Analysis and Modelling of Non-Steady Flow in Pipe and Channel Networks deals with flows in pipes and channel networks from the standpoints of hydraulics and modelling techniques and methods. These engineering problems occur in the course of the design and construction of hydroenergy plants, water-supply and other systems. In this book, the author presents his experience in solving these problems from the early 1970s to the present day. During this period new methods of solving hydraulic problems have evolved, due to the development of computers and numerical methods.

This book is accompanied by a website which hosts the author's software package, Simpip (an abbreviation of simulation of pipe flow) for solving non-steady pipe flow using the finite element method. The program also covers flows in channels. The book presents the numerical core of the SimpipCore program (written in Fortran).

Key features:

* Presents the theory and practice of modelling different flows in hydraulic networks

* Takes a systematic approach and addresses the topic from the fundamentals

* Presents numerical solutions based on finite element analysis

* Accompanied by a website hosting supporting material including the SimpipCore project as a standalone program

Analysis and Modelling of Non-Steady Flow in Pipe and Channel Networks is an ideal reference book for engineers, practitioners and graduate students across engineering disciplines.

Vinko Jovic, University of Split, Croatia
Vinko Jovic is a Professor and the Head of Department for Hydraulics and Hydromechanics in the Faculty of Civil Engineering at the University of Split.
His research interests also include numerical modelling. He has circa 60 published papers and has written two books in Croatian.

He is the creator of a software package called Simpip (an abbreviation of simulation of pipe flow) which is used for solving non-steady pipe flow using the finite element method.

Analysis and Modelling of Non-Steady Flow in Pipe and Channel Networks deals with flows in pipes and channel networks from the standpoints of hydraulics and modelling techniques and methods. These engineering problems occur in the course of the design and construction of hydroenergy plants, water-supply and other systems. In this book, the author presents his experience in solving these problems from the early 1970s to the present day. During this period new methods of solving hydraulic problems have evolved, due to the development of computers and numerical methods.

This book is accompanied by a website which hosts the author's software package, Simpip (an abbreviation of simulation of pipe flow) for solving non-steady pipe flow using the finite element method. The program also covers flows in channels. The book presents the numerical core of the SimpipCore program (written in Fortran).

Key features:

• Presents the theory and practice of modelling different flows in hydraulic networks
• Takes a systematic approach and addresses the topic from the fundamentals
• Presents numerical solutions based on finite element analysis
• Accompanied by a website hosting supporting material including the SimpipCore project as a standalone program

Analysis and Modelling of Non-Steady Flow in Pipe and Channel Networks is an ideal reference book for engineers, practitioners and graduate students across engineering disciplines.

Preface xiii

1 Hydraulic Networks 1

1.1 Finite element technique 1

1.1.1 Functional approximations 1

1.1.2 Discretization, finite element mesh 3

1.1.3 Approximate solution of differential equations 6

1.2 Unified hydraulic networks 21

1.3 Equation system 23

1.3.1 Elemental equations 23

1.3.2 Nodal equations 24

1.3.3 Fundamental system 25

1.4 Boundary conditions 28

1.4.1 Natural boundary conditions 28

1.4.2 Essential boundary conditions 30

1.5 Finite element matrix and vector 30

Reference 36

Further reading 36

2 Modelling of Incompressible Fluid Flow 37

2.1 Steady flow of an incompressible fluid 37

2.1.1 Equation of steady flow in pipes 37

2.1.2 Subroutine SteadyPipeMtx 40

2.1.3 Algorithms and procedures 42

2.1.4 Frontal procedure 45

2.1.5 Frontal solution of steady problem 51

2.1.6 Steady test example 57

2.2 Gradually varied flow in time 59

2.2.1 Time-dependent variability 59

2.2.2 Quasi non-steady model 60

2.2.3 Subroutine QuasiUnsteadyPipeMtx 61

2.2.4 Frontal solution of unsteady problem 63

2.2.5 Quasi-unsteady test example 65

2.3 Unsteady flow of an incompressible fluid 65

2.3.1 Dynamic equation 65

2.3.2 Subroutine RgdUnsteadyPipeMtx 68

2.3.3 Incompressible fluid acceleration 69

2.3.4 Acceleration test 72

2.3.5 Rigid test example 72

References 75

Further Reading 75

3 Natural Boundary Condition Objects 77

3.1 Tank object 77

3.1.1 Tank dimensioning 77

3.1.2 Tank model 79

3.1.3 Tank test examples 83

3.2 Storage 90

3.2.1 Storage equation 90

3.2.2 Fundamental system vector and matrix updating 91

3.3 Surge tank 91

3.3.1 Surge tank role in the hydropower plant 91

3.3.2 Surge tank types 94

3.3.3 Equations of oscillations in the supply system 99

3.3.4 Cylindrical surge tank 101

3.3.5 Model of a simple surge tank with upper and lower chamber 108

3.3.6 Differential surge tank model 112

3.3.7 Example 117

3.4 Vessel 121

3.4.1 Simple vessel 121

3.4.2 Vessel with air valves 124

3.4.3 Vessel model 126

3.4.4 Example 127

3.5 Air valves 128

3.5.1 Air valve positioning 128

3.5.2 Air valve model 133

3.6 Outlets 135

3.6.1 Discharge curves 135

3.6.2 Outlet model 137

Reference 138

Further reading 138

4 Water Hammer - Classic Theory 141

4.1 Description of the phenomenon 141

4.1.1 Travel of a surge wave following the sudden halt of a locomotive 141

4.1.2 Pressure wave propagation after sudden valve closure 141

4.1.3 Pressure increase due to a sudden flow arrest - the Joukowsky water hammer 143

4.2 Water hammer celerity 143

4.2.1 Relative movement of the coordinate system 143

4.2.2 Differential pressure and velocity changes at the water hammer front 145

4.2.3 Water hammer celerity in circular pipes 147

4.3 Water hammer phases 149

4.3.1 Sudden Flow Stop, Velocity Change V₀ 0 151

4.3.2 Sudden Pipe Filling, Velocity Change 0 V₀ 154

4.3.3 Sudden Filling of Blind Pipe, Velocity Change 0 V₀ 156

4.3.4 Sudden valve opening 159

4.3.5 Sudden forced inflow 161

4.4 Under-pressure and column separation 164

4.5 Influence of extreme friction 167

4.6 Gradual velocity changes 171

4.6.1 Gradual valve closing 171

4.6.2 Linear flow arrest 174

4.7 Influence of outflow area change 176

4.7.1 Graphic solution 178

4.7.2 Modified graphical procedure 179

4.8 Real closure laws 180

4.9 Water hammer propagation through branches 181

4.10 Complex pipelines 183

4.11 Wave kinematics 183

4.11.1 Wave functions 183

4.11.2 General solution 187

Reference 187

Further reading 187

5 Equations of Non-steady Flow in Pipes 189

5.1 Equation of state 189

5.1.1 p,T phase diagram 189

5.1.2 p,V phase diagram 190

5.2 Flow of …