CIVE 530 - OPEN-CHANNEL HYDRAULICS
LECTURE 1: INTRODUCTION

1.1 OPEN-CHANNEL FLOW VS. PIPE FLOW

Open-channel flow has free surface subject to atmospheric pressure.
Pipe flow (closed-conduit) flow has only hydraulic pressure.
Open-channel flow problems are more complicated than closed-conduit flow problems.
The free surface is likely to vary in space (varied) and time (unsteady).

Fig. 1-1 (Chow)

In pipes, the cross section is fixed; in channels it varies.
In pipes, the roughness may vary from smooth brass to corroded pipes.
In channels, roughness varies from plastic flumes (n = 0.008) to flood plains (n ≈ 0.1-0.2).
Furthermore, roughness may vary with the flow level.
Recent experience suggests that values of equivalent Manning roughness applicable to the wetlands of South Florida may be as high as n = 1.
In channels that move their bed, there may be ripples and dunes, or plane bed. (Page 553 of Engineering Hydrology)
Storm sewer flow looks like pipe flow, but it behaves like channel flow.

1.2 TYPES OF FLOW

Cross sections can be prismatic (artificial) or non-prismatic (natural).
The variables are:

Discharge Q,
flow area A,
mean velocity U [U = Q/A],
wetted perimeter P,
channel top width T,
hydraulic radius R [R = A/P],
hydraulic depth D [D = A/T],
channel width B,
channel bottom (bed) elevation z,
flow depth d (represented as y when it cannot be confused with stage).
stage y [y = z + d].

Open-channel flow can be classified as follows:

Steady or unsteady
Uniform or equilibrium
Gradually varied or rapidly varied
Spatially varied

The channel is prismatic if the cross-sectional area does not change along the channel.
The channel is natural if the cross-sectional area changes along the channel.
The flow is steady if the flow variables do not change in time.
The flow is unsteady if the flow variables change in time.
The flow is uniform if the flow is steady and the channel is prismatic.
The flow is equilibrium if the flow is steady and the channel is non-prismatic (natural channel).
The flow is gradually varied if the variables change little in space.
The flow is rapidly varied if the variables change quickly [in space or time].
The flow is spatially varied if Q changes in space.

Fig. 1-2 (Chow)

Steady uniform flow: steady flow in a prismatic channel.
Steady equilibrium flow: steady flow in a natural channel.
Unsteady uniform flow does not exist!
Steady gradually varied flow: water surface profiles, backwater computations.
Unsteady gradually varied flow: floods.
Steady rapidly varied flow: flow over spillways, the steady hydraulic jump. (Spillway photos)
Unsteady rapidly varied flow:

moving hydraulic jump,
surges,
roll waves,
tidal bores,
kinematic shocks.
Examples: Arizona roll waves, Archer tidal bores, Amazon tidal bore.

Fig. 19-3 (Chow)

1.3 STATE OF FLOW

Velocities
There are three typical velocities in open-channel flow:

The mean flow velocity (Manning, or Chezy, velocity)
The relative celerity of dynamic (short) waves (Lagrange celerity).
The relative celerity of kinematic (long) waves (Seddon celerity).

The Manning velocity is

U = (1/n) R^2/3 S^1/2

The Lagrange (dynamic wave) celerity is

c_d = U +/- (gd)^1/2

where d = flow depth.
The relative Lagrange celerity is

c_rd = (gd)^1/2

The Seddon (kinematic wave) celerity is

c_k = βU

The relative Seddon celerity is

c_rk = (β - 1)U

where β is the exponent of the discharge-area (or discharge-depth) rating.

Q = &alpha A ^β

Q = &alpha y ^β

For laminar sheet flow: β = 3
For turbulent flow in wide rectangular channels: β = 5/3 (Manning friction)
For turbulent flow in wide rectangular channels: β = 3/2 (Chezy friction)

The Froude number is the ratio of Manning velocity to the relative Lagrange celerity:

F = U/(gd)^1/2

Depending on the Froude number, the flow regime is:

subcritical (F < 1),
critical (F = 1), or
supercritical (F > 1).

The flow is subcritical when short waves travel faster than the mean velocity: upstream propagation of primary surface disturbances is possible (flow control is performed from downstream).
The flow is critical when short waves travel at the same speed as the mean velocity: primary surface disturbances remains stationary.
The flow is supercritical when short waves travel slower than the mean velocity: upstream propagation of primary surface disturbances is not possible (flow control is performed from upstream).
The Vedernikov number is the ratio of the relative Seddon celerity to the relative Lagrange celerity:

V = [(β - 1)U] / (gd)^1/2

Depending on the Vedernikov number, the flow regime is:

stable (V < 1),
neutral (V = 1), or
unstable (V > 1).

When β = 3 (laminar flow), V = 2 F; V/F = 2.
When β = 3/2 (Chezy turbulent flow), V= (1/2) F; V/F = 1/2.
When β = 5/3 (Manning turbulent flow), V= (2/3) F; V/F = 2/3.
The flow is stable when energy-transporting waves (dynamic) travel faster than mass-transporting waves (kinematic).
The flow is neutral when energy-transporting waves (dynamic) travel at the same speed than mass-transporting waves (kinematic).
The flow is unstable when mass-transporting waves (kinematic) travel faster than energy-transporting waves (dynamic).

La Joya Canal, Arequipa, Peru (2005).

Viscosities
There are three typical viscosities in open-channel flow:

The molecular (or kinematic) viscosity ν
The macroviscosity of the steady flow UL_o (L_o = macrolength scale such as flow depth or hydraulic radius)
The wave viscosity of the unsteady flow UL (L = wavelength of the sinusoidal perturbation to the steady uniform flow).

The Reynolds number R is the ratio of the macroviscosity (hydraulic viscosity of hydraulic diffusivity) to the molecular viscosity.

R = (UL_o) / ν

The flow is laminar if UL_o << ν (R less than 500)
The flow is transitional if UL_o ≈ ν (R between 500 and 2000)
The flow is turbulent if UL_o >> ν (R greater than 2000)

Fig. 1-5 (Chow)

The dimensionless wave number is defined as (2π × the ratio of macroviscosity to wave viscosity):

W = (2π/L) L_o

The flow is kinematic if W << 1
The flow is dynamic if W ≈ 1-10
The flow is inertial if W >> 10

Celerity (Ponce and Simons)

Summary:

F controls the direction of flow of primary energy disturbances.
V controls the external stability of the flow (whether mass surface waves can overtake energy surface waves).
R controls the internal stability of the flow.
W controls the predominance of mass or energy.

1.4 LAMINAR VS TURBULENT FLOW

The Darcy-Weisbach friction factor equation, applicable to pipe flow, is:

h_f = f (L /d_o) [U²/(2g)]

in which h_f = head loss, f = Darcy-Weisbach friction factor, d_o = pipe diameter; L = pipe length.
Since:

R = A/P = [(1/4) πd_o²] /(π d_o) = d_o/4

Then:

h_f = f L U²/(8gR)

The energy gradient, or energy slope, is:

S = h_f / L = f U²/(8gR) = (f/8) (D/R) F²

where D= hydraulic depth:

D = A / T

where T = top width, and F is the Froude number:

F² = U²/(gD)

In a hydraulically wide channel: D ≈ R. Therefore:

S = (f/8) F²

S = f' F²

where f' = f/8.
In laminar flow, the Darcy-Weisbach friction factor f is inversely proportional to the Reynolds number (f is in the range 0.04-0.40):

f = K/R_e

K depends only on channel shape: K = 24 for rectangular channels; K = 14 for triangular channels.
In turbulent flow, K is less dependent on R_e.
The Prandtl-von Karman equation:

1/(f)^1/2 = 2 log [R_e(f)^1/2] + 0.4

In turbulent flow, f is in the range 0.016-0.040.
f' is on the order 0.002-0.005.

Fig. 1-3 (Chow)

The friction equation, applicable to hydraulically wide channels, is:

S = f' F²

For slope S constant, an increase/decrease in f' causes a decrease/increase in F.
For friction f' constant, an increase/decrease in S causes an increase/decrease in F.
For Froude F constant, an increase/decrease in f' causes an increase/decrease in S.
Go to Chapter 2.

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