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lag. Peak flow attenuation. Recession limb. Rising limb. Outflow at x+ D x. c D t. time t. time t+ D t. x. Flood Routing definitions. Q(t). Inflow at x. t p. time. Flood Routing methods. Hydraulic Uses both dynamic and continuity equations

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Flood Routing definitions

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lag

Peak flow attenuation

Recession limb

Rising limb

Outflow at x+Dx

c Dt

time t

time t+Dt

x

Flood Routing definitions

Q(t)

Inflow at x

tp

time


Flood Routing methods

  • Hydraulic

    • Uses both dynamic and continuity equations

    • Allows backwater effects to be modelled

    • Solution advanced by timestep Dt

  • Hydrologic

    • Uses only continuity equation

    • Cannot model backwater effects

    • Solution advanced downstream by Dx


t+ t

t

A

Q

Q+Q

x

Kinematic Wave Equation

Continuity with no lateral inflow yields:

For quasi-uniform flow:

Substitute and separate variables to get wave eq.

or

where c = dQ/dA is wave celerity


Space-Time Coordinates

Time t

a Dx

Flow Q4 unknown

3

4

8

5

6

Nucleus

Dt

b Dt

7

1

2

Dx

Distance x


3

8

4

5

6

7

2

1

Continuity Around the Nucleus

bdt

adx


Generalized Muskingum equation

Let

and get Q4=f(Q1 , Q2 , Q3)

Collecting terms,

Setting b = 0.5 yields

where


or

Convert the Wave equation to a Diffusion equation

Diffusion coefficient is related to channel conveyance

Deriving the Diffusion equation

Non-centered finite difference scheme creates a numerical error


Compare the two equations for the diffusion coeff. D

f(a,b,D)=0 leads to multiple sets of (a,b) coordinates for any value of D.

Determine weighting coefficients


Numerical Stability Criteria

Condition for numerical stability is

Unstable


From parts 1 & 2

or

Limits for Dx and Dt

For b = 0.5

and

For very long channels, route hydrograph over multiple sub-reaches of length Dx=Length/N, N = 2,3,4...


From parts 1 & 2

or

or

From parts 2 & 3

Limits for Dx and Dt

For b = 0.5

and

For very long channels, route hydrograph over multiple sub-reaches of length Dx=Length/N, N=2,3,4...

For very short channels, use routing time-step equal to sub-multiple of hydrology time step, dt=Dt/N, N=2,3,4...


MIDUSS 98 Route Command


Estimated values of weighting coefficients

Details of last conduit design are displayed

Changes to Dx or Dt reported for information

User can change computed X or K values

MIDUSS 98 Route Command


Results of Route command


Calculating celerity


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