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

Assumption 1: incompressible flow

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 Coriolis Horizontal diffusion Vertical diffusion Horizontal pressure gradient Wave forcing

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

From moving to fixed frame of reference

Shallow Water Equations (3D)

Depth-averaged mass balance

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

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)

Tidal current modelling

Texel, NL

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 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 (

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

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

Use of 2D numerical model 1.Model construction 2.Hydraulic calibration 3.Morphological calibration: i.one-dimensional ii.two-dimensional 4.Verification 5.Application

Integrated numerical grids

Project ‘Cypress Creek, Texas, USA’

Study area

Study Area

Tropical Storm Allison, 2001

New FEMA Map, based on SOBEK

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

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

SOBEK model results

1998 Flooded Structures Summary, Computed vs. Observed Address Ponding in Inches Remarks Observed (1) Computed (2) Katy Hockley8 -inch9.6 -inchFinish Floor Unknown Katy Hockley14 -inch15.6 -inchFinish Floor Unknown Katy Hockley22 -inch22.8 -inchFinish Floor Unknown Sharp Rd3-inch4.8 -inchFinish Floor Unknown Sharp RdUnknown4.0 -inchFinish Floor Unknown Sharp Rd20 -inch20.4 -inchFinish Floor Unknown

Texel morphology

Real-life case: IJmuiden Harbour A B

Geological application: Wax delta Storms et al, 2007

Estuarine circulation See animations on

Lock exchange

Take home messages Go look for examples in your own field of interest Try to find peer-reviewed publications of the models you consider, don’t believe the brochures Don’t believe the prettiest picture Always assume that the model is wrong until proven otherwise