2.2 Flow Classification

With the basic terminology defined, important classifications of flow can be reviewed. Flow can be classified in the following ways:

Steady and Unsteady Flow

Flow is classified as steady or unsteady based on changes with respect to time. If depth and velocity do not vary with time, the flow regime is considered to be steady. If depth and velocity at a point vary with time, the flow regime is classified as unsteady.

Obviously, the real-world situation is unsteady flow; observations from the bank of a small stream for a significant time show that the depth and velocity vary. However, changes in depth and velocity at a given point normally occur very slowly, even during a flood event. The slow change in these variables often allows satisfactory solution of open channel hydraulic problems with the assumption of steady flow. Where depth and velocity change slowly with time but are significantly affected by floodplain or reservoir storage, hydrologic modeling is often used to assess these effects. Hydrologic routing is sometimes referred to as quasi-unsteady or simplified unsteady flow analysis. For simplified unsteady flow, a hydrology program such as HEC-1, or its successor HEC-HMS, is used to develop the peak discharges throughout the watershed, often with input from an open channel hydraulics program such as HEC-RAS. Chapters 5, 6, 7, and 8 further discuss the use of simplified unsteady flow and the development of the necessary parameters to apply this method. Chapter 14 also covers simplified unsteady flow analysis, as well as full unsteady flow.

Full unsteady flow analysis, also referred to as hydrodynamic modeling, involves the solution of the full equations of motion. Throughout this book, any discussion of HEC-RAS in steady flow analysis mode can include both steady flow and simplified unsteady flow analysis. References to unsteady flow analysis include only hydrodynamic hydraulic modeling. Figure 2.10 illustrates the difference between steady and unsteady flow.

Uniform and Varied Flow

Flow is classified as uniform or varied based on changes with respect to distance. A flow is uniform if flow velocity and depth at a given moment do not change with distance. In uniform flow, the channel invert profile, the water surface profile, and the energy grade line (friction slope) profile are all parallel. Varied flow means that the flow depth can change along the channel reach. These three profiles have different slopes and are nonparallel. Figure 2.11 illustrates the difference between these two types of flow classifications. Walking upstream or downstream along a channel of a small stream, one can observe that depth and velocity vary with distance (varied flow). Uniform and varied flow can be either steady or unsteady. However, unsteady uniform flow is nearly impossible to demonstrate outside of a laboratory, so steady uniform flow is the normal assumption used in this book and for actual hydraulic analysis problems for which steady uniform flow is applicable.

Although uniform flow seldom actually occurs in either man-made or natural channels, the assumption of uniform flow is often adequate, since it gives a reasonable estimate of the discharge conveyed for a given set of channel geometry and roughness conditions. However, it does not result in as precise or defensible a solution as the assumption of varied steady flow. Most small, relatively inexpensive structures, such as storm sewers and highway drainage channels, may be adequately designed with uniform flow assumptions. Larger, more expensive structures, such as the San Luis Canal in California, shown in Figure 2.12, require the assumption of gradually varied flow to design an adequate structure at a minimum cost. A uniform depth assumption would result in reaches of the canal that would be too large or too small, depending on if the actual depth was less than or more than normal depth, respectively.

Although this canal carries a steady flow in a man-made channel, variations in the canal slope will cause the actual depth to vary about normal depth.

Gradually and Rapidly Varied Flow

Depending on the rate of variation with respect to distance, flows can be classified as gradually varied or rapidly varied. For gradually varied flow, depth and velocity changes are small and gradual with distance; for rapidly varied flow, they are large and abrupt (Figure 2.13). Most situations are well represented by the assumption of gradually varied flow. Figure 2.14 shows a length of stream under gradually varied flow conditions where the water surface elevation decreases and the velocity increases as the flow encounters a small in-channel weir.

Rapidly varied flow typically occurs at hydraulic structures such as dam spillways, where flow depth and velocity change abruptly over relatively short distances. Bridge openings that severely constrict flow may also cause rapidly varied flow through the bridge opening. The occurrence of a hydraulic jump, where the flow abruptly changes from high velocity and relatively shallow flow to low velocity and large depth, is perhaps the most notable example of rapidly varied flow, as illustrated in Figure 2.13. Figure 2.15a shows flow passing over a small in-channel weir, with the depth becoming very shallow and the velocity increasing greatly. Figure 2.15b shows a hydraulic jump occurring a short distance downstream of the weir. Note the velocity variation from one side of the channel to the other in Figure 2.15b. Velocities are much smaller at the boundaries than in the middle of the channel. Flow upstream of the weir and downstream of the bridge in Figure 2.15b would be gradually varied, with rapidly varied flow occurring over the weir and in the hydraulic jump.

Figure 2.15 Rapidly varied flow (a) over a low weir and (b) in a hydraulic jump.

Subcritical and Supercritical Flow

A flow can be classified as subcritical or supercritical by comparing the ratio of inertial and gravitational forces at a stream location. The inertial forces are characterized by the velocity term (V²), and the gravitational forces are represented by the term gD. The ratio of the inertial forces to the gravitational forces is called the Froude number:


where	Fr	=	the Froude Number (dimensionless)
	V	=	the average velocity (ft/s, m/s)
	g	=	the gravitational constant (32.2 ft/s², 9.81 m/s²)
	D	=	the hydraulic depth (ft, m)

When Fr > 1, the flow is supercritical and inertial forces dominate. As a result, the channel velocity is high and the depth is low, with the flow described as rapid or shooting. Supercritical flow is generally associated with steeper slopes. Flow in a street gutter is often supercritical, due to the usual shallow depths of a few tenths of a ft (several cm) combined with a velocity of 1-2 ft/s (0.3-0.6 m/s). For Fr < 1, the flow is said to be subcritical, with gravitational forces dominant. Consequently, the flow has a relatively low velocity and high depth, and it may be described as calm or tranquil. This type of flow is generally associated with small channel slopes and is the most common type of flow in natural channels. However, a stream may have an average velocity of 10 ft/sec (3 m/s) and the flow would be subcritical if the hydraulic depth were 3.3 ft (1 m) or more. For Fr = 1, both the depth and the flow are said to be critical.

Critical flow is a transitional condition, where neither inertial nor gravitational forces dominate. Only a small change in velocity or depth causes the flow to change to subcritical or supercritical. Critical depth is further addressed in Section 2.6. The real-world condition is normally one of subcritical flow, and most river systems have Froude numbers less than 0.5, even during major floods. Similarly, man-made channels normally have subcritical flow.

Supercritical flow occurs most often in man-made channels; however, steep mountain streams can be supercritical, especially during floods. Figure 2.16a shows a steep, natural stream in Switzerland that appears to be in supercritical flow. Examples of supercritical flow include flow in street gutters during a rainfall event, flow down a spillway, and flow in steep concrete channels. Log rides in amusement park flumes are also good examples of supercritical flow.

Figure 2.16 (a) Supercritical flow in a natural channel and (b) flood bore, unsteady and rapidly varied, in the same channel.

Steady, uniform, gradually varied, or rapidly varied flow conditions for subcritical or supercritical flow regimes may be adequately addressed with the procedures outlined in this book. However, some flow situations require special methods outside the scope of this book. Figure 2.16b shows the same scene just a few minutes after the picture in Figure 2.16a was taken. This high-velocity flood wave (flood bore) is similar to what is expected from a dam-break flood event. For this flood wave, the discharge increased from about 175 to 21,200 ft³/s (5 to 600 m³/s) in a matter of seconds. Figure 2.16b thus represents an unsteady, rapidly varied flood event. Special analysis procedures, beyond the scope of this book, are required to estimate the speed and height of the leading edge of the flood wave. The movement of a dam-break type flood can be simulated with unsteady flow modeling, however.

Example 2.2 Computing hydraulic variables.

Water flows at a depth (y) of 4 ft in a trapezoidal channel with a bottom width (B_o) of 10 ft and side slopes of 1V:3H (z = 3). If the discharge is 400 ft³/s, determine the velocity, hydraulic depth, and Froude number.

Solution

From Table 2.1, the cross-sectional area of flow is the area of the trapezoidal-shaped channel:

Example 2.3 Computing hydraulic variables.

If the Froude number is 1.2 for the same hydraulic depth as Example 2.2, compute the velocity and discharge in the channel.

Solution

A much steeper slope would be required to pass this higher discharge, compared to the slope that resulted in the flow of 400 ft³/s.