Uniform Open Channel Flow

Manning’s Eqn for velocity or flow

€

v =

1

n

R

2 / 3

S S.I. units

v =

1 . 49

n

R

2 / 3

S English units

where n = Manning’s roughness

coefficient R = hydraulic radius = A/PS = channel slope

Q = flow rate (cfs) = v A

Brays Bayou

Concrete Channel

Uniform Open Channel Flow – Brays B.

Normal depth is function of flow rate, and geometry and slope. Can solve for flow rate if depth and geometry are known.

Critical depth is used to characterize channel flows -- based on addressing specific energy:

E = y + Q2/2gA2 where Q/A = q/y

Take dE/dy = (1 – q2/gy3) = 0.

For a rectangular channel bottom width b,

1. Emin = 3/2Yc for critical depth y = yc

2. yc/2 = Vc2/2g

3. yc = (Q2/gb2)1/3

In general for any channel, B = top width

(Q2/g) = (A3/B) at y = yc

Finally Fr = V/(gy)1/2 = Froude No.

Fr = 1 for critical flowFr < 1 for subcritical flowFr > 1 for supercritical flow

Critical Flow in Open Channels

Optimal Channels

Non-uniform Flow

Non-Uniform Open Channel Flow

With natural or man-made channels, the shape, size, and slope may vary along the stream length, x. In addition, velocity and flow rate may also vary with x.

H = z + y + α v

2

/ 2 g

( )

dH

dx

=

dz

dx

+

dy

dx

+

α

2 g

dv

2

dx

⎛

⎝

⎜

⎞

⎠

⎟

Where H = total energy headz = elevation head,

αv2/2g = velocity head

Thus,

Replace terms for various values of S and So. Let v = q/y = flow/unit width - solve for dy/dx

– S = − S

o

+

dy

dx

1 −

q

2

gy

3

⎡

⎣

⎢

⎤

⎦

⎥

since v = q / y

1

2 g

d

dx

v

2

[ ]=

1

2 g

d

dx

q

2

y

2

⎡

⎣

⎢

⎤

⎦

⎥

= −

q

2

g

1

y

3

⎡

⎣

⎢

⎤

⎦

⎥

dy

dx

Given the Fr number, we can solve for the slope of the water surface - dy/dx

Fr

2

= v

2

/ gy

( )

dy

dx

=

S

o

− S

1 − v

2

/ gy

=

S

o

− S

1 − Fr

2

where S = total energy slopeSo = bed slope, dy/dx = water surface slope

Note that the eqn blows up when Fr = 1 or So = S

Now apply Energy Eqn. for a reach of length L

y

1

+

v

1

2

2 g

⎡

⎣

⎢

⎤

⎦

⎥

= y

2

+

v

2

2

2 g

⎡

⎣

⎢

⎤

⎦

⎥

+ S − S

o

( )L

L =

y

1

+

v

1

2

2 g

⎡

⎣

⎢

⎤

⎦

⎥

− y

2

+

v

2

2

2 g

⎡

⎣

⎢

⎤

⎦

⎥

S − S

0

This Eqn is the basis for the Standard Step Method to compute water surface profiles in open channels

Backwater Profiles - Compute Numerically

Routine Backwater Calculations1. Select Y1 (starting depth)

2. Calculate A1 (cross sectional area)

3. Calculate P1 (wetted perimeter)

4. Calculate R1 = A1/P1

5. Calculate V1 = Q1/A1

6. Select Y2 (ending depth)

7. Calculate A2

8. Calculate P2

9. Calculate R2 = A2/P2

10. Calculate V2 = Q2/A2

Backwater Calculations (cont’d)

1. Prepare a table of values

2. Calculate Vm = (V1 + V2) / 2

3. Calculate Rm = (R1 + R2) / 2

4. Calculate Manning’s

5. Calculate L = ∆X from first equation

6. X = ∑∆Xi for each stream reach (SEE SPREADSHEET)

€

S =nVm

1.49Rm

23

⎛

⎝ ⎜ ⎜

⎞

⎠ ⎟ ⎟

2

€

L =

y1 + v12

2g

⎛

⎝ ⎜

⎞

⎠ ⎟−

y2 + v22

2g

⎛

⎝ ⎜

⎞

⎠ ⎟

S − S0

Watershed Hydraulics

Main Stream

Tributary

Cross Sections

Cross Sections

A

B

C

D

QA

QD

QC

QB

Bridge Section

Bridge

Floodplain

Brays Bayou-Typical Urban System

• Bridges cause unique problems in hydraulics

Piers, low chords, and top of road is considered

Expansion/contraction can cause hydraulic losses

Several cross sections are needed for a bridge

Critical in urban settings288 Crossing

The Floodplain

Top Width

Floodplain Determination

The Woodlands planners wanted to design the community to withstand a 100-year storm.

In doing this, they would attempt to minimize any changes to the existing, undeveloped floodplain as development proceeded through time.

The Woodlands

HEC RAS Cross Section

3-D Floodplain