Table Of ContentModeling and analysis of water-hammer in coaxial pipes
Pierluigi Cesana∗, Neal Bitter†
California Institute of Technology, Pasadena, CA 91125, USA
January 30, 2015
5
1
0
2
n Abstract
a
Thefluid-structureinteractionisstudiedforasystemcomposedoftwocoaxialpipes
J
inanannulargeometry,forbothhomogeneousisotropicmetalpipesandfiber-reinforced
9
(anisotropic)pipes. Multiplewaves,travelingatdifferentspeedsandamplitudes,result
2
whenaprojectileimpactsonthewaterfillingtheannularspacebetweenthepipes. Inthe
case of carbon fiber-reinforced plastic thin pipes we compute the wavespeeds, the fluid
]
n pressureandmechanicalstrainsasfunctionsofthefiberwindingangle. Thisgeneralizes
y thesingle-pipeanalysisofJ.H.You,andK.Inaba,Fluid-structureinteractioninwater-
d filled pipes of anisotropic composite materials, J. Fl. Str. 36 (2013). Comparison with
- a set of experimental measurements seems to validate our models and predictions.
u
fl Keywords: Fluid-structureinteraction;Water-hammer;homogeneousisotropicpipingma-
s. terials; carbon-fiber reinforced thin plastic tubes.
c
i
s
y 1 Introduction
h
p This article is part of a series of papers [20], [3], [4], [7], [13] devoted to the investigation of
[
water-hammerproblemsinfluid-filledpipes,bothfromtheexperimentalandtheoreticalper-
1 spective. Water-hammerexperimentsareaprototypemodelformanysituationsinindustrial
v andmilitaryapplications(e.g.,trans-oceanpipelinesandcommunicationnetworks)wherewe
3
have fluid-structure interaction and a consequent propagation of shock-waves. After the pi-
6 oneering work of Korteweg[11] (1878) and Joukowsky[8] (1900), who modeled water-hammer
4
7 waves by neglecting inertia and bending stiffness of the pipe, a more comprehensive inves-
0 tigation, developed by Skalak [14] in the Fifties, considered inertial effects both in the pipe
1. and the fluid, including longitudinal and bending stresses of the pipe. Skalak combined the
0 Shell Theory for the tube deformation and an acoustic model of the fluid motion. He shows
5 thereisacoexistenceoftwowavestravelingatdifferentspeeds: theprecursorwave(ofsmall
1
amplitude and of speed close the sound speed of the pipe wall) and the primary wave (of
:
v larger amplitude and lower speed). Additionally, a simplified four-equation one-dimensional
i model is derived based on the assumption that pressure and axial velocity of the fluid are
X
constant across cross-sections [14]. Later studies of Tijsseling [15]-[17] have regarded model-
r
a ing of isotropic thin pipes including an analysis of the effect of thickness on isotropic pipes
based on the four-equation model [17]. While all these papers consider the case of elastically
isotropic pipes, the investigation of anisotropy in water-filled pipes of composite materials
∗correspondingauthor,nowattheMathematicalInstitute,WoodstockRoad,OxfordOX26GG,England;
conductedtheoreticalanalysisofthemodel
†preparedexperimentalset-upandperformedexperimentalmeasurements
1
wasfirstobtainedin[20]wherestresswavepropagationisinvestigatedforasystemcomposed
of water-filled thin pipe with symmetric winding angles θ. In the same geometry, a plat-
±
form of numerical computations, based on the finite element method, was developed in [13]
todescribethefluid-structureinteractionduringshock-waveloadingofawater-filledcarbon-
reinforced plastic (CFRP) tube coupled with a solid-shell and a fluid solver. More complex
situationsinvolvesystems ofpipesmountedcoaxially wheretheannular regionsbetweenthe
pipescanbefilledwithfluid. Inthisscenario, Bu¨rmannhasconsideredthemodelingofnon-
stationaryflowofcompressiblefluidsinpipelineswithseveralflowsections[5]. Hisapproach
consists of reducing the system of partial differential equations governing the fluid-structure
interaction in coaxial pipes into a 1-dimensional problem by the Method of Characteristics.
Laterworkshaveappearedonthemodelingofsounddispersioninacylindricalviscouslayer
bounded by two elastic thin-walled shells [12] and of the wave propagation in coaxial pipes
filled with either fluid or a viscoelastic solid [6].
Motivated by the recent experimental effort of J. Shepherd’s group on the investigation
of the water-hammer in annular geometries [1], [2], [3], [4], we extend the modeling work
of [17] and [20] to investigate the propagation of stress waves inside an annular geometry
delimited by two water-filled coaxial pipes, in elastically isotropic and CFRP pipes. A pro-
jectile impact causes propagation of a water pressure wave causing the deformation of the
pipes. Positive extension in the radial direction of the outer pipe, accompanied by negative
extension (contraction) in the radial direction of the internal pipe, causes an increase in the
annular area thus activating the fluid-structure interaction mechanism.
The architecture of the paper is as follows. After reviewing the work of You and Inaba
on the modeling of elastically anisotropic pipes, we present the six-equation one-dimensional
model(Paragraph2.2)thatrulesthefluid-solidinteractioninatwo-pipesystem. InSection3
wecompareourtheoreticalfindingswithexperimentaldataobtainedduringaseriesofwater-
hammerexperiments. Finally,inthecaseoffiberreinforcedpipes,thewavepropagationand
the computation of hoop and axial strain are described in full detail in Paragraph 3.3.
r
ϕ r
ϕ
R1 R2
+θ
z
e1 −θ
e2
Figure 1: Schematic representation of coaxial thin pipes. LEFT: cross section. RIGHT:
lateral view. Notice that here we are referring to the case of CFRP pipes with winding
angles θ.
±
2 Thin pipes modeling
2.1 One-dimensional fluid-structure modeling
According to the technique of Tijsseling [17], one-dimensional governing equations for the
liquid and the pipes can be obtained upon averaging out the standard balance laws in the
radial direction. By adopting a cylindrical coordinates system, this approach is based upon
the assumption that the behavior of water velocity and pressure depend only on the spatial
2
Table 1: Notation. Here and in what follows subscript i is either set to be equal to 1 (in the
case in which we refer to the internal pipe) or 2 (external pipe).
C stiffnessmatrix(x,yandzcoor- u ,u two-dimensionaldisplacementcom-
i i,r i,z
dinates) ponentsofpipeialongr andz axis
SS compliance matrix (x,y and z u˙ ,u˙ two-dimensional velocity compo-
i i,r i,z
coordinates) nents of pipe i along r and z axis
r,ϕ,z cylindrical coordinates u˙ ,u˙ one-dimensional velocity compo-
i,r i,z
nents of pipe i along r and z axis
t time u˙ magnitude of u˙
i,z0 i,z
V volume fraction (fiber) θ fiber winding angle
f
p(r,z,t) two-dimensional fluid pressure V one-dimensionalaxialfluidvelocity
P(z,t) one-dimensional fluid pressure V magnitude of V
0
P magnitude of P v ,v two-dimensionalfluidvelocitycom-
0 r z
ponents along r and z directions
P pressure outside pipe 2 ε ,γ normal and shear strain in pipe i
out i i
P pressure inside pipe 1
in
R inner radius of pipe i c wavespeed
i
e thickness of pipe i σ ,τ normal and shear stress
i i i
ρ density (pipe i) σ ,σ ,σ two-dimensional stress components
i,t i,r i,ϕ i,z
along r,ϕ and z axis
ρ density of fluid σ one-dimensional axial stress
w i,z
K bulk modulus of fluid σ magnitude of σ
i,z0 i,z
E ,E effectiveYoung’smodulusalong E ,E Young’s modulus for matrix and
(1) (3) m f
transverse and longitudinal di- fiber
rections in a single ply
G effectiveshearmodulusinasin- G ,G shear modulus for matrix and fiber
31 m f
gle ply
ν ,ν Poisson’s ratio for matrix and ρ ,ρ density of matrix and fiber
m f m f
fiber
variable z. In what follows we define one-dimensional cross-averaged quantities and obtain
the corresponding field equations.
2.1.1 Governing equations for the fluid
The balance laws in the coordinate system (r,z) for the fluid read [15]
∂v ∂p
(2-d) Axial motion equation: ρ z + =0,
w ∂t ∂z
∂v ∂p
(2-d) Radial motion equation: ρ r + =0,
w ∂t ∂r
1 ∂p ∂v 1∂(rv )
(2-d) Continuity equation: + z + r =0.
K ∂t ∂z r ∂r
Here v (r,z,t) and v (r,z,t) are, respectively, the axial and radial velocity of the fluid and
z r
p(r,z,t) is the pressure; K is the bulk modulus of the fluid and ρ is the density of the
w
fluid. Wenowintroducethecross-sectionalaveraged(one-dimensional)velocityandpressure,
3
defined respectively as
1 (cid:90) R2
V(z,t):= 2πrv (r,z,t)dr, (2.1)
π(cid:0)R2 (R +e )2(cid:1) z
2− 1 1 R1+e1
1 (cid:90) R2
P(z,t):= 2πrp(r,z,t)dr. (2.2)
(cid:0) (cid:1)
π R2 (R +e )2
2− 1 1 R1+e1
We are in a position to introduce the one-dimensional equations of balance for the fluid,
which are
(1-d) Axial motion equation
∂V ∂P
ρ + =0 (2.3)
w ∂t ∂z
(1-d) Radial motion equation
(cid:12) (cid:12)
R2p(cid:12) (R +e )2p(cid:12)
1ρ R ∂vr(cid:12)(cid:12) + 2 (cid:12)r=R2− 1 1 (cid:12)r=R1+e1 P =0 (2.4)
2 w 2 dt (cid:12)r=R2 (cid:0)R22−(R1+e1)2(cid:1) −
(1-d) Continuity equation
1 ∂P ∂V 2 (cid:104) (cid:12) (cid:12) (cid:105)
K ∂t + ∂z + (R22−(R1+e1)2) R2vr(cid:12)r=R2 −(R1+e1)vr(cid:12)r=R1+e1 =0. (2.5)
We remark that Eq. (2.4) has been obtained by multiplying the two-dimensional radial
motionequationby2πr2, integratinginr fromR +e toR anddividingby2π(R2 (R +
1 1 2 2− 1
e )2). Here R is the internal radius and e the thickness of the internal pipe while R is the
1 1 1 2
internal radius and e the thickness of the external pipe (see Fig 1-LEFT). Moreover, in Eq.
2
(2.4) it is assumed that
∂v ∂v (cid:12) ∂v (cid:12)
r r =R r(cid:12) =(R +e ) r(cid:12) . (2.6)
dt 2 dt (cid:12)r=R2 1 1 dt (cid:12)r=R1+e1
This is consistent with the (2-d) Continuity equation under the hypothesis that K is large
andthattheaxialinflowv isconcentratedinthecentralaxisinthelimitR 0,e /R 0
z 1 1 1
→ →
[17].
2.1.2 Governing equations for the pipes
Letting i = 1,2, the equations of Axial motion and Radial motion in the pipes in the space
(r,z) are
∂u˙ ∂σ
(2-d) Axial motion equation: ρ i,z i,z =0,
i,t ∂t − ∂z
∂u˙ 1∂(rσ ) σ
(2-d) Radial motion equation: ρ i,t = i,r i,ϕ.
i,t ∂t r ∂r − r
Hereρ isthedensityofthepipe,σ (r,z,t),σ (r,z,t)andσ (r,z,t)aretheradial,axial
i,t i,r i,z i,ϕ
and hoop stress respectively and u˙ , u˙ are the radial and axial velocity respectively. By
i,r i,z
applying the cross-sectional average technique we obtain
4
(1-d) Axial motion equation
∂u˙ ∂σ
ρ i,z i,z =0, (2.7)
i,t ∂t − ∂z
(1-d) Radial motion equation
(cid:12) (cid:12)
(R +e )σ (cid:12) R σ (cid:12)
∂u˙ i i i,r(cid:12) i i,r(cid:12) 1
ρ i,r = Ri+ei Ri σ , (2.8)
i,t ∂t e (R +e /2) − e (R +e /2) − (R +e /2) i,ϕ
i i i i i i i i
where
1 (cid:90) Ri+ei
u˙ (z,t):= 2πru˙ (r,z,t)dr, (2.9)
i,z π(cid:0)(R +e )2 R2(cid:1) i,z
i i − i Ri
1 (cid:90) Ri+ei
u˙ (z,t):= 2πru˙ (r,z,t)dr, (2.10)
i,r π(cid:0)(R +e )2 R2(cid:1) i,r
i i − i Ri
1 (cid:90) Ri+ei
σ (z,t):= 2πrσ (r,z,t)dr, (2.11)
i,z π(cid:0)(R +e )2 R2(cid:1) i,z
i i − i Ri
are respectively the one-dimensional axial velocity, radial velocity and axial stress and
1 (cid:90) Ri+ei
σ := σ (r,z,t)dr. (2.12)
i,ϕ e i,ϕ
i Ri
2.1.3 Elastic properties of pipes
−1
By introducing the stiffness matrix C and the compliance matrix SS := C , the stress-
i i i
strain relation under the plane stress assumption reads, respectively, [20, Eqs. (13, 14)]
σ C C 0 ε
i,x i,11 i,13 i,x
σi,z = Ci,13 Ci,33 0 εi,z , (2.13)
τ 0 0 C γ
i,zx i,55 i,zx
ε S S 0 σ
i,x i,11 i,13 i,x
εi,z = Si,13 Si,33 0 σi,z .
γ 0 0 S τ
i,zx i,55 i,zx
In the case of elastically homogenous and isotropic pipes, tensor C reads
i
C C 0 E /(1 ν2 ) ν E /(1 ν2 ) 0
i,11 i,13 i,t − i,t i,t i,t − i,t
Ci,13 Ci,33 0 ≡ νi,tEi,t/(1−νi2,t) Ei,t/(1−νi2,t) 0 (2.14)
0 0 C 0 0 G
i,55 i,t
and, in turn
S S 0 1/E ν /E 0
i,11 i,13 i,t i,t i,t
−
Si,13 Si,33 0 = νi,t/Ei,t 1/Ei,t 0 (2.15)
−
0 0 S 0 0 1/G
i,55 i,t
5
y
z
+θ
θ
−
y x Figure 2: Schematic rep-
z
resentation of CFRP lay-
θ
− up structures as a com-
x bination of single uniaxial
y,2
z plies. Axes 1,2 and 3 are
3 +θ theprincipalaxesofasin-
1 gle ply.
x
whereE ,ν andG =E /(2+2ν )aretheYoung’smodulus,Poisson’sratioandshear
i,t i,t i,t i,t i,t
modulus of the material from which pipe i is made.
For anisotropic composite (fiber-reinforced) pipes the stiffness elements C are neces-
i,kl
sarilyafunctionofthegeometricandelasticpropertiesoffibersandofmatrix, includingthe
fiber winding angle θ. The difficulty in describing the elastic properties of fiber-reinforced
plastic thin pipes has been studied in [20] under the assumption that pipes are obtained by
rolling up a woven layer with symmetric angles θ. Each of these layers can be considered
±
as a lay-up structure of multiple plies of same thickness as shown in Figure 2. To keep the
notation simple, in what follows we drop the subscript i. Elastic moduli are computed as a
function of fiber volume fraction V and fiber and matrix elastic coefficients [9]
f
1 V (1 V ) 1 V (1 V )
E =E V +E (1 V ), = f + − f , = f + − f , (2.16)
(3) f f m − f E E E G G G
(1) f m 31 f m
ν =ν V +ν (1 V ), ρ =ρ V +ρ (1 V ) (2.17)
31 f f m f i,t f f m f
− −
where E ,G ,ν and ρ are, respectively, the Young’s modulus, shear modulus, Poisson’s
m m m m
ratio and density of the matrix. Then E ,G ,ν and ρ are, respectively, the fiber Young’s
f f f f
modulus, shear modulus, Poisson’s ratio and density of the fiber. The subscripts 3 and 1
indicate the longitudinal and transverse direction of a single ply (see Fig. 2). The Poisson’s
ratioν isdefinedastheratioofthecontractednormalstraininthedirection1tothenormal
31
strain in the direction 3, when a normal load is applied in the longitudinal direction. The
stiffness matrix for the composite is given by the volumetric average of the elastic stiffness
±θ
matrices from each single θ ply, denoted in what follows with C . Precisely, if all plies
±
have the same thickness, the stiffness matrix for a woven layer of pairs of θ plies is given
±
by
1 +θ −θ
C= (C +C ). (2.18)
2
±θ
The components of C read [20, Eq. (16)]
C+θ =C−θ =(cid:2)cos4(θ)E /E +sin4(θ)E /E (cid:3)+2sin2(θ)cos2(θ)(cid:2)ν E /E +2G (cid:3),
11 11 (1) (cid:93) (3) (cid:93) 31 (1) (cid:93) 31
C+θ =C−θ =[cos4(θ)E /E +sin4(θ)E /E ]+2sin2(θ)cos2(θ)(cid:2)ν E /E +2G (cid:3),
33 33 (3) (cid:93) (1) (cid:93) 31 (1) (cid:93) 31
C+θ =C−θ =[E /E +E /E 4G 2ν E /E ]sin2(θ)cos2(θ)+ν E /E ,
13 13 (3) (cid:93) (1) (cid:93)− 31− 31 (1) (cid:93) 31 (1) (cid:93)
C+θ =C−θ =[E /E 2ν E /E +E /E 4G ]cos2(θ)sin2(θ)+G ,
55 55 (1) (cid:93)− 31 (1) (cid:93) (3) (cid:93)− 31 31
6
with E = 1 ν2 E /E . Finally, the compliance elements read S = C /C , S =
(cid:93) − 31 (1) (3) 11 33 (cid:93) 33
2
C /C , S = C /C and C =C C C , where C are the elements of the matrix
11 (cid:93) 13 − 13 (cid:93) (cid:93) 11 33− 13 kl
C defined in (2.18).
2.2 Six-equation model
The axial and hoop strains in pipe i can be written as [20]
ε =S σ +S σ , ε =S σ +S σ , (2.19)
i,z i,13 i,ϕ i,33 i,z i,ϕ i,11 i,ϕ i,13 i,z
respectively. By using the strain-displacements relations
∂u
ε = i,z, (2.20)
i,z ∂z
bydifferentiatingintimeandbytakingthecross-sectionalaverage,Eq. (2.19)-LEFTbecomes
∂u˙ ∂σ ∂σ
i,z =S i,ϕ +S i,z (2.21)
∂z i,13 ∂t i,33 ∂t
where
2π (cid:90) Ri+ei
σ (z,t):= rσ (r,z,t)dr (2.22)
i,ϕ π(cid:0)(R +e )2 R2(cid:1) i,ϕ
i i − i Ri
is the one-dimensional (cross-averaged) hoop stress.
The radial displacement equation is obtained by plugging another strain-displacement
relation, which is,
u
ε = i,r (2.23)
i,ϕ r
into Eq. (2.19)-RIGHT yielding
u =rS σ +rS σ . (2.24)
i,r i,11 i,ϕ i,13 i,z
The equations of fluid and pipes are coupled by boundary conditions along the interfaces.
Indeed, at each fluid-solid interface, we equate the radial velocity and radial stress of the
fluid with those of the solid.
(cid:12) (cid:12) (cid:12) (cid:12)
σ2,r(cid:12)r=R2 =−p(cid:12)r=R2, u˙2,r(cid:12)r=R2 =vr(cid:12)r=R2,
(cid:12) (cid:12) (cid:12) (cid:12)
σ1,r(cid:12)(cid:12)r=R1+e1 =−p(cid:12)r=R1+e1, u˙1,r(cid:12)(cid:12)r=R1+e1 =vr(cid:12)r=R1+e1, (2.25)
σ2,r(cid:12)r=R2+e2 =−Pout =const., u˙2,r(cid:12)r=R2+e2 =Vrout =const. (=0m/s),
(cid:12) (cid:12)
σ1,r(cid:12)r=R1 =−Pin =const., u˙1,r(cid:12)r=R1 =Vrin =const. (=0m/s).
As in [17], we assume that the external and internal pressures in each pipe induce a hoop
stress which is constant in ϕ. Accordingly, we have [20], [18]
1 R2(R +e )2(P Pin) R2P (R +e )2P
σ = 1 1 1 − + 1 in− 1 1
1,ϕ −r2 2(R +e /2)e 2(R +e /2)e
1 1 1 1 1 1
(2.26)
1 R2(R +e )2(P P) R2P (R +e )2P
σ = 2 2 2 out− + 2 − 2 2 out.
2,ϕ −r2 2(R +e /2)e 2(R +e /2)e
2 2 2 2 2 2
7
By plugging Eqs. (2.25-Lines 1 and 2, RIGHT) into Eq. (2.5) we obtain
1 ∂P ∂V 2 (cid:104) (cid:12) (cid:12) (cid:105)
+ + R u˙ (cid:12) (R +e )u˙ (cid:12) =0. (2.27)
K ∂t ∂z (R22−(R1+e1)2) 2 2,r(cid:12)r=R2 − 1 1 1,r(cid:12)r=(R1+e1)
Now,assumingthatradialinertialforcesareignoredinbothfluidandpipesandthatthepipes
cross-sections remain plane for axial stretches (thus implying the independency of σ (z,t)
i,z
onr,especiallyinthinpipes)weobtainasimplifiedmodel. UponsubstitutionofEqs. (2.24)
and (2.26) into Eq. (2.27) and upon substitution of Eq. (2.26) into (2.21) (by replacing
(cid:12) (cid:12)
σ1,z(cid:12)r=R1+e1 =σ1,z, σ2,z(cid:12)r=R2 =σ2,z) we obtain new equations
∂P ∂V ∂σ ∂σ
m + +m 2,z m 1,z =0, (2.28)
21 ∂t ∂z 24 ∂t − 23 ∂t
and
∂u˙ ∂P ∂σ ∂u˙ ∂P ∂σ
1,z =m +S 1,z, 2,z =m +S 2,z (2.29)
∂z 51 ∂t 1,33 ∂t ∂z 61 ∂t 2,33 ∂t
where
(cid:110) (cid:104) (R +e )2+R2(cid:105) (cid:104) R2+(R +e )2(cid:105)(cid:111)
2 R2 S 2 2 2 +(R +e )2 S 1 1 1
1 2 2,11 2(R +e /2)e 1 1 1,11 2(R +e /2)e
m := + 2 2 2 1 1 1
21 K (R2 (R +e )2)
2− 1 1
1 2 (cid:104) R3 (R +e )3(cid:105)
+ S 2 +S 1 1
≈ K R2 (R +e )2 2,11 e 1,11 e
2− 1 1 2 1
2(R +e )2
m := S 1 1
23 1,13R2 (R +e )2 (2.30)
2− 1 1
2R2
m := S 2
24 2,13R2 (R +e )2
2− 1 1
m := S H
51 1,13 1
m := S H
61 2,13 2
with
(cid:16) e (cid:17)(cid:104) (R +e )2 (cid:105) 2R2 (R +e )2 (cid:16)R +e (cid:17)
H := ln 1+ 1 1 1 1 1 1 1 1
1 − R 2(R +e /2)e e (2R +e ) − 2(R +e /2)e ≈− e
1 1 1 1 1 1 1 1 1 1 1
(2.31)
(cid:16) e (cid:17)(cid:104) (R +e )2 (cid:105) 2R2 R2 R
H := ln 1+ 2 2 2 2 + 2 2.
2 R 2(R +e /2)e e (2R +e ) 2(R +e /2)e ≈ e
2 2 2 2 2 2 2 2 2 2 2
The simplified expressions in Eqs. (2.30) and (2.31) are obtained under the assumption
(e /(R +e )) 1, e /R 1. Note that it is not possible for the terms in Eq. ((2.30) to
1 1 1 2 2
(cid:28) (cid:28)
become singular, since the denominator becomes zero only when the annulus of water has
zero thickness. Summarizing, the six-equations model with the six unknowns(P,V,σ ,u˙ )
i,z i,z
for the two-pipe system read
Fluid (axial motion - continuity equation)
∂V ∂P
ρ z + =0, (2.32)
w ∂t ∂z
∂P ∂V ∂σ ∂σ
m + +m 2,z m 1,z =0, (2.33)
21 ∂t ∂z 24 ∂t − 23 ∂t
8
Pipes (axial motion - axial strain equation)
∂u˙ ∂σ
ρ i,z i,z =0, (2.34)
i,t ∂t − ∂z
∂u˙ ∂P ∂σ
i,z S H S i,z =0. (2.35)
∂z − i,13 i ∂t − i,33 ∂t
We seek solutions of (2.32-2.35) in the form of wave functions,
P =P f(z ct), V =V f(z ct), σ =σ f(z ct), u˙ =u˙ f(z ct), (2.36)
0 0 i,z i,z0 i,z i,z0
− − − −
where P ,V ,σ and u˙ are magnitudes and c is the wave speed. Substitution of (2.36)
0 0 i,z0 i,z0
into (2.32-2.35) leads to six linear homogeneous equations which we write in a compact form
m c 0 0 0 0 P 0
11 0
−
cm21 1 cm23 cm24 0 0 V0 0
− −
0 0 −m33 0 −c 0 σ1,z0 = 0 . (2.37)
0 0 0 m44 0 c σ2,z0 0
− −
cm51 0 cm53 0 1 0 u˙1,z0 0
cm 0 0 cm 0 1 u˙ 0
61 64 2,z0
(cid:124) (cid:123)(cid:122) (cid:125)
M
Tokeepauniformnotationwehavedefinedm =ρ−1, m =ρ−1, m =ρ−1, m =S
11 w 33 1,t 44 2,t 53 1,33
and m =S . Existence of non-trivial solutions to (2.37) requires the determinant of M
64 2,33
to be zero, yielding, in turn, the following dispersion relation:
(cid:0)m m m m m m m m m (cid:1)c6+
24 53 61 23 51 64 21 53 64
− −
(m m m +m m m +m m m m m m +m m m )c4+
21 33 64 21 44 53 23 44 51 24 33 61 11 53 64
−
( m m m m m m m m m )c2+m m m =0. (2.38)
21 33 44 11 33 64 11 44 53 11 33 44
− − −
Naturalfrequenciesofthesystemc ,withk =1,...,6,aretherootsof(2.38). Ingeneral,Eq.
k
(2.38)hastobesolvedbymeansofnumericalmethods. However,ifρ =ρ ,S =S
1,t 2,t 1,13 2,13
and S = S (e.g., if the pipes are composed of a matrix and fiber with the same
1,33 2,33
volume fraction and elastic properties) we can find exact solutions of Eq. (2.38) analytically.
Indeed, if we define p := m /m = (ρ S )−1, q := m /m = (ρ m )−1 and δ :=
33 53 1,t 1,33 11 21 w 21
(m m m m )/(m m ), with δ 0 then Eq. (2.38) reads
24 61 23 51 53 21
− ≥
(c2 p)2(c2 q)+δc4(c2 p)=0, (2.39)
− − − −
with roots
c1 = √p,
(cid:118)
(cid:117) (cid:115)
(cid:117) (p+q) (p+q)2 4pq(1 δ)
c = (cid:116) + − ,
2 2(1 δ) 4(1 δ)2 − 4(1 δ)2 (2.40)
− − −
(cid:118)
(cid:117) (cid:115)
(cid:117) (p+q) (p+q)2 4pq(1 δ)
c = (cid:116) − :=c
3 2(1 δ) − 4(1 δ)2 − 4(1 δ)2 w
− − −
c = c , c = c , c = c = c . Here c ,c and c are positive (forward traveling)
4 1 5 2 6 3 w 1 2 w
− − − −
while c ,c and c are negative (backward traveling) wave speeds. Since c is smaller than
4 5 6 w
c and c , we refer to it as the speed of the primary wave. Accordingly, we call c and c the
1 2 1 2
speeds of the precursor waves related to pipe 1 and 2, respectively.
9
2.2.1 Reconstruction of the physical quantities
Wearenowabletorecoverthemechanicalstraininthehoopandaxialdirectionsasfunctions
ofP,V,σ ,u˙ . ThankstoEq. (2.19)-RIGHTwecanwritetheone-dimensionalhoopstrain
i,z i,z
as follows
1 (cid:90) Ri+ei
ε =ε = 2πrε dr =S σ +S σ ,
i,hoop i,ϕ 2π(R +e /2)e i,ϕ i,11 i,ϕ i,13 i,z
i i i Ri
where σ has been obtained in (2.26) (with P =P =0). In turn, we have
i,ϕ in out
ε =S H P f(z ct)+S σ f(z ct). (2.41)
i,hoop i,11 i 0 i,13 i,z0
− −
The cross-sectional averaged axial strain can be obtained by Eqs. (2.20) and (2.35)
1 (cid:90) Ri+ei ∂u u˙
ε =ε = 2πrε dr = i,z = i,z0f(z ct)=
i,ax i,z 2π(R +e /2)e i,z ∂z − c −
i i i Ri
S H P f(z ct)+S σ f(z ct). (2.42)
i,13 i 0 i,33 i,z0
− −
Notice that here u = ( u˙ /c)F(z ct) follows from integrating the last equation in
i,z i,z0
− −
(2.36)overtimewithF(cid:48) =f. Then,fromthesystemofequations(2.37)wecaneasilyderive
the following relations
cS H P cS H P
u˙ = i,13 i 0 , σ = cρ i,13 i 0 , (2.43)
i,z0 ( 1+c2S ρ ) i,z0 − i,t( 1+c2S ρ )
i,33 i,t i,33 i,t
− −
and, by taking c=c =c ,
3 w
P =c ρ V . (2.44)
0 w w 0
Thanks to Eq. (2.44) we can express averaged hoop and axial stress dependent on either the
fluid velocity V or, upon inversion of Eq. (2.44), on the fluid pressure P . By plugging Eqs.
0 0
(2.43)-RIGHT and (2.44) into Eq. (2.41) and (2.42) with c=c we obtain
w
(cid:110) (cid:104) c S H (cid:105)(cid:111)
ε = S H +S c ρ w i,13 i c ρ V f(z c t), (2.45)
i,hoop i,11 i i,13 w i,t w w 0 w
− ( 1+c2S ρ ) −
− w i,33 i,t
S H
ε = i,13 i c ρ V f(z c t), (2.46)
i,ax w w 0 w
−( 1+c2S ρ ) −
− w i,33 i,t
S H
σ = ρ c2 i,13 i c ρ V .
i,z0 − i,t w( 1+c2S ρ ) w w 0
− w i,33 i,t
3 Water-hammer experiments
Armed with the set of analytic expressions from Section 2, we turn now to the simulation
of experimental measurements for the water-hammer experiment for a set of pipes including
homogeneous (isotropic) metal pipes and fiber-reinforced (anisotropic) pipes.
3.1 Experimental setup
The propagation of waves in the annular space between two pipes is studied experimentally
using the apparatus shown in Fig. 3. For all experiments, the outer pipe is a thick-walled
10
