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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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Flood routing definitions l.jpg

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 l.jpg

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


Kinematic wave equation l.jpg

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 l.jpg

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


Continuity around the nucleus l.jpg

3

8

4

5

6

7

2

1

Continuity Around the Nucleus

bdt

adx


Generalized muskingum equation l.jpg

Generalized Muskingum equation

Let

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

Collecting terms,

Setting b = 0.5 yields

where


Deriving the diffusion equation l.jpg

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


Determine weighting coefficients l.jpg

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 l.jpg

Numerical Stability Criteria

Condition for numerical stability is

Unstable


Limits for d x and d t l.jpg

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...


Limits for d x and d t11 l.jpg

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 l.jpg

MIDUSS 98 Route Command


Miduss 98 route command13 l.jpg

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 l.jpg

Results of Route command


Calculating celerity l.jpg

Calculating celerity


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