ecuanciones de mecanica de fluidos.pdf

8/10/2019 ecuanciones de mecanica de fluidos.pdf

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GENERALIZED WATER HAMMER ALGORITHM

FOR PIPING SYSTEMS WITH UNSTEADY

FRICTION

by

Jaime Surez Acua

A thesis submitted in partial fulfillment of the requirements for the degree of

MASTER OF SCIENCE

In

MECHANICAL ENGINEERING

UNIVERSITY OF PUERTO RICOMAYAGEZ CAMPUS

2005

Approved by:

________________________________Nellore S. Venkataraman, PhDMember, Graduate Committee

__________________Date

________________________________Vikram Raj Pandya, PhDMember, Graduate Committee

__________________Date

________________________________Walter Silva Araya, PhDPresident, Graduate Committee

__________________Date

________________________________Andres Calderon, PhDRepresentative of Graduate Studies

__________________Date

________________________________Paul Sundaram, PhDChairperson of the Department

__________________Date


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ABSTRACT

Unsteady flow in pipe networks is usually analyzed by means of one-dimensional

models, in which the energy dissipation is computed assuming the friction factor

dependent on the local and instantaneous Reynolds Number (quasi-steady model). This

model underestimates the friction forces and overestimates the persistence of oscillations

following the first one. Unsteady friction has great importance in pipe networks because

the maximum oscillation may occur after the first one.

Silva-Araya and Chaudhry developed (1993) a model to simulate unsteady friction in

transient flow which was tested in simple pipe valve systems only. The model

approximates the velocity profiles during the transient flow and computes the actual

energy dissipation. The ratio of the energy dissipation at any instant and the energy

dissipation obtained by the quasi-steady approximation is defined as the Energy

Dissipation Factor. This is a time-varying, non-dimensional parameter that is

incorporated in the friction term of the transient flow governing equations.

The objective of this study is to extend Silva-Araya and Chaudhrys unsteady friction

model for water hammer analysis to series and branching pipe systems. The conduits at

these systems may have different diameters, wall thickness, wall material, and/or friction

factors.

For the computation of the initial steady state conditions a subroutine that solves the

network flow equations using the Hardy Cross linear analysis is used. The subroutine

allows for the input of minor losses, use of either the Darcy-Weisbach or Hazen-Willians

pipe frictional loss. Either SI or English units are permissible.


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To create a user friendly Windows interface, the extended application was

developed using VBA (Visual Basic for Applications) in Excel for general program

applications.

The model was tested for three cases: single, series and branching piping systems.

The computed pressures are compared with the experimental data obtained for Silva

Araya (1993) for a single piping system case. For a series and a branching piping system

the computed results obtained by using unsteady friction are compared with the data

obtained by using steady friction model.


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RESUMEN

El flujo no permanente en redes de tuberas se analiza generalmente por medio de

modelos unidimensionales, en que la disipacin de energa se calcula asumiendo el factor

de friccin dependiente del Nmero de Reynolds local e instantneo (modelo cuasi-

permanente). Este modelo subestima las fuerzas de la friccin y sobreestima la

persistencia de las oscilaciones que siguen a la primera. La friccin en flujo no-

permanente tiene gran importancia en redes de tuberas, porque la oscilacin mxima

puede ocurrir despus de la primera.

Silva-Araya y Chaudhry desarrollaron (1993) un modelo para simular la friccin no

permanente en flujo transitorio, el cual se prob solamente en sistemas de tubo sencillo

vlvula. El modelo aproxima los perfiles de velocidad durante flujo transitorio y calcula

la disipacin de energa verdadera. La relacin entre la disipacin de energa en

cualquier instante y la disipacin de energa obtenida por la aproximacin cuasi-

permanente se define como el Factor de Disipacin de Energa. Este es un parmetro no-

dimensional que varia en el tiempo que se incorpora en el trmino de la friccin de flujo

transitorio que gobierna las ecuaciones.

El objetivo de este estudio es extender a sistemas de tuberas en serie y ramificadas,

el modelo de friccin no permanente de Silva-Araya y Chaudhry para el anlisis de Golpe


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de Ariete. Las tuberas en estos sistemas pueden tener diferentes dimetros, espesor de

pared, material de pared, y/o factores de friccin.

Para el cmputo del estado permanente inicial se utiliza una subrutina que resuelve

las ecuaciones de flujo de la red, usando el anlisis lineal de Hardy Cross. La subrutina

tiene en cuenta las prdidas menores de entrada, el uso de las perdidas por friccin en

tuberas de Darcy-Weisbach o la de Hazen-Willians. Son permisibles tanto el sistema

internacional como las unidades inglesas.

Para crear una comunicacin amigable con Windows, la aplicacin extendida se

desarroll utilizando VBA (Visual Basic for Applications) en Excel para las aplicaciones

generales del programa.

El modelo fue probado para tres casos: sistemas de tubera sencillos, en serie y

ramificados. Las presiones computadas se comparan con los datos experimentales

obtenidos para Silva Araya (1993) para el caso de sistemas de tubera sencillos. Para

sistemas de tubera en serie y ramificados los resultados calculados, obtenidos utilizando

friccin en flujo no-permanente se comparan con los datos obtenidos utilizando el

modelo de friccin permanente.


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To my mother Amanda and my wife Lilibeth.


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ACKNOWLEDGEMENTS

I want to express a sincere acknowledgement to my advisor, Dr. Walter Silva for his

motivation, time, and support during my research.

I wish to thank the Department of Mechanical Engineering at University of Puerto

Rico, Mayagez Campus for giving me the opportunity to carry out my master studies.

I wish to express my gratitude also to the Department of General Engineering at

University of Puerto Rico, Mayagez Campus for the financial assistance.

I would like to thank my mother and my wife, for their support, inspiration and love.


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Table of ContentsABSTRACT .................................................................................................................................................II

RESUMEN ................................................................................................................................................. IV

ACKNOWLEDGEMENTS ....................................................................................................................VII

TABLE OF CONTENTS........................................................................................................................VIII

TABLELIST ................................................................................................................................................X

FIGURELIST ............................................................................................................................................ XI

1 INTRODUCTION .............................................................................................................................2

1.1 MOTIVATION.................................................................................................................................31.2 LITERATURE REVIEW....................................................................................................................31.3 SUMMARY OF FOLLOWING CHAPTERS ..........................................................................................8

2 GOVERNINGEQUATIONS...........................................................................................................9

2.1 CLOSED CONDUIT TRANSIENT FLOW GOVERNING EQUATIONS ....................................................92.1.1 Continuity Equation.................................................................................................................92.1.2 Momentum Equation ...............................................................................................................9

2.2 ENERGYDISSIPATIONFUNCTION..............................................................................................112.3 MOMENTUMEQUATIONFORAXISYMMETRICPIPEFLOW .......................................................142.4 METHODOFCHARACTERISTICS ................................................................................................15

2.4.1 Characteristics Equations .....................................................................................................162.4.2 Boundary Conditions.............................................................................................................23

3 SYSTEMSPECIFICATION ...........................................................................................................37

3.1 SCHEMATICDIAGRAM...............................................................................................................373.2 NUMBERINGOFNODES .............................................................................................................383.3 NODETYPES...............................................................................................................................403.4 NUMBERINGOFCONDUITS .......................................................................................................41

4 METHODOLOGY ...........................................................................................................................42

4.1

COMPUTATIONOFTHEINITIALSTEADYSTATECONDITIONS ...................................................42

4.1.1 Generalized Network Equations............................................................................................434.1.2 Linearization of System Energy Equations............................................................................494.1.3 Hardy Cross Method .............................................................................................................50

4.2 GENERATIONOFTHECOMPUTATIONALGRID ..........................................................................534.3 TURBULENCEMODELING...........................................................................................................55

4.3.1 Turbulence Model for the Inner Region ................................................................................574.3.2 Outer Region Model and Blending Relation..........................................................................58


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4.4 INITIALVELOCITYPROFILE ........................................................................................................614.5 COMPUTATIONOFTHEPRESSUREGRADIENT ...........................................................................634.6 SHEARSTRESSDISTRIBUTION .....................................................................................................64

4.7 COMPUTATIONOFTHEVELOCITYPROFILE ...............................................................................654.7.1 Predictor Step........................................................................................................................664.7.2 Corrector Step .......................................................................................................................67

5 VERIFICATION...............................................................................................................................69

5.1 SINGLEPIPINGSYSTEM..............................................................................................................695.2 SERIESPIPINGSYSTEM ...............................................................................................................825.3 BRANCHINGPIPINGSYSTEM .....................................................................................................89

6 CONCLUSIONS ..............................................................................................................................94

APPENDIXA.VALUESOFTVS. TAUFORDIFFERENTTYPESOFVALVES ........................99

APPENDIX A1.BUTTERFLYVALVES...................................................................................................99APPENDIX A2.CIRCULARVALVES....................................................................................................101APPENDIX A3.BIPLANESVALVES.....................................................................................................103


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TableList

Tables Page

Table 3-1. Node Types..................................................................................................... 41Table 5-1. Parameters for Single Piping System. ............................................................ 71Table 5-2. Node type for Single Piping System. ............................................................. 71Table 5-3. Node type for Series Piping System............................................................... 83Table 5-4. Parameters for Series Piping System.............................................................. 83Table 5-5. Characteristics of the Pipes for Series Piping System.................................... 83Table 5-6. Node type for Branching Piping System. ....................................................... 90Table 5-7. Characteristics of the Pipes for Branching Piping System............................. 90


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FigureList

Figures Page

Figure 2-1. Coordinate System. ........................................................................................ 14Figure 2-2. Characteristic Lines inx-tPlane. ................................................................... 19Figure 2-3. Single Pipeline. .............................................................................................. 19Figure 2-4. Regions of Validity for a Single Pipeline. ..................................................... 20Figure 2-5. Constant-level Upstream Reservoir. .............................................................. 25Figure 2-6. Constant-level Downstream Reservoir. ......................................................... 28Figure 2-7. Dead End........................................................................................................ 29

Figure 2-8. Valve at Downstream End. ............................................................................ 31Figure 2-9. Series Junction. .............................................................................................. 33Figure 2-10. Branching Junction....................................................................................... 35Figure 2-11. Intermediate Valve ....................................................................................... 35Figure 3-1. Notation.......................................................................................................... 37Figure 3-2. Schematic Diagram of a Series System. ........................................................ 38Figure 3-3. Schematic Diagram of a Branching System................................................... 40Figure 4-1. Representative piping network: (a) assumed flow directions and numbering

scheme: (b) designated interior loops; (c) path between two fixed-grade nodes.(From [14])................................................................................................................ 44

Figure 4-2. Two piping networks: (a) tree watering system: P = 17, L = 0, j = 8, F = 10;

(b) water distribution system: P = 17, L = 4, J = 12, F = 2. (From Wood, 1981.).... 48Figure 4-3. Non-uniform grid. .......................................................................................... 53Figure 4-4. Regions of a Boundary Layer ........................................................................ 56Figure 4-5. Eddy Viscosity Distribution........................................................................... 60Figure 5-1. Schematic of Experimental Setup (Silva Araya, 1993) ................................. 70Figure 5-2. Characteristic Curve for Valve in Single Piping System. .............................. 71Figure 5-3. Pressure Head Histories for a Single Piping System Using Steady Friction. 73Figure 5-4. Pressure Head Histories for a Single Piping System Using Unsteady Friction.

................................................................................................................................... 76Figure 5-5. Velocity Profile in transient Flow at the Middle of the Pipe ......................... 78Figure 5-6. Energy Dissipation Factor in Transient Turbulent Flow................................ 81

Figure 5-7. Schematic of Series Piping System................................................................ 82Figure 5-8. Characteristic Curve for Valve in Series Piping System. .............................. 84Figure 5-9. Geometrically Expanding Grid Spacing for Series Piping System................ 85Figure 5-10. Eddy Viscosity distribution for Series Piping System. ................................ 85Figure 5-11. Initial Velocity Profile for Series Piping System......................................... 86Figure 5-12. Pressure Head Histories for a Series Piping System Using Steady and

Unsteady Friction...................................................................................................... 87


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Figure 5-13. Schematic of Branching Piping System....................................................... 89Figure 5-14. Eddy Viscosity distribution for Branching Piping System. ......................... 91Figure 5-15. Initial Velocity Profile for Branching Piping System. ................................. 91

Figure 5-16. Pressure Head Histories for a Branching Piping System Using Steady andUnsteady Friction...................................................................................................... 92


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1 INTRODUCTION

If sudden velocity and pressures variations are caused in a pipe system, for example

when pumps are shut off or valves are closed, a pressure wave develops which is

transmitted in the pipe at a certain velocity that is determined by fluid properties and the

pipe wall material. This phenomenon, called water hammer, can cause pipe and fittings

rupture. The intermediate stage flow, when the flow conditions are changed from one

steady state condition to another steady state, is called transient state flow or transient

flow; water hammer is a transient condition caused by sudden changes in flow velocity or

pressure.

Generally friction losses in the simulation of transient pipe flow are estimated by

using formulae derived for steady state flow conditions, this is known as the quasi-steady

approximation. The head loss during transient flow is equal to the head loss obtained for

steady uniform flow with an average velocity equal to the instantaneous transient velocity.

Although this approximation is enough to calculate the maximum pressures in the

absence of vapor cavitations or column separation, this is not accurate for the prediction

of the time history of pressure oscillations and the attenuation of pressure waves. An

accurate calculation of the damping effect due to unsteady friction losses is important forlong time simulations and for systems having multiple operations.

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1.1 Motivation

Water hammer analysis tools of the past have been noted for being difficult to use and

requiring extensive specialized knowledge. As a result, this critical aspect of piping

system design and operation has often been overlooked. Recently, pressure oscillations

caused serious damage in the important pipe conduction called the North Coast Super

aqueduct of Puerto Rico.

The present research extended Silva-Araya and Chaudhrys (1997) unsteady friction

model to series and branching pipe systems. The extended application was developed

using a Windows interface with VBA in Excel for general program applications. The

original research was tested in simple pipe valve systems only.

Piping systems engineers typically are not specialists in water hammer modeling.

With this work and fundamental knowledge they would be equipped with the proper tools

to design and analyze for water hammer transients in liquid systems.

1.2 Literature Review

Several researchers have proposed analytical and numerical approaches to model

energy losses in unsteady pipe flows. Historically, the most common method of

representing frictional resistance in unsteady flows has been the use of Darcy-Weisbach

steady-flow equation based on the mean velocity, considering that the shear stress at the

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wall is the same in steady-state flow condition than in transient condition. These models

fail to correctly predict the transient pressure head field beyond the first wave cycle.

Ohmi et al. (1980) studied the flow behavior in turbulent oscillatory flow and found

that the shear stress depends on the ratio of frequency of oscillation and the time-

averaged Reynolds number throw each cycle. When this ratio is small, the pressure

gradient and the wall shear stress are in phase with the mean velocity and the quasi-

steady approximation is applicable. As this ratio increases, the mean velocity delays from

the shear stress and the pressure gradient. Then the friction factor is different from its

values in quasi-steady conditions.

Experimental studies by Ramaprian and Tu (1979) showed that the turbulent structure

in oscillatory flow is different from that in equilibrium conditions, thereby increasing the

time-mean shear stress and the friction losses. Therefore, the energy dissipation in

oscillatory flow is greater than that computed by the quasi-steady approximation.

Vardy and Hwang (1991) adopted a five-region turbulence model to compute the

eddy viscosity distribution using empirical relations. In each region a different expression

was used to compute the eddy viscosity. This turbulence model is used to solve the

transient governing equations including axial and radial velocity components. The

equations are transformed by the method of characteristics and solved simultaneously.

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The equations must be solved at every time step at every cross section to obtain the

pressure head and the axial velocity. It was reported that this model is very time

consuming.

Giuseppe Pezzinga (1999) developed a quasi-two dimensional model for

axisymmetry unsteady-flow analysis in a pipe system. Taking into account the velocity

profile in the cross section, he used a two layer turbulence model, which divides the flow

into a viscous sub-layer and a turbulent sublayer, based on the mixing length hypothesis

in the turbulent zone and on Newtons law in the viscous sublayer.

Eichinger and Lein (1992) developed a numerical model to compute an unsteady

friction factor with the help of the turbulence model, developed by Nagano and Hishida

(1987), to compute the eddy viscosity and solve the transient pipe flow governing

equations.

Silva-Araya and Chaudhry (1993) developed an energy dissipation model for the

computation of laminar and turbulent unsteady friction losses and obtained instantaneous

velocity profiles to compute the Reynolds stresses in transient pipe flow. The energy

dissipation factor incorporated in the momentum equation is defined as the ratio of the

energy dissipated in transient flow to the energy dissipated according to the steady-state

approximation. This is a non-dimensional, time-varying parameter that modifies the

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friction term in the transient flow governing equations. For transient turbulent pipe flow a

mixing length eddy viscosity turbulence model proposed by Granville (1989) was chosen

for the inner region of the boundary layer and a constant eddy viscosity model proposed

by Kita et al., (1980) was used for the outer region.

Peralta Cooper (2001) replaced the mixing length model in Silva-Araya and

Chaudhrys model by the two-equation k - turbulence model proposed by Wilcox

(1988a, 1993b). The results were very similar to those obtained by using Granvilles

mixing length model. As a general conclusion, for most transient conditions, the use of

simple turbulence models provides similar results to two-equation models. The use of

steady-state turbulence models for unsteady flow is supported by physical measurements

of pressure oscillation. However, unsteady turbulence models are still an area where

research must be done.

Arangoitia Valdivia (2001) developed a mathematical model for the calculation of

water hammer considering the energy dissipation. He studied the water hammer produced

by the instantaneous closure of a valve, located at the end of a pipe line connected to a

constant head reservoir. The model consists of linearizing the equations of transient flow,

to establish an equation similar to the corresponding to an oscillatory mechanical system

and to use this equation as a boundary condition, in the method of characteristics.

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Zhao and Ghidaoui (2004) formulated, applied, and analyzed first and second order

explicit finite volume Godunov-type schemes for water hammer problems. The finite

volume formulation ensures that both schemes conserve mass and momentum and

produce physically realizable shock fronts. The implementation of boundary conditions,

such as, valves, pipe junctions, and reservoirs, within the Godunov approach is similar to

that of the method of characteristics approach. The schemes were applied to a system

consisting of a reservoir, a pipe, and a valve and to a system consisting of a reservoir, two

pipes in series, and a valve.

Trikha (1975) developed a procedure for simulating frequency-dependent friction in

transient laminar liquid flow by the method of characteristics. The procedure consists of

determining an approximate expression for frequency-dependent friction such that the use

of this expression requires much less computer storage than the use of the exact

expression proposed by Zielke (1967).

Schohl (1993) used a new approximation to the weighting function in Zilekes (1967)

equation in an improved implementation of Trikhas (1975) method for including

frequency dependent friction in transient laminar flow calculations. The new, five-termapproximation was fitted to the weighting function using a nonlinear least squares

approach.

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Vardy and Brown (2004) presented a method for evaluating wall shear stress from

known flow histories in unsteady pipe flows. The method builds on previous work by

Trikha, but has two important differences. One of these enables the method to be uses

with much larger integration time steps than are acceptable with Trikhas method. The

other, a general procedure for determining approximations to weighting functions,

enables it to be used at indefinitely small times. The method is applicable to both laminar

and turbulent flows.

Most of unsteady flow models with unsteady friction have been tested in simple pipe

systems, typically a valve-reservoir system. Recently, some attempts have been done to

apply then in pipe networks. The present work has extended the application of a quasi-

bidimensional model to branching and series systems.

1.3 Summary of Following Chapters

We first develop the necessary background theory in Chapter 1. Chapter 2 describes

the governing equations of the phenomenon. Chapter 3 explains how to describe the

system you are analyzing, how many components the system has, and of what type and

size they are. Chapter 4 describes the procedure followed to obtain the energy dissipation

factor as an integrated part of the transient analysis. Chapter 5 shows the verification of

the generalized algorithm, with the results obtained by using the unsteady friction model

for piping systems. Conclusions are presented in Chapter 6.

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2 GOVERNINGEQUATIONS

2.1 Closed Conduit Transient Flow Governing Equations

The dynamic equations in transients-state flow in closed conduits are the equations

for the conservation of mass and the momentum equation. These equations are a set of

hyperbolic partial differential equations. Using the Reynolds Transport Theorem and

assuming one-dimensional flow and elastic conduit with slightly compressible fluid,

which stretch or contracts with respect to time; these equations become (Chaudhry, 1987)

2.1.1 Continuity Equation

02 =

+

+

x

Va

pV

t

p 2.1

Where: p = pressure intensity, V = mean flow velocity, a = wave speed, = fluid

density, t= time,x= coordinate axis along conduit length.

2.1.2

Momentum Equation

0

2

sin1

=++

+

+

D

VfVg

x

p

x

VV

t

V

2.2

Where:g= acceleration due to gravity, = pipe angle with respect to the horizontal,


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D= pipe diameter,f = Darcy-Weisbach friction factor.

In most engineering applications, the convective acceleration terms, ( xpV / ) and

( ), are very small compared to the others terms and may be neglected. Therefore

by dropping these terms Equation 2.1 becomes:

xVV /

02 =

+

Va

t

p 2.3

Equation 2.2 becomes:

02

1=+

+

D

VfV

x

p

t

V

2.4

Expressing pressures in the pipeline in terms of the piezometric head, above a

specified datum as, and using the discharge, Q = VA, instead of the flow velocity and

considering the energy dissipation factor in the equation of momentum. Equation 2.3

becomes:

02

=

+

x

Q

gA

a

t

H 2.5


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Equation 2.4 is modified as:

02

=+

+

DA

QfQe

x

HgA

t

Qf 2.6

Where: Q= flow discharge, ef=energy dissipation factor,A= pipe cross sectional

area,H = piezometric head. f= Darcy- weisbach friction factor.

efis a non-dimensional and time dependent parameter to compute the variation of thefriction losses in space and time. This parameter is set equal to one if the quasi-steady

approach in transient pipe flow is used. The term in the energy equation that involves the

viscous and turbulent stresses is commonly called the dissipation function. A procedure

to compute the energy dissipation factor is described in the following paragraphs.

2.2 EnergyDissipationFunction

For boundary layer planar or axisymmetric flow, the expression to compute the

energy dissipation function, denoted by , is given by (White, 1991):

= ''

vur

u

r

u

2.7

Where: u = axial velocity of the fluid particle along the conduit length, r= radial


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coordinate measured from the pipe centerline, u/r = velocity gradient in the radial

direction, = dynamic viscosity of the fluid, = fluid density, ''vu = Reynolds

stresses. Equation 2.7 accounts the total shears stresses, viscous and turbulent stresses

and represents the energy dissipation per unit volume and per unit time.

Integration of Equation 2.7 across the conduit area from the wall to the pipe

centerline and multiplying by two can be estimated the dissipation integral. This result

accounts for the rate of energy dissipation per unit length in the conduit area.

Therefore:

= R

I rdrD 02 2.8

Where DI = dissipation integral. The dissipation of energy in transient flow can be

computed by integration the dissipation function across a pipe section. Evaluating

Equation 2.8 during an interval of time gives:

=2

1 0

2t

t

R

t rdrdtE 2.9

Silva Araya (1993) demonstrated that for steady-state the energy dissipation is given

by:


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Ag

fVD Is

2

3

= 2.10

WhereDLs= steady state dissipation integral,f = Darcy- weisbach friction factor,

V= mean velocity, = fluid density,A= pipe flow cross sectional area.

If Equation 2.10 is considered by an interval of time it becomes:

( 123

2 ttA

DfVEs = ) 2.11

Equation 2.9 can be used to compute the unsteady energy dissipation integral if the

velocity profile at any time during the computation and the Reynolds stresses are known.

Division of Equation 2.9 by Equation 2.11 gives the ratio, called the energy

dissipation factor, as defined by Silva Araya and Chaudhry (1997), of the energy

dissipated during the transient flow to that in the steady flow.

s

tf

E

Ee = 2.12

The energy dissipation factor is a function time-varying velocity gradient and the

Reynolds stresses for turbulent flow. To solve Equation 2.12 requires a procedure to

compute the instantaneous velocity distribution in transient flow condition and also a


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turbulence model. The next section describes the procedure implemented to find the

velocity distribution during the transient flow.

2.3 MomentumEquationforAxisymmetricPipeFlow

The velocity distribution in transient flow process for an axisymmetric

incompressible pipe flow is given by (White, 1991):

+

+

=

+

+

r

u

rr

u

x

p

r

uu

x

uu

t

urx

112

2

2.13

Where = kinematics viscosity of the fluid. The coordinate directions are defined in

Figure 2-1.

Figure 2-1. Coordinate System.


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Thus after time averaging the axisymmetric equation of motion, the following

assumptions were considered:

1. Incompressible flow and constant viscosity

2. The convective terms are not considered

3. The second order spatial derivative term is not considered.22 / xu

4. The velocities in the radial direction can be neglected

With these assumptions the momentum equation becomes, see appendix A.

+

+

=

+

r

u

rr

u

x

p

r

vu

t

u 112

2''

2.14

Where u = time averaged velocity in the axial direction, ''vu = Reynolds stress.

Equation 2.14 is the equation to compute the velocity distribution in turbulent transient

pipe flow.

2.4 MethodofCharacteristics

The equations describing the transient-state flow in closed conduits are quasi-linear,

hyperbolic, partial differential equations. A closed-form solution of these equations is not

available. However, by neglecting or linearizing the nonlinear term, various graphical and

analytical methods have been developed.


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The method of characteristics has become quite popular and is extensively used. For

the solution of one-dimensional, hydraulic transient problems (especially if the wave

speed is constant), this method has proven to be superior to other methods in several

aspects, such as correct simulation of steep wave fronts, illustration of wave propagation,

ease of programming, and efficiency of computations.

2.4.1 Characteristics Equations

Let us rewrite the momentum and continuity equations (Eqs. 2.5 and 2.6) as:

01 =+

+

= QRQ

x

HgA

t

QL 2.15

022 =

+

=

t

HgA

QaL 2.16

In which R =f/(2DA). Let us consider a linear combination of Eqs. 2.15 and 2.16:

L = L1+L2

Or

012 =+

+

+

+

QRQ

x

H

t

HgA

x

Qa

t

Q

2.17


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IfH = H(x,t) and Q = Q(x,t), then the total derivative

dt

dx

x

Q

t

Q

dt

dQ

+

= 2.18

and

dt

dxH

t

H

dt

dH

+

= 2.19

By defining the unknown multiplier as:

21a

dt

dx

== 2.20

i.e.,

a

1

= 2.21

And by using Equations. 2.18 and 2.19, Equation 2.17 can be written as

0=++ QRQdx

dH

a

gA

dt

dQ 2.22

If

adt

dx= 2.23


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and

0=+ QRQdx

dH

a

gA

dt

dQ 2.24

If

adt

dx= 2.25

Note that Equation 2.22 is valid if Equation 2.23 is satisfied and that Equation 2.24 is

valid if Equation 2.25 is satisfied.

In thex tplane, Equations 2.23 and 2.25 represent two straight lines having slopes

1/a. These are called characteristic lines. Mathematically, these lines divide the x t

plane in two regions, which may be dominated by two different kinds of solution, i.e., the

solution may be discontinuous along these lines. Physically, they represent the path

traversed by a disturbance. For example, a disturbance at point A (Figure 2-2) at time to

would reach pointP after time t.


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19

dt

Figure 2-2. Characteristic Lines inx-tPlane.

Considering a single pipeline (Figure 2-3) Equations 2.22 and 2.24 are valid along the

pipe length (i.e., for 0 < x < L) and special boundary conditions are required at the ends

(i.e., at x = 0 and at x = L) (Figure 2-4).

Figure 2-3. Single Pipeline.


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dt

Figure 2-4. Regions of Validity for a Single Pipeline.

Referring to Figure 2-2, knowing the values of Q and H at points A and B, their

values at pointPcan be determinates by solving Equations 2.22 and 2.24 as follows:

By multiplying the left-hand side of Equation 2.22 by dtand integrating, we obtain:

0=++ dtQdQRdHagA

dQP

A

P

A

P

A 2.26

The first two integral forms of Equation 2.26 are easily evaluated; however, the third

term is not, representing the friction losses, because the variation of Qwith respect to tis

not explicitly known. By using a first-order approximation, the integral of the third term

is evaluated as:


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21

tQRQttQRQdtQdQR AAApAAP

A= )( 2.27

In other words, we are saying that for the evaluation of this term, Qremains constant

form pointAto pointP. Hence, Equation 2.26 becomes:

0)( =++ AAAPAP QtQRHHa

gAQQ 2.28

Note that Equation 2.28 is exact except for the approximation of the friction term.

This first-order approximation usually yields satisfactory results for typical engineering

applications.

By proceeding similarly, we may write Equation 2.24 as:

0)( =+ BBBPBP QtQRHHa

gAQQ 2.29

By combining the known variables together, we may write Equation 2.28 as:

PaPP HCCQ = 2.30


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And Equation 2.29 as:

PanP HCCQ = 2.31

AAAAP QtQRHa

gAQC += 2.32

BBBBn QtQRHa

gAQC = 2.33

And

a

gACa = 2.34

Note that Equation 2.30 is valid along the positive characteristic lineAPand Equation

2.31 along the negative characteristic lineBP(Figure 2-2). The value of the constants CP

and Cn are known for each time step, and the constant Ca depends upon the conduit

properties. We will refer to Equation 2.30 as the positive characteristic equation and

Equation 2.31 as the negative characteristic equation. In Equations 2.30 and 2.31, we

have two unknowns, namely, HP and QP. The values of these unknowns can be

determined by simultaneously solving these equations.


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2.4.2

Boundary Conditions

Special boundary conditions are required to determine the transient-state head and

discharge at the boundaries. These are developed by solving Equations 2.22 or 2.24, or

both, simultaneously with the conditions imposed by the boundary. These conditions are

in terms of special relationships that define, at the boundary, the discharge, the head, or a

relation between them. Equation 2.22 is used for the downstream boundaries and

Equation 2.24 for the upstream boundaries.

We shall designate the upstream and downstream ends with reference to the initial-

flow direction even though the flow may reverse during the transient state. A section at

the upstream end of a conduit shall be numbered section 1 and the one at the downstream

end as sectionn+ 1, assuming the conduit is divided in nreaches. To specify variables at

different sections, we shall use two subscripts: The first subscript designates the conduit

number, and the second indicates the section number. For example, Qi,jrepresents flow at

thejth section of the ith conduit. For variables that have the same value at all sections of

a conduit, only one subscript will be used. For example, Cai refers to constant Ca

(Equation 2.34) for the ith conduit. Although CP and Cn may have different values at

different sections of a conduit, only one subscript will be used with them to indicate the

conduit number.


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2.4.2.1 Constant-Level Upstream Reservoir

In this case, it is assumed that the water level in the reservoir remains constant during

the transient-state conditions.

Let the entrance losses be given by the equation:

2

2

21,

i

Pe

gAkQh i= 2.35

In which kis the coefficient of entrance loss. Then, referring to Figure 2-5, we obtain:

2

2

2)1( 1,

1,

i

i

i gA

QkHH

P

resP += 2.36

In whichHres= height of the reservoir water surface above the datum.


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Figure 2-5. Constant-level Upstream Reservoir.

To develop the boundary condition, we have to solve this equation simultaneously

with the negative characteristic equation (Equation 2.31). Elimination of HPi,1 from

Equations 2.31 and 2.36 and simplification of the resulting equation yield:

0)(1,1,

21 =++ resanPP HCCQQk iiii 2.37

In which:

21 2

)1(

i

a

gA

kCk i

+= 2.38

By solving Equation 2.37 and neglecting the negative sign with the radical term, we

obtain:


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1

1

2

)(41111

1, k

HCCkQ

resan

Pi

+++

= 2.39

If the entrance losses as well as the velocity head are negligible, then:

resP HH i =1, 2.40

In whichHres= height of the reservoir water surface above the datum. Equation 2.31

for the upstream end thus becomes:

resanP HCCQ iii +=1, 2.41

2.4.2.2 Constant-Level Downstream Reservoir

If the head losses at the entrance to the reservoir are:

2

2

21,

i

P

egA

kQh ni += 2.42

Then referring to Figure 2-6:

2

2

2)1( 1,

1,

i

ni

ni gA

QkHH

P

resP

+

+= 2.43


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Elimination ofHpi,n+1from Equations 2.43 and 2.30 yields:

01,1,

22 =+ ++ resapPP HCCQQk iinini 2.44

In which:

22 2

)1(

i

a

gA

kCk i

= 2.45

By solving equation 2.44 and neglecting the positive sing with the radical term:

2

2

2

)(41111

1, k

HCCkQ

resap

P ni

=

+ 2.46

If the exit loss and the velocity head are negligible, then:

resP HH ni =+1, 2.47

Hence it follows from Equation 2.30 that:

resapP HCCQ iini =+1, 2.48


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Figure 2-6. Constant-level Downstream Reservoir.

2.4.2.3 Dead End at Downstream End

At the dead end (Figure 2-7), QPi,n+1 = 0. Hence, from the positive characteristic

equation (Equation 2.30), it follows that:

i

i

ni

a

p

PC

CH =

+1, 2.49


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Figure 2-7. Dead End.

2.4.2.4 Valve at Downstream

Steady-state flow through a valve can be written as:

1,1,2)(

++ = nini oovdo gHACQ 2.50

In which subscript o indicates steady-state conditions, Cd= coefficient of discharge,

Hoi,n+1= head upstream of the valve, andAv= area of the valve opening.

If we assume that the transient-state flow through a valve may be described by an

equation similar to the steady-flow equation, then:

1,1,2)(

++=

nini PvdP gHACQ 2.51


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Dividing Equation 2.51 by Equation 2.50, taking square of both sides and defining the

relative valve opening = (CdAv)/(CdAv)o, we obtain:

1,

1,

1,

1,

2

2

+

+

+

+=

ni

ni

ni

ni P

o

o

P HH

QQ

2.52

Substitution for Hp from the positive characteristic equation (Equation 2.30) into

Equation 2.52 yields:

01,1,

2 =+++ vpPvP

CCQCQinini

2.53

In which Cv= (Qoi,n+1)2/(CaHoi,n+1).Solving for Qpi,n+1and neglecting the negative

sign with the radical term:

vpvvP CCCCQ ini 45.02

1,++=

+ 2.54

NowHpi,n+1may be determined from Equation 2.30.

To compute the transient-state conditions for an opening or closing valve, versus t

curves (Figure 2-8b and c) may be specified either in a tabular form or by an algebraic

expression. Note that = 1 corresponds to a valve opening at which the flow through the

valve is Qoi,n+1under a head of Hoi,n+1.Values of tversus for different types of valves


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are given in Appendix A.

(a)

0

t

1

0

t

1

t (b) (c)

Figure 2-8. Valve at Downstream End.


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32

2.4.2.5 Orifice at Lower End

For an orifice, the opening remains constant. Therefore, the above equations may be

used with = 1.

2.4.2.6 Series Junction

A series junction is a junction of two conduits having different diameters, wall

thicknesses, wall materials, and/or friction factors. If the difference in the velocity heads

at sections (I, n+1) and (i+1, 1) (Figure 2-9) and the head losses at the junction are

neglected, then it follows from the energy equation that:

1,11, ++=

ini PP HH 2.55

The positive and negative characteristic equations for sections (I, n+1) and (i+1, 1)

are:

1,1, ++=

niiini PapP HCCQ 2.56

1,1111,1 ++++=

iiii PanP HCCQ 2.57

The continuity equation at the junction is:

1,11, ++=

ini PP QQ 2.58


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It follows from Equations 2.55 through 2.55 that:

1

1

1,

+

+

+ +

=

ii

ii

ni

aa

np

PCC

CCH 2.59

Now Hpi+1, 1, Qpi, n+1, and Qpi+1, 1 can be determined from Equations 2.55 through

2.57.

Figure 2-9. Series Junction.

2.4.2.7 Branching Junction

For the branching junction shown in Figure 2-10, the following equations can be

written:

1. Continuity equation:

1,21,11, ++++=

iini PPP QQQ 2.60


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2. Characteristic equations:

1,1, ++=

niiini PapP HCCQ 2.61

1,1111,1 +++++=

iiii PanP HCCQ 2.62

1,2221,2 +++++=

iiii PanP HCCQ 2.63

3. Equation for total head:

If the head losses at the junction are neglected and it is assumed that the velocity

heads in all conduits are equal, then it follows from the energy equation that:

1,21,11, +++==

iini PPP HHH 2.64

Simultaneous solution of Equations 2.60 through 2.64 yields:

21

21

1,

++

++

+ ++

=

iii

iii

ni

aaa

nnp

PCCC

CCCH 2.65

NowHpi+1, 1andHpi+2, 1can be determined from Equation 2.64; Qpi, n+1, Qpi+1, 1, and

Qpi+2, 1can be determined from Equations 2.61 through 2.63.


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Figure 2-10. Branching Junction.

2.4.2.8 Intermediate Valve

Figure 2-11 illustrates a valve positioned at the junction of two pipes that may have

different diameters, wall thicknesses, wall materials, and/or friction factors.

Figure 2-11. Intermediate Valve


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The positive and negative characteristic equations for sections (I, n+1) and (i+1, 1)

are given for Eqs. 2.56 and 2.57 respectively. The continuity equation at the junction is

given for Eq. 2.58.

It follows from Eqs. 2.56 through 2.58 that

1

11,

1,1

+

++

+

=

i

iniii

i

a

npap

pC

CHCCH 2.66

from Equation 2.30

i

nii

ni

a

pp

pC

QCH 1,

1,

+

+

= 2.67

And Qpi,n+1is computed for Eq. 2.54 for a valve at downstream. Values of tversus

for different types of valves are given in Appendix A.


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3 SYSTEMSPECIFICATION

This chapter describes how to specify the pipe system in the computer program.

3.1 SchematicDiagram

Figure 3.1 shows a system configuration. Liquid-conveying structures, such as pipes,

conduits or tunnels are conduitsand the storage facilities, devices or appurtenances, such

as reservoirs, tanks, valves or pipe junctions are the boundaries. The conduit cross

sections at the boundaries will be called nodes.

A square is used for a node and a circle for a conduit. The node number is written

inside the square and the conduit number inside the circle. The typeof each node will be

indicated by a numeral written inside a triangle (see Fig. 3-1). Each node type is

explained in this chapter.

Figure 3-1. Notation.


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3.2 NumberingofNodes

To identify the nodes in the system, they are numbered in a sequence starting at the

upstream end and, number the upstream node as one. Then, proceeding in the

downstream direction, the other nodes are numbered 2, 3, and so on. Fig. 3-2 illustrates

this procedure. The junction of two conduits is a node called a series junction. The

conduits joining at a series junction may have different diameters, wall thickness, wall

material, and/or friction factors. In a branching piping system having a tree-like structure

(Fig. 3-3), designate one pipeline as the main line, and the remaining pipelines as the

branch lines. The branch lines may have sub-branches and the sub-branches may be

further branched out.

Figure 3-2. Schematic Diagram of a Series System.


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Nodes on the main line are numbered like a series system, i.e., we start at the

upstream end and then proceed in the downstream direction. The nodes on each branch

line are then numbered one branch at a time. The numbering on the branch line starts at

the upstream end of the branch and then proceeds in the initial-steady-state-flow

direction. No particular sequence has to be followed to number the branches, i.e., you

decide which branch you want to number as branch no. 1, 2, 3, and so on.

Fig. 3-3 illustrates this numbering procedure for a branching system. We have

arbitrarily called one line as the main line and the remaining lines as branch lines. We

number the upstream node of the main line as 1 and number the remaining nodes on this

line proceeding in the downstream direction. The last node on the main line is 8. Then,

we select one of the branch lines as number 1 and number the nodes on this line as 9 to

11. Of the remaining two branch lines, we call one of them as branch number 2 and

number the nodes on this line as 12 to 14. Branch number 3 provides inflow to the main

line as compared to the other two lines in which flow is from the main line. We number

the upstream node on this line as 15 and the remaining two nodes on this line as 16 and

17.


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Figure 3-3. Schematic Diagram of a Branching System.

3.3 NodeTypes

Assign numbers to designate the type of a node. For example, an upstream reservoir

is node type 1, downstream reservoir as node type 2, and so on. Table 3-1 lists various

types of nodes and numbers to designate their type. Write in a triangle the number

designating the type of a node, i.e., the number in the triangle refers to the type of the

node, and the number in the square represents the number of the node.


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Node Type Node

1 Upstream Reservoir2 Downstream Reservoir

3 Series Junction4 Branching Junction5 Dead End6 Downstream Valve7 Orifice at Lower End8 Intermediate Valve

Table 3-1. Node Types

3.4 Numbering

of

Conduits

Different nodes are connected by conduits, and number these conduits to distinguish

them from another. It is not necessary to follow any particular sequence in numbering the

conduits as long as each conduit in the system in numbered once and the numbering of

the conduits starts at one. For example, the conduit between nodes 1 and 2 may be

numbered 1, 3, or any other number as long as it is not greater than the local number of

conduits in the system. The numbering of conduits should start at 1 and then continue

consecutively to the total number of conduits in the system. It does not matter if conduit

number 1 is located at the downstream end of the system, conduit no. 2 is located ate the

upstream end of the system, and conduit no. 3 is located as some intermediate location.


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4 METHODOLOGY

This chapter describes the procedure followed to obtain the energy dissipation factor

as an integrated part of the transient analysis in piping systems. The methodology

includes the application of the Hardy Cross method for the computation of the initial

steady state conditions and the use of a technique to generate the computational grid

without having an excessive number of grid points. The turbulence model is described

and details on the computation of the initial steady state velocity profile. The

approximation of the time dependent pressure gradient and the numerical scheme to solve

the axisymmetric momentum equation are also included.

4.1 Computationoftheinitialsteadystateconditions

For the computation of the initial steady state conditions a subroutine that solves the

network flow equations using the Hardy Cross linear analysis (adapted from [14]) is used.

The subroutine allows for the input of minor losses, use of either the Darcy-Weisbach or

Hazen-Willians pipe frictional loss, and the provision of rated power for pumps. Either SI

or English units are permissible.

It is necessary to know the flow direction in each pipe element at all times because

the required head summations depend on that flow direction. This process is handled by


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initially designating an assumed flow direction in each pipe element. As the calculations

proceed, the direction of the flow rate is positive if it is in the initial assumed direction

and negative if it reverses direction. Input geometric data require signs based on the

initial assumed flow directions. The subroutine requires input data specifying assigned

pipe numbers for the paths of pipe sections making up primary loops and pipe sections

between fixed-grade nodes. The pipe numbers are input with a positive sign if the

assumed flow direction is in the path direction and with a negative sign if it is opposite.

4.1.1 Generalized Network Equations

Networks of piping such as those shown in Figures 4.1 and 4.2 can be represented by

the following equations.

1. Continuity at thejth interior node:

0)( = ejj QQ 4.1

In which the subscript j refers to the pipes connected to a node, and Qe is the

external demand. The algebraic plus or minus sign convention pertains to the

assumed flow direction. Use the positive sign for flow into the junction, and the

negative sign for the flow out of the junction.


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Figure 4-1. Representative piping network: (a) assumed flow directions and

numbering scheme: (b) designated interior loops; (c) path between two fixed-

grade nodes. (From [14])

2. Energy balance around an interior loop:

= 0)( iiW 4.2

In which the subscript ipertains to the pipes that make up the loop. There will be


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a relation for each of the loops. Here it is assumed that there are not pumps

located in the interior of the network. The plus sign is uses if the flow in the

element is positive in the clockwise sense; otherwise, the minus sign is employed.

3. Energy balance along a unique path or pseudo loop connecting two fixed-grade

nodes:

( ) =+ 0)( HHW ipii 4.3

Where: His the difference in magnitude of the two fixed-grade nodes in the path

ordered in a clockwise fashion across the imaginary pipe in the pseudo loop. The term

(Hp)i is the head across a pump that could exist in the ith pipe element. If F is the

number of fixed-grade nodes, there will be (F 1) unique path equations. The plus and

minus signs in Eq. 4.3 follow the same argument given for Eq. 4.2.

LetPbe the number of pipe elements in the network, Jthe number of interior nodes,

andLthe number of interior loops. Then the following relation will hold if the network is

property defined:

P = J + L + F 1

In Fig. 4.1, J = 4, L = 2, F = 2, so that P = 4 + 2 + 2 1 = 7.

An additional necessary formulation is the relation between discharge and loss in


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each pipe; it is:

2

22 QgA

k

RQW x

+= 4.4

In whichRis the resistance coefficient, given by:

52

8

Dg

fLR

= 4.5

x = 2 for Darcy-Weisbach pipe frictional loss or x = 1.85 for Hazen-Willians pipe

frictional loss, and are the minor loss components associated with each pipe

element.

K

If the minor losses can be defined in terms of an equivalent length, Eq. 4.4 can be

replaced by:

xQRW = 4.6

In which the modified pipe resistance coefficient R is given by

( )[ ]52

8

Dg

LLfR e

+= 4.7


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Where:Leis the equivalent length.

An approximate pump head-discharge representation is given by the polynomial:

( ) 221 QaQaaQH op ++= 4.8

The coefficients ao, a1, and a2 are assumed known; typically, they can be found by

substituting three known data points from a specified pump curve and solving the threeresulting equations simultaneously. In place of defining the pump head-discharge curve,

an alternative means of including a pump in a line is to specify the useful power the

pump puts into the system. The useful, or actual, power is assumed to be constant

and allows H

fW.

pto be represented in the manner:

( )Q

WQH

f

p

.

= 4.9

This equation is particularly useful when the specific operating characteristics of a

pump are unknown.


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Figure 4-2. Two piping networks: (a) tree watering system: P = 17, L = 0, j =

8, F = 10; (b) water distribution system: P = 17, L = 4, J = 12, F = 2. (From

Wood, 1981.)


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4.1.2

Linearization of System Energy Equations

Equation 4.3 is a general relation that can be applied to any path or closed loop in a

network. If it is applied to a closed loop, His set equal to zero, and if no pump exists in

the path or loop, (Hp)iis equal to zero.

Define the function (Q) to contain the nonlinear terms W(Q) and HP(Q) in the form:

( ) ( ) ( ) ( )QHQRQHQWQ Px

P == 4.10

Equation 4.8 can be expanded in a Taylor series as:

( ) ( ) ( ) ( ) ...22

2

+++= oQ

o

Q

o QQdQ

dQQ

dQ

dQQ

oo

4.11

In which Qo is an estimate of Q. To approximate (Q) accurately, Qo should be

chosen so that the difference (Q - Qo) is numerically small. Retaining the first two terms

on the right-hand side of Eq. 4.11, and using Eq. 4.10, yields:

( ) ( ) ( o

Q

Px

ooP

x

o QQ

dQ

dHQRxQHQRQ

o

+ 1 ) 4.12

Note that the approximation to (Q)is now linear with respect to Q.


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The parameter Gis introduced as:

oQ

Px

odQ

dHQRxG = 1 4.13

Substituting Eq. 4.13 into Eq. 4.12 gives:

( ) ( ) ( )GQQQHQRQ ooPx

o +=

( )GQQHW oPoo += 4.14

In which Wo= W(Qo)andHpo=Hp(Qo).Finally, Eq. 4.14 is substituted into Eq. 4.3 to

produce the linearized loop or path energy equation:

( ) ( ) ( )[ ] =++ 0)( HGQQHW iioiipoioi 4.15

4.1.3

Hardy Cross Method

The Hardy Cross method of analysis is a simplified version of the iterative linear

analysis. In Eq. 4.15, let (Qo)Ibe the estimates of discharge from the previous iteration,

and let Qibe the new estimates of the discharge. Define a flow adjustment Q for each


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loop to be:

( )ioi

QQQ = 4.16

The adjustment is applied independently to all pipes in a given loop. Hence Eq. 4.15

can be written as

( ) ( ) =++ 0)( HGQHW iipoioi 4.17

Solving for Q, one has:

( ) ( ) ( )

=i

ipoio

G

HHWQ 4.18

For a closed loop in which no pumps or fixed-grade nodes are present, Eq. 4.18

reduces simply to the form:

( )( )

=i

io

G

WQ 4.19

The hardy Cross iterative solution is outlined in the following steps:

1. Assume an initial estimate of the flow distribution in the network that satisfies

continuity, Eq. 4.1. The closer the initial estimates are to the correct values, the


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fewer will be the iterations required for convergence. One guideline to use is that

in a pipe element as R increases, Qdecreases.

2. For each loop or path, evaluate Qwith Eq. 4.18 or 4.19. The numerators should

approach zero as the loops or paths become balanced.

3. Update the flows in each pipe in all loops and paths, that is, from Eq. 4.16

( ) += QQQ ioi 4.20

The term is used, since a given pipe may belong to more than one loop;

hence the correction will be the sum of corrections from all loops to which the

pipe is common.

Q

4. Repeat steps 2 and 3 until a desired accuracy is attained. One possible criterion to

use is

( )

i

ioi

Q

QQ 4.21

In which is an arbitrarily small number. Typically, 0.001<


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4.2 Generationofthecomputationalgrid

The skin friction in turbulent flow is greater than in laminar flow; therefore, the

velocity gradient is greater at the wall. To assure a good resolution of the velocity

gradient close to the wall it is very important to have small grid steps in this region. A

geometrically expanding grid spacing proposed by Cebeci and Smith (1974) was used in

this study. Figure 4.1 shows a typical grid. This was the same grid adopted by Silva-

Araya and Chaudhry (1997) to compute the velocity distribution during the transient

flow.

x Pipe Center Liner = R

r = 0Pipe Wall

Figure 4-3. Non-uniform grid.


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54

This technique is a geometric progression, in the radial direction, which have the

property that the ratio of lengths of any two adjacent intervals is a constant; that is:

1= jj Khh 4.22

Where the size of the jth -1 interval is, is the size of the jth interval, K is the

ratio of two successive intervals. The distance to the position in the radial direction is

given by the following formula:

1jh jh

)1/()1(1 = KKh j

j j = 1, 2, ...,J 4.23

Where: the length of the first cell and K is equal to the ratio of two successive

steps. The total number of pointsJwas calculated by the following formula:

1h

[ ]K

hRKJ

ln

)/)(1ln(1ln 1+= 4.24

Where: k = ratio of two successive steps, R = pipe radius, = length of the first

interval. The implementation of this grid generation technique was based in knowing the

length of the first interval, the pipe radius and the number of grid points (N).

1h

The constant expansion rate was computed using Newtons iteration method, which


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starts with an initial guess; the convergence is quadratic. Then the grid growth in the

radial direction until the grid points is equal to the pipe radius. In order to compare the

model results, it is important to have a similar grid; therefore this grid was similar to the

grid used by Silva-Araya and Chaudhry (1997).

4.3 Turbulencemodeling

The turbulent boundary layer can be considered as composed of an inner and outer

region. The inner region characteristics are strongly dependent on the conditions near the

wall and are affected by the pressure gradient and the shear stress distribution. This

region is subdivided into a very thin region near the wall, called the viscous sublayer,

where, where the flow is dominated by viscous effects. Farther from the wall the

turbulent stresses become more important. The transition between the viscous sublayer

and the fully turbulent region is called the buffer layer. Beyond this transition the

turbulent stresses dominate the flow and the average velocity profile can be estimated

using a logarithmic equation. In pipe flows the inner region includes most of the pipe

cross section. Figure 4.4 shows the regions of a turbulent boundary layer.


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0

5

10

15

20

25

0.1 1 10 100 1000 10000Y

+

u+

LogarithmicRegion

BufferRegion

ViscousSublayer

OuterRe ion

Figure 4-4. Regions of a Boundary Layer

As a general expression for the Reynolds stresses, Prandtl proposed the mixing length

theory (1925). According to this theory, the Reynolds shear stress is calculated as:

r

u

r

ulvu

= 2)''( 4.25

Where: lis the mixing length. The bar symbol on the velocity was dropped in the

previous equation.

The Reynolds stresses can also be modeled by using the eddy viscosity concept


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introduced by Boussinesq (1877). The basic assumption is that the turbulent stresses are

proportional to the velocity gradient. The same assumption is made in the mixing length

model. The coefficient of proportionality, , was called eddy viscosity and is defined as

=

r

uvu )''( 4.26

The close relation between the mixing length and the eddy viscosity concepts allowsconverting mixing length into eddy viscosity in many cases. A mixing-length model was

used for the inner region of the boundary layer. An eddy viscosity model was selected for

the outer region. The mixing-length values are converted into eddy viscosity to make

both models compatible.

4.3.1

Turbulence Model for the Inner Region

The following mixing length model, proposed by Granville (1989), and normalized

by using the wall variables, is used for the inner region of the boundary layer.

( )+++++ = 1/1 yekyl 4.27

Where , , , mixing length damping parametervlul /=+

vyuy /=+

w /=+ =+1


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(for flows with zero pressure gradient), =u friction velocity given by /wu = ,

=w wall shear stress, l = mixing length, v = kinematics viscosity, k von Karmans

constant andy= radial distance measured form the wall.

Granville (1990) derived the relation between the mixing length model (Eq. 4.27) and

the eddy viscosity as

( )

+= +++ 121

2

1 2 l 4.28

Where is the ratio of the eddy viscosity and the kinematics viscosity.v/ =+

4.3.2

Outer Region Model and Blending Relation

For the outer region the eddy viscosity is computed using a model proposed by Kita

et al. (1980). This is a constant eddy viscosity values across the outer region given by

++ = RCcout 4.29

Where Ccis slightly dependent on the Reynolds number.


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For the range 1 X 104< Re < 2 X 106the value of Cc is given by (Silva Araya and

Chaudhry, 1993).

( ) ( )2Relog0137.0Relog1390.04095.0 +=cC 4.30

If Re < 1 X 104then Cc= 0.07, if Re > 2 X 106then Cc= 0.075.

According to the criterion given by Kita et al (1980). the outer region starts when the

following condition if fulfilled

+=

++

077.011

Re154.0lim

cC

ky 4.31

WhereRe+= u+R / v andRis the pipe radius.

A smoother transition between the inner and outer eddy viscosities can be achieved

by using a blending relation. This merging is accomplished by an exponential function

given by:

( )++ ++ = outyyboute 4.32


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Where the exponent bis computed as

( )( )++

++

=

cout

cout

yy

Lnb

/ 4.33

The value of is linearly interpolated between the beginning of the falling limb of

the eddy viscosity distribution and

+c

out . is the value of y+cy

+corresponding to . Also

is linearly interpolated between and R

+c

+outy

+limy

+= uR/v. The corresponding value of

is . Figure 4.5 shows a typical eddy viscosity distribution with the inner, transition and

outer regions.

+out

Y+

+

Inner Region Transition OuterRegion

Figure 4-5. Eddy Viscosity Distribution.


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4.4 Initialvelocityprofile

For laminar flow the velocity profile is computed using the following expression

(White, 1991)

( 224

1rR

x

pu

=

) 4.34

Where R = pipe internal radius, xp / = pressure gradient, = dynamic viscosity.

No eddy viscosity computation is required.

For turbulent flow, the initial mean velocity distribution can be written in the form:

)( ++ = yfu 4.35

Where f(y+)is the profile across the pipe, including the sublayer, buffer, logarithmic

and outer regions.

If the shear stress, +, and the eddy viscosity distribution, +, are known, the velocity

profile can be obtained by the following equation (Silva Araya and Chaudhry, 1993).

+

+

+

++

+=

y

dyu0 1

4.36


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Where u+= 0 at y+ = 0. For a fully develop steady-state flow, the shear stress +,

varies linearly across the pipe section from 0 at the wall to 1 at the pipe centerline.

The gradients of the velocity profile are computed by using a forward-time, central-

spaced explicit scheme modified for a geometrically expanded grid. The partial

derivatives can be approximated as (Hoffman, 1989)

( )( )

+

++

r

uuu

r

u iii

11 1221

4.37

Where ii rr = /1 is the ratio of two successive grid spaces and .ii rrr =

1

In the previous expressions the subscript i refer to the i-th position inside the pipe

cross section. A second-order backward approximation is used for the gradient at the wall

( )( )21

1

2

22

)1(

1

+

+

NN

NN

w rr

uu

r

u

4.38

WhereNis the number of grid lines (the grid line at the wall is numberN).

Prior to the initiation of the transient flow, the dissipation integral is computed

substituting the derivatives of the initial flow profile into Equation 2.8. The integrated


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value is compared with the initial steady-state value given by Equation 2.10.

4.5 Computationofthepressuregradient

The computation of the pressure gradient needed in the Equation 3.14 is time

dependent and must be estimated from the local conditions. Given an approximation of

the velocity profile, for example the velocity profile at the previous time step, the

pressure gradient can be estimated from the momentum equation, expressed in a compact

form (Silva Araya and Chaudhry, 1997) as:

=

r

ur

rt

ur

x

prt

4.39

Where: +=t , = kinematics viscosity, = eddy viscosity. Multiplying

Equation 3.50 by 2 and integrating with respect to r and expressing it in a compact

form yields:

Rr

t

r

u

RtA

Q

x

p

=

=

21 4.40


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where tQ / can be approximated by the change in discharge from the previous

time step. Adjustments in the value of the pressure gradient may be required if the flow is

reduced drastically; but Equation 3.51 is a good initial approximation (Silva Araya and

Chaudhry, 1997).

4.6 Shearstressdistribution

Equation 2.14 can also be expressed in terms of the shear stress as:

( )r

r

rx

p

t

u

+

=

1 4.41

Where is the shear stress at a distance rfrom the pipe centerline.

Multiplying Equation 4.32 by rand integrating with respect to r, we obtain:

+

=

rr

x

prrdr

t

u

0

2

2 4.42

Dividing by the shear stress at the wall, the following non-dimensional for results:


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x

prrdr

t

u

r w

r

w

+

=

+

20 4.43

The time derivative is approximated as:

tMOC

ttt uu

t

u

+ 4.44

Where tMOCis the time step chosen to solve the characteristic equations.

For steady state flow this expression reduces to:

x

pRw

=

2 4.45

Equation 4.34 is used to update the shear stress distribution after every time step.

4.7 Computationofthevelocityprofile

This research adopted the procedure to compute the axisymmetric momentum

equation for the velocity profile across the unequally spaced grid, developed by Silva

Araya and Chaudhry, (1997). This procedure is reference as the McCormack scheme

predictor-corrector where is second order accurate in space and time scheme (Anderson

et al., 1984). In the predictor a step forward finite-differences approach is used for the


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spatial derivatives, and a backward finite differences is used in the corrector step. This

sequence was alternated during the computations from one time step to the next; as

reported by these researchers. Therefore:

4.7.1

Predictor Step

The approximations for the partial derivatives in this step are:

tuu

tu

n

ii

=

*

4.46

++

=

r

uu

r

u n

i

n

i 1 4.47

211

2

2

))(1(

))1((2

+

+

++=

r

uuu

r

u n

i

n

i

n

i

4.48

Where: , andii rrr = ++ 1 1 = ii rrr ii rr = + /1 . The superscript n refers to

the time level, the subscript irefers to the grid point, and the asterisk indicates a predicted

value.

Thus, the momentum equation is solved for (Silva Araya and Chaudhry, 1997) as*iu

211

1

1

*

))(1(

)1(2

1

+

++

+

+

+++

++

+=

r

uuut

tr

uu

dr

d

rt

x

puu

i

n

i

n

i

n

i

n

i

n

n

ii

4.49


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4.7.2

Corrector Step

The predicted values are used in a backward finite-difference approximation for the

first spatial derivative and are also used for the second spatial derivative. The

approximations are

t

uu

t

u n

ii

=

** 4.50

=

r

uu

r

u ii*

1*

4.51

2

*1

**1

2

2

))(1(

))1((2

+

+

++=

r

uuu

r

u iii

4.52

Where (**) indicates a corrected value. Therefore, the corrected velocity is:

2

*1

**1

*1

*1

**

))(1(

)1(2

1

+

+

+

+

+++

++

+=

r

uuut

tr

uu

dr

d

rt

x

puu

iii

ii

n

n

ii

4.53

Finally, the velocity for the next time step is given by:

2

***1 iin

i

uu

u

=+

4.54

The derivative of the eddy viscosity is obtained (Silva Araya, 1993) as:


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))(1(

)1(

1

122

1

ii

iii

rrdr

d

+

++=

+

4.55

The velocity distribution is computed at every section of the pipe. At the end of the,

method of characteristic (MOC) and for every time step the velocity profile is integrated

across the pipe section to compute the flow discharge. If the integrated discharge differs

from the one predicted by the MOC, the pressure gradient is slightly adjusted to satisfy

the continuity equation. The adjustment is done until both discharges differ by less than

5%.


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5 VERIFICATION

This chapter presents the results obtained by using the unsteady friction model for

piping systems. The generalized algorithm was analyzed for three cases: single, series

and branching piping systems. The computed pressures are compared with the

experimental data obtained for Silva Araya (1993) for a single piping system case. For a

series and a branching piping system the computed results obtained by using unsteady

friction are compared with the data obtained by using steady friction model.

5.1 SinglePipingSystem

The experimental data obtained by Silva Araya (1993) was used for the verification

of the generalized algorithm in a single piping system. The experimental apparatus used

is shown in Figure 5.1. The system was built at the Albrook Hydraulics Laboratory at

Washington State University.

The arrangement consists of a constant-head tank open to the atmosphere connected

by a horizontal pipe to a motorized butterfly valve located 32 m downstream of the tank.

A pump supplies water from the sump to the tank. At the valve, the water discharges into

another open tank and returns to the pump. Four pressure transducers were used to record

the pressure fluctuations for the valve closure in each section. These were located at X =


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L / 4, X = L / 2, X = 3L / 4 and X = L from the reservoir.

1 6

1

1 2

Figure 5-1. Schematic of Experimental Setup (Silva Araya, 1993)

The pipe material is commercial steel with an internal diameter of 0.101 m and a wall

thickness of 0.00635 m. The initial steady-state discharge was 0.0165 m3/s. This

corresponds to an average velocity of 20.02 m/s and an initial Reynolds number of 2.06

X 105. The relative roughness for this simulation was 0.001. Other parameters are listed

in Table 5.1. Table 5.2 shows the different types of nodes present in this system.

The transient conditions for this case were produced by a gradual closure of the

downstream valve. The valve closed completely in 1.085 sec. The curve of the effective

valve opening versus time curve used in the simulation is given in Figure 5.2


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Parameter Value

Initial Discharge 0.0165 m3/sUpstream Tank Head 2.65 mEntrance Loss Coefficient 1.44Friction Factor 0.0223Initial Valve Position 0= 0.62Wave Speed 1125 m/sDensity 999.1 Kg/m3

Dynamic Viscosity 1.06 X 10-3m2/s

Table 5-1. Parameters for Single Piping System.

Node Number Node Type

1 (1) Upstream Reservoir2 (6) Downstream Valve

Table 5-2. Node type for Single Piping System.

Effective Valve Opening

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

0 0.2 0.4 0.6 0.8 1 1.2

Time (s)

EffectiveOpening,

TAU

Figure 5-2. Characteristic Curve for Valve in Single Piping System.


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Figures 5.3 (a) to 5.3 (d) compare the measured pressures with the computed results

obtained by using steady friction. The maximum pressure is predicted accurately without

the unsteady friction model. The oscillations damp much more rapidly than the predicted

values. The shift in the frequency of the oscillations is due to the reduction of the wave

speed caused by the low pressures after the valve closure.


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H Vs T (Valve)

(20)

(10)

-

10

20

30

40

50

60

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2

Time (s)

PressureHead(m)

Experimental Data

Steady Friction

(a) At the Valve

H Vs T (x = 3L/4)

(20)

(10)

-

10

20

30

40

50

60

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2

Time (s)

Pressure

Hea

d(m)

Experimental Data

Steady Friction

(b) At x = 3L / 4

Figure 5-3. Pressure Head Histories for a Single Piping System Using Steady

Friction.

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