Hydrodynamic Simulation of St. Jones River Using FEM

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Hydrodynamic Simulation of St. Jones River Using
FEM

• Saiful Islam
• Luis Bermudez
• Lauren Shor
• Chun Xu
• Winter 2002

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Location of St. Jones River
East Coast USA
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Why ?

• Development of civilizations
• Nile - Egypt
• Amazon South America
• Yellow - China

Uses Agriculture Transportation Recreation Drinkin
g source
Necessity of being studied
Hydrodynamic simulation
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Hydrodynamic Simulation

• Understand and apply the theory of FEM for
  shallow waters
• Generate a very non uniform triangular grid of a
  water shed.
• Simulate hydrodynamic behavior (Tide Wave and
  profile velocities)

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Theory of FEM For Shallow Waters
Governing Equations for 2D shallow water
In conservative form
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Matrix Form of Governing Equations for 2D shallow
water
Non linear Hyperbolic Partial differential
p,q volumetric flow rate per unit width h
depth of the flow t time g ratio of weight to
mass Zo bed elevation from arbitrary datum n
Manning coefficient of roughness u,v velocity
components in x and y directions c (gh)1/2
celerity of elementary gravity waves
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FEM using Petrov-Galerkin formulation
How to solve the differential equation?

• Direct method
• Variational method
• Weighted residual method

• Collocation
• Subdomain
• Least Square method
• Galerkin

Approach using FEM
Converted into an integral equation
Galerkin use on differential equations of the
form Where L is the differential operator, u is
the dependent variable
Forcing error of approximation to 0
Set of r linearly Weighting function
Orthogonal Linear combination reach the
domain Interpolation function
Variation of this function Petrov Galerkin
Formulation
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Galerkin
Petrov-Galerkin
Better results to Galerkin method Dissipative
(oscillations of the solutions will decay with
time) Non dispersive (No wave components with
different frequencies traveling at different
speeds
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Generation Of The Grid
Digitizing
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Generate a Very Non-uniform Triangular Grid of a
Water Shed.
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Cross-Sectional Data
Lower station
Middle station
Upper Station
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75
50
25
X Y Z -75.67 39.34 0 -75.77
39.34 -6 -75.88 39.34 -2
Z
Elevation
X-section
X, distance
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Grid
Number of triangular elements 23 122 Number of
nodes 14 556
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Results
Elevation (m)
Flow profile of a wave as it goes upstream
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Results
Upstream Velocity Vectors
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Problems

• Digitizing an calibration of a big area
• Refinement of the mesh
• Sharp edges of the grid

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Conclusions

•           
• Numerical Methods, such as finite element methods
  help scientists and engineers to understand
  phenomena like flows in open channel. Governments
  and institutions can manage in a more efficient
  way a source like, water from rivers, so that the
  community will have the water they need for
  drinking, agriculture, cleaning and
  transportation among others.
• The grid created can be refined further and can
  be used for future studies of the St. Jones River
  by the Delaware National Estuarine.

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References

• Bathe, K, Finite Element Procedures, Prentice
  Hall, New Jersey, USA, 1996, Chapter 4
• Chaudhry, M.H., Open Channel Flow, Prentice Hall,
  1993, Chapter 16
• Katopodes, N. , A Dissipative Galerkin Scheme
  for Open- Channel Flow, Journal of Hydraulic
  Engineering, ASCE, Vol. 110, no.4, April 1984,
  pp. 450-466
• Katopodes, N. , Two Dimensional Surges and
  Shocks in Open Channels, Journal of Hydraulic
  Engineering, ASCE, Vol. 110, no.6, June 1984, pp.
  794-812

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Questions?