Calculo Analitico Flujo Trasiente

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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: [email protected]

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

187 © IWA Publishing 2004 Journal of Hydroinformatics | 06.3 | 2004



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.

188 Rodney J. Sobey | Analytical solutions for unsteady pipe flow Journal of Hydroinformatics | 06.3 | 2004



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




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.




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)




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)


transient internal forcing:

jntbn1nS2

LD

1

2a2V 01Htt0 n1,2,3,· · ·.

(21)


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




(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.




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.




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.


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


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.





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


(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




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.




the approach to the steady-state flow and in the timescale

of this transition.


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.




There is no zeroth-mode contribution. The modal coef-

ficients are

gn0, f |ne0

L

H 0 x

L Xn xd x (37)


transient internal forcing

jnt1nbnS2

LD

1

2a2H 01Htt0 n1,2,3,· · ·.

(38)


H x,tH 011 x

L2exp lt 2 ∑

n1

f |n3cosvnt l 2

vn

sinvnt4 Xn x

∑n1

3e0

t1

vn


(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


(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.




Figure 12 | WH3: start-up at 0≤t ≤20 sec.

Figure 13 | WH3: start-up at 90≤t ≤110 sec.




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.


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


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)




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


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.




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.




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.




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.




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.




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).


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