ANALYSIS A N D MODELLING OF NON-STEADY FLOW N PIPE A N D CHANNEL NETWORKS Vinko Jovic ANALYSIS AND MODELLING OF NON-STEADY FLOW IN PIPE AND CHANNEL NETWORKS ANALYSIS AND MODELLING OF NON-STEADY FLOW IN PIPE AND CHANNEL NETWORKS Vinko Jovi´c University of Split, Croatia A John Wiley & Sons, Ltd., Publication This edition first published 2013 C 2013 John Wiley & Sons, Ltd Registered office John Wiley & Sons Ltd, The Atrium, Southern Gate, Chichester, West Sussex, PO19 8SQ, United Kingdom For details of our global editorial offices, for customer services and for information about how to apply for permission to reuse the copyright material in this book please see our website at www.wiley.com The right of the author to be identified as the author of this work has been asserted in accordance with the Copyright, Designs and Patents Act 1988 All rights reserved No part of this publication may be reproduced, stored in a retrieval system, or transmitted, in any form or by any means, electronic, mechanical, photocopying, recording or otherwise, except as permitted by the UK Copyright, Designs and Patents Act 1988, without the prior permission of the publisher Wiley also publishes its books in a variety of electronic formats Some content that appears in print may not be available in electronic books Designations used by companies to distinguish their products are often claimed as trademarks All brand names and product names used in this book are trade names, service marks, trademarks or registered trademarks of their respective owners The publisher is not associated with any product or vendor mentioned in this book This publication is designed to provide accurate and authoritative information in regard to the subject matter covered It is sold on the understanding that the publisher is not engaged in rendering professional services If professional advice or other expert assistance is required, the services of a competent professional should be sought Library of Congress Cataloging-in-Publication Data Jovic, Vinko Analysis and modelling of non-steady flow in pipe and channel networks / Vinko Jovic pages cm Includes bibliographical references and index ISBN 978-1-118-53214-0 (hardback : alk paper) – ISBN 978-1-118-53686-5 (mobi) – ISBN 978-1-118-53687-2 (ebook/epdf) – ISBN 978-1-118-53688-9 (epub) – ISBN 978-1-118-53689-6 (wiley online library) Pipe–Hydrodynamics Hydrodynamics I Title TC174.J69 2013 621.8 672–dc23 2012039412 A catalogue record for this book is available from the British Library ISBN: 978-1-118-53214-0 Typeset in 9/11pt Times by Aptara Inc., New Delhi, India Contents Preface 1.1 1.2 1.3 1.4 1.5 2.1 2.2 2.3 xiii Hydraulic Networks Finite element technique 1.1.1 Functional approximations 1.1.2 Discretization, finite element mesh 1.1.3 Approximate solution of differential equations Unified hydraulic networks Equation system 1.3.1 Elemental equations 1.3.2 Nodal equations 1.3.3 Fundamental system Boundary conditions 1.4.1 Natural boundary conditions 1.4.2 Essential boundary conditions Finite element matrix and vector Reference Further reading 1 21 23 23 24 25 28 28 30 30 36 36 Modelling of Incompressible Fluid Flow Steady flow of an incompressible fluid 2.1.1 Equation of steady flow in pipes 2.1.2 Subroutine SteadyPipeMtx 2.1.3 Algorithms and procedures 2.1.4 Frontal procedure 2.1.5 Frontal solution of steady problem 2.1.6 Steady test example Gradually varied flow in time 2.2.1 Time-dependent variability 2.2.2 Quasi non-steady model 2.2.3 Subroutine QuasiUnsteadyPipeMtx 2.2.4 Frontal solution of unsteady problem 2.2.5 Quasi-unsteady test example Unsteady flow of an incompressible fluid 2.3.1 Dynamic equation 2.3.2 Subroutine RgdUnsteadyPipeMtx 2.3.3 Incompressible fluid acceleration 37 37 37 40 42 45 51 57 59 59 60 61 63 65 65 65 68 69 vi Contents 2.3.4 Acceleration test 2.3.5 Rigid test example References Further Reading 3.1 3.2 3.3 3.4 3.5 3.6 4.1 4.2 4.3 72 72 75 75 Natural Boundary Condition Objects Tank object 3.1.1 Tank dimensioning 3.1.2 Tank model 3.1.3 Tank test examples Storage 3.2.1 Storage equation 3.2.2 Fundamental system vector and matrix updating Surge tank 3.3.1 Surge tank role in the hydropower plant 3.3.2 Surge tank types 3.3.3 Equations of oscillations in the supply system 3.3.4 Cylindrical surge tank 3.3.5 Model of a simple surge tank with upper and lower chamber 3.3.6 Differential surge tank model 3.3.7 Example Vessel 3.4.1 Simple vessel 3.4.2 Vessel with air valves 3.4.3 Vessel model 3.4.4 Example Air valves 3.5.1 Air valve positioning 3.5.2 Air valve model Outlets 3.6.1 Discharge curves 3.6.2 Outlet model Reference Further reading 77 77 77 79 83 90 90 91 91 91 94 99 101 108 112 117 121 121 124 126 127 128 128 133 135 135 137 138 138 Water Hammer – Classic Theory Description of the phenomenon 4.1.1 Travel of a surge wave following the sudden halt of a locomotive 4.1.2 Pressure wave propagation after sudden valve closure 4.1.3 Pressure increase due to a sudden flow arrest – the Joukowsky water hammer Water hammer celerity 4.2.1 Relative movement of the coordinate system 4.2.2 Differential pressure and velocity changes at the water hammer front 4.2.3 Water hammer celerity in circular pipes Water hammer phases 4.3.1 Sudden flow stop, velocity change v0 → 4.3.2 Sudden pipe filling, velocity change → v0 4.3.3 Sudden filling of blind pipe, velocity change → v0 4.3.4 Sudden valve opening 4.3.5 Sudden forced inflow 141 141 141 141 143 143 143 145 147 149 151 154 156 159 161 Contents 4.4 4.5 4.6 4.7 4.8 4.9 4.10 4.11 5.1 5.2 5.3 5.4 5.5 5.6 5.7 5.8 vii Under-pressure and column separation Influence of extreme friction Gradual velocity changes 4.6.1 Gradual valve closing 4.6.2 Linear flow arrest Influence of outflow area change 4.7.1 Graphic solution 4.7.2 Modified graphical procedure Real closure laws Water hammer propagation through branches Complex pipelines Wave kinematics 4.11.1 Wave functions 4.11.2 General solution Reference Further reading 164 167 171 171 174 176 178 179 180 181 183 183 183 187 187 187 Equations of Non-steady Flow in Pipes Equation of state 5.1.1 p,T phase diagram 5.1.2 p,V phase diagram Flow of an ideal fluid in a streamtube 5.2.1 Flow kinematics along a streamtube 5.2.2 Flow dynamics along a streamtube The real flow velocity profile 5.3.1 Reynolds number, flow regimes 5.3.2 Velocity profile in the developed boundary layer 5.3.3 Calculations at the cross-section Control volume Mass conservation, equation of continuity 5.5.1 Integral form 5.5.2 Differential form 5.5.3 Elastic liquid 5.5.4 Compressible liquid Energy conservation law, the dynamic equation 5.6.1 Total energy of the control volume 5.6.2 Rate of change of internal energy 5.6.3 Rate of change of potential energy 5.6.4 Rate of change of kinetic energy 5.6.5 Power of normal forces 5.6.6 Power of resistance forces 5.6.7 Dynamic equation 5.6.8 Flow resistances, the dynamic equation discussion Flow models 5.7.1 Steady flow 5.7.2 Non-steady flow Characteristic equations 5.8.1 Elastic liquid 5.8.2 Compressible fluid 189 189 189 190 195 195 198 202 202 203 204 205 206 206 207 207 209 209 209 210 210 210 211 212 212 213 215 215 217 220 220 223 viii Contents 5.9 Analytical solutions 5.9.1 Linearization of equations – wave equations 5.9.2 Riemann general solution 5.9.3 Some analytical solutions of water hammer Reference Further reading 225 225 226 227 229 229 6.1 Modelling of Non-steady Flow of Compressible Liquid in Pipes Solution by the method of characteristics 6.1.1 Characteristic equations 6.1.2 Integration of characteristic equations, wave functions 6.1.3 Integration of characteristic equations, variables h, v 6.1.4 The water hammer is the pipe with no resistance 6.1.5 Water hammers in pipes with friction Subroutine UnsteadyPipeMtx 6.2.1 Subroutine FemUnsteadyPipeMtx 6.2.2 Subroutine ChtxUnsteadyPipeMtx Comparison tests 6.3.1 Test example 6.3.2 Conclusion Further reading 231 231 231 232 234 235 243 251 252 255 261 261 263 264 Valves and Joints Valves 7.1.1 Local energy head losses at valves 7.1.2 Valve status 7.1.3 Steady flow modelling 7.1.4 Non-steady flow modelling Joints 7.2.1 Energy head losses at joints 7.2.2 Steady flow modelling 7.2.3 Non-steady flow modelling Test example Reference Further reading 265 265 265 267 267 269 279 279 279 282 288 290 290 Pumping Units Introduction Euler’s equations of turbo engines Normal characteristics of the pump Dimensionless pump characteristics Pump specific speed Complete characteristics of turbo engine 8.6.1 Normal and abnormal operation 8.6.2 Presentation of turbo engine characteristics depending on the direction of rotation 8.6.3 Knapp circle diagram 8.6.4 Suter curves Drive engines 8.7.1 Asynchronous or induction motor 291 291 291 295 301 303 305 305 6.2 6.3 7.1 7.2 7.3 8.1 8.2 8.3 8.4 8.5 8.6 8.7 305 305 308 310 310 Contents 8.8 8.9 8.10 8.11 8.12 8.13 9.1 9.2 9.3 9.4 9.5 9.6 ix 8.7.2 Adjustment of rotational speed by frequency variation 8.7.3 Pumping unit operation Numerical model of pumping units 8.8.1 Normal pump operation 8.8.2 Reconstruction of complete characteristics from normal characteristics 8.8.3 Reconstruction of a hypothetic pumping unit 8.8.4 Reconstruction of the electric motor torque curve Pumping element matrices 8.9.1 Steady flow modelling 8.9.2 Unsteady flow modelling Examples of transient operation stage modelling 8.10.1 Test example (A) 8.10.2 Test example (B) 8.10.3 Test example (C) 8.10.4 Test example (D) Analysis of operation and types of protection against pressure excesses 8.11.1 Normal and accidental operation 8.11.2 Layout 8.11.3 Supply pipeline, suction basin 8.11.4 Pressure pipeline and pumping station 8.11.5 Booster station Something about protection of sewage pressure pipelines Pumping units in a pressurized system with no tank 8.13.1 Introduction 8.13.2 Pumping unit regulation by pressure switches 8.13.3 Hydrophor regulation 8.13.4 Pumping unit regulation by variable rotational speed Reference Further reading 311 312 314 314 318 321 322 323 323 327 333 334 336 339 341 345 345 345 346 348 350 353 355 355 355 358 360 362 362 Open Channel Flow Introduction Steady flow in a mildly sloping channel Uniform flow in a mildly sloping channel 9.3.1 Uniform flow velocity in open channel 9.3.2 Conveyance, discharge curve 9.3.3 Specific energy in a cross-section: Froude number 9.3.4 Uniform flow programming solution Non-uniform gradually varied flow 9.4.1 Non-uniform flow characteristics 9.4.2 Water level differential equation 9.4.3 Water level shapes in prismatic channels 9.4.4 Transitions between supercritical and subcritical flow, hydraulic jump 9.4.5 Water level shapes in a non-prismatic channel 9.4.6 Gradually varied flow programming solutions Sudden changes in cross-sections Steady flow modelling 9.6.1 Channel stretch discretization 9.6.2 Initialization of channel stretches 363 363 363 365 365 368 372 377 378 378 380 382 383 391 395 398 401 401 402 x 9.7 9.8 9.9 9.10 9.11 10 10.1 10.2 10.3 10.4 10.5 10.6 Contents 9.6.3 Subroutine SubCriticalSteadyChannelMtx 9.6.4 Subroutine SuperCriticalSteadyChannelMtx Wave kinematics in channels 9.7.1 Propagation of positive and negative waves 9.7.2 Velocity of the wave of finite amplitude 9.7.3 Elementary wave celerity 9.7.4 Shape of positive and negative waves 9.7.5 Standing wave – hydraulic jump 9.7.6 Wave propagation through transitional stretches Equations of non-steady flow in open channels 9.8.1 Continuity equation 9.8.2 Dynamic equation 9.8.3 Law of momentum conservation Equation of characteristics 9.9.1 Transformation of non-steady flow equations 9.9.2 Procedure of transformation into characteristics Initial and boundary conditions Non-steady flow modelling 9.11.1 Integration along characteristics 9.11.2 Matrix and vector of the channel finite element 9.11.3 Test examples References Further reading 404 406 407 407 407 409 411 412 413 414 414 416 417 422 422 423 424 425 425 427 431 434 435 Numerical Modelling in Karst Underground karst flows 10.1.1 Introduction 10.1.2 Investigation works in karst catchment 10.1.3 The main development forms of karst phenomena in the Dinaric area 10.1.4 The size of the catchment Conveyance of the karst channel system 10.2.1 Transformation of rainfall into spring hydrographs 10.2.2 Linear filtration law 10.2.3 Turbulent filtration law 10.2.4 Complex flow, channel flow, and filtration Modelling of karst channel flows 10.3.1 Karst channel finite elements 10.3.2 Subroutine SteadyKanalMtx 10.3.3 Subroutine UnsteadyKanalMtx 10.3.4 Tests Method of catchment discretization 10.4.1 Discretization of karst catchment channel system without diffuse flow 10.4.2 Equation of the underground accumulation of a karst sub-catchment Rainfall transformation 10.5.1 Uniform input hydrograph 10.5.2 Rainfall at the catchment Discretization of karst catchment with diffuse and channel flow References Further reading 437 437 437 437 438 443 446 446 447 449 451 453 453 454 456 458 463 463 466 468 468 473 474 477 477 Hydraulic Vibrations in Networks 513 SrcPnt=FindPoint(trim(Key)) if(SrcPnt.eq.0) stop 'No such point' if(InitVibra(SrcPnt,FromFreq,StepFreq,PhaseFreq,NoFreqSteps)) then k=1,NoFreqSteps if(.not.VibraSolver(k)) then call OutputMsg("Not allowed elements in model!") call CloseVibra exit endif enddo call CloseVibra endif end subroutine Vibrations It should be noted that the frequency is predetermined within the subroutine Vibrations, and that the calculation results are printed in a formatted form in the file *.vib for all points in the model configuration, which is appropriate for a smaller number of points only This program solution is made as a textbook example of vibration for the requirements of tests in the next section An interested reader can easily change the attached program sources at their own discretion, and link the precompiled SimpipCore project The entire calculation of hydraulic vibration is implemented in Fortran module VibraModule, logical function VibraSolver(kFreq)for which guidelines are shown here: module VibraModule use GlobalVars implicit none private ch,sh,GetBDC,IncVar private GetBDC,IncVar other specifications and definitions contains complex*16 function ch(x) ! complex function sinh(x) complex*16 x ch=(exp(x)+exp(-x))/2 end function ch complex*16 function sh(x) ! complex function sinh(x) complex*16 x sh=(exp(x)-exp(-x))/2 end function sh subroutine GetBDC end subroutine GetBDC subroutine InvVar end subroutine InvVar 514 Analysis and Modelling of Non-Steady Flow in Pipe and Channel Networks logical function VibraSolver(kFreq) use CplxSimpipFront implicit none integer kFreq real Whs,Area,Resist,dL complex*16 elem_mtx(2,2),elem_rhs(2) real*8 freq,Omega integer ixelem,ielem integer iconnect(2) freq = StartFreq+(kFreq-1)*FreqStep Omega=2*Pi_*freq call CplxIniFrnt Cmplx_dU=cmplx(0,0) AmplitudaH=0 call GetBDC(nPoints,Omega,Cmplx_Bdc) ixelem=1,size(elems) !SizeOfElems() ielem = front_order(ixelem) if(ielem.eq.0) cycle if(Elems(ielem).infront==0) cycle select case(Elems(ielem).tip) case (PIPE_OBJ) call GetPipeVibraInfo(ielem,Whs,Area,Resist,dL) call PipeVibraMtx(Omega,Whs,Area,Resist,dL,elem_mtx) case (JOINT_OBJ) case (VALVE_OBJ) case default VibraSolver=.false return endselect ! pass the element to the frontal procedure: iconnect(1)=Elems(ielem).n1 iconnect(2)=Elems(ielem).n2 call CplxFrontU(iconnect, ,Cmplx_Bdc,Cmplx_dU,LunFrn) enddo ! Backward frontal procedure call CplxFrontB(LunFrn,nPoints,Cmplx_dU) kVibraStage=kFreq call IncVar(freq,nPoints,Cmplx_dU,AmplitudaH) VibraSolver=.true end function VibraSolver other procedures end module VibraModule Hydraulic Vibrations in Networks 515 The function VibraSolver uses module CplxSimpipFront to solve vibrations, where a generated equation system is solved by the frontal method of assembly and elimination in complex number form: module CplxSimpipFront implicit none specifications and definitions private flact,fldea contains logical function OpenFrontalLun(FrontLen,lunfrn) end function OpenFrontalLun subroutine flact(noelnd,lelnod,nactiv,mxactv,kamo,lactiv) end subroutine flact subroutine fldea(nactiv,lactiv) end subroutine fldea subroutine CplxIniFrnt() end subroutine CplxIniFrnt subroutine CplxFrontU(iconnect, ,lun) end subroutine CplxFrontU subroutine CplxFrontB(lun, ,dU) end subroutine CplxFrontB end module CplxSimpipFront Well-known concepts and steps from the hydraulic solution are used and explained in the program solution SimpipCore The names of previous functions and subroutines can also be used, for example subroutine GetBDC or subroutine IncVar, because they can be declared private by attribute in modules where they are located All variables and procedures that are not declared with the attribute private are visible because they have the attribute public as implied 12.7 Illustrative examples Simple pipe vibration analysis Let us consider a simple pipe of length L = 500 m and inner diameter D = 500 mm filled by water, see Figure 12.1 The source of vibration is located at the end of the pipe The amplitude of the source is 1% of the prescribed Q value, that is l/s, and the phase angle for analyzed frequency response is After we select Option +Vibrations analysis, the SimpipCore program will ask us to “Enter point name:” of the vibration source location and computation starts The computation results will be stored in a file Vibra test a).vib 516 Analysis and Modelling of Non-Steady Flow in Pipe and Channel Networks test file: Vibra test a).simpip Points p0 100.00 D = 500 mm, s = 0.75 mm 0.00 k = 0.1 mm, steel, c = 1133.112 ms−1 Q = 200 l/s Δl = 25 m L = 500 m c = 2.266 s −1 L Figure 12.1 0 points p1 to p19 p20 500 0 Pipes c1 p0 p1 0.5 1.00E-03 0.0075 steel subdivision on 20 elements c20 p19 p20 0.5 1.00E-03 0.0075 steel Piezo p0 100 Charge p20 -0.200 Option +Vibrations Simple analysed pipe Figure 12.2 shows the piezometric head amplitudes for the location of the vibration source for a stage at time t = It is obvious that the pipe has resonant frequencies For a simple pipeline the fundamental circular frequency may be computed as ω0 = πc = 3.56 s−1 2L where L = 500 m and c = 1133.112 m/s For this simple system higher tones may be computed by multiplying the fundamental one by odd prime numbers Amplitude H′ ω0 = 3.56 s −1 ω1 = 10.68 s −1 10 000 5000 0 10 Freq ω[s −1] Figure 12.2 Nodal response vs excitation frequency 11 12 13 Hydraulic Vibrations in Networks 517 test file: Vibra test a).simpip 100.00 Points p1 0 p2 200 0 p3 500 0 p4 200 300 Pipe c1 0.5 0.0075 1.00E-03 steel p1 p2 c2 0.5 0.0075 1.00E-03 steel p2 p3 c3 0.5 0.0075 1.00E-03 steel p2 p4 Piezometric p3 100 Charge p1 -0.200 Option +Vibrations D = 500 mm, s = 0.75 mm k = 0.1 mm, steel, c = 1133.112 ms−1 L = 300 m L = 200 m L = 300 m Q = 200 l/s 0.00 Blind end Figure 12.3 Simple network The analysis has been performed in steps of frequencies 0.001 s−1 Using finer steps we may expect very high peaks of head responses, telling us that resonance will really occur It may be noticed that the vibrations are dumped (flow resistance effect), but the resonance may occur In order to make this clear, a computation with finer steps must be performed for a range of observed dangerous frequencies Simple network Let us consider forced vibrations in a simple branched system, see Figure 12.3 The system consists of a pipeline and a blind ended branch This example differs from the previous one because of the presence of the branch pipe of length 300 m Even for this simple branched system the resonant frequencies can be very hard to compute in advance Figure 12.4 shows the nodal head’s response vs excitation frequencies for the steady flow conditions in Amplitude H′ 10 000 1′ 2′ 3′ 4′ 5′ 6′ 5000 0 10 20 30 40 50 Freq ω [s −1] Figure 12.4 Nodal response vs excitation frequency 60 518 Analysis and Modelling of Non-Steady Flow in Pipe and Channel Networks Table 12.1 ω1 ω2 ω3 ω4 ω5 ω6 ω7 ω8 2.53 7.06 10.74 15.27 20.32 24.86 28.54 33.07 ω1 ω2 ω3 ω4 ω5 ω6 ω7 ω8 38.12 42.65 46.34 50.87 55.92 60.45 the main pipe (Q = 200 l/s and no branch flow) The analysis has been performed in steps of circular frequencies 0.001 × 2π s−1 from to 10 × 2π s−1 It can be seen that this system has eight resonant circular frequencies that are repeated after the eighth step, see Table 12.1 It should be noted that the size of the pipe element is not limited for the determination of resonant frequencies Unlike the first example where the pipe was subdivided into 20 elements, in the second example every pipe is considered a single pipe element This can be explained by the fact that time disappears as a variable from the developed harmonic state, that is it is a harmonic stationary state Reference Streeter, V.L and Wylie, E.B (1967) Hydraulic Transients, McGraw−Hill Book Co., New York, London, Sydney Further reading Fox, J.A (1977) Hydraulic Analysis of Unsteady Flow in Pipe Networks MacMillan Press, London Jovi´c, V (1987) Modelling of non−steady flow in pipe networks Proc 2nd Int Conf NUMETA ’87 Martinus Nijhoff Pub Swansea Jovi´c, V (1992) Modelling of hydraulic vibrations in network systems International Journal for Engineering Modelling 5: 11–17 Watters, G.Z (1984) Analysis and Control of Unsteady Flow in Pipe Networks Butterworths, Boston Index Abnormal operation, 305, 307, 321, 333 Absolute temperature, 165, 189, 191, 500 wave velocity, 423 Absorber, 349 Accelerating, 105 Accident, 56, 341, 344, 361 Accuracy norm, 55 testing, 54 Adiabatic process, 192, 193 Adjacent elements, 4, 11, 12 Air bubble, 130, 131, 391 density, 125, 132, 133 reference state, 124 tanks, 28 valve, 124–8, 130, 131, 133–5, 349, 350, 352, 359 Algebraic equation, 18, 26, 227, 223, 234, 426, 511 sign, 18, 25, 27, 32, 39, 46, 55, 58, 59, 106, 109, 392 Allocation, 51, 535 Amplification, 98, 105, 120 Amplitude complex, 506, 508–11 excitation, 512 finite, 143, 144, 407, 408, 411, 412 maximum, 102, 104 small, 107 Analysis approximate, 131 dimensionless, 301 harmonic, 506 numerical, 1, 28 simple, 171, 217 of small oscillations, 101 of vibrations, 505, 515 water hammer, 176 Analytical solutions, 225–8 Angular momentum, 291, 292, 295 velocity, 104, 293, 296, 301, 304, 305, 310, 312, 314, 318, 320–28, 331–40, 342 Approximation base, 2, 3, 10, 475 Area of influence, 232 Artificial source, 501, 502 Assembling algorithm, 5, 22, 25, 42, 495 Asymmetric losses, 80, 554 resistance, 99, 122, 125, 133 throttle, 94, 95, 97, 99, 100, 108, 128, 129, 553 Asynchronous motor, 310–12 Atmospheric pressure, 123–6, 133, 159, 164, 165, 191, 193–5, 207, 208, 218, 388, 443, 469, 500, 554 Automatic bypass, 333 Axial flow, 304, 307 Balance discharge, 77, 350 Balance of forces, 67, 165 Ball valve, 180, 266 Banach fixed point theorem, 28 Band width, 42–5, 48, 526 Basin groundwater, 437 suction, 344–6 stilling, 388 Basic variables, 58, 61, 239 Basis function, 4, 7, 10, 19, 20 vectors, 2, 4, 7, 20 Bend joint, 279, 280, 288 sharp, 280 Analysis and Modelling of Non-Steady Flow in Pipe and Channel Networks, First Edition Vinko Jovi´c © 2013 John Wiley & Sons, Ltd Published 2013 by John Wiley & Sons, Ltd 520 Bernoulli’s equation, 67, 300 Blade shape, 294 Blind pipe, 156–8, 183, 228, 511, 517 Booster station, 340, 342, 344, 346, 347–348, 350, 351, 352 Boundary conditions (in general), 7, 21, 45, 49, 53, 54, 59, 61, 72, 85, 228, 238, 239, 241, 312, 402, 403, 406, 411, 424, 426, 427, 430, 488, 489, 490, 495, 502, 508 downstream, 23, 384 essential, 30, 48, 53, 424 natural, 16, 17, 28–30, 48, 61, 424, 496, 501 prescribed, 63, 64 upstream, 406, 509 Boundary layer, 68, 203, 204, 206, 213, 363, 365, 366, 371, 390, 391, 398, 400, 505 Boussinesq coefficient, 204, 205, 210, 213, 418, 420 equation, 8, 10 Brakers, 312 Branch blind, 517, 518 channel, 401, 402, 404 hydraulic element, 21–4, 31, 45, 52, 64, 69, 485, 486 joint, 265, 279 nodal, 510, 511 pipe, 53, 54, 57, 64, 183, 521, 528 pump, 339 reflux, 277, 278 simple, 182, 517 valve, 270, 272, 273, 333 Breakdown, 52, 310, 311, 322, 323 Broad-crested, 398, 399, 400 Bulk modulus, 148, 149, 192, 194 Butterfly valve, 180, 266 Calculations of channel flow, 366 of the characteristics, 237 at the cross-section, 204 non-steady, 30, 31, 82 recursion, 240 resistance, 367 steady state, 29 unsteady state, 65, 72, 305, 312, 324, 327 Capacitive matrix, 10 Catchments, 437–9, 441, 443–7, 463–8, 470, 471, 473, 474, 476 Cavitation, 165, 300 Centrifugal force, 93, 291 pump, 291 Channel ascending, 439 branch, 442 Index closed, 369–71, 377 compound section, 371, 374–7 conveyance, 368–72, 416 of karst channel, 439, 440, 442, 446–54, 459, 461, 463, 465, 466, 468, 469, 471, 472 Coriolis coeff., 205, 210, 213, 365, 373–6, 387, 420 critical flow, 372–4, 383, 389, 390, 411 critical slope, 377, 383 descending, 440 finite element, 401, 402, 404–7, 427–33 of karst, 453–6, 458, 459, 463, 472, 474–7 flow and filtration, 451 flow velocity, 366, 367 mildly sloping, 363–5, 380 prismatic, 365, 382 roughness, 366, 367 subcritical flow, 382, 383, 424, 429, 431–3 supercritical flow, 383, 384, 393, 424, 430, 433 uniform flow, 365, 368, 377, 383 velocity, 365–7 steady flow, 363 unsteady flow, 414 Characteristics of pump complete, 305, 318 dimensionless, 301 IV quadrants, 305 normal, 295, 296, 299, 316, 318 normalized dimensionless, 314 Suter curves, 308 Characteristic equations, 220, 223–5, 231, 232, 234, 235, 256 Chez´y formula, 366, 367 Circular cross-section, 149, 202, 203, 214, 377 Circular frequency, 506, 508, 516, 518 Circular pipe, 65, 147, 202 Closed thermodynamic system, 123, 357 Colebrook–White equation, 38, 216, 367, 369 formula, 367 Column separation, 164–7 Comparison tests, 291 Complex closure laws, 180 Complex pipelines, 183 Compressibility modulus, 192 Compressible fluid, 93, 161, 183, 202, 209, 214–17, 223, 224 liquid, 196, 209, 231, 251, 267, 329 particle, 196 Computation procedure, 40 Computer accuracy, 55 Concentration, 479, 481, 483, 484, 487 Configuration compatible, elements, 11, 12, 22, 24, 52, 53, 327, 359, 480, 491, 509, 513 Index Conjugate depths, 385, 387, 388, 389, 413 Connectivity table, 5, 15, 42, 45–7, 49 Conservation law of (in general), 8, 19, 261, 264, 427, 432, 479–82 energy, 206, 209, 252, 253, 274, 285, 286, 329, 330, 428 mass, 207, 252, 274, 285, 330, 414, 415 momentum, 420 thermal flux, 9, 11, 12, 15, 17, 485 Consumption variation, 77, 359 Continuity equation differential form, 19, 20, 101, 207, 215, 415, 420, 422, 507 heat, 9, 11, 13, 14, 16–18 integral form, 20, 414 momentum, 482 nodal sum, 24, 35, 61, 78, 90, 100, 109, 113, 118, 122, 133, 354, 464, 510 relative motion, 144, 408, 412 special form, 415, 416 Contraction, 28, 135, 170, 176, 280, 355, 400 Control volume, 9, 197, 205, 206, 209, 210, 212, 213, 415, 418–20, 446, 481, 482 Convective-dispersive equation, 487, 488, 490, 501 Convective equation, 487, 488 Convective flow, 481 Convergence, 94 Coordinate vector, see Basis vector Coriolis coefficient, see Channel Coriolis coeff Critical depth, 372, 374, 377–90, 392, 395, 398, 399, 400, 402, 403 Critical discharge, 130, 131, 133 Critical flow, see Channel critical flow Critical point, 189, 190, 344 Critical slope, see Channel critical slope Critical temperature, 189 Cumulative conveyance, 452 Cylindrical surge tank, 94, 101, 106–8, 113, 117, 118 Daily balance, 77, 78, 79, 86 Dangerous underpressure, 125, 167 Destination list, 47, 48, 51 Differential form of conservation law, 481 continuity equation, 196, 207, 415, 419 dynamic equation, 417 elementary wave celerity, 409 equation of state, elastic liquid, 192 power, 213 water hammer celerity, 145 Diffuse flow, 438, 451, 463, 474, 475, 481, 482 Diffusion of momentum, 482 Discharge curve, 125, 133, 135, 136, 368–71, 389, 447, 511 Discrete coordinates, 236–8, 243–5 521 Discrete global system, 11 Discretization catchment discretization, 463–7 channel stretch, 401 finite element mesh, mesh of characteristics, 236, 244 Dispersion convective, 485 hydrodynamic, 483, 484 turbulent, 485 Divergence matrix, 10 Drawdown curve, 379, 380, 383, 384 Drive engines, 310 Duty point, 105, 296, 298, 354 Dynamic equation differential form, 417 energy head form, 416 integral energy form, 417 momentum conservation, 417 non-steady flow of compressible fluid, 217 non-steady flow of elastic liquid, 252 non-steady flow of incompressible fluid in compressible pipe, 219 non-steady flow of incompressible fluid in rigid pipe, 219 non-steady flow of liquid and saturated vapor, 218 quasy unsteady, 328 steady flow of compressible, 215, 216 steady flow of incompressible, 215 Dynamic equation of pumping unit, 104, 312 streamtube, 213 Dynamic equation – discussion, 213 Electrical motor, 310 Energy conservation law, 209, 253, 274, 285, 286, 329, 330, 428, 485 Equation of continuity compressible liquid, 209 differential form, 209 elastic liquid, 207, 208 incompressible, 215 nodal, 16 non-steady flow of compressible fluid, 219 non-steady flow of elastic liquid, 217 non-steady flow of incompressible fluid in compressible pipe, 219 non-steady flow of incompressible fluid in rigid pipe, 219 steady flow of compressible, 215 steady flow of incompressible, 215 Equation of oscillations, 102, 358 Equation of small oscillations, 101 Equation of state, 165, 189, 190, 191, 208, 217, 357 Euler’s equations, 291 522 Exact solutions, 487 Extreme friction, 167, 168, 171 Field density, 480–82 File front.f90, 48, 49, 80, 111, 117, 127, 135, 138, 560 Filtration law, 447, 449 Finite element mesh, 3, 21, 45, 52, 53, 64, 402, 477 Finite element technique, 1, 8, 10, 15, 22, 31, 42, 43, 44 Finite element union, 23 Finite volume, Fluid acceleration, 69, 70, 71, 161 Forced inflow, 69, 72, 161, 163, 164, 337 Forced vibrations, 505, 506, 517 Fortran module carst.f90, 456 channels.f90, 407 circular.f90, 377, 560 clssimpleparser.f90, 562 dbgraf.f90, 377, 560 globalvars.f90, 50 graphs.f90, 60 graphs.f90, 60, 560 ode.f90, 535 ODE.F90, 535 sdbgraf.f90, 377, 561 toleranca.f90, 57 trapez.f90, 377, 561 trapez.f90, 561 vibramodule.f90, 513 Frequency angular, 102 circular, 506, 508, 516 converter, 311, 339 current, 104, 311 domain, 506 variable, 312–14, 323, 333 variations, 311 Friction factor, 37, 214 resistance, 37, 69, 167, 168, 179, 183, 228, 243, 246–9, 279, 300 velocity, 484 Front discontinuity, 239 Front length, 48, 50, 51, 58 Frontal algorithm, 49 Frontal procedure, 45, 48, 51 Frontal solver, 49 Frontal technique, 45 FrontOrder, 53, 54, 64 FrontU, 53, 54, 80, 111, 117, 127, 135, 138 Froude number, 372, 374, 376, 382–6, 389, 391, 393, 410 Function aMoody, 38, 41, 254, 528 Functional approximation, Index Fundamental lemma, 1, 2, 7, 8, 10, 16, 19, 20 Fundamental system assembling, 42, 81 of equations, 25, 28 extension, 29, 30 initial, 30, 80 matrix, 30, 81, 91, 110, 111, 116, 126, 134, 135, 137, 467, 496 vector, 80, 82, 91, 110, 111, 116, 126, 133, 135, 137, 467, 496 Galerkin method, Gas phase, 167, 189 Gate valve, 135, 266, 511 General outlet, 136–8 General solution, 187, 226, 488 GGO theorem, Global direction, 327 Global equation, 5, 15, 19, 26, 28, 30 Global function, 4, Global node, 3, 4, 15, 21, 31 Gradually varied flow, 30, 59, 365, 379, 380, 395 Graph object, 30, 65, 86, 453 Graphic solution, 178 Gravity flow, 339 Groundwater, 350, 437, 441, 463 Hagen–Poisseuille law, 203, 204 Harmonic solution, 508, 509 Harmonic state, 506, 509, 518 Head increment norm, 54 Headrace tunnel, 91, 92–112, 117, 511, 545, 547, 548, 550 Heat conduction, 8, 13 Heat exchange, 487, 499 Heat production vector, 11 Heaviside step functions, 227 Homogeneous boundary conditions, Hydraulic efficiency, 292, 293 Hydraulic jump depths, 385 energy dissipation, 387, 388 length, 387 normal, 385, 387 phenomenon, 384, 386, 412, 413, 414, 421, 424 submerged, 388, 389 thrown away, 388 Hydrodynamic dispersion, 483, 484 Hydrophor equation, 357, 358 Hypothetic pumping unit, 321 Ideal gas, 191 Impeller, 291–5, 300–303, 305, 307 Implicit method, 532 Inadequate protection, 344 Index Incompressible flow, 37, 65, 68, 69, 70, 71, 72, 89, 215, 216, 219, 267, 272, 277, 278, 329 fluid, 37, 67, 69, 89, 161, 215, 216, 219, 231, 272, 277, 301, 329 liquid, 33, 267, 278 water, 72, 93, 99, 415 Increment of volume, 82 Individual gas constant, 191 Induction motor, see Asynchronous motor Inertia of pump, 312, 313, 315, 322 Initial conditions of equations, 7, 63, 232, 357, 424, 468, 476, 488, 502 hydrostatic, 72 prescribed, 63, 80, 82–6, 124, 232, 233, 267, 403 steady, 63, 84–9, 124, 184, 235, 238, 239, 241, 395–7 of the wave function, 244 Initialization of channel stretches, 402 Initialize tank, 85 Input data, 52 Input/output syntax, 53, 59, 65, 314, 561, 564 Integer function adddefpump, 322 addpump, 315, 322 addpumpdata, 315 calcvalvestatus, 267 frontlength, 51 getrefluxstatus, 267 lbandw, 43, 524, 526 unsrefluxstatus, 267 Integral transformation, 8, Integration parameter, 61, 264, 271, 273, 276, 432, 536 Interval halving, 378, 379 Inverse function, 377 2×2 matrix, 18 transform, 227 Jacobian matrix, 26, 28, 30, 31, 34, 430 Jaeger’s coefficient, 108 Joint bend, sharp bend, 280, 288, 289 finite element, 282 general, 281 transition, 280, 289 Joukowsky equation, 173 surge, 167, 171, 176 water hammer, 143, 151, 176 Karst flow, 437 Kinematic viscosity, 194, 202, 367, 449, 482, 520, 554 Kinetic energy, 198, 199, 205, 209–11 523 Kloss expression, 311, 322, 323 Knapp circle diagram, 305, 306, 308, 309, 318 Laminar boundary layer, 203 Laminar flow, 202–5, 366, 483 Liquid phase, 165–7, 189, 190, 192, 218 List of active nodes, 46–50 Loading torque, 104 Local node, 3, 4, 15, 21, 31 Local resistance, 265, 266, 279, 280, 289 Localized coordinate functions, Localized global functions, Logical variable fitpumpnormdata, 315–18 initializechannel, 398, 402, 403 speedtransients, 314, 324, 328, 332, 334 +speedtransients, 335–37, 339, 340, 342, 343, 344, 347 Logical function calcjointloss, 279 prefrontal, 50, 53 Longitudinal dispersion, 484, 487 Macro element, 401 Manning formula, 366, 368, 369, 380 Mass conservation, 206, 252, 274, 285, 330, 414, 415, 444, 446 Material derivative, 197, 198, 200, 202, 487 Matica, 440, 441 Matrix and vector updating, 80, 91, 110, 116, 126, 134, 135, 137 Matter, 189 Melting curve, 189 Mean value integral theorem, 18 Mesh of characteristics, 232, 235, 236, 240, 243, 425 elements, 3, 21, 45, 52, 53, 58, 64, 84–87, 477 Mildly sloping, 363–5, 380, 384, 398, 420 Modulus of elasticity, 148, 149, 194 Moment of inertia, 104, 312, 313, 315, 322 Moody chart, 39, 214 Motor torque, 310–23 Natural spline, 318 Navier–Stokes equation, 482, 483 Negative characteristic, 221, 225, 232, 234, 245, 255, 257, 425, 426 Negative index, 46, 49 Net positive suction head, 300, 301 Nominal point, 305, 314–16, 318, 322 Normal characteristics, 295, 301, 318 Normal depth, 369, 378–80, 382, 383, 461 NPSH curve, 301 Numerical diffusion, 498 Numerical integration, 61, 530 524 Operational variables, 323, 332, 334 Optimum working point, 298, 299 Options RUN, 66, 72, 73, 74, 84–90, 119, 129, 460, 462 INPUT, 58, 66, 73, 74, 84–88, 90, 119, 129 DEBUG, see www.wiley.com/go/jovic TRACE, see www.wiley.com/go/jovic DIGITS, 58, 66, 74 SPEED, see www.wiley.com/go/jovic THETA, see www.wiley.com/go/jovic +FEMPIPES, 251 +EXPLICITSUPERCRITICAL, 428 +CHECKVAPOUR, see www.wiley.com/go/jovic +NOPUMPCHECKVALVE, 334, 335 +SPEEDTRANSIENTS, 334–37, 339, 340, 342, 343, 347 +VIBRATIONS, 515–17 Oscillation period, 102 Outflow formula, 135 Outlet model, 137 Partial closure, 179, 401 Partial integration, 10, 16, 19 P´eclet number, 498 Penstock, 89, 91–4, 108, 180, 511 Pipe control volume, 206, 212, 213 Pipe finite element, 38, 522 Pipe segment, 205 Poiseuille’s formula, 194 Polar moment, 104, 312, 315, 322 Pole pairs, 310, 311 Polygonal graph, 59–61, 72, 241, 546, 547, 549, 550–54 Polje, 440 Positive characteristic, 220, 235, 245, 255–7, 426 Potential energy of particle, 199 Precipitation, 437, 466 Predictor-corrector, 532 Prefrontal procedure, 50, 53 Presjek (channel cross-section), 377 Pressure excesses analysis, 341 Pressure regulator, 180 Pressure wave propagation, 141 Primary variables, 80, 110, 113 Prismatic channel, 365, 410, 419, 431 Procedure buildmesh, 52, 53, 64, 559, 560 cplxfrontb, 514, 515 cplxfrontu, 514, 515 cplxinifrnt, 514, 515 femesh, 53 frontb, 51, 54, 64 frontu, 51, 53, 54, 61, 64, 80, 111, 117, 127, 135, 138, 468 getbdc, 53, 54, 61–4, 312, 560 Index inifrnt, 51–4, 64 input, 44, 52, 519, 521, 522, 547, 559 prefrontal, 50, 53, 468 sortelements, 53 steady, 53, 54, 63, 64, 267, 323, 402, 404, 456 steadyopenrefluxmtx, 269 Production density, 480, 481 Programs chtxfrictiondisc, 546 chtxtrenjerekurzija, 248 powerregulation, 120, 549–52 simplesteady, 43, 44, 519 solution simpipcore, 45, 48, 52, 57, 59, 91, 279, 402, 515, 559 surgetank, 118, 546, 547, 549, 553 test accuracy, 56 vessel, 128, 552–4, 557 Program units (modulus) all files, 560, 561, see www.wiley.com/go/jovic Progressive oscillations, 106 Propagation of positive and negative waves, 235, 407, 413 velocity, 176, 225, 231 through branches, 181 Pseudo-language, Pump finite element, 325, 327–30, 332 specific speed, 303, 304, 318 torque, 312, 313, 322, 324 in parallel, 297, 299 in series, 297 working point, 298, 322 Pumping plants, 291 Pumping station, 127, 334, 339, 344–8, 350, 352, 359– Pumping units, 291, 299, 312–15, 324, 327, 329, 339, 343–45, 350, 354, 356, 359, 360 Quasi non-steady, 60, 61, 66, 88 Quasi-unsteady, 61–3, 65, 72, 90 QuasiUnsteady, 64, 69, 251, 328 Radial flow, 319 Rainfall transformation, 468 Real closure laws, 180 Real function toleranca(x, ndigits), 56 Real*8 function amoody, 38, 528 calcnominalspeed, 324 fekanal, 454, 457 pgarea, 60 pgperim, 60 pgstatic, 60 pgval, 60 Index pumpdehhodeo, 317 pumpdehhodeq, 317 pumpdemmodeo, 317 pumpdemmodeq, 317 pumphead, 320 pumphho, 317 pumpmmo, 317 pumptorque, 320 suterhead, 318 sutertorque, 318 Rectangular channel, 373, 385 Recursion, 237, 241, 242, 249 Recursive calculation, 240, 248–50 function fp(x, t), 241, 242, 248 function fm(x, t), 241, 242, 248 program chtxrekurzija, 241, 242 Reflection conditions, 240 Reflux preventer, 266–78, 313, 327, 333, 334, 336, 337, 339, 347 Regulation algorithm, 359 device, 180, 354 limit, 354, 357 valve, 135, 136, 180, 333, 344, 346 Regulator, 93, 180, 359 Relative motion, 143, 144, 165, 407, 408, 412 Relief valves, 28, 126, 136, 344, 347, 350 Riemann general solution, 226 Rigid fluid flow, 67, 68 Rounding error, 55 Runge–Kutta method, 118, 120, 128, 461, 533 Runge–Kutta–Fehlberg, 533 Saturated liquid, 191, 300 liquid/vapor, 218 vapor, 165–7, 191–5, 218, 300, 345 water, 166, 167, 192–4, 500 Schnyder–Bergeron method, 178 Secondary process, 479 variables of the tank, 80 variables of the surgetank, 110 variables of the vessel, 126 Separation of boundary layer, 398, 505 column, 164–7 variables, 70, 450, 451, 459 Sewage pressure pipeline, 350–53 Shape function, 4, 17, 19, 475, 477, 490, 491 Sharp bend, 280 Simpipcore parser, 65 Simpipcore user manual, 57 525 Small oscillations, 101, 102 Solver banded, 44, 524 frontal, 49, 50, 53, 431 vibrasolver, 512–15 Speed governor, 98, 105, 106, 120 of sound, 192, 193, 195, 209, 216–18, 220 Spillway aeration, 390, 391 Standing wave, 412, 413 Starter, 312, 347 Starting torque, 322 Stator blades, 93, 307 Storage (simpip object), 90 Subcritical flow, see Channel subcritical flow Submersible, 350 Subroutine airvalvenode, 135 assembly, 42, 44, 524, 525 bandsol, 43, 524, 526, 527 calcairvalvevars, 135 calcdiffertankvars, 114 calcoutletq, 137 calcsimpletankvars, 110 calcsurgetankvars, 110, 114 calctankvars, 80 calcvesselvars, 126 carstnode, 468 chtxunsteadypipemtx, 251, 255, 261 defpumptopump, 322 differtanknode, 117 explsupercriticalchannelmtx, 428, 429 femunsteadypipemtx, 428, 429 fitconstrainedpoly, 317 fitpumpnormdata, 315–18 flact, 51, 515 fldea, 51, 515 frontb, 51, 54, 64 frontu, 51, 53, 54, 61, 64, 80, 111, 117, 127, 135, 138, 468, 515 gateoutlet, 138 gateoutletq, 137 generaloutlet, 138 generaloutletq, 137 getbdc, 53, 54, 61, 63, 64, 312–15, 560 incvar, 54, 64, 80, 110, 113, 126, 137, 255, 406, 430, 431, 513–15, 560 inifrnt, 51, 53, 54, 64 marklastnode, 50 motor, 323 nonsteadyjointmtx, 282, 285 outletnode, 138 preljevoutlet, 138 preljevoutletq, 137 526 Subroutine (Continued ) pumpmtx, 333 quasiunsteadypipemtx, 61, 63, 64, 69, 251 qucloserefluxmtx, 277 quopenitrefluxmtx, 277 quopenrefluxmtx, 277 qupumpmtx, 328, 329 qurefluxmtx, 277 quunsteadyjointmtx, 282, 283 quvalvemtx, 270, 277 reliefoutlet, 138 reliefoutletq, 137 rgdcloserefluxmtx, 278 rgdopenitrefluxmtx, 278 rgdopenrefluxmtx, 278 rgdpumpmtx 329 rgdrefluxmtx, 277 rgdunsteadyjointmtx, 282, 284 rgdunsteadypipemtx, 68, 69, 251 rgdvalvemtx, 270, 272, 278 rstvar, 63, 64 savesolutionstage, 63, 64 setspeeds, 312, 323, 333 setsuterdata, 318 simpletanknode, 111 steady, 53, 54, 63, 267, 279, 323, 402, 404, 456 steadygeneralvalvemtx, 267, 268 steadyjointmtx, 279, 281 steadykanalmtx, 454, 456 steadyopengeneralvalvemtx, 269 steadyopenvalvemtx, 268 steadypipemtx, 40, 42, 53, 54, 63, 522, 523 steadypumpmtx, 323, 325 steadyrefluxmtx, 267, 269 steadyvalvemtx, 267 subcriticalchannelmtx, 428–30 subcriticalsteadychannelmtx, 404 supercriticalchannelmtx, 429, 430, 431 supercriticalsteadychannelmtx, 404–7 surgetanknode, 111, 117 tanknode, 80 unscloserefluxmtx, 278 unsopenitrefluxmtx, 279 unsopenrefluxmtx, 278 unspumpmtx, 330 unsrefluxmtx, 277, 278 unsteady, 64, 69, 269, 282, 327, 428, 458, 559–61 unsteadyjointmtx, 282 unsteadykanalmtx, 456, 458 unsteadypipemtx, 69, 251 unsteadypumpmtx, 327, 328 Index unsteadyrefluxmtx, 269, 277 unsteadyvalvemtx, 269 unsvalvemtx, 270, 273, 278, 279 vesselnode, 127 Substance, 189, 190, 479 Suction basin, 344–6 head, 300, 301 pipeline, 298, 299, 300, 344 pressure, 300 Sudden acceleration, 161 changes, 239, 279, 280, 398, 400 closing, 93, 166, 167, 227, 228, 239–41 filling, 156, 228, 432 forced inflow, 161, 337 opening, 72 stop, 102, 128, 141, 153 Supercritical flow, see Channel supercritical flow Surge tank differential or Jonson’s, 97 simple, 94, 95, 101 tank stability, 95, 104, 106, 108, 117, 550 with a Ventouri transition, 95 with air throttle, 95, 96 with asymmetric throttle, 95 with the gallery, 95, 96 with upper and lower gallery, 97 Suter curves, 308, 309, 318, 322 Switchyard, 92 Synchronous speed, 310, 311 System assembling, 5, 42 consumption, 59, 89, 359 resistances, 297 Table of element connections, Tailrace mains, 91 tunnel, 97 Tank dimensioning, 77 model, 79, 83 object, 77 secondary variables, 80 Thermodynamic coordinates, 165, 189 equilibrium, 189 forces, 444 process, 125, 191, 200, 216 system, 123 Thoma’s criterion, 108, 119 Torque characteristic of the motor, 310 of a pump, 310 Index Total energy of a particle, 198 Transient electromagnetic, 312 electromechanical, 312 flow, 203 hydraulic, 312, 366, 367 operation, 333 phenomena, 312, 314, 333, 337 region, 203 state, 288, 305, 308, 312–14, 322, 324, 334, 337, 339, 341, 344 velocity, 293 Triple line, 189 point, 189, 190 Turbine efficiency, 105 speed governor, 105, 291, 305, 335 operation, 105 power, 105, 106, 551 specific speed, 304 torque, 104 Turbo engine discharge, 292 impeller, 293 Turbulent diffusion, 483, 484 rough flow, 39, 203, 366 smooth flow, 39, 203, 366 transient flow, 39, 203, 366 Underground accumulation, 466, 467 channel, 440, 447, 448, 450, 452, 466 karst flow, 437 Unified hydraulic networks, 21 Unsteady flow modelling, 69, 327, 456 solution, 64 Valve ball, 266 butterfly, 266 closure, 141, 142, 170, 176, 178, 180, 181 finite element, 266, 268, 274, 276, 327 gate, 135, 266, 511 reflux, 266, 354 status, 267, 268, 270–73, 276 Vapor bubble, 165 density, 193, 194 527 pressure, 165–7, 191–5, 218, 300, 345 region, 191–5, 218 Variable consumption, 89, 353, 354 h, v, 223 kstage, 63 p, v, 222 runVibrations, 512 speedtransients, 314, 324, 333, 337, 339, 347 voltage, 312–14, 323, 333 frequency, 312–14, 323, 332, 333 Vessel model, 126 simple, 121 with air valves, 124 Vibrations forced, 505, 517 harmonic, 506 hydraulic, 505, 506, 512 Vogt parameter, 108 Volume balance, 30, 61, 77, 78, 126, 133, 445 Volume change, 113, 197, 201, 207, 427, 446 Vrulja (singular), vrulje (plural), 441, 442 Water column, 165–7, 176, 355, 360, 469 consumption, 59, 78, 360 density, 131, 193, 194, 357, 485, 500 Water hammer absorber, 349 analysis, 176 analytical solutions, 225, 227, 228 celerity, 143, 145, 147, 149–51, 161, 181, 208, 218, 225, 246, 274, 285, 330, 355, 356, 360 cycle, 153, 176 phases, 149, 163, 241, 347 reduction, 176, 333 Wave celerity, 141, 144, 409, 410, 416 equation, 19, 225, 226 kinematics, 183, 394, 407 propagation, 130, 141, 159, 160, 167, 231, 239, 413 Wave function of characteristics, 222, 225, 231, 232, 335–48, 258, 423, 426 negative wave, 184, 185, 236 positive wave, 185, 226, 235, 238, 246 Weak formulation, 8, 10, 12, 16–19, 28, 475 Weighted residuals method, Wetted surface, 365, 419 Wooden impeller, 291, 292 .. .ANALYSIS AND MODELLING OF NON- STEADY FLOW IN PIPE AND CHANNEL NETWORKS ANALYSIS AND MODELLING OF NON- STEADY FLOW IN PIPE AND CHANNEL NETWORKS Vinko Jovi´c University of Split, Croatia... flows in pipes and channel networks from both the standpoint of hydraulics and of modelling techniques and methods These classical engineering problems occur in the course of the design and construction... 7.1.3 Steady flow modelling 7.1.4 Non- steady flow modelling Joints 7.2.1 Energy head losses at joints 7.2.2 Steady flow modelling 7.2.3 Non- steady flow modelling Test example Reference Further reading