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The effect of viscosity on free vibrations of submerged fluid-filled spherical shells
Journal o[Sottnd and Vibration (1981) 77(1), 101-125
T H E EFFECT OF VISCOSITY O N FREE V I B R A T I O N S OF SUBMERGED FLUID-FILLED SPHERICAL SHELLS T. C. Su Department of Civil Engineering, Texas A & M University, College Station, Texas 77843, U.S.A. (Received 14 April 1980, and in revised form 8 January 1981) In order to clarify the effect of fluid viscosity on the vibration of submerged elastic shells, the axisymmetric free oscillations of a fluid-filled spherical shell immersed in a sound field are studied. The dynamic response of the shell is determined by the classical normal mode method, while a boundary layer approximation is employed for the fluid medium. In the absence of viscosity, the shell motion is always damped due to the compressibility of the fluid outside the shell. It is shown that, except for the appearance of natural frequencies with a large damping component, the presence of surrounding fluid outside a fluid-filled shell produces only small changes in the real part of the frequency spectra. The analysis of the influence of viscosity reveals that the viscosity has essentially no effect on the frequencies of shells of moderate thickness. However, the viscous damping is predominant for the non-radiating modes of a fluid-filled submerged shell and the damping is due solely to viscosity for all modes if the outer fluid is assumed incompressible. Moreover, for very thin shells, viscous effects are noticeable on both frequencies and total damping, as the frequency is close to the upper branch of the in oactto frequency. It also is found that in the presence of compressible flow outside the shell, the damping effect of viscosity can be either positive or negative. Thus, the damping is reduced by fluid viscosity for certain cases. This implies that both the damping of the shell motion and the acoustic radiation damping are reduced by fluid viscosity, since the viscous dissipation damping must remain positive. It also is noted that in the case of negative viscous damping, the inviscid damping is always predominant and the viscous contribution is rather insignificant. A detailed examination indicates that the viscosity of the contained fluid will, in general, produce a damping effect, except near the valley of the negative portion of the damping curve where the viscosity of the inner fluid will reduce the damping even further. On the other hand, the viscosity of the surrounding fluid will, in general, increase the damping when frequency of vibration is less than the critical frequency, which is somewhat above the upper branch of the in vacuo frequency. The damping is reduced by the viscosity of external fluid for frequencies associated with large inviscid damping, or when the frequency of vibration is greater than the critical frequency. 1. INTRODUCTION An extraordinary amount of research has been performed in recent years on problems involving dynamic interactions between elastic structures and the surrounding or contained fluid medium, particularly for structures having large areas of contact with the adjacent fluid, such as plates and shells. Free vibrations of elastic shells submerged in an acoustic medium have been investigated by several authors [1, 2]. Free vibrations of fluid-filled shells also have received a good deal of attention [3-6]. In all these investigations, however, the effect of viscosity has been omitted. It is well known that although for m a n y practical applications the viscosity is vanishingly small and that small amplitude oscillatory fluid motion is potential flow to a first approximation, near the surface of the oscillating shell there exists a thin layer in which the flow is rotational. This rotational flow regime, in which the viscous effect is noticeable, 101 0022-460x/81/130101 +25 $02.00/0 O 1981 Academic Press Inc. (London) Limited
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has a direct influence on the hydrodynamic forces acting on the shell and consequently might have noticeable effects on the vibration characteristics of shells. The objective of this study is to clarify the effect of viscosity on fluid-structure interactions by investigating the axisymmetric vibrations of a fluid-filled elastic shell submerged in a compressible fluid medium. This is an extension of previous studies by the author of submerged empty shells and fluid-filled shells h~ vacuo [7-9]. Based on the classical bending theory, the dynamic response of the shell is determined by the normal mode method. For the fluid medium, the linearized Navier-Stokes equations for adiabatic compressible fluid are further simplified by boundary layer approximations. From this, the first order viscous effects are evaluated. 2. BASIC EQUATIONS 2.1. BASIC EQUATIONS FOR SHELL Based on the classical bending theory of spherical shells, the mode shapes for axisymmetric, torsionless vibration of the shell may be expressed by Legendre polynomials Pk(COS 0) and their derivatives. Except for k = 0, there exist two natural radian frequencies, (.ok and ta~,, for each value of k. The mode shapes for both frequencies are identical, the ratios of the tangential to radial displacements are different. The natural frequencies are determined by the frequency equation as given in reference [10], for example. The mode shapes corresponding to to~, are designated by W~, and V~, and are given by W~, = Pk(COS 0),
V~, = C'k(d/dO)Pk(COS O),
(1)
where
(1 + vs)(1 + e~) +
es(k
-
1)(k + 2)
C'k=(l+es)(kE+k_l+v,)_(i_ v,2 )C~,kRIc,) , 2,
(2)
vs is the Poisson's ratio, cs = .4E/p,, E is the Young's modulus of the shell material, p, is the density of the shell material, e~ = h2/12R 2, and h and R are the thickness and radius of the shell, respectively (see Figure 1). The mode shapes corresponding to (.ok are designated by Wk and Vk and can be obtained from the given expressions by omitting the primed superscripts. It should be noted, however, that cok, Wk and Vk do not exist for k = 0. w
Figure 1. Shell geometry.
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With the notation given, the dynamic response of the submerged shell can then be .~xpressed in terms of normal modes: V = ~. [~k(t)Vk(O)+ri~k(t)V~k(O)], k=1
rloWo + k-I
[rlk(t)~Vk(O)+~l'k(t) wk(0)], '
(3)
~vhere r/k and r/~, are normal co-ordinates, determined by the equations of motion
+~Ok'Ok= 9. , -
~k "i-tOk ~tk =
(r~Vk--p~Wk) dA/21rR2p~hMk, (TsVrk--psW'k) dA/2zrR2p, hM'k.
(4)
2rrREp,hMk and 2zrR2pJZM'k are the generalized masses, and % and p~ are tangential and normal hydrodynamic forces which must be determined from the following governing equations for the fluid field. 2.2. BASIC EQUATIONS FOR THE FLUID MEDIUM The general equations for a viscous, compressible, heat-conducting fluid can be found, Eor example, in the book by Moore [ll'l. In the present study, it is assumed that the amplitude of oscillation, a, is small in comparison with both the radius of the shell, R, and the depth of penetration of the rotational flow, 8. Furthermore, since the oscillations are ~mall, the relative changes in the fluid density, pressure, temperature, and transport parameters also are small. Therefore, these equations can be linearized to yield the [ollowing equations for the fluid: :ontinuity equation:
Op'/at + Po div u = O;
(5)
momentum equation (with volume force absent): po Ou/Ot = -grad p' + tzoV2u + (~ro+ tXo/3) grad div u;
(6)
energy equation (without external heating): 0s'
-~- = - ( l / n o ) div ( q / T ) - (1/aoT2)(q. grad T) + (1/ao)Cl,dT.
(7)
Here p, p, and T are the density, pressure, and temperature of the fluid, the subscript 0 represents the constant equilibrium value of a variable and a prime a small variation of the quantity concerned; u is the velocity vector; s is the entropy per unit mass; and q and ~r are the heat conduction vector and the rate of dissipation of energy per unit volume, respectively; ~'o= Ao+3Z/.tois the volume or bulk viscosity, and Ao is the second viscosity. It can be shown that for small oscillations the last two terms in equation (7), which correspond to the entropy increase due to heat conduction and viscous dissipation, are of secondary importance and can be neglected. Furthermore, it is assumed that no heat conduction takes place across the fluid-shell interface: i.e., the shell is insulated. This implies that the thermal boundary layer is not present and consequently the temperature would change appreciably only over the distance of the order of R. It can then be shown, by estimating the order of magnitude of the remaining terms in equation (7), that the
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change in entropy in its dimensionless form , s'/cp is of the order of ( T ' / T ) ( a / R ) x (cd~/k)-t(pa[2R/lt) -t. Therefore, the change in entropy, s', also is of second order importance provided that the reciprocal of the product of the Reynolds number,
p(al2)R/iz, and the Prandtl number, cplt/k, is at most of order 1 (cp is the specific heat at constant pressure). This condition can be fulfilled if the frequency ~ is sufficiently high. Thus, the energy equation is irrelevant and the system can be considered to be adiabatic to the first order. For adiabatic flow, the relationship between p' and p' can be given simply as p' = (op/Ooo),p' = c~oo ',
(8)
where Co is the constant equilibrium value of the sound speed in the fluid medium. The basic equations for adiabatic compressible fluid can therefore be summarized as follows: the continuity equation (5), the momentum equation (6), the p' and p' relationship (8), and the viscous stress tensor ,r = A0(div u)I + i~o(def u).
(9)
Taking the divergence of equation (6) and substituting equations (5) and (8) into the result, gives an uncoupled equation for p', (1/C2o) O2p'/Ot 2 = V2p'+ (1/C~po)(~bto + ~o)V2(ap'/Ot),
(10)
while the substitution of equations (5) and (8) into equation (6) yields the following equation for u:
po au/at = -(1 + [((o + l~o/3)/poc2o] o/ot)Vp' +/.t oVZu.
(11)
Once the pressure equation (10) is solved for a given boundary condition, the momentum equation (11) becomes a non-homogeneous vector diffusion equation for u. The solution for u and p, together with the stress strain relationship (9), completely prescribes the hydrodynamic forces acting on the shell. 2.3. B O U N D A R Y CONDITIONS At the fluid-structure interface, it is generally assumed that the local velocity of the fluid is equal to the local velocity of the solid with which it is in contact (non-slip condition). Consequently, the boundary conditions at the fluid-shell interface can be written as
o,(R, O, t) = W,,(R, O, t),
vo(R, O, t) = V,,(R, O, t),
(12, 13)
where v, and vo are the radial and tangential components of the Velocity vector u, respectively. In addition, it is required that the velocity remain finite as r ~ 0 and the solution must satisfy the Sommerfeld radiation condition as r--~ co. 3. METHOD OF SOLUTION 3.1. T H E B O U N D A R Y L A Y E R A P P R O X I M A T I O N S Sufficiently near the shell, the solution of the boundary layer equation is shown to be a good approximation to the solution of the Navier-Stokes equations provided that the viscosity is small, or, more precisely, that 8 / R - f ~ - ~ < < 1. Since the characteristic frequency associated with the structure is usually relatively high for most applications, the analysis based on boundary layer equations will yield sufficiently accurate solutions. For the potential flow region, tZo, 8o ~, 0 and the governing equations (10) and (11) reduce to
SUBMERGED FLUID-FILLED SHELL VIBRATIONS
105
Eulerian forinT: (1/c ~) a2p/Ot 2 = V2p,
p au/Ot = - V p .
(14, 15)
Within the boundary layer, however, the effect of viscosity is significant. A boundary ayer theory [12] will reduce the 8-momentum component of equation (11) to the form, calid only within a thin layer near the spherical shell, p Ooo/Ot = - ( l / R )
ap/oal,-R +~ o2ve/Or2,
(16)
r162 the inner boundary condition vo = V,, at r = R
(17)
and the outer boundary condition OO = DO iaviscidlr-R- at OO =
(R
-
r ) / ( v i / f l ) 1/2 >>1,
OOinviscialr-R§ at ( r - R ) / ( v o / f l )
t/2 >>1.
(18) (19)
Based on the boundary layer approximation, the pressure p of the fluid is determined from the inviscid flow theory for the entire flow field. The same is true for both the tangential velocity component, oe, and the radial velocity component, v,, outside the boundary layer. Near the solid surface, however, vo is governed by the boundary layer equation (16) and v, is determined from the continuity equation once oo is obtained. 3 . 2 FREQUENCY EQUATIONS
The solution based on these boundary layer approximations may be found in references [8] and [9] for the flow fields outside and inside the shell, respectively. Also given in these references are expressions for the hydrodynamic forces acting on the shell surfaces at
W
:
t
p
r'-R
"
o
Ir~R"
\,L-
Figure 2. Hydrodynamic forces acting on the surface of the shell.
r = R § a n d r = R - . The total hydrodynamic forces acting on the shell surface can then be obtained by superpositioning (see Figures 1 and 2):
ps = PI,=R§
-,
r, = 7 , o l , - , r - ~,ol,-R-.
Substituting these expressions for the hydrodynamic forces into shell equations of motion (4), and writing the normal co-ordinates ~Tkas r/k = qk e it~t (with a similar expression for ~7~,),one obtains for harmonic oscillations the following governing equations for the axisymmetric vibrations of a fluid-filled spherical shell submerged in a compressible fluid t Here, and henceforth, we omit for brevity the suffix in Po and go and the superscript in p',
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SU
medium. For k ~>1, one has
~,TJ - 0 ~-ff, ) J - T L qk [ _,. [1"1R~2
T, t,-ff-, , -,-~-, , -
o~ -~, ] j=o, (20a)
k-if/,
2
hL~, c, /
2
\ c, /
wheret
2 ~k =
glk[
Po Co H + P i
2k +---~ h
p, R/2 2
Co
.112
.
2
CI
~2k = Ao[ Ck --R--~c~H][ C'k - RI2C~H] + A,[ A=(l_i)~/~k(k+l) R 2k+l h
ci j ]
p, R.Q J'
1 ( c~_~O)
-R--~c'jj['lrCk' 112
C'Ri.J1j, 2
Mk, mk =Mg
P(R-'~c~) ,
Mk
(21)
and the expression for ~J3k is similar to that for r except that Ck must be replaced by C~. The functions H and J appearing in equations (21) are defined by (2),/nR\
t2)/-QR\ I
where h~ ) and j, are spherical Bessel functions of the third kind and of the first kind respectively. The primes in expressions (22) and (23) represent differentiation of a function with respect to its argument. For k = 0, qo does not exist and q~ is given by
q~o[(~roR~2 (.og~2 ~ (fflRI2 ] h L\ C~ /
\ c, /
_2R1__1__( po co H + Z _ ~clj ) . ~o- h Mo\ p, Rfl p,
J=O'
o\'-~,'-,]
(24)
Equations (20) and the first of equations (24) are a set of homogeneous algebraic equations. Requiring that non-trivial solutions exist for the displacements (qk/h) and (q'k/h) yields the frequency equations
(tO'kR/C,)2- (I2R/c,) 2- ~o(l'2R/c,) 2 = 0
(25)
for k = 0 and 4
\ Cs I
+
2
\ Cs I t
+
t
2
\
Cs I
\
2
Cs /
1'
i- Here, and henceforth, the suffices " i " and " o " indicate those quantities associated with the fluid inside and outside the shell, respectively.
SUBMERGED
FLUID-FILLED
107
SHELL VIBRATIONS a
/ +4s3kmk[(~-)2--(X+ClJk)(-~)2]+ 2qbkr
2} = 0
(26)
for k >/1. As mentioned previously, the viscosity effect appeared only through the functions elk, 4S2k and dX3kwhich vanish for inviscid fluid or for breathing vibrations (i.e., k = 0). One may immediately recover the frequency equations for the inviscid flow case by dropping terms involving IPlk, ~02k and IP3k. In general, the frequency equations (25) and (26), being non-linear equations involving transcendental functions of complex arguments, cannot be easily solved analytically. However, it can be shown that for up to moderate values of k, [~/lk[, [I//2k[ and 1r are of the order (v/Rcs) 1/2 and are small compared with the remaining inviscid hydrodynamic pressure terms. Therefore, the solutions of these frequency equations can be expressed in terms of the inviscid natural frequency/20, in a closed form, by Newton's method. The results are
/2R/cs = ~2oR/c,
(27)
for k = 0 and ~R Cs
Cs
t'|kCl)k
(~Jlk + 03k --21//:k) +
+<,~,<,,,,,r(.,-,o,/2_(,.<,~,<~ ,, c . , , c . , ~} i l l / , ,,-2--,, ,-77-,)
,-77-,, J ,
'),,
, c,,
01k
~(~o.i~ { , <,., , c. , <,.,
,,. ,
..,
, c, ,
" [., (<,,,<.~/"+(<,-'~
iki/j
. . , , T J ,,
(28)
for k ~> 1, where ~k, ~Oikand qbk (= d~k/d(.QR/cs)) are evaluated at 17 =/20. Therefore once the natural frequencies for the vibrations in an inviscid fluid are known, corresponding frequencies in a viscous fluid can be readily computed. This first order iteration of Newton's method provides sufficient accuracy consistent with the boundary layer approximation.
4. NATURAL FREQUENCIES FOR THE VIBRATIONS IN AN INVISCID FLUID 4.1. N U M E R I C A L M E T t t O D S - - O N E P A R A M E T E R A N A L Y S I S In the presence of outer fluid, the solution for/20 must be complex which leads to searching for roots in two-dimensional space [13]. Moreover, for a submerged shell filled with compressible fluid, frequency equations are no longer polynomial equations. Since the frequency equations then contain the Bessel's functions of the first kind which involve circular transcendental functions, there exists an infinite number of roots for each mode. The search of these roots is not necessarily an easy task. Since only the roots with small damping components are of significant interest for the free oscillations of the submerged shell, attention in this work has been focussed on the search for such roots. Consequently,
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the complex frequency/20 can be written as 12o = a0o(1 + ie),
(29)
where e << 1 and/'2o is the real part of the complex frequency/20. Substituting expression (29) into the frequency equation for the inviscid flow case, expressing each function in terms of its Taylor's series expansiont, and then dropping higher order terms in e, one can write the resultant equations in the form (Dl I + eD12) + i(D21 + ~'D22) = 0,
where D~I, etc., are real functions of/~0. Setting the real and imaginary parts to zero and eliminating e gives a single frequency equation involving only ~0: F(~o) = DIID22-DI2D21 = 0.
(30)
After .00 has been found, e is then given by e = -DEI/D22.
(31)
Therefore, instead of searching for roots in two-dimensional space directly, one can solve equation (30), which involves only one parameter/~o, and evaluate e from equation (31) to obtain an initial estimate of/~o. If e is sufficiently small, this initial estimate should be sufficiently close to a root. Then, iterative methods which are valid for complex roots, such as Newton's method, could be used to obtain the solution in a few.iterations. In this study, the method of t:,lse position has been used for the root search of equation (30) to obtain an initial estimate of/2o to start the calculation. Newton's method was then used to obtain the complex roots in the frequency equations. A double precision algorithm was used and/20 was obtained such that the modulus of the left-hand side of the frequency equations is less than 10 -1~ for k = 0 and k ~>0, respectively. This "one parameter analysis" was used by the author in the previous study of submerged empty shells [7, 14]. Improvement, however, is made by using the output as an initial estimate to obtain refined solutions by Newton's method. Thus, the validity of the solution obtained is not limited to the roots with small damping. 4.2. SOLUTION OF INVISCID FLOW PROBLEMS For comparison purposes, computation was made first for the vibrations of submerged empty shells. Corresponding to a steel shell in water at I0~ co/c, = 0.292, po/p, = 0.127, and ),, = 0.3 were used and h / R = 0.03 was selected. Frequencies obtained are tabulated in reference [13]. For each mode, two of these frequencies possess small damping components. The solid lines in Figure 3 are a plot:~ of the real parts of those two branches of frequencies. It shows that these radian frequencies of the submerged shell are lower than the corresponding frequencies h, vacuo (shown as dash-dot lines) due to an added mass effect. For the lower branch this effect is more pronounced, while for the upper branch the added mass eilects gradually approach zero as mode number increases. Plotted as solid dots are the frequencies possessing large damping components. Since the numerical method used in the present study could guarantee the finding of only those frequencies possessing small damping, these highly damped modes presented in this paper are incomplete. Nevertheless, they represent a class of natural frequency and therefore are t Although the expansion is invalid at the singular points ]~, = 0, it should be pointed out that these singular points correspond to the natural frequencies of a fluid-filled rigid shell and therefore are excluded in the present study of shell vibration. :I: It is to be noted that this and all other spectra plotted in this paper are discrete: i.e., only those points corresponding to integial V:~luesof the--mode-number k are iahysically meaningful.
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109
g o r
0
I
Z
3
4
5
6
7
8
9
I0
Mode number, k
Figure 3. Frequency spectrum for an e m p t y steel shell s u b m e r g e d in water.
, Real c o m p o n e n t s ; - - . - - ,
in vacuo.
included in the presentation. Solutions of this third family of modes were first given by Hayek and DiMaggio [2] for k =2. Complete numerical solutions can be found in reference [15]. The dash-dot lines (which sometimes merge with the solid lines) in Figure 4 are plots of damping components of the complex frequency spectrum l~oR/c,. It is seen that ~o is positive, indicating positive damping. Furthermore, for the lower branch Bo, l~oR/cs is monotonically decreasing for 1 0 > k > 1. However, it decreases less rapidly for k~>8. These curves clearly indicate that l~oR/cs will cease to be a monotonic function for large values of k. In the vicinity of k = 8, the damping is extremely small and thus shell motion is almost steady state (non-radiating). Further explanation of non-radiating modes can be found in reference [7] or [14]. While the case of small damping has been studied previously [7, 8, 14], the refined computation makes available natural frequencies with large damping components as presented in Figures 3 and 4 (shown as solid dots). Computation also was made for a steel shell in petroleum at 10~ in which c,/cs = 0.267, po/ps = 0.114 were used. Results presented in reference [13] exhibit essentially the same features as were previously discussed in the case in water. For the vibrations of fluid-filled shells in vacuo, natural frequencies are real. Thus the frequency equations for the inviscid flow case can be solved directly by methods of false
110
T. C. S U
2
I
I
i
I
I
I
I
I
I
I0 0
5 2 i0-1 5 2 10-2 5
li:
2 i0-3 5
o
E
2 10-4
o
5 (3. E D
2
eo
10-5
\
\
5 2 10-6
'\
5
\
2 10-7
-5x10-6 --10-5 2
II
I
I
2
3
I "~1~
I
4
6
Mode
5 number,
I
I
=
7
8
9
-SxlO -5 I0
k
Figure 4. Damping components of the natural frequencies for an empty shell submerged in water. Due to compressibility; - - - , due to viscosity; , total.
-----,
position. For comparison purposes, solutions are presented by the solid lines in Figures 5(a) and (b) for a steel shell filled with water and petroleum, respectively. These figures show a well-known feature, namely that the natural frequencies of a fluid-filled elastic shell follow very closely either those of a fluid-filled rigid shell (dash lines in these figures, defined by j~ = 0) or the in v a c u o natural frequencies [5, 6, 9]. The natural frequencies for the fluid-filled spherical shell submerged in an acoustic medium were then computed. As shown in reference [13] the natural frequencies for the fluid-filled submerged shells are complex. The computed natural frequencies are tabulated in reference [13] for a water-filled and a petroleum-filled steel shell submerged in water, and a water-filled shell submerged in petroleum, respectively. For all the cases studied, it is noted that the damping component is always positive, and thus the shell vibrations are damped. This damping is associated solely with the effect of compressibility. The real part of the complex frequencies are plotted in Figure 6(a), for a water-filled spherical shell in water, in Figure 6(b), for a petroleum-filled spherical shell in water, and Figure 6(c), for a water-filled spherical shell in petroleum. The properties of fluids and shells are the same as given in the previous discussion. The petroleum is chosen for its high viscosity. Figures 6(a)-(e) show that, except for those few frequencies that exhibit large damping (shown as solid dots), the frequency spectra for a fluid-filled spherical shell submerged in
SUBMERGED
FLUID-FILLED
SHELL
111
VIBRATIONS
Ill
I0
8
,g c
o E
0~ o rr
0
I
2
3
4
5
6
7
8
9
100
I
2
3
4
5
6
7
8
9
10
Mode number, k (o)
(b)
F i g u r e 5. F r e q u e n c y s p e c t r u m for (a) a w a t e r - f i l l e d shell a n d (b) a p e t r o l e u m - f i l l e d shell in vacuo.
, Steel
shell;---, rigidshell.
an acoustic medium follow closely those of a fluid-filled shell in v a c u o . Comparisons of Figures 6(a) or (c) and (b) with Figures 5(a) and (b) further reveal that the change in the real part of the frequency spectra due to the presence of the outer fluid is almost indiscernible except in the vicinity of to~, for small values of k and near tOk where the added mass effect due to the outer fluid is pronounced. Also, as would be expected, there exists little difference between the frequency spectra for a water-filled shell in water (Figure 6(a)) and in petroleum (Figure 6(c)). The damping components corresponding to the two lower branches of the frequency spectra (i.e., curves Be and B1 in Figure 6(a)) are plotted in Figure 7 for a water-filled shell submerged in water. This B~ branch, which is located near the first zero of j~,, does not have a counterpart in the case of a submerged empty shell and should not be confused with the Bt branch (membrane modes) there. For the lowest branch Be, the inviscid damping (represented by a dash-dot line) of intermediate modes is very small. Therefore, there exists a non-radiating phenomenon for intermediate modes of the lowest branch of the frequency spectra. Moreover, the magnitude of damping is smaller than in the case of the empty shell, as can easily be seen from Figures 7 and 4. Thus, the inviscid damping
112
T . C . SU
_o |
qu
\
u
,n
.E (x/
\
.
o
o
2
E o
m
rJ.~_
~. 0 8 =.M
u *m
o
\\ \
_o"E o o
a~
.o= o~
if)
"~
0
N r-.
-"
0
0")
~)
~
~0
s3/~o~.~ JO S~/H~
u")
~
'SlUauodtuo'~ IDaN
)0
(%/
--
o
SUBMERGED FLUID-FILLED SHELL VIBRATIONS 2 i0-1 .5 2
I
I
[
[
1
I
I
I
f
I
"7
I 9
I0
113
Bi
iO-Z 5
\
2 10-3
\
5 2
\
10-4 5
%
2 10-5
\
5 2 o o. E
8
5
\
2
o. E o
i
10-6
10-7
\
5 2 io-a
5 2 10-9
I !
I 2
I 3
I 4
I ,5
t 6
I 7
I 8
Mode number, k
Figure 7. Damping components of the natural frequencies for a water-filled steel shell in water--lower branches. - - . - - , Due to compressibility; - - - , due to viscosity; . . , total.
of the lowest branch of the frequency spectra of submerged shells is further reduced by the presence of the interior fluid. This is reasonable since the radian frequency is reduced by the added mass effect of the inner fluid so that acoustic wave lengths are increased and the non-radiating phenomenon is more pronounced according to a mechanism described in references [7] and [14]. On the other hand, with inner fluid, the damping of the additional BI branch is quite large, and the energy is rapidly radiated to the far field. For each mode it is noted that the damping components possess two peak values. One occurs on BI branch, and the other occurs in the vicinity of in vacuo frequencies to~, as shown in Figure 8. Besides, extremely large damping also occurs at isolated solid dots in the frequency spectra. Also shown in these typical curves is the feature that for each mode between these two peak values that appear on B1 the branch and near to~,, there is a minimum damping. This minimum damping occurs in the" vicinity of 9 2 tOk ( = ~/(mktO k + O kt 2 )/(tNk dr 1 ) ) which is very close to to~,. Moreover, damping for every mode approaches asymptotic to a line of constant slope at high frequencies. Similar plots corresponding to a petroleum-filled shell in water and water-filled shell in petroleum, respectively, are given in reference [13]. Those plots exhibit the same features as Figures 7 and 8.
114
T.
I0 0
I
I
C.
SU
I
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Figure 8. D a m p i n g c o m p o n e n t s of the natural f r e q u e n c i e s for a w a t e r - f i l l e d steel shell in w a t e r - - t y p i c a l modes. , T o t a l d a m p i n g ; - - - , viscous d a m p i n g .
5. EFFECT OF VISCOSITY In this section, the analytical expression (28) is evaluated numerically to obtain the natural frequencies of viscous flow problems, which can be written in the following form:
[2R
- - ~ - - ~ Cs
4 c,
,(
,
q-if-, --ff-, )'
where/2o is the natural frequency in the absence of fluid viscosity and ~ and/'~o are small perturbations due to the viscosity effect. As expected, the results indicate that ~ <<~o and the effect of viscosity on the real components of the natural frequencies are indiscernible for shells with moderate h/R ratio. Consequently, the radian frequencies remain practically unchanged with or without fluid viscosity. On the other hand, since/~o is small, /~o is relatively more significant or even predominant in some cases. It should be emphasized, however, that while ~o represents perturbations on the total damping
SUBMERGED FLUID-FILLEDSHELL VIBRATIONS
9
115
component due to the viscous effect, no distinction will be made, in the present study, as to what extent it contributes to the acoustic radiation damping or to the viscous dissipation. The natural frequencies of free vibrations of an empty spherical steel shell submerged in water and petroleum are tabulated in reference [13]. In the numerical computations (viRes) 1/2 = 2"93 x 10 -s and 1.05 x 10 -3 are used for water and petroleum respectively, corresponding to a shell with a radius R = 0.3048 m (1 ft). It is found that for both the branches Bo and BI, the viscosity of the outer fluid will cause the frequency to decrease, while for other frequencies, the viscosity of the outer fluid will cause the frequency to increase. In all cases, however, its contribution to /~ is indiscernible. Therefore, the frequency ~ is the same as given in the inviscid problem. More significant effects due to fluid viscosity are observed on the imaginary components of the natural frequencies. The damping component of frequency spectra is given in Figure 4, where viscous damping is shown as a dashed line, inviscid damping as dash--dot lines, and the total damping as solid lines. These curves indicate clearly that viscous damping is predominant for the intermediate modes of the lower frequency branch Bo, which corresponds to the region of non-radiating modes for the inviscid fluid solution. The figure also indicates that for the upper frequency branch BI, the viscous damping remains small compared with the damping due to compressibility. However, it may be noted that the former tends to increase whereas the latter tends to decrease with the mode number k. It is likely that the viscosity will have certain effects on the total damping for large values of k. In addition to the frequency branches Bo and BI mentioned above, there are other [requencies with large inviscid damping. Their viscous damping is found to be negative. Thus, the viscosity has the effect of reducing damping. However, total damping remains positive, since the inviscid positive damping is several orders of magnitude larger than the :lamping contribution due to viscosity. It should be remembered that the viscous damping represents the perturbation on the total damping component due to the effect of fluid r It is by no means all dissipative, since the viscosity could bring a change to the tcoustic radiation damping as well. This is particularly obvious for the case of negative r damping. In this case, the acoustic radiation damping must be reduced by fluid Jiscosity since the viscous dissipation should always remain positive. The effect of viscosity on free oscillation of fluid-filled spherical shells in vacuo has been audied [9]. Previous studies revealed that the fluid viscosity always produces a damping :ffect on the shell motion: i.e., /~ =/'~o ~>0. Moreover, the change on the real and maginary components of each natural frequency due to viscosity are equal in magnitude ~ut opposite in sign: that is,/~v = -/'~. Therefore, the frequency of a vibrating spherical .;hell will decrease with the increase of viscosity of the contained fluid. Typical curves on tamping components for the mode number k = 3, 4 given in Figure 9 reveal that viscous lamping reaches a peak value in the vicinity of hi vactto frequencies to~, and then drops 'apidly to a minimum. It is noted that the minimum damping is so small that the shell notion is almost steady state (non-dissipating). For the investigation of the effect of viscosity on free vibrations of submerged fluid-filled pherical shells, computations have been made for water-filled and petroleum-filled shells ubmerged in water, and a water-filled shell submerged in petroleum. Results of these :omputations are tabulated in reference [13]. Unless mentioned specifically, an h / R ratio ff 0.03 is assumed for this discussion. From the computed frequencies, it is again noted that the effect of viscosity on the real :omponents of the complex frequencies is very small. Therefore, the real components of he frequencies of a submerged fluid-filled shell remain practically unchanged with or
116
T . C. S U
10-4 5 2 10-5
k=4
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2 10-7 5
E
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IO-tO 5 2
i0~ 0
I
2
3
4
5
6
7
8
9
I0
II
Rodion frequency, -~R/cs
Figure 9. Damping components o[ the natural frequencies for a water-filled steel shell in vacuo---typical modes.
without fluid viscosity, and the results presented in Figures 6(a)-(c) are good for the present case. Significant effects due to fluid viscosity are observed on the imaginary components of the natural frequencies for some cases. The solid lines in Figure 7 are the plots of the total damping components for the Bo and B1 branches, which are the sums of viscous damping (broken lines) and the inviscid damping (dash--dot lines) due to the compressibility of the outer fluid. It is obvious that the viscous damping predominates in the non-radiating modes. It also is noted from Figure 7 that viscosity may have certain effects on total damping of the B~ branch for larger k. This becomes obvious for the case of petroleumfilled shells submerged in water [13]. The viscous damping for higher branches are plotted in Figure 8 as dashed lines. In the figure, the viscous damping of typical modes k = 3 and 4 are shown. It is noted that one of the peak values of viscous damping occurs in the vicinity of in v a c u o frequencies to~, and then drops rapidly to a minimum such as the one shown in Figure 9 for the case with absence of the outer fluid. It should be noted, however, that in the presence of the outer fluid the viscous damping is dropped to a negative value for some cases, so that the total damping is reduced by the effect of viscosity. These negative damping modes are indicated by circles in Figures 6(a)-(c). Nevertheless, a comparison with the inviscid damping shows that the negative viscous damping is small in magnitude and is insignificant in its contribution to total damping. It is noted that, for the higher frequency branches, the total damping of a submerged fluid-filled shell, in general, remains practically unaffected by the presence of fluid viscosity. Exceptions occur in the vicinity of O3k where the inviscid damping assumes its minimum value and viscous damping is close to its peak value and
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is positive. Therefore, in the vicinity of a3k, the viscous damping may have a significant contribution in the total damping, which is evident from Figure 8. Negative viscous damping is found also in all isolated frequencies between branch Bo and branch BI. Those frequencies are always associated with very large inviscid damping; therefore, viscous damping is relatively insignificant. All evidence nevertheless seems to indicate that reducing damping due to viscosity is related to the combined actions of the fluid viscosity and compressibility of the outer fluid. This will be elucidated in a later discussion of the effect of compressibility. Comparison of Figures 6(a) and (c) further indicates that with the increase of outer fluid viscosity, modes with negative viscous damping spread, while a comparison of Figures 6(a) and (b) indicates that with the increase of interior fluid viscosity, the number of modes with negative viscous damping is reduced. To study the effects of outer-fluid viscosity and inner-fluid viscosity separately for a fluid-filled submerged shell, computations were made for the case of water-filled shells in water, one with viscosity of the inner fluid set to zero and the other with viscosity of the outer fluid ignored. Frequencies computed are tabulated in reference [13]. Viscous damping values for typical modes are plotted in Figures 10(a) and (b) for the inviscid inner fluid case and for the inviscid outer fluid case, respectively. Also plotted in both figures for comparison purposes are the cases previously discussed with viscous fluid both inside and outside the shell. Figure 10(a) clearly indicates that the viscosity of the contained fluid in general will produce a damping effect, except near the valley of the negative portion of the damping curve where the viscosity of the inner fluid will reduce the damping even further. From Figure 10(b), it is noted that the viscosity of the surrounding fluid generally will increase the damping when the radian frequency of vibration is less than a critical frequency which is somewhat above the in oacuo frequency ~o~,, and the damping is reduced with the viscosity of outer fluid when the frequency of vibration is greater than the critical frequency or for a frequency with large inviscid damping. 6. EFFECT OF COMPRESSIBILITY To clarify the influence of fluid compressibility on the viscous damping, various cases of incompressible flow have been examined. Limiting cases of incompressible flow have been discussed by DiMaggio [16] and Hayek and DiMaggio [2] for fluid-filled shells and submerged shells, respectively. Both studies are concerned with inviscid fluid. As noted in previous studies (see referenc.e [7] or [14]), the validity of the incompressible flow analysis can be assured only for low frequency oscillations with 1-2<
H = - ( . O R / c o ) [ 1 / ( k + 1)]
(32, 33)
corresponding to the presence of incompressible flow inside and outside the shell, respectively. First one can consider a compressible-fluid-filled shell submerged in an incompressiblefluid medium and determine the influence of the compressibility of the outer fluid on the characteristics of natural frequencies. It can be analytically shown that, in the presence of the incompressible outer fluid, the change of the real and imaginary components of each natural frequency due to viscosity are equal in magnitude and opposite in sign. The imaginary part of the natural frequency, due solely to viscosity, is always positive. Thus, the fluid viscosity will produce a damping effect on the vibration and will reduce its frequency [13].
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From the result given above, it is obvious that the negative viscous damping is due to the effect of compressibility of the outer fluid, since the viscous damping is always positive when the outer flow is assumed incompressible. This is consistent with our physical intuition since when the outer flow is assumed incompressible, the acoustic radiation damping is always zero and the only damping mechanism left is the viscous dissipation. The occurrei~ce of negative viscous damping in this case would imply a negative viscous dissipation, which is, of course, not possible. Computations have been made for water-filled steel shells in water and for water-filled steel shells in petroleum. In both cases the outer flow is assumed incompressible and the computed inviscid damping, in general, is of the order 10 -17, which can be considered zero. The computed viscous damping is always positive and the change of the real and imaginary components of each natural frequency is equal in magnitude and opposite in sign as predicted in the analysis. It is also noted that, except for those frequencies involving large inviscid damping which no longer exist when the outer flow is assumed incompressible, the real components of natural frequencies dominated by interior fluid remain practically unchanged whether or not the fluid outside the shell is compressible. Whereas the damping components of typical modes plotted in Figures 1 l(a) and (b) indicate that when the outside fluid is assumed incompressible the viscous damping generally increases somewhat, compared with the compressible flow case, if the frequency is greater than a critical frequency which is somewhat above the in vacuo frequency ta~,, and vice versa if the frequency is less than the critical frequency. It is also noted that the damping for a water-filled shell immersed in incompressible water is, in general, larger than the corresponding case when the outer fluid is absent (see Figure 9). An exception occurs for the non-dissipating mode of k = 2, in which damping is smaller with the presence of incompressible flow outside the shell. This is noted to be the case even if the outside fluid is of high viscosity, such as petroleum. The limiting case of incompressible-fluid-filled shells submerged in a compressible-fluid medium also has been investigated. The inviscid frequency equation can be written as a polynomial equation in this case so that there exists only a finite number of roots for each mode. The computed frequencies are listed and plotted in reference [13]. It is noted that the spectra of radian frequencies for an incompressible-fluid-filled shell submerged in a compressible-fluid medium follows closely that of the empty shell in a compressible fluid or incompressible-fluid-filled shell hz vacuo. T h e frequency is generally lower in the present case due to the additional added mass effect of the inner fluid. This is particularly true for the lower branch of the frequency spectra and for the lower modes of the upper branch. Breathing vibration (k = 0) is not possible due to the presence of the incompressible fluid inside the shell. As to the damping components, the inviscid damping is reduced in the presence of the incompressible inner fluid. Also noted is that damping due to viscosity alone decreases for k ~<6 of the lower branch of the frequency spectra. Viscous damping is higher with the incompressible viscous fluid inside the shell for k > 6 in the lower branch and all k's in the upper branch. Moreover a frequency associated with large inviscid damping is obtained. The viscous damping of this highly damped mode is again negative, so that total damping is reduced by fluid viscosity. Thus, the compressibility of the inner fluid is not essential, while that of the outer fluid is essential for the existence of the reduced damping by viscosity. 7. EFFECT OF SHELL THICKNESS The computed results discussed so far are limited to thin shells of a moderate h / R ratio 0.03. One can now examine the viscous effect on the fluid-filled submerged shell when
SUBMERGED
FLUID-FILLED
SHELL
VIBRATIONS
121
the h / R ratio is further reduced. In the following discussion, the fluids are assumed viscous and compressible. Computations have been carried out for a water-filled shell with h / R = 0.003 submerged in water and water-filled shells with h / R = 0.003 and h / R = 0.0003, respectively, submerged in petroleum.
Q:
o E
8 13 n"
0
I
2
3
4
5
6
7
8
9
I0
Mode number, k
Figure 12. Frequencyspectrumfor a water-filledsteel shellwith h/R = 0.003 submergedin water (O, mode with negativeviscousdamping). Figure 12 shows plots of the real components of the natural frequency spectra. This figure in general, confirms the previous conclusion that, except for those few frequencies that exhibit very large inviscid damping, the frequency spectra for a fluid-filled spherical shell submerged in an acoustic medium follow closely those of a fluid-filled shell hi vacuo. It should be noted, however, while the natural frequencies of a fluid-filled elastic shell in vacuo follow very closely either those of a fluid-filled rigid shell (broken lines in Figure 5(a)) or the in vacuo natural frequencies, the degree of closeness decreases with h / R as shown in reference [9]. As expected, Figure 12 indicates that frequency spectra deviate from those of fluid-filled rigid shells for the in vacuo natural frequencies as the h / R ratio decreases. Added mass effects also are more pronounced for the lowest branch for thinner shells. The inviscid damping for typical modes is tabulated in reference [13]. The damping exhibits essentially the same features as compared to the case of the h / R ratio equal to
122
T.C. SU
0.03: that is, the low damping (non-radiating) for the lowest branch of frequency spectra Be, peak damping on the BI branch and in the vicinity of the in uacuo frequency ~o~,,and a minimum damping near 03k which is below but close to co~ Quantitatively, there are significant differences for varying h/R ratios. Inviscid damping for shells with small h/R ratios is in general large and shell motions will be rapidly damped, except for non-radiating modes on the Be branch and minimum damping near 03k. For the water-filled shells submerged in petroleum with h/R ratios equal to 0.0003, inviscid damping computed in most frequencies is almost constantly 0.33. However, minimum damping near 03 remains small and damping on the Be branch is even lower as h/R decreases. Lower damping on the Be branch is expected since the frequency there is lower for smaller h/R, so that for certain modes the acoustic wave number is less than the wave number of the vibrating surface, resulting in the existence of the non-radiating modes [7, 14]. The viscous effect is expected to be more pronounced for very thin shells, since the perturbation term in expression (28) which takes account of viscous effects is explicitly proportional to R/h through the functions $ik as defined in expression (21). In the computation of water-filled shells in petroleum with h/R = 0.0003, noticeable changes iO 0
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Figure 13. Damping components of natural frequencies for a water-filled steel shell with water. , Total damping; - - - , viscous damping.
h/R =
0 . 0 0 3 in
SUBMERGED
FLUID-FILLED
SHELL VIBRATIONS
123
of some frequencies occurred. Table 12 in reference [13] summarizes the computed results for typical modes in this case. It is noted that the viscous effect is most significant near to~,, where substantial reduction of radian frequencies could result through viscosity effects. The damping due to viscosity is plotted in Figure 13 in dotted lines. Negative viscous damping occurs more frequently for thinner shells, which is to be expected as a result of pronounced viscous effects and large inviscid damping associated with the thinner shells. However, due to large inviscid damping, the damping reduction by viscous action is insignificant and the total damping, plotted in solid lines, is hardly affected by the viscosity. However, on the Bo branch where the non-radiating mode exists and inviscid damping is extremely small, the viscous damping, which is positive there, predominates. Therefore, the shell motion also will be damped, although it remains unclear whether the damping is due mainly to the viscous dissipation or to the acoustic radiation to the far field by the viscous action. It also is noted that near O3k,where minimum inviscid damping occurred, the viscous damping is at its maximum and is predominating. 8. CONCLUSION In order to clarify the effect of viscosity on fluid-structure interaction, the investigation of axisymmetric vibrations of a fluid-filled spherical elastic shell submerged in a compressible fluid medium has been carried out by use of boundary layer theory. To elucidate the findings, various limiting cases have been studied, including the inviscid fluid limit, the incompressible flow limit and the thin shell limit. In the absence of viscosity, it is found that, in general, no real natural frequencies can exist for free vibrations of a fluid-filled submerged shell. Thus, the shell motion is always damped and the damping is due to the compressibility of the fluid outside the shell. Moreover, the natural frequencies generally follow closely those of a fluid-filled shell hz vacuo, which in turn follow closely either those of a fluid-filled rigid shell or the in v a c u o natural frequencies. Therefore, the presence of the surrounding fluid outside a fluid-filled shell produces only insignificant changes in the real part of the frequency spectra. It is noted, however, that between OJk and first zero of j~, there exist natural frequencies with large damping components due to the presence of compressible fluid outside the shell. The damping components for the lowest branch of the frequency spectra are very small and there exists a non-radiating phenomenon for intermediate modes of this branch which is more pronounced in the presence of interior fluid. This is due to the lower frequencies and therefore increasing acoustic wave lengths associatedwith the presence of the interior fluid. The damping components for each mode possess two peak values, where efficient sound radiation is possible. One appears on the B~ branch which is close to the first zero of j~, and the other occurs in the vicinity of in vacuo frequencies oJ~,. Besides those non-radiating modes, minimum damping occurs in the vicinity of a~k, which is very close to but lower than a~,. The analysis of viscosity reveals that the viscosity has essentially no effect on the real component of natural frequencies for shells with moderate h / R ratios. Consequently, the radian frequencies remain practically unchanged with or without fluid viscosity. However, the viscosity did produce noticeable effects on the damping part of the natural frequencies. It is found that the viscous damping is predominant for the non-radiating modes of a fluid-filled submerged shell. For all modes of a fluid-filled shell, either h~ vacuo or in an incompressible fluid medium, the damping is due to viscosity only. The viscous damping reaches its peak value when the radian frequency is close to the in vacuo frequency a~, while a minimum of inviscid damping occurred nearby, and noticeable effects on total
124
T.C. SU
damping due to viscosity are observed. This is particularly true as viscous damping becomes predominant there for very thin shells. Unlike the case of a fluid-filled shell hz vacuo in which the viscous damping is always positive and the viscous contributions to the real and imaginary components of the natural frequencies are equal in magnitude and opposite in sign, it is found that, in the presence of compressible flow outside the shell, the damping effect of viscosity can be either positive or negative. Thus, the total damping is reduced by fluid viscosity for certain cases, although in these cases of negative viscous damping the inviscid damping is always predominant and the viscous contribution is rather insignificant. It is further noted that for the modes possessing negative viscous damping, both the damping of the shell motion and the acoustic radiation damping are reduced by fluid viscosity, since the viscous dissipation damping must remain positive. Moreover, the study indicates that the viscosity of the contained fluid will produce a damping effect except near the valley of the negative portion of the damping curve where the viscosity of the inner fluid will reduce the damping even further. The viscosity of the surrounding fluid will, in general, increase the damping when the radian frequency of vibration is less than a critical frequency which is somewhat above the hz vactto frequency to~,, and damping is reduced with the viscosity of the outer fluid when either the radian frequency of vibration is greater than the critical frequency or frequencies associated with large inviscid damping. The dual role played by the viscosity is rather curious. A study has been carried out for limiting cases of incompressible flow to elucidate the role played by fluid compressibility in this reduced damping phenomenon. It is shown that the viscous damping is always positive and the viscous effect on the real and imaginary components of the natural frequencies are equal in magnitude and opposite in sign in the case of a fluid-filled shell submerged in an incompressible fluid medium. Therefore, inviscid damping is essential for the occurrence of reduced damping by viscosity. Studies of very thin shells show that viscous effects become more noticeable as the h / R ratio decreases. Both real and imaginary components of natural frequencies can be affected by viscosity in some cases.
REFERENCES 1. M. C. JUNGER 1952 Journal o[Applied Mechanics 19, 439-445. Vibrations of elastic shells in a fluid medium and the associated radiation of sound. 2. S. HAYEK and F. L. DIMAGGIO 1970 International Journal of Solids and Structures 6, 333-351. Complex natural frequencies of vibrating submerged spheroidal shells. 3. A. E. H. L o v e 1888 Proceedings of the London Mathematical Society 19, 170-207. The free and forced vibrations of an elastic spherical shell containing a given mass of liquid. 4. P. M. MORSE and H. FESHBACH 1953 Methods of Theoretical Physics, Part II. New York: McGraw-Hill Book Company. See pp. 1469-1472. 5. R. RAND and F. L. DIMAGGIO 1967 Journal of the Acoustieal Society of America 42, 1278-1286. Vibrations of fluid-filled spherical and spheroidal shells. 6. A. E. ENGIN and Y. K. LIU 1970 Journal of Biomechani'cs 3, 11-22. Axisymmetrie response of a fluid-filled spherical shell in free vibrations. 7. T. C. Su 1974 Eng. Sc. D. Dissertation, Columbia University, New York. The effect of viscosity on the dynamics of a spherical shell immersed in a fluid medium. 8. T. C. Stl and Y. K. LOll 1975 Jottrnal of Enghzeeringfor Industry, Transactions of the American Society of Mechanical Engineers 97, 1338-1344. The effect of viscosity on the dynamics of a submerged spherical shell. 9. T. C. So 1981 Journal of Sound and Vibration 74, 205-220. The effect of viscosity on free oscillations of fluid-filled spherical shells. 10. S. HAYEK 1966 Journal of the Acoustical Society of America 40, 342-348. Vibration of a spherical shell in an acoustic medium.
SUBMERGED FLUID-FILLED SIIELL VIBRATIONS
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11. F. K. MOORE (editor) 1959 Theory of LamhzarFlow. Princeton University Press. See pp. 3 4 46. 12. K. STEWARTSON 1964 The Theory of Latnhzar Boundary Layers in Compressible Fhdds. Oxford University Press. See pp. 15-22. 13. T. C. SU 1980 Ocean Engineerhzg Program, Texas A&M University, TEES Report No. COE-222. Natural frequencies of submerged fluid-filled spherical shells. 14. Y. K. L o u and T. C. S o 1978 Journal of the Acoustical Society of America 63, 1402-1408. Free oscillations of a submerged spherical shell. 15. C. A. FELLIPA and T. L. GEERS 1980 Journal of the Acoustical Society of America 67, 1427-1431. Axisymmetric free vibration of a submerged spherical shell. 16. F. L. DIMAGGIO 1975 Shock and Vibration Digest 7, 5-12. Dynamic response of fluid-filled shells.
T H E EFFECT OF VISCOSITY O N FREE V I B R A T I O N S OF SUBMERGED FLUID-FILLED SPHERICAL SHELLS T. C. Su Department of Civil Engineering, Texas A & M University, College Station, Texas 77843, U.S.A. (Received 14 April 1980, and in revised form 8 January 1981) In order to clarify the effect of fluid viscosity on the vibration of submerged elastic shells, the axisymmetric free oscillations of a fluid-filled spherical shell immersed in a sound field are studied. The dynamic response of the shell is determined by the classical normal mode method, while a boundary layer approximation is employed for the fluid medium. In the absence of viscosity, the shell motion is always damped due to the compressibility of the fluid outside the shell. It is shown that, except for the appearance of natural frequencies with a large damping component, the presence of surrounding fluid outside a fluid-filled shell produces only small changes in the real part of the frequency spectra. The analysis of the influence of viscosity reveals that the viscosity has essentially no effect on the frequencies of shells of moderate thickness. However, the viscous damping is predominant for the non-radiating modes of a fluid-filled submerged shell and the damping is due solely to viscosity for all modes if the outer fluid is assumed incompressible. Moreover, for very thin shells, viscous effects are noticeable on both frequencies and total damping, as the frequency is close to the upper branch of the in oactto frequency. It also is found that in the presence of compressible flow outside the shell, the damping effect of viscosity can be either positive or negative. Thus, the damping is reduced by fluid viscosity for certain cases. This implies that both the damping of the shell motion and the acoustic radiation damping are reduced by fluid viscosity, since the viscous dissipation damping must remain positive. It also is noted that in the case of negative viscous damping, the inviscid damping is always predominant and the viscous contribution is rather insignificant. A detailed examination indicates that the viscosity of the contained fluid will, in general, produce a damping effect, except near the valley of the negative portion of the damping curve where the viscosity of the inner fluid will reduce the damping even further. On the other hand, the viscosity of the surrounding fluid will, in general, increase the damping when frequency of vibration is less than the critical frequency, which is somewhat above the upper branch of the in vacuo frequency. The damping is reduced by the viscosity of external fluid for frequencies associated with large inviscid damping, or when the frequency of vibration is greater than the critical frequency. 1. INTRODUCTION An extraordinary amount of research has been performed in recent years on problems involving dynamic interactions between elastic structures and the surrounding or contained fluid medium, particularly for structures having large areas of contact with the adjacent fluid, such as plates and shells. Free vibrations of elastic shells submerged in an acoustic medium have been investigated by several authors [1, 2]. Free vibrations of fluid-filled shells also have received a good deal of attention [3-6]. In all these investigations, however, the effect of viscosity has been omitted. It is well known that although for m a n y practical applications the viscosity is vanishingly small and that small amplitude oscillatory fluid motion is potential flow to a first approximation, near the surface of the oscillating shell there exists a thin layer in which the flow is rotational. This rotational flow regime, in which the viscous effect is noticeable, 101 0022-460x/81/130101 +25 $02.00/0 O 1981 Academic Press Inc. (London) Limited
102
T . C . SU
has a direct influence on the hydrodynamic forces acting on the shell and consequently might have noticeable effects on the vibration characteristics of shells. The objective of this study is to clarify the effect of viscosity on fluid-structure interactions by investigating the axisymmetric vibrations of a fluid-filled elastic shell submerged in a compressible fluid medium. This is an extension of previous studies by the author of submerged empty shells and fluid-filled shells h~ vacuo [7-9]. Based on the classical bending theory, the dynamic response of the shell is determined by the normal mode method. For the fluid medium, the linearized Navier-Stokes equations for adiabatic compressible fluid are further simplified by boundary layer approximations. From this, the first order viscous effects are evaluated. 2. BASIC EQUATIONS 2.1. BASIC EQUATIONS FOR SHELL Based on the classical bending theory of spherical shells, the mode shapes for axisymmetric, torsionless vibration of the shell may be expressed by Legendre polynomials Pk(COS 0) and their derivatives. Except for k = 0, there exist two natural radian frequencies, (.ok and ta~,, for each value of k. The mode shapes for both frequencies are identical, the ratios of the tangential to radial displacements are different. The natural frequencies are determined by the frequency equation as given in reference [10], for example. The mode shapes corresponding to to~, are designated by W~, and V~, and are given by W~, = Pk(COS 0),
V~, = C'k(d/dO)Pk(COS O),
(1)
where
(1 + vs)(1 + e~) +
es(k
-
1)(k + 2)
C'k=(l+es)(kE+k_l+v,)_(i_ v,2 )C~,kRIc,) , 2,
(2)
vs is the Poisson's ratio, cs = .4E/p,, E is the Young's modulus of the shell material, p, is the density of the shell material, e~ = h2/12R 2, and h and R are the thickness and radius of the shell, respectively (see Figure 1). The mode shapes corresponding to (.ok are designated by Wk and Vk and can be obtained from the given expressions by omitting the primed superscripts. It should be noted, however, that cok, Wk and Vk do not exist for k = 0. w
Figure 1. Shell geometry.
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103
With the notation given, the dynamic response of the submerged shell can then be .~xpressed in terms of normal modes: V = ~. [~k(t)Vk(O)+ri~k(t)V~k(O)], k=1
rloWo + k-I
[rlk(t)~Vk(O)+~l'k(t) wk(0)], '
(3)
~vhere r/k and r/~, are normal co-ordinates, determined by the equations of motion
+~Ok'Ok= 9. , -
~k "i-tOk ~tk =
(r~Vk--p~Wk) dA/21rR2p~hMk, (TsVrk--psW'k) dA/2zrR2p, hM'k.
(4)
2rrREp,hMk and 2zrR2pJZM'k are the generalized masses, and % and p~ are tangential and normal hydrodynamic forces which must be determined from the following governing equations for the fluid field. 2.2. BASIC EQUATIONS FOR THE FLUID MEDIUM The general equations for a viscous, compressible, heat-conducting fluid can be found, Eor example, in the book by Moore [ll'l. In the present study, it is assumed that the amplitude of oscillation, a, is small in comparison with both the radius of the shell, R, and the depth of penetration of the rotational flow, 8. Furthermore, since the oscillations are ~mall, the relative changes in the fluid density, pressure, temperature, and transport parameters also are small. Therefore, these equations can be linearized to yield the [ollowing equations for the fluid: :ontinuity equation:
Op'/at + Po div u = O;
(5)
momentum equation (with volume force absent): po Ou/Ot = -grad p' + tzoV2u + (~ro+ tXo/3) grad div u;
(6)
energy equation (without external heating): 0s'
-~- = - ( l / n o ) div ( q / T ) - (1/aoT2)(q. grad T) + (1/ao)Cl,dT.
(7)
Here p, p, and T are the density, pressure, and temperature of the fluid, the subscript 0 represents the constant equilibrium value of a variable and a prime a small variation of the quantity concerned; u is the velocity vector; s is the entropy per unit mass; and q and ~r are the heat conduction vector and the rate of dissipation of energy per unit volume, respectively; ~'o= Ao+3Z/.tois the volume or bulk viscosity, and Ao is the second viscosity. It can be shown that for small oscillations the last two terms in equation (7), which correspond to the entropy increase due to heat conduction and viscous dissipation, are of secondary importance and can be neglected. Furthermore, it is assumed that no heat conduction takes place across the fluid-shell interface: i.e., the shell is insulated. This implies that the thermal boundary layer is not present and consequently the temperature would change appreciably only over the distance of the order of R. It can then be shown, by estimating the order of magnitude of the remaining terms in equation (7), that the
104
T, C. SU
change in entropy in its dimensionless form , s'/cp is of the order of ( T ' / T ) ( a / R ) x (cd~/k)-t(pa[2R/lt) -t. Therefore, the change in entropy, s', also is of second order importance provided that the reciprocal of the product of the Reynolds number,
p(al2)R/iz, and the Prandtl number, cplt/k, is at most of order 1 (cp is the specific heat at constant pressure). This condition can be fulfilled if the frequency ~ is sufficiently high. Thus, the energy equation is irrelevant and the system can be considered to be adiabatic to the first order. For adiabatic flow, the relationship between p' and p' can be given simply as p' = (op/Ooo),p' = c~oo ',
(8)
where Co is the constant equilibrium value of the sound speed in the fluid medium. The basic equations for adiabatic compressible fluid can therefore be summarized as follows: the continuity equation (5), the momentum equation (6), the p' and p' relationship (8), and the viscous stress tensor ,r = A0(div u)I + i~o(def u).
(9)
Taking the divergence of equation (6) and substituting equations (5) and (8) into the result, gives an uncoupled equation for p', (1/C2o) O2p'/Ot 2 = V2p'+ (1/C~po)(~bto + ~o)V2(ap'/Ot),
(10)
while the substitution of equations (5) and (8) into equation (6) yields the following equation for u:
po au/at = -(1 + [((o + l~o/3)/poc2o] o/ot)Vp' +/.t oVZu.
(11)
Once the pressure equation (10) is solved for a given boundary condition, the momentum equation (11) becomes a non-homogeneous vector diffusion equation for u. The solution for u and p, together with the stress strain relationship (9), completely prescribes the hydrodynamic forces acting on the shell. 2.3. B O U N D A R Y CONDITIONS At the fluid-structure interface, it is generally assumed that the local velocity of the fluid is equal to the local velocity of the solid with which it is in contact (non-slip condition). Consequently, the boundary conditions at the fluid-shell interface can be written as
o,(R, O, t) = W,,(R, O, t),
vo(R, O, t) = V,,(R, O, t),
(12, 13)
where v, and vo are the radial and tangential components of the Velocity vector u, respectively. In addition, it is required that the velocity remain finite as r ~ 0 and the solution must satisfy the Sommerfeld radiation condition as r--~ co. 3. METHOD OF SOLUTION 3.1. T H E B O U N D A R Y L A Y E R A P P R O X I M A T I O N S Sufficiently near the shell, the solution of the boundary layer equation is shown to be a good approximation to the solution of the Navier-Stokes equations provided that the viscosity is small, or, more precisely, that 8 / R - f ~ - ~ < < 1. Since the characteristic frequency associated with the structure is usually relatively high for most applications, the analysis based on boundary layer equations will yield sufficiently accurate solutions. For the potential flow region, tZo, 8o ~, 0 and the governing equations (10) and (11) reduce to
SUBMERGED FLUID-FILLED SHELL VIBRATIONS
105
Eulerian forinT: (1/c ~) a2p/Ot 2 = V2p,
p au/Ot = - V p .
(14, 15)
Within the boundary layer, however, the effect of viscosity is significant. A boundary ayer theory [12] will reduce the 8-momentum component of equation (11) to the form, calid only within a thin layer near the spherical shell, p Ooo/Ot = - ( l / R )
ap/oal,-R +~ o2ve/Or2,
(16)
r162 the inner boundary condition vo = V,, at r = R
(17)
and the outer boundary condition OO = DO iaviscidlr-R- at OO =
(R
-
r ) / ( v i / f l ) 1/2 >>1,
OOinviscialr-R§ at ( r - R ) / ( v o / f l )
t/2 >>1.
(18) (19)
Based on the boundary layer approximation, the pressure p of the fluid is determined from the inviscid flow theory for the entire flow field. The same is true for both the tangential velocity component, oe, and the radial velocity component, v,, outside the boundary layer. Near the solid surface, however, vo is governed by the boundary layer equation (16) and v, is determined from the continuity equation once oo is obtained. 3 . 2 FREQUENCY EQUATIONS
The solution based on these boundary layer approximations may be found in references [8] and [9] for the flow fields outside and inside the shell, respectively. Also given in these references are expressions for the hydrodynamic forces acting on the shell surfaces at
W
:
t
p
r'-R
"
o
Ir~R"
\,L-
Figure 2. Hydrodynamic forces acting on the surface of the shell.
r = R § a n d r = R - . The total hydrodynamic forces acting on the shell surface can then be obtained by superpositioning (see Figures 1 and 2):
ps = PI,=R§
-,
r, = 7 , o l , - , r - ~,ol,-R-.
Substituting these expressions for the hydrodynamic forces into shell equations of motion (4), and writing the normal co-ordinates ~Tkas r/k = qk e it~t (with a similar expression for ~7~,),one obtains for harmonic oscillations the following governing equations for the axisymmetric vibrations of a fluid-filled spherical shell submerged in a compressible fluid t Here, and henceforth, we omit for brevity the suffix in Po and go and the superscript in p',
106
T.C.
SU
medium. For k ~>1, one has
~,TJ - 0 ~-ff, ) J - T L qk [ _,. [1"1R~2
T, t,-ff-, , -,-~-, , -
o~ -~, ] j=o, (20a)
k-if/,
2
hL~, c, /
2
\ c, /
wheret
2 ~k =
glk[
Po Co H + P i
2k +---~ h
p, R/2 2
Co
.112
.
2
CI
~2k = Ao[ Ck --R--~c~H][ C'k - RI2C~H] + A,[ A=(l_i)~/~k(k+l) R 2k+l h
ci j ]
p, R.Q J'
1 ( c~_~O)
-R--~c'jj['lrCk' 112
C'Ri.J1j, 2
Mk, mk =Mg
P(R-'~c~) ,
Mk
(21)
and the expression for ~J3k is similar to that for r except that Ck must be replaced by C~. The functions H and J appearing in equations (21) are defined by (2),/nR\
t2)/-QR\ I
where h~ ) and j, are spherical Bessel functions of the third kind and of the first kind respectively. The primes in expressions (22) and (23) represent differentiation of a function with respect to its argument. For k = 0, qo does not exist and q~ is given by
q~o[(~roR~2 (.og~2 ~ (fflRI2 ] h L\ C~ /
\ c, /
_2R1__1__( po co H + Z _ ~clj ) . ~o- h Mo\ p, Rfl p,
J=O'
o\'-~,'-,]
(24)
Equations (20) and the first of equations (24) are a set of homogeneous algebraic equations. Requiring that non-trivial solutions exist for the displacements (qk/h) and (q'k/h) yields the frequency equations
(tO'kR/C,)2- (I2R/c,) 2- ~o(l'2R/c,) 2 = 0
(25)
for k = 0 and 4
\ Cs I
+
2
\ Cs I t
+
t
2
\
Cs I
\
2
Cs /
1'
i- Here, and henceforth, the suffices " i " and " o " indicate those quantities associated with the fluid inside and outside the shell, respectively.
SUBMERGED
FLUID-FILLED
107
SHELL VIBRATIONS a
/ +4s3kmk[(~-)2--(X+ClJk)(-~)2]+ 2qbkr
2} = 0
(26)
for k >/1. As mentioned previously, the viscosity effect appeared only through the functions elk, 4S2k and dX3kwhich vanish for inviscid fluid or for breathing vibrations (i.e., k = 0). One may immediately recover the frequency equations for the inviscid flow case by dropping terms involving IPlk, ~02k and IP3k. In general, the frequency equations (25) and (26), being non-linear equations involving transcendental functions of complex arguments, cannot be easily solved analytically. However, it can be shown that for up to moderate values of k, [~/lk[, [I//2k[ and 1r are of the order (v/Rcs) 1/2 and are small compared with the remaining inviscid hydrodynamic pressure terms. Therefore, the solutions of these frequency equations can be expressed in terms of the inviscid natural frequency/20, in a closed form, by Newton's method. The results are
/2R/cs = ~2oR/c,
(27)
for k = 0 and ~R Cs
Cs
t'|kCl)k
(~Jlk + 03k --21//:k) +
+<,~,<,,,,,r(.,-,o,/2_(,.<,~,<~ ,, c . , , c . , ~} i l l / , ,,-2--,, ,-77-,)
,-77-,, J ,
'),,
, c,,
01k
~(~o.i~ { , <,., , c. , <,.,
,,. ,
..,
, c, ,
" [., (<,,,<.~/"+(<,-'~
iki/j
. . , , T J ,,
(28)
for k ~> 1, where ~k, ~Oikand qbk (= d~k/d(.QR/cs)) are evaluated at 17 =/20. Therefore once the natural frequencies for the vibrations in an inviscid fluid are known, corresponding frequencies in a viscous fluid can be readily computed. This first order iteration of Newton's method provides sufficient accuracy consistent with the boundary layer approximation.
4. NATURAL FREQUENCIES FOR THE VIBRATIONS IN AN INVISCID FLUID 4.1. N U M E R I C A L M E T t t O D S - - O N E P A R A M E T E R A N A L Y S I S In the presence of outer fluid, the solution for/20 must be complex which leads to searching for roots in two-dimensional space [13]. Moreover, for a submerged shell filled with compressible fluid, frequency equations are no longer polynomial equations. Since the frequency equations then contain the Bessel's functions of the first kind which involve circular transcendental functions, there exists an infinite number of roots for each mode. The search of these roots is not necessarily an easy task. Since only the roots with small damping components are of significant interest for the free oscillations of the submerged shell, attention in this work has been focussed on the search for such roots. Consequently,
108
T . C . SU
the complex frequency/20 can be written as 12o = a0o(1 + ie),
(29)
where e << 1 and/'2o is the real part of the complex frequency/20. Substituting expression (29) into the frequency equation for the inviscid flow case, expressing each function in terms of its Taylor's series expansiont, and then dropping higher order terms in e, one can write the resultant equations in the form (Dl I + eD12) + i(D21 + ~'D22) = 0,
where D~I, etc., are real functions of/~0. Setting the real and imaginary parts to zero and eliminating e gives a single frequency equation involving only ~0: F(~o) = DIID22-DI2D21 = 0.
(30)
After .00 has been found, e is then given by e = -DEI/D22.
(31)
Therefore, instead of searching for roots in two-dimensional space directly, one can solve equation (30), which involves only one parameter/~o, and evaluate e from equation (31) to obtain an initial estimate of/~o. If e is sufficiently small, this initial estimate should be sufficiently close to a root. Then, iterative methods which are valid for complex roots, such as Newton's method, could be used to obtain the solution in a few.iterations. In this study, the method of t:,lse position has been used for the root search of equation (30) to obtain an initial estimate of/2o to start the calculation. Newton's method was then used to obtain the complex roots in the frequency equations. A double precision algorithm was used and/20 was obtained such that the modulus of the left-hand side of the frequency equations is less than 10 -1~ for k = 0 and k ~>0, respectively. This "one parameter analysis" was used by the author in the previous study of submerged empty shells [7, 14]. Improvement, however, is made by using the output as an initial estimate to obtain refined solutions by Newton's method. Thus, the validity of the solution obtained is not limited to the roots with small damping. 4.2. SOLUTION OF INVISCID FLOW PROBLEMS For comparison purposes, computation was made first for the vibrations of submerged empty shells. Corresponding to a steel shell in water at I0~ co/c, = 0.292, po/p, = 0.127, and ),, = 0.3 were used and h / R = 0.03 was selected. Frequencies obtained are tabulated in reference [13]. For each mode, two of these frequencies possess small damping components. The solid lines in Figure 3 are a plot:~ of the real parts of those two branches of frequencies. It shows that these radian frequencies of the submerged shell are lower than the corresponding frequencies h, vacuo (shown as dash-dot lines) due to an added mass effect. For the lower branch this effect is more pronounced, while for the upper branch the added mass eilects gradually approach zero as mode number increases. Plotted as solid dots are the frequencies possessing large damping components. Since the numerical method used in the present study could guarantee the finding of only those frequencies possessing small damping, these highly damped modes presented in this paper are incomplete. Nevertheless, they represent a class of natural frequency and therefore are t Although the expansion is invalid at the singular points ]~, = 0, it should be pointed out that these singular points correspond to the natural frequencies of a fluid-filled rigid shell and therefore are excluded in the present study of shell vibration. :I: It is to be noted that this and all other spectra plotted in this paper are discrete: i.e., only those points corresponding to integial V:~luesof the--mode-number k are iahysically meaningful.
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109
g o r
0
I
Z
3
4
5
6
7
8
9
I0
Mode number, k
Figure 3. Frequency spectrum for an e m p t y steel shell s u b m e r g e d in water.
, Real c o m p o n e n t s ; - - . - - ,
in vacuo.
included in the presentation. Solutions of this third family of modes were first given by Hayek and DiMaggio [2] for k =2. Complete numerical solutions can be found in reference [15]. The dash-dot lines (which sometimes merge with the solid lines) in Figure 4 are plots of damping components of the complex frequency spectrum l~oR/c,. It is seen that ~o is positive, indicating positive damping. Furthermore, for the lower branch Bo, l~oR/cs is monotonically decreasing for 1 0 > k > 1. However, it decreases less rapidly for k~>8. These curves clearly indicate that l~oR/cs will cease to be a monotonic function for large values of k. In the vicinity of k = 8, the damping is extremely small and thus shell motion is almost steady state (non-radiating). Further explanation of non-radiating modes can be found in reference [7] or [14]. While the case of small damping has been studied previously [7, 8, 14], the refined computation makes available natural frequencies with large damping components as presented in Figures 3 and 4 (shown as solid dots). Computation also was made for a steel shell in petroleum at 10~ in which c,/cs = 0.267, po/ps = 0.114 were used. Results presented in reference [13] exhibit essentially the same features as were previously discussed in the case in water. For the vibrations of fluid-filled shells in vacuo, natural frequencies are real. Thus the frequency equations for the inviscid flow case can be solved directly by methods of false
110
T. C. S U
2
I
I
i
I
I
I
I
I
I
I0 0
5 2 i0-1 5 2 10-2 5
li:
2 i0-3 5
o
E
2 10-4
o
5 (3. E D
2
eo
10-5
\
\
5 2 10-6
'\
5
\
2 10-7
-5x10-6 --10-5 2
II
I
I
2
3
I "~1~
I
4
6
Mode
5 number,
I
I
=
7
8
9
-SxlO -5 I0
k
Figure 4. Damping components of the natural frequencies for an empty shell submerged in water. Due to compressibility; - - - , due to viscosity; , total.
-----,
position. For comparison purposes, solutions are presented by the solid lines in Figures 5(a) and (b) for a steel shell filled with water and petroleum, respectively. These figures show a well-known feature, namely that the natural frequencies of a fluid-filled elastic shell follow very closely either those of a fluid-filled rigid shell (dash lines in these figures, defined by j~ = 0) or the in v a c u o natural frequencies [5, 6, 9]. The natural frequencies for the fluid-filled spherical shell submerged in an acoustic medium were then computed. As shown in reference [13] the natural frequencies for the fluid-filled submerged shells are complex. The computed natural frequencies are tabulated in reference [13] for a water-filled and a petroleum-filled steel shell submerged in water, and a water-filled shell submerged in petroleum, respectively. For all the cases studied, it is noted that the damping component is always positive, and thus the shell vibrations are damped. This damping is associated solely with the effect of compressibility. The real part of the complex frequencies are plotted in Figure 6(a), for a water-filled spherical shell in water, in Figure 6(b), for a petroleum-filled spherical shell in water, and Figure 6(c), for a water-filled spherical shell in petroleum. The properties of fluids and shells are the same as given in the previous discussion. The petroleum is chosen for its high viscosity. Figures 6(a)-(e) show that, except for those few frequencies that exhibit large damping (shown as solid dots), the frequency spectra for a fluid-filled spherical shell submerged in
SUBMERGED
FLUID-FILLED
SHELL
111
VIBRATIONS
Ill
I0
8
,g c
o E
0~ o rr
0
I
2
3
4
5
6
7
8
9
100
I
2
3
4
5
6
7
8
9
10
Mode number, k (o)
(b)
F i g u r e 5. F r e q u e n c y s p e c t r u m for (a) a w a t e r - f i l l e d shell a n d (b) a p e t r o l e u m - f i l l e d shell in vacuo.
, Steel
shell;---, rigidshell.
an acoustic medium follow closely those of a fluid-filled shell in v a c u o . Comparisons of Figures 6(a) or (c) and (b) with Figures 5(a) and (b) further reveal that the change in the real part of the frequency spectra due to the presence of the outer fluid is almost indiscernible except in the vicinity of to~, for small values of k and near tOk where the added mass effect due to the outer fluid is pronounced. Also, as would be expected, there exists little difference between the frequency spectra for a water-filled shell in water (Figure 6(a)) and in petroleum (Figure 6(c)). The damping components corresponding to the two lower branches of the frequency spectra (i.e., curves Be and B1 in Figure 6(a)) are plotted in Figure 7 for a water-filled shell submerged in water. This B~ branch, which is located near the first zero of j~,, does not have a counterpart in the case of a submerged empty shell and should not be confused with the Bt branch (membrane modes) there. For the lowest branch Be, the inviscid damping (represented by a dash-dot line) of intermediate modes is very small. Therefore, there exists a non-radiating phenomenon for intermediate modes of the lowest branch of the frequency spectra. Moreover, the magnitude of damping is smaller than in the case of the empty shell, as can easily be seen from Figures 7 and 4. Thus, the inviscid damping
112
T . C . SU
_o |
qu
\
u
,n
.E (x/
\
.
o
o
2
E o
m
rJ.~_
~. 0 8 =.M
u *m
o
\\ \
_o"E o o
a~
.o= o~
if)
"~
0
N r-.
-"
0
0")
~)
~
~0
s3/~o~.~ JO S~/H~
u")
~
'SlUauodtuo'~ IDaN
)0
(%/
--
o
SUBMERGED FLUID-FILLED SHELL VIBRATIONS 2 i0-1 .5 2
I
I
[
[
1
I
I
I
f
I
"7
I 9
I0
113
Bi
iO-Z 5
\
2 10-3
\
5 2
\
10-4 5
%
2 10-5
\
5 2 o o. E
8
5
\
2
o. E o
i
10-6
10-7
\
5 2 io-a
5 2 10-9
I !
I 2
I 3
I 4
I ,5
t 6
I 7
I 8
Mode number, k
Figure 7. Damping components of the natural frequencies for a water-filled steel shell in water--lower branches. - - . - - , Due to compressibility; - - - , due to viscosity; . . , total.
of the lowest branch of the frequency spectra of submerged shells is further reduced by the presence of the interior fluid. This is reasonable since the radian frequency is reduced by the added mass effect of the inner fluid so that acoustic wave lengths are increased and the non-radiating phenomenon is more pronounced according to a mechanism described in references [7] and [14]. On the other hand, with inner fluid, the damping of the additional BI branch is quite large, and the energy is rapidly radiated to the far field. For each mode it is noted that the damping components possess two peak values. One occurs on BI branch, and the other occurs in the vicinity of in vacuo frequencies to~, as shown in Figure 8. Besides, extremely large damping also occurs at isolated solid dots in the frequency spectra. Also shown in these typical curves is the feature that for each mode between these two peak values that appear on B1 the branch and near to~,, there is a minimum damping. This minimum damping occurs in the" vicinity of 9 2 tOk ( = ~/(mktO k + O kt 2 )/(tNk dr 1 ) ) which is very close to to~,. Moreover, damping for every mode approaches asymptotic to a line of constant slope at high frequencies. Similar plots corresponding to a petroleum-filled shell in water and water-filled shell in petroleum, respectively, are given in reference [13]. Those plots exhibit the same features as Figures 7 and 8.
114
T.
I0 0
I
I
C.
SU
I
I
I
I
I
I
I
I
5 2 i0-1 5 2 10-2 5 2 10-3 5 2 -
--3 o
.
10-4
,o, al
5
c c
g E o u
0"0 ,~r ",% ,/" ,',,,
iIIi I| ~
0 tI
2 oe
"o'o, ,o1,,
,
--
~-.
b
10-5
i
t l
5
l
o. E
I
2
l
10-6 5
,,.9;O:_o0-oO--o--o
p~ i a
2
ii
10-7
"~
t/
0 k=4 _
I !
_10-7
2
!
,l
5 _10-6 2
0
I
I
I
I
I
1
I
I
f
I
2
3
4
5
6
7
8
9
I I0
5 --iO-fl II
R a d i a n f r e q u e n c y , ~"~R/c s
Figure 8. D a m p i n g c o m p o n e n t s of the natural f r e q u e n c i e s for a w a t e r - f i l l e d steel shell in w a t e r - - t y p i c a l modes. , T o t a l d a m p i n g ; - - - , viscous d a m p i n g .
5. EFFECT OF VISCOSITY In this section, the analytical expression (28) is evaluated numerically to obtain the natural frequencies of viscous flow problems, which can be written in the following form:
[2R
- - ~ - - ~ Cs
4 c,
,(
,
q-if-, --ff-, )'
where/2o is the natural frequency in the absence of fluid viscosity and ~ and/'~o are small perturbations due to the viscosity effect. As expected, the results indicate that ~ <<~o and the effect of viscosity on the real components of the natural frequencies are indiscernible for shells with moderate h/R ratio. Consequently, the radian frequencies remain practically unchanged with or without fluid viscosity. On the other hand, since/~o is small, /~o is relatively more significant or even predominant in some cases. It should be emphasized, however, that while ~o represents perturbations on the total damping
SUBMERGED FLUID-FILLEDSHELL VIBRATIONS
9
115
component due to the viscous effect, no distinction will be made, in the present study, as to what extent it contributes to the acoustic radiation damping or to the viscous dissipation. The natural frequencies of free vibrations of an empty spherical steel shell submerged in water and petroleum are tabulated in reference [13]. In the numerical computations (viRes) 1/2 = 2"93 x 10 -s and 1.05 x 10 -3 are used for water and petroleum respectively, corresponding to a shell with a radius R = 0.3048 m (1 ft). It is found that for both the branches Bo and BI, the viscosity of the outer fluid will cause the frequency to decrease, while for other frequencies, the viscosity of the outer fluid will cause the frequency to increase. In all cases, however, its contribution to /~ is indiscernible. Therefore, the frequency ~ is the same as given in the inviscid problem. More significant effects due to fluid viscosity are observed on the imaginary components of the natural frequencies. The damping component of frequency spectra is given in Figure 4, where viscous damping is shown as a dashed line, inviscid damping as dash--dot lines, and the total damping as solid lines. These curves indicate clearly that viscous damping is predominant for the intermediate modes of the lower frequency branch Bo, which corresponds to the region of non-radiating modes for the inviscid fluid solution. The figure also indicates that for the upper frequency branch BI, the viscous damping remains small compared with the damping due to compressibility. However, it may be noted that the former tends to increase whereas the latter tends to decrease with the mode number k. It is likely that the viscosity will have certain effects on the total damping for large values of k. In addition to the frequency branches Bo and BI mentioned above, there are other [requencies with large inviscid damping. Their viscous damping is found to be negative. Thus, the viscosity has the effect of reducing damping. However, total damping remains positive, since the inviscid positive damping is several orders of magnitude larger than the :lamping contribution due to viscosity. It should be remembered that the viscous damping represents the perturbation on the total damping component due to the effect of fluid r It is by no means all dissipative, since the viscosity could bring a change to the tcoustic radiation damping as well. This is particularly obvious for the case of negative r damping. In this case, the acoustic radiation damping must be reduced by fluid Jiscosity since the viscous dissipation should always remain positive. The effect of viscosity on free oscillation of fluid-filled spherical shells in vacuo has been audied [9]. Previous studies revealed that the fluid viscosity always produces a damping :ffect on the shell motion: i.e., /~ =/'~o ~>0. Moreover, the change on the real and maginary components of each natural frequency due to viscosity are equal in magnitude ~ut opposite in sign: that is,/~v = -/'~. Therefore, the frequency of a vibrating spherical .;hell will decrease with the increase of viscosity of the contained fluid. Typical curves on tamping components for the mode number k = 3, 4 given in Figure 9 reveal that viscous lamping reaches a peak value in the vicinity of hi vactto frequencies to~, and then drops 'apidly to a minimum. It is noted that the minimum damping is so small that the shell notion is almost steady state (non-dissipating). For the investigation of the effect of viscosity on free vibrations of submerged fluid-filled pherical shells, computations have been made for water-filled and petroleum-filled shells ubmerged in water, and a water-filled shell submerged in petroleum. Results of these :omputations are tabulated in reference [13]. Unless mentioned specifically, an h / R ratio ff 0.03 is assumed for this discussion. From the computed frequencies, it is again noted that the effect of viscosity on the real :omponents of the complex frequencies is very small. Therefore, the real components of he frequencies of a submerged fluid-filled shell remain practically unchanged with or
116
T . C. S U
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Figure 9. Damping components o[ the natural frequencies for a water-filled steel shell in vacuo---typical modes.
without fluid viscosity, and the results presented in Figures 6(a)-(c) are good for the present case. Significant effects due to fluid viscosity are observed on the imaginary components of the natural frequencies for some cases. The solid lines in Figure 7 are the plots of the total damping components for the Bo and B1 branches, which are the sums of viscous damping (broken lines) and the inviscid damping (dash--dot lines) due to the compressibility of the outer fluid. It is obvious that the viscous damping predominates in the non-radiating modes. It also is noted from Figure 7 that viscosity may have certain effects on total damping of the B~ branch for larger k. This becomes obvious for the case of petroleumfilled shells submerged in water [13]. The viscous damping for higher branches are plotted in Figure 8 as dashed lines. In the figure, the viscous damping of typical modes k = 3 and 4 are shown. It is noted that one of the peak values of viscous damping occurs in the vicinity of in v a c u o frequencies to~, and then drops rapidly to a minimum such as the one shown in Figure 9 for the case with absence of the outer fluid. It should be noted, however, that in the presence of the outer fluid the viscous damping is dropped to a negative value for some cases, so that the total damping is reduced by the effect of viscosity. These negative damping modes are indicated by circles in Figures 6(a)-(c). Nevertheless, a comparison with the inviscid damping shows that the negative viscous damping is small in magnitude and is insignificant in its contribution to total damping. It is noted that, for the higher frequency branches, the total damping of a submerged fluid-filled shell, in general, remains practically unaffected by the presence of fluid viscosity. Exceptions occur in the vicinity of O3k where the inviscid damping assumes its minimum value and viscous damping is close to its peak value and
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is positive. Therefore, in the vicinity of a3k, the viscous damping may have a significant contribution in the total damping, which is evident from Figure 8. Negative viscous damping is found also in all isolated frequencies between branch Bo and branch BI. Those frequencies are always associated with very large inviscid damping; therefore, viscous damping is relatively insignificant. All evidence nevertheless seems to indicate that reducing damping due to viscosity is related to the combined actions of the fluid viscosity and compressibility of the outer fluid. This will be elucidated in a later discussion of the effect of compressibility. Comparison of Figures 6(a) and (c) further indicates that with the increase of outer fluid viscosity, modes with negative viscous damping spread, while a comparison of Figures 6(a) and (b) indicates that with the increase of interior fluid viscosity, the number of modes with negative viscous damping is reduced. To study the effects of outer-fluid viscosity and inner-fluid viscosity separately for a fluid-filled submerged shell, computations were made for the case of water-filled shells in water, one with viscosity of the inner fluid set to zero and the other with viscosity of the outer fluid ignored. Frequencies computed are tabulated in reference [13]. Viscous damping values for typical modes are plotted in Figures 10(a) and (b) for the inviscid inner fluid case and for the inviscid outer fluid case, respectively. Also plotted in both figures for comparison purposes are the cases previously discussed with viscous fluid both inside and outside the shell. Figure 10(a) clearly indicates that the viscosity of the contained fluid in general will produce a damping effect, except near the valley of the negative portion of the damping curve where the viscosity of the inner fluid will reduce the damping even further. From Figure 10(b), it is noted that the viscosity of the surrounding fluid generally will increase the damping when the radian frequency of vibration is less than a critical frequency which is somewhat above the in oacuo frequency ~o~,, and the damping is reduced with the viscosity of outer fluid when the frequency of vibration is greater than the critical frequency or for a frequency with large inviscid damping. 6. EFFECT OF COMPRESSIBILITY To clarify the influence of fluid compressibility on the viscous damping, various cases of incompressible flow have been examined. Limiting cases of incompressible flow have been discussed by DiMaggio [16] and Hayek and DiMaggio [2] for fluid-filled shells and submerged shells, respectively. Both studies are concerned with inviscid fluid. As noted in previous studies (see referenc.e [7] or [14]), the validity of the incompressible flow analysis can be assured only for low frequency oscillations with 1-2<
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corresponding to the presence of incompressible flow inside and outside the shell, respectively. First one can consider a compressible-fluid-filled shell submerged in an incompressiblefluid medium and determine the influence of the compressibility of the outer fluid on the characteristics of natural frequencies. It can be analytically shown that, in the presence of the incompressible outer fluid, the change of the real and imaginary components of each natural frequency due to viscosity are equal in magnitude and opposite in sign. The imaginary part of the natural frequency, due solely to viscosity, is always positive. Thus, the fluid viscosity will produce a damping effect on the vibration and will reduce its frequency [13].
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From the result given above, it is obvious that the negative viscous damping is due to the effect of compressibility of the outer fluid, since the viscous damping is always positive when the outer flow is assumed incompressible. This is consistent with our physical intuition since when the outer flow is assumed incompressible, the acoustic radiation damping is always zero and the only damping mechanism left is the viscous dissipation. The occurrei~ce of negative viscous damping in this case would imply a negative viscous dissipation, which is, of course, not possible. Computations have been made for water-filled steel shells in water and for water-filled steel shells in petroleum. In both cases the outer flow is assumed incompressible and the computed inviscid damping, in general, is of the order 10 -17, which can be considered zero. The computed viscous damping is always positive and the change of the real and imaginary components of each natural frequency is equal in magnitude and opposite in sign as predicted in the analysis. It is also noted that, except for those frequencies involving large inviscid damping which no longer exist when the outer flow is assumed incompressible, the real components of natural frequencies dominated by interior fluid remain practically unchanged whether or not the fluid outside the shell is compressible. Whereas the damping components of typical modes plotted in Figures 1 l(a) and (b) indicate that when the outside fluid is assumed incompressible the viscous damping generally increases somewhat, compared with the compressible flow case, if the frequency is greater than a critical frequency which is somewhat above the in vacuo frequency ta~,, and vice versa if the frequency is less than the critical frequency. It is also noted that the damping for a water-filled shell immersed in incompressible water is, in general, larger than the corresponding case when the outer fluid is absent (see Figure 9). An exception occurs for the non-dissipating mode of k = 2, in which damping is smaller with the presence of incompressible flow outside the shell. This is noted to be the case even if the outside fluid is of high viscosity, such as petroleum. The limiting case of incompressible-fluid-filled shells submerged in a compressible-fluid medium also has been investigated. The inviscid frequency equation can be written as a polynomial equation in this case so that there exists only a finite number of roots for each mode. The computed frequencies are listed and plotted in reference [13]. It is noted that the spectra of radian frequencies for an incompressible-fluid-filled shell submerged in a compressible-fluid medium follows closely that of the empty shell in a compressible fluid or incompressible-fluid-filled shell hz vacuo. T h e frequency is generally lower in the present case due to the additional added mass effect of the inner fluid. This is particularly true for the lower branch of the frequency spectra and for the lower modes of the upper branch. Breathing vibration (k = 0) is not possible due to the presence of the incompressible fluid inside the shell. As to the damping components, the inviscid damping is reduced in the presence of the incompressible inner fluid. Also noted is that damping due to viscosity alone decreases for k ~<6 of the lower branch of the frequency spectra. Viscous damping is higher with the incompressible viscous fluid inside the shell for k > 6 in the lower branch and all k's in the upper branch. Moreover a frequency associated with large inviscid damping is obtained. The viscous damping of this highly damped mode is again negative, so that total damping is reduced by fluid viscosity. Thus, the compressibility of the inner fluid is not essential, while that of the outer fluid is essential for the existence of the reduced damping by viscosity. 7. EFFECT OF SHELL THICKNESS The computed results discussed so far are limited to thin shells of a moderate h / R ratio 0.03. One can now examine the viscous effect on the fluid-filled submerged shell when
SUBMERGED
FLUID-FILLED
SHELL
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121
the h / R ratio is further reduced. In the following discussion, the fluids are assumed viscous and compressible. Computations have been carried out for a water-filled shell with h / R = 0.003 submerged in water and water-filled shells with h / R = 0.003 and h / R = 0.0003, respectively, submerged in petroleum.
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Figure 12. Frequencyspectrumfor a water-filledsteel shellwith h/R = 0.003 submergedin water (O, mode with negativeviscousdamping). Figure 12 shows plots of the real components of the natural frequency spectra. This figure in general, confirms the previous conclusion that, except for those few frequencies that exhibit very large inviscid damping, the frequency spectra for a fluid-filled spherical shell submerged in an acoustic medium follow closely those of a fluid-filled shell hi vacuo. It should be noted, however, while the natural frequencies of a fluid-filled elastic shell in vacuo follow very closely either those of a fluid-filled rigid shell (broken lines in Figure 5(a)) or the in vacuo natural frequencies, the degree of closeness decreases with h / R as shown in reference [9]. As expected, Figure 12 indicates that frequency spectra deviate from those of fluid-filled rigid shells for the in vacuo natural frequencies as the h / R ratio decreases. Added mass effects also are more pronounced for the lowest branch for thinner shells. The inviscid damping for typical modes is tabulated in reference [13]. The damping exhibits essentially the same features as compared to the case of the h / R ratio equal to
122
T.C. SU
0.03: that is, the low damping (non-radiating) for the lowest branch of frequency spectra Be, peak damping on the BI branch and in the vicinity of the in uacuo frequency ~o~,,and a minimum damping near 03k which is below but close to co~ Quantitatively, there are significant differences for varying h/R ratios. Inviscid damping for shells with small h/R ratios is in general large and shell motions will be rapidly damped, except for non-radiating modes on the Be branch and minimum damping near 03k. For the water-filled shells submerged in petroleum with h/R ratios equal to 0.0003, inviscid damping computed in most frequencies is almost constantly 0.33. However, minimum damping near 03 remains small and damping on the Be branch is even lower as h/R decreases. Lower damping on the Be branch is expected since the frequency there is lower for smaller h/R, so that for certain modes the acoustic wave number is less than the wave number of the vibrating surface, resulting in the existence of the non-radiating modes [7, 14]. The viscous effect is expected to be more pronounced for very thin shells, since the perturbation term in expression (28) which takes account of viscous effects is explicitly proportional to R/h through the functions $ik as defined in expression (21). In the computation of water-filled shells in petroleum with h/R = 0.0003, noticeable changes iO 0
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0 . 0 0 3 in
SUBMERGED
FLUID-FILLED
SHELL VIBRATIONS
123
of some frequencies occurred. Table 12 in reference [13] summarizes the computed results for typical modes in this case. It is noted that the viscous effect is most significant near to~,, where substantial reduction of radian frequencies could result through viscosity effects. The damping due to viscosity is plotted in Figure 13 in dotted lines. Negative viscous damping occurs more frequently for thinner shells, which is to be expected as a result of pronounced viscous effects and large inviscid damping associated with the thinner shells. However, due to large inviscid damping, the damping reduction by viscous action is insignificant and the total damping, plotted in solid lines, is hardly affected by the viscosity. However, on the Bo branch where the non-radiating mode exists and inviscid damping is extremely small, the viscous damping, which is positive there, predominates. Therefore, the shell motion also will be damped, although it remains unclear whether the damping is due mainly to the viscous dissipation or to the acoustic radiation to the far field by the viscous action. It also is noted that near O3k,where minimum inviscid damping occurred, the viscous damping is at its maximum and is predominating. 8. CONCLUSION In order to clarify the effect of viscosity on fluid-structure interaction, the investigation of axisymmetric vibrations of a fluid-filled spherical elastic shell submerged in a compressible fluid medium has been carried out by use of boundary layer theory. To elucidate the findings, various limiting cases have been studied, including the inviscid fluid limit, the incompressible flow limit and the thin shell limit. In the absence of viscosity, it is found that, in general, no real natural frequencies can exist for free vibrations of a fluid-filled submerged shell. Thus, the shell motion is always damped and the damping is due to the compressibility of the fluid outside the shell. Moreover, the natural frequencies generally follow closely those of a fluid-filled shell hz vacuo, which in turn follow closely either those of a fluid-filled rigid shell or the in v a c u o natural frequencies. Therefore, the presence of the surrounding fluid outside a fluid-filled shell produces only insignificant changes in the real part of the frequency spectra. It is noted, however, that between OJk and first zero of j~, there exist natural frequencies with large damping components due to the presence of compressible fluid outside the shell. The damping components for the lowest branch of the frequency spectra are very small and there exists a non-radiating phenomenon for intermediate modes of this branch which is more pronounced in the presence of interior fluid. This is due to the lower frequencies and therefore increasing acoustic wave lengths associatedwith the presence of the interior fluid. The damping components for each mode possess two peak values, where efficient sound radiation is possible. One appears on the B~ branch which is close to the first zero of j~, and the other occurs in the vicinity of in vacuo frequencies oJ~,. Besides those non-radiating modes, minimum damping occurs in the vicinity of a~k, which is very close to but lower than a~,. The analysis of viscosity reveals that the viscosity has essentially no effect on the real component of natural frequencies for shells with moderate h / R ratios. Consequently, the radian frequencies remain practically unchanged with or without fluid viscosity. However, the viscosity did produce noticeable effects on the damping part of the natural frequencies. It is found that the viscous damping is predominant for the non-radiating modes of a fluid-filled submerged shell. For all modes of a fluid-filled shell, either h~ vacuo or in an incompressible fluid medium, the damping is due to viscosity only. The viscous damping reaches its peak value when the radian frequency is close to the in vacuo frequency a~, while a minimum of inviscid damping occurred nearby, and noticeable effects on total
124
T.C. SU
damping due to viscosity are observed. This is particularly true as viscous damping becomes predominant there for very thin shells. Unlike the case of a fluid-filled shell hz vacuo in which the viscous damping is always positive and the viscous contributions to the real and imaginary components of the natural frequencies are equal in magnitude and opposite in sign, it is found that, in the presence of compressible flow outside the shell, the damping effect of viscosity can be either positive or negative. Thus, the total damping is reduced by fluid viscosity for certain cases, although in these cases of negative viscous damping the inviscid damping is always predominant and the viscous contribution is rather insignificant. It is further noted that for the modes possessing negative viscous damping, both the damping of the shell motion and the acoustic radiation damping are reduced by fluid viscosity, since the viscous dissipation damping must remain positive. Moreover, the study indicates that the viscosity of the contained fluid will produce a damping effect except near the valley of the negative portion of the damping curve where the viscosity of the inner fluid will reduce the damping even further. The viscosity of the surrounding fluid will, in general, increase the damping when the radian frequency of vibration is less than a critical frequency which is somewhat above the hz vactto frequency to~,, and damping is reduced with the viscosity of the outer fluid when either the radian frequency of vibration is greater than the critical frequency or frequencies associated with large inviscid damping. The dual role played by the viscosity is rather curious. A study has been carried out for limiting cases of incompressible flow to elucidate the role played by fluid compressibility in this reduced damping phenomenon. It is shown that the viscous damping is always positive and the viscous effect on the real and imaginary components of the natural frequencies are equal in magnitude and opposite in sign in the case of a fluid-filled shell submerged in an incompressible fluid medium. Therefore, inviscid damping is essential for the occurrence of reduced damping by viscosity. Studies of very thin shells show that viscous effects become more noticeable as the h / R ratio decreases. Both real and imaginary components of natural frequencies can be affected by viscosity in some cases.
REFERENCES 1. M. C. JUNGER 1952 Journal o[Applied Mechanics 19, 439-445. Vibrations of elastic shells in a fluid medium and the associated radiation of sound. 2. S. HAYEK and F. L. DIMAGGIO 1970 International Journal of Solids and Structures 6, 333-351. Complex natural frequencies of vibrating submerged spheroidal shells. 3. A. E. H. L o v e 1888 Proceedings of the London Mathematical Society 19, 170-207. The free and forced vibrations of an elastic spherical shell containing a given mass of liquid. 4. P. M. MORSE and H. FESHBACH 1953 Methods of Theoretical Physics, Part II. New York: McGraw-Hill Book Company. See pp. 1469-1472. 5. R. RAND and F. L. DIMAGGIO 1967 Journal of the Acoustieal Society of America 42, 1278-1286. Vibrations of fluid-filled spherical and spheroidal shells. 6. A. E. ENGIN and Y. K. LIU 1970 Journal of Biomechani'cs 3, 11-22. Axisymmetrie response of a fluid-filled spherical shell in free vibrations. 7. T. C. Su 1974 Eng. Sc. D. Dissertation, Columbia University, New York. The effect of viscosity on the dynamics of a spherical shell immersed in a fluid medium. 8. T. C. Stl and Y. K. LOll 1975 Jottrnal of Enghzeeringfor Industry, Transactions of the American Society of Mechanical Engineers 97, 1338-1344. The effect of viscosity on the dynamics of a submerged spherical shell. 9. T. C. So 1981 Journal of Sound and Vibration 74, 205-220. The effect of viscosity on free oscillations of fluid-filled spherical shells. 10. S. HAYEK 1966 Journal of the Acoustical Society of America 40, 342-348. Vibration of a spherical shell in an acoustic medium.
SUBMERGED FLUID-FILLED SIIELL VIBRATIONS
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11. F. K. MOORE (editor) 1959 Theory of LamhzarFlow. Princeton University Press. See pp. 3 4 46. 12. K. STEWARTSON 1964 The Theory of Latnhzar Boundary Layers in Compressible Fhdds. Oxford University Press. See pp. 15-22. 13. T. C. SU 1980 Ocean Engineerhzg Program, Texas A&M University, TEES Report No. COE-222. Natural frequencies of submerged fluid-filled spherical shells. 14. Y. K. L o u and T. C. S o 1978 Journal of the Acoustical Society of America 63, 1402-1408. Free oscillations of a submerged spherical shell. 15. C. A. FELLIPA and T. L. GEERS 1980 Journal of the Acoustical Society of America 67, 1427-1431. Axisymmetric free vibration of a submerged spherical shell. 16. F. L. DIMAGGIO 1975 Shock and Vibration Digest 7, 5-12. Dynamic response of fluid-filled shells.