Applications of the Bernoulli equation to open channel flows
3.3 FROUDE NUMBER .1 Definition
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-
Fig. 3.9 Overflow above a broad-crested weir (weir height: 0.067 m, channel width: 0.25 m, crest length: 0.42 m):
(a) flow from the left to the right - note the supercritical flow downstream of the weir crest; (b) unduiar flow above the weir crest, flow from the right to the left.
where (icharac is a characteristic dimension (see notes below). For open channel flows, the rele- vant characteristic length is the flow depth d. For a rectangular channel, the Froude number is defined as:
Fr = gd
Rectangular channel (3.23a)
3.3 Froude number 39
Notes
1. The Froude number is used for scaling open channel flows and free-surface flows.
2. The length (icharac is the characteristic geometric dimension: internal diameter of pipe for pipe flows, flow depth for open channel flow in a rectangular channel, ship draught, hydrofoil thickness, etc.
3. For a channel of irregular cross-sectional shape, the Froude number is usually defined as:
Fr= . ^ (3.23b) where Kis the mean flow velocity, A is the cross-sectional area and B is the free-surface width.
For a horizontal channel, equation (3.23b) implies Fr = I at critical flow conditions (see Section 3.2.3, in this chapter).
4. Some authors use the notation:
^ V'^ pV^A inertial force Fr = = —^ oc
^^charac P^^^charac W e i g h t
3.3.2 Similarity and Froude number
In problems of fluid flow there are always at least four parameters involved: pressure difference AP, velocity V, length d and fluid density p. In open channel hydraulics another important parameter is the gravity acceleration g.
Considering a closed pipe circuit, and provided that the liquid remains liquid (i.e. no cavita- tion), the piezometric pressure (P + pgz) is merely affected by the gravity effect as the pressure P is free to take any value at all. In an open channel, through the existence of a fi*ee surface where the pressure must take a prescribed value Patm? the gravity can influence the flow pattern. For Patm fixed, the gravity determines the piezometric pressure and hence the flow properties. In any flow situations where the gravity effects are significant, the appropriate dimensionless number (i.e.
the Froude number) must be considered. In practice, model studies of open channel flows and hydraulic structures are performed using a Froude similitude. That is, the Froude number is the same for the model and the prototype (see Part III).
Notes
1. For open channel flow (one-phase and one-component flow) the Froude number plays a very important part in the appropriate flow equations. This has been already noted at critical flow con- ditions defined SiS d = d^ for which Fr = I.
2. For two-phases (e.g. ice-water flows) or two-components (e.g. air-water flows) open channel flows, the Froude number might not be a significant number.
3. If the viscosity fi plays an effective part, the Reynolds number is another relevant parameter in dimensional analysis (see Part III).
Application
A prototype channel is 1000 m long and 12 m wide. The channel is rectangular and the flow condi- tions are: d = 3m and Q = 15m^/s. What would be the size and discharge of a 1/25 scale model using a Froude similitude?
Solution
On the 1/25 scale model, all the geometric dimensions (including the flow depth) are scaled by a | factor 1/25:
Length Width Flow depth
-^model ~ ^prototype'-^-^
^model ~ -^prototype'-^-^
"model ~ "prototype'^-^
The flow velocity on the model is deduced by a Froude similitude. The Froude similitude implies that the Froude number is the same on both model and prototype:
Froude similitude It yields:
Velocity
For the discharge, the continuity equation Q = VdB Discharge As a result the model characteristics are:
Length Width Flow depth Velocity Discharge Froude similitude
-''''model -'^''prototype
''^model ~ '^prototype'V-^-^
provides the scaling ratio:
C^model ~ ^prototype'V-^*^
^model = 4 0 m
^modei = 0.25 m
^model = 0 . 1 2 m
^modei = 0.083 m/s emodei = 0.0048 m^/s
^''model ~ ^Vototype "= 0 . 0 7 7
3.3.3 Critical conditions and wave celerity
Considering an oscillatory wave', the velocity C or celerity^ of waves of length / is given by:
C
" - j t ' ^ ' ^ l ^ I (3.24)
v^here d is the flow^ depth and tanh is the hyperbolic tangent flinction (Henderson, 1966: p. 37;
Liggett, 1994: p. 394). For a long v^ave of small amplitude (i.e. d/l <$C 1) this expression becomes:
A similar result can be deduced from the continuity equation applied to a weak free-surface gravity wave of height A J <^ J in an open channel (Streeter and Wylie, 1981:p. 514).The wave celerity equals:
C=-sjgd (3.25) where C is the celerity of a small free-surface disturbance in open channel flow.
The Froude number may be vmtten in a form analogous to the Sarrau-Mach number (see Section 3.3.4) in gas flow, as the ratio of the flow velocity divided by the celerity of small disturbances:
Fr=^ (3.26)
^The term progressive wave can be also used (e.g. Liggett, 1994).
^The speed of a disturbance is called celerity (e.g. sound celerity, wave celerity).
3.3 Froude number 41 where C = J^ is the celerity of small waves (or small disturbance) at the free surface in open channels.
For Froude or Sarrau-Mach numbers less than unity, disturbances at a point are propagated to all parts of the flow; however, for Froude and Sarrau-Mach numbers of greater than unity, disturbances propagate downstream^ only. In other words, in a supercritical flow, small disturb- ances propagate only in the downstream flow direction. While in subcritical flows, small waves can propagate in both the upstream and downstream flow directions.
Note
Oscillatory waves (i.e. progressive waves) are characterized by no net mass transfer.
3.3.4 Analogy with compressible flow
In compressible flows, the pressure and the fluid density depend on the velocity magnitude rela- tive to the celerity of sound in the fluid Csound- The compressibility effects are often expressed in term of the Sarrau-Mach number Ma = F/Csound- Both the Sarrau-Mach number and the Froude number are expressed as the ratio of the fluid velocity over the celerity of a disturbance (celerity of sound and celerity of small wave respectively).
Dimensional analysis shows that dynamic similarity in compressible flows is achieved with equality of both the Sarrau-Mach and Reynolds numbers, and equal value of the specific heat ratio.
The propagation of pressure waves (i.e. sound waves) in a compressible fluid is compar- able to the movement of small amplitude waves on the surface of an open channel flow. It was shown (e.g. Thompson, 1972; Liggett, 1994) that the combination of motion equation for two- dimensional compressible flow with the state equation produces the same basic equation as for open channel flow (of incompressible fluid) in which the gas density is identified with the flow depth (i.e. free-surface position). Such a result is obtained however assuming: an inviscid flow, a hydrostatic pressure gradient (and zero channel slope), and the ratio of specific heat y must equal 2.
The formal analogy and correspondence of flow parameters are summarized in the following table:
Basic parameters
Other parameters
Flow analogies Basic assumptions
Open channel flow Flow depth d Velocity V d^
d^y-'^
Jgd
Froude number Gravity acceleration g Channel width B Hydraulic jump Oblique shock wave"^
Inviscid flow
Hydrostatic pressure gradient
Compressible flow Gas density p Velocity V Absolute pressure P Absolute temperature T Sound celerity C Sarrau-Mach number 1/2 (P/p-^) Flow area A Normal shock wave Oblique shock wave 7 = 2
^Downstream means in the flow direction.
'^Also called oblique jump or diagonal jump.
Application
The study of two-dimensional supercritical flow in open channel is very similar to the study of supersonic gas flow Liggett (1994) developed the complete set of flow equations. The analogy was applied with some success during the early laboratory studies of supersonic flows.
Notes
1. At the beginning of high-speed aerodynamics (i.e. first half of the 20th century), compressible flows were investigated experimentally in open channels using water. For example, the propagation of oblique shock waves in supersonic (compressible) flows was deduced from the propagation of oblique shock waves at the fi*ee surface of supercritical open channel flows. Interestingly the celerity C in open channel flow is slow (compared to the sound celerity) and it can be easily observed.
With the development of high-speed wind tunnels in the 1940s and 1950s, some compressible flow experimental results were later applied to open channel flow situations. Nowadays the ana- logy is seldom applied because of limitations.
2. The main limitations of the compressible flow/open channel flow analogy are
(a) The ratio of specific heat must equal 2. For real gases the maximum possible value for y is 5/3 (see Appendix Al.l). For air, y = 1.4. The difference in specific heat ratio (between the analogy and real gases) implies that the analogy can only be approximate.
(b) The accuracy of fi*ee-surface measurements is disturbed by surface tension effects and the presence of capillary waves at the free surface.
(c) Other limitations of the analogy include the hydraulic jump case. The hydraulic jump is ana- logue to a normal shock wave. Both processes characterize a flow discontinuity with energy dissipation (i.e. irreversible energy loss). But in a hydraulic jump, the ratio of the sequent depths (i.e. upstream and downstream depth) is not identical to the density ratio across a normal shock wave (except for Fr = 1).
3.3.5 Critical flows and controls
Occurrence of critical flow - control section
For an open channel flow, the basic equations are the continuity equation (i.e. conservation of mass), and the motion equation or the Bernoulli equation which derives from the Navier-Stokes equation. The occurrence of critical flow conditions provide one additional equation:
V = 4sd Flat rectangular channel (3.27) in addition to the continuity equation and the Bernoulli equation. These three conditions fix all
the flow properties at the location where critical flow occurs, called the control section.
For a given discharge, the flow depth and velocity are fixed at a control section, independently of the upstream and downstream flow conditions. For a rectangular channel, it yields:
(3.28)
V=V, = \\g^ (3.29) where B is the channel width.
3.3 Froude number 43 Corollary: at a control section the discharge can be calculated once the depth is known: e.g. the critical depth meter.
Hydraulic structures that cause critical flow (e.g. gates and weirs) are used as control sections for open channel flows. Examples of control sections include: spillway crest, weir, gate, overfall, etc. (e.g. Henderson, 1966: Chapter 6). Some hydraulic structures are built specifically to create critical flow conditions (i.e. critical depth meter). Such structures provide means to record the flow rates simply by measuring the critical flow depth: e.g. gauging stations, broad-crested weir, sharp-crested weirs (e.g. Rouse, 1938: pp. 319-326; Henderson, 1966: pp. 210-214; Bos, 1976;
Bos etal, 1991).
A control section 'controls' the upstream flow if it is subcritical and controls also the down- stream flow if it is supercritical. A classical example is the sluice gate (Fig. 3.6).
Upstream and downstream controls
In subcritical flow a disturbance travelling at a celerity C can move upstream and downstream because the wave celerity C is larger than the flow velocity r(i.e. V/C < 1). A control mechanism (e.g. sluice gate) can make its influence on the flow upstream of the control. Any small change in the downstream flow conditions affects tranquil (subcritical) flows. Therefore subcritical flows are controlled by downstream conditions. This is called a downstream flow control.
Conversely, a disturbance cannot travel upstream in a supercritical flow because the celerity is less than the flow velocity (i.e. V/C > 1). Hence supercritical flows can only be controlled from upstream (i.e. upstream flow control). All rapid flows are controlled by the upstream flow conditions.
Discussion
All supercritical flow computations (i.e. 'backwater' calculations) must be started at the upstream end of the channel (e.g. gate and critical flow). Tranquil flow computations are started at the down- stream end of the channel and are carried upstream.
Considering a channel with a steep slope upstream followed by a mild slope downstream (see definitions in Chapter 5), critical flow conditions occur at the change of slope. Computations must proceed at the same time from the upstream end (supercritical flow) and from the downstream end (subcritical flow). Near the break in grade, there is a transition from a supercritical flow to a sub- critical flow. This transition is called a hydraulic jump. It is a flow discontinuity that is solved by applying the momentum equation across the jump (see Chapter 4).
Application: influence of the channel width
For an incompressible open channel flow, the differential form of the continuity equation (Q = VA = constant) along a streamline in the ^-direction gives:
V^+A^=0 (3.30a)
OS OS
where ^ is the cross-sectional area. Equation (3.30a) is valid for any shape of cross-section. For a channel of rectangular cross-section, it becomes:
q^+B^=Q (3.30b) 3^ ds
where B is the channel surface width and q is the discharge per unit width {q = QIB).
For a rectangular and horizontal channel (ZQ constant and H fixed) the differentiation of the Bernoulli equation is:
^ - j i ^ + . J L ^ = 0 (3.31) ds g(p ds gd^ ds
Introducing the Froude number and using the continuity equation (3.30) it yields:
^l-Fr')^=Fr'^^ (3.32) In a horizontal channel, equation (3.32) provides a mean to predict the flow depth variation
associated with an increase or a decrease in channel width.
Considering an upstream subcritical flow (i.e. Fr < I) the flow depth decreases if the chan- nel width B decreases: i.e. dd/ds < 0 for 3^/3^ < 0. As a result the Froude number increases and when Fr = I critical flow conditions occur for a channel width B^^^^.
Considering an upstream supercritical flow (i.e. Fr> 1), a channel contraction (i.e. dB/ds < 0) induces an increase of flow depth. As a result the Froude number decreases and when Fr = I critical flow conditions occur for a particular downstream channel width ^min-
With both types of upstream flow (i.e. sub- and supercritical flow), a constriction which is severe enough may induce critical flow conditions at the throat. The characteristic channel width 5inin for which critical flow conditions occur is deduced from the Bernoulli equation:
R - Q
-^min 8 ^ 3 (3.33)
where E^ is the upstream specific energy. Note that equation (3.33) is valid for horizontal chan- nel of rectangular cross-section, ^min is the minimum channel width of the contracted section for the appearance of critical flow conditions. For B > ^min? the channel contraction does not induce critical flow and it is not a control section. With B < B^^^ critical flow takes place, and the flow conditions at the control section may affect (i.e. modify) the upstream flow conditions.
Notes
1. At the location of critical flow conditions, the flow is sometimes referred to as 'choking'.
2. Considering a channel contraction such as the critical flow conditions are reached (i.e. B = ^mm) the flow downstream of that contraction will tend to be subcritical if there is a downstream con- trol or supercritical in absence of downstream control (Henderson, 1966: pp. 47-49).
3. The symbol used for the channel width may be:
B channel free-surface width (m) (e.g. Henderson, 1966),
W channel bottom width (m), also commonly used for rectangular channel width.
3.4 PROPERTIES OF COMMON OPEN-CHANNEL SHAPES