Flood Routing definitions

Q(t)Peak flow attenuation

time

lag

Inflow at x

x

time t time t+tc t

tp

Recession limbRising limb Outflow at x+x

Flood Routing methods

Hydraulic Uses both dynamic and continuity equations Allows backwater effects to be modelled Solution advanced by timestep t

Hydrologic Uses only continuity equation Cannot model backwater effects Solution advanced downstream by x

Kinematic Wave Equation

0

t

A

x

Q

dA

dQ

A

QthatsoWLfQ

)(

t

Q

ct

Q

dQ

dA

x

Q

1

Continuity with no lateral inflow yields:

For quasi-uniform flow:

Substitute and separate variables to get wave eq.

01

t

Q

cx

Qor

Q Q+Q

x

t+ t

t A

where c = dQ/dA is wave celerity

Space-Time Coordinates

Time t

Distance x

8

7

65

43

21

t

x

x

tNucleus

Flow Q4 unknown

Continuity Around the Nucleus

x

QQ

x

Q

56

t

QQ

t

Q

78

438

217

246

135

1

1

1

1

QQQ

QQQ

QQQ

QQQ

07856

QQQQx

tc0

1

t

Q

cx

Q

8

7

65

43

21

dx

dt

Generalized Muskingum equation

011

11

2143

1324

QQQQ

QQQQ

xtc

Let

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

1

11

1

4

3

2

1

33221144

C

C

C

C

QCQCQCQC

Collecting terms,

where

KtX

KtX

KtX

KtX

21

2

21

2

Setting = 0.5 yields

Deriving the Diffusion equation

01

t

Q

cx

QNon-centered finite difference scheme creates a numerical error

2

21

x

QD

t

Q

cx

Q

Convert the Wave equation

to a Diffusion equation

2

2

2

3

2

1

x

Q

dhdKQ

K

t

Q

cx

Q

Diffusion coefficient is

related to channel conveyance

012212

12

2

x

Qx

t

Q

cx

Q or

Determine weighting coefficients

xshwhere

dhdQh

Qff

f

1221

122122

x

dhdQs

QD

f

Compare the two equations for the diffusion coeff. D

5.0;2

1 dh

dQh

Q

f

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

Numerical Stability Criteria

1

1

x

tc

Unstable

Condition for numerical stability is

Limits for x and t

122x

tcFor = 0.5 x

D

2

12 and

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

From parts 1 & 2

tcDxx

tc

x

D

22

1 or

Limits for x and t

122x

tcFor = 0.5 x

D

2

12 and

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

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

From parts 2 & 3 c

Dxt

x

D

x

tc 2212

or

From parts 1 & 2

tcDxx

tc

x

D

22

1 or

MIDUSS 98 Route Command

MIDUSS 98 Route Command

Details of last conduit design are displayed

Changes to x or t reported for information

User can change computed X or K values

Estimated values of weighting coefficients

Results of Route command

Calculating celerity

2Q