US Army Corps
of Engineers
Hydrologic Engineering Center
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
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
Energy Principles
he
Y2
Y1
Z2
Z1
Datum
Z 2  Y2 
 2 V22
2g
 Z1  Y1 
1 V12
2g
 he
Momentum Equation
Fx = m a
2
1
P2
 W

Wx
Z2
Datum
Ff
P1
L
Z1
Momentum Equation
P2 - P1 + Wx - Ff = Q  Δ 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
Δ Vx = Change in velocity from 2 to 1 in X direction
Momentum Equation – Forces
Pressure: P  AY
Weight:
Wx  W sin 
 A1  A2 
Wx   
 L S0
2


Friction:
Ff   p L
Where:
 A1  A2 
Ff   
 Sf L
 2 
   RSf
Q
1 V1   2 V2 
Mass x acceleration: ma 
g
Energy vs. Momentum
• Energy – Internal energy dissipation
represented by loss term, Sf (Manning’s n)
• Momentum – External boundary shear forces
represented by friction term, Sf (Manning’s n)
Unsteady Flow Equations
Momentum Equation:
2

(
α
Q 
Q /A)  gA( h   )  0
t
x
x S o S f
Continuity Equation:
Q + A = 0
x t
Steady Flow Equations
Energy (momentum) Equation:
( αQ2/A)
+ gA( h - So + S f ) = 0
x
x
Continuity Equation:
Q = VA
Numerical Solution
Friction slope averaging Steady: Average conveyance
Unsteady: Average friction slope
Average conveyance Eq.
 Q1  Q2 
Sf 

 K1  K 2 
2
Average friction slope Eq.
Sf 
Sf1  Sf 2
2
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.
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
Courant Number
1/ 2
 gA 
Cr   
 B 
T
x
For best results, the Cr should be near
1.0
Courant Number Example
• Depth ~ 10 ft.
• Cross section spacing (x) of ~ 1000 ft.
• Requires computational time step (t) about
1 minute
Finite Difference Modeling
Considerations
1.Stability of the computations.
2. Numerical accuracy of the
computations.
3. Resolution of input hydrographs.
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)
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.
Expansion/Contraction Coeffs.
• Not used in the momentum formulation
(RAS-unsteady)
• Should be in the data, however, for use
with steady flow analysis
Data Requirements
(Flow and Boundary Conditions)
Steady: Discharge (Q) at each cross
section.
Unsteady: Inflow hydrograph(s) which
are routed by the model.
Calibration Strategy – Targets
Steady: Match observed water surface (or
EGL) elevations.
Unsteady: As above, along with timing,
hydrograph shape, computed flow
distribution in networks.
Calibration Strategy Adjustments
Steady: Manning’s n
Unsteady: n and volume (storage); make
adjustments throughout range of flows in
hydrograph. Add/subtract flows if
necessary.
Flow Accounting
Q
Observed
outflow 2
Inflow
Observed
outflow 1
Computed
outflow
Time
Application Strategy
1. Check with range of steady flows
Rough stage calibration.
Possible supercritical flow locations.
Modeling of hydraulic structures.
2. Prepare hydrographs (boundary
conditions)
Upstream flows
Tributary (local flows)
Ungaged/unmodeled flows
Downstream (rating curve?)
3. Calibration
Manning’s n affects both stage and timing.
Storage areas can be very important.
Fine tuning via conveyance adjustment.
QUESTIONS?