A-to-Z Guide to Thermodynamics,
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In contrast to pipe flows, open channel flows are characterized by a free surface which is exposed to the atmosphere. The pressure on this boundary thus remains approximately constant irrespective of any changes in the water depth and the flow velocity. These free-surface flows occur commonly in engineering practice, and include both large-scale geophysical flows (rivers and estuaries) and small-scale man-made flows (irrigational channels, drainage channels and sewers). Although many of the traditional examples are of primary interest to the civil engineer, the underlying theory of open channel hydraulics is appropriate to any free-surface flow. In general, these flows may be laminar or turbulent, steady or unsteady, and uniform or varied. However, as in the case of pipe flow, this general class of problems may be subdivided into two distinct groups. The first involves significant changes in the water depth over relatively short channel lengths. These are classified as "rapidly varying flows," and are largely unaffected by shear forces. In contrast, the second group involves less rapid changes in the water depth (occurring over longer distances), and are classified as "gradually varying flows." This latter group may be significantly affected by shear forces.

Most elementary open channel hydraulics is based upon three underlying assumptions: (a) the fluid is homogeneous and incompressible; (b) the flow is steady; and (c) the pressure distribution is hydrostatic at all control sections. Although there are important exceptions, notably the inhomogeneity caused by the entrainment of air in a high-speed flow or the unsteadiness associated with the propagation of flood waves or tidal bores, these assumptions are widely applicable and lead to important simplifications of the conservation equations. In particular, if the pressure distribution is assumed hydrostatic at all control sections, this implies that the streamlines are straight, parallel, and approximately horizontal, and that there are no pressure gradients due to the curvature of the flow. In this case the hydraulic gradient line, or piezometric line, is the same for all streamtubes and is co-incident with the free surface. This result accounts for the wide application of the energy line — hydraulic gradient line as a means of describing an open channel flow.

To emphasize the importance of the unconstrained free-surface boundary, a transitional flow involving a step in a rectangular channel is considered ( Figure 1a).

If the volume flow per unit width is , mass conservation defines the velocity (u) in terms of the water depth (d) such that = u1d1 = u2d2. Furthermore, if the energy head (E) is expressed in terms of the specific head (or "specific energy") measured relative to the channel bed, E may be defined in terms of d and alone: where the final term is an alternative representation of the velocity head. Figure 1b concerns the variation of E and d for a given value of , and demonstrates that if E and are fixed there are (in general) two potential solutions for d. In the present case if there are no energy losses, Bernoulli's Theorem gives E2 = E1 – Δz, and thus the state (E2, d2) could be represented by point B or point B′ on the specific energy curve. Figure 1a. Transitional flow. Figure 1b. Specific energy curve (not to scale).

The depths d2 and d′ both represent physically realistic solutions, and are often referred to as "alternate depths" corresponding to two different flow regimes. Although the specific energy in each case is the same (E2), point B′ corresponds to a deep slow flow whereas point B′ describes a shallow fast flow.

Point C on the specific energy curve (Figure 1b) defines the common boundary of these two flow regimes, and represents the so-called critical flow. This state, which is usually defined in terms of a critical depth (dc), represents the minimum specific energy for a given volume discharge. If the water depth increases above the critical depth (d > dc) the specific energy also increases due to the flow-work associated with the hydrostatic pressure. In contrast, if the water depth reduces below the critical depth (d < dc) the kinetic energy (or velocity head) accounts for the increase in the specific energy. The critical flow may also be interpreted as producing the maximum flow for a given specific energy.

If dE/dy = 0 at the critical limit, it follows that the critical depth (dc) and the critical velocity (uc) are given by: These definitions allow the classification of the flow regimes noted above. If d > dc (or u < uc) the regime is described as subcritical (or subundal) flow; whereas if d < dc (or u > uc) supercritical (or superundal) flow is said to occur. A close analogy exists between these definitions of an open channel flow and the distinction of subsonic or supersonic flow in a compressible fluid. Indeed, this analogy can be further extended since the critical velocity uc defines the speed of a surface wave in water of depth dc As a result the Froude Number (Fr), defined by Fr2 = u/(gd), defines the ratio of the free stream velocity to the surface wave velocity. In the context of open channel flows Fr < 1 implies subcritical flow, Fr > 1 supercritical flow, and Fr = 1 critical flow. This approach is directly analogous to the Mach Number (M) description of a compressible flow. This defines the ratio of the gas velocity to the sonic velocity, such that M < 1 implies subsonic flow and M > 1 supersonic flow.

Returning to the transition problem (Figure 1a), the development of the water surface over the downstream step is dependent upon the "accessibility" of the two flow regimes. The specific energy curve (Figure 1b) provides guidance in this respect. If the upstream state is defined by A, and the discharge per unit width is constant, any changes must take place along the E-d curve shown on Figure 1b. To move from A to B is clearly possible, but to move from B to B′ requires a reduction in the specific energy below E2, and thus cannot be justified in the present transition.

As a result, the specific energy curve suggests that if the upstream flow is subcritical (point A), an increase in the bed elevation will produce a reduction in the water depth from d1 to d2 Not only is this result somewhat unexpected, the accessibility of the various flow regimes is dependent upon the upstream condition. For example, if the initial conditions were described by A′ (rather than A) the upstream regime would be supercritical. In this case, a similar accessibility argument suggests that an increase in the bed elevation would produce an increase in the water depth from to , with the final energy state represented by B′ on the E-d curve. This notion of the accessibility of the open channel flow regimes is analogous to a similar process in thermodynamic theory, in which the accessibility of various gas states is not only dependent upon the change in the energy level, but also on the required entropy change.

We have already noted that the distinction between subcritical and supercritical flow is dependent upon the velocity of a surface wave or disturbance. This has important implications for the control of any open channel flow, and in particular the estimation of water levels for a given volume discharge. If a flow is supercritical, a disturbance at the water surface is unable to travel upstream (relative to a stationary observer) because the velocity of the flow exceeds the wave velocity (Fr > 1). As a result, all supercritical flows are controlled from upstream, and may be considered "blind" to any changes which arise downstream. In contrast, subcritical flows (Fr < 1) are such that a surface disturbance can either travel upstream or downstream, and as a result these flows are typically controlled from downstream.

In its simplest form, a control structure is designed to change the water depth to (or through) the critical depth (dc), so that the discharge is fixed relative to the depth. In practice, most control structures accelerate a subcritical flow, through the critical regime, to produce a shallow fast supercritical flow. The most common examples of such structures include sluice gates and Weirs (Figure 2a and b). Figure 3. Open channel control structures.

To include more than one effective control within an open channel, the supercritical flow produced by an upstream control must be reconverted to a subcritical flow. This is usually achieved by a hydraulic jump (or stationary bore) in which the characteristics of the subcritical flow are determined by a second downstream control. These events are associated with large energy losses, and are often used as an effective means of dissipating unwanted kinetic energy downstream of an overflow (spillway) or underflow (sluice gate) structure. The hydraulic jump is in many respects analogous to a shock wave arising within a compressible flow. For example, whereas the hydraulic jump provides a transition from supercritical to subcritical flow, the shock wave involves a transition from supersonic to subsonic flow. In both cases there is a critical velocity below which these transitions cannot occur, and both processes involve an increase in entropy. Indeed, in the case of a hydraulic jump the increase in entropy per unit mass is proportional to the cube of the depth change, whereas in a shock wave this increase is proportional to the cube of the pressure difference (provided this is small). Specialist texts describing open channel flow are given by Chow (1959), Henderson (1966), Francis and Minton (1984) and Townson (1991).

#### REFERENCES

Chow, V. Y. (1959) Open Channel Hydraulics. McGraw Hill Inc., New York.

Francis, J. D. R. & Minton, P. (1984) Civil Engineering Hydraulics. Arnold, London.

Henderson, F. M. (1966) Open Channel Flow. Macmillan, London.

Townson, J. M. (1991) Free Surface Hydraulics. Unwin Hyman, London.

#### References

1. Chow, V. Y. (1959) Open Channel Hydraulics. McGraw Hill Inc., New York.
2. Francis, J. D. R. & Minton, P. (1984) Civil Engineering Hydraulics. Arnold, London.
3. Henderson, F. M. (1966) Open Channel Flow. Macmillan, London.
4. Townson, J. M. (1991) Free Surface Hydraulics. Unwin Hyman, London.