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Analytical solutions for unsteady pipe flow
Rodney J. Sobey
Rodney J. Sobey
Department of Civil and Environmental Engineering
Imperial College London,
London SW7 2AZ,
UK
E-mail: r.j.sobey@imperial.ac.uk
ABSTRACT
A sequence of analytical solutions explore the spectrum of response patterns expected for unsteady
elastic-compressible flow in pipes. Complete analytical details of the solutions are provided, together
with specific suggestions for an associated set of analytical benchmark tests. Illustrations of
predicted response patterns provide the basis for a discussion of many significant physical aspects
and their representation in discrete numerical codes. An evaluation of the incompressible flow
approximation completes the discussion.
Key words | analytical solution, benchmark problems, pipe flow, unsteady flow, water hammer,
wave equation
INTRODUCTION
Numerical modeling is the tool of choice in studies of
unsteady flow in pipes. Yet there remains a useful role for
analytical solutions to schematic problems. Analytical
solutions have value in both classroom instruction on
unsteady pipe flow and in the confirmation of numerical
codes.In classroom instruction, unsteady pipe flow is often
an engineering student’s first significant exposure to
unsteady flow. The mathematical sophistication and
physical complexity introduce a leap in conceptual chal-
lenges. Analytical solutions can provide a rapid and con-
venient introduction to the spectrum of response patterns.
In numerical model evaluation, analytical solutions
can provide a rapid measure of physical and code credi-
bility that approaches the value of extensive field or
laboratory experiments. In rational model evaluation,
experimental measurements and analytical solutions have
a genuinely complementary role. Measurements have
certain reality, but analytical solutions can provide rapid
and detailed response patterns across the complete space
and time spectrum.
A sequence of well-defined analytical benchmark
problems is proposed. These benchmark problems are
analytical in the sense that each problem has an exact
analytical solution. Analytical solutions alone have
absolute credibility. A numerical code must be modified to
exactly match the context of an analytical solution. But
then the numerical and analytical solutions should match
exactly. Any differences can be attributed to the code.
The attention to benchmark problems directly
addresses numerical model evaluation, but each of thefour problems has intrinsic value in classroom instruction.
In fact, three of the solutions were initially established as
instructional illustrations.
This paper will introduce a sequence of analytical
benchmark problems that are appropriate for numerical
codes for unsteady pipe flow. It will begin by adapting a
general analytical solution (Sobey 2002a) for unsteady
channel flow. It will then define a sequence of application
problems that explore both the underlying physical pro-
cess and the interaction with the operational context of a
numerical model. Numerical models of unsteady pipe flow
are boundary driven, and detailed attention is given to
response patterns associated with a wide range of com-
mon boundary conditions. For each problem, the com-
plete analytical solution is given for both head H( x,t) and
flow V ( x,t), in a manner immediately suitable for coding.
The problems include the response to sudden valve
closure, the impact of valve closure over a finite time,
start-up transients and the evolution to steady state and
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finally the response to periodic forcing. This final
problem provides an opportunity for a comparative evalu-
ation of the rigid water column or incompressible flowapproximation.
FIELD EQUATIONS FOR UNSTEADY PIPE FLOW
The immediate response to rapid changes in pipe flow is
taken up by the elastic compressibility of both the fluid
and the pipe walls. Unsteady and compressible flow fol-
lows the cross-section-integrated mass and momentum
conservation equations (Li 1983):
∂H
∂tV
∂H
∂ xa2
g
∂V
∂ x0
∂V
∂tV
∂V
∂ x g
∂H
∂ x t0 P
r A(1)
in which x is local position, t is local time, H( x,t) is the
local elevation of the hydraulic grade line to a fixed
horizontal datum, V ( x,t) is the local cross-section-
averaged flow velocity, a = [(∂( A)/∂ p)/ A] − 1/2 is the
speed of an elastic wave in the composite fluid–
pipe system, g is the gravitational acceleration, A( x,t) is
the local pipe cross-section area and P( x,t) is the
local pipe perimeter. The boundary shear t0( x,t) is esti-
mated from a friction model, typically Darcy–Weisbach
or Manning.
These conservation equations are readily established
by imposing unsteady mass and momentum conservation
to a finite control volume of length x along the pipe and
taking the calculus limit as x goes to zero. Adopting an
equation of state for the fluid–pipe composite in the form
1 r A
D
Dt r A 1
ra2D r
Dt(2)
and writing pressure as p = g(H − z) leads directly to
Equations (1).
A conceptually useful approximation to Equations (1)
is the linearized pipe flow equations:
∂H
∂ta2
g
∂V
∂ x0
∂V
∂t g
∂H
∂ x lV (3)
in which the small advective acceleration terms are
neglected, and the Darcy–Weisbach or Manning approxi-
mation for the boundary resistance is replaced by a linear
approximation lV , in which l is a constant friction factor.
These equations continue to represent the major processes
influencing unsteady flow in pipes, though in a less
complete manner.
Except for the advective accelerations, these linear-
ized equations retain all the complicated hyperbolic phys-
ics of unsteady compressible pipe flow. In addition, the
linearization does not invalidate a numerical algorithmchoice that was based on the complete Equations (1).
A numerical solution to Equations (3) imposes almost
identical challenges.
While linear friction is certainly a compromise, it must
be recalled that quadratic friction also is not entirely
satisfactory. The utility of the linear approximation will be
enhanced by realistic estimates of l. Equating the linear
and Darcy–Weisbach estimates gives
l
f zV z
2D (4)
in which f is the Darcy–Weisbach friction factor. In the
linearized Equation (3b), l must be constant. A suitable
predictive equation for l would be
l f
2DKV L (5)
where KV L is a suitable averaged velocity scale over the
flow.
Eliminating V by cross-differentiation amongEquations (3) gives the generalized wave equation
∂2H
∂t2 a2∂2H
∂ x2 l∂H
∂t(6)
in which H( x,t) is the dependent variable. When the
coefficients a and l are constant, this PDE is linear and
useful analytical solutions are possible.
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Similarly eliminating H by cross-differentiation gives
the same generalized wave equation in V :
∂2V
∂t2a2 ∂2V
∂ x2 l∂V
∂t. (7)
The details are very similar.
GENERAL ANALYTICAL SOLUTION
Quite general analytical solutions to a non-homogeneous
variation on Equation (6) have been established by
Sobey (2002a). Adapting that solution to the context of
unsteady pipe flow, the general analytical solution for
H( x,t) is
H ( x,t)mxba1exp(2 m x)cos(kxvt 1)
a2exp( m x)cos(kxvt 2)
3 f |0 g0
l1exps2 lt+
1
le0
t
12exps2 lt tj0s tdd t4X
0 x
∑n1
3 f |
nexp
s lt/2
dcosv
nt
gn l f |n/2
vn
exps2 lt/2dsinvnt
e0
t1
vn
exps2 lt t/2dsinvnt tjn td t4 Xn( x) (8)
This general solution includes the zeroth-order free modes
(see the appendix), which are non-zero only for gradient
or Neumann boundary conditions at both ends (Type 4 in
Sobey (2002a)). The dispersion relationship, relating spaceand time periodicities, has two forms:
kv
aF1√ 1 l v2
2G
1
2
, vn√ bn2a2 l 22 (9)
for the forced and free modes, respectively. These forms are
special cases of the same generalized dispersion relation-
ship. Sobey (2002a) gives the complete details, including the
definition of the solution parameters, the water surface slope
m and the datum level b for the steady-state flow, the forced
modes amplitudes a1, a2 and phases 1, 2, the free mode
eigenfunctionsXn( x) and the definition of the functions jn(t)from the transient boundary conditions, and parameters f ˆ
n
and gn from the initial conditions.
Terms 1–4 are contributed by periodic boundary con-
ditions. Terms 1 and 2 describe the steady pipe flow.
Terms 3 and 4 are the forced modes. Modes in the bound-
ary conditions will appear in the response. These forced
modes decay as they evolve in space at the rate m = lv/
(2a2k).
The balance of the solution are the free mode
responses at the eigenmodes, excited by both the initial
conditions and non-periodic transient boundary forcing.Within the summation, terms 8 and 9 are contributed by
the initial conditions and term 10 by non-periodic bound-
ary conditions, such as valve closure. These free modes
decay as they evolve in time at the rate l/2. The zeroth-
mode contributions, terms 5–7, may appear only where
there are gradient or Neumann boundary conditions at
both ends. Terms 5 and 6 are contributed by the initial
conditions and term 7 by non-periodic boundary con-
ditions. Except for f ˆ0, the free modes decay at the rate l.
The dispersion relationship, Equation (9), relates
space and time periodicities in both the free and forcedmodes. Without friction, it has the classical wave form
v = ak.
The general analytical solution for V ( x,t) of Equations
(7) is
V x,tba1exps2 m xcosskxvt 1a2exps m xcosskx
vt 2
3 f |0 g0
l1exps2 lt
1
le0
t
12exps2 lt tj0s tdd t4 X0 x∑
n1
3 f ˆnexps lt/2dcosvnt
gn l f |n/2
vn
exps2 lt/2dsinvnt
e0
t1
vn
exps2 lt t/2dsinvnt tjn tdd t4 X
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in which b, a1, 1, a2, 2, f ˆn, gn, jn and Xn( x) are re-defined
in relation to V . But they must be carefully coordinated to
be exactly consistent with the conditions imposed for thev( x,t) solution. The dispersion relationship remains
unchanged, Equation (9). Again, complete details are
given in Sobey (2002a), supplemented by the appendix.
Equations (3) describe an initial, boundary value
problem. The solutions for H and V change with both the
initial conditions and the boundary conditions.
CODE MODIFICATIONS
Analytical solutions can be used for numerical code evalu-
ation in either of two ways.
(i) Make no changes to the numerical code. An average
value for l would be adopted; a also needs to be
constant, but it is usually already in numerical
codes. The analytical and numerical solutions
should have trend agreement, but they will not be
identical. Such a comparison is valuable, but not
absolute.
(ii) Modify the numerical code to be a solution to thelinearized equations. The numerical code would be
modified to be a numerical solution of the linearized
equations. A comparison of analytical and numerical
solutions should then be absolute. The necessary
code modifications certainly provide the opportunity
for coding error, but access to an exact solution
should facilitate the identification of any such errors.
Both modes have value. The latter is absolute and must
be preferred. However, the code modifications must be
carefully undertaken. Two changes are necessary:
(i) Omit the advective acceleration terms V ¤H/∂ x in the
mass equation and V ¤V /∂ x in the momentum
equation. These terms are nonlinear and are often a
problem in numerical codes, like finite difference
and finite element, that approximate Equations (1)
as simultaneous linear algebraic equations in the
nodal H and V . In most situations, these terms are
small contributors to the conservation balances
and many codes have an existing option to exclude
them.
(ii) Change the friction term from t0 P/ A to lV in the
momentum equation. Many codes have an optional
choice of friction formula, either Darcy–Weisbach
or Manning. An additional option would not be
difficult to include.
ANALYTICAL BENCHMARK TESTS
A sequence of analytical benchmark tests has been
designed to spotlight the physically and numerically sig-
nificant response patterns that are expected to be within
the predictive capabilities of cross-section-integrated
models for unsteady flow in pipes. These problems are
intended as a supplement, not a substitute, for thesequence of numerical test problems. It is anticipated that
the analytical problems identified here will be of primary
benefit in initial model development and in the evaluation
of user-specific variations or subsequent versions that
introduce new physical, geometrical, numerical or graphi-
cal capabilities. They have complementary value in
classroom instruction.
Each test has a limited objective, seeking to focus on
crucial problems in relative isolation. All problems assume
a single length of pipe, for which a and l are known. The
pipe (Figure 1) extends from xF at F to xL at L. Each of the
following problems have been given a WH identifier,
suggesting their relevance to codes principally intended
for water-hammer-related problems. The specific problem
descriptions are listed in tables, which also include a
recommendation, tOutput, for the time resolution of
analytical and numerical solutions. This is the time
resolution adopted in all the subsequent response pattern
illustrations.
Figure 1 | Single pipe.
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WH1: SUDDEN VALVE CLOSURE FROM STEADY
FLOW
Sudden valve closure from steady flow is the classic water-
hammer problem. At times t < t0 (Figure 2), steady flow is
established at a velocity V 0 from the reservoir at x = 0 to
the free discharge to the atmosphere at x = L. At time
t = t0, the valve at x = L is suddenly closed, leading to a
sudden head rise of aV 0/ g at the valve. This sudden change
in the head is propagated throughout the system and is
slowly attenuated by friction.
At steady flow, Equations (3) become
V constant≠
V 0 and 0 g
SH 0
L
D lV 0 (11)
so that
V 0 gH 0
lL. (12)
Problem WH1, outlined in Table 1, explores the
response to sudden valve closure at t0. The initial con-
ditions are steady flow, with a constant velocity V 0 and
an hydraulic grade line that falls linearly from H0 at x = 0
to zero at x = L. The solution will evolve toward quies-
cent conditions in the pipe, but at early times the
response is dominated by the sudden large head rise
(H = aV 0/ g) at time t0 at x = L, by the propagation of
this step disturbance upstream at the elastic wave speed
a and by sequential reflections from the reservoir and the
closed valve.
In the analytical solution for H, the initial and bound-
ary conditions are
H x,0 f xH 0S1 x
LD ,
∂H
∂tU x,0
g xa2
g
∂V
∂ xU x,0
0
H xF ,tH 0
∂H
∂tU xL ,t
1
gS∂V
∂t lV DU
xL ,t
V 0
gdtt0 l1Htt0 (13)
where d(t) is the Dirac delta function and H(t) is theHeaviside unit step function. The eigenvalues and eigen-
functions are, accordingly,
bnn1 2p
L , Xn xS2
LD
1
2sinbn x n1,2,3,· · ·.
(14)There is no zeroth-mode contribution. The modal coeffi-
cients are
gn0, f |ne0
L
H 0 x
L Xn xd x n1,2,3,· · · (15)
and the transient boundary conditions at xL lead to the
transient internal forcing:
jnt1nS2
LD
1
2 a2V 0
gdtto l1Hstto
n1,2,3,·· ·. (16)
Figure 2 | Sudden closure.
Table 1 | WH1: Sudden valve closure from steady flow
x F x L a t 0 t F ∆t Output t L
0 5000 m 1000 m/s 2 s 0 1 s 30 s
IC H x,0H 0 S1 x
LD V ( x,0)=V 0
BC H( xF ,t>t0)=H 0 V ( xL,t>t0)=0
H 0=10 m, l=0.02 s11, V 0=gH 0/( lL)
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The analytical solution is accordingly
H x,tH 0exp lt 2∑n1
f |n3cosvnt l/2
vn
sinvnt4 Xn x
∑n1
3e0
t1
vn
exp lt t 2sinvnt tjn tdd t4 Xn x.
(17)
It is assumed here, and subsequently, that definite integral
expressions in both x and t are easily evaluated and need
not be pursued. Engineering software platforms often have
a computer algebra capability, which will easily accommo-date simple analytical integrations of this nature. The
same capability could often be used to confirm that each
analytical solution for H( x,t) and V ( x,t) does indeed satisfy
the field equations, the initial conditions and the boundary
conditions. All the application code used in the prep-
aration of illustrations WH1–WH4 adopted this confir-
mation step.
For V , the initial and boundary conditions are
V x,0 f xV 0 ,∂V
∂t U x,0
g x0
∂V
∂ xU xF ,t
g
a2
∂H
∂tU xF,td≠0, Vs xL,td≠V0f12Htt0.
(18)
The eigenvalues do not change, but the eigenfunctions
become
Xn xS2
LD
1
2cosbn x n1,2,3,· · ·. (19)
The modal coefficients are
gn=0, f |ne0
L
V 0 Xn xd x n1,2,3,· · · (20)
and the transient boundary conditions at xL lead to the
transient internal forcing:
jntbn1nS2
LD
1
2a2V 01Htt0 n1,2,3,· · ·.
(21)
The analytical solution is accordingly
V x,texp lt/2∑n1
f |n
3cosvnt
l/2
vnsinvnt
4 Xn x
∑n1
3e0
t1
v0
exp lt t 2sinvnt tjn td t4 Xn x.
(22)
The nature of the response is clear from Equations (17)
and (22). The solution is entirely free-mode transients,
each of which is a standing wave mode at the discrete
system eigenmodes. The influence of these transients
decays exponentially with time at the rate l/2, dictated bythe pipe friction.
In Figure 3, sudden valve closure at t0 forces a step
change in head from 0 to H = aV 0/ g and in flow velocity
from V 0 to 0. These step changes are propagated back
along the pipe at speed a. Behind the step change the flow
is reduced to zero.
The propagation of these initial step changes are seen
particularly clearly in Figure 4, which superimposes longi-
tudinal profiles along the pipe at times 1 sec, before valve
closure at t0, and 3, 4 and 5 sec, all after valve closure at t0.
The spatial resolution of the plot is 100 m. The H profile inparticular shows that the H step change in head climbs
up the initial hydraulic grade line which slopes down from
the reservoir to the valve, a feature that is somewhat
disguised in Figure 3.
The step changes reach the reservoir at time t0 + L/a
where the head is constant at H0. Both the head and
flow velocity steps are reflected from the reservoir, the
head as a continuing H step and the flow velocity at
− V 0. Behind the reflected step, the flow is now towards
the reservoir. There is a further reflection from the closed
valve at time t0 + 2L/a from the reservoir at time t0 + 3L/
a and from the closed valve at time t0 + 4L/a. This cycle
is then repeated with a period of 4L/a = 20 sec. These
changes are especially clear in Figure 5, which shows the
time evolution at distributed x locations along the pipe.
A moderately slow decay with friction can be seen in
the head and flow magnitudes at time t0 + 4L/a. But
it is clear that the dominant dynamic influence is
step disturbance propagation at a speed a. Equations
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(17) and (22) show that the time scale for frictional
decay is
T f 2
l(23)
which is 100 sec for the present problem. Friction will
suppress the sudden changes in head and flow, but not
sufficiently rapidly to mitigate the full impact throughout
the pipe.
Figure 3 | WH1: initial response to sudden valve closure.
Figure 4 | WH1: propagation of step changes in head and flow velocity from valve closure.
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At the valve x = xL, the initial step change in head
from 0 to H and in flow velocity from V 0 to 0 must be
represented by numerous eigenmodes. The wavelengths
Ln = 2p/bn of the eigenmodes are 20,000, 6,667, 4,000, . . .,
m for n = 1,2,3, . . .. Including only a few modes will not
capture the step change. For the WH1 problem, the sum-
mations were truncated at M = 50 eigenmodes, followingthe 0.01 convergence criterion adopted elsewhere (Sobey
2002b). The impact of this summation truncation can be
seen most clearly in the ‘Gibbs oscillations’ immediately
before and after the steps in Figure 4. These oscillations
are expected: they will be mitigated by significantly
increasing M but eliminated only by a summation over an
infinite number of eigenmodes.
The eigenmode amplitudes are shown in Figure 6:
numerous higher wavenumbers bn and higher frequencies
vn are included. The eigenmodes decay very slowly, for V
especially in this problem. In an analytical solution, this isan inconvenience, but not a problem.
For a numerical solution, it can be a problem.
Numerical codes have a distinct spatial resolution x, and
mostly also a distinct temporal resolution t. This finite
resolution imposes (Bath 1974) a Nyquist limit of
b N p
D x, and vN=
p
Dt(24)
respectively on wavenumbers and frequencies that can be
resolved by a numerical model. For the present problem, a
typical x might be of the order of 20 m, and a typical t
might be of the order of 0.02 sec. These correspond to
Nyquist limits of 0.31 m − 1 and 314 sec − 1, respectively.
No wavenumbers or frequencies above these limits can be
represented. In problem WH1, the first 50 free-modewavenumbers and frequencies are well within these
Nyquist limits; in other problems they may not be. Ana-
lytical and physical activity above these Nyquist limits
would be aliased or folded to wavenumbers and fre-
quencies at or just below the respective Nyquist limits. In
numerical models, these are manifested as so-called ‘‘2 x’’
Figure 5 | WH1: time evolution of step changes in head and flow velocity from valve closure.
Figure 6 | WH1: eigenmodes in analytical solution.
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oscillations, oscillations at the Nyquist limit, most notably
in the H response.
WH2: VALVE CLOSURE OVER FINITE TIME
Sudden valve closure immediately imposes the very sig-
nificant head rise aV 0/ g. Gradual valve closure is a com-
mon expedient to mitigate this difficulty. Problem WH2,
outlined in Table 2, explores the response to valve closure
from t0 sinusoidally over a duration tC . At the same time, it
switches the orientation of the problem (see Figure 7): for
a comparison with numerical code, this will exercise a V
boundary condition at xF and a H boundary condition at
xL, the reverse of problem WH1.
In the analytical solution for H, the initial and bound-
ary conditions are
H x,0 f xH L x
L ,
∂H
∂tU x,0
g x0
∂H
∂ xU xF ,t
1
g S∂V
∂t lV D U xF ,t
5 lV 0
gfor t%t0
pV 0
2gtC
sinptt0
tC
lV 0
2gF1cos
ptt0
tC G for t0tt0
0 for tRt0tC
H xL ,tH L. (25)
The eigenvalues and eigenfunctions are accordingly
bnn1 2n
L , Xn xS2
LD
1
2
cosbn x n1,2,3,···. (26)
There is no zeroth-mode contribution. The modal
coefficients are
gn=0, f |ne0
L
H LS x
L1D Xn xd x n1,2,3,· · · (27)
and the transient boundary conditions at xF lead to thetransient internal forcing
jntS2
LD
1
2 a2V 0
g
5 l for t%t0
p0
2tC
sinptt0
tC
l
2F1cos
ptt0
tC G for t0tt0tC
0 for tRt0tC
(28)for n = 1,2,3, . . .. The analytical solution is accordingly
H x,tH 0exp lt 2∑n1
f |n3cosvnt l 2
vn
sinvnt4 Xn x
∑n1
3e0
t
1
v0
exp lt t 2sinvnt tjn td t4 Xn x. (29)
Table 2 | WH2: Value closure over finite time
x F x L a t 0 t F ∆t Output t L
0 5000 m 1000 m/s 2 s 0 1 s 30 s
IC H x,0H L x
L V ( x,0)=V 0
BC V xF ,t5V 0 for t%t0
1
2 F1cosptt0
tC G for t0tt0tC
0 for tRt0tC
H ( xL,t)=H L
H L=10 m, l=0.02 s11, V 0= gH L/( lL); tC =5, 15, 25 s.
Figure 7 | WH2: steady flow prior to start of valve closure.
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For V , the initial and boundary conditions are
V x,0 f xV 0 ,
∂V
∂tU x,0 g x0
V xF ,t5V 0 for t%t0
1
2V 0F1cos
ptt0
tC G for t0tt0tC
0 for tRt0tC
∂V
∂tU xF ,t
g
a2
∂H
∂tU xL ,t
0. (30)
The eigenvalues do not change, but the eigenfunctions
become
Xn xS2
LD
1
2sinbn x n1,2,3,· · ·. (31)
There are no zeroth-mode contributions. The modal
coefficients are
gn0, f |ne0
L
V 0 Xn xd x n1,2,3,· · · (32)
but the transient boundary conditions at xL lead to thetransient internal forcing
jnt
S2
LD
1
2a2V 05
bn for t%t0
1
2bnF1cos
ptt0
tC G for t0tt0tC
0 for tRt0tC
(33)
for n = 1,2,3, . . .. The analytical solution is accordingly
V x,texp lt 2∑n1
f |n3cosvnt l 2
vn
sinvnt4 Xn x
∑n1
3e0
t1
vn
exp lt t 2sinvnt tjn td t4 Xn x.
(34)
The nature of the response, Equations (29) and (34), is
formally similar to the WH1 solution, as expected. But the
algebraic and numerical details differ significantly. Thesummations were truncated at M = 50 eigenmodes, to
meet the 0.01 convergence criterion. The response for
valve closure over tC = 5 sec, equal to the propagation time
L/a of a disturbance over the length of the pipe, is shown
in Figure 8.
The sudden changes are smoothed by the sinusoidal
closure, but the response pattern retains most of the
features exhibited by sudden valve closure. Propagation
of the disturbance at speed a, multiple reflections from
the pipe ends, a multiple-reflection periodicity of 4L/a
and gradual attenuation by friction all remain apparent.In WH1, the duration of the peak head rise was
2L/a = 10 sec at the valve end. For closure over tC = L/a,
the peak head rise remains at the valve, the magnitude
remains unchanged but the duration is reduced to about
5 sec.
The response to longer closure times, tC = 3L/a and
5L/a, are shown in Figures 9 and 10, respectively, where
the surface plots have been extended to tL = 60 sec to
capture the evolving pattern.
All suggestions of a step response are now gone, but
the maximum head at the valve is only slowly attenuated.For tC = 0 (problem WH1), the global Hmax is 122.1 m; for
tC = 5, 15 and 25 sec, the global Hmax is progressively
attenuated to 108.8 m, 87.0 m and 51.4 m, respectively.
The continuing periodicity at 4L/a (20 sec) is seen
most clearly in the time histories for H at the valve and for
V at the reservoir. Extending the closure time provides the
opportunity for friction to be influential, but inertia
remains the dominant process at these relatively short
times.
WH3: START-UP AND EVOLUTION TO STEADY
STATE
Problem WH3, outlined in Figure 11 and Table 3, explores
start-up transients and the evolution to steady state. The
numerical value of this problem somewhat duplicates
problem WH1. The start-up problem nonetheless has con-
siderable instructional value, especially in the manner of
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Figure 8 | WH2: response to valve closure over t C=L/ a=5 sec.
Figure 9 | WH2: response to valve closure over t C=3L/ a=15 sec.
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the approach to the steady-state flow and in the timescale
of this transition.
In the analytical solution for H, the initial and bound-
ary conditions are
H( x,0) = f ( x) = H0, V ( x,0) = g( x) = 0 (35)
H( xF ,t) = H0, H( xL,t) = H0[1 − H(t − t0)].
The eigenvalues and eigenfunctions are accordingly
bnnp
L , Xn xS2
LD
1
2sinbn x n1,2,3,·· ·. (36)
Figure 10 | WH2: response to valve closure over t C=5L/ a=25 sec.
Figure 11 | WH3: static conditions prior to start-up.
Table 3 | WH3: Start-up and evolution to steady state
x F x L a t 0 t F t Output t L
0 5000 m 1000 m/s 2 sec 0 1 sec 30 sec
IC H ( x, 0) = H 0 V ( x,0) = 0
BC H ( xF , t) = H 0 H ( xL, t > t0) = 0
H0=10 m, l=0.02 sec−1.
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There is no zeroth-mode contribution. The modal coef-
ficients are
gn0, f |ne0
L
H 0 x
L Xn xd x (37)
and the transient boundary conditions at xL lead to the
transient internal forcing
jnt1nbnS2
LD
1
2a2H 01Htt0 n1,2,3,· · ·.
(38)
The analytical solution is accordingly
H x,tH 011 x
L2exp lt 2 ∑
n1
f |n3cosvnt l 2
vn
sinvnt4 Xn x
∑n1
3e0
t1
vn
exp lt t 2sinvnt tjn td t4 Xn x.
(39)For V , the initial and boundary conditions are
V x,0 f x0,∂V
∂tU x,0
g x0
∂V
∂tU xF ,t
g
a2
∂H
∂tU xF ,t
0
∂V
∂tU xL ,t
g
a2
∂H
∂tU xL ,t
gH 0
a2 dtt0. (40)
For Type 4 boundary conditions (gradient boundaryconditions at both ends), the free modes are augmented
by a zeroth mode. The eigenvalues and eigenfunctions
are
b00, X0 xS1
LD
1
2
bnnp
L , Xn xS2
LD
1
2cosbn x n1,2,3,· · ·. (41)
The gn and f ˆn modal coefficients are all zero, except
for
f |oe0
L
V X0 xd x (42)
where V N
= gH0/ lL from domain integration of the
momentum equation.
The transient boundary conditions at xL lead to the
transient internal forcing
jnt1nS2
LD
1
2 gH 0dtt0 n1,2,3,· · · (43)
which includes the zeroth-mode contribution. The
analytical solution is accordingly
V x,tV 3 f |01
le0
t
1exp lt tj0 td t4 X0 x
∑n1
3e0
t1
vn
exp lt t 2sinvnt tjn td t4 Xn x.
(44
)The transient response, Equations (39) and (44), is
again entirely free modes. The summations were truncated
at M = 100 eigenmodes to meet the 0.01 convergence
criterion.
The evolution of the response is most transparent in
the V response, Figure 12. The immediate response to the
sudden valve opening at x = L is a sudden velocity of
V = gH0/a, which propagates up the pipe to the reservoir,
initiating a sequence of velocity steps from the reservoir
and the valve. At each reflection, the flow velocity is
augmented by a velocity of the order of V , climbing
toward the steady state flow V 0 = gH0/ lL.
Without friction, the method of characteristics gives
the velocity step as exactly gH0/a every reflection (i.e.
every L/a sec), suggesting a time to steady state of the
order of (V 0/V )(L/a) = 1/ l = 50 sec. The influence of
friction, Figure 13, is a progressive decline in the magni-
tude of this velocity step and a much slower approach to
the steady state flow.
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Figure 12 | WH3: start-up at 0≤t ≤20 sec.
Figure 13 | WH3: start-up at 90≤t ≤110 sec.
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The H response necessarily mirrors the V response.
There is a gradual transition from a horizontal Hydraulic
Grade Line to the linearly falling Hydraulic Grade Line atsteady state, through a sequence of propagating and
reflecting step changes in H. At about 100 sec (Figure 13),
steady state is clearly approached but not yet reached.
WH4: RESPONSE TO SUDDEN PERIODIC FORCING
The final problem, WH4, outlined in Table 4, explores
the unsteady response to sudden periodic forcing,
such as a seiche in the reservoir at x = 0 in Figure 2.Additionally, this problem provides an opportunity to
evaluate the limits of applicability of the incompressible
or rigid-column approximation to unsteady flow in
pipes.
In the analytical solution for H, the initial and
boundary conditions are
H ( x,0) f xH 0S1 x
LD ,
∂H
∂tU
( x,0)
g( x)a2
g
∂V
∂ xU
( x,0)
0
H ( xF ,t)H 0a0sinvt, H ( xL ,t)0 (45)
where a0 is the amplitude and v = 2p/T the frequency of
the reservoir seiche.
For the forced mode, boundary condition matching
gives b = H0, m = − H0/L and
31 0 1 0
0 1 0 1
C S C SS C S C 43
a1cos 1
a1sin 1
a2cos 2
a2sin 2 4
30
a0
00
4(46)
in which C + = exp( + mL)cos kL, C − = exp( − mL)cos kL,
S + = exp( + mL)sin kL, S − = exp( − mL)sin kL and m = lv/
(2a2k). Equation (46) is a linear equation system, which
may be solved directly for a1cos 1 to a2sin 2. a1, a2, 1
and 2 are then immediately available.
For the free modes, the eigenvalues and eigenfunc-
tions are
b0np
L , Xn( x)S2
LD
1
2sinbn x n1,2,3,·· ·. (47)
There is no zeroth-mode contribution. The modal coef-
ficients are
f |ne0
L
[a1exp( m x)cos(kx 1)a2exp( m x)
cos(kx 2)]Xn( x)dx
gne0
L
v[a1exp( m x)sin(kx 1)a2
exp( m x)sin(kx 2)]Xn( x)dx (48)
for n = 1,2,3, . . .. The boundary forcing is periodic and
transient internal forcing, jn(t), is zero.
The analytical solution is accordingly
H ( x,t)H 0S1 x
LDa1exp( m x)cos(kxvt 2)a2
exp( m x)cos(kxvt 2)∑n1
F f |nexp( lt/2)cosvnt
gn l f
ˆn/2
vn
exp( lt/2)sinvntG Xn( x). (49
For V , the initial and boundary conditions are
V ( x,0) f ( x)V 0 ,∂V
∂tU
( x,0)
g( x)0
∂V
∂ xU
( xF ,t)
g
a2
∂H
∂tU
( xF ,t)
ga0v
a2 cosvt
∂V
∂ xU
( xL ,t)
g
a2
∂H
∂tU
( xL ,t)
0. (50)
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For the forced mode, boundary condition matching gives
3 m k m k
k m k m
kS mC kC mS kS mC kC mS
kC mS kS mC kC mS kS mC 4
3
a1cos 1
a1sin 1
a2cos 2
a2sin 2 4
3 ga0v/a2
0
00
4. (51)
As for H, Equation (51) is a linear equation system, which
may be solved directly for a1cos 1 to a2sin 2. a1, a2, 1
and 2 are then immediately available. For gradient or
Neumann boundary conditions at both ends (Type 4 in
Sobey 2002a), b cannot be defined by boundary condition
matching. A supplementary integral momentum flux
condition over the entire domain establishes that b = V 0.
For Type 4 boundary conditions, the free modes
are augmented by a zeroth mode. The eigenvalues and
eigenfunctions are
b00, X0( x)S1
LD
1
2
bnnp
L , X0( x)S2
LD
1
2cosbn x n1,2,3·· ·. (52)
The modal coefficients are
f |ne0
L
[a1exp( m x)cos(kx 1)a2exp( m x)
cos(kx 2)]Xn( x)dx
gne0
L
v[a1exp( m x)sin(kx 1)a2
exp( m x)sin(kx 2)]Xn( x)dx (53)
for n = 0,1,2, . . ., which includes the zeroth mode. The
analytical solution for V ( x,t) is then
V ( x,t)V 0a1exp( m x)cos(kxvt 1)a2exp( m x)
cos(kxvt 2)F f |0 g0
l(1exp( ll))G
exp( lt/2)∑n1
F f |ncosvnt gn l f |n/2
vn
sinvntG Xn( x). (54)
The nature of the response, Equations (49) and (54),now includes both a forced mode (the first three terms
of each equation) and free modes. The forced mode
responds at the frequency v of the sustained forcing,
Equation (45b). The free modes respond at the eigen-
modes, vn in time and bn in space and decay exponen-
tially with time at a rate dictated by the friction
coefficient l. The free mode summations were truncated
at M = 11 eigenmodes, to meet the 0.01 convergence
criterion.
The initial response to forcing with period
2p/v = 60 sec is shown in Figure 14. The response is
dominated by the initial steady flow field, which persists as
the time-averaged flow, H = H0(1 − x/L), V = V 0. In Figure
14, this time-averaged flow has been subtracted to show
the detail of the response much more clearly.
The frictional response time T f is 100 sec (see Equa-
tion (23)) and the decay of the free modes is clearly seen in
the evolving clarity of the forced mode response at period
60 sec.
Table 4 | WH4: Response to sudden periodic forcing
x F x L a t 0 t F ∆t Output t L
0 5000 m 1000 m/s 0 s 0 T/20 s 2T s
IC H x,0H 0 S1 x
LD V ( x,0)=V 0
BC H( xF ,t>t0)=H 0+a0sin vt H ( xL,t)=0
H 0=10 m, l=0.02 s11, V 0=gH 0/( lL), a0=1 m, T =2/v=60 s.
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A common simplifying approximation to unsteady
flow in pipes is the incompressible flow or rigid water
column approximation. This asserts that the system
response time is sufficiently slow that elastic changes in
the mass density of the fluid and in the cross-section area
A of the pipe are not significant. The context of problem
WH4 provides an opportunity to explore the value of this
approximation. Where pipe-fluid compressibility is not
significant, the equation of state for the pipe-fluid com-
posite (see Equation (2)) becomes
1 A
D Dt
(pA)=0 (55)
and the unsteady pipe flow equations (see Equations 1)
become
∂V
∂ x0
∂V
∂tV
∂V
∂ x g
∂H
∂ x to P
r A(56)
or
∂V
∂ x0
∂V
∂t g
∂H
∂ x lV (57)
in a linear approximation.
Under the same initial and boundary conditions as for
the complete compressible problem (Equations (45)),
Equations (57) are solved to give
V rigid(t)V 0 ga0
L(v2 l2)(v(e ltcosvt) lsinvt)
H rigid( x,t)L x
gSdV rigid
dt lV rigidD . (58)
This incompressible approximation is shown in Figure 15
for the identical conditions as the complete compressible
solution in Figure 14.
Figure 14 | WH4: initial differential response to 60 sec period forcing.
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The differences appear very small. Figures 16 and 17
focus on these differences:
H = H( x,t) − Hrigid( x,t), V = V ( x,t) − V rigid( x,t). (59)
Figure 16, for the initial 120 sec, is dominated by the
free mode transients. Their magnitude is relatively
small, being a response to the ∂H/∂t but not H dis-
continuity in the forcing at x = 0. Their contribution to
the response is dwarfed by the forced mode, even near
t = 0. As seen clearly in the H response, these free
modes decay with time at the frictional timescale
T f = 60 sec. The V response suggests a significant
residual component at period 60 sec, corresponding to
the forcing frequency v. This residual component at the
forcing frequency is quite clear in Figure 17 at a much
later time. This persistent residual difference between
the compressible flow solution and the incompressible
flow approximation results directly from the incom-
pressible flow approximation. Note in particular the V
solutions for compressible flow (Equation (54)) and
incompressible flow (Equation (58a)). The forced-mode
part of Equation (54) has a spatial structure, through
the forced wave motion, cos(kx . . .), and through the
frictional attenuation, exp( ± m x). In the incompressible
flow approximation, there is no spatial structure. It is
the intrinsic spatial structure in the forced mode that is
seen in Figure 17. Some small residual free mode
response can be identified in the H trace, but this does
decay with time.
The response pattern at the forcing frequency persists
for all time. There is a difference between the complete
compressible flow solution and incompressible flow
approximation, but the magnitude is quite small (note the
relative magnitudes in Figure 17), and those magnitudes
decrease as the period T of the forcing becomes very much
longer than the period T 1 = 2p/v1 ( = 10 sec here) of the
dominant free mode.
Figure 15 | WH4: initial differential response to 60 sec period forcing, using incompressible flow approximation.
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CONCLUSIONS
Analytical solutions have a useful role in the evaluation of numerical codes for unsteady compressible flow in pipes.
Analytical solutions alone have absolute credibility. They
provide a measure of physical and code credibility that is
not otherwise available to numerical codes. The necessary
code modifications are outlined.
Analytical solutions have an equally important
complementary role in instruction, by providing focused
illustrations of the nature and magnitude of unsteady
response patterns in a number of conceptually challenging
contexts.
A sequence of four analytical benchmark problems areoutlined:
WH1: sudden valve closure from steady flow
WH2: valve closure over finite time
WH3: start-up and evolution to steady state
WH4: response to sudden periodic forcing.
Each case includes the full details of the analytical solu-
tion, an illustration of the predicted response pattern and
a discussion of the significant physical and numerical
aspects.
Collectively, these analytical solutions provide the
framework for a wide-ranging confirmation of numerical
codes for flood and tide propagation.
The utility of the incompressible flow approximation
to unsteady pipe flow is finally considered. A linear
analytical solution is established for conditions equivalent
to problem WH4. It is shown that the difference from
the complete linear compressible solution is small
where the period T of the forcing is much longer than the
period T 1 = 2p/v1 ( = 10 sec here) of the dominant free
mode.
Figure 16 | WH4: difference between complete compressible flow solution and incompressible flow approximation for 60 sec period forcing for times 0< t<120 sec.
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ACKNOWLEDGEMENTSThis study was initiated and largely completed over a
sabbatical leave during 2002 at the Department of
Civil and Environmental Engineering, Imperial College
London.
APPENDIX
Zeroth-mode Green’s function
In the terminology of Equation (6), the complete general
solution for H in Sobey (2002a) has the generic form
H( x,t) = H ˜ ( x,t) + H8( x,t) + H9( x,t) (60)
where H ˜ ( x,t) is the forced mode solution (Equation (6)
with non-homogeneous but periodic boundary con-
ditions), H′( x,t) is the free mode solution (Equation (6)
with potential non-homogeneous internal forcing ( x,t)
but homogeneous boundary conditions) and H″ ( x,t) is the
residual transient solution (Equation (6) with non-
homogeneous and non-periodic boundary conditions and
quiescent initial conditions).
The solution for H′( x,t) in Sobey (2002a) excluded the
zeroth-mode contribution to the particular solution
Pnte0
t
Gn(t, t) |n td t where |nte0
L
x,t Xn xd x.
(61)
G0(t) potentially exists only for problems with gradient or
Neumann boundary conditions at both ends (Type 4 in
Sobey (2002a)).
Following again the method of variation of par-
ameters, the zeroth-mode Green’s function is
G0(t, t)1
l(1exp( l(t t))). (62)
Figure 17 | WH4: difference between complete compressible flow solution and incompressible flow approximation for 60 sec period forcing for times 300< t<420 sec.
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Including a possible zeroth mode, the complete solution to
the transient problem becomes
H ( x,t)∑n0
[H n(t) Pn(t)]Xn( x)
3 f |0 g0
l(1exp( lt))
1
le0
t
(1exp( l(t t))) |0( t)d t4 X0( x)
∑n0
3 f |nexp( lt/2)cosvnt
gn l f |n/2
vn
exp( lt/2)
sinvnte0
t1
vn
exp( l(t t)/2)sinvn(t t) |n( t)d t4 Xn( x).
(63)
While internal forcing ( x,t), and hence |n( t), has no role
in unsteady pipe flow, transient boundary conditions
become equivalent internal forcing jn(t) and transient boundary conditions have a major role in unsteady pipe
flow.
The complete solution for V ( x,t) is formally identical.
REFERENCES
Båth, M. 1974 Spectral Analysis in Geophysics. Elsevier, Amsterdam.Li, W. H. 1983 Fluid Mechanics in Water Resources Engineering.
Allyn and Bacon, Boston, MA.
Sobey, R. J. 2002a Analytical solution of non-homogeneous wave
equation. Coastal Engng J. 44 (1), 1–24.
Sobey, R. J. 2002b Analytical solutions for flood and tide codes.
Coastal Engng J. 44 (1), 25–52 (see also errata 44, 281).
207 Rodney J. Sobey | Analytical solutions for unsteady pipe flow Journal of Hydroinformatics | 06.3 | 2004
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