Applications of the Bernoulli equation to open channel flows
3.2 APPLICATION OF THE BERNOULLI EQUATION - SPECIFIC ENERGY .1 Bernoulli equation
3.2.3 Specific energy Definition
The specific energy E is defined as:
E{y) = ^ ^ + ' ^ + {z{y)-z,) (3.16) v^here P is the pressure, v is the velocity, z is the elevation and z^ is the bed elevation. The spe-
cific energy is similar to the energy per unit mass, measured with the channel bottom as the datum. The specific energy changes along a channel because of changes of bottom elevation and energy losses (e.g. friction loss).
The mean specific energy is defined as:
E = H-z^ (3.17a)
where //is the mean total head. For diflat channel, assuming a hydrostatic pressure distribution (i.e. P = pgd) and an uniform velocity distribution, it yields:
E = d + — Flat channel and hydrostatic pressure distribution (3.17b) where d is the flow depth and Fis the mean flow velocity.
Notes
It must be emphasized that, even if the total head H is constant, the specific energy varies with the bed elevation z^ (or bed altitude) as:
H = E^z,
For a non-uniform velocity distribution, assuming a hydrostatic pressure distribution, the mean specific energy as used in the energy equation becomes:
E = d cos 0 + a — 2g
where Q is the channel slope, a is the Coriolis coefficient and Fis the mean flow velocity (F = QIA).
For a horizontal channel, the Bernoulli equation implies that the specific energy is constant.
This statement is true only within the assumptions of the Bernoulli equation: i.e. for an incom- pressible, frictionless and steady flow along a streamline.
For a rectangular channel it is convenient to combine the continuity equation and the specific energy definition. Using the total discharge Q, the expression of the specific energy becomes (for a flat channel):
^ = ^ + r - r (3.17c) Igd^B^
where B is the free-surface width.
Analysis of the specific energy
Relationship flow depth versus specific energy
The specific energy is usually studied as a fimction of the flow depth d. It is convenient to plot the relationship d =f(E) as shown in Fig. 3.5. In rectangular channels there is only one specific energy-flow depth curve for a given discharge per unit width Q/B.
For a tranquil and slow flow, the velocity is small and the flow depth is large. The kinetic energy term F^/2g is very small and the specific energy tends to the flow depth d (i.e. asymp- tote E = d).
For a rapid flow (e.g. a torrent), the velocity is large and, by continuity, the flow depth is small. The pressure term PIpg (i.e. flow depth) is small compared to the kinetic energy term.
The specific energy term tends to an infinite value when d tends to zero (i.e. asymptote d — 0).
At any cross-section, the specific energy has a unique value. For a given value of specific energy and a given flow rate, there may be zero, one or two possible flow depths (Fig. 3.5, Appendix Al .4).
3.2 Application of the Bernoulli equation - specific energy 31
E/dr.
^ Fig. 3.5 Dimensionless specific energy 6 curve for a flat rectangular channel
(equation (3.20)).
Critical flow conditions
For a constant discharge Q and a given cross-section, the relationship E —f{d) indicates the existence of a minimum specific energy (Fig. 3.5). The flow conditions {d^, V^, such that the mean specific energy is minimum, are called the critical flow conditions. They take place for:
dd / ( g constant) 0
The critical flow conditions may be expressed in terms of the discharge Q and geometry of the channel (cross-section A, free-surface width B) after transformation of equation (3.16). For a rectangular flat channel of constant width, the minimum specific energy E^i^ and the critical flow depth are respectively:
^ m i „ = | 4 (3.18)
QL
The specific energy can be rewritten in dimensionless term (for a flat channel) as:
(3.19)
(3.20)
Equation (3.20) is plotted in Fig. 3.5. Note that equation (3.20) is an unique curve: it is valid for any discharge.
The relationship specific energy versus flow depth indicates two trends (Fig. 3.5). When the flow depth is greater than the critical depth (i.e. d/d^ > 1), an increase in specific energy causes an
increase in depth. For d>dc, the flow is termed subcritical. When the flow depth is less than the critical depth, an increase in specific energy causes a decrease in depth. The flow is supercritical.
If the flow is critical (i.e. the specific energy is minimum), small changes in specific energy cause large changes in depth. In practice, critical flow over a long reach of channel is unstable.
The definition of specific energy and critical flow conditions are summarized in the table below for flat channels of irregular cross-section (area A) and channels of rectangular cross- section (width 5 ) :
Variable Channel of irregular cross-section Rectangular channel Specific energy E
dE_
dd
dd' limE limE
^ . ^
1 -
gA' 2gA'
gA' 3B^
Infinite d
dB_
dd
Critical depth d^
Critical velocity V^
Minimum specific energy E^^^
Froude number Fr
BE dd
ft
2 5, V
I A
1 -Fr^
fs
2 '
Jgd
1 -Fr^
Notes: A: cross-sectional area; B: free-surface width; Q: total discharge; q: discharge per unit width.
Notes
1. In open channel flow, t h e Froude n u m b e r is defined such as it equals 1 for critical flow condi- tions. T h e corollary is that critical flow conditions are reached if F r = 1.
2. A general dimensionless expression of the specific energy is:
d, dA 2
3.2 Application of the Bernoulli equation - specific energy 33
3.
For a flat channel it yields:
The ratio AIB of cross-sectional E area over
^
d.
[i.^=)
free-surface width is called sometimes the mean depth.
Application of the specific energy
For a frictionless flow in a horizontal channel, the specific energy is constant along the channel.
The specific energy concept can be applied to predict the flow under a sluice gate as a fiinction of the gate operation (Fig. 3.6). For a given gate opening, a specific discharge (q = QIB) takes place and there is only one specific energy/flow depth curve.
Critical flow conditions
Fig. 3.6 Specific energy diagrams for flow under a sluice gate: (a) sluice gate; (b) larger gate opening for the sanfie flow rate;
(c) constant specific energy and larger flow rate [q" > q).
The upstream and downstream values of the specific energy are equal (by application of the Bernoulli equation). For this value of specific energy, there are two possible values of the flow depth: one subcritical depth (i.e. upstream depth) and one supercritical depth (i.e. downstream depth). If the value of the specific energy is known, the upstream and downstream flow depths are deduced from the specific energy curve (Fig. 3.6a).
If the specific energy is not known a priori, the knowledge of the upstream flow depth and of the flow rate fixes the specific energy, and the downstream flow depth is deduced by graphical solution.
If the discharge remains constant but for a larger gate opening (Fig. 3.6b), the upstream depth is smaller and the downstream flow depth is larger.
In Fig. 3.6c, the specific energy is the same as in Fig. 3.6a but the flow rate is larger. The upstream and downstream flow depths are located on a curve corresponding to the larger flow rate. The graphical solution of the specific energy/flow depth relationship indicates that the upstream flow depth is smaller and the downstream depth is larger than for the case in Fig. 3.6a.
This implies that the gate opening in Fig. 3.6c must be larger than in Fig. 3.6a to compensate the larger flow rate.
Note
A sluice gate is a device used for regulating flow in open channels.
Discussion
Change in specific energy associated with a fixed discharge
Considering a rectangular channel. Figs 3.5 and 3.6 show the relationship flow depth versus spe- cific energy. For a specific energy such as £ > E^^, the two possible depths are called the alternate depths (Appendix A 1.4).
Let us consider now a change in bed elevation. For a constant total head H, the specific energy decreases when the channel bottom rises. For a tranquil flow (i.e. Fr <\ ovd> d^), a decrease in specific energy implies a decrease in flow depth as:
- | - 1 - F r ^ > 0 (3.21) oa
For rapid flow (i.e. Fr> \ or d < d^), the relation is inverted: a decrease of specific energy implies an increase of flow depth. The results are summarized in the following table:
Flow depth Fr Type of flow Bed raised Bed lowered d> dc < 1 slow, tranquil, fluvial d decreases d increases
Subcritical
d < dc > 1 fast, shooting, torrential d increases d decreases Supercritical
Figure 3.7 shows practical applications of the specific energy concept to flow transitions in open channel. For a change of bed elevation, the downstream flow conditions (E2, t/2) can be deduced directly from the specific energy/flow depth curve. They are fiinctions of the upstream flow con- ditions (Ey di) and the change of bed elevation Az^.
3.2 Application of the Bernoulli equation - specific energy 35
Notes
1. Examples of application of the relationship E versus d are shown in Figs 3.6 and 3.7: sluice gate and transition with bed elevation.
2. Rouse (1946: p. 139) illustrated with nice photographs the effects of change in bed elevation for subcritical (d > d^ and supercritical {d < d^ flows.
E, E2
iAZn •^ d
E2 £ i d
iAZn
Fig. 3.7 Specific energy and transition problem.
1 dIE
0.8
0.6
0.4
0.2 ^
Critical flow conditions
ql\~gE^ Fig. 3.8 Dimensionless discharge-depth curve for given specific energy in a horizontal rec- 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 tangular channel (equation (3.22b)).
Case of fixed specific energy
For a fixed specific energy E, the relationship flow rate versus flow depth in a rectangular flat channel is:
Q = Bd^2g(E - d) (3.22a) where d is the flow depth and B is the channel width. In dimensionless terms, it becomes:
4
B^gE' # > - ! (3.22b)
Equation (3.22b) is plotted in Fig. 3.8.
Figure 3.8 indicates that there is a maximum discharge g^ax such as the problem has a real solu- tion. It can be shown that, for a fixed specific energy E, the maximum discharge is obtained for critical flow conditions (i.e. d = d^ (see notes below). Results are detailed in the following table:
E constant General section Rectangular channel
Critical depth d^ • E - 2B^ ' 3
Maximum discharge Qn- i^max 1 6 r> Qm.. = B,\\-\gE
Notes
1. The expression of the maximum discharge for a given total head was first formulated by the Frenchman IB. Belanger for the flow over a broad-crested weir (Belanger, 1849). He showed in particular that maximum flow rate is achieved for dQ/dd = 0 (if the streamlines are parallel to the weir crest).
3.3 Froude number 37 2. Assuming a rectangular channel of constant width, equation (3.17) can be rewritten as:
Q = Bd-^lgiE - d) (3.22) For a constant specific energy, the maximum discharge is obtained for dQ/dd = 0, i.e.:
2gE-3gd _^
4^g{E-d) or
E=-d 2
Such a relationship between specific energy and flow depth is obtained only and only at critical flow conditions (i.e. equation (3.18), Fig. 3.5). The result implies that £" = E^nm and d = d^ (critical flow depth) for Q = gj^ax-