River Mechanics and Introduction to Unsteady Flow Equations

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River Mechanics and Introduction to Unsteady Flow
Equations

• Objective Present key items for switching from
  HEC-RAS steady flow analysis to HEC-RAS unsteady
  flow simulation.

Michael Gee, Ph.D, PE Senior Hydraulic Engineer
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Steady vs. Unsteady

• Difference in handling friction and other losses
• Difference in numerical solution algorithm
• Difference in computation of X-Sec properties
• Difference in handling non-flow areas
• Difference in flow and boundary condition data
  requirements
• Difference in calibration strategy
• Difference in application strategy

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Energy Principles
he
Energy Grade Line
Water Surface
Y2
Channel Bottom
Y1
Z2
Z1
Datum
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Momentum Equation
?Fx m a
2
1
P2
W
?
?
Wx
Ff
P1
L
Z2
Z1
Datum
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Momentum Equation
P2 - P1 Wx - Ff Q ? ? Vx
Where P Hydrostatic Pressure
Wx Force due to weight of water in X
direction Ff Force due to external friction
from 2 to 1 Q Discharge ? Density of
water ? Vx Change in velocity from 2 to 1 in X
direction
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Momentum Equation Forces
Pressure
Weight
Friction
Where
Mass x acceleration
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Energy vs. Momentum

• Energy Internal energy dissipation represented
  by loss term, Sf (Mannings n)
• Momentum External boundary shear forces
  represented by friction term, Sf (Mannings n)

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Unsteady Flow Equations
Momentum Equation
Continuity Equation
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Steady Flow Equations
Energy (momentum) Equation
Continuity Equation
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Numerical Solution
Friction slope averaging - Steady Average
conveyance Unsteady Average friction slope
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Average conveyance Eq.
Average friction slope Eq.
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Numerical Solution
Algorithms used - Steady Iterative convergence
section-by-section for each flow. Unsteady
Matrix solution for flow and stage simultaneously
at all sections each time step.
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Numerical Solution of the Unsteady Flow Equations
CONVERGENCE The state of tending to a unique
solution. A given scheme is convergent if an
increasingly finer computational grid leads to a
more accurate solution. STABILITY (NUMERICAL
or COMPUTATIONAL) The ability of a scheme to
control the propagation or growth of small
perturbations introduced in the calculations. A
scheme is unstable if it allows the growth of
error to eventually obliterate the true
solution. Ref River Hydraulics EM
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Courant Number
For best results, the Cr should be near 1.0
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Courant Number Example

• Depth 10 ft.
• Cross section spacing (?x) of 1000 ft.
• Requires computational time step (?t) about 1
  minute

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Finite Difference Modeling Considerations

• Stability of the computations.
• 2. Numerical accuracy of the computations.
• 3. Resolution of input hydrographs.

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Pre-Computation of Hydraulic Properties(CSECT or
HTAB)
Steady Compute exact hydraulic properties at a
section for each trial water surface elevation
from the GR points, n-values,etc. Unsteady
Hydraulic properties are pre-computed for all
possible water surface elevations at each cross
section (HTAB)
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Non-Flow Areas
Steady ineffective areas may or may not be
occupied by water. Unsteady All areas
containing water (even if not moving) must be
included.
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Expansion/Contraction Coeffs.

• Not used in the momentum formulation
  (RAS-unsteady)
• Should be in the data, however, for use with
  steady flow analysis

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Data Requirements(Flow and Boundary Conditions)
Steady Discharge (Q) at each cross
section. Unsteady Inflow hydrograph(s) which
are routed by the model.
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Calibration Strategy Targets
Steady Match observed water surface (or EGL)
elevations. Unsteady As above, along with
timing, hydrograph shape, computed flow
distribution in networks.
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Calibration Strategy - Adjustments
Steady Mannings n Unsteady n and volume
(storage) make adjustments throughout range of
flows in hydrograph. Add/subtract flows if
necessary.
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Flow Accounting
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Application Strategy

• 1. Check with range of steady flows
• Rough stage calibration.
• Possible supercritical flow locations.
• Modeling of hydraulic structures.

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2. Prepare hydrographs (boundary
conditions) Upstream flows Tributary (local
flows) Ungaged/unmodeled flows Downstream
(rating curve?)
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3. Calibration Mannings n affects both stage
and timing. Storage areas can be very
important. Fine tuning via conveyance adjustment.
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QUESTIONS?