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Free Surface Hydrodynamics 2DH and 3D Shallow Water Equations

Free Surface Hydrodynamics 2DH and 3D Shallow Water Equations. Prof. Dano Roelvink. Contents. Main assumptions and derivation from Navier-Stokes Equations Some simple limit cases (A bit on) numerical models Typical applications. Momentum balance. Mass balance.

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Free Surface Hydrodynamics 2DH and 3D Shallow Water Equations

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  1. Free Surface Hydrodynamics2DH and 3D Shallow Water Equations Prof. Dano Roelvink

  2. Contents • Main assumptions and derivation from Navier-Stokes Equations • Some simple limit cases • (A bit on) numerical models • Typical applications

  3. Momentum balance

  4. Mass balance

  5. Assumption 1: incompressible flow

  6. Averaging momentum balance over short timescales • Turbulence • Reynolds stresses • Approximated by turbulent shear stresses

  7. Shallow water approximation • Horizontal scales >> vertical scales • Vertical velocities << horizontal velocities • Neglect vertical acceleration

  8. Hydrostatic pressure • Inhomogeneous (density not constant): • Homogeneous (density constant):

  9. Shallow Water Equations (3D) Acceleration Horizontal diffusion Horizontal pressure gradient Wave forcing Vertical diffusion Coriolis

  10. Boundary conditions Bottom (z=-d) Surface ( )

  11. From moving to fixed frame of reference

  12. Shallow Water Equations (3D)

  13. Depth-averaged mass balance

  14. Depth-averaged momentum balance Atmospheric pressure Wind shear stress Acceleration Bed shear stress Wave forcing Advection Coriolis Water level gradient Horizontal diffusion

  15. Limit case: stationary, uniform flow Question: given Chezy law, how can you compute velocity u?

  16. Limit case: 1D tidal wave • Very long tidal wave in deep channel From continuity eq.

  17. Shallow water wave celerity • Introduce sinusoidal solutions:

  18. How to use it • Period T is given (approx. 12 hrs) • Celerity c depends only on water depth • Velocity u depends on water depth and tidal amplitude • Example: given water depth of 20 m, tidal amplitude of 1 m, estimate celerity and amplitude of velocity

  19. Limit case: 1D St Venant equations • Neglect v velocity and all gradients with y

  20. Limit case: backwater curve • St Venant + stationary: neglect d/dt

  21. Limit case: stationary wind setup • Wind exerts surface shear stress • If there is a closed boundary , the cross-shore velocity goes to zero • Wind stress term is compensated by surface slope term

  22. Setup question • Wind shear stress is 1 N/m2 • Length of sea or lake is 100 km • Water depth is 10 m • How big is water level difference • Is it different for a lake or a sea?

  23. 3D limit case: vertical profile of uniform, stationary flow • Shear stress term balances pressure gradient term • Pressure gradient given by surface slope term: • Parabolic viscosity distribution • Solution: logarithmic profile: (Derivation in lecture notes)

  24. Why these analyses if you have numerical models? • Numerical models can be wrong • Need to understand the outcome • Need to be able to check at least the order of magnitude of the outcome

  25. Numerical models • Grid types • Rectilinear, curvilinear, unstructured • Discretization • Finite difference, finite volume, finite elements • Solution methods • Implicit vs explicit • Explicit: hard stability criterion

  26. Delta Delft-UNSTRUC Hydrodynamic Model • Currently under development for Delta • New hybrid grid • 3-dimensional, ocean-to-river • Will house: • hydrodynamics • salinity • temperature • sediment • phytoplankton • bivalves 18

  27. Applications • Tidal current modelling (Texel, Singapore) • Storm surge prediction (Hurricane Ike, North Sea) • Detailed river modelling (Rhine branches) • Flooding (USA) • Water quality modelling • Morphology modelling (IJmuiden)

  28. Tidal current modelling

  29. Texel, NL

  30. Example: Hurricane Ike • A hydrodynamic model has been set up with the Delft3D system running in 2D mode. The hurricane track used in this model was downloaded from http://weather.unisys.com/hurricane/ . • The model predicts surge levels of more than 5 metres above mean sea level in both San Antonio Bay and Matagorda Bay. • To synthesize the hurricane, the in-house Wind Enhanced Scheme (WES) was used. The WES scheme was originally developed by the UK Meteorological Office based on Holland’s model (Holland, 1975). • The model resolution is 2 km and the bathymetry and land height originates from one minute GEBCO gridded data (http://www.gebco.net/data_and_products/gridded_bathymetry_data

  31. Detailed modelling Rhine branches Dutch Rhine branches Measures: • Dredging • Channel narrowing bygroyne extension • Measures to correct bend profiles Waal Rotterdam Ruhrgebiet (main German industrial and urban area)

  32. 2D numerical model Rhine branches: 2 bifurcations 5 domains, to be extended to Duisburg

  33. Use of 2D numerical model • Model construction • Hydraulic calibration • Morphologicalcalibration: • one-dimensional • two-dimensional • Verification • Application

  34. Integrated numerical grids

  35. Project ‘Cypress Creek, Texas, USA’

  36. Study area

  37. Study area Study Area

  38. Tropical Storm Allison, 2001

  39. New FEMA Map, based on SOBEK

  40. Integrated SOBEK 1D-2D model FEMA 1% Floodplain Boundary HEC-RAS Cross Section Flow Node HEC-HMS

  41. Raw 1-ft LiDAR Input data: LiDAR data, … Bare Earth 15-ft LiDAR

  42. SOBEK model results

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