Distributed Flow Routing

Surface Water Hydrology, Spring 2005Reading: 9.1, 9.2, 10.1, 10.2

Venkatesh Merwade, Center for Research in Water Resources

Outline

• Flow routing

• Flow equations for Distributed Flow Routing (St. Venant equations)– Continuity– Momentum

• Dynamic Wave Routing– Finite difference scheme

Flow Routing

• Definition: procedure to determine the flow hydrograph at a point on a watercourse from a known hydrograph(s) upstream

• Why do we route flows?– Account for changes in flow

hydrograph as a flood wave passes downstream

• This helps in – Accounting for storages– Studying the attenuation of

flood peaks

Flow Routing Types

• Lumped (Hydrologic)– Flow is calculated as function of time, no

spatial variability– Governed by continuity equation and

flow/storage relationship

• Distributed (Hydraulic)– Flow is calculated as a function of space and

time– Governed by continuity and momentum

equations

Flow routing in channels

• Distributed Routing• St. Venant equations

– Continuity equation

– Momentum Equation

0

t

A

x

Q

What are all these terms, and where are they coming from?

0)(11 2

fo SSgx

yg

A

Q

xAt

Q

A

Assumptions for St. Venant Equations

• Flow is one-dimensional

• Hydrostatic pressure prevails and vertical accelerations are negligible

• Streamline curvature is small.

• Bottom slope of the channel is small.

• Manning’s equation is used to describe resistance effects

• The fluid is incompressible

Continuity Equation

dxx

QQ

x

Q

t

Adx

)(

Q = inflow to the control volume

q = lateral inflow

Elevation View

Plan View

Rate of change of flow with distance

Outflow from the C.V.

Change in mass

Reynolds transport theorem

....

.0scvc

dAVddt

d

Continuity Equation (2)

0

qt

A

x

Q

0)(

t

y

x

Vy

0

t

y

x

Vy

x

yV

Conservation form

Non-conservation form (velocity is dependent variable)

Momentum Equation

• From Newton’s 2nd Law: • Net force = time rate of change of momentum

....

.scvc

dAVVdVdt

dF

Sum of forces on the C.V.

Momentum stored within the C.V

Momentum flow across the C. S.

Forces acting on the C.V.

Elevation View

Plan View

• Fg = Gravity force due to weight of water in the C.V.

• Ff = friction force due to shear stress along the bottom and sides of the C.V.

• Fe = contraction/expansion force due to abrupt changes in the channel cross-section

• Fw = wind shear force due to frictional resistance of wind at the water surface

• Fp = unbalanced pressure forces due to hydrostatic forces on the left and right hand side of the C.V. and pressure force exerted by banks

Momentum Equation

....

.scvc

dAVVdVdt

dF

Sum of forces on the C.V.

Momentum stored within the C.V

Momentum flow across the C. S.

0)(11 2

fo SSgx

yg

A

Q

xAt

Q

A

0)(

fo SSgx

yg

x

VV

t

V

0)(11 2

fo SSgx

yg

A

Q

xAt

Q

A

Momentum Equation(2)

Local acceleration term

Convective acceleration term

Pressure force term

Gravity force term

Friction force term

Kinematic Wave

Diffusion Wave

Dynamic Wave

Momentum Equation (3)

fo SSx

y

x

V

g

V

t

V

g

1

Steady, uniform flow

Steady, non-uniform flow

Unsteady, non-uniform flow

Applications of different forms of momentum equation

• Kinematic wave: when gravity forces and friction forces balance each other (steep slope channels with no back water effects)

• Diffusion wave: when pressure forces are important in addition to gravity and frictional forces

• Dynamic wave: when both inertial and pressure forces are important and backwater effects are not negligible (mild slope channels with downstream control)

0)(

fo SSgx

yg

x

VV

t

V

Dynamic Wave Routing

Flow in natural channels is unsteady, nonuniform with junctions, tributaries, variable cross-sections, variable resistances, variable depths, etc etc.

Solving St. Venant equations• Analytical

– Solved by integrating partial differential equations– Applicable to only a few special simple cases of kinematic waves

• Numerical– Finite difference approximation– Calculations are performed on a

grid placed over the (x,t) plane– Flow and water surface elevation

are obtained for incremental time and distances along the channel

x-t plane for finite differences calculations

∆x ∆x

∆t h0, Q0, t0 h1, Q1, t0 h2, Q2, t0

h0, Q0, t1 h1, Q1, t1 h2, Q2, t2

∆x ∆x

∆t

i, ji-1, j i+1, j

i-1, j+1 i-1, j+1 i+1, j+1

x-t plane

Cross-sectional view in x-t plane

Finite Difference Approximations

• Explicit• Implicit

t

uu

t

u ji

ji

ji

11

x

uu

x

u ji

ji

ji

211

Temporal derivative

Spatial derivative

t

uuuu

t

u ji

ji

ji

ji

21

11

1

Temporal derivative

x

uu

x

uu

x

u ji

ji

ji

ji

1

111 )1(

Spatial derivative

Spatial derivative is written using terms on known time line

Spatial and temporal derivatives use unknown time lines for computation

Example