Comparison of MOC and Lax FDE for simulating transients in Pipe Flows

International Research Journal of Engineering and Technology (IRJET) e-ISSN: 2395 -0056
Volume: 04 Issue: 03 | Mar -2017 www.irjet.net p-ISSN: 2395-0072
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Comparison of MOC and Lax FDE for simulating transients in Pipe
Flows
2
1 PhD Research Scholar, Deptt. of Civil Engineering, Assam Down Town University, Guwahati, Assam, India.
2 Professor, Deptt. of Civil Engineering, Assam Down Town University, Guwahati, Assam, India
---------------------------------------------------------------------***--------------------------------------------------------------------
Abstract -The method of characteristic (MOC) approach
transforms the water hammer partial differential
equations into ordinary differential equations along
characteristic lines. The fixed-grid MOC is the most
accepted procedure for solving the water hammer
equations and has the attributes of being simple to code,
efficient, accurate and provides the analysts with full
control over the grid selection. Some authors are of the
opinion that Lax Finite Difference Explicit method
provides more convincing results for solving unsteady
transient situations in pipe flow. Here an approach is
made to compare the MOC and Lax FDE scheme of
discretizationforhydraulic transientgoverningequation,
with the help of MATLAB as the programming tool and
finally Lax FDE scheme is observed to be more effective.
Key Words: Hydraulic pipe transients,waterhammer,valve,
numerical model, discharge, velocity, pressure etc.
1. INTRODUCTION:
The variation in discharge and pressure head can be studied
by solving the governing equations for hydraulic transients
in a pipe using the Method of Characteristics for
discretization of the partial differential equationsandalso by
Lax Finite Difference Explicit method. Due to the non-
linearity of the governing equations, various numerical
approaches have been developed for pipeline transient
calculations, which include the Method of Characteristics
(MOC), Finite Difference (FD) and Finite Volume (FV) etc.
Among these methods, MOC proved to be the most popular
among the water hammer analysts. In fact, out of the 14
commercially available water hammer software packages
found on the world wide web, 11 are based on MOC, two are
based on implicit FD method [11]. After theFiniteDifference
Equations (FDE) are obtained, the numerical models are
developed using MATLAB. The models are then validated
using lab data. Chudhury M.H. [13] advocated comparative
effectiveness of Lax FDE method over MOC, which has been
analyzed and observed here.
2. Literature Review:
The basic unsteady flow equations along pipe due to closing
of the valve near the turbine are non-linear and hence its
analytical solution is not possible. Watt C.S.et al (1980)[1]
have solved for rise of pressure by MOC for only 1.2 seconds
and the transient friction values have not been considered.
Goldberg D.E. and Wylie B.(1983)[2] used theinterpolations
in time, rather than the more widely used spatial
interpolations, demonstrates several benefits in the
application of the method of characteristics (MOC) to wave
problems in hydraulics. Chudhury M.H. and Hussaini
M.Y.(1985)[3] solved the water hammer equations by
MacCormack, Lambda, and Gabutti explicit FD schemes.
Sibetheros I. A. et al. (1991) [4] investigated the method of
characteristics (MOC) with spline polynomials for
interpolations requiredinnumerical waterhammeranalysis
for a frictionless horizontal pipe. Silva-Arya W.F.and
Choudhury M.H.(1997)[5] solved the hyperbolic part of the
governing equation by MoC in one dimensional formandthe
parabolic part of the equation by FD in quasi-two-
dimensional form. Pezzinga G. (1999) [6] presented both
quasi 2-D and 1-D unsteady flow analysis in pipe and pipe
networks using finite difference implicitscheme. Pezzinga G.
(2000) [7] also worked to evaluate the unsteady flow
resistance by MoC. He used Darcy-Weisback formula for
friction and solved for head oscillationsupto4secondsonly.
Damping with constant friction factor is presented but not
much pronounced, as the solution time was very small.
Bergant A. et al (2001) [8] incorporated two unsteady
friction models proposed by Zielke W. (1968) [9] and
Brunone B. et al.(1991)[10] into MOC water hammer
analysis. Zhao M. and Ghidaoui M.S. (2004)[11] formulated,
applied and analyzed first and second –order explicit finite
volume (FV) Godunov-type schemes for water hammer
problems. They have compared both the FV schemes with
MoC considering space line interpolation for threetestcases
with and without friction for Courant numbers 1,
0.5.0.1.They modeled the wall friction using the formula of
Brunone B. et al (1991) [10]. It has been found that the First
order FV Gadunov scheme produces identical results with
MoC considering space line interpolation. They advocated
that although different approaches such as FV, MOC, FD and
finite element (FE) provide an entirely different framework
for conceptualizing and representing the physicsoftheflow,
the schemes that result from different approaches can be
similar and even identical. BarrD.I.H.(1980)[12]formulated
Er. Milanjit Bhattacharyya1, Dr. Mimi Das Saikia

friction losses, while Chudhury M.H. (1994)[13] advocated
comparative effectiveness of Lax FDE method over MOC.
Saikia M.D. and Sarma A.K.. (2006)[14] also compared Lax
FDE method in their approach and found compatibleresults.
3. GOVERNING EQUATION :
The basic equations of continuity and momentum in
unsteady flow along pipe due to closing of the valve near the
turbine may be written as:
Continuity:
0
2

x
Q
gA
a
t
H




………………..(1)
Momentum:
0
2
1
2
 QQ
gDA
f
t
Q
gAx
H




………………(2)
Where, H= pressure head, A = area of pipe or conduit,
a=velocity of pressure wave, Q= discharge, g= acceleration
due to gravity, t = time, f =friction factor, D= diameterofpipe
or conduit x = distance along the pipe.
4. METHOD OF CHARACTERISTIC (MOC):
Method of characteristic (MOC) is the method which is used
to solve the governing equation of the flow of fluid through
the pipe. In this method the non-linear second order partial
differential equation is converted into a second order
ordinary differential equation (ODE). The ODE is then
discretized to form the algebraic equation, which is then
solved numerically using a computer program.
The discretized equations thus obtained are as follows:-
     j
k
j
k
j
k
j
k
j
k
j
k
j
k
j
k
j
k QQQQ
gDA
taf
QQ
gA
a
HHH 111121111
1
422
1





     j
k
j
k
j
k
j
k
j
k
j
k
j
k
j
k
j
k QQQQ
gDA
taf
HH
a
gA
QQQ 111121111
1
422
1





5. LAXFINITEDIFFERENCEEXPLICITMETHOD
(LAX FDE)
Chaudhury25 claims that Lax explicit method yields
satisfactory results in nonlinear partial difference
equation with smaller time step provided initial and
boundary conditions are correctly imposed. Although
smaller time step apparently would increase the
volume of computation time, much iteration needed in
implicit method is saved leading to a net decrease in
time. Hence, Lax Diffusivemethodhasbeenconsidered
for comparison.
In Lax finite differenceexplicitmethodtheequation (1)
and (2) have been converted to:
   j
k
j
k
j
k
j
k
j
k QQ
xgA
ta
HHH 112
2
11
1
2
1
2
1






     j
k
j
k
j
k
j
k
j
k
j
k
j
k
j
k
j
k QQQQ
gA
tf
HH
x
tgA
QQQ 11111111
1
).(
822
1








6. BARR’S FRICTION EQUATION
(UNSTEADY/VARIABLE FRICTION
EQUATION)
The friction factor f in the above equation is replaced
by the following Barr’s explicit approximations which
covers full range of flow conditions, from laminar to
turbulent.
  
    










kDkDRR
RR
f ee
ee
/7.3
1
/29/1
7/log518.4/log02.5
log2
1
7.052.0
1010
10
Where,
f = friction factor
k = sand roughness coefficient
D = Diameter of pipe
Re = Reynold’s number
7. IMPLEMENTATION OF DEVELOPED
NUMERICAL MODEL TO THE SIMILAR
PROBLEM AS MENTIONED BY SAIKIA M.D.
AND SARMA A.K. (2006)[14]
Fig -1: Schematic representation of water hammer
situation without surge tank (considering 4 sections of
the pipe)
The numerical model is implemented to the data given by
Saikia M.D. and Sarma A.K. (2006)[14]. The pipe is divided
into 4 sections of equal length, which means there are 5
locations for the calculations. The lab data is given as
follows:-
Length of the pipe = 12,000 ft
Discharge = 20 ft3/sec
Initial Pressure Head at the different locations:
Location 1 (Reservoir end) = 600 ft
Location 2 = 587.5 ft
Location 3 = 565 ft

Location 4 = 547.5 ft
Location 5 (Valve end) = 530 ft
Diameter of pipe = 2 ft
Area of valve opening = 3.1416 ft2
Surface roughness coefficient = 0.007093 ft
Kinematic Viscosity = 0.000001 ft2/sec
Coefficient of discharge = 0.90
Velocity of pressure wave = 3000 ft/sec
Fig -2: Pressure Head vs time at pipe position, x=5(from
Numerical Model by Saikia M.D. and Sarma A.K.)
Fig -3: Pressure head vs time at pipe position, x =5
(Developed Numerical Model by MOC and using data
from Saikia M.D. and Sarma A.K.)
Fig -4: Discharge vs time at pipe position, x =4 (from
Numerical Model by Saikia M.D. and Sarma A.K.)
Fig -5: Discharge v/s time at pipe position. x =4
(Developed Numerical Model by MOC method and using
data from Saikia M.D. and Sarma A.K.)
From the above analysis it is found that the developed
numerical model with MOC using Barr’s friction equation is
in excellent agreement with the resultsobtainedby Saikia M.
D. and and Sarma A. K (2006)[14]. Therefore our algorithm
can be used to compare different numerical models to solve
hydraulic transient in pipe flow without surge tank.
For the comparision between the two numerical methods
viz. MOC and Lax FDE we have taken the hydraulictrainsient
case with the input parameters from the quoted reference,
Saikia M.D. and Sarma A. K.(2006)[14].
Therefore the output data using MOC as a numerical scheme
with variable friction is plotted as shown in below.

Fig -6: Pressure head vs time at pipe position, x =5
data from Saikia M.D. and Sarma A. K.)
Fig -7: Discharge vs time at pipe position, x =4
data from Saikia M.D. and Sarma A.K.)
Now applying the developed numerical model usingLaxFDE
method to the hydraulic transient case discussed above the
following results are obtained and plotted as shown
graphically. In this case we have considered Barr’s
unsteady/variable friction equation to calculate the friction
factor.
Fig -8: Graph for variation of pressure head vs time (at
pipe position, x=5) (by developed numerical model
using LAX FDE method and using data from Saikia M.D.
and Sarma A.K.)
Fig -9: Graph for variation of Discharge vs. time (at pipe
position, x= 4) (by developed numerical model using
LAX FDE method and using data from Saikia M.D. and
Sarma A.K.)
8. CONCLUSIONS
As seen from the above analysis the Lax FDE proves to be
more advantageous than MOC for simulating transients in
pipe .More over the damping effect of the fluctuations is
more evident if we use Lax FDE method compared to MOC
method.Hence Lax FDE method is much better numerical
method to calculate hydraulic transient fluctuations with
surge tank in case of pipe flow.
REFERENCES
[1] C.S Watt., J.M. Hobbs and A.P Boldy. 1980. Hydraulic
Transients Following Valve Closure. Journal Hy. Div.
ASCE. Vol. 106(10): 1627-1640.
[2] Goldberg D.E. and Wylie B. 1983. Characteristics
Method Using Time-Line Interpolations. Journal Hy.
Div. ASCE. Vol. 109(5): 670-683.
[3] Chudhury M.H.,and Hussaini M.Y.. 1985. Second-order
accurate explicit finite–difference schemes for water
hammer analysis. Journal of fluid Eng. Vol. 107. pp.
523-529.
[4] Sibetheros I.A., Holley E.R. and Branski J.M.. 1991.
Spline Interpolations for Water Hammer Analysis.
Journal of Hydraulic Engineering. Vol. 117(10):1332-
1351.
[5] Silva-Arya W.F., and Chaudhury M.H.. 1997.
Computation of energy dissipation in transient flow.
Journal Hydraulic Engineering, ASCE. Vol. l123(2):
108-115.
[6] Pezzinga G. 1999. Quasi-2D Model for Unsteady Flow
in pipe networks. Journal of Hydraulic Engineering,
ASCE. Vol. 125(7): 666-685.
[7] Pezzinga G.. 2000. Evaluation of Unsteady Flow
Resistance by quasi-2D or 1D Models. Journal of
Hydraulic Engineering. Vol. l126(10): 778-785.
[8] Bergant A., Simpson A.R. and Vitkovsky J. 2001.
Developments in unsteady pipe flow friction

modeling. Journal of Hydraulic Research. Vol. 39(3):
249-257.
[9] Zielke W. 1968. Frequency -dependent Friction in
Transient pipe flow. Journal of Basic Eng, ASME. Vol. l
90(9): 109-115.
[10] Brunone B., Golia U.M.and Greco M.. 1991. Some
remarks on the momentum equation for fast
transients. Proc. Int. Conf. on Hydraulic transients
with water column separation, IAHR, Valencia, Spain.
Pp. 201-209.
[11] Zhao M., and Ghidaoui M.S.. 2004. Godunov-Type
Solutions for Water Hammer Flows. Journal of
hydraulic Engineering, ASCE. Vol. l13 (4): 341-348.
[12] Barr D.I.H.. 1980. The transition from laminar to
turbulent flow. Proc .Instn Civ .Engrs, Part 2. pp. 555-
562.
[13] Chudhury M.H.. 1994. Open Channel Flow. Prentice-
Hall of India Pvt. Ltd., New Delhi, India.
[14] Saikia M.D. and Sarma A.K.. 2006.Simulationof Water
Hammer Flows with Unsteady Friction Factor. ARPN
Journal of Engg & Applied Sciences, Vol.1, No.4, ISSN
1819-6608.

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