium and the perturbations around the static equilibrium are obtained by application of the finite element method. 3.1. AN:d~YS~SOF THE STATIC EQUILIBRIUM In order to discretize the equations governing the static deformations, the following variational statement is utilized /=~lfO¢' {&/~A~l(r/~, ~/~) + 6rf~A~3(rl'~, T/~)} d~ = 0,
(19)
where 6r/~ and 6~/~ are the variations in the steady-state displacements r/~ and ~/~, while Ai°l(rl~, rl~) and Ai°s(rl~, rl~) represent the left-hand sides of equations (13) and (14), respectively; n is the number of finite elements utilized in the discretization and ~, is the length of the ith element. The solutions for ~/~ and r/~ are sought in the form
rl°l = [N,e]{q°},
~1~= [N3e]{q°},
(20)
where [N~] and [N3e] are two matrices of interpolation functions of the space co-ordinate ¢, and {q,~) is the element in-plane displacement vector dependent only on time. It may be shown that a cubic interpolation model for ~7~ and linear interpolation for ~/~ can guarantee convergence of the finite element scheme. Thus, one can proceed in the same manner as for the out-of-plane motion in the inextensible case to obtain a matrix equation governing the static equilibrium of an element as follows: [K°](qT) = {F,°},
(21)
where
[/¢,1 = [Ao]-l~[{[I~] + 0([As] + [11,1 ~) + 021t81) +~¢{[I81 - 0([1121 + [I12V) + 021111} +(n: + t~2){[19] + 0([I12] - [1131) -
02[~4]}
-~a2{[I~] + [Ii01 + 0([216] + [114] - [11~]) -
02[l, sl}l[Ao] -',
{F,-°} = [Aol-rr[o(II, o + ~2){Fa} - ~t~2(0{F2} + {F3}) + {F,}].
(22) (23)
The integrals [Ix] to [I23], (F~}, etc. and [Ao] are given in Appendix A. In deriving equation (21) the assumption has been made that the pressure in the external fluid is constant while the internal pressure varies linearly along the centreline. Thus
I]p = Ileo - ~t~2~,
(24)
= ~.L/2Di,
(25)
where
Although the perturbation equations are identical in the two cases of inextensible and extensible pipes, the steady-statecombinedforce 1-1° is slightlydifferent in the two cases. This, however, does not change the dynamicalbehaviour significantly,
250
A. K. MISRA ET AL.
L and /9, being the length and internal diameter of the curved pipe, while ~. is the frictional resistance coefficient for turbulent flow in a curved pipe; )~ is the nondimensionalized frictional resistance coefficient. The resistance coefficient ~ for a curved pipe is somewhat larger than that for a straight pipe (~.0), and according to the theory of Schlichting [2] is given by )~ = )~o[1 + 0.075 Re°2S(Di/2Ro)°5],
(26)
where Re is the Reynolds number ( U D J v ) , and Ro is the radius of curvature of the pipe segment. Equation (21) corresponds to a single finite element. Such equations for all the elements are assembled to form the global equation of static equilibrium, which is then solved numerically. 3.2. ANALYSIS OF MOTION AROUND THE STATIC EQUILIBRIUM
Similarly to the analysis of the static equilibrium equations, the variational statement used for the finite element model of in-plane perturbations is {•IllAi1(1"]1 , 1"]~) -b dir/3Ai3(r/1 , ~/~)} de = 0,
i=1 Jo
(27)
where 8r/~' and 6t/~" are the variations in the dimensionless in-plane displacement perturbations, while Ai*l(~l'~, ~1~) and Ai*3(r/{, ~/~) represent the left-hand sides of equations (15) and (17), respectively; n and ~, have the same meaning as in equation (19). Proceeding as in Section 3.1, one obtains the matrix differential equation governing the motion of a typical element as follows: [M,]{/ji} + [Di](q;) + [Ki](qi )
-----{0},
(28)
where [M,] = [Ao]-lr{(1 + fia)[I1] + (1 +/~)[14]}[do] -1, [oil = [dol-lT[fll/2tt{2([15] d- 01114]) -q- ([•20] -- [114]T)} "[- m[II] qt_ •[I411[Ao1-1,
(29) (30)
[K~] = [Aol-IT[{[I3] --k 0([115 ] --]-[/151 T) -~- 02118]}
+zg{[Is] -
0([112 ] "]- [/12] T) "[- 02111]}
+ ~ ( q { [ I 7 ] - 0([15] 7- - [12o]) - 02[I14]} -[--C2{[122] -- O([lll] T -- [121]) -- 021123)}) +U2([I9] + 0([112] -- [113]) - 02[I4]} +a1{[I9] + 0([112] -- [113]) - 02[14]} + bl{[I5] + 0[I14]}
+a2{[I10] + 0([116] - [1/7]) - 02[I18]} + b2{[I,l] +
0[I23]}l[Ao] -1.
(31)
The integrals [11] to [123] as well as [Ao] are given in Appendix A. The coefficients al, a2, etc. are associated with linear interpolation of the static parameters as follows: H ° = al +
a2~,
I -I°' = bl +
b2~,
~'
+ Orl~ = cl + c2~.
(32)
Hence, in terms of the values of H ° at nodes ] and ] + 1, a~ = H ° ],
a2=(H °
l+l-H°l)/~e,
(33)
CURVED PIPES TRANSPORTINGFLUID. PART n
251
where ~e is the length of the element in question. Similar expressions hold good for bl, bz and ca, c2. Equation (28) holds good for a single element. Such equations for all the elements are assembled to form the global equation of motion, which is then converted to an eigenvalue problem that is solved numerically. 4. RESULTS AND DISCUSSION This Section presents results for the static displacement field and the combined force as obtained by using the method outlined in Section 3.1 and the frequencies of in-plane motion obtained from the analysis of Section 3.2. The in-plane frequencies are nondimensionalized in two different ways:
ooi = f2,LZ[(Mt + Mf )IEI] m,
o)* = coil O2,
(34)
where Q, are the dimensional circular frequencies and the subscript i denotes in-plane motion. The second form of nondimensionalization is done to facilitate comparison with Chen's results. These nondimensional frequencies are obtained as functions of either t2 defined in equation (5) or ti* = ti/0. 4.1. STEADY-STATE CONFIGURATION The stressed configurations of a clamped-clamped semi-circular pipe conveying fluid are shown in Figures 2 and 3 for various flow velocities. In the calculations, it was assumed that y = 0 (i.e., no gravity load) and sO= 104. The deformations are exaggerated in the figures for clarity; the magnification factor is indicated in each case. Figure 2 corresponds to the case of inviscid flow (~ = 0). It may be noted that the ff:3zr
- = ~ ,
U
F ~ L7:3-2 ~
t
&/
Figure 2. Static equilibrium configuration of a clamped-clamped semi-circular pipe conveying inviscid fluid for sg = 104 and (a) t~ = 2Jr, 2.5at, 3~r; (b) ~ = 3.2~r, 3.6rL 4:r. The dashed line represents the undeformed pipe. Deformation is magnified by a factor of 28 in (a) and 30 in (b).
A. K. MISRA E T A L ,
252
F ~=2,5~
t
u
,
Figure 3. Static equilibrium configuration of a damped-clamped semi-circular pipe conveying viscous fluid for ~ / = 104 and ~ = 2~, 2.5~:, 3~. The dashed line represents the undeformed pipe. Deformation is magnified by a factor of 25.
stressed shape is symmetric; this is because the fluid force acting on the pipe is only the centrifugal force, which is symmetric. When the flow velocity increases, the symmetrical deformation away from the initial unstressed shape increases gradually. In the case of viscous flow, on the other hand, the stressed shape is not symmetric (Figure 3), since the pressure caused by the frictional resistance varies along the length of the pipe. It may be noted that the deformations in both the cases of inviscid and viscous flows are fairly small (less than 5%) even for very large flow velocities (up to t~ = 4~). It is also interesting to note that, beyond a certain flow velocity, the stressed configuration changes to a higher mode: from the extensional mode to a bending mode (see the cases of t~ = 4at for inviscid flow and ~ = 3~ for viscous flow). The stressed configuration of the pipe was obtained for several flow velocities between 3.6n and 4n for the inviscid case and it was found that the change occurs around 3.9¢r. Why this change occurs is not quite clear. The global stiffness matrix however, remains positive definite throughout, indicating no buckling in this flow velocity range. The distribution of the static combined force H ° along the semi-circular clampedclamped pipe for various internal flow velocities is shown in Figure 4. The intricate shape of the curves is related to the static deformation of the pipe (Figures 2 and 3); the variation of II ° with s / L is more complicated in the viscous-flow case, because the static deformation is of more complex form (Figure 3 versus Figure 2). For an inextensible pipe, there is no difference between the values of 17° for inviscid and viscous flows [1]; in both cases, it is equal to -t~ 2. This is shown as a dashed line. For an extensible pipe, however, there is a difference between viscous and inviscid results (shown as solid lines). The difference is small for low flow velocities, but becomes significant at high velocities. It may also be noticed that, if the internal flow is inviscid, the values of H ° for inextensible and extensible pipes are fairly close; the maximum difference between the two results is less than 10%, except for a = 3Jr, when the difference can be as high as 30% near the middle of the pipe. 4.2. IN-PLANE DYNAMICS
Using the static results obtained in Section 4.1, the coefficients al, az, etc. given by equations (32) are determined and then substituted into the expression for the dynamic
CURVED PIPESTRANSPORTINGFLUID. PARTII -30
253
(a Inviscldflow 1
-40
Viscousflow7
v,sco
-50 -20
2
, '
, '
, '
I t
i
/
(b)
70 ~ j /
-
J~
o E L)
o_ -f20 "6 --140
l
i
tnvmc, flow Inwscld
L--
'
r
e~
--160 ~
"~
I
I-
vscous7 ! (d)
-
• Invlsc~dflow
-250
-
300
I
0
0"25
I
0.50
I
0"75
s/L
Figure 4. Distribution of static combined pressure-tension force along a clamped-clamped semi-circular pipe conveying fluid for M= 104 and (a) t~*=2, (b) ~*=3, (c) t~*=4 and (d) 5" =5. - - - , inextensible case; - - , extensible case.
stiffness matrix [Ki] defined by equation (31). Calculations are then conducted to compute the in-plane eigenfrequencies of the dynamic system and to assess its stability. 4.2.1.
Convergence study
Figure 5 shows the convergence, with increasing number of finite elements, of the lowest three natural frequencies associated with t~ = 0, for various values of ~/. It may be seen that convergence is very slow, and that it is affected by the slenderness parameter ~/(i.e., AtLZ/l). For a small number of elements (ten or so), the results for different values of ~/are very different. For a large number of elements (forty or so), the results are comparable. In the curved beam theory used in this work, it is assumed that the length of the pipe is large in comparison with the radius of the pipe. This implies that s4 must be large; however, calculations with large ~1 result in high computational cost. Therefore, a value of ~/ that provides a reasonable trade-off between cost and accuracy is used in the calculations to be presented: ~1 = 104. It may be of interest to note that for the inextensible case, less than ten elements lead to convergence of the results [1], as opposed to more than thirty elements required for the present extensible case. Thus, the extensible analysis is computationally more expensive.
254
A.
140 [ - - T
[
l
j~
]
MISRA E T A L .
K
i-r~a
200
)
l
J~ I
[
r
f
! 120 - 180
3
~d
= 4 x 10 s
>r,
~ =4×1°~
I00
160
~,
~ .o
~.~
~ 6o _~=10, ~......_ z5
~ o . . . .
~0
~=25x
20
I 9
I
I
~
.
~
.
_____
-
JooF
103
I J 17
J
~
* ~ .
^_
'--t.L*°
i
I 25
J
I
J
1 J 3:5
%
i
- - _ _ _
,
Total number of e l e m e n t s 260
,\,
,
I
'
~
,
i
~
I
~
I (~)
240
=o 2 2 0 3
g
~
*
200
= 104
I
E
£
180
E 160
.~ = 2 . 5 x 10 a
140
~ 9
J
i
r r 17
~
~
r t 25
J
J
I r 53
i
J 41
Total number of e l e m e n t s
Figure 5. Convergence of in-plane natural frequencies of a clamped-clamped semi-circular pipe for different values of ~ and t2 = H = 0: (a) lowest natural frequency, (b) second natural frequency, (c) third natural frequency.
255
CURVED PIPES TRANSPORTING FLUID. PART II
4.2.2. In-plane dynamics of an extensible clamped-clamped semi-circularpipe This section presents results with several variants of the theory: in one, the steady-state combined force H ° is neglected; in the second variant, H ° is taken into account, but the initial deformations are assumed to be negligible, i.e., the terms involving ~/(~/]' + &/~) in equations (13) and (14) are set to zero; in the third variant both II ° and ~/(r/]' + 0~/;) are non-zero and this is considered to be the complete theory. The first variant is recognized as physically not realizable, but is considered for comparison. The calculations are conducted for a system with fl = 0.5, 7 =/3, = / ~ = A = zX= 0, ..~ ----10 4 (see equations (5) for the definitions of these parameters). Figure 6 shows the results obtained when the internal fluid is inviscid. It may be noted that for fi -< 3at, the effect of the sO0/]' + 0~/~) terms is not very important. This is so because the static deformations are not very large, as was observed earlier. However, for fi > 3~, static deformation effects become slightly more pronounced, in the second and third modes particularly, reflecting slightly greater departure from the unstressed state of the pipe. The most important feature of Figure 6 is the fact that extensible theory, properly taking into account the steady-state combined force H ° predicts that no instability occurs for a clamped-clamped curved pipe, similarly to the modified inextensible theory presented in Part I of this study [1]. The frequencies predicted by the extensible theory and the modified inextensible theory are quite close and they change very slightly with the flow velocity, unlike the case of H ° = 0 when the system loses stability by buckling. This leads to the conclusion that it is the steady-state forces, rather than the steady-state deformations, which are primarily responsible for the inherent stability of fluid-conveying clamped-clamped curved pipes. Hill and Davis [3] and Doll and Mote [4, 5] have also presented extensible theories and reached the same general conclusion, namely that curved pipes with clamped or 24
l
i
I
I
2O - - ' ~ " ~ . - - ' - ' ~ ' - ~
i_~-
•
>t,
~6
~.
Thnrd mode
12
~.~.
E c5
Second
mode
4 ~.
0
I I
I 2
F~rst mode
N. I~ 3
i 4
Dumenslonless flow veloc#y, tT*= tT/Tr
Figure 6. Dimensionless in-plane frequencies of a clamped-clamped semi-circular pipe conveying inviscid fluid as functions of the dimensionless flow velocity for fl =0.5, ~ / = 104, ]t=fla =fla = A = ~ = 0 . - -
I-[° = ~ ( n ~ '
+ 0 n ; ) = 0; - -
-, ri ° 4: 0,
~(,7~' +
on;) =
0;
n ° :~ 0, ~ ( n ~ '
+ o n ; ) 4: 0.
256
A . K . MISRA E T A L .
Present study
O
....
Doll end Mote (t974)
--'--
Hill ond Davis (1974)
I
2
3
4
Figure 7. Comparison of the in-plane fundamental frequency of a clamped-clamped semi-circular pipe conveying inviscid fluid as a function of the dimensionless flow velocity, obtained from the theories of Doll and Mote (fl =0.5, ~¢ = 1.58 × 104), Hill and Davis ~fl = 0-43, ~¢ = 1.4 × 105) and this study (fl = 0.5,
~=10). otherwise supported ends do not lose stability when subjected to internal flow. Figure 7 compares the results obtained by these two investigators with those obtained by the present theory [including H ° and ~/(~/~' + Oq~) terms] with the assumption that the fluid is inviscid. It may be noted that the general character of the solutions is similar in all three cases, although the results are not identical. Hill and Davis' equations of motion are perhaps the closest to those utilized here ar,d the results from these two theories are therefore close. Some parameters are different however, namely fl = 0.43 and ~¢ = 1.4 x 105 as compared to 0-5 and 104, respectively, in the present case. Hill and Davis, similarly t o the present theory, considered motions about the deformed initial state, calculated in a linearized fashion. On the other hand, Doll and Mote calculated the deformed state by a more sophisticated approach, involving a cumulative application of the linearized equations. Their fl was the same as in the present calculations and ~¢ was 1.6 zr2 x 103. (Note that this is so, despite what appears in their published work (fl = 1), due to a typographical error [6].) It should be noted that Doll and Mote and Hill and Davis effectively considered inviscid flow. The present work, however, also considers viscous flow. It may be seen in Figure 8 that the frictional effects are not very pronounced for the first mode, but they are important for the higher modes. The reason for this is not understood at present. The important point, however, is that even for viscous flow, clamped-damped curved pipes do not lose stability. 4.2.3. In-plane dynamics when the downstream end slides axially As mentioned earlier in Section 1, there are cases of practical interest when one end of the pipe is clamped as far as the transverse motion is concerned, but is free to slide axially. Calculations have been conducted for such a system with parameters identical to those in the case of a clamped-clamped pipe, and for viscous flow. The variation of the first three in-plane frequencies of a senti-circular pipe with
257
CURVED PIPES TRANSPORTINGFLUID. PART II 24 - -
~
p
i
i
2O TNrd mode
,g" 3
g~
"~. 12
~
'
~
"
-
Second mode - - ~
-
i
~.., "~" ~ " ~ .
4f
FLrs~ m o d e
--" '~"""" .
\.
I
0
I
I
1~
2
I
3
4
Figure 8. Dimensionless in-plane frequencies of a clamped-clamped semi-circular pipe conveying viscous fluid as functions of the dimensionless flow velocity with the same parameters as in Figure 6. - n ° = ~ ( f l ~ ' + o ~ ; ) = 0 ; - - - , n°#=o, ~ ( ~ i ' + 0 ~ ; ) = 0 ; , n°:eo, ~ ( ~ ' + 0 n ; ) ~ 0 .
- - r ~
i
i
i
(b i
J
[
~
iJ~
]
r
i
(o)
~1"04
0.8
'~"kO
"~8 3 r d mode
f 2 n d mode
~
0-4
o.9
3
113rd mode
I°
.o5
I'1
2"
0
~(3"99
E t--,i
0
3#
0
"•.8
~---~- Value of E*
"1"0 -0-4-
_-----2 nd mode 1.0
J l s t mode
I
I, 1.03
q
-0"8 f -I-2
1-04 --7 344jVolues
Ist mode I 0-5
of E* I
0
2
4 Re (o~*)
I
9
1.05 - Flutter threshold
/
I
1
0
0-5
I'0
1.5
2-0
2.5
E*
Figure 9. Argand diagram of in-plane motion of a clamped-sliding semi-circular pipe conveying viscous fluid as functions of the flow velocity for ~ = 0.5, ~ = 104 (a) Argand diagram, (b) real part of the frequencies. The parameters are the same as in Figure 6; ri ° 4: 0, ~/(r/]' + 0r/~) ~ 0.
258
A.K.
M1SRA E T A L .
increasing nondimensional flow velocity t~* is shown in Figure 9(a) and (b). Figure 9(a) shows the Argand diagram, while in Figure 9(b) the real part of the frequencies is plotted against ti*. The effect of low flow velocity is to damp the motion in all three modes; but when the flow velocity is sufficiently high (ti* = 0.99), the first mode loses stability by flutter [Figure 9(a)]. When the flow velocity is increased further, the pipe buckles in the first mode at ti* = 1.05 and subsequently in the second mode at ti* =- 1.1 [Figure 9(b)]. The variation of the first three in-plane frequencies of a quarter-circular pipe (0 = n/4) with the nondimensional flow velocity is shown in Figure 10. All the parameters, except 0, are the same as in Figure 9. The pipe in this case also loses stability by flutter, but in the second mode and at a much higher flow velocity (a* = 2.91). It appears that, qualitatively, the dynamical behaviour of a clamped-sliding pipe is similar to that of a cantilever pipe, although the sequence of instabilities may not be the same. It is interesting to compare the results obtained for the clamped-sliding pipe using the extensible and the modified inextensible theories. It was noted that, in the case of the latter, the combined force H ° and hence the frequencies depend on whether the force-balance approach or Castigliano's theorem is used and also on the nature of the sliding end. Sliding boundary conditions can be represented by any of the following three ways: (i) 6 = q~ + Oil3 = O, My = 0; (ii) t/; = 0, My = 0; and (iii) r/; = 0, My GO. The critical flow velocity is dependent on the type of the sliding end. 1"2
T
~
J~-
r
7
T
,2.91 -- -Ist mode
1.0 0.8 0.6 T o
E H
Vcfluesof ff *
2-~
J
0.4 0.2
~'9~rd
225 _
~_-8 0
mode
f~ I -
0
o
-0.2
0
Znd mode
-0.4 2.91 -0.6
0
--
J
2
__
l
4
d
I
l
6
8 IO-I
IO
Re (w)x
J
12
14
Figure 10. Dimensionless in-plane frequencies of a clamped-sliding quarter-circular pipe conveying viscous fluid as functions of the flow velocity for fl = 0.5, ~¢ = l04. The parameters are the same as in Figure 6; 17° ~: 0, ~ ( n ~ ' + 0n.~) ~ 0.
259
CURVED PIPES TRANSPORTING FLUID. PART II
- - ~ T T T 1"5I I'0 "'j2ndm°de I~3rdm°de
T
"[ T-_[~ ~,
N 4th mode 3 Values of ~*
1.4~ 5 0.5
E
F
O\ 0 Ist mode
0
4
6
-0.5 -I-0 -
0
4
L 8
1 . L . _ _ ~ 12 t6 Re(w*)
20
Figure 11. Argand diagram of in-plane motion of an inextensible semi-circular clamped-axially sliding pipe of type (i), defined in the last paragraph of the preceding page, for fl = 0.5.
Figure 11 shows the Argand diagram of in-plane motion of an inextensible semi-circular clamped-sliding pipe for fl = 0.5. The sliding boundary condition is of type (i) and the force balance approach was used to determine 1-I°. It may be noted that the pipe loses stability by buckling in the first mode, as opposed to flutter predicted by the extensible theory. The critical flow velocity here (t~* a little larger than 1.5) is higher than that in the extensible case (/~* - 0.99). If the inextensible pipe has sliding ends of type (ii) or (iii), again it loses stability by buckling. The important thing to note is that, unlike the case of truly clamped-clamped (or pinned-pinned) pipes, the presence of a sliding end is sufficient to permit an instability to occur. 5. CONCLUSION The dynamics and stability of extensible curved pipes conveying fluid have been studied in this paper. Pipes clamped at both ends as well as pipes that can slide at the downstream end were considered. The initial shape of an extensible pipe (i.e., at t~ = 0) changes under the action of the centrifugal forces acting on the system at any given t~, and if the flow is realistically considered to be viscous, by the action of frictional forces also. For inviscid flow, the stressed shape changes symmetrically vis-h-vis the unstressed shape of a clampedclamped pipe; but this change is asymmetrical if the flowing fluid is viscous. The deformations however, are fairly small for both types of flow.
260
A . K . MISRA E T A L .
For an inextensible pipe, there is no difference between the values of steady-state combined force H ° for inviscid and viscous flows; in both cases, it is equal to -t~ 2. For an extensible pipe, however, there is a difference between viscous and inviscid results, especially at high flow velocities. The equations of motion generally depend on those of static equilibrium for any given flow velocity, which must be solved first and then used for the solution of the equations of motion. This is true for the in-plane equations of motion, whereas, at least with the approximations introduced in this paper, out-of-plane motions are unaffected. Indeed, the interesting finding was made that the out-of-plane motions are identical to those of the modified inextensible theory of Part I [1]. Hence, attention in this paper is confined to the in-plane motion. It was observed that as long as the steady-state forces are taken into account, it does not make any difference whether the terms involving the initial deformations in the equations of motion are considered or neglected. The results obtained are qualitatively the same and differ very little quantitatively. They both lead to the same conclusion that the frequencies of curved pipes clamped at both ends do not change greatly with increasing flow velocity, and the system thus never loses stability. The results compare quite well with the extensible theories of Hill and Davis and Doll and Mote, and the modified inextensible theory presented in Part I of this work. Since the differences in the frequencies as obtained from the modified inextensible and extensible theories are small, the former may be used instead of the latter (at least for pipes with both ends supported) for any practical purposes. The reason for doing so is that obtaining solutions by the extensible theory is rather expensive. To achieve convergent results by the extensible theory, many finite elements are necessary--easily more than 30--this number increasing with ~ / = AtL2/L This contrasts with similar convergence being achieved by the modified inextensible theory with 5 to 10 elements. The above discussion, of course, applies provided that the stressed centreline remains close to the unstressed one. If it does not, then the extensible theory must be used. If one end of the pipe is allowed to slide axially (even though it is clamped as far as the transverse motion is concerned), then there is a drastic change in the stability behaviour of the pipe. At sufficiently high flow velocity, the pipe becomes unstable by flutter, in the first mode for a semi-circular pipe and in the second mode for a quarter-circular pipe. This is an entirely new observation, not previously reported by any of the earlier investigators. It may be noted that no results are presented for cantilevered pipes in this paper. This is because, for extensible cantilevered pipes, the static deformations may be large and hence, should be determined by solving the nonlinear equations of static equilibrium, and not by solving the linearized equations as was done in this paper. ACKNOWLEDGEMENT The authors gratefully acknowledge the support given to this research by the Natural Sciences and Engineering Research Council of Canada, Le Fonds FCAR of Qu6bec and McGill University. REFERENCES
1. A. K. MmRA, M. P. PA~r~oussIs and K. S. VAN 1988 On the dynamics of curved pipes transporting fluid. Part I: Inextensible theory. Journal of Fluids and Structures 2, 221-244. 2. H. SCHLICHT~NO1960 Boundary Layer Theory. New York: McGraw-Hill. 3. J. L. HILLand C. G. DAvis 1974 The effect of initial forces on the hydroelastic vibration and stability of planar curved tubes. Journal of Applied Mechanics 41, 355-359. 4. R. W. DoLL and C. D. MOTE, JR 1974 The dynamic formulation and the finite element
CURVED PIPES TRANSPORTING FLUID. PART II
261
analysis of curved and twisted tubes transporting fluids. Report to the National Sciences Foundation, Dept of Mechanical Engineering, University of California, Berkeley. 5. R. W. DOLL and C. D. MOTE, JR 1974 On the dynamic analysis of curved and twisted cylinders transporting fluids. Journal of Pressure Vessel Technology 98, 143-150. 6. M. P. PnYDoussis 1986 Discussion on "Application of the transfer matrix method to non-conservative systems involving fluid flow in curved pipes" by C. Dupuis and J. Rousselet. Journal of Sound and Vibration H I , 167-168. APPENDIX
A
The integrals [11]-[1=3],{F~)-{F4} and [Ao] appearing in equations (22)-(23) and (29)-(31) are defined as follows:
[/11= L~"[@2]r[(p2]d~',
[/2] = ffe [(P2]'r[~2] d~,
[/3]= L e° [¢d"'[,P=]dC,
[/4] = f0~e[(~4]T[(j)4]d~,
[]51 = L e~ [¢21~[¢=] ' dC,
[/6] = Lce [(Pz]"r[~b4] d~,
[/71= L Ce[(~2]tT[~)4] 'dC,
[18] = L e~[~4]'T[(p41 ' d~,
[/9] ---~L ee [q~)2]T[(~)2]" d~',
[11o1 f f C[,~=]~[¢=]" dC,
[/11]=
=
fo:eC[,~d~[¢d' dC.
[/12] = foe° [q~zlT[(])4]' rig,
[]ld = foe"[q~4V[ed'dC,
[1141= ffe [@2]'[4)41d,~,
[]15] = L ~e[~2]'tT[(]~4] ' d~',
[],61= f f C[,/,dq~,4]' tiC,
[/17]= foe"C[,/..V[@d'dC, [/181 = L ee C[@4]r[q~4] d,~,
where
[1201 = fO~eC[~)4]T[(~4] 'dC/,
[]~11 = f f ¢[@,V[,/,~[' dC,
[122] = L ~e C[I~)2]tT[~4] 'dC,
[]2~]= Le"C[,Pdq¢~] d4,,
[q)2l = [1, ¢, C=, ~3, O, O] and 0 0 0
0 0 0
0 0 1
0 0 0
C~
C~
o
o
2Ce
3C~
0
0
o
o
1
Co
[q~=lTdC,
{5) =
[~)4] TdC,
fo fo
(~} = -
I 0
[.4o1=
{F3} -
fo fo
1
0 0 1 ~ 0 0
(F~} =
[qL] = [0, O, O, O, 1, C];
1 0
C[q,:]" dC, ~(O~o[¢~1" + ~o[,P41 ") OC.
ON THE DYNAMICS OF C U R V E D PIPES TRANSPORTING FLUID PART II: EXTENSIBLE THEORY A. K.
MISRA,M. P. PAIOOUSSISAND K. S. VAN
Department of Mechanical Engineering, McGill University, Montreal, Quebec, Canada H3A 2K6 (Received 16 June 1987 and in revised form 20 January 1988) The paper treats the effect of internal flow on the statics, dynamics and stability of extensible curved pipes. The cases of pipes clamped at both ends and pipes clamped at the upstream end but free to slide axially at the downstream end are considered. For pipes clamped at both ends, the stressed shape is symmetrical for inviscid flow but asymmetrical for viscous flow; there is also a significant difference between the steady-state combined tension and pressure force calculated for the two flows. The stability behaviour of a clamped-clamped curved pipe, however, does not depend on whether viscosity is taken into account or not; the natural frequencies do not change much with increasing flow velocity and the system thus never loses stability. This conclusion is consistent with that obtained in the first part of this work using a modified inextensible theory and with the predictions of other extensible theories. There is, however, a substantial change in the stability behaviour if the downstream end of the pipe is allowed to slide axially, even though it is still clamped as far as transverse displacement is concerned. The pipe then loses stability by flutter at sufficiently high flow velocities. 1. I N T R O D U C T I O N IN THE FIRSTPART of this work [1], the centreline of the pipe has been assumed to be inextensible, i.e., the centreline strain in the pipe is taken to be zero. Two variants of the inextensible theory were considered: in the first case, the steady-state forces induced by the pressure and centrifugal forces were neglected, while in the second, they were taken into account. There was significant difference in the results obtained from the two theories, at least for pipes supported at both ends. The pipes lose stability by buckling (divergence) at a sufficiently high flow velocity according to the first theory; however, when the steady-state forces are taken into account, in the second version of the inextensible theory, the pipes remain stable for all flow velocities. The objective of this paper is to present an analysis of the problem without assuming that the centreline of the pipe is inextensible. This analysis is called here the extensible theory. The results obtained from this theory are compared with the results yielded by the two inextensible theories, leading to the final recommendations as to the most appropriate theory for the problem at hand. This paper also presents results for a combination of boundary conditions not considered so far by any previous investigation, i.e., when one end is completely clamped, while the other end is clamped in terms of lateral constraint but free to slide axially. This configuration is of practical importance, since in many situations, in order to allow for thermal stress relief, it is not desirable to use totally clamped supports at the two ends of the pipe, using instead a sliding support. 0889-9746/88/030245 + 17 $03.00
(~) 1988 Academic Press Lmalted
246
n . K . MISRA E T A L .
2. G O V E R N I N G E Q U A T I O N S O F M O T I O N The system under consideration, as in Part I, consists of a curved pipe of length L, having a uniform cross-sectional area A,, mass per unit length Mr, flexural rigidity E1 and shear rigidity GJ (Figure 1). The pipe is initially in a plane having an arbitrary centreline shape. It conveys a stream of fluid of mass Mi per unit length. The flow is assumed to be a plug flow with constant velocity U. Strictly speaking, for an extensible pipe, the area of cross-section changes during the oscillations of the pipe. Hence, the flow velocity varies slightly with time. This variation however, is likely to have a small effect on the dynamics of the pipe and is neglected in this paper. The equations governing the motion of the combined pipe-fluid system, valid for both inextensible and extensible pipes, have been derived in the first part of this work [1]. In nondimensionalized form they are In~° + on;') + [ n ( n ~ + o n , ) ] ' + o n
"
iv
-
]/O/Xo + U- : ( n "i -Jr
073' -3I- 8 )
+ [2/31ati(0; + 003) + k01 + (1 +/3.)#, 1 = 0,
(1)
tokyo + C,2n'~] + [2/3~atiO; + zxO2 + (1 +/3a)#2] = 0,
(2)
.(,72 - ow") - A 0 ( V / ' + One) + ( n n ~ ) ' -
- n ' + o(n~' + on;) + o n ( n ; + ons) + rO~o+ a:o(n; + on,)] - [/Suzti(//; - 8//1) + zX//, + (1 +/~.)#,1 = 0, amed)centrehne
j~
The stramed centrehne
2"0
g
"0
t
x
g/ Figure 1. Kinematics of the system; definition of coordinate systems and coordinates used.
(3)
CURVED PIPES TRANSPORTING FLUID. PART II
247
[ o ( o ~ - ~ ) - A(W" + 0o~)l + [o~bl = o,
(4)
where th = u / L ,
~h=v/L,
"~- U L ( M f / E I ) 1/2, A=GJ/EI,
~13=w/L,
~a = M a / ( M f Ai"Mr),
¢r=IJ(Mr+M,)L
y = (Mr + My - A o p p ) g L 3 / E I , l i = (Aipi - AoPe - Qz)L2/E1,
2,
= t [ E I / ( M t + MI)L4] ~/2,
~=s/L,
~a m_l~la/(Mf ..~ Mt),
M=A,L2/L
= Md(M~ +
O=L/Ro,
M,), (5)
lip = ( A t p i -- A o p ¢ ) L 2 / E I , A = c L 2 / [ E I ( M f + Mr)] 1/2 and
F~ = A e / c .
In the above, u and v are transverse deformations of the pipe in and normal to the plane of the pipe, respectively; w is the longitudinal displacement, ~p is the twist of the tube; s is the distance of a point on the pipe measured along the centreline; M, and c are the added mass per unit length and the coefficient of viscous damping due to the surrounding fluid, associated with transverse motion; /14a and g play similar roles in longitudinal motion; p; and Pe are the pressure in the internal and external fluids, respectively; Qz is the axial force; 0Lxo, etc. are the direction cosines of the gravity acceleration vector. Prime and dot denote differentiation with respect to ~ and 3, respectively. Unlike the inextensible case, the longitudinal strain e = ( O w / a s ) - (u/Ro) --/:O. Thus, u and w, or equivalently ~/a and t/3, are not directly related to each other in the present case. Furthermore, the axial force Qz is given by e z = E A t e = E A t 3ss
;
(6)
hence, using equation (5), the dimensionless pressure-tension combined force 11 may be expressed as II = lip - M(t/; - 001 ).
(7)
Once rh and t/3 are known, YI can be determined from equation (7). It may be noted that this procedure can be used only for the extensible case, since the last term vanishes if the pipe is inextensible.
2.1. EQUATIONS GOVERNING THE STATIC EQUILIBRIUM OF THE PIPE As in the inextensible case, each of the displacements and twist ~/1, t/2, ~/3, and ~p can be imagined to consist of a steady or static part and a perturbation about the static part, i.e., ~1 ~__n l q_ ?]~ '
7j2_. ~ _l_ ?]~,
~j3 = ?]; ..1_/j~ '
~]) = /po q_ ~/),,
(8)
the superscripts ° and * denoting the steady and perturbed parts, respectively. The equations governing the steady part are obtained from equations (1)-(4) by deleting the time-dependent terms and are given by (t/~'v + 00~') + [II°0/~ ' + 0~7~)]' + 0II ° - yO~xo+ aZ(~lT ' + 0~/~' + 0) = 0,
(9)
0l~ iv - 0~p°") - A0(~0 °" + 0~/~") + (II%/~')' - 7¢~y0+ t/zr/~" = 0,
(10)
- n °' + o(,71" + o,7~") + o n ° ( n l ' + 0 7 ; ) + ~,O~zo+ a2o(,11 ' + o n ; ) = o,
(11)
0 ( 0 ~ , ° - r/~") - A O p ° " + 0r/~") = 0.
(12)
248
A.K. MISRA E T
AL,
It may be noted that equations (9) and (11) governing the static in-plane displacements are decoupled from the equations of out-of-plane static equilibrium (10) and (12). Furthermore, if the gravity effect is negligible (i.e., y = 0), or if the pipe initially lies in a vertical plane (i.e., 0% = 0), then the static out-of-plane deformations r/~ and ~po vanish. All the cases considered in this work satisfy this requirement and hence, ~/~ and ~p° are always zero. Using equation (7), the in-plane static equilibrium equations (9) and (11) can be rewritten after linearization as (,7~~ + 0,~") + [n~(,7~' + 0,7~)1' - s~0(,7~' - 0,7~) + a 2 ( ~ " + 0 ~ ' ) + 0 ( n ~ + ~2) _ ro~ ° = 0, -~(~;"-
(13)
0,7~') - 0(,7~" + 0,7~") - onp(,7?' + 0,7;) - 0242(,77' + 0,7;) + n ; - rO~zo= o.
(14) Once the in-plane static displacements have been evaluated from equations (13) and (14), the steady combined force II ° can be determined from equation (7). In the derivation of equations (13) and (14), it has been assumed that the static displacements are small. This assumption is validated later by examining the results. 2.2. GOVERNING EQUATIONS OF MOTION ABOUT THE EQUILIBRIUM POSITION
Substituting equation (8) into the equations of motion (1)-(4), subtracting the static equilibrium equations (9)-(12), using equation (7) and neglecting the second and higher order pertrubation terms, one obtains
- 0 s O ( n ; ' - 0 ~ t ) + a z ( n t " + o n ; ' ) + 2fl~/2~i(0 ~, + o 0 ; ) + A 0 t + (1 + ~ a ) # t = 0, ('7~ ' ~ -
0~,*") - 0 A ( ~ , * " + 0'7~") + [ n ° , 7 ~ ' ] ' - ~ [ , ? ~ , ( , ~ ,
(15)
- 0,7~)1'
+t32t/f" + 2fll/2~i1~ ' + A//~ + (1 + fl,)i)~' = O,
(16)
o ( , 7 t ' + 0 ~ " ) + o n o ( o ~ , + 0,7~) - 0~(,7~' + o ~ ) ( n ~ ' - o,7t)
-/~//~' - (1 + fia)#~ = O, 0(0~p* - t/~") - A(tp*" + 0r/~'") + o~)* = O,
(17)
(18)
where equations (15) and (17) govern in-plane motion, while equations (16) and (18) correspond to out-of-plane motion. Similarly to the case of the static equilibrium equations, the in-plane and out-of-plane perturbation equations may be solved separately; however, the out-of-plane perturbations depend on the in-plane static displacements through FI°. It may be noted that the out-of-plane perturbation equations (16) and (18) in the present extensible case are identical to those for an inextensible pipe [1] provided that t/~ = 0 and that the steady state pressure-tension effects are taken into account. As has been mentioned earlier in Section 2.1, the former condition is satisfied if the pipe initially lies in a vertical plane or if the gravity effects are negligible. Since all the cases considered in the present work satisfy these conditions, out-of-plane motion need not
CURVED PIPES TRANSPORTINGFLUID. PAR']?II
249
be analysed again,t and from here on efforts are directed towards the analysis of in-plane motion only. 3. FINITE E L E M E N T ANALYSIS OF IN-PLANE M O T I O N Solutions of the equations governing the static equilibrium and the perturbations around the static equilibrium are obtained by application of the finite element method. 3.1. AN:d~YS~SOF THE STATIC EQUILIBRIUM In order to discretize the equations governing the static deformations, the following variational statement is utilized /=~lfO¢' {&/~A~l(r/~, ~/~) + 6rf~A~3(rl'~, T/~)} d~ = 0,
(19)
where 6r/~ and 6~/~ are the variations in the steady-state displacements r/~ and ~/~, while Ai°l(rl~, rl~) and Ai°s(rl~, rl~) represent the left-hand sides of equations (13) and (14), respectively; n is the number of finite elements utilized in the discretization and ~, is the length of the ith element. The solutions for ~/~ and r/~ are sought in the form
rl°l = [N,e]{q°},
~1~= [N3e]{q°},
(20)
where [N~] and [N3e] are two matrices of interpolation functions of the space co-ordinate ¢, and {q,~) is the element in-plane displacement vector dependent only on time. It may be shown that a cubic interpolation model for ~7~ and linear interpolation for ~/~ can guarantee convergence of the finite element scheme. Thus, one can proceed in the same manner as for the out-of-plane motion in the inextensible case to obtain a matrix equation governing the static equilibrium of an element as follows: [K°](qT) = {F,°},
(21)
where
[/¢,1 = [Ao]-l~[{[I~] + 0([As] + [11,1 ~) + 021t81) +~¢{[I81 - 0([1121 + [I12V) + 021111} +(n: + t~2){[19] + 0([I12] - [1131) -
02[~4]}
-~a2{[I~] + [Ii01 + 0([216] + [114] - [11~]) -
02[l, sl}l[Ao] -',
{F,-°} = [Aol-rr[o(II, o + ~2){Fa} - ~t~2(0{F2} + {F3}) + {F,}].
(22) (23)
The integrals [Ix] to [I23], (F~}, etc. and [Ao] are given in Appendix A. In deriving equation (21) the assumption has been made that the pressure in the external fluid is constant while the internal pressure varies linearly along the centreline. Thus
I]p = Ileo - ~t~2~,
(24)
= ~.L/2Di,
(25)
where
Although the perturbation equations are identical in the two cases of inextensible and extensible pipes, the steady-statecombinedforce 1-1° is slightlydifferent in the two cases. This, however, does not change the dynamicalbehaviour significantly,
250
A. K. MISRA ET AL.
L and /9, being the length and internal diameter of the curved pipe, while ~. is the frictional resistance coefficient for turbulent flow in a curved pipe; )~ is the nondimensionalized frictional resistance coefficient. The resistance coefficient ~ for a curved pipe is somewhat larger than that for a straight pipe (~.0), and according to the theory of Schlichting [2] is given by )~ = )~o[1 + 0.075 Re°2S(Di/2Ro)°5],
(26)
where Re is the Reynolds number ( U D J v ) , and Ro is the radius of curvature of the pipe segment. Equation (21) corresponds to a single finite element. Such equations for all the elements are assembled to form the global equation of static equilibrium, which is then solved numerically. 3.2. ANALYSIS OF MOTION AROUND THE STATIC EQUILIBRIUM
Similarly to the analysis of the static equilibrium equations, the variational statement used for the finite element model of in-plane perturbations is {•IllAi1(1"]1 , 1"]~) -b dir/3Ai3(r/1 , ~/~)} de = 0,
i=1 Jo
(27)
where 8r/~' and 6t/~" are the variations in the dimensionless in-plane displacement perturbations, while Ai*l(~l'~, ~1~) and Ai*3(r/{, ~/~) represent the left-hand sides of equations (15) and (17), respectively; n and ~, have the same meaning as in equation (19). Proceeding as in Section 3.1, one obtains the matrix differential equation governing the motion of a typical element as follows: [M,]{/ji} + [Di](q;) + [Ki](qi )
-----{0},
(28)
where [M,] = [Ao]-lr{(1 + fia)[I1] + (1 +/~)[14]}[do] -1, [oil = [dol-lT[fll/2tt{2([15] d- 01114]) -q- ([•20] -- [114]T)} "[- m[II] qt_ •[I411[Ao1-1,
(29) (30)
[K~] = [Aol-IT[{[I3] --k 0([115 ] --]-[/151 T) -~- 02118]}
+zg{[Is] -
0([112 ] "]- [/12] T) "[- 02111]}
+ ~ ( q { [ I 7 ] - 0([15] 7- - [12o]) - 02[I14]} -[--C2{[122] -- O([lll] T -- [121]) -- 021123)}) +U2([I9] + 0([112] -- [113]) - 02[I4]} +a1{[I9] + 0([112] -- [113]) - 02[14]} + bl{[I5] + 0[I14]}
+a2{[I10] + 0([116] - [1/7]) - 02[I18]} + b2{[I,l] +
0[I23]}l[Ao] -1.
(31)
The integrals [11] to [123] as well as [Ao] are given in Appendix A. The coefficients al, a2, etc. are associated with linear interpolation of the static parameters as follows: H ° = al +
a2~,
I -I°' = bl +
b2~,
~'
+ Orl~ = cl + c2~.
(32)
Hence, in terms of the values of H ° at nodes ] and ] + 1, a~ = H ° ],
a2=(H °
l+l-H°l)/~e,
(33)
CURVED PIPES TRANSPORTINGFLUID. PART n
251
where ~e is the length of the element in question. Similar expressions hold good for bl, bz and ca, c2. Equation (28) holds good for a single element. Such equations for all the elements are assembled to form the global equation of motion, which is then converted to an eigenvalue problem that is solved numerically. 4. RESULTS AND DISCUSSION This Section presents results for the static displacement field and the combined force as obtained by using the method outlined in Section 3.1 and the frequencies of in-plane motion obtained from the analysis of Section 3.2. The in-plane frequencies are nondimensionalized in two different ways:
ooi = f2,LZ[(Mt + Mf )IEI] m,
o)* = coil O2,
(34)
where Q, are the dimensional circular frequencies and the subscript i denotes in-plane motion. The second form of nondimensionalization is done to facilitate comparison with Chen's results. These nondimensional frequencies are obtained as functions of either t2 defined in equation (5) or ti* = ti/0. 4.1. STEADY-STATE CONFIGURATION The stressed configurations of a clamped-clamped semi-circular pipe conveying fluid are shown in Figures 2 and 3 for various flow velocities. In the calculations, it was assumed that y = 0 (i.e., no gravity load) and sO= 104. The deformations are exaggerated in the figures for clarity; the magnification factor is indicated in each case. Figure 2 corresponds to the case of inviscid flow (~ = 0). It may be noted that the ff:3zr
- = ~ ,
U
F ~ L7:3-2 ~
t
&/
Figure 2. Static equilibrium configuration of a clamped-clamped semi-circular pipe conveying inviscid fluid for sg = 104 and (a) t~ = 2Jr, 2.5at, 3~r; (b) ~ = 3.2~r, 3.6rL 4:r. The dashed line represents the undeformed pipe. Deformation is magnified by a factor of 28 in (a) and 30 in (b).
A. K. MISRA E T A L ,
252
F ~=2,5~
t
u
,
Figure 3. Static equilibrium configuration of a damped-clamped semi-circular pipe conveying viscous fluid for ~ / = 104 and ~ = 2~, 2.5~:, 3~. The dashed line represents the undeformed pipe. Deformation is magnified by a factor of 25.
stressed shape is symmetric; this is because the fluid force acting on the pipe is only the centrifugal force, which is symmetric. When the flow velocity increases, the symmetrical deformation away from the initial unstressed shape increases gradually. In the case of viscous flow, on the other hand, the stressed shape is not symmetric (Figure 3), since the pressure caused by the frictional resistance varies along the length of the pipe. It may be noted that the deformations in both the cases of inviscid and viscous flows are fairly small (less than 5%) even for very large flow velocities (up to t~ = 4~). It is also interesting to note that, beyond a certain flow velocity, the stressed configuration changes to a higher mode: from the extensional mode to a bending mode (see the cases of t~ = 4at for inviscid flow and ~ = 3~ for viscous flow). The stressed configuration of the pipe was obtained for several flow velocities between 3.6n and 4n for the inviscid case and it was found that the change occurs around 3.9¢r. Why this change occurs is not quite clear. The global stiffness matrix however, remains positive definite throughout, indicating no buckling in this flow velocity range. The distribution of the static combined force H ° along the semi-circular clampedclamped pipe for various internal flow velocities is shown in Figure 4. The intricate shape of the curves is related to the static deformation of the pipe (Figures 2 and 3); the variation of II ° with s / L is more complicated in the viscous-flow case, because the static deformation is of more complex form (Figure 3 versus Figure 2). For an inextensible pipe, there is no difference between the values of 17° for inviscid and viscous flows [1]; in both cases, it is equal to -t~ 2. This is shown as a dashed line. For an extensible pipe, however, there is a difference between viscous and inviscid results (shown as solid lines). The difference is small for low flow velocities, but becomes significant at high velocities. It may also be noticed that, if the internal flow is inviscid, the values of H ° for inextensible and extensible pipes are fairly close; the maximum difference between the two results is less than 10%, except for a = 3Jr, when the difference can be as high as 30% near the middle of the pipe. 4.2. IN-PLANE DYNAMICS
Using the static results obtained in Section 4.1, the coefficients al, az, etc. given by equations (32) are determined and then substituted into the expression for the dynamic
CURVED PIPESTRANSPORTINGFLUID. PARTII -30
253
(a Inviscldflow 1
-40
Viscousflow7
v,sco
-50 -20
2
, '
, '
, '
I t
i
/
(b)
70 ~ j /
-
J~
o E L)
o_ -f20 "6 --140
l
i
tnvmc, flow Inwscld
L--
'
r
e~
--160 ~
"~
I
I-
vscous7 ! (d)
-
• Invlsc~dflow
-250
-
300
I
0
0"25
I
0.50
I
0"75
s/L
Figure 4. Distribution of static combined pressure-tension force along a clamped-clamped semi-circular pipe conveying fluid for M= 104 and (a) t~*=2, (b) ~*=3, (c) t~*=4 and (d) 5" =5. - - - , inextensible case; - - , extensible case.
stiffness matrix [Ki] defined by equation (31). Calculations are then conducted to compute the in-plane eigenfrequencies of the dynamic system and to assess its stability. 4.2.1.
Convergence study
Figure 5 shows the convergence, with increasing number of finite elements, of the lowest three natural frequencies associated with t~ = 0, for various values of ~/. It may be seen that convergence is very slow, and that it is affected by the slenderness parameter ~/(i.e., AtLZ/l). For a small number of elements (ten or so), the results for different values of ~/are very different. For a large number of elements (forty or so), the results are comparable. In the curved beam theory used in this work, it is assumed that the length of the pipe is large in comparison with the radius of the pipe. This implies that s4 must be large; however, calculations with large ~1 result in high computational cost. Therefore, a value of ~/ that provides a reasonable trade-off between cost and accuracy is used in the calculations to be presented: ~1 = 104. It may be of interest to note that for the inextensible case, less than ten elements lead to convergence of the results [1], as opposed to more than thirty elements required for the present extensible case. Thus, the extensible analysis is computationally more expensive.
254
A.
140 [ - - T
[
l
j~
]
MISRA E T A L .
K
i-r~a
200
)
l
J~ I
[
r
f
! 120 - 180
3
~d
= 4 x 10 s
>r,
~ =4×1°~
I00
160
~,
~ .o
~.~
~ 6o _~=10, ~......_ z5
~ o . . . .
~0
~=25x
20
I 9
I
I
~
.
~
.
_____
-
JooF
103
I J 17
J
~
* ~ .
^_
'--t.L*°
i
I 25
J
I
J
1 J 3:5
%
i
- - _ _ _
,
Total number of e l e m e n t s 260
,\,
,
I
'
~
,
i
~
I
~
I (~)
240
=o 2 2 0 3
g
~
*
200
= 104
I
E
£
180
E 160
.~ = 2 . 5 x 10 a
140
~ 9
J
i
r r 17
~
~
r t 25
J
J
I r 53
i
J 41
Total number of e l e m e n t s
Figure 5. Convergence of in-plane natural frequencies of a clamped-clamped semi-circular pipe for different values of ~ and t2 = H = 0: (a) lowest natural frequency, (b) second natural frequency, (c) third natural frequency.
255
CURVED PIPES TRANSPORTING FLUID. PART II
4.2.2. In-plane dynamics of an extensible clamped-clamped semi-circularpipe This section presents results with several variants of the theory: in one, the steady-state combined force H ° is neglected; in the second variant, H ° is taken into account, but the initial deformations are assumed to be negligible, i.e., the terms involving ~/(~/]' + &/~) in equations (13) and (14) are set to zero; in the third variant both II ° and ~/(r/]' + 0~/;) are non-zero and this is considered to be the complete theory. The first variant is recognized as physically not realizable, but is considered for comparison. The calculations are conducted for a system with fl = 0.5, 7 =/3, = / ~ = A = zX= 0, ..~ ----10 4 (see equations (5) for the definitions of these parameters). Figure 6 shows the results obtained when the internal fluid is inviscid. It may be noted that for fi -< 3at, the effect of the sO0/]' + 0~/~) terms is not very important. This is so because the static deformations are not very large, as was observed earlier. However, for fi > 3~, static deformation effects become slightly more pronounced, in the second and third modes particularly, reflecting slightly greater departure from the unstressed state of the pipe. The most important feature of Figure 6 is the fact that extensible theory, properly taking into account the steady-state combined force H ° predicts that no instability occurs for a clamped-clamped curved pipe, similarly to the modified inextensible theory presented in Part I of this study [1]. The frequencies predicted by the extensible theory and the modified inextensible theory are quite close and they change very slightly with the flow velocity, unlike the case of H ° = 0 when the system loses stability by buckling. This leads to the conclusion that it is the steady-state forces, rather than the steady-state deformations, which are primarily responsible for the inherent stability of fluid-conveying clamped-clamped curved pipes. Hill and Davis [3] and Doll and Mote [4, 5] have also presented extensible theories and reached the same general conclusion, namely that curved pipes with clamped or 24
l
i
I
I
2O - - ' ~ " ~ . - - ' - ' ~ ' - ~
i_~-
•
>t,
~6
~.
Thnrd mode
12
~.~.
E c5
Second
mode
4 ~.
0
I I
I 2
F~rst mode
N. I~ 3
i 4
Dumenslonless flow veloc#y, tT*= tT/Tr
Figure 6. Dimensionless in-plane frequencies of a clamped-clamped semi-circular pipe conveying inviscid fluid as functions of the dimensionless flow velocity for fl =0.5, ~ / = 104, ]t=fla =fla = A = ~ = 0 . - -
I-[° = ~ ( n ~ '
+ 0 n ; ) = 0; - -
-, ri ° 4: 0,
~(,7~' +
on;) =
0;
n ° :~ 0, ~ ( n ~ '
+ o n ; ) 4: 0.
256
A . K . MISRA E T A L .
Present study
O
....
Doll end Mote (t974)
--'--
Hill ond Davis (1974)
I
2
3
4
Figure 7. Comparison of the in-plane fundamental frequency of a clamped-clamped semi-circular pipe conveying inviscid fluid as a function of the dimensionless flow velocity, obtained from the theories of Doll and Mote (fl =0.5, ~¢ = 1.58 × 104), Hill and Davis ~fl = 0-43, ~¢ = 1.4 × 105) and this study (fl = 0.5,
~=10). otherwise supported ends do not lose stability when subjected to internal flow. Figure 7 compares the results obtained by these two investigators with those obtained by the present theory [including H ° and ~/(~/~' + Oq~) terms] with the assumption that the fluid is inviscid. It may be noted that the general character of the solutions is similar in all three cases, although the results are not identical. Hill and Davis' equations of motion are perhaps the closest to those utilized here ar,d the results from these two theories are therefore close. Some parameters are different however, namely fl = 0.43 and ~¢ = 1.4 x 105 as compared to 0-5 and 104, respectively, in the present case. Hill and Davis, similarly t o the present theory, considered motions about the deformed initial state, calculated in a linearized fashion. On the other hand, Doll and Mote calculated the deformed state by a more sophisticated approach, involving a cumulative application of the linearized equations. Their fl was the same as in the present calculations and ~¢ was 1.6 zr2 x 103. (Note that this is so, despite what appears in their published work (fl = 1), due to a typographical error [6].) It should be noted that Doll and Mote and Hill and Davis effectively considered inviscid flow. The present work, however, also considers viscous flow. It may be seen in Figure 8 that the frictional effects are not very pronounced for the first mode, but they are important for the higher modes. The reason for this is not understood at present. The important point, however, is that even for viscous flow, clamped-damped curved pipes do not lose stability. 4.2.3. In-plane dynamics when the downstream end slides axially As mentioned earlier in Section 1, there are cases of practical interest when one end of the pipe is clamped as far as the transverse motion is concerned, but is free to slide axially. Calculations have been conducted for such a system with parameters identical to those in the case of a clamped-clamped pipe, and for viscous flow. The variation of the first three in-plane frequencies of a senti-circular pipe with
257
CURVED PIPES TRANSPORTINGFLUID. PART II 24 - -
~
p
i
i
2O TNrd mode
,g" 3
g~
"~. 12
~
'
~
"
-
Second mode - - ~
-
i
~.., "~" ~ " ~ .
4f
FLrs~ m o d e
--" '~"""" .
\.
I
0
I
I
1~
2
I
3
4
Figure 8. Dimensionless in-plane frequencies of a clamped-clamped semi-circular pipe conveying viscous fluid as functions of the dimensionless flow velocity with the same parameters as in Figure 6. - n ° = ~ ( f l ~ ' + o ~ ; ) = 0 ; - - - , n°#=o, ~ ( ~ i ' + 0 ~ ; ) = 0 ; , n°:eo, ~ ( ~ ' + 0 n ; ) ~ 0 .
- - r ~
i
i
i
(b i
J
[
~
iJ~
]
r
i
(o)
~1"04
0.8
'~"kO
"~8 3 r d mode
f 2 n d mode
~
0-4
o.9
3
113rd mode
I°
.o5
I'1
2"
0
~(3"99
E t--,i
0
3#
0
"•.8
~---~- Value of E*
"1"0 -0-4-
_-----2 nd mode 1.0
J l s t mode
I
I, 1.03
q
-0"8 f -I-2
1-04 --7 344jVolues
Ist mode I 0-5
of E* I
0
2
4 Re (o~*)
I
9
1.05 - Flutter threshold
/
I
1
0
0-5
I'0
1.5
2-0
2.5
E*
Figure 9. Argand diagram of in-plane motion of a clamped-sliding semi-circular pipe conveying viscous fluid as functions of the flow velocity for ~ = 0.5, ~ = 104 (a) Argand diagram, (b) real part of the frequencies. The parameters are the same as in Figure 6; ri ° 4: 0, ~/(r/]' + 0r/~) ~ 0.
258
A.K.
M1SRA E T A L .
increasing nondimensional flow velocity t~* is shown in Figure 9(a) and (b). Figure 9(a) shows the Argand diagram, while in Figure 9(b) the real part of the frequencies is plotted against ti*. The effect of low flow velocity is to damp the motion in all three modes; but when the flow velocity is sufficiently high (ti* = 0.99), the first mode loses stability by flutter [Figure 9(a)]. When the flow velocity is increased further, the pipe buckles in the first mode at ti* = 1.05 and subsequently in the second mode at ti* =- 1.1 [Figure 9(b)]. The variation of the first three in-plane frequencies of a quarter-circular pipe (0 = n/4) with the nondimensional flow velocity is shown in Figure 10. All the parameters, except 0, are the same as in Figure 9. The pipe in this case also loses stability by flutter, but in the second mode and at a much higher flow velocity (a* = 2.91). It appears that, qualitatively, the dynamical behaviour of a clamped-sliding pipe is similar to that of a cantilever pipe, although the sequence of instabilities may not be the same. It is interesting to compare the results obtained for the clamped-sliding pipe using the extensible and the modified inextensible theories. It was noted that, in the case of the latter, the combined force H ° and hence the frequencies depend on whether the force-balance approach or Castigliano's theorem is used and also on the nature of the sliding end. Sliding boundary conditions can be represented by any of the following three ways: (i) 6 = q~ + Oil3 = O, My = 0; (ii) t/; = 0, My = 0; and (iii) r/; = 0, My GO. The critical flow velocity is dependent on the type of the sliding end. 1"2
T
~
J~-
r
7
T
,2.91 -- -Ist mode
1.0 0.8 0.6 T o
E H
Vcfluesof ff *
2-~
J
0.4 0.2
~'9~rd
225 _
~_-8 0
mode
f~ I -
0
o
-0.2
0
Znd mode
-0.4 2.91 -0.6
0
--
J
2
__
l
4
d
I
l
6
8 IO-I
IO
Re (w)x
J
12
14
Figure 10. Dimensionless in-plane frequencies of a clamped-sliding quarter-circular pipe conveying viscous fluid as functions of the flow velocity for fl = 0.5, ~¢ = l04. The parameters are the same as in Figure 6; 17° ~: 0, ~ ( n ~ ' + 0n.~) ~ 0.
259
CURVED PIPES TRANSPORTING FLUID. PART II
- - ~ T T T 1"5I I'0 "'j2ndm°de I~3rdm°de
T
"[ T-_[~ ~,
N 4th mode 3 Values of ~*
1.4~ 5 0.5
E
F
O\ 0 Ist mode
0
4
6
-0.5 -I-0 -
0
4
L 8
1 . L . _ _ ~ 12 t6 Re(w*)
20
Figure 11. Argand diagram of in-plane motion of an inextensible semi-circular clamped-axially sliding pipe of type (i), defined in the last paragraph of the preceding page, for fl = 0.5.
Figure 11 shows the Argand diagram of in-plane motion of an inextensible semi-circular clamped-sliding pipe for fl = 0.5. The sliding boundary condition is of type (i) and the force balance approach was used to determine 1-I°. It may be noted that the pipe loses stability by buckling in the first mode, as opposed to flutter predicted by the extensible theory. The critical flow velocity here (t~* a little larger than 1.5) is higher than that in the extensible case (/~* - 0.99). If the inextensible pipe has sliding ends of type (ii) or (iii), again it loses stability by buckling. The important thing to note is that, unlike the case of truly clamped-clamped (or pinned-pinned) pipes, the presence of a sliding end is sufficient to permit an instability to occur. 5. CONCLUSION The dynamics and stability of extensible curved pipes conveying fluid have been studied in this paper. Pipes clamped at both ends as well as pipes that can slide at the downstream end were considered. The initial shape of an extensible pipe (i.e., at t~ = 0) changes under the action of the centrifugal forces acting on the system at any given t~, and if the flow is realistically considered to be viscous, by the action of frictional forces also. For inviscid flow, the stressed shape changes symmetrically vis-h-vis the unstressed shape of a clampedclamped pipe; but this change is asymmetrical if the flowing fluid is viscous. The deformations however, are fairly small for both types of flow.
260
A . K . MISRA E T A L .
For an inextensible pipe, there is no difference between the values of steady-state combined force H ° for inviscid and viscous flows; in both cases, it is equal to -t~ 2. For an extensible pipe, however, there is a difference between viscous and inviscid results, especially at high flow velocities. The equations of motion generally depend on those of static equilibrium for any given flow velocity, which must be solved first and then used for the solution of the equations of motion. This is true for the in-plane equations of motion, whereas, at least with the approximations introduced in this paper, out-of-plane motions are unaffected. Indeed, the interesting finding was made that the out-of-plane motions are identical to those of the modified inextensible theory of Part I [1]. Hence, attention in this paper is confined to the in-plane motion. It was observed that as long as the steady-state forces are taken into account, it does not make any difference whether the terms involving the initial deformations in the equations of motion are considered or neglected. The results obtained are qualitatively the same and differ very little quantitatively. They both lead to the same conclusion that the frequencies of curved pipes clamped at both ends do not change greatly with increasing flow velocity, and the system thus never loses stability. The results compare quite well with the extensible theories of Hill and Davis and Doll and Mote, and the modified inextensible theory presented in Part I of this work. Since the differences in the frequencies as obtained from the modified inextensible and extensible theories are small, the former may be used instead of the latter (at least for pipes with both ends supported) for any practical purposes. The reason for doing so is that obtaining solutions by the extensible theory is rather expensive. To achieve convergent results by the extensible theory, many finite elements are necessary--easily more than 30--this number increasing with ~ / = AtL2/L This contrasts with similar convergence being achieved by the modified inextensible theory with 5 to 10 elements. The above discussion, of course, applies provided that the stressed centreline remains close to the unstressed one. If it does not, then the extensible theory must be used. If one end of the pipe is allowed to slide axially (even though it is clamped as far as the transverse motion is concerned), then there is a drastic change in the stability behaviour of the pipe. At sufficiently high flow velocity, the pipe becomes unstable by flutter, in the first mode for a semi-circular pipe and in the second mode for a quarter-circular pipe. This is an entirely new observation, not previously reported by any of the earlier investigators. It may be noted that no results are presented for cantilevered pipes in this paper. This is because, for extensible cantilevered pipes, the static deformations may be large and hence, should be determined by solving the nonlinear equations of static equilibrium, and not by solving the linearized equations as was done in this paper. ACKNOWLEDGEMENT The authors gratefully acknowledge the support given to this research by the Natural Sciences and Engineering Research Council of Canada, Le Fonds FCAR of Qu6bec and McGill University. REFERENCES
1. A. K. MmRA, M. P. PA~r~oussIs and K. S. VAN 1988 On the dynamics of curved pipes transporting fluid. Part I: Inextensible theory. Journal of Fluids and Structures 2, 221-244. 2. H. SCHLICHT~NO1960 Boundary Layer Theory. New York: McGraw-Hill. 3. J. L. HILLand C. G. DAvis 1974 The effect of initial forces on the hydroelastic vibration and stability of planar curved tubes. Journal of Applied Mechanics 41, 355-359. 4. R. W. DoLL and C. D. MOTE, JR 1974 The dynamic formulation and the finite element
CURVED PIPES TRANSPORTING FLUID. PART II
261
analysis of curved and twisted tubes transporting fluids. Report to the National Sciences Foundation, Dept of Mechanical Engineering, University of California, Berkeley. 5. R. W. DOLL and C. D. MOTE, JR 1974 On the dynamic analysis of curved and twisted cylinders transporting fluids. Journal of Pressure Vessel Technology 98, 143-150. 6. M. P. PnYDoussis 1986 Discussion on "Application of the transfer matrix method to non-conservative systems involving fluid flow in curved pipes" by C. Dupuis and J. Rousselet. Journal of Sound and Vibration H I , 167-168. APPENDIX
A
The integrals [11]-[1=3],{F~)-{F4} and [Ao] appearing in equations (22)-(23) and (29)-(31) are defined as follows:
[/11= L~"[@2]r[(p2]d~',
[/2] = ffe [(P2]'r[~2] d~,
[/3]= L e° [¢d"'[,P=]dC,
[/4] = f0~e[(~4]T[(j)4]d~,
[]51 = L e~ [¢21~[¢=] ' dC,
[/6] = Lce [(Pz]"r[~b4] d~,
[/71= L Ce[(~2]tT[~)4] 'dC,
[18] = L e~[~4]'T[(p41 ' d~,
[/9] ---~L ee [q~)2]T[(~)2]" d~',
[11o1 f f C[,~=]~[¢=]" dC,
[/11]=
=
fo:eC[,~d~[¢d' dC.
[/12] = foe° [q~zlT[(])4]' rig,
[]ld = foe"[q~4V[ed'dC,
[1141= ffe [@2]'[4)41d,~,
[]15] = L ~e[~2]'tT[(]~4] ' d~',
[],61= f f C[,/,dq~,4]' tiC,
[/17]= foe"C[,/..V[@d'dC, [/181 = L ee C[@4]r[q~4] d,~,
where
[1201 = fO~eC[~)4]T[(~4] 'dC/,
[]~11 = f f ¢[@,V[,/,~[' dC,
[122] = L ~e C[I~)2]tT[~4] 'dC,
[]2~]= Le"C[,Pdq¢~] d4,,
[q)2l = [1, ¢, C=, ~3, O, O] and 0 0 0
0 0 0
0 0 1
0 0 0
C~
C~
o
o
2Ce
3C~
0
0
o
o
1
Co
[q~=lTdC,
{5) =
[~)4] TdC,
fo fo
(~} = -
I 0
[.4o1=
{F3} -
fo fo
1
0 0 1 ~ 0 0
(F~} =
[qL] = [0, O, O, O, 1, C];
1 0
C[q,:]" dC, ~(O~o[¢~1" + ~o[,P41 ") OC.