5. IJMCAR - A Numerical and Analytical Approach for Solvin

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A NUMERICAL AND ANALYTICAL APPROACH FOR SOLVING

NONLINEAR WATER HAMMER EQUATIONS

JAIPAL 1, RAKESH C. BHADULA 2 & V. N. KALA 3 1 Associate Professor, Department of Mathematics, D.B.S. (P.G.) College Dehradun, Uttarakhand, India

2Department of Mathematics Uttaranchal University, Dehradun, Uttarakhand, India 3Department of Applied Science, G. B Pant Engineering College, Garhwal, Uttarakhand, India

ABSTRACT

The nonlinear equations of motions and equation of continuity for water hammer problem in the pipe line

systems are solved numerically by applying explicit central difference method and an attempt is made to solve these

nonlinear equations analytically with suitable initial and boundary conditions. The dependence of flow and pressure

head in pipe on the valve closure time at different points in the pipe is discussed. The negative flow and very high

pressure head is observed at some points in the pipe which is due to upstream wave during sudden closer of valve.

KEYWORDS: Equation of Continuity for Water, To Upstream Wave

Received: Nov 04, 2015; Accepted: Nov 16, 2015; Published: Nov 21, 2015; Paper Id.: IJMCARDEC20155

INTRODUCTION

Water hammer is high pressure wave generated by a sudden changes of velocity in closed pipe line. It has

been studied for many decades but till today there is not efficient method available to predict the exact location of

water hammer in pipe lines. Masashi and syuuzi (1984) solve equation using series solution method and compared

the result with finite difference method. Brunone et al. (1995) developed 2-D model and considered Rapid

damping of the pressure peaks after the end of a complete closure maneuver, are closely linked to the shape of the

cross-sectional velocity distributions and their Variability in time. Yao et al. (2014) described the attenuation of

water hammer pressure wave with time varying valve closure by using an asymptotic analysis. They examined the

effect of flow reversal on the pressure wave attenuation through comparison with a similar method applied to the

water hammer generated during flow establishment the flow reversal Ghidaoui (2004) reviewed the relation

between straight equation and speeds in single as well as two phase transient flow by using order of magnitude

analysis. Ghidaui et al. (2001) performed linear stability analysis of base flow velocity profiles for laminar and

turbulent water hammer flows. The base flow velocity profile are determined analytically. Where the transient is

generated by instantaneous reduction in flow rate at the downstream end of a single pipe system .The presence of

inflection points in the base flow velocity profile and the large velocity gradient near the pipe wall are the sources

of flow instability. Delgodo et al. (2014) discussed the uncertainties associated with the hydraulic transient

modeling of raising pipe system with and without surge protection.They developed a one dimensional hydraulic

transient model based on the classic water hammer theory and solved by the method of characteristics (MOC). The

transient test with and without the hydropnuematic vessel connected to the system for different flow rates were

observed by them. Hariri Asli et al.(2012) defined an eulerian based computational model compared with

Original A

rticle International Journal of Mathematics and Computer Applications Research (IJMCAR) ISSN(P): 2249-6955; ISSN(E): 2249-8060 Vol. 5, Issue 6, Dec 2015, 45-56 © TJPRC Pvt. Ltd

46 Jaipal, Rakesh C. Bhadula & V. N. Kala

Impact Factor (JCC): 4.6257 NAAS Rating: 3.80

regression of the relationship between the dependent and in dependent variables for water hammer surge wave in pipe.

Riedelmeier et al. (2014) constructed the analysis of effects of fluid-structure interaction (FSI) during water hammer in

piping systems, attest facility and showed resonance experiments on movable bends in two piping system configuration

focused on junction coupling were carried out. Ronghe Wang et al. (2014) derived the hydraulic calculating equation based

on the method of characteristics by considering the pipe head loss and node cavitations .Edwards et. Al.(2014) presented

his method to asses transient modeling and analysis errors which occurs due to demand aggregating and uncertainty. They

investigated the effect of aggregating demand from the small network of residues and found to be most significant when

the number and distribution of the demands was randomly varied. G Pezzinga et al. (2014) showed 2-D features of

viscoelastic model of pipe transients and given an innovative approach to analyze transients in a pressurized polymeric

pipe. In fact on one side the proposed 2-D Kelvin-Voigt model calibrated by using the pressure traces has been checked by

considering the instantaneous profiles of the axial component of the local velocity measured by means of an ultrasonic

Doppler velocitimeter on the other hand he made comparisons with results given by a 2-D model in which an elastic

behavior has been assumed for pipe material. I.A.Sibetheros et al.(1991) investigated the method of characteristics with

spline polynomials for interpolations for numerical water hammer analysis for a frictionless horizontal pipe. Chyr Pyng

Liou et al.(2014) presented a method which improve the accuracy of the approximation commonly used with the friction

term in the water hammer equation. Alexandre Kepler Soares et al. (2008) investigated on the analysis of hydraulic

transient in polyvinyl chloride (PVC) pipe line. They emphasized the importance of pipe wall viscoelasticity in hydraulic

transients in PVC pipeline. Simpson (1986) showed a range of short duration pressure pulses measured in a reservoir

upward sloping pipeline valve system. Due to upward slope of the pipe vapor cavity was confined to be adjacent to the

valve with no distributed cavitations along the pipe.Don J.Wood (2005) presented both MOC and the WCM to slope the

transient flow problem. He stated that the MOC and WCM are both capable of accurately solving for transient pressures

and flow in the water pipeline networks including the effect of pipe friction. Anton Bergant et al.(2008) showed that

describe unsteady friction, cavitations including Column Separation and trapped air pockets, fluid structure interaction,

pipe wall viscoelasticity and leakages and blockages in transient pipe flow. Their models are based on method of

characteristics. David E. Goldberg (1983) presented solution of broad cross section of time varying wave problems in

hydraulics, MOC based methods are likely to continue to see extension use. They showed extent of potential application of

the implicit time-line method also deserves more attention. Gottlieb et al. (1981) investigated numerical model with

experimental result. They considered four different configurations of a steel and a plastic pipeline. Extremely high-pressure

peaks were recorded immediately upon collapse of the vapor cavity. The pressure then dropped to about 40% of the

pressure peak level and maintained this level for twice of l/a seconds. They showed the presence of peaks resembled the

pressure peaks associated with the implosion of gas bubbles in pumps. Martin (1983) presented transient cavitations in as

reservoir-pipe-valve system. The water contained a minimal amount of dissolved gas. They emphasized on limited

cavitations. They showed in experimental results that the maximum pressure may exceed the Joukowsky pressure rise in

the form of short duration pressure pulse. They observed that the reservoir pressure was rising during the experiment

because the tank was to small.Kim et al. (2015) predicted that water hammer following the tripping of pumps lead to over

pressure and negative pressure. Chen et al. (2015) used the method of lines to obtain a finite dimensional ordinary

differential equations model based on the original partial differential equations system for the problem of mitigating water

hammer during valve closure. They concluded that simulation results demonstrating the capability of optimal boundary

control to reduce flow fluctuation. An analytical solution of the nonlinear water hammer equations is still awaited. In this

paper we have solved nonlinear water hammer equations numerically as well as analytically. However we have made some

A Numerical and Analytical Approach for Solving 47 Nonlinear Water Hammer Equations


assumptions to obtain the analytical solution of nonlinear water hammer equations without deviating the fundamental

natures of the model.

MATHEMATICAL MODEL

The unsteady state equation of continuity and equation of motion together with convective terms and nonlinear

friction term can be given as-

2

sin 0H H a V

V Vt x g x

θ∂ ∂ ∂+ − + =∂ ∂ ∂

(1)

sin 02

Q Q H fQ gA gA Q Q

t x t DAθ∂ ∂ ∂+ + + + =

∂ ∂ ∂ (2)

Neglecting convective terms in equation (1) and taking θ =0 for horizontal pipe then continuity equation becomes

2H QgA a

t x

∂ ∂= −∂ ∂

(3)

and equation of motion becomes

02

Q H fgA Q Q

t t DA

∂ ∂+ + =∂ ∂

(4)

On differentiating partially equation (3) with respect to x and equation (4) with respect to t partially respectively

and then subtracting them, we get

2 22

2 20

2

QQ f Q QQ Q a

t DA t t x

∂∂ ∂ ∂+ + − = ∂ ∂ ∂ ∂ (5)

Applying explicit central difference method to solve equation (5), putting

1 12

2 2

2k k ki i iQ Q QQ

t t

+ −− +∂ =∂ ∆

(6.1)

21 1

2 2

2k k ki i iQ Q QQ

x x+ −− +∂ =

∂ ∆ (6.2)

1 1

2

k ki iQ QQ

t t

+ −−∂ =∂ ∆

(6.3)

in equation (5) we get



( )1 11 1 1 1

2

2 1 12

2

2 2

20

k k kk k k k ki i iki i i i i

i

k k ki i i

Q Q QQ Q Q Q Qc Q

t t t

Q Q Qa

x

+ −+ − + −

+ −

− − + − + + ∆ ∆ ∆

− +− = ∆

(7)

( ) ( )( )

1 1 1 1 1 11

1 1

2

2 0

k k k k k k k k ki i i i i i i i i

k k ki i i

Q Q Q r Q Q Q Q Q Q

r Q Q Q

+ − + − + −

+ −

− + + − + −

− − + = (8)

Where

2a t

rx

∆ = ∆ , 1 4

c tr

∆= (9)

The initial conditions are

( ) 0,0Q x Q= or ( ) 0,0iQ x Q= (initial flow in the pipe) (10.i)

0

0t

Q

t =

∂ = ∂ (for steady flow) (10.ii)

The boundary conditions are

( ) 00,Q t Q= (11.i)

( ) 0, 1c

tQ L t Q

t

= −

where ct is valve closure time (11.ii)

0( ,0)H x H= , 0

0t

H

t =

∂ = ∂ (11.iii)

Similarly the equation of continuity becomes

( )1 12 1 1

k k k ki i i iH H r Q Q+ −

+ −= − − (12)

Where 2

2

a tr

gA x

− ∆=∆

Analytical Solution for Water Hammer Equations-

Using Q Q Q

t Q t

∂ ∂=∂ ∂

in equation (5), we get

2 22

12 2

Q Q Qa Q a

t t x

∂ ∂ ∂+ =∂ ∂ ∂

(13)



where 1

fa

DA=

Transformation of coordinates

Taking coordinates transformation as

1a tτ = , 1a xz

a= ; (14)

then equation (13) is reduced to

2 2

2 2

Q Q QQ

zτ τ∂ ∂ ∂+ =∂ ∂ ∂

(15)

Following Polyanin and Zaitsev 2004, the analytical solution of equation (15) is obtained and given by

1 11 22

1

( )2 tanh( )

1

A z AQ c c

A

τ−= ± +−

(16)

Putting equation (16) in equation (3) and solving, we get

1 1 12 32

1 1

2 ( )tanh( )

1

ac A z AH c c

gAA A

τ−= ± + +−

(17)

where 3c is arbitrary constant.

The values of 1c , 2c , 3c and 1A are obtained using initial and boundary conditions.

RESULTS AND DISCUSSIONS

The numerical values for flow Q and pressure head H are obtained from the difference equations (8) and (12)

respectively with the help of MATLAB and are shown graphically from figure 1 to figure 7. In figure 1 the variation of

water flow in the pipe together with axial distance x and time t is shown. Valve closure time is taken 0.1 sec. The another

sample parameter are taken as wave velocity c=1000 m/s, pipe radius r is 0.3 m, initial pressure head 0H =600 m and

initial flow 0Q =1.4 3 / secm .



0

0.5

1

0

0.5

1-20

-15

-10

-5

0

5

t/T

Variation of flow pattern in pipe

x/L

-18

-16

-14

-12

-10

-8

-6

-4

-2

0

2

Figure 1: Variation of Flow in Pipe with x and t

The variation of flow with time and distance can be clearly observed form figure 1. Initially the variation of flow

with x is very small and at t = 0.1 s and x = 100 it becomes zero but it changes rapidly as time increases. The value of

flow at some middle points in the pipe is larger than the initial value 0Q . This is feasible since as the valve is going to close

then there is some back flow causing greater values of Q at mixing points. As time exceeds the value of flow rate

increases in the upstream direction (i.e. negative value of Q ) which is true since the valve is closed and there is no chance

for downstream flow.

0

0.5

1

0

0.5

1-5

0

5

10

15

t/T

Variation of pressure head in pipe due to closure of valve

x/L

0

2

4

6

8

10

12

Figure 2: Variation of Pressure Head In Pipe with x And t

The variation of pressure head with respect to time and axial distance is shown in figure 2. The pressure head in

the pipe increases as time increases. We have observed that at ct t= there is negative transient pressure head which may

cause cavitations or column separation. After that transient pressure head increases as time increases and is maximum at

1t

T= i.e. 2t T s= = . We observed maxH = 7800 m .



0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1-0.7

-0.6

-0.5

-0.4

-0.3

-0.2

-0.1

0

0.1 Flow pattern when valve is fully closed

Non

dim

ensi

onal

flo

w in

pip

e

Nondimensional distance x along pipe length

Figure 3: Variation of Flow with x at ct t=

The flow pattern in pipe when the valve is fully closed is shown in figure 3. Clearly when the valve is fully closed

at the end of pipe the flow is 0 while in the starting it is slightly greater than the initial flow due to upstream wave and after

that the flow is negative and nonlinear.

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1-1

-0.8

-0.6

-0.4

-0.2

0

0.2

0.4

0.6

0.8

1

Nondimensional time

Non

dim

ensi

onal

flo

w

Flow pattern at a fixed point x at different time

Figure 4: Variation of Flow with t at x =0.9

To study the flow pattern at a fixed position with time, figure 4 is plotted at x =0.9. It is clear from the figure that

flow decreases first and then it increases for a very short time and then it continuously decreases with a slight fluctuation.

After some time when valve is about half closed flow becomes negative.

0 0.2 0.4 0.6 0.8 1-2

0

2

4

6

8

10Pressure when valve is partially closed

Non

dim

ensi

on

al P

ress

ure

hea

d

Nondimensional distance x

Figure 5: Variation of Pressure with x at some Fixed Time



The variation of pressure in the pipe in axial direction is shown in figure 5. Initially the pressure head decreases

and reaches to a negative value and after that it increases continuously and become constant near the valve.

0 0.2 0.4 0.6 0.8 10

2

4

6

8

10

12

14X: 1Y: 13

Pressure head at valve with different time

Nondimensional time

No

nd

imen

sio

nal

pre

ssu

re h

ead

Figure 6: Variation of Pressure with t at Fixed Point x

The negative value of transient pressure head my cause cavitations or column separation. In figure 6 the variation

of pressure head at valve i.e. at L with time is shown. Initially for a very short time the pressure appear to be constant after

that it increases very fast as time increases. The initial pressure head in the pipe was 600 m while it is more than 7800 m

at 2t s= ,where the valve closure time is 0.1 s . This very high pressure head i.e. more than 13 times the initial head may

cause bursting of pipe.

The comparison of pressure head obtain by numerical method and analytical method for nonlinear model is shown

in figure 7.

0 20 40 60 80 1000

1000

2000

3000

4000

5000

6000

7000

8000Comparision of pressure head by numerical and analytical methods

Axial distance

P

ress

ure

hea

d

Figure 7: Comparison of Pressure Head by Numerical and Analytical Method



0

50

100

00.5

11.5

2-60

-40

-20

0

20

x

Variation of flow in the pipe due to closure of valve

t -45

-40

-35

-30

-25

-20

-15

-10

-5

0

Figure 8: Variation of Flow in the Pipe with Axial Distance and Time

0

50

100

00.5

11.5

20

2000

4000

6000

8000

x

Variation of pressure head in the pipe due to closure of valve

t

1000

2000

3000

4000

5000

6000

7000

Figure 9: Variation of pressure due to closure of valve

Analytical solutions of the nonlinear model for water hammer problem are shown by the equations (16) and (17).

In the above solution there is restriction that 1 1A ≠ ± ,0. To draw the graph for the analytical solution we have taken 1A =

2.5 and result are shown from figures 7 to 9. The variation of flow in the pipe with x and t is shown in figure 8. We

observed that flow is positive before complete closure of valve while it becomes negative after fully closing of valve which

is good agreement with the practical situation. The variation of pressure head in the pipe with time and axial distance is

shown in figure 9. It is clear from the figure that pressure head increases as time increases and it is maximum at the valve.

CONCLUSIONS

Numerical as well as analytical approach to solve water hammer equations are given here by applying suitable

initial and boundary conditions. The results obtained from both these methods are in good agreement with practical

situations. The model can be used to find the location of maximum pressure head in the pipe so that necessary treatment



can be made prior to the damage of pipe line due to water hammer.

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5. IJMCAR - A Numerical and Analytical Approach for Solvin