Free Surface Hydrodynamics 2DH and 3D Shallow Water Equations

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 gtgt vertical scales
• Vertical velocities ltlt 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
(No Transcript)
28
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)

29
Tidal current modelling
30
Texel, NL
31
(No Transcript)
32
(No Transcript)
33
(No Transcript)
34
(No Transcript)
35
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 Hollands 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_produc
  ts/gridded_bathymetry_data

36
(No Transcript)
37
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)
38
2D numerical model
Rhine branches 2 bifurcations
5 domains, to be extended to Duisburg
39
(No Transcript)
40
Use of 2D numerical model

• Model construction
• Hydraulic calibration
• Morphologicalcalibration
• one-dimensional
• two-dimensional
• Verification
• Application

41
(No Transcript)
42
Integrated numerical grids


43
Project Cypress Creek, Texas, USA
44
Study area
45
Study area
Study Area
46
Tropical Storm Allison, 2001
47
New FEMA Map, based on SOBEK
48
Integrated SOBEK 1D-2D model
FEMA 1 Floodplain Boundary
HEC-RAS Cross Section
Flow Node HEC-HMS
49
Input data LiDAR data,
Bare Earth 15-ft LiDAR

• Raw 1-ft LiDAR

50
SOBEK model results
51
1998 Flooded Structures Summary, Computed vs.
Observed
Address Ponding in Inches Ponding in Inches Remarks
Address Observed (1) Computed (2) Remarks
10502 Katy Hockley 8 -inch 9.6 -inch Finish Floor Unknown
10866 Katy Hockley 14 -inch 15.6 -inch Finish Floor Unknown
10870 Katy Hockley 22 -inch 22.8 -inch Finish Floor Unknown
26253 Sharp Rd 3-inch 4.8 -inch Finish Floor Unknown
26257 Sharp Rd Unknown 4.0 -inch Finish Floor Unknown
27010 Sharp Rd 20 -inch 20.4 -inch Finish Floor Unknown
52
Texel morphology
53
Real-life case IJmuiden Harbour
A
B
54
Geological application Wax delta
Storms et al, 2007
55
Estuarine circulation

• See animations on www.openearth.nl

56
Lock exchange
57
Take home messages

• Go look for examples in your own field of
  interest
• Try to find peer-reviewed publications of the
  models you consider, dont believe the brochures
• Dont believe the prettiest picture
• Always assume that the model is wrong until
  proven otherwise