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LUNCLASSIFIED
~~ DEPARTMENTr OF CIVIL ENGINEERING
WC-
AN EXTENSION OFTH4E THEORY OF WATER, HAMMER
by
Richard
AFSWP No. 922Office of Naval Research
Contract No. Nonr-266(o8)Technical Report No. 15
CU-15-55.ONR-266(OS) -CE
March 195 5
' umba inivrt
in the (Ittg of Ntem Vork
DEPARTMENT OF CIVIL ENGINEERINGAND ENGINEERING MECHANICS
C0
AN EXTENSION OFTHE THEORY OF WATER HAMMER
by
Richard Skalak
AFSWP No. 922
Office of Naval Research
Contract No. Nonr-266(08)
Technical Report No. 15
CU- 15-55-ONR-266(08) -CE
March 1955
WY The American Society of Mechanical Etr29 WEST 39TH STREET, NEW YORK 18, NEW YORK
AN EXTENSION OF THE THEORY OF WATER HAMMER
R. SkalaxAsst. Prof. of Civil Engrg.
Dept. of Civil Engrg. & Engrg. Mechanics
Columbia University
School of Engineering
New York 27, N. Y.
Contributed by the Hydraulic Division for presentation at the ASME Diamond
Jubilee Spring Meeting, Baltimore, Md. - April 18-21, 1955. (Manuscript
received at ASME Headquarters May 19, 1954.)
Written discussion on this paper will be accepted up to May 20, 1955.
(Copies will be available until Feb. 3, 1956)
ADVANCE COPY: Released for general publication upon presentation.
Decision on publication of this paper in our ASME journal had not been takenwhen this pamphlet was prepared. Discussion is printed only if the paper ispublished in an ASME journal.
Printed in U.S A Price: 50 cents per copy
(25 cents to ASME members)
All 19,T" "SION OF T1E TIEOEY OF -'AT3h HAXIP2
Ly h. Skalak
NOl 7ENCLATURE
The following nomenclature is used in the paper:
c Velocity of sound in fluid
Ce Joukowsky water hari.er velocity (See eq. 1)
K Bulk modulus of fluid
a Radius of tube
h Thickness of tube wall
E Young's modulus of tube wall material
p Pressure in fluid
v xial velocity of fluid
z Distance axially along the tube
t Tim
u Axial displacement of tube
w Radial deflection of tube wall (+ outward)
Poo Initial density of fluid
r Radial distance
4Velocity potential
m Mass of tube per unit surface area
7 Poisson's ratio for tube wall
C0 Velocity of sound in tube wallCoW o ( -V-
q Force per unit area on tube wall
t, Xk Transform variables
n, s Integers
N See eq. (23)
IL See eq. (24)
J o,Jl Bessel functions
oIF
D See eq. (30)
F See eq. (36)
G See eq. (37)
B See eq. (38)
- 2 -
CI; Arbitrary constant velocities Is H Integrals. See eq. (53)
01102 Phase velocities L Length of wave front2
Z: See eq. (48) R - -0
Variable of integration c1zSee eq. (50 A =
1. Introduction
The usual theory of water hammer which is used to compute the magnitude and
velocity of pressure waves in a pipe filled with an elastic fluid was given by
Joukowsky [8] in 1898. This theory predicts that pressure waves travel without
change of shape at a velocity oe:
CCe= C 2 (1)
In the derivation of this formula it is assumed that the pressure is uniform
across any section of the pipe and that the deflection of the pipe is equal to the
static deflection due to the instantaneous pressure in the fluid. The first
assumption neglects, in effect, the inertial forces associated with radial motion
of the fluid. The second neglects the mass of the pipe wall and the longitudinal
and bending stresses in the pipe wall.
For the case of an infinite train of sinusoidal pressure waves in a tube filled
with fluid, much work has been done without resorting to the above assumptions. In
1878 Korteweg [9] showed that the phase velocity of sinusoidal waves varies with
the wave-length when the radial inertia of the fluid and the mass of the pipe wall
are taken into account. Korteweg also gave the expression for ce, eq. (1) as an
approximation of the phase velocity for long wave-lengths. The work of Korteweg
was extended in 1898 by Lamb [10] who considered the pipe as an elastic membrane
thus including the effect of longitudinal stresses in the pipe wall. Lamb showed
there are two finite phase velocities as the wave length approaches infinity.
One of these is very close to Korteweg's result ce and the other is very nearly the
velocity of sound in the material of the tube wall. Recent papers by Jacobi [6]
and Thomson [1143 include the flexural stiffness of the pipe wall. The inclusion
of the bending stresses affects the phase velocities appreciably only for very shortwave-lengths.
r ' the r- ,-r .
In the present paper water hammer waves are considered without some of the
simplifications made in Lhe Joukowsky theory. The equations of motion for a
thin cylindrical tube are used, cf. FlUgge (11],
with one additional term for the effect of rotatory inertia. The fluid is assumed
to be elastic and inviscid but no assumptions are made regarding its motion except
that it is small.
Only the case where the velocity of sound in the material of the pipe wall in
greater than in the fluid will be considered. Much of the analysis would be valid
in the opposite case, but the relative speeds of the various waves would differ.
2. A Formal Solution by Integral Transforms
obiainecL in the form of a. do~te i+egral.
In this section a formal solution of a typical water hammer problem isA
The solution is reduced to terms of single real integrals whose evaluation is
discussed in the next section.
The problem considered is to find the moticn of a thin cylindrical tube filled
with fluid starting from given initial conditions. The conditions, shown in Fig. 1,
are that for t - 0:
p-O, v a 0 for z > 0
P Pop0 v- vo for z e- 0 (2)
u 0 0, w a 0, for all z
The velocity v0 is assumed to be related to the pressure po0 by:
This is the velocity that would exist behind a plane shock wave of pressure p0 in
an infinite body of fluid. Thus the initial conditions for the fluid correspond toa step shock wave moving in the positive direction. An external pressure equal top0 is assumed acting on the outside of the tube for z < 0 to prevent any motion of
the tube at z - -oo , This external pressure is assumed to remain on the tube for
z < 0 for all t.
These initial conditions are intended to simulate the situation arising when
a pipe leading from a re-servoir is filled with fluid at rest below reservoir
pressure and a valve at the rpservoir end is suddenly opened. The part of the tube
left of the origin in Fig. 1 is a convenient device for supplying fluid and energyin place of a reservoir. It is expected that the particular means of generating awater hammer wave will not essentially affect its characteristics when the wave hasprogressed far enough down the pipe.
The motion of the fluid will be described by a velocity potential which must
satisfy the equation:
V 2 + "' (4)
where the subscripts r and z denote derivatives.
The motion of the tube due to a radial force per unit area, q (z, t), isgoverned by two equations:
___w~z y+ 1120o 12C - + 2!a UUz- -j-2 + 1 f z z U(5 )
LL U w + 1- w o (6)ae+ailed.
A derivation of these equations is given in [12], They agree with the equationsgiven by Flhgge [11] except thaot the rotatory inertia term Wzz has been added above.
The fact that the fluid and tube are expected to remain in contact is e np'esedby:
w €] (7)I =r J.o.
The pressure exerted on the tube wall by the fluid in:
r [ o)a (8)
This is the value of q to be used for z ;, 0 in eq. (5). For z < O, the external
pressure - p0 must be added.
It is convenient to consider the solution of the problem as the sum of two parts:
The first part is the shock wave that would be produced by the initial
conditions if the tube were rigid. The tube displacements w1 , ul, for this part
are zero. The velocity and pressure vl, pl, for the fluid are:
V,--Vo' I for Z > = 0Vp I, o~ 2 >C V% (9)
PI POj P I = 0
The second part is the response of the tube and fluid to a progressive load
pressure ps equal to the pressure produced by the shock wave of the first part. This
load pressure acts on the tube wall radially outward. Its magnitude is Po for
0 < z < ct and zero elsewhere.
Inasmuch as wl, ul, vl, etc. are known, the problem reduces to finding the
the response w2. u2 , V2, etc. of the tube and fluid to the load pressure ps" For
this second part of the solution the system is initially at rest and the governing
equations are (4), (5), (6), (7) with:
'2 = [a] _ 2Ps (10)
The solution is begun by transforming the governing equations and eliminatingthe velocity potential 42" The remaining transformed equations are solved simultane-
ously and the inversions of the transformations then yield the solution.
With respect to z a Fourier transform, cf. (15] p. 42p is used. Defining thetransform of a function g (z, t) with respect to z by:
CO
-00then the inversion is:
00S(z't t) ) L I (12)
-00
With respect to time a Laplace transform is usedp but the tranform variable
adopted is-t- times the usual transform variable. This results in a certain
synetry with the rirst transform and convenience in later analysis. Defining the
transform of a function (, t) with respect to t by:
00
then the inversion is: 0
-oo- L o.VW = =a~.tCX (114)
-CO- C f
The double transform of derivatives of g (s, t) may be expressed in terms of the
double transform of g (z, t) by the usual partial integration procedure, (13] p. 27.
Using the fact that initial velocities and displacements are zero, the transform of
the derivatives involved are:
Using q2 as given by eq, (10), the doubw transformation of eqs. (o1), (5), (6)and (7) yields: =
rr r$2r 2C (16)
s !2 2 'W 2 +I 2 SI W (4- 2. + + L 2.
+ P5 (17)
- + t Lk- - W2 W2 = 3h (18)0
r] &
The double transform are functions of k and SL only except for
2 hc s 2 I ) Freemeq. (16) it isfounad that the rdepenenc of 2involves a Bessel function of the first kind. The function pa in known. Solving
eqs. (16), (17), (18), (19 for ,2 , 121 2 and inrtig yields:
- 0 -
00 L c C 2 mo
cO oo1 2. / - h. Lfl.tZ + r1.. a (21)
___ LJ0(- -- )eL t-oo O- (22
+~ +* c'o h2']
a 0 aLa J~iCt i e2 (214)
Sa (a ,- I ,
P, 0- (22)
The evaluation of the radial deflection wifl be considered first. It istypical of the quantities of interest. The inner or fl integral of eq. (20) may beevaluated by Cauohy,'s residue theorem applied in the complex £i plane. Durin thisprocess is regarded as a constant real parameter since the outer integral involvesonly real values of
The factor in the denominator of the integrand in eq. (20) produces a simple
because (c + ) A is finite at this point. There are poles at the points for which
the remaining bracket is zero, i.e., where:
1 (25)
L2~ Nj
For any real this equation has an infinite number of roots rL n* These roots
have an interesting physical significance. They are the circular frequencies of the
modes of free vibration of the tube and fluid system having a wave-length X -
Alternatively the ratios - are the phase velocities cn of progressive sinusoidal
waves of wave-length 2f. These facts may be developed by considering the governing
equations (4), (5), (6) and (7) with q in eq. (5) equal to the fluid pressure only.
A solution is assumed of the form:
w - w e 1 + (26)
where the prime quantities are constants and f(r) is an unknown function of r.
Substituting eqs. (26) in eqs. (4), (5), (6) and (7) the constants and f(r) may be
eliminated, leading to exactly the transcendentaI eq. (25) relating I and fl .
For real values of f all the roots L n of eq. (25) are also real. If any of
the JL n were complex or imaginary the corresponding free vibration would grow or decay
indefinitely in time. Since the system has no means of dissipating or radiating energy,
this is not possible physically.
Fig. 2 shows the S vs t values for a typical case in which the velocity of sound
in the tube wall is greater than in the fluid, i.e., c0 > t, It is found that for any
value of there is one root 1 , for which "A - f ) is negative. This root corre-
sponds to a phase velocity less than c, the velocity of sound in the fluid. There are
also an infinite number of roots for which (c 2 " 2) is positive. These correspond
to phase velocities greater than c. Fig. 3 shows the same data as in Fig. 2 plotted as
phase velocity vs wave-length.
The fact that two sets ofStn curves pass through the origin in Fig. 2 shows that
there are two modes which have finite phase velocities as the wave length approaches
infinity and the frequency approaches zero. This was also shown by Lamb (10), p. 7.
It is mentioned here because in a recent paper [i1), p. 933, it is erroneously statshd
that there is a cutoff frequency below which only one mode may be propagated.
The above discussion shows that the poles of the integrand of eq. (20) are at
SL - 0 and 11 - /l n, all on the r.al SL axis. The path of integration is below
this axis and it will be closed in the usual manner by adding a semicircle of
infinite radius in the upper half of the SI plane. The integral over the semicircle
approaches zero as the radius approaches infinity. The contribution of the pole at
- 0 to w2 is:(O
co
a ---2- -, L (27)I [ K Y2+i Zy~ 11 -- t C
The contributions of the nY poles to w. may be expressed as:
W . O eL 21RL~ z e _ (28)
The sum indicated is over the -L n roots for each value of * The values of the
SL as functions of t are given by Fig, 2. Eq. (28) may be written in the form
L~~j + 1
29 r s al f D (29)
where P~ 0 2I Cl~ 2(~ 2 ~ J2 +(0
+C~_____ +(- + +
The integral in eq. (29) covers all four quadrants of the siC. curves in Fig, 2,9but it may be expressed as an integral over the first quadrant values only because
D is even in a and even in and the -Ln curves are symmetric about both ams.
Thus:
Spn ( i -CLt) S t( + t)}~ L
.L C
where the summation is over the .L. n curves in only the first quadrant of Fig. 2. This
concludes the frzral solution for w. The evaluation of the integrals remaining in
60q4 (27)and (31) will be considered in the next section.
instead of evaluatLi, u2 and 2' certain quantities of interest which are
derivatives of u2 and 2 will be computed directly. These are:
Pressure in the fluid:602 (32)
Axial velocity of the fluid: b t - =(
Longitudinal strain in the tube; :b L
Expressions for P20 V2 and uz2 are derived from the double integrals for u2 and 4 2'
eqs. (21) and (22), through application of eq. (15). The inner integral in S may
be evaluated in each case by using Cauchy's residue theorem as in the case of w2.
The results are:
P2 = P O O C Ie(' [0 n n +SI) (33)
-00 0
(O S in ( 1-At) sin(+: (34)S2I -L . + r I L.( +)
where:=
(37
+(38)
and f(l ) stands for the denoi~naor of the interad in eq. (27).
- Ii -
The solution of the problem originally posed at the beginning of this section
is given formally by:
w UW2a " '2b Eqs. (27) and (31)
P Pl * P2 Eqs. (9) and (33) (39)
V l * v2 Eqs. (9) and (34)
uz a uz2 Eq. (35)
3. Evaluation of the Solution for Large Values of Iz and t.
In the formal solution indicated by eqs. (39) the integrations with respect towould be difficult to carry out in general because the IL n values are known asfunctions of t only through the transcendental equation (25). However, some informa-tion may be obtained by considering approximations of the integrals for large values
of I Ind t.
Consider the part of w given by eq. (31) and, in particular, values of z and t
such that:
z -c' t (40)
where c' is an arbitrarily selected velocity. Substituting in eq. (31):
00
1 -andIt is found by inspection of D, that the functions ( - D D
c) care bounded for all of the P_ n curves for all [ except for the lowest two curves, A,and ft 20 at the point - 0. The distinction between the lowest two and the
remaining sL n curves arises because only the IL 1 and -a 2 curves pass through the
origin. A physical interpretation of this fact is that only these two lowest modes
have finite phase velocities as the wave-length increases indefinitely.
For the higher - n branches, n > 2, the integral (4i) may be approximated by
the method of stationary phase (7J p. 505, for large values of t. This method shows
- 12 -
these integrals to be of the order of
The integrals over the SL 1 and SL 2 curves will be broken into two parts; one
part from I =O to f = 6 and a second part from E to co where f- is aSMaI
positive number. By the stationary phase theorem the portions from f to oo are again
0 ) Collecting these results, eq. (41) becomes:
Po ii_(_c_-) sn( C'j+n)t1 dt +O(L) (42)
JAL (-9.-fC Jo
Since E may be chosen indefinitely small, the functions in the integrand may
be approximated by an appropriate number of terms of their expansions about 0 0.
Consider the first term of the integral with A' - XL which may be written:6
The ~i deomnao W1 s~(-fL ) I A (143)
The denominator of the integrand approaches a constant time approaceszero. The constant is:
where: c- -= Ly
=Ur Dos, 4PO C t + 2.C [ir C 1 h2A 2e 1 22j
The constant c is the phase velocity of the slowst waves for wavelengthsapproaching infinity. It is very close to the Joukowsk velocity e . An ejxmssion
for c1 is derived in the appendix.
In the numerator of eq. (43) S11 is replaced by the first two terms of its
expansion about O 0:
,)I= (46)
This expansion is developed in the appendix. It is necessary to use two terms
of the expansion because the first term alone would result in a vanishing numerator
for the case c' n c 1 .
Using eqs. (44) and (46) in (43) yields%pO0w = PO - a ' + l 3 at (47)
0
where z' is defined by:
zI= (C' - cI) t (48)
The new coordinate 21 is the location of any point relative to an origin moving atvelocity c1 . In eq. (47) the upper limit has been changed from E to cv , The
integral thus added is 0(-) by the stationary phase theorem, and is negligible
in the present approximation.
For z' - O, the integral in eq. (47) has a constant value of provided d1 ispositive which is the case here. For points where 1 1 is Lare, the termcontaining 3 may be neglected. Then the integral in eq. (47) becomes ! i for ! s'*For points near z' - 0 it is convenient to change the variable of integration from
to n IS '. Then:
W:= PC +I (49)
0where: a,'t (EQ)
From eq. (49) it may be seen that for w1 constant must be constant, say o0
Then:
This shows that points for which wI is constant continually move away from the pointz' 0 0. To suvnarize, wT is a wave having constant deflection at the point 2' W 0
i.e., moving with velocity c I . At points far ahead and far behind this wave frontpoint the deflection is also constant, but the wave front continually spreads out
- i/I -
around the point a' 0. Fig. 4 shows the general shape of the wave,
If the same reasoning that has been used to evaluate wI is applied to the
remaining terms of eq. (42) similar results are obtained. Thus for large t:
2 b I-H, * 12 - H (52)
where: 00
IrM(n- -I Dn(53)
The constants e2, D2 , d2 wh.ich appear in H2 and 12 are defined in the same
way as Cl, D1, dl in eqs. (45) and (46) by changing all subscripts from 1 to 2 in
these equations. The velocity c2 is very nearly the velocity of sound in the tube
wall, co.
The term T is the deflection w, discussed above. The H1 term is a wave of
the same velocity and shape as wT but of smaller Alitude and traveling in the
negative direction.
The H2 and 12 terms are also waves of the same shape as wTO but traveling at
the velocity c2 . These precursor waves are found to be very smll compared to the
main water haumer waves 13 HI which travel at velccity Cl.
The complete deflection w is the sum of wbO, eq. (52) and w2., eq. (27)9 Thedeflection w2a is independent of time and its value for large values of Izi =a beapproximated by the saw kind of reasoning as used for w~b for large values of t,
The result is:
h2.20 2 E [a + Pa 4~L l for Zi (54.)
me term 2h will be neglected compared to unity since on4 thin-12a (I - )
walled tubes are considered, Thus:
2
\A + o for (55)C2o- 2 Eh
The procedures used to evaluate w may also be used to obtain approximations of
p, v and u for large I j and t starting with the general expressions, eq#. (39),The results are: w + p O - H +I H or ± :
- 2Eh _
op-Zec~ ((I+ -N2_)v- r _1, ,
LL C1
2. -(I, ,))
These equations lead to four wave fronts in each case as discussed for v.
Numerical results for a typical case of water in a thin steel tube are shown inFig. 5. In the present approximation for large t, the pressure and fluid velocity
are again independent of r, i.e., they are constant across any section.
In order to estimate the length over which a given wave front extends, theslope wz will be computed at the points which move with constant velocity CI or 02.The difference in w before and after a wave front divided by the slope at the wave
front point will be indicative of the length of a wave front.
The double integral for wz is derived from w, eq. (20). The first integral may
be evaluated and the result folded by the aw proced.. es used for v. For a pointmoving so that C - cI t:
(57
o
PU Vc~ ac ' +L +I-, z-H o
2 E iC
- 16 -
The function .( in bounded for all and the first integral in eq. (57)
is 0 (1) by the Iiemann-Lebeague lemmas [16] p. 172.
In the seconi integral the integrand is also bounded, This integral may be
approximated by the method of stationary phase. The dominant part of the integral
will come from the vicinity of stationary phase points which are defined by:
C±n)= 0 L~e. C,(58)
In general such points give a contribution which is 0 -- ), but if:
Sc,-±n = o
then the contribution to the integral from the vicinity of this point is 0
[5] p. 40. Eq. (58) and (59) are both satisfied on the IL 1 curve atv 0 "O t
Hence this contribution will predominate over all others for large values of to
To evaluate the dominant term, the expression for -a 1 eq. (k6), is substituited In
eq. (57). Hence for large t: OD
Wo PO c0s (S t)
w = Po -L 4. (60)
0Changing the variable of integration to t)
00
0
This integral (61) is known, (5] p. 41. Hence:
W 3 (-") (62)
Now the change of w from a location far behind the point z - c t to a locationof + he point
far aheadA Z CI t is by eq. (49):
- 17 -
&w - (63)
Defining the "nominal length" h of the wave front to be aw divided by wz:
'' 3 J-
Computation of the "nominal length", L2 , of the wave fronts moving at velocity o2
results in the same formula with dl replaced by d2.
Some numerical values of L, and L2 are given in Table 1. It may be noted that
although the length of the wave front is long compared to the diameter of the pipe
it is short compared to the distance from the origin that the wave has traveled.
The above computations of the lengths of wave fronts hold also for p. v and
u because they are described by the same integralsk 1 1, Hls 120 H 2 .
4. An Extension of Joukowsky's Method
Some of the results derived by use of integral transforms m also be derived
by a much simpler procedure which is an extension of Joukowsky's method. The
extension consists of taking into account the longitudinal stresses and longitudinal
inertia of the pipe wall. This makes possible the precursor type wav which is
essentially a tension wave in the pipe wall.
The following derivation follows the general plan of Joukowsky'a paper (8].
The pressure and axial velocity of the fluid are assumed to be uniform over amy
cross-section of the pipe. The fluid is assumed compressiblej its bulk modulus
being K. The equation of continuity my then be written in the form t
V I _ P 2 bw (65)- I K T 6t a 6
- 18 -
The equations of motion of the fluid reduce to:
-v bp (66)
So far, these are the equations usually used. The new element is the use of the
equations of motion of the tube considered as an elastic membrame instead of a ring
of no mass. These equations are derived from eqs. (5) and (6) by dropping all terms
arising from bending stiffness and rotatory inertia and substituting p for q:
- - - - (67)a*2 MC 2.
"U w 0 (68)
In eq. (67) the term -i has been omitted because the effect of radial inertia of the
tube and of the fluid is neglected as in the Joukovsky theory.
No attempt will be made to solve the equations (65), (66), (67) and (68) ingeneral. It is sufficient for the purpose at hand to show that the system pernits
solutions which are waves of arbitrary shape moving at either of the velocities aI or
c2 without dispersion. A solution is assumed of the form:
w W, wf f(Z + Zt)
p = p f (f Z t) (69)
v1 = vL,' f (I + -Et)
Ut =LIti(+-Z)where the primed quantities are constants and ; is an unknown constant velocity. The
constants u.L, u' and 3 are related by the requirement:
& t 1 11 (70)
Substituting the assumed form (69) into eqs. (65), (66), (67), (68) and (70), there
result five linear homogeneous equations involving the five primed constants. In order
that these equations have a solution different from zero it is necessary for the
determinant of the coefficients to be zero. This condition yields an equation
- !12 -
involving ; which can be satisfied only if takes on the values - c1 or c2. Thus
the velocities at which waves may travel without dispersion in the present considera-
tion are the same as the pr:'ncipal velocities derived in the integral transform
method,
A further point of interest is that the ratio of the change in w to the change
in p across a wave front is the same in the extended Joukowsky theory and the
transform solution. The same is true of any two of the quantities w, p, v, Uz, etc.
This is shown by solving for the ratios of interest in each theory with c or c2
as the case may be.
These results are of interest because they indicate the significance of the
assumptions made in the uoukowsky theory. It shows that c differs from c becauseIonJittitdl 1 .Odf the. pipe. e
the Joukowsky theory neglects longitudinal stresses and inertia. It also showsA A
that the dispersion predicted by the transform method is due to the effects of the
bending stiffness and of the radial inertia of the pipe and fluid. The effect of
radial inertia is much more important. After the wave disperses somewhat, bending
effects are negligible.
5. Other Initial Conditions
The integral transform method may be applied to certain other initial conditions
beside those assumed in Sections 2 and 3. The problem and results shown in Fig. 6
illustrate a second example. The initial conditions are:
v v for z <0 v=-v for s >00 0
p u - w = ut - wt -'0 for all z.
Under these conditions each half of the pipe simulates the water hamer problem in
which an established flow is suddenly cut off by an instantaneous valve closure.
Since by symmetry no fluid crosses the plane z - 0, each half of the pipe acts as
a closed valve as far as the other half is concerned.
The magnitudes and velocities of the waves may also be established on the basis
of the simolified procedure of Section 4. In general, this will be the simpler
method. It does not predict dispersion, but it may be assumed that the dispersion
characteristics are the same in any case.
6. Conclusions
The theory developed herein shows the following results which are not contained
in the usual theory:
1. If the initial conditions specify a sharp wave front of pressure, this "ront
will gradually disperse over a finite length as it proceeds. Thisdispersion continues Indefinitely so that no steady wave shape is ever
reached.
2. Two waves will be generated in general; one which is similar to the
Joukowsky result and one which is essentially a tension wave in the pipe
wall. The pressure change due to this precursor is very small.
0~.3. AfterAsufficiently long time has elapsed since the generation of a wave
there will be one point in each wave front which has a constant deflection
and fluid pressure and moves at a constant velocity. This velocity is
very close to the Joukowsky velocity for the main pressure wave. For the
precursor this velocity is slightly less than the ve.ocity of sound in
the pipe wall.
4. The bending stresses in the pipe wall have very little effect on the
characteristic velocities and the rates of dispersion after a sufficient
time has elapsed. Such stresses can only be significant in the early
stages of the motion in regions where the wave front is sharp so that the
variation of pressure and deflection is very rapid along the length of
the pipe.
5. The relations between fluid velocity, pressure and hoop stress given by theJoukowsky theory are substantially correct, but there are also longitudinal
stresses in the pipe not predicted by the usual theory.
6. Available experimental datais in very close agreement with the predictions
of the Joukowsky theory as to the pressure rise and velocity of the main
pressure wave. Inasmuch as the analysis presented in this paper gives
essentially the same results as the Joukowsky theory for these items, it isconfirmed experimentally. However, there is no experimental data available
on the precursor wave or the rate of dispersion predicted herein.
ACKNOWLEDGEMENTS
The author is grateful to Professor H. H. Bleich for his
advice and encouragement throughout the preparation of this
paper and the thesis on which it is based. This research was
partially supported by the Office of Naval Research under
Contract Nonr 266(08).
L 115L I 0 GRA PT
(1) Allievi, L., Allgemenie theorie uber die veranderliche Bewegung desWassers in Leitungen, translated from Italian into German by R. Dubsand V. Bataillard, Julius Springer, Berlin, 1909.
(2) American Society of Mechanical Engineers, Symposium on Water Hamer, 1933.
(3) Biot, M. A., "Propagation of Elastic Waves in a Cylindrical Bore ContainingFluid," Journal of Applied Physics, Vol. 23, 1952, pp. 997-1055.
(4) Fay R. D., "Waves in liquid-filled Cylinders," Journal of the AcousticalSociety of America, Vol. 24, No. 5, Sept. 1952, pp. 459-462.
(5) Havelock, T. H., The Propagation of Disturbances in Dispersive Media,Cambridge University Press, 1914.
(6) Jacobi, W. J., "Propagation of Sound Waves along Liquid Cylinders,"Journal of the Acoustical Society of America, Vol. 21, No. 2, 1949,pp. 120-127.
(7) Jeffreys, H. and Jeffreys, B. S., Methods of Mathematical Physics,Cambridge University Press, 2nd Ed., 1950.
(8) Joukowsky, N., "Uber den ydraulischen Stoss in Wasserleitungsrohren,"Yimoires de l'Academie Imperiale des Sciences de St. Petersbourg, 8 serie,
Vol. 9, fNo. 5, 1900.
(9) Korteweg, D. J., "Ueber die Fortpflanzungsgeschwindigkeit des Schallesin elastisches Rohren," Anrlalen der Physik und Chemie, 9 Floge, Band 5,1878, pp. -1h2.
- 22 -
(10) Lamb, H., "On the Velocity of Sound in a Tube, as Affected by the Elasticityof the Walls "'Manchester Literary and Philosophical Society, Memoirs andProc., Vol. 42, No. 9, 1898.
(11) Flhgge, W., Statik und Dynamik der Schalen, Julius Springer, Berlin, 1934,Reprinted by Edwards Brothers, Ann Arbor, Mich., 1943.
(12) Skalak, R., An Extension of the Theory of Water Hammer, Doctoral thesis,Columbia University, 1954.
(13) Sneddon, I. N., Fourier Transforms, McGraw-Hill, 1951.
(14) Thomson, W. T., "Transmission of Pressure Waves in Liquid File 4 -.Proc. of the First U. S. National Congress of Applied Mechanics, ASME,1951, pp. 927-933.
(15) Titchmarsh, E. C., Introduction to the Theory of Fourier Integrals,Oxford University Press, 1937.
(16) Whittaker, E. T. and Watson, G. N., A Course in Modern Analysis,Cambridge University Press, 4th Ed., 1952.
APPenDIX
Expansion of -a 1 and n 2 About " 0
The values -f 1' .I 2 are defined as the lowest two roots of eq. (25). From
the numerical results represented by Fig. 2, it is expected that both - 1 and 912approach zero as approaches zero. Hence, Al 1 and -n- 2 havean expansion of
the form:
= .(71)
where only odd powers occur because the -- are odd functions of ( . The values
of ; and d are determined by substituting this expansion in eq. (25) and equating
the coefficient of each power of I appearing to zero.
1
The terms - and L in the transcendental equation (25) are given in full byN
eqs. (23) and (24). Expanding the Bessel functions, t
may be written in the form;
- \[)- 5, C 4 +-) J (72)++ ....]-
8~~ 9
In order to be able to clear eq. (25) of fractions, the quotient of the two
infinite series in eq. (72) is written as a single series:
S, 0& a(7.3)
Using this value the transcendental eq. (25) may be put in the form:
M 0, 0- I C*'L
+
-6- - ,--o
Now the assumed expansion given by eq. (71) is substituted above for -i. • The
result is a series of the form:
it+±K '+ b3 f + = (75)
Each coefficient b of this series must be zero since the value 1I is a rootn hof the transcendental eq. (25). Equating b I to zero and neglecting - compared
to unity yields an equation involving ; but not d. This is interpreted as an
equation on the yet unknown ;. The roots are:
C = CT 2AR , + R + Rt (\ -- V') A AR + , + ( - -/ 1)(2A+ "1 (76)L2 (2 Ai-?s)
where _ (07)IC7 MoO (77)
Equating b2 to zero yields an equation involving both ; and d. This equation
is considered to determine two values of d, one for c - cI and one for c - 2.
The solution of the equation b2 0 for d yields:
C L T- (78J
_ 4V
-v I
t r74
-777---
-4 a
- 25-
52The ro dire an IMfInafmber of/
addit!"l| branches in i his &red.
4
3
2/
I =
o 2 4- G 8 10 12 14-FIG. 3 C. --
__ - I..
/\ - -,oo---.,, v,,,,Jou ,,1
Qorr
----------- .6
_____LI -20 -16 -12 -8 -0 o 4.
Fig. 4. Form of Tjplcol Wave Front
P-0 -- -. P.O
V v-v V. 949 jV. 9V t i 0
Iiiiial ouid1ow.hio~ p~.~v.~'*",Iitial Corikl~ions
No+* P. V. __. P.
5 9 mabt Cat
C t No*.e.- siCC
6404 *.01jp io a l
I~~ I --Tn-o 1
f- la-z -
S40.+ t c3 fppNotto CAa.
Va .(so ,49 p£n foi_ .07 P . I tuh fil. .ith F5q P. .00wes ofwP.pm o selhb i wt
waij m tr V t~.. C , fs~ i t~ ce o l w v~ r ± ~ oeo ntain o lw
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