FLO2D Model Theory and Numerical Stability Criteria

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FLO-2D Model Theory and Numerical Stability
Criteria
Jim OBrien FLO-2D Software Inc.
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Focus of this Session

• Model Theory
• Stability Criteria
• How the Model Works
• Model Assumptions and Limitations

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Modeling Theory Momentum Equation

• Momentum equation choice of
• Kinematic Wave (no depressions)
• Diffusive wave (including pressure head)
• Dynamic wave (include acceleration terms)

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Momentum Equation
Sf So
- ?h/?x
- ( Vx /g)? Vx/?x
- (1/g) ? Vx/ ?t
convective acceleration
local acceleration
Kinematic wave
Diffusive wave
Full dynamic wave
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Relative Magnitude of the Momentum Equation Terms
Bed Pressure
Convective Local
Slope Gradient Acceleration
Acceleration
Equation Term So
?y/?x V?V/g?x
?V/g?t Magnitude (m/m) 0.005 0.0001
0.000035 0.000005 Essentially steady,
uniform flow
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Overland Flow Continuity Equation
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Routing Algorithm Highlights

• Dynamic wave momentum equation
• Central difference finite difference routing
  scheme
• Uses Newton-Raphson numerical method to determine
  the roots for the second-order, non-linear,
  partial differential equation
• Timesteps increment/decrement based on numerical
  stability criteria
• Separate channel floodplain stability criteria
• Unlimited array sizes (Data Input Manual List)

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Finite Difference Explicit Numerical Scheme

• Subject to strict numerical stability criteria
• When stability criteria is exceeded numerical
  surging

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Key to Numerical Stability and Speed

• Vol / t small
• Change in a grid element volume per timestep
  should be small.
• Steep rising hydrographs and small grid elements
  cause problems

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Numerical Stability Criteriaor how to control
the timestep

• Three Methods
• 1. Specify percent change in depth (DEPTOL)
  from previous timestep.

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2. Courant Criteria

• Control the variable timesteps

Courant coefficient (C 1 fixed). Hard-wired.
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3. Dynamic Wave Stability Criteria

• ? WAVEMAX in TOLER.DAT

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Numerical surging results if the stability
criteria is not appropriate

• You assign only DEPTOL and WAVEMAX in TOLER.DAT
• Look for
• High velocities
• Spikes in the computed hydrographs

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How FLO-2D Works
Velocity across the boundary is computed by the
momentum equation.
Linear estimate of the flow depth to compute the
velocity. Average depth, slope and n-values
across grid boundary.
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How FLO-2D Works
Discharges are computed across the grid element
boundaries.
Psuedo 2-D model During a timestep, the
discharge flux in all 8-directions for each grid
element are calculated one a time.
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How FLO-2D Works
If the numerical stability criteria is exceeded
for any grid element,
Whoops!!!
all the hydraulic computations for that timestep
are wiped out and the timestep is decreased.
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Are the results believable? Rectangular Flume

• 200 ft wide x 3000 ft long
• n 0.05
• Discharge 8,000 cfs
• slope 0.003 16 ft per mile
• What is the steady, uniform flow depth and
  velocity?

200 ft
X
3,000 ft
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Steady, uniform flow
v1
v2
d1
d2
sta 1
sta 2
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Momentum Equation
Sf So
- ?h/?x
- ( Vx /g)? Vx/?x
- (1/g) ? Vx/ ?t
convective acceleration
local acceleration
Kinematic wave
Diffusive wave
Full dynamic wave
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Computation Results

• Flow area A 1,365.53 ft2
• Flow depth d 6.83 ft
• Flow velocity v 5.86 fps

Run Model
Run MAXPLOT
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Model Assumptions and Limitations

• Primary assumptions and limitations involve
  spatial and temporal resolution of the grid
  system
• Each grid element is represented by a single
  elevation, n-value, flow depth
• Steady flow for the duration of the timestep
• Hydrostatic pressure distribution
• Channel grid elements are represented by one
  channel geometry and roughness
• 1-dimensional channel flow (no secondary
  currents, no vertical velocity distributions)
• Rapidly varying flow such as hydraulic jumps or
  shock waves are not simulated

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Whats coming up? Editing Components with
the GDS