Analysis and Modelling of Non-Steady Flow in Pipe and Channel Networks

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 an ideal fluid in a streamtube 195

5.2.1 Flow kinematics along a streamtube 195

5.2.2 Flow dynamics along a streamtube 198

5.3 The real flow velocity profile 202

5.3.1 Reynolds number, flow regimes 202

5.3.2 Velocity profile in the developed boundary layer 203

5.3.3 Calculations at the cross-section 204

5.4 Control volume 205

5.5 Mass conservation, equation of continuity 206

5.5.1 Integral form 206

5.5.2 Differential form 207

5.5.3 Elastic liquid 207

5.5.4 Compressible liquid 209

5.6 Energy conservation law, the dynamic equation 209

5.6.1 Total energy of the control volume 209

5.6.2 Rate of change of internal energy 210

5.6.3 Rate of change of potential energy 210

5.6.4 Rate of change of kinetic energy 210

5.6.5 Power of normal forces 211

5.6.6 Power of resistance forces 212

5.6.7 Dynamic equation 212

5.6.8 Flow resistances, the dynamic equation discussion 213

5.7 Flow models 215

5.7.1 Steady flow 215

5.7.2 Non-steady flow 217

5.8 Characteristic equations 220

5.8.1 Elastic liquid 220

5.8.2 Compressible fluid 223

5.9 Analytical solutions 225

5.9.1 Linearization of equations - wave equations 225

5.9.2 Riemann general solution 226

5.9.3 Some analytical solutions of water hammer 227

Reference 229

Further reading 229

6 Modelling of Non-steady Flow of Compressible Liquid in Pipes 231

6.1 Solution by the method of characteristics 231

6.1.1 Characteristic equations 231

6.1.2 Integration of characteristic equations, wave functions 232

6.1.3 Integration of Characteristic Equations, Variables H, V 234

6.1.4 The water hammer is the pipe with no resistance 235

6.1.5 Water hammers in pipes with friction 243

6.2 Subroutine UnsteadyPipeMtx 251

6.2.1 Subroutine FemUnsteadyPipeMtx 252

6.2.2 Subroutine ChtxUnsteadyPipeMtx 255

6.3 Comparison tests 261

6.3.1 Test example 261

6.3.2 Conclusion 263

Further reading 264

7 Valves and Joints 265

7.1 Valves 265

7.1.1 Local energy head losses at valves 265

7.1.2 Valve status 267

7.1.3 Steady flow modelling 267

7.1.4 Non-steady flow modelling 269

7.2 Joints 279

7.2.1 Energy head losses at joints 279

7.2.2 Steady flow modelling 279

7.2.3 Non-steady flow modelling 282

7.3 Test example 288

Reference 290

Further reading 290

8 Pumping Units 291

8.1 Introduction 291

8.2 Euler's equations of turbo engines 291

8.3 Normal characteristics of the pump 295

8.4 Dimensionless pump characteristics 301

8.5 Pump specific speed 303

8.6 Complete characteristics of turbo engine 305

8.6.1 Normal and abnormal operation 305

8.6.2 Presentation of turbo engine characteristics depending on the direction of rotation 305

8.6.3 Knapp circle diagram 305

8.6.4 Suter curves 308

8.7 Drive engines 310

8.7.1 Asynchronous or induction motor 310

8.7.2 Adjustment of rotational speed by frequency variation 311

8.7.3 Pumping unit operation 312

8.8 Numerical model of pumping units 314

8.8.1 Normal pump operation 314

8.8.2 Reconstruction of complete characteristics from normal characteristics 318

8.8.3 Reconstruction of a hypothetic pumping unit 321

8.8.4 Reconstruction of the electric motor torque curve 322

8.9 Pumping element matrices 323

8.9.1 Steady flow modelling 323

8.9.2 Unsteady flow modelling 327

8.10 Examples of transient operation stage modelling 333

8.10.1 Test example (A) 334

8.10.2 Test example (B) 336

8.10.3 Test example (C) 339

8.10.4 Test example (D) 341

8.11 Analysis of operation and types of protection against pressure excesses 345

8.11.1 Normal and accidental operation 345

8.11.2 Layout 345

8.11.3 Supply pipeline, suction basin 346

8.11.4 Pressure pipeline and pumping station 348

8.11.5 Booster station 350

8.12 Something about protection of sewage pressure pipelines 353

8.13 Pumping units in a pressurized system with no tank 355

8.13.1 Introduction 355

8.13.2 Pumping unit regulation by pressure switches 355

8.13.3 Hydrophor regulation 358

8.13.4 Pumping unit regulation by variable rotational speed 360

Reference 362

Further reading 362

9 Open Channel Flow 363

9.1 Introduction 363

9.2 Steady flow in a mildly sloping channel 363

9.3 Uniform flow in a mildly sloping channel 365

9.3.1 Uniform flow velocity in open channel 365

9.3.2 Conveyance, discharge curve 368

9.3.3 Specific energy in a cross-section: Froude number 372

9.3.4 Uniform flow programming solution 377

9.4 Non-uniform gradually varied flow 378

9.4.1 Non-uniform flow characteristics 378

9.4.2 Water level differential equation 380

9.4.3 Water level shapes in prismatic channels 382

9.4.4 Transitions between supercritical and subcritical flow, hydraulic jump 383

9.4.5 Water level shapes in a non-prismatic channel 391

9.4.6 Gradually varied flow programming solutions 395

9.5 Sudden changes in cross-sections 398

9.6 Steady flow modelling 401

9.6.1 Channel stretch discretization 401

9.6.2 Initialization of channel stretches 402

9.6.3 Subroutine SubCriticalSteadyChannelMtx 404

9.6.4 Subroutine SuperCriticalSteadyChannelMtx 406

9.7 Wave kinematics in channels 407

9.7.1 Propagation of positive and negative waves 407

9.7.2 Velocity of the wave of finite amplitude 407

9.7.3 Elementary wave celerity 409

9.7.4 Shape of positive and negative waves 411

9.7.5 Standing wave - hydraulic jump 412

9.7.6 Wave propagation through transitional stretches 413

9.8 Equations of non-steady flow in open channels 414

9.8.1 Continuity equation 414

9.8.2 Dynamic equation 416

9.8.3 Law of momentum conservation 417

9.9 Equation of characteristics 422

9.9.1 Transformation of non-steady flow equations 422

9.9.2 Procedure of transformation into characteristics 423

9.10 Initial and boundary conditions 424

9.11 Non-steady flow modelling 425

9.11.1 Integration along characteristics 425

9.11.2 Matrix and vector of the channel finite element 427

9.11.3 Test examples 431

References 434

Further reading 435

10 Numerical Modelling in Karst 437

10.1 Underground karst flows 437

10.1.1 Introduction 437

10.1.2 Investigation works in karst catchment 437

10.1.3 The main development forms of karst phenomena in the Dinaric area 438

10.1.4 The size of the catchment 443

10.2 Conveyance of the karst channel system 446

10.2.1 Transformation of rainfall into spring hydrographs 446

10.2.2 Linear filtration law 447

10.2.3 Turbulent filtration law 449

10.2.4 Complex flow, channel flow, and filtration 451

10.3 Modelling of karst channel flows 453

10.3.1 Karst channel finite elements 453

10.3.2 Subroutine SteadyKanalMtx 454

10.3.3 Subroutine UnsteadyKanalMtx 456

10.3.4 Tests 458

10.4 Method of catchment discretization 463

10.4.1 Discretization of karst catchment channel system without diffuse flow 463

10.4.2 Equation of the underground accumulation of a karst sub-catchment 466

10.5 Rainfall transformation 468

10.5.1 Uniform input hydrograph 468

10.5.2 Rainfall at the catchment 473

10.6 Discretization of karst catchment with diffuse and channel flow 474

References 477

Further reading 477

11 Convective-dispersive Flows 479

11.1 Introduction 479

11.2 A reminder of continuum mechanics 479

11.3 Hydrodynamic dispersion 483

11.4 Equations of convective-dispersive heat transfer 485

11.5 Exact solutions of convective-dispersive equation 487

11.5.1 Convective equation 487

11.5.2 Convective-dispersive equation 488

11.5.3 Transformation of the convective-dispersive equation 490

11.6 Numerical modelling in a hydraulic network 490

11.6.1 The selection of solution basis, shape functions 490

11.6.2 Elemental equations: equation integration on the finite element 492

11.6.3 Nodal equations 495

11.6.4 Boundary conditions 495

11.6.5 Matrix and vector of finite element 496

11.6.6 Numeric solution test 497

11.6.7 Heat exchange of water table 499

11.6.8 Equilibrium temperature and linearization 500

11.6.9 Temperature disturbance caused by artificial sources 501

References 503

Further reading 503

12 Hydraulic Vibrations in Networks 505

12.1 Introduction 505

12.2 Vibration equations of a pipe element 506

12.3 Harmonic solution for the pipe element 508

12.4 Harmonic solutions in the network 509

12.5 Vibration source modelling 512

12.6 Hints to implementation in SimpipCore 512

12.7 Illustrative examples 515

Reference 518

Further reading 518

Index 519