Analysis of Unsteady Pipe Flow Using Modified FEM

8/13/2019 Analysis of Unsteady Pipe Flow Using Modified FEM

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COMMUNICATIONS IN NUMERICAL METHODS IN ENGINEERINGCommun. Numer. Meth. Engng 2005; 21:183–199

Published online 24 December 2004 in Wiley InterScience (www.interscience.wiley.com). DOI: 10.1002/cnm.741

Analysis of unsteady pipe ow using the modiednite element method

Romuald Szymkiewicz1;∗;† and Marek Mitosek 2

1Faculty of Civil and Environmental Engineering; Gdansk University of Technology; 80-952 Gdansk ; Poland 2Faculty of Environmental Engineering; Warsaw University of Technology; 00-653 Warsaw; Poland

SUMMARY

A modied nite element method is proposed to solve the unsteady pipe ow equations. This approachyields a six-point implicit scheme with two weighting parameters. An accuracy analysis carried outusing the modied equation approach showed that the proposed scheme has higher accuracy comparedto other methods. A comparison of experimental data and the results of numerical solution showed thatthe required damping and smoothing of a pressure wave can be obtained when numerical diusion is produced by the applied method. It suggests that the physical dissipation process observed in the wa-ter hammer phenomenon is not represented properly in its mathematical model. Therefore, the classicalsystem of water hammer equations seems to be incomplete. Copyright ? 2004 John Wiley & Sons, Ltd.

KEY WORDS: pipe ow; water hammer; nite element method; numerical diusion

1. INTRODUCTION

Taking into account the elasticity of pipe walls, the compressibility of liquid, the hydrauliclosses due to pipe friction, assuming uniform distribution of pressure and velocity over thecross-section area and that the pipeline is always completely lled with water, Parmakian [1]

proposed the following mathematical model of the water hammer phenomenon:

@V

@t + V

@V

@x + g

@H

@x +

f

2 D V |V | = 0 (1)

@H

@t + V

@H

@x +

c2

g

@V

@x = 0 (2)

∗Correspondence to: R. Szymkiewicz, Faculty of Civil and Environmental Engineering, Gdansk University of Technology, 80-952 Gdansk, Poland.

†E-mail: [email protected]

Received 1 December 2003Copyright ? 2004 John Wiley & Sons, Ltd. Accepted 2 November 2004


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184 R. SZYMKIEWICZ AND M. MITOSEK

where x is the space co-ordinate, t the time, V the velocity in the pipe, H the piezometrichead, f the friction factor, D the inside diameter of the pipe, g the acceleration of gravity,and c the velocity of pressure wave.

The pressure wave velocity is expressed as follows:

c = 1

1

K +

D

Ee

(3)

where is the density of the liquid, K the bulk modulus of elasticity of the liquid, E themodulus of elasticity of pipe-wall material, and e the thickness of pipe wall.

Equation (1) is the momentum equation and Equation (2) the equation of continuity.They form a rst-order hyperbolic system of partial dierential equations, for which aninitial–boundary problem is formulated. It is solved in domain: 06 x6 L, t ¿0 (where L is thelength of the pipeline). Proper initial and boundary conditions have to be imposed. Steadyow assumed at t = 0 enables to determine the functions V ( x; t =0) and H ( x; t =0) over theentire pipeline (06 x6 L). Since from each boundary, only one characteristic enters the do-main of solution, at x = 0 and x = L one function, V (t ) or H (t ), has to be imposed. For giveninitial and boundary conditions this system is solved numerically.

The water hammer equations can be solved by all methods suitable for hyperbolic equations.The method of characteristics (MOC) is most commonly used. It was proposed by Streeter and Lai [2] as well as by Evangelisti [3]. Its description was given by Almeida and Koelle[4], Goldberg and Wylie [5], Wylie and Streeter [6] as well. Usually MOC is applied for axed grid. Consequently, this approach needs an interpolation in space or in time.

To obtain a stable solution, the CFL condition has to be satised. For the Courant number equal to unity, MOC ensures an exact solution, whereas for other values it produces numericaldiusion. The dissipation error is introduced by linear interpolation between the nodes used to

calculate the values of function V and H at the points of intersections of characteristics. Thenumerical diusion gives rise to smoothing and damping of the pressure wave. One can showthat in this case the coecient of numerical diusion is equal to the one corresponding to thewell-known pure advection equation solved by the nite dierence method using the up-windscheme [7]. Therefore one should remember that the numerical diusion generated by MOCincreases with the increase in distance step and with the decrease in Courant number.

To improve MOC, Sibetheros et al. [8] made an attempt to apply an interpolation by thespline functions. Another approach to nd a suitable method to solve the water hammer equa-tions was proposed by Chaudhry and Hussaini [9]. It concerned the nite-dierence schemesof second order of accuracy. Both mentioned approaches do not represent remarkable advan-tages. Consequently, MOC with linear interpolation in space or in time is still most commonlyused to solve the water hammer equations.

Taking into account the experience resulting from the solution of the hyperbolic equations by various numerical schemes, Fletcher [10] inferred the general guideline that rst-order formulae for derivatives should be avoided. A rst-order representation for the advectiveterm in the governing equation will generate spatial derivatives of second order and higher in the modied equation, which is the equivalent governing equation actually solved. Theintroduction of spurious second or third derivatives can change the character of the solutionsignicantly. Similar recommendation is given by Leonard and Drummond [11].

Copyright ? 2004 John Wiley & Sons, Ltd. Commun. Numer. Meth. Engng 2005; 21:183–199


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ANALYSIS OF UNSTEADY PIPE FLOW USING MODIFIED FEM 185

As far as the MOC with linear interpolation in space is considered, for the Courant number less than unity, it represents an approximation of rst order of accuracy with regard to x aswell as to t . The only way of achieving accurate solutions is to use small time step and aCourant number close to 1. This requirement can be fullled easily for a simple pipe only.

When the pipe’s diameter varies, the wave celerity varies as well. In such a case, it is dicultto keep constant Courant number close to 1 and consequently this method will always producenumerical diusion.

To avoid the aforementioned problems, the modied nite element method (FEM) can beapplied. This approach proposed to solve the 1D unsteady ow and transport equations inopen channel [12] can be successfully applied to solve the water hammer equations as well.The modication deals with the process of integration and leads to a more general form of algebraic equations approximating the governing equations. Its particular cases are the standardnite element method and the well-known nite dierence schemes.

Usually the accuracy analysis is carried out in a classical way. It deals with the so-calledcoecients of convergence, which characterize the amplitude and phase errors. However,these coecients explain only general tendencies of the applied numerical method. As theyare related to the wavelengths, information resulting from such an analysis cannot be useddirectly to assess the numerical errors generated while solving the dierential equations. For

better assessment of the numerical errors the so-called modied equation approach seems to be very useful. This approach proposed by Warming and Hyett [13] is particularly suitable for the hyperbolic equations. Such analysis provides a large amount of information and suggeststhe way of increasing the solution accuracy. Proper choice of values of the two weighting

parameters ensures third-order accuracy of the proposed method.

2. SOLUTION OF THE WATER HAMMER EQUATIONS BY THE MODIFIED FEM

According to the Galerkin procedure, described in detail by Zienkiewicz [14], the solution of Equations (1) and (2) has to satisfy the following condition:

L0

(fa; : : :)N d x = M −1k =1

xk +1 xk

(fa; : : :)N d x = 0 (4)

where is the symbolic representation of Equations (1) and (2), fa the approximation of anyfunction f( x; t ) occurring in Equations (1) and (2), N( x) the vector of linear basis functions, L the length of pipeline, k the index of node, and M the number of grid points.

In the standard approach the function f( x; t ) is approximated as follows:

fa( x; t ) = M k =1

fk (t ) N k ( x) (5)

where fk (t ) represents nodal value of function f( x; t ).



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When linear basis functions are applied in each element, only the two following integralsin Equation (4) exist

I 1 = xk +1 xk

fa( x; t ) N k ( x) d x = 23 fk (t ) +

1

3 fk +1(t ) xk

2 (6a)

I 2 =

xk +1 xk

fa( x; t ) N k +1( x) d x =

1

3 fk (t ) +

2

3 fk +1(t )

xk

2 (6b)

where xk is the distance between the nodes.In this case, the standard approach can be modied. Namely, the integral of the product of

approximation of function and basis function in an element can be expressed as a product of certain average value of the function in the element and the integral of the basis function inthis element. Therefore,

I 1 = xk +1

xk

fa( x; t ) N k ( x) d x = fc(t ) xk +1

xk

N k ( x) d x = fc(t ) xk

2

(7a)

I 2 =

xk +1 xk

fa( x; t ) N k +1( x) d x = fc(t )

xk +1 xk

N k +1( x) d x = fc(t ) xk

2 (7b)

The weighted average fc(t ) in the considered element can be expressed using the followingformula:

for Equation (7a): fc(t ) = !fk (t ) + (1 −!)fk +1(t ) (8a)

for Equation (7b): fc(t ) = ( 1 −!)fk (t ) + !fk +1(t ) (8b)

where ! is the weighting parameter ranging from 0 to 1.Consequently Equations (7a) and (7b) can be rewritten as follows:

I 1 = (!fk (t ) + (1 −!)fk +1(t )) xk

2 (9a)

I 2 = ((1 −!)fk (t ) + !fk +1(t )) xk

2 (9b)

This concept applied for open channel unsteady ow and transport equations [12] can bedeveloped for the water hammer equations as well. Calculation of one integral in expression(4) over an element of length xk = xk +1 − xk is made in the following way:

X k +1

X k @V c

@t

+ 1

2

@V 2a

@x

+ g @H a

@x

− f2 D

V |V |c N k d x

=

!

dV k dt

+ (1 −!) dV k +1

dt

xk

2 +

1

4 (−V 2k + V

2k +1)

+ g

2 (− H k + H k +1) +

xk 4 D

(!fk V k |V k | + (1 −!)fk +1V k +1|V k +1|) (10)



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X k +1 X k

@H c@t

+ V c@H a@x

+ c2

g

@V a@x

N k d x

=! d H k

dt + (1 −!) d H k +1

dt

xk

2

+ 1

2 (!V k + (1 −!)V k +1)(− H k + H k +1) +

c2

2g(−V k + V k +1) (11)

X k +1 X k

@V c@t

+ 1

2

@V 2a@x

+ g @H a@x

+

f

2 D V |V |

c

N k +1 d x

= (1 −!) dV k dt

+ ! dV k +1

dt xk

2

+ 1

4

(−V 2k + V 2

k +1)

+ g

2 (− H k + H k +1) +

xk 4 D

((1 −!)fk V k |V k | + !fk +1V k +1|V k +1|) (12)

X k +1 X k

@H c@t

+ V c@H a@x

+ c2

g

@V a@x

N k +1 d x

=

(1 −!)

d H k dt

+ ! d H k +1

dt

xk

2 +

1

2((1 −!)V k + !V k +1)(− H k + H k +1)

+ c2

2g(−V k + V k +1) (13)

In these expressions, the subscript a denotes approximation according to formula (5) whilethe subscript c denotes approximation by Equations (8a) and (8b). It means that the pro-

posed modication concerns an alternative way of approximation of equations except for theterms containing the spatial derivatives. The derivatives of rst order with regard to x areapproximated using the standard approach.

When all integrals in each element are summarized according to Equation (4), the globalsystem of ordinary dierential equations over time is obtained. It has the followingform:

S dXdt

+ CX = 0 (14)

where S is the constant matrix, symmetrical and banded, C the variable matrix, asymmetricaland banded, X= (V 1; H 1; V 2; H 2; : : : ; V M ; H M )

T the vector of unknowns set up from nodal values

of V and H , dXdt

=

dV 1dt ; d H 1

dt ; : : : ; dV M

dt ; d H M

dt

Tthe vector of time derivatives, and T the symbol of

transposition.



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The matrices S and C have dimensions of (2 M )× (2 M ) with bandwidth equal to 7.The following two-level method is applied to time integration:

X j+1 =X j + t

dX j+1dt + (1 − )

dX jdt

(15)

where is the weighting parameter ranging from 0 to 1, t the time step, and j the indexof time level.

It leads to the system of non-linear algebraic equations

(S + tC j+1)X j+1 = (S− t (1 − )C j)X j (16)

which has to be completed by imposed boundary conditions. They are as follows:

• at the beginning of the pipe (k = 1), a constant piezometric head H 1 = const is imposed,• at the end of the pipe (k = M ), the velocity of outowing water is calculated dependingon the gate opening [2]; in this paper, it is assumed that the gate is closed immediatelyso V M (t ) = 0 for t¿0.

The presented method involves two weighting parameters ! and . Taking != 23

, the FEMin its standard form is obtained, whereas for != 1, this method coincides with the nitedierence one. When = 1

2, the method becomes the Crank–Nicolson scheme (equivalent to

the implicit trapezoidal rule), whereas with = 1, it is the implicit Euler scheme (Figure 1).To solve the non-linear system, an iterative method, for example Newton’s one, has to beused.

∆ t

∆x ∆x

t

t

t j

j+1

xx x xk 1 k k+1−

(1 ) t− θ ∆

θ∆t

1

2

−ω 1

2

−ω

ω

Figure 1. Numerical grid for the modied nite element method.



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3. STABILITY ANALYSIS AND NUMERICAL ERRORS GENERATED BY THE FEM

Stability analysis is carried out for the simplied linear form of the governing equations whichis as follows:

@V

@t + g

@H

@x = 0 (17)

@H

@t +

c2

g

@V

@x = 0 (18)

Approximation of this system on the grid points presented in Figure 1 with x = const yieldsthe following algebraic equations:

(1 −!)V j

+1k −1 − V

jk −1

t + 2!

V j+1

k − V jk

t + (1 −!)

V j+1

k +1 − V jk +1

t

+ g

x

− H j

+1k −1 + H

j+1k +1

+ (1 − )

g

x

− H jk −1 + H

jk +1

= 0 (19)

(1 −!) H j

+1k −1 − H

jk −1

t + 2!

H j+1

k − H jk

t + (1 −!)

H j+1

k +1 − H jk +1

t

+ c2

g x

− V j+1k −1 + V

j+1k +1

+ (1 − )

c2

g x

− V jk −1 + V

jk +1

= 0 (20)

The stability analysis was carried out by the Neumann method [15]. The so-called ampli-

cation matrix, being a result of this approach, has the following eigenvalue:

= 1 − 4 R2

4 R22 + 1 +

2 R

4 R22 + 1 i (21)

with

R=

C r tan

m x

2

1 + (2!− 1)tan2m x

2

(22)

where is the eigenvalue of amplication matrix, i = (−1)1= 2 the imaginary unit, C r = ct= xthe Courant number and m the wave number.

The condition of numerical stability [15]:||61 (23)

in this case takes the following form:

||=

1 −

2− 1

2 + 1= (4 R2)

1= 261 (24)



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1.5

1.0

0.5

0 2 4 6 8 10 12 14 16 18 20

Θ = 1

Θ = 0.67

Θ = 0.5

Θ= 0.4

ω = 0.5; C = 1r

λ I I

N

Figure 2. Amplitude portrait for dierent .

It has to be satised for any wave number m. This takes place when

¿ 12

(25)

and

!¿12

(26)

As the relation (24) is satised for any Courant number, the above relations ensure uncondi-tional stability.

The conclusions presented above can be illustrated by the graphs of the eigenvalue modulusof amplication matrix (24) called the amplitude portrait [7]. Since m x = 2=N ( N = number of computational intervals x per wavelength corresponding to examined component of Fourier series), the modulus of eigenvalue is a function of N : ||= f( N ). In Figures 2 and 3,one can notice that for C r = 1 and =

12

, ||= 1. It means that the scheme does not change the

wave amplitude. For ¡ 12

, one obtains ||¿1. In this case the wave amplitude is amplied

and consequently the scheme becomes unstable, whereas for ¿ 12

, the amplitude is damped.Errors generated by any numerical method can be examined more exactly by the modied

equation approach [10]. The solution of the hyperbolic equation has a form of waves that

are dened by their amplitudes and phase celerities. The numerical methods applied to obtainthis solution should not change the wave parameters. A method that changes the amplitudeis called dissipative whereas a method changing the phase celerity is called dispersive. Bothmentioned phenomena are caused by truncation error while approximating derivatives.

To carry out an accuracy analysis using the modied equation approach all nodal values of functions V and H in the algebraic equations that approximate Equations (17) and (18), arereplaced by Taylor series expansion around the node (k; j + 1) (Figure 1). Next, the obtained



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I λ I

0 2 4 6 8 10 12 14 16 18 20

N

1.0

0.5

ω Θ= 0.5; = 0.67

rr

r

C = 0.5

C = 2

C = 1

Figure 3. Amplitude portrait for dierent Courant numbers.

relations are rearranged so that they contain only spatial derivatives. Finally the followingsystem of equations is derived:

@V

@t + g

@H

@x = n

@2V

@x2 + n

@3 H

@x3 + · · · (27)

@H

@t +

c2

g

@V

@x = n

@2 H

@x2 + n

@3V

@x3 + · · · (28)

The approximation of Equations (17) and (18) by the FEM modies them to the systemof dierential equations with an innite number of terms. It is well known that for smoothfunction the total value of the right-hand side is dominated by its rst term. It is known aswell that in the modied equation, the even-order derivatives are associated with dissipationwhereas the odd-order ones—with dispersion. Consequently, the type of error that dominates

the numerical solution is determined by the rst term of the right-hand side of the modiedequation.

In Equations (27) and (28) vn is the coecient of numerical diusion given by expression

n =− 1

2

c xC r (29)

whereas n and

n are the coecients of numerical dispersion dened as follows:

n = g x2

6 ((2 − 3!) − (2 − 3)C r ) (30)

n = c2 x2

6g ((2 − 3!) − (2 − 3)C r ) (31)

The modied equations can be used to demonstrate the consistency and the order of accuracyof the applied method and to deduce practically all information on its numerical properties.

From relation (29) the following results:

• FEM does not generate any numerical diusion for = 12

, i.e. when it represents anapproximation of the second order with regard to t . It can be shown, that the terms of



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0 2 4 6 8 10 12 14 16 18 20

N

0

0.2

0.4

0.6

0.8

1

Rω = 1

ω = 2/3

ω from (32)

Figure 4. Phase portrait for dierent !.

higher order than 2, which have been omitted here, disappear for C r = 1. In this case themethod ensures an exact solution, whereas for C r = 1 unphysical oscillations can appear in the solution because of dispersivity (n =0 and n;

n =0).• FEM is unstable for ¡ 1

2. In this case vn¡0. It means that the initial-value problem for

the system of Equations (27) and (28) is ill-posed and consequently its solution doesnot exist.

• FEM ensures a stable solution for ¿ 12

, because with vn¿0 the initial-boundary problemfor Equations (27) and (28) is well posed and its solution exists always. However it contains a dissipation error, because in this case the method produces a numericaldiusion. It increases with the increase in x and C r giving rise to smooth solution andreduction of the pressure gradients.

The FEM is of second order of accuracy with regard to x as well as with regard to timet for = 1

2 only. For another value of it is of rst order of accuracy with regard to t .

However, in this case, the order of accuracy can be increased. Namely, for = 12

and with

! = 2

3 −

C 2r 6

(32)

the terms of second and third order in the system of modied Equations (27) and (28) arecancelled. It means that in this case the proposed version of FEM ensures an accuracy of thirdorder with regard to x as well as to t . This holds for C r 61 only because of the condition of stability (26).

The advantages of the proposed approach are illustrated in Figures 4 and 5 where the so-

called phase portraits are presented. They are represented by the convergence coecient R( N ) being the ratio of numerical and exact wave celerities. The diagrams presented in Figure 4 for various ! indicate that the presented modication of FEM signicantly improves its numerical

properties. Namely, one can notice that for ! dened by Equation (32), the wave speed isaected only for the shortest waves. There is remarkable dierence on comparing with != 2

3

(standard FEM) and != 1 (the nite dierence method). Consequently, the proposed methodis able to ensure better accuracy of the solution especially when the strong gradients occur.



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0 2 4 6 8 10 12 14 16 18 20

N

0

0.2

0.4

0.6

0.8

1

C = 1

C = 0.75

C = 0.25

r

r

r

R

Figure 5. Phase portrait for dierent Courant numbers.

90

80

70

60

50

40

30

20

10

0

0 1 2 3 4 5 6 7 8 9

t [s]

∆ Θx = 5 m, C = 1, = 0.5

∆ Θx = 5 m, C = 2, = 1.0

∆x Θ= 10 m, C = 2, = 1.0

r

r

r

H [ m ]

Figure 6. Computed head oscillations by the modied FEM for simplehypothetical pipeline at downstream end.

The remarks presented above were conrmed by numerical tests carried out for the hypo-thetical pipeline. In the pipeline of length L = 500m, the initial velocity was V 0 = 0:4 m= s. The

piezometric head at reservoirs was H r = 50 m = const, whereas the wave speed was 1000m= s.The valve was closed immediately. The calculated pressures at the end of the pipeline with!= 1

2 for dierent values of , C r and x are presented in Figure 6. For =

12

and C r = 1,an exact solution of the linear water hammer equations was obtained. For increasing valueof , C r and x the function H (t ) becomes more and more smooth and simultaneously itsamplitude is strongly damped. The only reason for this process is the numerical diusiongenerated by the applied method.



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4. PHYSICAL MODEL AND EXPERIMENTAL DATA

In order to obtain reliable experimental data, an installation for physical experiment was used.To eliminate the interaction of various factors inuencing the water hammer phenomenon, the

simplest cases were studied.An experimental apparatus for investigating the water hammer events in the pipelines is

shown in Figure 7.It is composed of a straight pipeline (1), pressurized tank (2) and a ball valve (4) mounted

at the end of the pipe. The valve was closed manually. The valve closure time T c, measuredto a thousandth of a second, was from 0.018 to 0 :025 s in all tests. The measurements andfurther analysis of pressure head characteristics H (t ) referred to a simple water hammer inwhich the wave reection time (water hammer period) T was always greater than the valueof closure time.

The pressure was recorded by means of a measuring system consisting of strain gauges(5), extensometer amplier (6) and a computer (7) with AD= DA (20 MHz) card. The signalfrom the transducer was sampled with a frequency of 2000 Hz. The gauges, with measuring

ranges up to 1.2 and 2 MPa (with measurement uncertainty ±0:5%), had linear performancecharacteristics in the whole range of the measured pressure with a correlation coecientnot less than 0.999. The pressure transducers are located at two points along the pipeline(Figure 7).

The pressure p(t ) was recorded during the process of the water hammer for dierentvelocities V 0 of steady water ow (varied from 0.3 to 2 m= s in the experiments). Thevelocity V 0 in the pipeline was measured by the volumetric method (with measurementuncertainty ±1%).

The tested pipes (1) were fed from a reservoir (2), which itself was lled from the pipenetwork. The water pressure p s at the installation was constant (p s = 0:65±0:01 MPa). The

pressure was reduced by means of a pressure-reducing valve (3). The stream pressure valuep0 at the pipe outlet and the water ow velocity V 0 were adjusted in order to avoid col-umn separation events at the water hammer [16–18]. After an initial steady state had beenestablished, a transient event was initiated by a rapid valve closure.

The experiments were carried out for various pipelines having various parameters as length,diameter, initial velocity, etc. In this paper, we would focus attention on the simplest case of the water hammer, namely in steel pipeline having constant diameter only. The characteristics

QP

9

oL1 D, e

Vo

Ps

3 2

5

6 7

5 4

8

Figure 7. Experimental installation.



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Table I. Steel pipe characteristics.

Parameter Value

Length L (m) 72.0

Inside diameter D (m) 0.042Thickness of pipe wall e (m) 0.003Roughness height k (m) 0.00008Wave’s velocity c (m= s) 1245.0Time of closure T c (s) 0.021Initial pressure head H 0 (m) 51.0Initial velocity V 0 (m= s) 0.408

H

[ m ]

t [s]

0 1 2 3 4 5 6 7 8

120

100

80

60

40

20

0

Figure 8. Observed head oscillations for single steel pipeline at its downstream end.

of this pipeline are described in detail in Table I. The experiments carried out for numerouscases allowed us to record 10 sets of data representing the head oscillations H (t ) at thedownstream end and at the mid-length of pipe. To check the obtained results and to eliminatethe accidental errors, the experiments were repeated many times.

The typical graph of H (t ) is presented in Figure 8 as an example of acquired data. For each experiment, a strong damping of pressure wave was observed. Very quickly, after severalseconds, the amplitudes became insignicant and the hydrostatic state was reached. Simulta-neously the wave front became smoother and consequently the function H (t ) lost its initiallysharp form.

5. NUMERICAL TESTS AND DISCUSSION OF OBTAINED RESULTS

The calculations of the water hammer phenomenon were carried out by the proposed method,with the friction factor given by Colebrook–White formula. The results obtained for the setof data that minimized the numerical diusion ( x = 0:5 m, t = 0:00025 s, C r = 0:62) are



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presented in Figure 9(a). Taking into account the applied values of x and C r , one canevaluate the strength of the introduced numerical diusion. To obtain the results presentedin Figure 9(a), an insignicant numerical diusion n = 3:86 m

2= s was introduced. The goalof this introduction was to eliminate the unphysical oscillations resulting from non-linearity

of the Equations (1) and (2). One can notice an essential disagreement with the observed pressure oscillations. From experimental results it is shown that the oscillations are relativelyquickly damped, and that after 8 s they practically disappear. Conversely, in the results of computation, such strong damping is not observed.

It is well known that for the friction factor given by the Colebrook–White formula, thecoincidence is observed at the beginning of the water hammer phenomenon, only for the rstcycle of pressure wave. Then the experimental data and calculations dier more and morewith time.

Generally, the observed damping of pressure wave amplitude is more intensive comparedwith the calculated one. This disagreement is considered as a result of an inadequate estimationof the shear stress for unsteady ow using the equation for the steady state [19–22]. Itis accepted commonly that the steady-state equation for friction is not able to reproduceaccurately the shear stress during transient ow [23]. For this reason, the eort of manyauthors is oriented to improve the evaluation of the shear stress for unsteady pipe ow. For example, an improvement of equation for friction has been proposed by Zielke [22], Vardyand Hwang [24], Brunone et al. [25], Pezzinga [21], Axworthy et al. [19], Pezzinga [21] andothers as well. Usually, the proposed improvement deals with modication of the equation for friction. However, the algebraic term representing the friction process in Equation (1) is notable to smooth the oscillations. Therefore, it seems that an improvement of the water hammer equations cannot be focused on the modication of the friction term only. As long as thewater hammer equations preserve the hyperbolic character, they will be unable to smooth thehead oscillations.

The numerical experiment conrms the remarks presented above. For the set of parameters

that minimized the numerical errors, the friction factor f resulting from Colebrook–Whiteformula was multiplied by constant value K . Therefore, in Equation (1), a modied frictionfactor equal to K ·f was used. This approach being the simplest way of improvement of thefriction factor is considered here only to show the results of such idea. Usually, the proposedimprovements have more sophisticated form. Obviously, one can expect that the damping of head oscillations increases with increase in K . The results achieved with K = 10 are presentedin Figure 9(b). As one can notice, the head oscillations are damped better but without anysmoothing. Consequently, the calculated function H (t ) keeping its sharp form diers from theobserved one.

The observed and computed pressure oscillations can be well matched by introducing nu-merical diusion. As it was shown earlier, the numerical diusion produced by FEM increaseswith the increase in x, C r and . Very good agreement was obtained for ! = 0:525, = 0:850,

x = 3:6 m and C r = 0:865. For these data, the coecient of numerical diusion is equal tovn = 1357 m2= s. In Figure 9(c), the results of computations are compared with the ones pro-

vided by physical experiment. Note, that the numerical diusion introduced into the solutionensures the damping of amplitude of pressure wave and its smoothing simultaneously. Onecan add that the numerical diusion is able to ensure an excellent agreement even for the fric-tion factor f = 0. These results allow us to deduce the nature of desired physical mechanismin the water hammer equations.



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0 1 2 3 4 5 6 7 8

t [s]

0

20

40

60

80

100

120

0 1 2 3 4 5 6 7 8

0

20

40

60

80

100

120

0 1 2 3 4 5 6 7 8

0

20

40

60

80

100

120

H [ m ]

H [ m ]

H [ m ]

(a)

t [s](b)

t [s](c)

Figure 9. Computed and observed head oscillations for single steel pipeline at downstream end: (a) for friction factor f by Colebrook–White formula; (b) for friction factor f multiplied by K =10; (c) for

friction factor f by Colebrook–White formula with numerical diusion.



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It seems that, apart from friction expressed in Equation (1) by algebraic terms another factor takes part in the process of wave damping as well. As it was shown the friction termis able to reduce the wave amplitude only. An excellent adjustment ensured by numericaldissipation suggests that the lacking process should be expressed by the diusive term that is

able to ensure such smoothing.As far as an improvement of the friction factor f is considered it should be remembered

that the required damping can be achieved by an interaction of the numerical diusion and theshear stress. Therefore, it is possible to obtain a satisfying agreement between the experimentaldata and the calculations for their various combinations. Since the quality of obtained solutionis determined by the physical process of friction as well as by the numerical error, it is of essential importance to separate both eects. For this reason, to evaluate the real meaning of any introduced improvement of the equation for the shear stress, the numerical errors should

be evaluated. A satisfying agreement of the experimental data and calculations cannot beconsidered as a sucient proof as long as the numerical and physical eects in the improvedsolution are not separated. In the other case, the conclusions may be misleading. This seemsto be a very important problem while analysing unsteady pipe ow.

6. CONCLUSIONS

The nite element method in the version obtained by modifying the standard approach appliedin the Galerkin procedure was proposed to solve the unsteady pipe ow equations. Finallya six-point implicit scheme for xed mesh with two weighting parameters was obtained. Itensures an approximation of third order with regard to x and to t for C r 61. Consequentlythe numerical solution is more accurate compared with the one given by standard version of the nite element and dierence method.

The results of solution of the water hammer equations by the proposed method compared

with the experimental data showed an essential inuence of the numerical factors such asthe time step t , the space interval x and the weighting parameters on the quality of theresults. The nature of this inuence was explained by the accuracy analysis carried out usingthe modied equations approach. From the point of view of the theory of the numericalmethods it seems to be a paradox that the highly accurate methods give the results thatdisagree with measurements. Conversely, large numerical diusion produced by the methods,

being a poor approximation, ensures satisfying agreement. Since in the water hammer process,an important role is played by the mechanism of physical dissipation, which is not representedin Equations (1) and (2), therefore, in practice, to obtain a satisfying adjustment, the lackingeect of this mechanism is reproduced by numerical dissipation. It seems the water hammer equations should contain another additional mechanism of physical dissipation, which would

be able to ensure both eects simultaneously–damping and smoothing of the pressure wave.

REFERENCES

1. Parmakian J. Water Hammer Analysis. Prentice-Hall, Inc: New York, 1955.2. Streeter VL, Lai Ch. Water hammer analysis including uid friction. Journal of Hydraulics Division (ASCE)

1962; 88(HY3):79–111.3. Evangelisti G. Water hammer analysis by the method of characteristics. L’Energia Elettrica 1969; 46(10):

673–692.



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4. Almeida AB, Koelle E. Fluid Transients in Pipe Networks. Elsevier Applied Science: London, 1992.5. Goldberg DE, Wylie EB. Characteristics method using time-line interpolations. Journal of Hydraulic

Engineering (ASCE) 1983; 109(5):670–683.6. Wylie EB, Streeter VL. Fluid Transient in Systems. Prentice-Hall, Inc.: Englewood Clis, New York, 1993.7. Abbott MB, Basco DR. Computational Fluid Dynamics. Longman Scientic and Technical: Essex, UK, 1989.

8. Sibetheros IA, Holley ER, Branski JM. Spline interpolation for water hammer analysis. Journal of HydraulicEngineering (ASCE) 1991; 117(10):1332–1351.

9. Chaudhry MH, Hussaini MY. Second order accurate explicit nite-dierence schemes for water hammer analysis.Journal of Fluid Engineering 1985; 107(4):523–529.

10. Fletcher CA. Computational Techniques for Fluid Mechanics, vol. 1. Springer: Berlin, Germany, 1991.11. Leonard BP, Drummond JE. Why you should not use ‘hybrid’, ‘power-law’ or related exponential schemes

for convective modelling—there are much better alternatives. International Journal for Numerical Methods inFluids 1995; 20:421–442.

12. Szymkiewicz R. Method to solve 1D unsteady transport and ow equations. Journal of Hydraulics Engineering(ASCE) 1995; 121(5):396–403.

13. Warming RF, Hyett BJ. The modied equation approach to the stability and accuracy analysis of nite-dierencemethods. Journal of Computational Physics 1974; 14:159–179.

14. Zienkiewicz OC. The Finite Element Method in Engineering Science. McGraw-Hill: London, 1972.15. Potter D. Computational Physics. Wiley: London, 1973.16. Bergant A, Simpson AR. Pipeline column separation ow regime. Journal of Hydraulic Engineering (ASCE)

1999; 125(8):835–848.

17. Mitosek M. Study of cavitation due to water hammer in plastic pipes. Plastics, Rubber Composites Processingand Application 1997; 26(7):324–329.

18. Mitosek M. Study of transient vapour cavitation in series pipe systems. Journal of Hydraulic Engineering(ASCE) 2000; 126(12):904–911.

19. Axworthy DH, Ghidaoui MS, McInnis DA. Extended thermodynamics derivation of energy dissipation inunsteady pipe ow. Journal of Hydraulic Engineering (ASCE) 2000; 12(4):276–287.

20. Pezzinga G. Quasi-2D model for unsteady ow in pipe networks. Journal of Hydraulic Engineering (ASCE)1999; 125(7):676–685.

21. Pezzinga G. Evaluation of unsteady ow resistances by quasi—2D or 1D models. Journal of HydraulicEngineering (ASCE) 2000; 126(10):778–785.

22. Zielke W. Frequency dependent friction in transient pipe ow. Journal of Basic Engineering (ASME) 1968;90(1):109–115.

23. Elansary AS, Silva W, Chaudhry MH. Numerical and experimental investigation of transient pipe ow. Journal of Hydraulic Research 1994; 32(5):689–706.

24. Vardy AE, Hwang K. A characteristic model of transient friction. Journal of Hydraulic Research 1991;229(5):669–684.

25. Brunone B, Golia UM, Greco M. Some remarks on the momentum equation for fast transients. Proceedings of the International Meeting on Hydraulic Transients and Water Column Separation. 9th Round Table, IAHR:Valencia, Spain, 1991; pp. 140–148.


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Analysis of Unsteady Pipe Flow Using Modified FEM

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