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1-D Dynamic Modelling

Mathematical Background. 1-D Dynamic Modelling. Fundamental Basis. MIKE 11. Modelling of unsteady flow is based on three fundamental elements: A differential relationship expressing the physical laws A finite difference scheme producing a system of algebraic equations

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1-D Dynamic Modelling

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  1. Mathematical Background 1-DDynamic Modelling

  2. Fundamental Basis MIKE 11 • Modelling of unsteady flow is based on three fundamental elements: • A differential relationship expressing the physical laws • A finite difference scheme producing a system of algebraic equations • A mathematical algorithm to solve these equations

  3. Fundamental Basis MIKE 11 PHYSICAL SYSTEM River Network Flood Plains Structures PHYSICAL LAWS Conservation of Mass Conservation of Momentum SCHEMATIZE Represent by a simple Equivalent System DISCRETIZE Express as a Finite Difference Relation BOUNDARIES NUMERICAL MODEL OUTPUTS

  4. Saint-Venant Equations MIKE 11 • General Assumptions: • Incompressible and homogenous fluid • Flow is mainly one-dimensional, (i.e. uniform velocity & WL horizontal in cross-section) • Bottom slope is small • Small longitudinal variation of cross-sectional parameters • Hydrostatic pressure distribution. Continuity Equation (Conservation of Mass) Momentum Equation (Conservation of Momentum) (Newton’s 2’nd Law)

  5. Conservation of Mass MIKE 11 Net increase of Mass from Time1 to Time2 = Net Mass Flux into control volume (Time1 to Time2) + Net Mass Flux out of control volume (Time1 to Time2) I.e.: And:

  6. Conservation of Momentum MIKE 11 Net increase of Momentum from Time1 to Time2 = Net Momentum Flux into control volume (Time1 to Time2) + Sum of external forces acting over the same time G

  7. Conservation of Momentum MIKE 11 Momentum = Mass per unit length * Velocity Momentum Flux = Momentum * velocity Pressure Force = Hydrostatic Pressure P Friction Force = Force due to Bed Resistance Gravity Force = Contribution in X-direction Momentum = Momentum Flux + Pressure - Friction + Gravity

  8. Conservation of Momentum MIKE 11 Momentum: Momentum Flux Pressure Term: Friction Term: Gravity Term:

  9. Differential Equations MIKE 11 Wave Approximations: Kinematic Wave Diffusive Wave Fully Dynamic Wave

  10. Kinematic Wave MIKE 11 Includes: 1. Bed Friction Term 2. Gravity Term Applications: + Steep Rivers - Backwater Effects NOT applicable - Tidal Flows NOT applicable

  11. Diffusive Wave MIKE 11 Includes: 1. Hydrostatic Gradient Term 2. Bed Friction Term 3. Gravity Term Applications: + Relatively Steady Backwater Effects + Slowly Propagating Flood Waves - Tidal Flows NOT applicable

  12. Fully Dynamic Wave MIKE 11 Includes: 1. Acceleration Term 2. Hydrostatic Gradient Term 3. Bed Friction Term 4. Gravity Term Applications: + Fast Transients + Tidal Flows + Rapidly changing backwater effects + Flood waves

  13. High Order Fully Dynamic Wave MIKE 11 Includes: 1. Acceleration Term 2. Hydrostatic Gradient Term 3. Bed Friction Term (Modified compared to Fully Dynamic Wave) 4. Gravity Term Applications: + Fast Transients + Tidal Flows + Rapidly changing backwater effects + Flood waves + Steep Channels

  14. Solution Scheme MIKE 11 Implicit Abbot-Ionescu 6-point scheme

  15. Solution Scheme MIKE 11 Implicit Abbot-Ionescu 6-point scheme t unknown n+1 dt n known Q / h Q / h h/ Q dx dx X 0 j-1 j+1 j 0

  16. Solution Scheme MIKE 11 Solution method Double Sweep algorithm Nodal point solution Grid point solution Matrix bandwidth minimization

  17. MIKE 11 Model Data Requirements • Solution of governing flow equations requires detailed descriptions of: • Catchment Delineation • River and Floodplain Topography • Hydrometric Data for Boundary Conditions • Hydrometric Data for Calibration / Validation • Man-made Interventions

  18. MIKE 11 Stability Given: Initial Conditions and Finite Difference Approximation which is consistent Then: Stability is the necessary and sufficient condition for convergence Stability analysis can only be done for linear differential eq. Explicit methods: Conditionally stable (Cr < 1)Implicit methods: Unconditionally stable Courant Number: Example: D=10;V=1; dX=1000

  19. MIKE 11 Boundary Conditions Discharge, Q : Upstream of River Lateral Inflow Closed End (Q=0) Discharge Control Pump Water Level, h : Downstream River boundary Outlet in Sea (tide, wind) Water level control Q/h Boundary : Downstream Boundary (Never upstr.) Critical Outflow from Model Q Q Q h or Q/h In general, Boundaries should be located where key investigation area is not directly affected by boundary condition!

  20. MIKE 11 Initial Conditions Always specify h and Q for simulation: • Possibilities: • Specify manually (in HD Parameter Editor) • Select from HOTSTART file • Automatically calculated (Steady state approach) Safest to Start with Lower Levels. Never initialize a Flood problem with floodwaters in the flood plains.

  21. MIKE 11 Data Needs Reliable Data required: ‘GARBAGE IN = GARBAGE OUT’ Topography Data: Width, Area, Volume of inundated plains Schematization of Model Aerial/Satellite/Radar images of flood extents Reservoir data (control strategy, spillway etc.) Cross section data DATUM - Same reference level for all data! Hydraulic Data: Stage & Discharge hydrographs Rating Curves Peak Water level during significant events Used for Boundary conditions and Calibration

  22. MIKE 11 Calibration Adjustment of Model parameters to obtain agreement between simulated and measures values. • Items: • Reservoirs/storage area - storage volume must be correct • Unsteady flow - agreement (simulated & measured) - usually adjust roughness parameters • Equivalent longitudinal conveyance - longitudinal profile shows obvious errors • Main features : • Timing of Peak • Value of Peak • Shape of Hydrograph • Accuracy: • No quantitative criterion can be given (very much dependent on data quality) • Each case is unique

  23. MIKE 11 Calibration Main parameter to Modify during Calibration process: River Bed Roughness. • Modification of River Bed Roughness in MIKE 11: • Relative resistance (variation with cross section Width) • Resistance factor (variation with Water level) • Resistance number (longitudinal variation) • Time Series (seasonal variation)

  24. MIKE 11 Verification Verify Model’s Performance - VERY IMPORTANT ! Do not use data from Calibration period! Actions to perform before application of Model: 1) Setup of River Model2) Calibration (preferably data from several periods) 3) Verification (do not use data from Calibration period)4) Application (‘production runs’)

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