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The effect of viscosity on free oscillations of fluid-filled spherical shells
Journnl
of Sound
THE
and Vibration (1981) 74(2), 205-220
EFFECT
OF VISCOSITY FLUID-FILLED
ON FREE OSCILLATIONS SPHERICAL
OF
SHELLS
T. C. Su Department of Cioil Engineering, Texas A&M Uniwrsity, College Station, Texas 77843, 1J.S.A (Receiaed 19 February 1980, crnd in rerised form 9 June 19801
In order to clarify the effect of fluid viscosity on the vibrations of elastic shells, the axisymmetric free oscillations of a fluid-filled spherical shell 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 compressible viscous fluid medium. The study shows that the fluid viscosity in general will produce a damping effect on the shell motion. It is also shown that the change on the real and imaginary components of each natural frequency due to viscosity are equal in magnitude but opposite in sign. Therefore, the circular frequency of a vibrating spherical shell will decrease with the increase of viscosity of the contained fluid. The study further reveals that viscous damping reaches a peak value in the vicinity of the upper branch in LWCIIOfrequencies and then drops rapidly to a minimum. It is noted that the minimum damping is so small that the shell motion is almost steady state (non-dissipating).
1. INTRODUCTION
free vibrations of a compressible fluid contained in a spherical shell have been investigated by several authors. Rayleigh [I] first solved the problem of axisymmetric vibrations of a fluid contained in a rigid shell. Extensional axisymmetric vibrations of an elastic spherical shell in vucuo were first studied by Lamb [2]. Applying the method of Lamb [2,3], Love [4] treated shells by exact elasticity theory and solved the problem of vibrations of a spherical shell containing an incompressible fluid. More recently, the solution for vibrations of a fluid-filled spherical membrane has appeared in the book by Morse and Feshbach [S]. More detailed treatments of extensional axisymmetric vibrations of spherical fluid-filled shells has been given by Rand and DiMaggio [6]. Their work has been extended to include the effect of the shell’s bending by Engin and Liu [7]. In all these investigations, however, the effect of viscosity has been omitted. It is well known that although for many practical applications the viscosity is vanishingly small and motion of the fluid executing small oscillations 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, where the viscous effect is noticeable. may be of great consequence to the hydrodynamic forces acting on the shell and consequently may have noticeable effects on vibration characteristics of the shells. The objective of the present study is to clarify the effect of viscosity on fluid/structure interactions by investigating the axisymmetric vibrations of a fluid-filled elastic shell. This is an extension of previous study on submerged shells by the author [B, 91. Based on the classical bending theory, the dynamic response of the shell is determined by the normal mode method. For the fluid mediums, the linearized Navier-Stokes equations for an adiabatic compressible fluid are derived from the general Navier-Stokes equations for a viscous, compressible, heat-conducting fluid. Moreover, for most applications, where the
The
0022-460X/81
/020205
+ 16 $02.00/0
$)) 198 1 Academic
Press Inc. (London
I I.mlited
206
T. C. SU
ratio of the depth of the extent to which the influence of viscosity penetrates into the fluid and the radius of the shell is small, the Navier-Stokes equations are further simplified by boundary layer approximations. From this, the first order viscous effects are evaluated for free oscillations of an elastic spherical shell and its contained fluid. 2. FORMULATION
OF THE PROBLEM
2.1. BASIC EQUATIONS FOR SHELL The dynamic response of the shell may be determined by the normal mode method. 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 f3) and their derivatives. Except for k = 0, there exist two natural frequencies, wk and w;, for each value of k. The mode shapes for both frequencies are identical but the ratios of the tangential to radial displacements are different. The natural frequencies are determined by the frequency equation as given by Hayek [lo], for example. The mode shapes corresponding to o’k are designated by W; and V; and are given by wl, = Pk(cos 8),
vl, = c; (d/d@Pk(cos #),
(1)
where ”
(1 + v,)(l+ E,) + s,(k - l)(k +2) =(1+e,)(k2+k-1+v,)-(l-v~)(~~R/c,)2’
(2)
Here v,~is Poisson’s ratio, cs is the sound speed of the shell material, F, = h2/12R2, and h and R are the thickness and radius of the shell, respectively (see Figure 1).
Figure
1. Shell geometry.
The mode shapes corresponding to Wkare designated by wk and vk and can be obtained from the given expressions by merely omitting the primed superscripts. It should be noted, however, that wk, wk, and Vk do not exist for k = 0. With the notation given, the dynamic response of the submerged shell can then be expressed in terms of normal modes: v=r/bvb
+k;,
w=dlwb
+ f k=l
[77k(I)vk(~)+rl;(l)v;(8)1,
[rlk(c)Wk(B)+77~(t)W;(8)],
(3)
FREE
VIBRATIONS
where qk and q; are normal co-ordinates, ;ik +&hk ;ii,
+w;*&
OF FLUID-FILLED
determined
=js
(?vk
-pswk)
=I{
(rsVI( -p,WI,)
SHELLS
207
by the equations of motion
dA/2-nR”pshMk,
dA/2rR2p,hAl;,
131
in which 2nR’&hM, and 2xR2p,hM; are the generalized masses, and 7S and p5 are tangential and norma! hydrodynamic forces which must be determined from the 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, for example, in the book by Moore [ll]. However, solving the general equation is extremely difficult due to the presence of non-linear convective terms. In order that the convective terms should be negligible, it is necessary that the amplitude of oscillation, a, be small in comparison with both the radius of the shell, R, and the depth of penetration of the rotational flow, S. This is assumed to be the case. Furthermore, since the oscillations are small, the relative changes in the fluid density, pressure, temperature, and transport parameters are also small. Therefore, these equations can be linearized to yield the following governing equations for the fluid: continuity equation: @‘/at + p. div II = 0;
15)
momentum equation (with volume force absent): ~o(&B/c%)= -grad p’+ ~OV2u+(~~+ ~43) grad div u. energy equation (without external heat addition): ds’lat = -(~/PO) div (q/T) -(llpoT2)(q.
grad T) +(llpd@,/T,
(71
where p, p, and T are the density, pressure, and temperatures of the fluid, the subscript 0 represents the constant equilibrium value of a variable while the primed notation represents the small variation of that variable, u is the velocity vector, s is the entropy per unit mass, and q and Of are the heat conduction vector and the rate of dissipation of energy per unit volume, respectively; LO= Ao+$.L~ is the volume or bulk viscosity, and ho 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 the 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 whence 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 change in entropy in its dimensionless form, s’/cp is Of the order of (T’/T)(a/R)(c,p/k )-’ :x: Thus the change in entropy, s’ , is also of second order importance provided (wflR/w )-I. that the reciprocal of the product of the Reynolds number, p(aR)R/p, and the Prandtl number, c+/k, is at most of order 1 (c, is the specific heat at constant pressure). This condition can be fulfilled if the frequency R 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’= (@Jl&%)sP = C:,P’,
(8)
208
T. C. SU
where co is the constant equilibrium value of the sound speed in the fluid medium. The basic equations for an adiabatic compressible fluid can therefore be summarized as follows: the continuity equation (5), the momentum equation (6), the p’and p’relationship (S), and the viscous stress tensor T = ho(div u)I+ po(def u).
(9)
Taking the divergence of equation (6) and combining the result with-equations (S), one obtains an uncoupled equation for p’, (1/C~)~2p’/i)t2=v2p’+(1/C~~O)(~~O+~0)v2(~p’/~f) while the substitution equation for u:
of equations
(5) and (8) into equation
(10)
(6) yields the following
po$= _ l+5u+y3 5 Vp’ f jlov2u. POCO at
>
(
(5) and
(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 solutions for u and p, together with the stress strain relationship (9), completely prescribe the hydrodynamic forces acting on the shell. 2.3. BOUNDARY 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 (the non-slip condition). Consequently, the boundary conditions at the fluid-shell interface can be written as u,(R 8, t) = W,(R
8, f),
Q(R 8, t) = V,(R
8, f),
(12713)
where t’, and ve are the radial and tangential components of the velocity vector u, respectively. In addition, it is required that the velocity components are finite at the origin. 3. SMALL VISCOSITY ANALYSIS 3.1.
THE
BOUNDARY
LAYER
EQUATIONS
The influence of small viscosity will be examined later, and one can therefore begin with the linear equation (11). After taking the curl of both sides, the term curl grad p vanishes identically, giving p&)/at) curl u = j&J2 curl u.
(14)
This indicates that the vorticity, curl u, satisfies a heat conduction equation. Therefore, it decreases exponentially toward the interior of the fluid. In other words, the motion of the fluid caused by the oscillations of the shell is rotational only in a certain layer near the wall, while at larger distances it rapidly changes to potential flow. The depth of penetration of the rotational flow is of the order S -Jv/n, where Y is called the kinematic viscosity and is equal to po/po. For a fluid medium of vanishingly small viscosity, two important limiting cases are possible here. For the potential flow region, wo, lo --, 0, and the governing equations (10) and (11) reduce to the Eulerian formsi (1/C2)a2p/at2 + Here, and henceforth,
for brevity
= v2p,
paujat = -vp.
the suffices in pO and pLg and the superscript
(15,161 in p’ are omitted.
FREE
VIBRATIONS
OF
FLUID-FILLED
SHELLS
209
Within the boundary layer, however, the effect of viscosity is significant. It is necessary to retain the viscous term in the Navier-Stokes equations. This can be accomplished by a double limiting procedure, a/& + CCas k, [ + 0, known as Prandtl boundary layer theory [ 121. For axisymmetric motions, this will reduce the 8-momentum component of equation (11) to the form, valid only within a thin layer near the spherical shell, pavO/i)t= -(l/R)(i~p/M)l,=~ with the inner boundary
condition L’fj= v,,
and the outer
boundary
(17)
+p3’vOlilr2,
atr=R
(18)
condition 1’0= L’8 ,nvlscidJr=R at (R -r)/(v/R)“‘>>
1.
(19)
It is known that the change in pressure across the layer is small and is determined by the f+ctionless flow. The bulk viscosity lo has disappeared from the boundary layer equations, since it contributes a term of order U(V/~U~~). The effect of curvature is also irrelevant since for finite curvature, it has appeared only in the convective terms in the substantial derivative, which have been neglected in the present study. Sufficiently near the boundary, 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, S/R -Jv/flR’<< 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. 3.2. 3.2.1.
BOUNDARY
Flowfield
LAYER
SOLUTIONS
(R-ct
Based on the classical boundary layer approximation discussed in the preceding section, 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, tie, and the radial velocity component, t’,, outside of the boundary layer. Near the solid surface, however, z’~ is governed by the boundary layer equation (17) and C, is determined from the continuity equation once us has been obtained. Since the pressure, p, satisfies the wave equation (lS), and is regular at the origin, its solution may be written as, for harmonic oscillations, p
=eiRr? A,,j,(ar)P,,(x),
(20)
n=O
where cy = n/c, x r cos 8,0 is the frequency of vibration, A, is an arbitrary constant to be determined by the boundary conditions at the shell’s surface, and the j,(ar) are spherical Bessel functions of the first kind. Substituting expression (20) into equation (16), one obtains the following solutions for L’, and vH, i
3c
u, = i
C, A, j Xar)P,k) n-
eiR’,
X y A,tj,I(ar)s(x) I10= Rpr ,,Lo
i:~(z)=di,(z)/dz,
e’“‘,
(21) (221
while the substitution of expression (20) into the boundary layer equation (17) and matching the inviscid solution (21) by enforcing the outer boundary condition (19), yields
210
T. C. SU
the following expression for ue near the shell surface (r = R), dPn(X) -ice
iot ’
(23)
where g is the principal value of (iL?/v)“’ and the D,‘s are arbitrary constants. Substitution for p' from equation (8) and vg from equation (23), and the use of expression (20), then gives v,, from the continuity equation (5),,as +I)$
1
P,,(,y)eiR’.
(24)
The tangential component of the hydrodynamic force of the inner fluid acting on the shell is then given by, according to the boundary layer approximations,
Tr8
I
rzR
due
=‘r
I,_R
cc
=pg
RR
dP,*(xl
,,;oDnedee
ior
.
(25)
The unknown constants A,, and D, in these equations can be expressed in terms of the normal co-ordinates nk and nk, by enforcing the non-slip boundary conditions at r = R. For harmonic oscillations, the normal co-ordinates qk can be written as q)7k = qk eiRr (with a similar expression for 7;). With the aid of expressions (20) and (25), one can obtain the following expressions for the hydrodynamic forces acting on the shell surface (r = R j:
+q:
[
n(n -tl) l----
gR
J = j,(aR)/jL(aR). 3.2.2.
JPn(x),
(26)
(28)
Shell response
Substituting the expressions obtained for the hydrodynamic forces into the shell equations of motion (4), one obtains the following governing equations for the axisymmetric vibrations of a spherical shell submerged in a compressible fluid medium for k 2 1:
(2%)
(2%)
FREE
VIBRATIONS
OF FLUID-FILLED
211
SHELLS
where
P*k=A[ck-~J][c;-~J]. v5k=n[ck-~J]2, .1=(1-i)J2
-k(k+l) R 1 ~h~(&)1’2~(+)1’2,
mk=z,
(30)
and the expression for Plr3kis similar to that for PI, except that Ck must be replaced For k = 0, q. does not exist and 4;) is given by
~[(~)2-(~)2_@o(~)2]=o,
by C;i.
131)
where
It should be noted that the viscosity effect has appeared only through the functions !Prk, 1Y2k,and Pjk. For an inviscid fluid or for breathing vibrations (i.e., k = 0), these quantities vanish as would be expected. 3.3, FREQUENCY
EQUATIONS
Both equations (29) and equations (31) are a set of homogeneous algebraic equations. Requiring that non-trivial solutions exist for the displacement (qk/h) and (q;/h) yields the frequency equations (w;R/c.J~ - (f2R/c.J2 - @o(flR/c,)’ = 0
(32)
for k = 0 and (f?R/~,)~[l f Qj,rnk + @k]- (nR/c,)2{~,m,(u,R/c,)‘+
~k(~~Rlc,)~+[(~kRIc,)~
+
~~~~/C.~~*~~+~W~~/C,~~~~~~/~,~~-~~~/~.~~*~~~~~~~~~/~,~~
-
(1 + mA)(f?R/c,)*l+
+2@kP2kmk(f2R/c,,)2}
9gkmk[(wkRIcs)2 - Cl+
%)(f2RI~,)~l
= 0
(33)
for ksl. As mentioned previously, the viscosity effect has appeared only through the functions qIk, Pzk, and tYXk,which vanish for an inviscid fluid. One may immediately recover the following frequency equations for the inviscid flow case: (&R/c,)
- (~&R/c,)~ - @&A,R/cJ2
= 0
(341
for k = 0, and WWIcJ2
-
~w~~l~,~‘lC~.n,,~l~,~*bdJVcJ21
+ @~(~dVc,)2(m~+ l)[K41Rlc,)~- (&RIc,)~I = 0
(35)
for k s 1, where 3: = (m,w2, +uhz)/(mk + 1)
(36)
212
T. C. SU
and 0, is the free oscillation frequency when viscosity is absent and (Pk is a function of (30) evaluated at 0 = 0,. In general, the frequency equations (32) and (33), 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, 1?Vlkl, 1F,, 1and 1lP3k 1are of order (v/ Rc,)~” 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 Ro, in an analytical form, by Newton’s method. The results are OR/c, as defined by equation
flR,c,
= f&R/c,
(37)
for k = 0 and OR/c,
=(~noR/c,)-(mk~k(~0RI~,)‘(tylk+1Y3k--22k)
+
(~,R,c,)~(W,,[(~~OR,C,)~
-
bDV~s)~l
~w~~/c,~21~~,~~~~,,~,cs~2 x Wk[mk(~~R,c,)~+ k4NcA21+ K4VA2 + ~~1Rl4~1~
+ ‘J’~~/~K~oR,c~~~
-
-Gc~~R,c,)~(w;R/c,)~ -bn,&~~R,c,)~
+
+ (0oRlc,Pb,kh
+ WZ,RlcJ4
(w;R,c,)~I(~oR,c~)‘}}
(38)
for k 2 1, where &, ‘Pik, and &( = d&/d(flR/c,)) are evaluated at 0 = 0,. Therefore, once the natural frequencies for the vibrations in an inviscid fluid are known, the corresponding frequencies in a viscous fluid can be readily computed. It should be pointed out that the above formulae were obtained from the first order iteration of Newton’s method, which provides sufficient accuracy for consistency with the boundary layer approximation. 4. NATURAL FREQUENCIES FOR THE VIBRATIONS IN AN INVISCID FLUID The method of false position was used for the root searching of equations (34) and (35). The solutions of these equations are presented by the solid lines+ in Figures 2(a)-(d) for a steel shell filled with water, petroleum, ethyl alcohol (C2HsOH), and carbon tetrachloride (CCL), respectively. The values of c/c, and p/ps are given in Table 1. Broken lines in these figures, defined by j ;(L&R/c) = 0, are plots of the frequency spectra for the fluid in a rigid spherical shell. The frequency in uucuo is plotted in Figure 3 for comparison purposes. Figures 2(a)-(d) indicate that the natural frequencies of a fluid-filled elastic shell follow very closely either those of a fluid-filled rigid shell (broken lines in these figures, defined by j; = 0) or the in Z~UCUO natural frequencies. This has been recognized by several authors [6, ,7] and it can be proved analytically as follows. From equations (34), (35), (30) and (28), and with the aid of the identities [(2k + 1)/2]M;
=1
for k = 0,
[(2k + 1)/2][MJ(l+
mk)]= 1
for k 2 1,
one can write (39) + It is to be noted that this and all other spectra plotted in this paper are discrete: corresponding to integral values of the mode number k are physically meaningful.
i.e., only those
points
0
Flgurc 2 Frequency
IO
012345678 shell, (ci an cth~l-alcr)hol-filled
Mode number k shell and 1d13 carbon-tetrachlori[i~-~lled
9 to shell.
---, Steel shell
214
T. C. SU TABLE
1
Physical constants used in the numerical computations Physical constants of fluids (at 10°C)
c/c, 0.292 0.267 0.242 0.195
Water Petroleum Ethyl alcohol Carbon tetrachloride
(~/Rc,)‘~~’
PIPS
0.127 0.114 0.103 0.207
2.93 x 1.0.5 x 3.42 x 2*14x
1o-5 10T3 1O-5 lo-’
Physical and geometrical constants of steel shell cS = 49.7 x 10’ m/s (16.3 x lo3 ft/s) kg/m3 (15.25 Ibf s2/ft4) v, = 0.3 h/R = 0.03 R = 0.3048 m (1 ft)
ps =7871.65
I,
/
I1
I
12345676
I
I
I
9
IO
Mode number. k Figure 3. Natural
frequencies
of a steel shell in UCJCUCJ with h/R = 0.03.
- - --,
for &R/c,
FREE
VIBRATIONS
OF FLUID-FILLED
SHELLS
(%WG)~ -(w&c,)~ U&R/C,)*
215
c4*1
for k = 0 and k z 1, respectively. For¬ close to in ~a~uu frequencies, it is seen that the last factor on the right-hand side of equation (39) and the last two factors in equation (40) are at least of order of one. Consequently, h ps cs f&R 1 _____-R p c c, O(1)
for all values of k. For practical purposes, the value of (h/R)(p,/p)(c,/c)
IjI
- _--&R .I cs I
Furthermore,
141) is of order of one. Therefore,
1 O(1)’
it can be shown from expressions (34) and (35) that the smallest value of
f&R/c not close to in U~CKUfrequencies must be greater than the first zeros of jd&R/c) = 0 for k = 0 or j;(f&R/c) = 0 for k 2 1. Consequently, with the values of L-,/Cused in the present study, these f&R/c, must be at least of order of one. Hence,
jI I-_J.’
1 O(1)’
A careful examination of the Bessel function as plotted in reference [13] reveals that this is only possible if j’(&R/c) is small. Therefore, the natural frequencies of a fluid-filled elastic shell, if not close to in oucuo frequencies, must follow closely those of a fluid-filled rigid shell. The degree of closeness increases with (h/R)(p,/p)(c,/c) as can be shown from expression (41). Moreover, it can also be shown that the spacings of the frequency spectra are proportional to c/c,. These confirm the results presented in Figures 2(a)-(d). Figure 4 is a plot of frequency spectra for a water-filled steel shell with a smaller h/R ratio. From this it is clear that as h/R decreases, the natural frequencies begin to deviate from either the in t’acuo frequencies or frequencies of a fluid-filled rigid shell. More computation has been carried out and results are tabulated in a report by Su [ 143. 5. EFFECTS OF VISCOSITY Natural frequencies for the vibrations in a viscous fluid are given by equation (38) for is generally complex and can be written as the sum of real and imaginary parts: a= fi +ifi. Since the natural frequencies of an inviscid-fluidfilled shell are always real, the damping component fi is solely due to the effect of viscosity. It can now be shown that the viscosity will produce a damping effect on the shell motion: i.e., fi > 0. Moreover, the change on the real and imaginary components of each natural frequency due to viscosity are equal in magnitude but opposite in sign: that is, fi = Q0 - 6. Therefore, the radian frequency of a vibrating spherical shell will decrease due to the viscosity of the contained fluid. First, it can be noted that by using expression (30) and after some mathematical manipulations, the numerator of the perturbed term in expression (38) can be expressed as
k 2 1. Thus, the natural frequency
A (%Rlc,)3 W,R/c,+ x
GhNc,~*lh + 1)
I(ck-~J).[(~)2-(~)2]+m,(c~-~~)[(~)2-(~)2]]2,
216
T. C. SU
9
3
2
0
I2
345676 Mode number, k
9
IO
Figure 4. Frequency spectrum for a water-filled thin shell with h/R = 0.003. Key as Figure 2.
while the term
which appears in the denominator rewritten as 2{(&R/cs)“[1
+
of the perturbed
@k(mk
+
1)1-
term in expression
(38), can be
h’k~/d*h’;~h)*~
by virtue of expression (35). Moreover, from the definition of C& in expression (36), the remainder in the denominator of the perturbed term in expression (38) is equal to (%R/c,)3(W
+ l>&{(%R/c,)*
- &R/c,)*).
Thus equation (38) can be put in the form RR/c,
= (&R/c,)
+ mk,(C; x (mk
-{A(fhR/cs)3/h -
+
l)){(ck
WU~>J)WOWG~*
+ lXW,Rlc,~*-
-
WRWW&NG~*
- (~kR/cJ*l~*/{~ h&~/d”~*~k~,
-
b.dVc,)*1
+ MMVG>~
(42)
FREE
VIBRATIONS
OF FLUID-FILLED
217
SHELLS
where 0 = 21(n,Rlc,)4C1 + @k(rnk +
III- (w~Rlc,)2(w;Rlc,)2}[(noR/c~)2 - kdvCJ21
may be rewritten as 0 = 2(WkRIC.~)2[(~oRIC.~)2 -
b%wcs~21” +2mkb3&C,)* - bkwc,~21
x {[o&~IC,~2- hJvcs~212 + (wk~Ics~2HdvG~2- hQ~/dII by first eliminating Gk with the aid of expression (35) and then expressing w;’ in terms of WE and Wi by virtue of expression (36) and some mathematical manipulations. It is then obvious that 0 is a positive quantity since & > wk G=0 and mk > 0. Moreover, bk
=
d@k/d(f&&/cs)
2
c,
= c 2k
=
RZp
(c,/c‘)d~k/d(RoR/c,),
d h Mk z d(&R/c,)
1’
1
jk(GRIc) [ (RoR/c) j;(&R/c)
which is also a positive quantity, since it can be shown analytically that d j,(x) dx [ xj;(x)
-
1
=~{“[“l(‘“*‘““‘i”‘l’*‘““;‘““~o
for x >o,
provided k is a positive integer. Proof of this result is given in the Appendix. Thus the perturbed term in 12R/cs, which is caused by the fluid viscosity, takes the form [(-,I) times a positive quantity], and can be expressed as [(-1 + i) times a positive quantity] by the definition of n given in expression (30). Therefore, it is concluded that the viscosity of the fluid within the shell will cause a change of frequencies of the free vibration. The change of the real and imaginary components of each natural frequency are equal in magnitude and opposite in sign. The imaginary part of the natural frequency, solely due to viscosity, is always positive. Thus, the fluid viscosity will produce a damping effect on the vibration and will reduce its radian frequency. Numerical computations have been carried out for a steel shell of hf R = 0.03 filled with water, petroleum, ethyl alcohol and carbon tetrachloride, respectively, and for a steel shell of smaller h/R ratios filled with water or petroleum. The petroleum was chosen for its high viscosity. The numerical results of these computations are summarized in reference [ 141. In the numerical computations physical constants listed in Table 1 are used. For the case of h/R = 0.03, it is noted that although the viscosity of the fluid will reduce the radian frequency fi, its contribution to the frequency is indiscernible. Therefore, the radian frequency fi is essentially the same as 0, given in Figures 2(a)-(d) for the inviscid flow problem. Some noticeable changes on the radian frequency may occur due to the viscosity of fluid for very small values of h/R, such as h/R = O-0003 for a fluid-filled steel shell, as indicated from data in reference [14]. It also should be noted that the damping component fi of the natural frequency for a viscous-fluid-filled shell is always positive. The numerical results confirm that the change on the real and imaginary components of each natural frequency due to viscosity are equal in magnitude but opposite in sign. The damping distributions plotted in Figures 2(a)-(d) and 4 further indicate that larger damping occurs in the neighborhood of d - &,, which is consistent with the form of expression (42), since the denominator of the perturbed term in expression (42) approaches zero as 0, approaches &. Typical curves of damping components for the mode numbers k = 3 and 4 given in Figure 5 reveal that viscous damping reaches a peak value in the vicinity of the in uac~u frequencies w ; and then drops rapidly to a minimum. It can be noted that the minimum damping is so small that the shell motion is almost steady state (non-dissipating).
218
T. C. SIJ
2 10-S 5
2 lo-‘0 5
6
01234567 Radian
frequency,
9
IO
II
h/c,
Figure 5. Damping components of the natural frequencies for a water-filled steel shell, typical modes.
A study of the balance of energy has been conducted by Su [14]. It is shown that due to the damping effect of the fluid viscosity, the total energy of both the interior potential flow and the shell decreases while the energy is transferred into the boundary layer. 6. CONCLUSION In order to clarify the effect of viscosity on fluid/structure interaction, the investigation of axisymmetric vibrations of a compressible fluid contained in a spherical elastic shell has been carried out on the basis of boundary layer theory. In the absence of viscosity, it has been shown that the natural frequencies are real and follow very closely either those of a fluid-filled rigid shell or the in uucuo natural frequencies. The degree of closeness increases with (h/R)(p,/p)(c,/c). The analysis of viscosity reveals, that, for free vibrations, the viscosity will in general produce a damping effect on the shell motion. Computations reveal that viscous damping reaches a peak value in the vicinity of the in V~CUOfrequencies w; and then drops rapidly to a minimum. It is noted that the minimum damping is so small that the shell motion is almost steady state (non-dissipating). Moreover, it has been shown that the change in the real and imaginary components of each natural frequency due to viscosity are equal in magnitude but opposite in sign. Therefore, the radian frequency of a vibrating spherical shell will be reduced due to the viscosity of the contained fluid. However, the reduction is almost indiscernible and the viscosity has essentially no effect on the real components of the natural frequencies, provided that h/R is not extremely small. From a study of.the energy balance it has been concluded that due to the viscous damping an energy transfer
FREE VIBRATIONS
219
OF FLUID-FILI.ED SHELLS
takes place, with the total energy of both the interior potential while energy is pumped into the boundary layer.
flow and the shell
decreasing
REFERENCES RAYLEIGH 1872 Proceedings of the London Marhematical Society 4, 93-103. On the vibrations of a gas contained within a rigid spherical envelope. H. LAMB 1882 Proceedingsof the London MathematicalSociety 14,.50-56. On the vibrations of a spherical shell. H. LAMB 1882 Proceedings of the London Il4athematicalSociety 13, 189-212. On the vibrations of an elastic sphere. A. E. H. LOVE 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. 1953 Methods of Theoretical Physics, Part II. New York: P. M. MORSE and H. FESHBACH McGraw-Hill Book Company. See pp. 1469-1472. 1967 JournaloftheAcousticalSocietyofAmerica 42, 1278-1286. R. RAND andF. DIMAGGIO Vibrations of fluid-filled spherical and spheroidal shells. A. E. ENGIN and Y. K. LIU 1970 JournalofBiomechanics 3,l l-22. Axisymmetric response of a fluid-filled spherical shell in free vibrations. 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. T. C. SU and Y. K. LOU 1975 Journal of Engineering for Industry, Transactions of the American Society of Mechanical Engineers 97, 1338-1344. The effect of viscosity on the dynamics of a submerged spherical shell. 1966 Journal of the Acoustical Society of America 40, 342-348. Vibration of a S. HAYEK spherical shell in an acoustic medium. F. K. MOORE (editor) 19.59 Theory of Laminar Flovv. Princeton University Press. See pp. 34-46. K. STEWARTSON 1964 The Theory of Laminar Boundary Layers in Compressible F1uid.c. Oxford University Press. See pp. 1.5-22. M. ABRAMOWITZ and I.A. STEGUN 1965 Handbook OfMathematical Functions. New York: Dover. See p. 438. T. C. SU 1980 TEES Report No. COE-217, Ocean Engineering Program, Texas A&M University. Natural frequencies of fluid-filled spherical shells.
1. LORD 2. 3. 4. 5. 6. 7. 8. 9.
10. 11. 12. 13. 14.
APPENDIX: AN INEQUALITY INVOLVING SPHERICAL BESSEL FUNCTIONS Here, the proof of the inequality involving the spherical Bessel function of the first kind of order n, j,(x) is presented. It is to be shown that x(ji’ -j,jE)+aj,jl,>O
(Al)
for all values of II = 0, 1,2,3, . . . , )a/ s 1 and x > 0. From the definition of jn(x), it can be seen that j,{xj,“+2xj~,+[x2--n(n+1)]j,z}=0.
(A21
Multiplying both sides of expression (Al) by x and replacing the -x2j,,jz {2xj,,jA+ [x2- n (n + l)]ji} shows that expression (Al) is equivalent to
term by
x2j:,2+(2+(Y)Xj.j:,+[X2-n(n+l)]j2,)0, which may be rewritten as (xj~/j.)‘+(2+cu)(xjL/j.)+[x2-n(n
+l)]>O,
or
(A3J
220
T. C. SU
If x2 > n(n + 1) + (1 + a/2)‘, the proof of the inequality (A3) is trivial, whereas for x2 G n (n + 1) + (1 + a/2)2, the inequality (A3) can be verified by introducing a new variable 2 = (xjL/j,)+(l+cu/2),
(A4)
which satisfies the differential equation xz’+(l+a)Z+(1+LY/2)(1+CY)+[22-n(n+1)+x2-(1+a/2)2]=O
(AS
by virtue of expressions (A2). Thus, the inequality (A3) is equivalent to T(x)=z2-n(n+1)-(1+a/2)*+x2>0.
(A6)
From the definitions of z and r, it is obvious that z(x), and thus r(x), is a piecewise continuous function of x with discontinuities which occur at the zero of j,. However, in the vicinity of these discontinuities, r(x) is always positive. Furthermore, as x + O’, t -+ . As it has these special properties, it is clear n(l+a)+x*>Ofor~~~~l,andn=0,12,... that T(x) will be a positive definite function of x if T’(x) > 0 whenever r(x) = 0. If f(x) is assumed to vanish at a certain point x = x0, then r:,-[n(n+l)+(l+~/2)*-X~]=o,
(A7)
and from expression (A5) x,2~,+(1+cu)2o+(1+a/2)(1+(Y)=0,
(A8)
where z. = 2(x0) and zb = z’(xo). With the use of expression (A6), T’(x0) = 2(zozb +x0). By using expressions (A7) and (A8), this can be reduced to T’(xo) = (2/xo)[(l+&
-(l +cu)(l +cu/2)zo+x;].
(A%
One can then proceed to show that (1+a)*:,-_(1+~)(1+~/2)zo+x~>0,
(AlO)
where
z0=*Jn(n +l)+(l+cx/2)2-x~
(Al la, b)
for 0 < xi s II (n + 1) + (1 + u/2)*. For negative zo, the proof of expression (AlO) is trivial, whereas for positive .zo, the inequality (AlO) can be verified by showing that [(1+cu)f~+X~]2>(1+a)2(l+~/2)2Z~. By virtue of expressions (A7) and (Alla), that
and a rearrangement
of terms, it can be shown
[(1+ cY,z:, + x;12 - (1 +a)*(1 + (w/2)2Zo =(1+cu)2z~[n(n+1)]+(1-(u2)[n(n+1)+(1+(u/2)2]xo+(u2X~, which is greater than zero, for ]CX] G 1. Consequently, completes the proof.
r(x)
is positive definite. This
of Sound
THE
and Vibration (1981) 74(2), 205-220
EFFECT
OF VISCOSITY FLUID-FILLED
ON FREE OSCILLATIONS SPHERICAL
OF
SHELLS
T. C. Su Department of Cioil Engineering, Texas A&M Uniwrsity, College Station, Texas 77843, 1J.S.A (Receiaed 19 February 1980, crnd in rerised form 9 June 19801
In order to clarify the effect of fluid viscosity on the vibrations of elastic shells, the axisymmetric free oscillations of a fluid-filled spherical shell 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 compressible viscous fluid medium. The study shows that the fluid viscosity in general will produce a damping effect on the shell motion. It is also shown that the change on the real and imaginary components of each natural frequency due to viscosity are equal in magnitude but opposite in sign. Therefore, the circular frequency of a vibrating spherical shell will decrease with the increase of viscosity of the contained fluid. The study further reveals that viscous damping reaches a peak value in the vicinity of the upper branch in LWCIIOfrequencies and then drops rapidly to a minimum. It is noted that the minimum damping is so small that the shell motion is almost steady state (non-dissipating).
1. INTRODUCTION
free vibrations of a compressible fluid contained in a spherical shell have been investigated by several authors. Rayleigh [I] first solved the problem of axisymmetric vibrations of a fluid contained in a rigid shell. Extensional axisymmetric vibrations of an elastic spherical shell in vucuo were first studied by Lamb [2]. Applying the method of Lamb [2,3], Love [4] treated shells by exact elasticity theory and solved the problem of vibrations of a spherical shell containing an incompressible fluid. More recently, the solution for vibrations of a fluid-filled spherical membrane has appeared in the book by Morse and Feshbach [S]. More detailed treatments of extensional axisymmetric vibrations of spherical fluid-filled shells has been given by Rand and DiMaggio [6]. Their work has been extended to include the effect of the shell’s bending by Engin and Liu [7]. In all these investigations, however, the effect of viscosity has been omitted. It is well known that although for many practical applications the viscosity is vanishingly small and motion of the fluid executing small oscillations 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, where the viscous effect is noticeable. may be of great consequence to the hydrodynamic forces acting on the shell and consequently may have noticeable effects on vibration characteristics of the shells. The objective of the present study is to clarify the effect of viscosity on fluid/structure interactions by investigating the axisymmetric vibrations of a fluid-filled elastic shell. This is an extension of previous study on submerged shells by the author [B, 91. Based on the classical bending theory, the dynamic response of the shell is determined by the normal mode method. For the fluid mediums, the linearized Navier-Stokes equations for an adiabatic compressible fluid are derived from the general Navier-Stokes equations for a viscous, compressible, heat-conducting fluid. Moreover, for most applications, where the
The
0022-460X/81
/020205
+ 16 $02.00/0
$)) 198 1 Academic
Press Inc. (London
I I.mlited
206
T. C. SU
ratio of the depth of the extent to which the influence of viscosity penetrates into the fluid and the radius of the shell is small, the Navier-Stokes equations are further simplified by boundary layer approximations. From this, the first order viscous effects are evaluated for free oscillations of an elastic spherical shell and its contained fluid. 2. FORMULATION
OF THE PROBLEM
2.1. BASIC EQUATIONS FOR SHELL The dynamic response of the shell may be determined by the normal mode method. 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 f3) and their derivatives. Except for k = 0, there exist two natural frequencies, wk and w;, for each value of k. The mode shapes for both frequencies are identical but the ratios of the tangential to radial displacements are different. The natural frequencies are determined by the frequency equation as given by Hayek [lo], for example. The mode shapes corresponding to o’k are designated by W; and V; and are given by wl, = Pk(cos 8),
vl, = c; (d/d@Pk(cos #),
(1)
where ”
(1 + v,)(l+ E,) + s,(k - l)(k +2) =(1+e,)(k2+k-1+v,)-(l-v~)(~~R/c,)2’
(2)
Here v,~is Poisson’s ratio, cs is the sound speed of the shell material, F, = h2/12R2, and h and R are the thickness and radius of the shell, respectively (see Figure 1).
Figure
1. Shell geometry.
The mode shapes corresponding to Wkare designated by wk and vk and can be obtained from the given expressions by merely omitting the primed superscripts. It should be noted, however, that wk, wk, and Vk do not exist for k = 0. With the notation given, the dynamic response of the submerged shell can then be expressed in terms of normal modes: v=r/bvb
+k;,
w=dlwb
+ f k=l
[77k(I)vk(~)+rl;(l)v;(8)1,
[rlk(c)Wk(B)+77~(t)W;(8)],
(3)
FREE
VIBRATIONS
where qk and q; are normal co-ordinates, ;ik +&hk ;ii,
+w;*&
OF FLUID-FILLED
determined
=js
(?vk
-pswk)
=I{
(rsVI( -p,WI,)
SHELLS
207
by the equations of motion
dA/2-nR”pshMk,
dA/2rR2p,hAl;,
131
in which 2nR’&hM, and 2xR2p,hM; are the generalized masses, and 7S and p5 are tangential and norma! hydrodynamic forces which must be determined from the 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, for example, in the book by Moore [ll]. However, solving the general equation is extremely difficult due to the presence of non-linear convective terms. In order that the convective terms should be negligible, it is necessary that the amplitude of oscillation, a, be small in comparison with both the radius of the shell, R, and the depth of penetration of the rotational flow, S. This is assumed to be the case. Furthermore, since the oscillations are small, the relative changes in the fluid density, pressure, temperature, and transport parameters are also small. Therefore, these equations can be linearized to yield the following governing equations for the fluid: continuity equation: @‘/at + p. div II = 0;
15)
momentum equation (with volume force absent): ~o(&B/c%)= -grad p’+ ~OV2u+(~~+ ~43) grad div u. energy equation (without external heat addition): ds’lat = -(~/PO) div (q/T) -(llpoT2)(q.
grad T) +(llpd@,/T,
(71
where p, p, and T are the density, pressure, and temperatures of the fluid, the subscript 0 represents the constant equilibrium value of a variable while the primed notation represents the small variation of that variable, u is the velocity vector, s is the entropy per unit mass, and q and Of are the heat conduction vector and the rate of dissipation of energy per unit volume, respectively; LO= Ao+$.L~ is the volume or bulk viscosity, and ho 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 the 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 whence 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 change in entropy in its dimensionless form, s’/cp is Of the order of (T’/T)(a/R)(c,p/k )-’ :x: Thus the change in entropy, s’ , is also of second order importance provided (wflR/w )-I. that the reciprocal of the product of the Reynolds number, p(aR)R/p, and the Prandtl number, c+/k, is at most of order 1 (c, is the specific heat at constant pressure). This condition can be fulfilled if the frequency R 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’= (@Jl&%)sP = C:,P’,
(8)
208
T. C. SU
where co is the constant equilibrium value of the sound speed in the fluid medium. The basic equations for an adiabatic compressible fluid can therefore be summarized as follows: the continuity equation (5), the momentum equation (6), the p’and p’relationship (S), and the viscous stress tensor T = ho(div u)I+ po(def u).
(9)
Taking the divergence of equation (6) and combining the result with-equations (S), one obtains an uncoupled equation for p’, (1/C~)~2p’/i)t2=v2p’+(1/C~~O)(~~O+~0)v2(~p’/~f) while the substitution equation for u:
of equations
(5) and (8) into equation
(10)
(6) yields the following
po$= _ l+5u+y3 5 Vp’ f jlov2u. POCO at
>
(
(5) and
(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 solutions for u and p, together with the stress strain relationship (9), completely prescribe the hydrodynamic forces acting on the shell. 2.3. BOUNDARY 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 (the non-slip condition). Consequently, the boundary conditions at the fluid-shell interface can be written as u,(R 8, t) = W,(R
8, f),
Q(R 8, t) = V,(R
8, f),
(12713)
where t’, and ve are the radial and tangential components of the velocity vector u, respectively. In addition, it is required that the velocity components are finite at the origin. 3. SMALL VISCOSITY ANALYSIS 3.1.
THE
BOUNDARY
LAYER
EQUATIONS
The influence of small viscosity will be examined later, and one can therefore begin with the linear equation (11). After taking the curl of both sides, the term curl grad p vanishes identically, giving p&)/at) curl u = j&J2 curl u.
(14)
This indicates that the vorticity, curl u, satisfies a heat conduction equation. Therefore, it decreases exponentially toward the interior of the fluid. In other words, the motion of the fluid caused by the oscillations of the shell is rotational only in a certain layer near the wall, while at larger distances it rapidly changes to potential flow. The depth of penetration of the rotational flow is of the order S -Jv/n, where Y is called the kinematic viscosity and is equal to po/po. For a fluid medium of vanishingly small viscosity, two important limiting cases are possible here. For the potential flow region, wo, lo --, 0, and the governing equations (10) and (11) reduce to the Eulerian formsi (1/C2)a2p/at2 + Here, and henceforth,
for brevity
= v2p,
paujat = -vp.
the suffices in pO and pLg and the superscript
(15,161 in p’ are omitted.
FREE
VIBRATIONS
OF
FLUID-FILLED
SHELLS
209
Within the boundary layer, however, the effect of viscosity is significant. It is necessary to retain the viscous term in the Navier-Stokes equations. This can be accomplished by a double limiting procedure, a/& + CCas k, [ + 0, known as Prandtl boundary layer theory [ 121. For axisymmetric motions, this will reduce the 8-momentum component of equation (11) to the form, valid only within a thin layer near the spherical shell, pavO/i)t= -(l/R)(i~p/M)l,=~ with the inner boundary
condition L’fj= v,,
and the outer
boundary
(17)
+p3’vOlilr2,
atr=R
(18)
condition 1’0= L’8 ,nvlscidJr=R at (R -r)/(v/R)“‘>>
1.
(19)
It is known that the change in pressure across the layer is small and is determined by the f+ctionless flow. The bulk viscosity lo has disappeared from the boundary layer equations, since it contributes a term of order U(V/~U~~). The effect of curvature is also irrelevant since for finite curvature, it has appeared only in the convective terms in the substantial derivative, which have been neglected in the present study. Sufficiently near the boundary, 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, S/R -Jv/flR’<< 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. 3.2. 3.2.1.
BOUNDARY
Flowfield
LAYER
SOLUTIONS
(R-ct
Based on the classical boundary layer approximation discussed in the preceding section, 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, tie, and the radial velocity component, t’,, outside of the boundary layer. Near the solid surface, however, z’~ is governed by the boundary layer equation (17) and C, is determined from the continuity equation once us has been obtained. Since the pressure, p, satisfies the wave equation (lS), and is regular at the origin, its solution may be written as, for harmonic oscillations, p
=eiRr? A,,j,(ar)P,,(x),
(20)
n=O
where cy = n/c, x r cos 8,0 is the frequency of vibration, A, is an arbitrary constant to be determined by the boundary conditions at the shell’s surface, and the j,(ar) are spherical Bessel functions of the first kind. Substituting expression (20) into equation (16), one obtains the following solutions for L’, and vH, i
3c
u, = i
C, A, j Xar)P,k) n-
eiR’,
X y A,tj,I(ar)s(x) I10= Rpr ,,Lo
i:~(z)=di,(z)/dz,
e’“‘,
(21) (221
while the substitution of expression (20) into the boundary layer equation (17) and matching the inviscid solution (21) by enforcing the outer boundary condition (19), yields
210
T. C. SU
the following expression for ue near the shell surface (r = R), dPn(X) -ice
iot ’
(23)
where g is the principal value of (iL?/v)“’ and the D,‘s are arbitrary constants. Substitution for p' from equation (8) and vg from equation (23), and the use of expression (20), then gives v,, from the continuity equation (5),,as +I)$
1
P,,(,y)eiR’.
(24)
The tangential component of the hydrodynamic force of the inner fluid acting on the shell is then given by, according to the boundary layer approximations,
Tr8
I
rzR
due
=‘r
I,_R
cc
=pg
RR
dP,*(xl
,,;oDnedee
ior
.
(25)
The unknown constants A,, and D, in these equations can be expressed in terms of the normal co-ordinates nk and nk, by enforcing the non-slip boundary conditions at r = R. For harmonic oscillations, the normal co-ordinates qk can be written as q)7k = qk eiRr (with a similar expression for 7;). With the aid of expressions (20) and (25), one can obtain the following expressions for the hydrodynamic forces acting on the shell surface (r = R j:
+q:
[
n(n -tl) l----
gR
J = j,(aR)/jL(aR). 3.2.2.
JPn(x),
(26)
(28)
Shell response
Substituting the expressions obtained for the hydrodynamic forces into the shell equations of motion (4), one obtains the following governing equations for the axisymmetric vibrations of a spherical shell submerged in a compressible fluid medium for k 2 1:
(2%)
(2%)
FREE
VIBRATIONS
OF FLUID-FILLED
211
SHELLS
where
P*k=A[ck-~J][c;-~J]. v5k=n[ck-~J]2, .1=(1-i)J2
-k(k+l) R 1 ~h~(&)1’2~(+)1’2,
mk=z,
(30)
and the expression for Plr3kis similar to that for PI, except that Ck must be replaced For k = 0, q. does not exist and 4;) is given by
~[(~)2-(~)2_@o(~)2]=o,
by C;i.
131)
where
It should be noted that the viscosity effect has appeared only through the functions !Prk, 1Y2k,and Pjk. For an inviscid fluid or for breathing vibrations (i.e., k = 0), these quantities vanish as would be expected. 3.3, FREQUENCY
EQUATIONS
Both equations (29) and equations (31) are a set of homogeneous algebraic equations. Requiring that non-trivial solutions exist for the displacement (qk/h) and (q;/h) yields the frequency equations (w;R/c.J~ - (f2R/c.J2 - @o(flR/c,)’ = 0
(32)
for k = 0 and (f?R/~,)~[l f Qj,rnk + @k]- (nR/c,)2{~,m,(u,R/c,)‘+
~k(~~Rlc,)~+[(~kRIc,)~
+
~~~~/C.~~*~~+~W~~/C,~~~~~~/~,~~-~~~/~.~~*~~~~~~~~~/~,~~
-
(1 + mA)(f?R/c,)*l+
+2@kP2kmk(f2R/c,,)2}
9gkmk[(wkRIcs)2 - Cl+
%)(f2RI~,)~l
= 0
(33)
for ksl. As mentioned previously, the viscosity effect has appeared only through the functions qIk, Pzk, and tYXk,which vanish for an inviscid fluid. One may immediately recover the following frequency equations for the inviscid flow case: (&R/c,)
- (~&R/c,)~ - @&A,R/cJ2
= 0
(341
for k = 0, and WWIcJ2
-
~w~~l~,~‘lC~.n,,~l~,~*bdJVcJ21
+ @~(~dVc,)2(m~+ l)[K41Rlc,)~- (&RIc,)~I = 0
(35)
for k s 1, where 3: = (m,w2, +uhz)/(mk + 1)
(36)
212
T. C. SU
and 0, is the free oscillation frequency when viscosity is absent and (Pk is a function of (30) evaluated at 0 = 0,. In general, the frequency equations (32) and (33), 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, 1?Vlkl, 1F,, 1and 1lP3k 1are of order (v/ Rc,)~” 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 Ro, in an analytical form, by Newton’s method. The results are OR/c, as defined by equation
flR,c,
= f&R/c,
(37)
for k = 0 and OR/c,
=(~noR/c,)-(mk~k(~0RI~,)‘(tylk+1Y3k--22k)
+
(~,R,c,)~(W,,[(~~OR,C,)~
-
bDV~s)~l
~w~~/c,~21~~,~~~~,,~,cs~2 x Wk[mk(~~R,c,)~+ k4NcA21+ K4VA2 + ~~1Rl4~1~
+ ‘J’~~/~K~oR,c~~~
-
-Gc~~R,c,)~(w;R/c,)~ -bn,&~~R,c,)~
+
+ (0oRlc,Pb,kh
+ WZ,RlcJ4
(w;R,c,)~I(~oR,c~)‘}}
(38)
for k 2 1, where &, ‘Pik, and &( = d&/d(flR/c,)) are evaluated at 0 = 0,. Therefore, once the natural frequencies for the vibrations in an inviscid fluid are known, the corresponding frequencies in a viscous fluid can be readily computed. It should be pointed out that the above formulae were obtained from the first order iteration of Newton’s method, which provides sufficient accuracy for consistency with the boundary layer approximation. 4. NATURAL FREQUENCIES FOR THE VIBRATIONS IN AN INVISCID FLUID The method of false position was used for the root searching of equations (34) and (35). The solutions of these equations are presented by the solid lines+ in Figures 2(a)-(d) for a steel shell filled with water, petroleum, ethyl alcohol (C2HsOH), and carbon tetrachloride (CCL), respectively. The values of c/c, and p/ps are given in Table 1. Broken lines in these figures, defined by j ;(L&R/c) = 0, are plots of the frequency spectra for the fluid in a rigid spherical shell. The frequency in uucuo is plotted in Figure 3 for comparison purposes. Figures 2(a)-(d) indicate that the natural frequencies of a fluid-filled elastic shell follow very closely either those of a fluid-filled rigid shell (broken lines in these figures, defined by j; = 0) or the in Z~UCUO natural frequencies. This has been recognized by several authors [6, ,7] and it can be proved analytically as follows. From equations (34), (35), (30) and (28), and with the aid of the identities [(2k + 1)/2]M;
=1
for k = 0,
[(2k + 1)/2][MJ(l+
mk)]= 1
for k 2 1,
one can write (39) + It is to be noted that this and all other spectra plotted in this paper are discrete: corresponding to integral values of the mode number k are physically meaningful.
i.e., only those
points
0
Flgurc 2 Frequency
IO
012345678 shell, (ci an cth~l-alcr)hol-filled
Mode number k shell and 1d13 carbon-tetrachlori[i~-~lled
9 to shell.
---, Steel shell
214
T. C. SU TABLE
1
Physical constants used in the numerical computations Physical constants of fluids (at 10°C)
c/c, 0.292 0.267 0.242 0.195
Water Petroleum Ethyl alcohol Carbon tetrachloride
(~/Rc,)‘~~’
PIPS
0.127 0.114 0.103 0.207
2.93 x 1.0.5 x 3.42 x 2*14x
1o-5 10T3 1O-5 lo-’
Physical and geometrical constants of steel shell cS = 49.7 x 10’ m/s (16.3 x lo3 ft/s) kg/m3 (15.25 Ibf s2/ft4) v, = 0.3 h/R = 0.03 R = 0.3048 m (1 ft)
ps =7871.65
I,
/
I1
I
12345676
I
I
I
9
IO
Mode number. k Figure 3. Natural
frequencies
of a steel shell in UCJCUCJ with h/R = 0.03.
- - --,
for &R/c,
FREE
VIBRATIONS
OF FLUID-FILLED
SHELLS
(%WG)~ -(w&c,)~ U&R/C,)*
215
c4*1
for k = 0 and k z 1, respectively. For¬ close to in ~a~uu frequencies, it is seen that the last factor on the right-hand side of equation (39) and the last two factors in equation (40) are at least of order of one. Consequently, h ps cs f&R 1 _____-R p c c, O(1)
for all values of k. For practical purposes, the value of (h/R)(p,/p)(c,/c)
IjI
- _--&R .I cs I
Furthermore,
141) is of order of one. Therefore,
1 O(1)’
it can be shown from expressions (34) and (35) that the smallest value of
f&R/c not close to in U~CKUfrequencies must be greater than the first zeros of jd&R/c) = 0 for k = 0 or j;(f&R/c) = 0 for k 2 1. Consequently, with the values of L-,/Cused in the present study, these f&R/c, must be at least of order of one. Hence,
jI I-_J.’
1 O(1)’
A careful examination of the Bessel function as plotted in reference [13] reveals that this is only possible if j’(&R/c) is small. Therefore, the natural frequencies of a fluid-filled elastic shell, if not close to in oucuo frequencies, must follow closely those of a fluid-filled rigid shell. The degree of closeness increases with (h/R)(p,/p)(c,/c) as can be shown from expression (41). Moreover, it can also be shown that the spacings of the frequency spectra are proportional to c/c,. These confirm the results presented in Figures 2(a)-(d). Figure 4 is a plot of frequency spectra for a water-filled steel shell with a smaller h/R ratio. From this it is clear that as h/R decreases, the natural frequencies begin to deviate from either the in t’acuo frequencies or frequencies of a fluid-filled rigid shell. More computation has been carried out and results are tabulated in a report by Su [ 143. 5. EFFECTS OF VISCOSITY Natural frequencies for the vibrations in a viscous fluid are given by equation (38) for is generally complex and can be written as the sum of real and imaginary parts: a= fi +ifi. Since the natural frequencies of an inviscid-fluidfilled shell are always real, the damping component fi is solely due to the effect of viscosity. It can now be shown that the viscosity will produce a damping effect on the shell motion: i.e., fi > 0. Moreover, the change on the real and imaginary components of each natural frequency due to viscosity are equal in magnitude but opposite in sign: that is, fi = Q0 - 6. Therefore, the radian frequency of a vibrating spherical shell will decrease due to the viscosity of the contained fluid. First, it can be noted that by using expression (30) and after some mathematical manipulations, the numerator of the perturbed term in expression (38) can be expressed as
k 2 1. Thus, the natural frequency
A (%Rlc,)3 W,R/c,+ x
GhNc,~*lh + 1)
I(ck-~J).[(~)2-(~)2]+m,(c~-~~)[(~)2-(~)2]]2,
216
T. C. SU
9
3
2
0
I2
345676 Mode number, k
9
IO
Figure 4. Frequency spectrum for a water-filled thin shell with h/R = 0.003. Key as Figure 2.
while the term
which appears in the denominator rewritten as 2{(&R/cs)“[1
+
of the perturbed
@k(mk
+
1)1-
term in expression
(38), can be
h’k~/d*h’;~h)*~
by virtue of expression (35). Moreover, from the definition of C& in expression (36), the remainder in the denominator of the perturbed term in expression (38) is equal to (%R/c,)3(W
+ l>&{(%R/c,)*
- &R/c,)*).
Thus equation (38) can be put in the form RR/c,
= (&R/c,)
+ mk,(C; x (mk
-{A(fhR/cs)3/h -
+
l)){(ck
WU~>J)WOWG~*
+ lXW,Rlc,~*-
-
WRWW&NG~*
- (~kR/cJ*l~*/{~ h&~/d”~*~k~,
-
b.dVc,)*1
+ MMVG>~
(42)
FREE
VIBRATIONS
OF FLUID-FILLED
217
SHELLS
where 0 = 21(n,Rlc,)4C1 + @k(rnk +
III- (w~Rlc,)2(w;Rlc,)2}[(noR/c~)2 - kdvCJ21
may be rewritten as 0 = 2(WkRIC.~)2[(~oRIC.~)2 -
b%wcs~21” +2mkb3&C,)* - bkwc,~21
x {[o&~IC,~2- hJvcs~212 + (wk~Ics~2HdvG~2- hQ~/dII by first eliminating Gk with the aid of expression (35) and then expressing w;’ in terms of WE and Wi by virtue of expression (36) and some mathematical manipulations. It is then obvious that 0 is a positive quantity since & > wk G=0 and mk > 0. Moreover, bk
=
d@k/d(f&&/cs)
2
c,
= c 2k
=
RZp
(c,/c‘)d~k/d(RoR/c,),
d h Mk z d(&R/c,)
1’
1
jk(GRIc) [ (RoR/c) j;(&R/c)
which is also a positive quantity, since it can be shown analytically that d j,(x) dx [ xj;(x)
-
1
=~{“[“l(‘“*‘““‘i”‘l’*‘““;‘““~o
for x >o,
provided k is a positive integer. Proof of this result is given in the Appendix. Thus the perturbed term in 12R/cs, which is caused by the fluid viscosity, takes the form [(-,I) times a positive quantity], and can be expressed as [(-1 + i) times a positive quantity] by the definition of n given in expression (30). Therefore, it is concluded that the viscosity of the fluid within the shell will cause a change of frequencies of the free vibration. The change of the real and imaginary components of each natural frequency are equal in magnitude and opposite in sign. The imaginary part of the natural frequency, solely due to viscosity, is always positive. Thus, the fluid viscosity will produce a damping effect on the vibration and will reduce its radian frequency. Numerical computations have been carried out for a steel shell of hf R = 0.03 filled with water, petroleum, ethyl alcohol and carbon tetrachloride, respectively, and for a steel shell of smaller h/R ratios filled with water or petroleum. The petroleum was chosen for its high viscosity. The numerical results of these computations are summarized in reference [ 141. In the numerical computations physical constants listed in Table 1 are used. For the case of h/R = 0.03, it is noted that although the viscosity of the fluid will reduce the radian frequency fi, its contribution to the frequency is indiscernible. Therefore, the radian frequency fi is essentially the same as 0, given in Figures 2(a)-(d) for the inviscid flow problem. Some noticeable changes on the radian frequency may occur due to the viscosity of fluid for very small values of h/R, such as h/R = O-0003 for a fluid-filled steel shell, as indicated from data in reference [14]. It also should be noted that the damping component fi of the natural frequency for a viscous-fluid-filled shell is always positive. The numerical results confirm that the change on the real and imaginary components of each natural frequency due to viscosity are equal in magnitude but opposite in sign. The damping distributions plotted in Figures 2(a)-(d) and 4 further indicate that larger damping occurs in the neighborhood of d - &,, which is consistent with the form of expression (42), since the denominator of the perturbed term in expression (42) approaches zero as 0, approaches &. Typical curves of damping components for the mode numbers k = 3 and 4 given in Figure 5 reveal that viscous damping reaches a peak value in the vicinity of the in uac~u frequencies w ; and then drops rapidly to a minimum. It can be noted that the minimum damping is so small that the shell motion is almost steady state (non-dissipating).
218
T. C. SIJ
2 10-S 5
2 lo-‘0 5
6
01234567 Radian
frequency,
9
IO
II
h/c,
Figure 5. Damping components of the natural frequencies for a water-filled steel shell, typical modes.
A study of the balance of energy has been conducted by Su [14]. It is shown that due to the damping effect of the fluid viscosity, the total energy of both the interior potential flow and the shell decreases while the energy is transferred into the boundary layer. 6. CONCLUSION In order to clarify the effect of viscosity on fluid/structure interaction, the investigation of axisymmetric vibrations of a compressible fluid contained in a spherical elastic shell has been carried out on the basis of boundary layer theory. In the absence of viscosity, it has been shown that the natural frequencies are real and follow very closely either those of a fluid-filled rigid shell or the in uucuo natural frequencies. The degree of closeness increases with (h/R)(p,/p)(c,/c). The analysis of viscosity reveals, that, for free vibrations, the viscosity will in general produce a damping effect on the shell motion. Computations reveal that viscous damping reaches a peak value in the vicinity of the in V~CUOfrequencies w; and then drops rapidly to a minimum. It is noted that the minimum damping is so small that the shell motion is almost steady state (non-dissipating). Moreover, it has been shown that the change in the real and imaginary components of each natural frequency due to viscosity are equal in magnitude but opposite in sign. Therefore, the radian frequency of a vibrating spherical shell will be reduced due to the viscosity of the contained fluid. However, the reduction is almost indiscernible and the viscosity has essentially no effect on the real components of the natural frequencies, provided that h/R is not extremely small. From a study of.the energy balance it has been concluded that due to the viscous damping an energy transfer
FREE VIBRATIONS
219
OF FLUID-FILI.ED SHELLS
takes place, with the total energy of both the interior potential while energy is pumped into the boundary layer.
flow and the shell
decreasing
REFERENCES RAYLEIGH 1872 Proceedings of the London Marhematical Society 4, 93-103. On the vibrations of a gas contained within a rigid spherical envelope. H. LAMB 1882 Proceedingsof the London MathematicalSociety 14,.50-56. On the vibrations of a spherical shell. H. LAMB 1882 Proceedings of the London Il4athematicalSociety 13, 189-212. On the vibrations of an elastic sphere. A. E. H. LOVE 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. 1953 Methods of Theoretical Physics, Part II. New York: P. M. MORSE and H. FESHBACH McGraw-Hill Book Company. See pp. 1469-1472. 1967 JournaloftheAcousticalSocietyofAmerica 42, 1278-1286. R. RAND andF. DIMAGGIO Vibrations of fluid-filled spherical and spheroidal shells. A. E. ENGIN and Y. K. LIU 1970 JournalofBiomechanics 3,l l-22. Axisymmetric response of a fluid-filled spherical shell in free vibrations. 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. T. C. SU and Y. K. LOU 1975 Journal of Engineering for Industry, Transactions of the American Society of Mechanical Engineers 97, 1338-1344. The effect of viscosity on the dynamics of a submerged spherical shell. 1966 Journal of the Acoustical Society of America 40, 342-348. Vibration of a S. HAYEK spherical shell in an acoustic medium. F. K. MOORE (editor) 19.59 Theory of Laminar Flovv. Princeton University Press. See pp. 34-46. K. STEWARTSON 1964 The Theory of Laminar Boundary Layers in Compressible F1uid.c. Oxford University Press. See pp. 1.5-22. M. ABRAMOWITZ and I.A. STEGUN 1965 Handbook OfMathematical Functions. New York: Dover. See p. 438. T. C. SU 1980 TEES Report No. COE-217, Ocean Engineering Program, Texas A&M University. Natural frequencies of fluid-filled spherical shells.
1. LORD 2. 3. 4. 5. 6. 7. 8. 9.
10. 11. 12. 13. 14.
APPENDIX: AN INEQUALITY INVOLVING SPHERICAL BESSEL FUNCTIONS Here, the proof of the inequality involving the spherical Bessel function of the first kind of order n, j,(x) is presented. It is to be shown that x(ji’ -j,jE)+aj,jl,>O
(Al)
for all values of II = 0, 1,2,3, . . . , )a/ s 1 and x > 0. From the definition of jn(x), it can be seen that j,{xj,“+2xj~,+[x2--n(n+1)]j,z}=0.
(A21
Multiplying both sides of expression (Al) by x and replacing the -x2j,,jz {2xj,,jA+ [x2- n (n + l)]ji} shows that expression (Al) is equivalent to
term by
x2j:,2+(2+(Y)Xj.j:,+[X2-n(n+l)]j2,)0, which may be rewritten as (xj~/j.)‘+(2+cu)(xjL/j.)+[x2-n(n
+l)]>O,
or
(A3J
220
T. C. SU
If x2 > n(n + 1) + (1 + a/2)‘, the proof of the inequality (A3) is trivial, whereas for x2 G n (n + 1) + (1 + a/2)2, the inequality (A3) can be verified by introducing a new variable 2 = (xjL/j,)+(l+cu/2),
(A4)
which satisfies the differential equation xz’+(l+a)Z+(1+LY/2)(1+CY)+[22-n(n+1)+x2-(1+a/2)2]=O
(AS
by virtue of expressions (A2). Thus, the inequality (A3) is equivalent to T(x)=z2-n(n+1)-(1+a/2)*+x2>0.
(A6)
From the definitions of z and r, it is obvious that z(x), and thus r(x), is a piecewise continuous function of x with discontinuities which occur at the zero of j,. However, in the vicinity of these discontinuities, r(x) is always positive. Furthermore, as x + O’, t -+ . As it has these special properties, it is clear n(l+a)+x*>Ofor~~~~l,andn=0,12,... that T(x) will be a positive definite function of x if T’(x) > 0 whenever r(x) = 0. If f(x) is assumed to vanish at a certain point x = x0, then r:,-[n(n+l)+(l+~/2)*-X~]=o,
(A7)
and from expression (A5) x,2~,+(1+cu)2o+(1+a/2)(1+(Y)=0,
(A8)
where z. = 2(x0) and zb = z’(xo). With the use of expression (A6), T’(x0) = 2(zozb +x0). By using expressions (A7) and (A8), this can be reduced to T’(xo) = (2/xo)[(l+&
-(l +cu)(l +cu/2)zo+x;].
(A%
One can then proceed to show that (1+a)*:,-_(1+~)(1+~/2)zo+x~>0,
(AlO)
where
z0=*Jn(n +l)+(l+cx/2)2-x~
(Al la, b)
for 0 < xi s II (n + 1) + (1 + u/2)*. For negative zo, the proof of expression (AlO) is trivial, whereas for positive .zo, the inequality (AlO) can be verified by showing that [(1+cu)f~+X~]2>(1+a)2(l+~/2)2Z~. By virtue of expressions (A7) and (Alla), that
and a rearrangement
of terms, it can be shown
[(1+ cY,z:, + x;12 - (1 +a)*(1 + (w/2)2Zo =(1+cu)2z~[n(n+1)]+(1-(u2)[n(n+1)+(1+(u/2)2]xo+(u2X~, which is greater than zero, for ]CX] G 1. Consequently, completes the proof.
r(x)
is positive definite. This