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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 Hydrodynamics2DH 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
Averaging momentum balance over short timescales • Turbulence • Reynolds stresses • Approximated by turbulent shear stresses
Shallow water approximation • Horizontal scales >> vertical scales • Vertical velocities << horizontal velocities • Neglect vertical acceleration
Hydrostatic pressure • Inhomogeneous (density not constant): • Homogeneous (density constant):
Shallow Water Equations (3D) Acceleration Horizontal diffusion Horizontal pressure gradient Wave forcing Vertical diffusion Coriolis
Boundary conditions Bottom (z=-d) Surface ( )
Depth-averaged momentum balance Atmospheric pressure Wind shear stress Acceleration Bed shear stress Wave forcing Advection Coriolis Water level gradient Horizontal diffusion
Limit case: stationary, uniform flow Question: given Chezy law, how can you compute velocity u?
Limit case: 1D tidal wave • Very long tidal wave in deep channel From continuity eq.
Shallow water wave celerity • Introduce sinusoidal solutions:
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
Limit case: 1D St Venant equations • Neglect v velocity and all gradients with y
Limit case: backwater curve • St Venant + stationary: neglect d/dt
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
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?
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)
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
Numerical models • Grid types • Rectilinear, curvilinear, unstructured • Discretization • Finite difference, finite volume, finite elements • Solution methods • Implicit vs explicit • Explicit: hard stability criterion
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
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)
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
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)
2D numerical model Rhine branches: 2 bifurcations 5 domains, to be extended to Duisburg
Use of 2D numerical model • Model construction • Hydraulic calibration • Morphologicalcalibration: • one-dimensional • two-dimensional • Verification • Application
Study area Study Area
Integrated SOBEK 1D-2D model FEMA 1% Floodplain Boundary HEC-RAS Cross Section Flow Node HEC-HMS
Raw 1-ft LiDAR Input data: LiDAR data, … Bare Earth 15-ft LiDAR