1.1 OPEN-CHANNEL FLOW
Open-channel flow has a free surface and it is, therefore, subject to atmospheric pressure
In closed-conduit flow, the cross section is fixed by the pipe boundaries. On the other hand, in open-channel flow, the flow cross section is not fixed, varying with the flow. In closed-conduit flow, the roughness varies from smooth brass to corroded pipes; in open-channel flow, it varies from acrylic glass or lucite® (a very smooth type of plastic), to that of natural stream channels and their neighboring flood plains.
In closed-conduit flow, the
hydraulic pressure at the center of the pipe defines the hydraulic grade line (HGL in Fig. 1-2).
The hydraulic pressure (head of water) above the centerline of the pipe is referred to as the piezometric head.
The energy grade line includes the velocity head
In open-channel flow, the flow depth measured above the
channel bottom defines the water surface elevation,
which is equivalent to the hydraulic grade line of closed-conduit flow; see Fig. 1-3.
The total energy grade line includes the velocity head
Note the difference between closed-conduit and open-channel flow. In closed-conduit flow, water will rise in a piezometer tube up the level where it defines the hydraulic grade line associated with the hydraulic pressure in the conduit. On the other hand, in open-channel flow, the water surface is the hydraulic grade line, which is at atmospheric pressure.
1.2 TYPES OF FLOW
There are two general types of open-channel cross sections:
Artificial, or human-made channels, are usually prismatic, featuring a constant shape and size, at least for a certain length of channel. Conversely, natural channels are typically nonprismatic, i.e., the shape and size of the cross section varies along the channel. Artificial channels are also referred to as canals.
Several geometric and hydraulic properties help describe an open channel (Fig. 1-4). These are:
In prismatic channels, the flow depth d is often referred to as y, particularly when it cannot be confused with stage. Also, the channel side slope is often referred to as z H : 1 V, particularly when it cannot be confused with bed elevation.
Open-channel flow may be classified as follows:
The flow is steady when the hydraulic variables (discharge, flow area, mean velocity, flow depth, and so on) do not vary in time. Conversely, the flow is unsteady when the hydraulic variables vary in time and space. Steady flow is relatively simple to calculate, compared to unsteady flow.
The flow is uniform when the channel is prismatic and the hydraulic variables (Q, A, V, d, and so on) are constant in time and space. The flow is in equilibrium when the channel is nonprismatic and the hydraulic variables are approximately constant in time and space. The calculation of uniform flow is relatively straight forward when compared to that of other states of flow.
The flow is gradually varied when the discharge Q is constant but the other hydraulic variables (A, V, d, and so on) vary gradually in space. Under gradually varied flow, the pressure distribution in the vertical direction, normal to the flow, is very close to hydrostatic, i.e., proportional to the flow depth.
The flow is rapidly varied when the discharge is constant but the other hydraulic variables (A, V, d, and so on) vary rapidly in space, in such a way that a hydrostatic pressure distribution cannot be assumed in the vertical direction normal to the flow. While the calculation of gradually varied flow is somewhat involved but doable, the calculation of rapidly varied flow is generally more complex, in practice being based on empirical formulas, for lack of a theoretical solution.
The flow is spatially varied when the discharge Q varies in space only, i.e., along the channel, typically due to lateral inflow or outflow.
Steady uniform flow occurs in a prismatic channel (Fig. 1-5); steady equilibrium flow occurs in a nonprismatic channel. Unsteady uniform flow does not exist in nature, because the flow cannot be uniform and unsteady at the same time. The word "unsteady" implies nonequilibrium; thus, unsteady equilibrium flow does not exist.
Steady gradually varied flow is represented by the water surface profiles, also referred to as backwater (or drawdown) computations (Chapter 7). Unsteady gradually varied flow is the calculation of flood flows, or flood routing (Chapter 10).
Steady rapidly varied flow is represented by the flow over spillways or the hydraulic jump. Unsteady rapidly varied flow is represented by the moving hydraulic jump, surges, roll waves, kinematic shocks, and tidal bores. Figure 1-6 shows a train of roll waves in a steep irrigation canal.
Spatially varied flow occurs in an artificial canal when the discharge is varying along the channel, due to lateral water extractions or channel overflow.
1.3 STATE OF FLOW
The state of open-channel flow may be described in terms of certain characteristic velocities and
Velocity is the ratio of length (distance) over time, with units
There are three characteristic velocities in open-channel flow:
The mean velocity of the steady uniform flow using the Manning equation (SI units) is:
in which n = Manning's friction coefficient, R = hydraulic radius, and S = friction slope.
The mean velocity of the steady uniform flow using the Chezy equation is:
in which C = Chezy friction coefficient.
In general, four forces are active in a control volume in open-channel flow. These forces are due to friction, gravity, the pressure (flow depth) gradient, and inertia. Kinematic waves are those where the momentum balance is expressed in terms of the frictional and gravitational forces only (Lighthill and Whitham, 1955).
in which β = exponent of the discharge-flow area rating, defined as follows:
Dynamic waves are those where the momentum balance is expressed in terms of the pressure-gradient and inertial forces only. The celerity of dynamic waves is:
in which g = gravitational acceleration, and D = hydraulic depth, D = A /T.
From Eq. 1-3, the relative celerity of kinematic waves is:
From Eq. 1-5, the (absolute value of the) relative celerity of dynamic waves is:
For rectangular channels, for which D = d, or for hydraulically wide channels, for which D ≅ d, the relative celerity of dynamic waves is:
Equation 1-8 is known as the Lagrange (relative) celerity equation, after Lagrange (1788), who first derived it.
The Froude number is defined as follows (Chow, 1959):
The Froude number characterizes the condition of:
Under subcritical flow, surface waves (perturbations) can travel upstream, because their upstream celerity -w is greater than the mean flow velocity u.
Under critical flow, surface waves (perturbations) remain stationary, because their (absolute) celerity w is equal to the mean flow velocity u.
Under supercritical flow, surface waves (perturbations) can travel downstream only, because their upstream celerity -w is smaller than the mean flow velocity u.
The Vedernikov number characterizes the following states of flow:
Under stable flow, the relative kinematic wave celerity v is smaller than the relative dynamic wave celerity w and, therefore, surface waves (perturbations) are able to attenuate (dissipate).
Under neutrally stable flow, the relative kinematic wave celerity v is equal to the relative dynamic wave celerity w and, therefore, surface waves (perturbations) neither attenuate nor amplify. Amplification amounts to negative dissipation.
Under unstable flow, the relative kinematic wave celerity v is greater than the relative dynamic wave celerity w. Therefore, surface waves (perturbations) are subject to amplification. In practice, the condition V ≥ 1 leads to the development of roll waves, a train of waves that travel downstream, typically in artificial channels of steep slope (Cornish, 1907) (Fig. 1-7).
The three velocities u, v, and w lead to only two independent velocity ratios, the Froude (Eq. 1-8) and Vedernikov (Eq. 1-9) numbers. The third ratio:
is the dimensionless relative kinematic wave celerity, equal to the exponent of the discharge-area rating minus 1. Thus, it is seen that the exponent β in Eq. 1-4 is a function of both the Froude and Vedernikov numbers.
The value of β varies with the type of friction regime (laminar, transitional, or turbulent; and turbulent Manning or Chezy) and cross-sectional shape. For laminar flow, β = 3. For turbulent flow, under Manning friction: 1 ≤ β ≤ 5/3, depending on the shape of the cross section. For turbulent flow, under Chezy friction: 1 ≤ β ≤ 3/2, depending on the shape of the cross section.
There are three asymptotic cross-sectional shapes in open channels:
Neutrally stable flow
For neutral stability: V = 1. Therefore, from Eq. 1-11, the Froude number corresponding to neutrally stable flow is:
Table 1-1 shows values of Fns for selected values of β. It is seen that as β varies from β = 3 (laminar flow) to β = 1 (inherently stable channel), the value of Fns varies from Fns = 1/2 to Fns = ∞. In other words, as β ⇒ 1, Fns ⇒ ∞.
In practice, since friction has effectively a lower bound, the Froude number is restricted to an upper bound, which is seldom likely to exceed F = 25. Therefore, in most cases, a value of β = 1.04 would be already stable for practical purposes.
As shown in Table 1-1, values of β in open-channel
and overland flow are limited in the range
There are three characteristic viscosities in open-channel flow:
The kinematic viscosity ν of the fluid varies as a function of temperature (Appendix A).
The concept of hydraulic diffusivity νh is due to Hayami (1951). Hayami combined the governing equations of open-channel flow (Chapter 10) to develop a single convection-diffusion equation, i.e., an equation describing the convection (first-order process) and diffusion (second-order process) of a flood wave. The hydraulic diffusivity is defined as follows:
in which qo = equilibrium unit-width discharge, and So = friction (energy) slope. It is seen that flood wave diffusion is directly proportional to unit-width discharge and inversely proportional to friction (energy) slope.
Equation 1-13 can be expressed in terms of velocity and flow depth as follows:
A related value of diffusivity, which is independent of slope, is:
In general, for an arbitrary cross-sectional shape:
in which Ro = hydraulic radius.
In kinematic wave theory, the characteristic reach length is defined as follows (Lighthill and Whitham, 1955):
in which Lo is the length of channel in which the equilibrium flow drops a head equal to its depth. Thus, in terms of the characteristic reach length, the hydraulic diffusivity is:
In a manner resembling the hydraulic diffusivity, the wave diffusivity is conveniently defined as follows:
in which L = wavelength of the disturbance.
The Reynolds number R is (Chow, 1959):
The Reynolds number R describes the flow regime as either:
Under steady flow conditions in open-channel flow, laminar flow occurs for R ≤ 500 and turbulent flow for R > 2000. Transitional flow occurs in the intermediate range: 500 < R ≤ 2000. Under unsteady flow, the mixed laminar-turbulent flow described in Table 1-1 is akin to transitional flow, featuring a comparable range of Reynolds numbers.
In practice, most open-channel flow cases are in the turbulent regime. Conversely, most overland flow cases (i.e., free-surface plane flow) are in the laminar or mixed laminar-turbulent regime.
The dimensionless wavenumber σ is defined as follows (Ponce and Simons, 1977):
The wavenumber σ describes the dimensionless length scale of the wave, as shown in Fig. 1-11, in terms of: (a) kinematic waves, (b) dyamic waves, and (c) mixed kinematic-dynamic waves. Figure 11 is applicable for the case of Chezy friction in hydraulically wide channels.
Under kinematic flow, depicted on the left side of Fig. 1-11:
Under dynamic flow, depicted on the right side of Fig. 1-11:
Under mixed kinematic-dynamic flow, depicted by the middle section of Fig. 1-11:
Dynamic hydraulic diffusivity
For low Vedernikov numbers, V ⇒ 0, the dynamic hydraulic diffusivity reduces to the expression for kinematic hydraulic diffusivity, i.e., Eq. 1-18. Conversely, for high Vedernikov numbers, V ⇒ 1, and the dynamic hydraulic diffusivity vanishes. Under this flow condition, the total absence of wave attenuation is conducive in the development of roll waves (Figs. 1-6 and 1-7).
1.4 FLOW REGIMES
The flow regimes in open-channel flow are:
The flow regimes are characterized by the Reynolds number R, Eq. 1-20. In open-channel flow, the laminar regime prevails for R ≤ 500, the transitional regime for 500 < R ≤ 2000, and the turbulent regime for R > 2000.
The flow regimes vary with roughness of the channel surface. Figure 1-12 shows the relation between Reynolds number R and Darcy-Weisbach friction factor f for flow in smooth channels. Figure 1-13 shows the relation between Reynolds number R and Darcy-Weisbach friction factor f for flow in rough channels.
The Darcy-Weisbach friction formula, developed in connection with flow in pipes, is:
in which hf = frictional head loss; f = Darcy-Weisbach friction factor; L = length of the pipe; do = pipe diameter; V = mean flow velocity in the pipe; and g = gravitational acceleration.
The examination of Figs. 1-12 and 1-13 enables the following conclusions:
Note that the Prandtl-von Karman equation can be expressed in explicit form as follows:
The Darcy-Weisbach formula, Eq. 1-23, is strictly applicable to closed-conduit (pipe) flow. In pipe flow, the characteristic frictional length is the pipe diameter do. On the other hand, in open-channel flow, the characteristic frictional length is the hydraulic radius R, i.e., the ratio of flow area to wetted perimeter:
Since the flow area of a circular pipe (flowing full) is A = π do2/4, and the wetted perimeter is P = π do, it follows that the hydraulic radius is equal to 1/4 of the pipe diameter or, conversely, that the diameter is equal to 4 times the hydraulic radius. Therefore, the Darcy-Weisbach formula applicable to open-channel flow is:
in which V = mean flow velocity in the channel.
In open-channel flow, the energy slope, which under steady flow is the same as friction, bed, or bottom slope, is:
For any arbitrary cross-sectional shape, the Froude number is:
in which D = hydraulic depth: D = A /T.
Equation 1-30 can be expressed in terms of the Froude number as follows:
Equation 1-32 states the proportionality between energy slope and Froude number, with the proportionality factor being a function of the Darcy-Weisbach friction factor and the shape factor D /R.
For a hydraulically wide channel, for which D ≅ R, Eq. 1-32 reduces to:
Thus, for a hydraulically wide channel, the proportionality factor between energy slope and Froude number is only a function of the Darcy-Weisbach friction factor. In essence, for application to open-channel flow, a modified Darcy-Weisbach friction factor f, equal to 1/8 of the conventional Darcy-Weisbach friction factor f is applicable. The modified Darcy-Weisbach equation for open-channel flow is:
Table 1-2 shows approximate values of Darcy-Weisbach friction factor f and corresponding modified friction factor f for selected values of R in the turbulent range.
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