ANALYSIS AND MODELLING OF NON-STEADY FLOW IN PIPE AND CHANNEL NETWORKS -- Contents -- Preface -- 1 Hydraulic Networks -- 1.1 Finite element technique -- 1.1.1 Functional approximations -- 1.1.2 Discretization, finite element mesh -- 1.1.3 Approximate solution of differential equations -- 1.2 Unified hydraulic networks -- 1.3 Equation system -- 1.3.1 Elemental equations -- 1.3.2 Nodal equations -- 1.3.3 Fundamental system -- 1.4 Boundary conditions -- 1.4.1 Natural boundary conditions -- 1.4.2 Essential boundary conditions -- 1.5 Finite element matrix and vector -- Reference -- Further reading -- 2 Modelling of Incompressible Fluid Flow -- 2.1 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 -- 2.2 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 -- 2.3 Unsteady flow of an incompressible fluid -- 2.3.1 Dynamic equation -- 2.3.2 Subroutine RgdUnsteadyPipeMtx -- 2.3.3 Incompressible fluid acceleration -- 2.3.4 Acceleration test -- 2.3.5 Rigid test example -- References -- Further Reading -- 3 Natural Boundary Condition Objects -- 3.1 Tank object -- 3.1.1 Tank dimensioning -- 3.1.2 Tank model -- 3.1.3 Tank test examples -- 3.2 Storage -- 3.2.1 Storage equation -- 3.2.2 Fundamental system vector and matrix updating -- 3.3 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 -- 3.4 Vessel -- 3.4.1 Simple vessel -- 3.4.2 Vessel with air valves -- 3.4.3 Vessel model -- 3.4.4 Example -- 3.5 Air valves -- 3.5.1 Air valve positioning -- 3.5.2 Air valve model -- 3.6 Outlets -- 3.6.1 Discharge curves -- 3.6.2 Outlet model -- Reference -- Further reading -- 4 Water Hammer - Classic Theory -- 4.1 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 -- 4.2 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 -- 4.3 Water hammer phases -- 4.3.1 Sudden flow stop, velocity change v0 ? 0 -- 4.3.2 Sudden pipe filling, velocity change 0 ? v0 -- 4.3.3 Sudden filling of blind pipe, velocity change 0 ? v0 -- 4.3.4 Sudden valve opening -- 4.3.5 Sudden forced inflow -- 4.4 Under-pressure and column separation -- 4.5 Influence of extreme friction -- 4.6 Gradual velocity changes -- 4.6.1 Gradual valve closing -- 4.6.2 Linear flow arrest -- 4.7 Influence of outflow area change -- 4.7.1 Graphic solution -- 4.7.2 Modified graphical procedure -- 4.8 Real closure laws -- 4.9 Water hammer propagation through branches -- 4.10 Complex pipelines -- 4.11 Wave kinematics -- 4.11.1 Wave functions -- 4.11.2 General solution -- Reference -- Further reading -- 5 Equations of Non-steady Flow in Pipes -- 5.1 Equation of state -- 5.1.1 p,T phase diagram -- 5.1.2 p,V phase diagram -- 5.2 Flow of an ideal fluid in a streamtube -- 5.2.1 Flow kinematics along a streamtube -- 5.2.2 Flow dynamics along a streamtube -- 5.3 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 -- 5.4 Control volume -- 5.5 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 -- 5.6 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 -- 5.7 Flow models -- 5.7.1 Steady flow -- 5.7.2 Non-steady flow -- 5.8 Characteristic equations -- 5.8.1 Elastic liquid -- 5.8.2 Compressible fluid -- 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 -- 6 Modelling of Non-steady Flow of Compressible Liquid in Pipes -- 6.1 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 -- 6.2 Subroutine UnsteadyPipeMtx -- 6.2.1 Subroutine FemUnsteadyPipeMtx -- 6.2.2 Subroutine ChtxUnsteadyPipeMtx -- 6.3 Comparison tests -- 6.3.1 Test example -- 6.3.2 Conclusion -- Further reading -- 7 Valves and Joints -- 7.1 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 -- 7.2 Joints -- 7.2.1 Energy head losses at joints -- 7.2.2 Steady flow modelling.

7.2.3 Non-steady flow modelling -- 7.3 Test example -- Reference -- Further reading -- 8 Pumping Units -- 8.1 Introduction -- 8.2 Euler's equations of turbo engines -- 8.3 Normal characteristics of the pump -- 8.4 Dimensionless pump characteristics -- 8.5 Pump specific speed -- 8.6 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 -- 8.7 Drive engines -- 8.7.1 Asynchronous or induction motor -- 8.7.2 Adjustment of rotational speed by frequency variation -- 8.7.3 Pumping unit operation -- 8.8 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 -- 8.9 Pumping element matrices -- 8.9.1 Steady flow modelling -- 8.9.2 Unsteady flow modelling -- 8.10 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) -- 8.11 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 -- 8.12 Something about protection of sewage pressure pipelines -- 8.13 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 -- 9 Open Channel Flow -- 9.1 Introduction -- 9.2 Steady flow in a mildly sloping channel.

9.3 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 -- 9.4 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 -- 9.5 Sudden changes in cross-sections -- 9.6 Steady flow modelling -- 9.6.1 Channel stretch discretization -- 9.6.2 Initialization of channel stretches -- 9.6.3 Subroutine SubCriticalSteadyChannelMtx -- 9.6.4 Subroutine SuperCriticalSteadyChannelMtx -- 9.7 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 -- 9.8 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 -- 9.9 Equation of characteristics -- 9.9.1 Transformation of non-steady flow equations -- 9.9.2 Procedure of transformation into characteristics -- 9.10 Initial and boundary conditions -- 9.11 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 -- 10 Numerical Modelling in Karst -- 10.1 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.