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FHWA > Engineering > Hydraulics > Introduction to Highway Hydraulics 
Introduction to Highway HydraulicsChapter 4  OpenChannel FlowOpenchannel flow is more complex than closedconduit flow flowing full because the water surface is determined by the mechanics of motion. In addition, if the bottom boundary is movable (alluvial boundary) another complexity is introduced. When the channel is mobile, the resistance to flow is a function of the flow. In this chapter the concepts and equations for the simplest flow condition (steady, uniform flow) will be described, as well as the bedform conditions that occur in an alluvial channel. Flow conditions and equations for solving problems of increasing flow complexity will be given. The onedimensional method will be used in the descriptions of the equations. Alluvial channels are channels formed in material that has been and can be transported by the flow. They are commonly made up of bed material composed of sand, gravel, and cobblesized material. These materials are important in drainage design because they affect resistance to flow and erosion. Concrete channels and culverts may have an alluvial boundary because of deposition of bed material in the invert. 4.2.2 Bedforms in Sand Channels The predominant material in sandbed streams ranges from coarse silt to sand. There may be finer or coarser material in the bed, but the dominant size will be sand (50 percent or more). In sandbed streams, the bed material is easily eroded and continually being moved and shaped by the flow. Interaction between the flow of water sediment mixture and the sand bed creates different bed configurations which change the resistance to flow, velocity, water surface elevation, and sediment transport. Consequently, it is necessary to understand what bedforms will be present so that the resistance to flow can be estimated and flood stages, depth of flow, and water surface profiles can be computed in order to design drainage channels. Flow in alluvial channels is divided into two regimes separated by a transition zone. Forms of bed roughness in sand channels are shown in Figure 4.1. The flow regimes are:
Resistance to flow for the different bedforms and coarser bed material will be given later in this section. For information on sediment transport and additional information on bedforms the reader is referred to HDS 6 (Richardson et al. 2001). At high flows, most sandbed stream channels shift from a dune bed to a transition or a plane bed configuration. If the slope is steep antidune flow may occur. Resistance to flow is then decreased to one half to one third of that preceding the shift in bedform. The increase in velocity and corresponding decrease in depth may increase erosion and scour around bridge piers and abutments and increase the required size of riprap. If flow transitions to antidune flow significant wave action may occur. At low flow, coarse alluvial bed material may not move, but at moderate or large flows, the material may become mobile. With the movement of coarsebed material, large bars may form which will be residual at low flow. These bars can redirect flow and cause bank erosion, scour holes, and clog drainage channels. Resistance to flow for coarsebed material is caused by the grain roughness of the material and the form loss caused by the bars. However, coarsebed material in drainage channels can have a beneficial effect by decreasing erosion by armoring of the bed. Information on armoring is given in HEC20 (Lagasse et al. 2001) and HDS 6 (Richardson et al. 2001). The determination of Manning's n for coarsebed material is given later. In steady, uniform openchannel flow, there are no accelerations, streamlines are straight and parallel, and the pressure distribution is hydrostatic. The slope of the water surface S_{w}, the bed surface S_{o}, and the energy gradient S_{f} are equal (Figure 3.2). It is the simplest flow condition to analyze. Steady uniform flow is an idealized concept for openchannel flow and is difficult to obtain even in laboratory flumes. For many applications, the flow is essentially steady and changes in width, depth, or direction (resulting in nonuniform flow) are so small that the flow can be considered uniform. In other cases, the changes occur over such a long distance the flow is a gradually varied flow. Depth in steady uniform flow is called the normal depth and the symbol for it is given the subscript o as in Y_{o}. Velocity (V) is often given the same subscripts, i.e., V_{o}. Other variables of interest for steady uniform flow are (1) the discharge (Q), (2) the velocity distribution v_{y} in the vertical, (3) the headloss h_{L} through the reach, and (4) the shear stress, both local and at the bed τ_{o}. All these variables are interrelated. In the following section, engineering equations will be given along with example problems for obtaining values for these variables. 4.3.1 Manning's Equation for Mean Velocity and Discharge Water flows in a sloping drainage channel because of the force of gravity. Flow is resisted by the friction between the water and wetted surface of the channel. The quantity of water flowing (Q), the depth of flow (y), and the velocity of flow (V) depend upon the channel shape, roughness (n), and slope (S_{0}). Various equations have been devised to determine the velocity and discharge in open channels. A useful equation is the one that is named for Robert Manning, an Irish engineer. The Manning's equation for the velocity of flow in open channels is: (4.1)
where:
Over many decades, typical Manning's n values have been compiled allowing an engineer to estimate the appropriate value by knowing the general nature of the channel boundaries. Most hydraulics textbooks and drainage design manuals provide tables of typical Manning's n values. An abbreviated list of such Manning's roughness coefficients is given in Appendix B, Table B.2. Several pictorial guides are also available showing the Manning's n value for different types of channels and floodplains (Barnes 1967 and Acrement and Schneider 1984). Special considerations exist for very steep channels (Jarrett 1985). A numerical approach for n value estimates consists of the selection of a base roughness value for a straight, uniform, and smooth channel in the materials involved, and then adding values for the channel under consideration: (4.2)n = (n_{0} + n_{1} + n_{2} + n_{3} +n_{4}) m_{5} where:
A discussion of this method and coefficients can be found in Cowan (1956) and Chow (1959). This method may be useful for natural channels, but has limited application for most roadway drainage design work. For rock riprap channels the Manning's n is often described as some function of the rock size. Several equations are provided in HEC15, including: (4.3)n = (K_{u})(y^{1/6}) / (2.25 + 5.23 log (y/D_{50})) where:
This equation is valid for y /D_{50} ranging from 1.5 to 185 which should be typical of most conditions encountered in roadside and other small channels. For conditions outside this range see HEC15. Roughness characteristics on the floodplain are complicated by the presence of vegetation, natural and artificial irregularities, buildings, undefined direction of flow, varying slopes, and other complexities. Resistance factors reflecting these effects must be selected largely on the basis of past experience with similar conditions. In general, resistance to flow is large on the floodplains. In some instances, conditions are further complicated by deposition of sediment and development of dunes and bars which affect resistance to flow and direction of flow. The presence of ice affects channel roughness and resistance to flow in various ways. When an ice cover occurs, the open channel is more nearly comparable to a closed conduit. There is an added shear stress developed between the flowing water and ice cover. This surface shear is much larger than the normal shear stresses developed at the air water interface. The ice water interface is not always smooth. In many instances, the underside of the ice is deformed so that it resembles ripples or dunes observed on the bed of sandbed channels. This may cause overall resistance to flow in the channel to be further increased. With total or partial ice cover, the drag of ice retards flow, decreasing the average velocity and increasing the depth. The hydraulic radius, R, is a shape factor that depends only upon the channel dimensions and the depth of flow. It is computed by the equation: (4.4)R = A / P where:
The discharge (Q) is determined from the equation of continuity (see Chapter 3). The equation is: Q = V A where:
By combining Equations 4.1 and 4.4, Manning's equation can be used to compute discharge directly: (4.5)
In some computations, it is convenient to group the crosssectional properties into a term called conveyance, K, (4.6)
then Q = K S^{1/2} When a channel cross section is irregular in shape such as one with a relatively narrow deep main channel and wide shallow overbank area, the cross section must be subdivided and the flow computed separately for the main channel and overbank area. The same procedure is used when different parts of the cross section have different roughness coefficients. In computing the hydraulic radius of the subsections, the water depth common to the two adjacent subsections is not counted as wetted perimeter (see Example Problem 4.3). Conveyance can be computed and a curve drawn for any channel cross section. The area and hydraulic radius are computed for various assumed depths and the corresponding value of K is computed from the equation. Values of conveyance are plotted against the depths of flow and a smooth curve connecting the plotted points is the conveyance curve. If the section was subdivided, the conveyance of each subsection (K_{a}, K_{b},...K_{n}) is computed and the total conveyance of the channel is the sum of the conveyances of the subsections. Discharge can then be computed using Equation 4.7 Example Problem 4.3 illustrates a conveyance curve for a compound cross section. The concept of channel conveyance is useful when computing the distribution of overbank flood flows in the stream cross section and the distribution through the openings in a proposed stream crossing. The discharge through each opening can be assumed to have the same ratio to the total discharge as the ratio of conveyance of the opening bears to the total conveyance of the channel. 4.3.2 Aids in the Solution of Manning's Equation Equations for the computation of Area, A, wetted perimeter, P, and hydraulic radius, R, in rectangular and trapezoidal channels (Figure 4.2) are: (4.8)A = By + Zy^{2} (4.9)(4.10)
Variables are defined in Figure 4.2.
Example Problem 4.1 (SI Units) Given: Trapezoidal earth channel B = 2 m, sideslope 1V:2H, S = 0.003 m/m, normal depth y = 0.5 m, n = 0.02. Find: Velocity (V) and discharge (Q) Solution: SI units; K_{u} = 1
V = (1/0.02) (0.35^{2/3}) (0.003^{1/2}) = 1.36 m/s Q = V A =1.36 (2(0.5) + 2(0.5)^{2}) = 2.0 m^{3}/s Example Problem 4.1 (English Units) Given: Trapezoidal earth channel B = 6.5 ft, sideslope 1V:2H, S = 0.003 ft/ft, normal depth y = 1.6 ft, n = 0.02. Find: Velocity (V) and discharge (Q) Solution: English units; K_{u} = 1.49
V = (1.49/0.02) (1.14^{2/3}) (0.003^{1/2}) = 4.45 ft/s Q = A V = (6.5(1.6) + 2 (1.6)^{2}) 4.45 = 69.06 ft^{3}/s Example Problem 4.2 (SI Units) Given: A concrete trapezoidal channel B = 1.5 m, sideslopes = 1V:2H, n = 0.013, slope = 0.002, Q = 3 m^{3}/s Find: Depth (y) and velocity (v) Solution: 1. Use Manning's equation where K_{u} = 1 and relationships for A and R are A = By + Zy^{2} = 1.5 y + 2y^{2}
substitute A and R into Manning's equation 2. Trial and error solution for y to find a depth where Q = 3 m^{3}/s Try y = 0.70; = 4.03 m^{3}/s Since 4.03 > 3.0, the assumed value for y is too large. Try a smaller value such as 0.60. Try y = 0.6; = 2.96 m^{3}/s since 2.96 = 3.0, the assumed value for y is okay. Therefore, use y = 0.60 m and use continuity to find the velocity (V = Q/A) V = 3.0 / (0.9 + 0.72) = 1.85 m/s Example Problem 4.2 (English Units) Given: A concrete trapezoidal channel B = 5.0 ft, sideslopes = 1V:2H, n = 0.013, slope = 0.002, Q = 105 ft^{3}/s Find: Depth (y) and velocity (v) Solution: 1. Use Manning's equation where K_{u} = 1.49 and relationships for A and R are A = B y + Z y^{2} = 5y + 2y^{2}
substitute A and R into Manning's equation 2. Trial and error solution for y to find a depth where Q = 105 ft^{3}/s Try y = 2.5 Try y = 2.0 109 is close to 105, try 1.96 for y Since 1.96 ft gave a Q of 105.11, the y of 1.96 is good, therefore use y = 1.96 and use continuity to find the velocity (V = Q/A). Example Problem 4.3 (SI Units) Given: A compound channel as illustrated, with an n value of 0.03, a longitudinal slope of 0.002 m/m and sideslopes of 1V:1H. Find: Discharge (Q) Solution:
Q = K S^{1/2} = 3037.1 (0.002)^{1/2} = 135.8 m^{3}/s, say 136 m^{3}/s This result matches the correct discharge value for a 2 m flow depth as calculated above in item 3. Example Problem 4.3 (English Units) Given: A compound channel as illustrated, with an n value of 0.03, a longitudinal slope of 0.002 ft/ft and sideslopes of 1V:1H. Find: Discharge (Q) Solution:
Q = K S^{1/2} = 107655 (0.002)^{1/2} = 4814 ft^{3}/s This result matches closely to the 4,812 ft^{3}/s calculated in item 3. The small difference is the result of rounding in the conveyance calculations. There are times in the design of highway drainage facilities that knowledge of the velocity distribution in the vertical is needed (e.g., the design of riprap for scour and erosion control). As a result of boundary roughness, the velocity varies vertically from some minimum value along the bed to a maximum value near the water surface (Figure 4.3). In this section, the Einstein form of the KarmanPrandtl velocity distribution in the vertical and mean velocity equations will be given for steady uniform flow (Einstein 1950). For their derivation the reader is referred to any standard fluid mechanics text or HDS 6 (Richardson et al. 2001).
The equations for velocity distribution (v) and mean velocity (V) can be written in the following dimensionless forms: (4.11)(4.12)
Shear stress is the force water exerts on the bed and bank of a channel as it flows over them. The following equations can be used to determine the shear stress on the boundary of the channel that results from the force of flowing water. For the derivations of these equations refer to fluid mechanics texts or HDS 6 (Richardson et al. 2001). The first equation (Equation 4.13) is an exact equation, giving the average shear stress over the wetted perimeter. The next equations are semiempirical and result from solving the KarmanPrandtl velocity equation.
τ_{o} = γ R S_{o} where:
(4.15)
where τ_{o} is the shear stress at a point in the flow, N/m^{2} (lb/ft^{2}), v_{1} and v_{2} are point velocities in the vertical at y_{1} and y_{2}, respectively; V is the mean velocity in the vertical with a depth of y_{o}; and the other terms have been defined previously. Example Problem 4.4 (SI Units) Determine the shear stress along the wetted perimeter of a trapezoidal channel. Also determine the shear stress on a particle along the bottom of the same channel. Given: Trapezoidal channel as illustrated with S_{o} = 0.005, γ = 9800 N/m^{3}, V = 1.8 m/s, D_{84} = 0.15m Find: (1) τ_{o} along wetted perimeter Solution: (1) the shear stress along the wetted perimeter is given by τ_{o} = γ R S_{0} where:
(2) shear stress along the bottom at a point is Example Problem 4.4 (English Units) Determine the shear stress along the wetted perimeter of a trapezoidal channel. Also determine the shear stress on a particle along the bottom of the same channel. Given:Trapezoidal channel as illustrated with S_{o} = 0.005, γ = 62.4 lb/ft^{3}, V = 5.9 ft/s, D_{84} = 0.49 ft Find: (1) τ_{o} along wetted perimeter Solution: (1) the shear stress along the wetted perimeter is given by τ_{o} = γ R S_{0} where:
(2) the shear stress along the bottom at a point is 4.3.5 Froude Number and Relationship to Subcritical, Critical, and Supercritical Flow An extremely important dimensionless parameter in openchannel flow is the Froude Number, defined as the ratio of the inertia forces to the gravity forces. It is normally expressed as: (4.16)
where:
V and y can be the mean velocity and depth in a channel or the velocity and depth in the vertical. If the former are used, then the Froude Number is for the average flow conditions in the channel. If the latter are used, then it is the Froude Number for that vertical at a specific location in the cross section. The Froude Number uniquely describes the flow pattern in openchannel flow. For example, in alluvial channel flow with sandbed material, ripples and dunes only form when the Froude Number is less than 1.0 (subcritical flow); whereas, antidunes only form when the Froude Number is greater than 1.0. Plane bed formation is independent of the Froude Number. The Froude Number is the scaling parameter that is used in modeling openchannel flow structures in the laboratory. When the Froude Number is 1.0, the flow is critical; values of the Froude Number greater than 1.0 indicate supercritical or rapid flow and smaller than 1.0 indicate subcritical or tranquil flow. Velocity and depth at critical flow are called the critical velocity and critical depth. Channel slope which produces critical depth and critical velocity is the critical slope. The change from supercritical to subcritical flow is often abrupt (particularly if the Froude Number is larger than 2.0) resulting in a phenomenon known as the hydraulic jump. Critical depth and velocity for a particular discharge are only dependent on channel size and shape and are independent of channel slope and roughness. Critical slope depends upon the channel roughness, channel geometry, and discharge. For a given critical depth and velocity, the critical slope for a particular roughness can be computed by Manning's equation. Supercritical flow is difficult to control because abrupt changes in alignment or in cross section produce waves which travel downstream, alternating from side to side, sometimes causing the water to overtop the channel sides. Changes in channel shape, slope, alignment, or roughness cannot be reflected upstream. In supercritical flow, the control of the flow is located upstream. Supercritical flow is common in steep flumes, channels, and mountain streams. Subcritical flow is relatively easy to control for flows with Froude Numbers less than 0.8. Changes in channel shape, slope, alignment, and roughness affect the flow for small distances upstream. The control in subcritical flow is located downstream. Subcritical flow is common in channels, flumes and streams located in the plains regions and valleys where slopes are relatively flat. Critical depth is important in hydraulic analysis because it is always a hydraulic control. The flow must pass through critical depth in going from subcritical flow to supercritical or going from supercritical flow to subcritical. Although, in the latter case a hydraulic jump usually occurs. Typical locations of critical depth are:
Location and magnitude of critical depth and the determination of critical slope for a cross section of a given shape, size, and roughness are important in channel design and analysis. The equations for determining the critical depth are provided in the discussion of specific discharge and specific energy in steady rapidly varied flow (Section 4.6). Unsteady flows of interest to the highway drainage engineer or designer are:
Waves are an important consideration in bridge hydraulics when designing slope protection of embankments and dikes, and channel improvements. In the following paragraphs, only the basic one dimensional analysis of waves and surges is presented. Other aspects of waves are presented in other sections. For shallow water waves (long waves  Figure 4.5) where the normal depth (y_{o}) is small in comparison to the wave length, the basic equation for the celerity (velocity of the wave relative to the velocity of the flow) is given by: (4.17)
Note that the celerity of a shallow water wave of small amplitude is the same as the denominator of the Froude Number. (4.18)
As explained in the discussion of the Froude Number (Section 4.3.5), when Fr < 1 (subcritical or tranquil), a small amplitude wave moves upstream. When Fr > 1 (supercritical or rapid flow), a small amplitude wave moves downstream and when Fr = 1 (critical flow), a small amplitude wave is stationary. The fact that waves or surges cannot move upstream when the Froude Number is equal to or greater than 1.0 is important to remember when determining when the stage discharge relation at a cross section can be affected by downstream conditions.
A surge is a rapid increase in the depth of flow (Figure 4.6). A surge may result from the sudden release of water from a dam or an incoming tide. The lifting of a gate in a channel not only causes a positive surge to move downstream, it also causes a negative surge to move upstream (Figure 4.6). As it moves upstream, a negative surge quickly flattens out. See HDS 6 for more detail and the basic surge equation (Richardson et al. 2001). Steady nonuniform flow occurs when the quantity of water (discharge) remains constant, but the depth of flow, velocity, or cross section changes from section to section. From the continuity equation, the relation of all cross sections will be: (4.19)Q = A_{1} V_{1} = A_{2} V_{2} = A_{n} V_{n} Velocity in steady nonuniform flow can be computed using Manning's equation if the change in velocity from section to section is small so that the effect of acceleration is small. The hydraulic design engineer needs a knowledge of nonuniform flow in order to determine the behavior of the flowing water when changes in channel resistance, size, cross section, shape, or slope occur. Typical examples might include determining water surface elevation changes in a channel of constant slope that goes through a short transition from a concrete trapezoidal cross section (with a low Manning's n) to a larger grasslined trapezoidal cross section (a high Manning's n), or a stream with constant slope and Manning's n that is a long distance upstream of a culvert that constricts the flow.
These two situations define two basic cases of steady nonuniform flow. The first case is for relatively short distances (a few meters (feet) to several hundred meters (feet)) where accelerations are more important than friction. This case is called STEADY RAPIDLY VARIED FLOW. The effect of friction, if it is important, is taken into account by subdividing the distance into shorter segments and using Manning's equation along these shorter segments. The second case is for long distances (hundreds to thousands of meters (feet)), where friction losses are more important than accelerations. This case is called GRADUALLY VARIED FLOW. Method of analysis and equations for these two cases will be given in the next two sections. 4.6 Steady Rapidly Varied Flow Steady flow through relatively short transitions where the flow is uniform before and after the transition can be analyzed using the energy equation. Energy loss due to friction may be neglected, at least as a first approximation. Refinement of the analysis can be made in a second step by including friction loss. For example, the water surface elevation through a transition is determined using the energy equation and then modified by determining the friction loss effects on velocity and depth in short subsections through the transition. However, energy losses resulting from flow separation cannot be neglected, and transitions where separation may occur need special treatment which may include model studies. Contracting flows (converging streamlines) are less susceptible to separation than expanding flows. Also, any time a transition changes velocity and depth such that the Froude Number approaches unity, problems such as waves, blockage, or choking of the flow may occur. If the approaching flow is supercritical, a hydraulic jump may result. Transitions for supercritical flow are discussed in the next section. Transitions are used to contract or expand a channel width (Figure 4.7a), to increase or decrease bottom elevation (Figure 4.7b), or to change both the width and bottom elevation. The analysis or design of transitions is aided by the use of the depth of flow and velocity head terms in the energy equation (see Chapter 3). The sum of the two terms is called the specific energy or specific head, H, and defined as: (4.20)
where:
The specific energy, H, is the height of the total energy above the channel bed. The relationship between the three terms in the specific energy equation, q, y, and H, are evaluated by considering q constant and determining the relationship between H and y (specific energy diagram) or considering H constant and determining the relationship between q and y (specific discharge diagram). These diagrams for a given discharge or energy are then used in the design or analysis of transitions or flow through bridges. They are explained in the next two sections.
4.6.2 Specific Energy Diagram and Evaluation of Critical Depth For a given q, Equation 4.20 can be solved for various values of H and y. When y is plotted as a function of H, Figure 4.8 is obtained. There are two possible depths called alternate depths for any H larger than a specific minimum. Thus, for specific energy larger than the minimum, the flow may have a large depth with small velocity or small depth with large velocity. Flow for a given unit discharge (q) cannot occur with specific energy less than the minimum. Single depth of flow at the minimum specific energy is called the critical depth, y_{c}, and the corresponding velocity, the critical velocity, V_{c} = q/y_{c}. The relation for y_{c} and V_{c} for a given q (for a rectangular channel) is: (4.21)
Note that for critical flow: (4.22)(4.23)
Thus, flow at minimum specific energy has a Froude Number equal to 1. Flows with velocities larger than critical (Fr > 1) are called rapid or supercritical and flow with velocities smaller than critical (Fr < 1) are called tranquil or subcritical. These flow conditions are illustrated in Figure 4.9 where a rise in the bed causes a decrease in depth when the flow is tranquil and an increase in depth when the flow is rapid. Furthermore there is a maximum rise in the bed for a given H_{1} where the given rate of flow is physically possible. If the rise in the bed is increased beyond Δz_{max} for H_{min} then the approaching flow depth y_{1} would have to increase (increasing H) or the flow would have to be decreased. Thus, for a given flow in a channel, a rise in the bed level can occur up to a Δz_{max} without causing backwater.
Distinguishing between the types of flow and how the water surface reacts with changes in cross section is important in channel design; thus, the location of critical depth and the determination of critical slope for a cross section of given shape, size, and roughness becomes necessary. Equations for direct solution of the critical depth are available for several prismatic shapes (Brater and King 1976); however, some of these equations were not derived for use in the metric system. For any channel section, regular or irregular, critical depth may be found by a trialanderror solution of the following equation: (4.24)
where A_{c} and T_{c} are the area and topwidth at critical flow. Nomographs are available to solve this equation, and are particularly useful for circular sections (Figures 4.10a and b). An expression for the critical velocity (V_{c}) of any cross section at critical flow conditions is: (4.25)(4.26) where: y_{c} = A_{c}/ T_{c} Uniform flow within about 10 percent of the critical depth is unstable and should be avoided in design. The reason for unstable flow can be seen by referring to the specific head diagram (Figure 4.8). As the flow approaches the critical depth from either limb of the curve, a very small change in energy is required for the depth to abruptly change to the alternate depth on the opposite limb of the specific head curve. If the unstable flow region cannot be avoided in design, the least favorable type of flow should be assumed for the design.
4.6.3 Specific Discharge Diagram Equation 4.20 can be rearranged to determine q, the unit discharge, as a function of H, the specific energy, and y, the depth of flow. (4.27)
For a constant H, q can be solved as a function of y and the specific discharge diagram will result (Figure 4.11).
For any discharge smaller than a specific maximum q for the given H, two depths of flow are possible. The depth at maximum q for a given specific energy (H) is the critical depth (y_{c}) and the velocity is the critical velocity (V_{c}). (4.28)
and (4.29)
Example Problem 4.5 (SI Units) Given: Determine the critical depth in a trapezoidal shaped swale with z = 1, given a discharge of 9.2 m^{3}/s and a bottom width, B = 6 m. Also, determine the critical velocity. Find: Critical depth Velocity at critical depth Solution: For a Q of 9.2 m^{3}/s A_{c}^{3} / T_{c} = Q^{2} / g A_{c}^{3} / T_{c} = (9.2)^{2} / 9.81 = 8.63 For a trapezoidal channel area substituting A = y (B + Z y) and T = B + 2Z y gives 8.63 = [y (6 + y)]^{3} / (6 + 2y) A trial and error solution yields y = 0.6 m. V_{c} = (g y_{c})^{1/2} y_{c} = A / T = [0.6 (6 + 0.6)] / (6 + 1.2) = 0.55 m V_{c} = [9.81 (0.55)]^{1/2} = 2.3 m/s Example Problem 4.5 (English Units) Given: Determine the critical depth in a trapezoidal shaped swale with z = 1, given a discharge of 325 ft^{3}/s and a bottom width, B = 20 ft. Also, determine the critical velocity. Find: Critical depth Velocity at critical depth Solution: For a Q of 325 ft^{3}/s A_{c}^{3} / T_{c} = Q^{2} / g A_{c}^{3} / T_{c} = 325^{2} / 32.2 = 3280.28 For a trapezoidal channel area substituting A = y (B + Z y) and T = B + 2Z y gives 3280 = [y (20 + y)]^{3} /(20 + 2y) A = 1.95 (20 + (1) 1.95) = 42.80 ft^{2} T = 20 + 2 (1) (1.95) = 23.90 ft A trial and error solution yields y = 1.95 ft. V_{c} = (g y_{c})^{1/2} y_{c} = A / T = 42.80 / 23.9 = 1.79 ft V_{c} = [32.2 (1.79)]^{1/2} = 7.59 ft/s (end example problem) For maximum discharge at constant H, the Froude Number is 1.0, and the flow is critical. The relation between y_{c}, V_{c}, H, and q_{max} for a constant H is: (4.30)
Flow conditions for constant specific energy for a width contraction are illustrated in Figure 4.12 assuming no geometrical effects such as eccentricity, skew, piers, scour, and expansion. Contraction causes a decrease in flow depth when the flow is tranquil and an increase when the flow is rapid. The maximum possible contraction without causing backwater effects occurs when the Froude Number is 1.0, the discharge per unit of width q is a maximum, and y_{c} is 2/3 H. A further decrease in width will cause backwater. That is, an increase in depth upstream will occur to produce a larger specific energy and increase y_{c} in order to get the flow through the decreased width. The flow in Figure 4.12 can go from point A to C and then either back to D or down to E depending on the downstream boundary conditions. An increase in slope of the bed downstream from C and no separation would allow the flow to follow the line A to C to E. Similarly, the flow can go from B to C and back to E or up to D depending on boundary conditions. Figure 4.12 is drawn with the side boundary forming a smooth streamline. If the contraction was due to bridge abutments, the upstream flow would follow a natural streamline to the point of maximum constriction, but then downstream, the flow would probably separate. Tranquil approach flow could follow line A to C but the downstream flow probably would not follow either line C to D or C to E, but would have an undulating hydraulic jump. There would be interaction of the flow in the separation zone and considerable energy would be lost. If the slope downstream of the abutments was the same as upstream, then the flow could not be sustained with this amount of energy loss. Backwater would occur, increasing the depth in the constriction and upstream, until the flow could go through the constriction and establish uniform flow downstream.
A hydraulic jump will occur when the flow velocity (V_{1}) is rapid or supercritical and the slope is decreased to a slope for subcritical flow, or an obstruction such as an energy dissipator is placed in the flow. The supercritical depth is changed to a subcritical depth, called the sequent depth. Depending on the magnitude of the Froude Number, a considerable amount of energy is changed to heat. The larger the Froude Number, the more energy that is lost. The existence of a jump assumes adequate tailwater conditions exist. Many engineers/ designers assume that a jump will always occur when a change from a steep grade to a flat grade is encountered, such as near the outlet end of a culvert (i.e., a brokenback culvert). The jump will only occur with adequate downstream tailwater to maintain the sequent depth just below the culvert grade break. Without adequate tailwater, the jump will be swept downstream out of the culvert, causing a potentially large scour hole at the culvert outlet. The relation between the supercritical depth and the sequent depth for a rectangular flat channel is: (4.31)
The corresponding energy loss in a hydraulic jump is the difference between the two specific energies. It can be shown that this headloss is: (4.32)
Equation 4.32 has been experimentally verified along with the dependence of the jump length L_{j} and energy dissipation (headloss h_{L}) on the Froude Number of the approaching flow (Fr_{1}). The results of these experiments are given in Figure 4.13. When the Froude Number for rapid flow is less than 2.0, an undulating jump with large surface waves is produced. The waves are propagated for a considerable distance downstream. In addition, when the Froude Number of the approaching flow is less than 3.0, the energy dissipation of the jump is not large and jets of high velocity flow can exist for some distance downstream of the jump. These waves and jets can cause erosion a considerable distance downstream of the jump. For larger values of the Froude Number, the rate of energy dissipation in the jump is very large and Figure 4.13 is recommended. Example Problem 4.6 (SI Units) Given: A hydraulic jump occurs in a 5m wide rectangular channel at a flow depth of 0.5m. Determine the downstream water surface elevation needed to cause the jump. Also calculate the headloss due to the jump. Given Q = 20 m^{3}/s.
Find: (1) Determine the required downstream WSEL to initiate a jump Solution: (1) Using the formula for a hydraulic jump (Equation 4.31), find y_{2}. From continuity V_{1} = Q / A_{1} = 20 / [5 (0.5)] = 8 m/s Fr = V_{1} / (g y_{1})^{1/2} = 8 / [9.81 (0.5)]^{1/2} = 3.6 (2) Find the headloss, h_{L}, across the jump (Equation 4.32) h_{L} = (2.3  0.5)^{3} / [4 (2.3) (0.5)] = 1.27 m Example Problem 4.6 (English Units) Given: A hydraulic jump occurs in a 16.4 ft wide rectangular channel at a flow depth of 1.64 ft. Determine the downstream water surface elevation needed to cause the jump. Also calculate the headloss due to the jump. Given Q = 700 ft^{3}/s. Find: (1) Determine the required downstream WSEL to initiate a jump Solution: (1) Using the formula for a hydraulic jump (Equation 4.31), find y_{2}. From continuity V_{1} = Q / A_{1} = 700 / [1.64 (16.4)] = 26.03 ft/s Fr = V_{1} / (g y_{1})^{1/2} = 26.03 / [32.2 (1.64)]^{1/2} = 3.58 (2) Find the headloss, h_{L}, across the jump (Equation 4.32) h_{L} = (7.52  1.64)^{3} / [4 (7.52) (1.64)] = 4.12 ft 4.6.5 Subcritical Flow in Bends When subcritical flow goes around a bend, the water surface is elevated on the outside of the bend and lowered on the inside of the bend (Figure 4.14). The approximate difference in elevation (ΔZ) between the water surface along the sides of the curved channel can be found by the following equations. (4.33)
where:
In channel design, superelevation is accounted for by adding ΔZ/2 to the normal depth to define the maximum water surface depth at the outside of the bend. This equation gives values of ΔZ somewhat lower than will occur in natural channels because of the assumption of uniform velocity and uniform curvature, but the computed value will be generally less than 10 percent in error. Equation 4.34 (HDS 6, Richardson et al. 2001) is more accurate, but the difference in superelevation obtained by using the two equations is small, and in alluvial channels the resulting erosion of the concave bank and deposition on the convex bank leads to further error in computing superelevation. (4.34)
It is recommended that Equation 4.33 be used to compute superelevation in alluvial channels. For lined canals with strong curvature and large velocities, superelevation should be computed using Equation 4.34. Other problems introduced by curved alignment of channel in subcritical flow include spiral flow, changes in velocity distribution, and increased friction losses within the curved channel as contrasted with the straight channel. For more information on flowaround bends see Rouse (1950), Chow (1959), or Richardson et al. (2001). 4.6.6 Supercritical Flow in Bends Changes in alignment of supercritical flow are difficult to make. Water traveling at supercritical velocities around bends builds up waves which may "climb out" of the channel. Waves that are set up may continue downstream for a long distance. Also, sharp changes in alignment may set up a hydraulic jump with the flow overtopping the banks. Changes in alignment, whenever possible, should be made near the upper end of the section before the supercritical velocity has developed. If a change in alignment is necessary in a channel carrying supercritical flow, the channel should be rectangular in cross section, and preferably enclosed. On small chutes, experiments have shown that an angular variation (α) of rectangular flow boundaries (expansion) should not exceed that produced by the equation: (4.35)
Changes in alignment of open channels can and should be designed to reduce the wave action, resulting from the change in direction in flow (see Richardson et al. 2001). Often designs involving supercritical flow should be model tested to develop the best design, or even a design that will work. Example Problem 4.7 (SI Units) Given: During high runoff, a 2.0 m deep mountain stream flows near bank full with a normal depth and velocity of 1.8m and 3.4 m/s, respectively. At a sharp bend r_{o} = 12 m, r_{c} = 10 m, r_{i} = 8 m. Will flow overtop the bend? Find: ΔZ Solution: Use the superelevation formula
The water surface raises approximately (0.47/2) m on the outside of the bend and lowers by that same amount on the inside of the bend. The maximum flow depth in the bend will be: y_{outside} = 1.8 + (0.47 / 2) = 2.04 m which is greater than the channel depth (2.0 m) and overtopping will occur. Example Problem 4.7 (English Units) Given: During high runoff, a 6.56 ft deep mountain stream flows near bank full with a normal depth and velocity of 5.91 ft and 11.15 ft/s, respectively. At a sharp bend r_{o} = 39.37 ft, r_{c} = 32.81 ft, r_{i} = 26.25 ft. Will flow overtop the bend? Find: ΔZ Solution: Use the superelevation formula The water surface raises approximately (1.54/2) ft on the outside of the bend and lowers by that same amount on the inside of the bend. The maximum flow depth in the bend will be y_{outside} = 5.9 + (1.54 / 2) = 6.67 ft which is greater than the channel depth (6.56 ft) and overtopping will occur. In this section, a second type of steady nonuniform flow is considered, gradually varied flow. In gradually varied flow, changes in depth and velocity take place slowly over large distances, resistance to flow dominates, and acceleration forces are neglected. Analysis of gradually varied flow involves: (1) the determination of the general characteristics of the water surface; and (2) the elevation of the water surface or depth of flow. In gradually varied flow, the actual flow depth, y, is either larger or smaller than the normal depth, y_{o}, and either larger or smaller than the critical depth, y_{c}. The water surface profiles, which are often called backwater curves, depend on the magnitude of the actual depth of flow, y, in relation to the normal depth, y_{o}, and the critical depth, y_{c}. Normal depth, yo, is the depth of flow that would exist for steady uniform flow as determined using Manning's velocity equation, and the critical depth is the depth of flow when the Froude Number equals 1.0. Reasons for the depth being different than the normal depth are changes in slope of the bed, changes in cross section, obstruction to flow, and imbalances between gravitational forces accelerating the flow and shear forces retarding the flow. In working with gradually varied flow, the first step is to determine what type of water surface profile would exist. The second step is to perform the numerical computations. An excellent gradually flow reference is Chow (1959). 4.7.2 Types of Water Surface Profiles The types of water surface profiles are obtained by analyzing the change of the various terms in the gradually varied flow equation: (4.36)
The slope of the water surface dy/dx depends on the slope of the bed S_{o}, the ratio of the normal depth y_{o} to the actual depth y and the ratio of the critical depth y_{c} to the actual depth y. The difference between flow resistance for steady uniform flow no to flow resistance for steady nonuniform flow n is small and the ratio is taken as 1.0. With n = n_{o}, there are 12 types of water surface profiles. The 12 types are subdivided into 5 classes which depend on the bed slope. These are illustrated in Figure 4.15 and summarized in Table 4.1.
When y —> y_{c}, the assumption that acceleration forces can be neglected no longer holds. Equation 4.36 indicates that dy/dx is perpendicular to the bed slope when y —> y_{c}. For locations close to the cross section where flow is critical, a distance from 3 to 20 m, (10 to 65 ft) curvilinear flow analysis and experimentation must be used to determine the actual values of y. When analyzing long distances, 30 to 100 m or longer, (100 to 300 ft or longer) one can assume qualitatively that y reaches y_{c}. In general, when the flow is rapid (Fr > 1), the flow cannot become tranquil or subcritical without a hydraulic jump occurring. In contrast, subcritical flow can become rapid, or supercritical, (cross the critical depth line). This is illustrated in Figure 4.16. When there is a change in cross section or slope or an obstruction to the flow, the qualitative analysis of the flow profile depends on locating the control points, determining the type of water surface profile upstream and downstream of the control points, and then sketching these profiles. It must be remembered that when flow is supercritical (Fr > 1), the control depth is upstream and the water surface profile analysis proceeds in the downstream direction. When flow is subcritical (Fr < 1), the control depth is downstream and the computations must proceed upstream. Water surface profiles that result from a change in slope of the bed are illustrated in Figure 4.16.
Note:
To determine if the hydraulic jump occurs on the steep or mild slope, calculate the sequent depth (y_{2}) for the y_{1} depth using the hydraulic jump equation. If y_{2} from the hydraulic jump is larger than the normal depth y_{0} from Manning's equation on the mild slope, then there will be an M3 curve on the mild slope until the y_{2} equals the depth that corresponds to the initial depth needed for the jump to occur. If y_{2} is smaller than the depth that would balance with the downstream depth, the jump will occur on the steep slope and an S1 curve will occur to connect with the normal depth at the control section. Example Problem 4.8 (SI Units) Given: A 5 m wide, rectangular channel goes from a very steep grade to a mild slope. The design discharge is 24.8 m^{3}/s and the normal depth and velocity on the steep slope were calculated to be 0.33 m and 15 m/s, respectively. On the mild slope, the normal depth and velocity were calculated to be 2.96 m and 1.68 m/s, respectively. Determine the type of flow occurring in both channels. If a hydraulic jump occurs, evaluate the depth downstream of the hydraulic jump, the location of the jump, and the water surface profile classification. Find: 1. Find the critical depth, y_{c}, on the steep slope. q = Q/B where B is the channel width
On the steep slope, the normal depth is 0.33 m. Since y < y_{c}, supercritical flow occurs on the steep slope. Note that the unit discharge (q) is the same for the mild slope and hence, y_{c}, is the same for the steep and mild slope sections. On the mild slope, the normal depth is 2.96 m. Since y > y_{c}, subcritical flow occurs on the mild slope. Therefore, a hydraulic jump should occur. 2. Next, determine if the jump will occur on the steep slope or on the mild slope. To determine if the hydraulic jump occurs on the steep or mild slope, calculate the sequent depth (y_{2}) for the steep slope y_{1} depth using the hydraulic jump equation. If y_{2} from the hydraulic jump is larger than the normal depth y_{0} from Manning's equation on the mild slope, then there will be an M3 curve on the mild slope until the y_{2} equals the critical depth. If y_{2} is smaller than y_{0} on the mild slope, then the jump may occur on the steep slope and an S1 curve will occur to connect with the normal depth at the control section.
Compare parameters and determine the type of water surface classification. Since the sequent depth (y_{2}) is greater than the mild slope normal depth (i.e., 3.73 > 2.96), the mild slope channel will have an M3 curve until the hydraulic jump occurs. Example Problem 4.8 (English Units) Given: A 16 ft wide, rectangular channel goes from a very steep grade to a mild slope. The design discharge is 875 ft^{3}/s and the normal depth and velocity on the steep slope were calculated to be 1.0 ft and 54.7 ft/s, respectively. On the mild slope, the normal depth and velocity were calculated to be 9.71 ft and 5.6 ft/s, respectively. Determine the type of flow occurring in both channels. If a hydraulic jump occurs, evaluate the depth downstream of the hydraulic jump, the location of the jump, and the water surface profile classification. Find: 1. Find the critical depth, yc, on the steep slope. q = Q/B where B is the channel width
On the steep slope, the normal depth is 1.0 ft. Since y < y_{c}, supercritical flow occurs on the steep slope. Note that the unit discharge (q) is the same for the mild slope and hence, y_{c}, is the same for the steep and mild slope sections. On the mild slope, the normal depth is 9.71 ft. Since y > y_{c}, subcritical flow occurs on the mild slope. Therefore, a hydraulic jump should occur. 2. Next, determine if the jump will occur on the steep slope or on the mild slope. To determine if the hydraulic jump occurs on the steep or mild slope, calculate the sequent depth (y_{2}) for the steep slope y_{1} depth using the hydraulic jump equation. If y_{2} from the hydraulic jump is larger than the normal depth y_{0} from Manning's equation on the mild slope, then there will be an M3 curve on the mild slope until the y_{2} equals the critical depth. If y_{2} is smaller than y_{0} on the mild slope, then the jump may occur on the steep slope and an S1 curve will occur to connect with the normal depth at the control section.
Compare parameters and determine the type of water surface classification. Since the sequent depth (y_{2}) is greater than the mild slope normal depth (i.e., 13.14 > 9.71), the mild slope channel will have an M3 curve until the hydraulic jump occurs. The standard step method is a simple computational procedure to determine the water surface profile in gradually varied flow. Prior knowledge of the type of water surface profile as determined in the preceding section would be useful to determine whether the analysis should proceed up or downstream. The standard step method is derived from the energy equation. The equation is: where:
The above equation is used in the standard step method. An example of the use of the standard step method is given Chow (1959) and Richardson et al. (2001). Although computer programs (such as HECRAS) are commonly used to compute water surface profiles, it is recommended that a qualitative sketch of the water surface profiles be made using the information given in the preceding section. This is particularly useful in complicated profiles where the channel slopes change from steep to mild or mild to steep.

Contact:Veronica Ghelardi 

Updated: 08/29/2012 