7.1  EQUATION OF GRADUALLY VARIED FLOW

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The flow is gradually varied when the discharge Q is constant but the other hydraulic variables (A, V, D, R, P, and so on) vary gradually in space. The basic assumptions of gradually varied flow are:

1. The flow is steady, i.e., none of the hydraulic variables vary in time.

2. The streamlines are essentially parallel; thus, the pressure distribution in the vertical direction is hydrostatic, i.e., proportional to the flow depth.

3. The head loss is the same as that corresponding to uniform flow; therefore, the uniform flow formula may be used to evaluate the energy slope.

4. The value of Manning's n is the same as that of uniform flow.

1. The slope of the channel is small.

2. The pressure correction factor cosθ ≅ 1.

3. There is negligible air entrainment.

4. The conveyance is an exponential function of the flow depth (except for the case of circular culverts).

5. The roughness (Manning's n) is independent of the flow depth (only an approximation) and is constant throughout the reach under consideration.

Fig. 7-1  Definition sketch for energy in open-channel flow.

Under gradually varied flow, the gradient of hydraulic head is (Fig. 7-1):

dH         d                      V 2 _____  =  ___ ( z  +  y  +  _____ )  =  - Sf   dx         dx                      2g

(7-1)

The negative sign in front of the friction slope S_f is required because the flow direction is from left to right, while the derivative is taken from right to left, by convention. By definition, the friction slope is:

hf                    Sf  =  _____            ΔL

(7-2)

in which ΔL = length of the channel reach under consideration.

The gradient of specific energy is:

dE         d               V 2             dz _____  =  ___ ( y  +  _____ )  =  - ____  -  Sf   dx         dx              2g              dx

(7-3)

The gradient of the channel bed, or channel slope (bottom slope), is:

dz          z2 - z1                     _____  =  ________   dx             ΔL

(7-4)

dz          z1 - z2                      -  _____  =  ________  =  So       dx             ΔL

(7-5)

Therefore, the gradient of specific energy is:

dE         d               V 2               _____  =  ___ ( y  +  _____ )  =  So  -  Sf   dx         dx              2g

(7-6)

Under steady flow: Q = V A = constant. Therefore:

d                 Q 2               ____ ( y  +  _______ )  =  So  -  Sf  dx              2g A2

(7-7)

dy           d           Q 2               _____  +  _____   ( _______ )  =  So  -  Sf   dx           dx        2g A2

(7-8)

dy            Q 2       dA          _____  -  ( ______ ) _____  =  So  -  Sf   dx           g A3       dx

(7-9)

dy            Q 2       dA     dy          _____  -  ( ______ ) _____ _____   =  So  -  Sf   dx           g A3       dy     dx

(7-10)

Using Eq. 3-11:

dy            Q 2 T       dy          _____  -  ( ________ ) _____   =  So  -  Sf   dx             g A3        dx

(7-11)

Therefore, the flow-depth gradient is:

dy                    So  -  Sf               _____  =  _______________________   dx          1  -  [(Q 2 T ) / (g A3)]

(7-12)

The friction slope based on the Chezy equation (Eqs. 5-10 and 2-4) is:

Q 2                Sf  =  ____________            C 2 A2 R

(7-13)

Since R = A / P :

Q 2 P                Sf  =  _________            C 2 A3

(7-14)

Substituting Eq. 7-14 into Eq. 7-12, the flow-depth gradient is:

dy           So  -  [(Q 2 P ) / (C 2 A3)]               _____  =  ___________________________   dx              1  -  [(Q 2 T ) / (g A3)]

(7-15)

dy           So  -  (g/C 2) (P / T ) [(Q 2 T ) / (g A3)]               _____  =  _______________________________________   dx                       1  -  [(Q 2 T ) / (g A3)]

(7-16)

The square of the Froude number is (Eq. 3-12):

Q 2 T                F 2  =  _________                g A3

(7-17)

Substituting Eq. 7-17 into Eq. 7-16:

dy           So  -  (g/C 2) (P / T ) F 2               _____  =  _________________________   dx                       1  -  F 2

(7-18)

Substituting Eq. 5-12 into Eq. 7-18:

dy           So  -  f (P / T ) F 2               _____  =  _____________________   dx                    1  -  F 2

(7-19)

Therefore, the depth gradient (dy/dx) is a function of:

1. Channel slope S_o,

2. Friction coefficient f,

3. Ratio of wetted perimeter to top width P / T, and

4. Froude number.

For dy/dx = 0, Eq. 7-19 reduces to a statement of uniform flow:

So  =  f (P / T ) F 2

(7-20)

For F = 1, Eq. 7-20 reduces to a statement of critical uniform flow:

So  =  f (Pc / Tc )  =  Sc

(7-21)

in which S_c = critical slope, i.e., the channel slope for which the flow is critical.

In terms of critical slope (Eq. 7-21), the flow-depth gradient is:

dy           So  -  (P / T ) (Tc / Pc ) Sc F 2               _____  =  ________________________________   dx                            1  -  F 2

(7-22)

For (P / T ) ≅ (P_c / T_c ), i.e., for a constant ratio (P / T) with flow depth, Eq. 7-22 reduces to:

dy           So  -  Sc F 2               _____  =  _______________   dx               1  -  F 2

(7-23)

For conciseness, the flow-depth gradient can be written as:

dy                          Sy  =  _____             dx

(7-24)

Substituting Eq. 7-24 into Eq. 7-23, the flow-depth gradient is:

Sy           (So / Sc)  -  F 2       ____  =  __________________  Sc                  1  -  F 2

(7-25)

Equation 7-25 (or 7-23) is the steady gradually varied flow equation (Fig. 7-2). The depth gradient S_y is a function only of: (1) channel slope S_o, (2) critical slope S_c, and (3) Froude number F.

Fig. 7-2  Steady gradually varied flow.

7.2  CHARACTERISTICS OF FLOW PROFILES

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In Eq. 7-25, the sign of the left-hand side (LHS) is that of S_y (numerator), since S_c (denominator) is always positive (friction is always positive). The sign of S_y (i.e, the sign of the LHS) may be one of three possibilities:

• A positive value, leading to RETARDED FLOW (BACKWATER),

• A zero value, leading to UNIFORM FLOW (NORMAL), or

• A negative value, leading to ACCELERATED FLOW (DRAWDOWN).

In the right-hand side (RHS) of Eq. 7-25, there are three possibilities for the numerator (USDA Soil Conservation Service, 1971):

• S_o / S_c > F ², leading to SUBNORMAL FLOW,

• S_o / S_c = F ², leading to NORMAL FLOW, or

• S_o / S_c < F ², leading to SUPERNORMAL FLOW.

There are three possibilities for the denominator:

• 1 > F ², leading to SUBCRITICAL FLOW,

• 1 = F ², leading to CRITICAL FLOW, or

• 1 < F ², leading to SUPERCRITICAL FLOW.

Given the above inequalities, there arise three types (or families) of water-surface profiles, shown in Table 7-1. The total number of profiles is 12. A summary is shown in Table 7-2.

Type I In the Type I family, the flow is subnormal/subcritical. Therefore, the rule is:                              1  >  F 2  <  (So / Sc )                                     (7-26) which is the same as:                              (So / Sc )  > <  1                                     (7-27) Equation 7-27 states that So may be lesser than, equal to, or greater than Sc . This gives rise to three types of profiles: M1:  So < Sc Fig. 7-3  Definition sketch for M1 water-surface profile. The depth gradient Sy varies from asymptotic to So (i.e., asymptotic to the horizontal) at the downstream end to asymptotic to zero (i.e., asymptotic to the normal depth) at the upstream end. C1:  So = Sc Fig. 7-4  Definition sketch for C1 water-surface profile. The depth gradient Sy varies from Sc at the downstream end to Sc at the upstream end, i.e., the water-surface profile is a horizontal line. S1:  So > Sc Fig. 7-5  Definition sketch for S1 water-surface profile. The depth gradient Sy varies from asymptotic to So (i.e., asymptotic to the horizontal) at the downstream end to asymptotic to + ∞ (i.e., asymptotic to the hydraulic jump) at the upstream end. Since                              (So / Sc )  >  F 2                                     (7-28) and                              F 2  >  0                                     (7-29) it follows that                              (So / Sc )  >  0                                     (7-30) Thus:                              So  >  0                                     (7-31) Therefore, no horizontal (H) or adverse (A) profiles are possible in the Type I family of water-surface profiles.

Type II A In the Type II A family, the flow is subnormal/supercritical. Therefore, the rule is:                              1  <  F 2  <  (So / Sc )                                     (7-32) which is the same as:                              (So / Sc )  >  1                                     (7-33) Equation 7-33 states that So may only be greater than Sc . This gives rise to only one profile: S2:  So > Sc Fig. 7-6  Definition sketch for S2 water-surface profile. The depth gradient Sy varies from - ∞ (i.e., asymptotic to an abrupt slope break) at the upstream end to asymptotic to zero (i.e., asymptotic to the normal depth) at the downstream end. Since                              (So / Sc )  >  F 2                                     (7-34) and                              F 2  >  0                                     (7-35) it follows that                              (So / Sc )  >  0                                     (7-36) Thus:                              So  >  0                                     (7-37) Therefore, no horizontal (H) or adverse (A) profiles are possible in the Type II A family of water-surface profiles.

Type II B In the Type II B family, the flow is supernormal/subcritical. Therefore, the rule is:                              1  >  F 2  >  (So / Sc )                                     (7-38) which is the same as:                              (So / Sc )  <  1                                     (7-39) Equation 7-39 states that So may be lesser than Sc , equal to 0, or lesser than 0. This gives rise to three types of profiles: M2:  0 <  So < Sc Fig. 7-7  Definition sketch for M2 water-surface profile. The depth gradient Sy varies from - ∞ (i.e., abrupt slope break) at the downstream end to asymptotic to zero (i.e., asymptotic to the normal depth) at the upstream end. H2: 0 =  So < Sc Fig. 7-8  Definition sketch for H2 water-surface profile. The depth gradient Sy varies from - ∞ (i.e., abrupt slope break) at the downstream end to asymptotic to 0 (to headwater) at the upstream end. A2:  So < 0 < Sc Fig. 7-9  Definition sketch for A2 water-surface profile. The depth gradient Sy varies from - ∞ (i.e., abrupt slope break) at the downstream end to asymptotic to < 0 (to headwater) at the upstream end.

Type III In the Type III family, the flow is supernormal/supercritical. Therefore, the rule is:                              1  <  F 2  >  (So / Sc )                                     (7-44) which is the same as:                              (So / Sc )  > <  1                                     (7-45) Equation 7-45 states that So may be lesser than, equal to, or greater than Sc . This gives rise to five types of profiles: S3:  So > Sc Fig. 7-10  Definition sketch for S3 water-surface profile. The depth gradient Sy varies from asymptotic to Sc at the upstream end to asymptotic to zero (i.e., asymptotic to the normal depth) at the downstream end. C3:  So = Sc Fig. 7-11  Definition sketch for C3 water-surface profile. The depth gradient Sy varies from Sc at the downstream end to Sc at the upstream end, i.e., the water-surface profile is a horizontal line. M3:  0 <  So < Sc Fig. 7-12  Definition sketch for M3 water-surface profile. The depth gradient Sy varies from asymptotic to Sc at the upstream end to asymptotic to + ∞ (i.e., asymptotic to the hydraulic jump) at the downstream end. H3:  0 =  So < Sc Fig. 7-13  Definition sketch for H3 water-surface profile. The depth gradient Sy varies from asymptotic to Sc at the upstream end to asymptotic to + ∞ (i.e., asymptotic to the hydraulic jump) at the downstream end. A3:  So = Sc Fig. 7-14  Definition sketch for A3 water-surface profile. The depth gradient Sy varies from asymptotic to Sc at the upstream end to asymptotic to + ∞ (i.e., asymptotic to the hydraulic jump) at the downstream end.

Figure 7-15 shows a graphical representation of flow-depth gradients in water-surface profile computations. The arrows indicate the direction of computation.

Fig. 7-15  Graphical representation of flow-depth gradient ranges in water-surface profile computations.

7.3  LIMITS TO WATER SURFACE PROFILES

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The flow-depth gradients vary between five (5) limits (Fig. 7-15):

1. The channel slope S_o

2. The critical slope S_c

3. Zero.

4. + ∞

5. - ∞

The theoretical limits to the water-surface profiles may be analyzed using Eq. 7-25, repeated here in a slightly different form:

So  -  Sc F 2       Sy  =  ______________                1  -  F 2

(7-46)

Operating in Eq. 7-46:

Sy (1  -  F 2)  =  So  -  Sc F 2

(7-47)

So  -  Sy       F 2  =  ____________              Sc  -  Sy

(7-48)

For uniform (normal) flow: S_y  =  0, and Eq. 7-48 reduces to:

So  =  Sc F 2

(7-49)

For gradually varied flow: S_y  ≠  0, and Eq. 7-48 is subject to three (3) cases:

ONE F 2 > 0 So > Sy  and  Sc > Sy The following inequality is satisfied: So > Sy < Sc So < Sy  and  Sc < Sy The following inequality is satisfied: So < Sy > Sc It is concluded that Sy has to be either less than both So and Sc, or greater than both.

TWO F 2 = 0 This leads to:  So = Sy             z1 - z2       So  =  _________                 L (7-51)             y2 - y1       Sy  =  _________                 L (7-52) Combining Eqs. 7-51 and 7-52:                    z1 + y1  =  z2 + y2                  (7-53) Equation 7-53 depicts a true reservoir (Fig. 7-15). Fig. 7-16  True reservoir condition.

THREE F 2 < 0:  Since F ≥ 0, this condition is impossible. The following inequality is NOT satisfied:  So > Sy > Sc The following inequality is NOT satisfied:  So < Sy < Sc It is concluded that Sy cannot be less than So and greater than Sc , or less than Sc and greater than So . Thus, Sy has to be less than both So AND Sc, or greater than both (See Case 1).

Uses of water-surface profiles

Table 7-3 and Fig. 7-17 show the typical occurrence of mild water-surface profiles.

Fig. 7-17  Typical occurrence of mild profiles.

Table 7-4 and Fig. 7-18 show the typical occurrence of steep water-surface profiles.

Fig. 7-18  Typical occurrence of steep profiles.

7.4  METHODOLOGIES

[Direct Step Method Example]   [Questions]   [Problems]   [References]   •   [Top]   [Equation of Gradually Varied Flow]   [Characteristics of Flow Profiles]   [Limits to Water Surface Profiles]

There are two ways to calculate water-surface profiles:

1. The direct step method.

2. The standard step method.

The direct step method is applicable to prismatic channels, while the standard step method is applicable to any channel, prismatic and nonprismatic (Table 7-5). The direct step method is direct, readily amenable to use with a spreadsheet, and relatively straight forward in its solution. The standard step method is iterative and complex in its solution. In practice, the standard step method is represented by the Hydrologic Engineering Center's River Analysis System, referred to as HEC-RAS (U.S. Army Corps of Engineers, 2014).

The direct step method applies particularly where data is scarce and resources are limited. The standard step method applies for comprehensive projects. The use of a widely accepted government program such as HEC-RAS enhances credibility.

The required number of cross sections in the standard step method increases with the channel slope. Steeper channels may require more cross sections. Lesser cross sectional variability results in more reliable and accurate results. Note that extensive two- and three-dimensional flow features may not be accurately represented in the one-dimensional water-surface profile model.

7.5  DIRECT STEP METHOD EXAMPLE

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Calculation of M₂ and S₂ profiles, upstream and downstream of a change in grade, from mild to steep

Input data:

• Discharge Q = 2000 m³

• Bottom width b = 100 m.

• Side slope 2 H : 1 V

• Upstream channel slope = 0.0001

• Upstream channel Manning's n = 0.025

• Downstream channel slope = 0.03

• Downstream channel Manning's n = 0.045

Solution

Calculate the normal depth and velocity, and critical depth and velocity, in the upstream and downstream channels. Use ONLINE CHANNEL 05.

For the upstream channel:

Column [12]:  The cumulative channel length, i.e., the cumulative sum of ΔL increments.

Derivation of the formula for the channel length increment ΔL With reference to Fig. 7-19, the average friction slope is:                   hf       Sf ave  =  ______                   ΔL (7-54) Fig. 7-19  Definition sketch for the calculation of channel length increment ΔL.                    H1 + z1  =  H2 + z2  +  Sf ave ΔL                  (7-55)                    z1 - z2  =  H2 - H1  +  Sf ave ΔL                  (7-56)                    z1 - z2  -  Sf ave ΔL  =  H2 - H1                  (7-57)             z1 - z2       So  =  _________                ΔL (7-58)                    So ΔL  -  Sf ave ΔL  =  H2 - H1                  (7-59)                    (So  -  Sf ave) ΔL  =  H2 - H1                  (7-60)                  H2 - H1       ΔL  =  ______________              So  -  Sf ave (7-61)                     ΔH       ΔL  =  ______________              So  -  Sf ave (7-62) Equation 7-62 enables the calculation of the channel length increment ΔL, i.e., Col. 11 of Table 7-6. When ΔL is negative, the calculation moves upstream, as in the M2 profile (Table 7-6). Conversely, when ΔL is positive, the calculation moves downstream, as in the S2 profile (Table 7-7).

Table 7-6  Calculation of M2 water-surface profile (So = 0.0001).
[1]	[2]	[3]	[4]	[5]	[6]	[7]	[8]	[9]	[10]	[11]	[12]
y	A	V	V 2/(2g )	H	P	R	Sf	Sf ave	ΔH	ΔL	∑ ΔL
	(1)	(2)	(3)	(4)	(5)	(6)	(7)	(8)	(9)	Box
3.364	359.033	5.570	1.581	4.945	115.044	3.1208	0.00425	---	---	---	0
4.000	432.000	4.630	1.092	5.092	117.888	3.664	0.00237	0.00331	0.147	-45.794	-45.794
5.000	550.000	3.636	0.674	5.674	122.360	4.495	0.00111	0.00174	0.528	-354.878	-400.672
6.000	672.000	2.976	0.451	6.451	126.833	5.298	0.00060	0.000855	0.777	-1029.139	-1429.811
7.000
8.000
9.000
10.000
10.097							0.0001

In the direct step method, the accuracy of the computation depends on the size of depth interval (Col. [1] of Table 7-6). The smaller the interval, the more accurate the computation will be. This is because the average friction slope for a subreach (Col. [9]) is the arithmetic average of the friction slopes at the grid points (linear assumption). In practice, a computation using a computer would have a much finer interval than that shown in Table 7-6.

The calculation of the S₂ water-surface profile is shown in Table 7-7.

Table 7-7  Calculation of S2 water-surface profile (So = 0.03).
[1]	[2]	[3]	[4]	[5]	[6]	[7]	[8]	[9]	[10]	[11]	[12]
y	A	V	V 2/(2g )	H	P	R	Sf	Sf ave	ΔH	ΔL	∑ ΔL
	(1)	(2)	(3)	(4)	(5)	(6)	(7)	(8)	(9)	Box
3.364	359.033	5.570	1.581	4.945	115.044	3.1208	0.00425	---	---	---	0
3.300	351.780	5.685	1.647	4.9475	114.758	3.0654	0.0147	0.01425	0.002	0.127	0.127
3.200	340.48	5.874	1.759	4.9586	114.311	2.979	0.0163	0.0155	0.0111	0.765	0.892
3.100	329.22	6.075	1.881	4.981	113.864	2.891	0.0181	0.0172	0.0224	1.750	2.642
3.000
2.900
2.800
2.700
2.669							0.03

Online calculations

The M₂ water-surface profiles may be calculated online using ONLINE_WSPROFILES_22 (Fig. 7-20). Using the number of computational intervals n = 100, and number of tabular output intervals m = 100, the length of the M₂ water-surface profile is calculated to be:   ∑ ΔL = 147,691.5 m.

Fig. 7-20  Definition sketch for M2 water-surface profile.

The S₂ water-surface profiles may be calculated online using ONLINE_WSPROFILES_25 (Fig. 7-21). Using the number of computational intervals n = 100, and number of tabular output intervals m = 100, the length of the S₂ water-surface profile is calculated to be:   ∑ ΔL = 152.02 m.

Fig. 7-21  Definition sketch for S2 water-surface profile.

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QUESTIONS

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1. What is steady gradually varied flow?

2. What is retarded flow?

3. What is accelerated flow?

4. What is subnormal flow?

5. What is supernormal flow?

6. How many profiles are possible in steady gradually varied flow?

7. What is the rule for the Type I family of water-surface profiles?

8. What is the rule for the Type III family of water-surface profiles?

9. Which water-surface profiles are completely horizontal?

10. What are the five limits to the water surface profiles?

11. What is the typical application of the M₁ water-surface profile?

12. What is the typical application of the S₁ water-surface profile?

13. What is the typical application of the S₃ water-surface profile?

14. What is the main difference between the direct-step and standard-step methods of water-surface profile computations?

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PROBLEMS

[References]   •   [Top]   [Equation of Gradually Varied Flow]   [Characteristics of Flow Profiles]   [Limits to Water Surface Profiles]   [Methodologies]   [Direct Step Method Example]   [Questions]

1. A perennial stream has the following properties:   discharge Q = 30 m³/s, bottom width b = 55 m, side slope z = 2, bottom slope S_o = 0.0004, and Manning's n = 0.035. A 2-m high diversion dam is planned on the stream to raise the head for an irrigation canal (Fig. 7-22).

   • Calculate the normal depth.

   • Calculate the total length of the M₁ water-surface profile. Use n = 400 and m = 400.

   • Calculate the partial length of the M₁ profile, from the dam to a location upstream where the normal depth is exceeded by 1%.

   Use ONLINE CHANNEL 01 and ONLINE WSPROFILES 21.

   Fig. 7-22  A diversion dam.

2. A perennial stream has the following properties:   discharge Q = 1000 ft³/s, bottom width b = 150 m, side slope z = 2, bottom slope S_o = 0.00038, and Manning's n = 0.035. A 6-ft high diversion dam is planned on the stream to raise the head for an irrigation canal.

   • Calculate the normal depth.

   • Calculate the total length of the M₁ water-surface profile. Use n = 400 and m = 400.

   • Calculate the partial length of the M₁ profile, from the dam to a location upstream where the normal depth is exceeded by 1%.

   Use ONLINE CHANNEL 01 and ONLINE WSPROFILES 21.

3. A mild stream flows into a steep channel, producing an M₂ upstream of the brink. The hydraulic conditions in the channel are:   Q = 28 m³/s, bottom width b = 12 m, side slope z = 2.5, bottom slope S_o = 0.0007, and Manning's n = 0.04. Assume critical depth near the change in slope.

   • Calculate critical and normal depth.

   • Calculate the length of the M₂ water-surface profile using n = 100 and m = 100.

   • Calculate the length of the M₂ water-surface profile using n = 200 and m = 200.

   • Calculate the length of the M₂ water-surface profile using n = 400 and m = 400.

   • Comment on the results of b, c, and d.

   • Determine the design channel length at 99% of the normal depth. Use n = 400.

     Use ONLINE CHANNEL 05 and ONLINE WSPROFILES 22.

4. A mild stream flows into a steep channel, producing an M₂ upstream of the brink. The hydraulic conditions in the channel are:   Q = 500 ft³/s, bottom width b = 30 ft, side slope z = 1.5, bottom slope S_o = 0.00075, and Manning's n = 0.04. Assume critical depth near the change in slope.

   • Calculate critical and normal depth.

   • Calculate the length of the M₂ water-surface profile using n = 100 and m = 100.

   • Calculate the length of the M₂ water-surface profile using n = 200 and m = 200.

   • Calculate the length of the M₂ water-surface profile using n = 400 and m = 400.

   • Comment on the results of b, c, and d.

   • Determine the design channel length at 99% of the normal depth. Use n = 400.

     Use ONLINE CHANNEL 05 and ONLINE WSPROFILES 22.

5. An overflow spillway flows into a mild channel, producing a hydraulic jump. The channel is rectangular, with Q = 3 m³/s, bottom width b = 8 m, and Manning's n = 0.015. The flow depth at the toe of the spillway is 0.1 m and the approximate slope at the toe of the spillway is 0.1. The slope of the [mild] downstream channel, which functions as a stilling basin, is 0.0001. Calculate:

   • the critical depth,

   • the length L_tc from the toe of the spillway to critical depth downstream,

   • the Froude number at the toe of the spillway,

   • the Froude number downstream of the hydraulic jump,

   • the sequent depth y₂ (the normal depth downstream of the hydraulic jump),

   • the length L_j of the jump, assuming L_j = 6.2 y₂

   • the minimum length of the stilling basin L_sb = L_tc + L_j

   Use ONLINE CHANNEL 02 and ONLINE WSPROFILES 23. In the latter, use n = 100 and m = 100.

6. An overflow spillway flows into a mild channel, producing a hydraulic jump. The channel is rectangular, with Q = 100 ft³/s, bottom width b = 20 ft, and Manning's n = 0.015. The flow depth at the toe of the spillway is 0.4 ft and the approximate slope at the toe of the spillway is 0.1. The slope of the [mild] downstream channel, which functions as a stilling basin, is 0.0001. Calculate:

   • the critical depth,

   • the length L_tc from the toe of the spillway to critical depth downstream,

   • the Froude number at the toe of the spillway,

   • the Froude number downstream of the hydraulic jump,

   • the sequent depth y₂ (the normal depth downstream of the hydraulic jump),

   • the length L_j of the jump, assuming L_j = 6.2 y₂

   • the minimum length of the stilling basin L_sb = L_tc + L_j

   Use ONLINE CHANNEL 02 and ONLINE WSPROFILES 23. In the latter, use n = 100 and m = 100.

7. A diversion dam of height H = 1.8 m is planned on a steep stream with bottom slope S_o = 0.035. A hydraulic jump is expected upstream of the dam. Identify the type of water surface profile. Using ONLINE CALC, calculate the length of the water surface profile, from the location of the diversion dam, in the upstream direction, to the [downstream end of the] hydraulic jump. The channel has Q = 4 m³/s, bottom width b = 3 m, side slope z = 1, and Manning's n = 0.03. What are the sequent depths? What is the Froude number of the upstream flow? Use m = 100 and n = 100. Verify the sequent depth y₂ using ONLINE CHANNEL 11.

8. A mild channel enters into a steep channel of slope S_o = 0.03. Identify the type of water surface profile in the steep channel. Using ONLINE CALC, calculate the normal depth in the steep channel, and the length of the water surface profile to within 2% of normal depth. The channel has Q = 3 m³/s, bottom width b = 5 m, side slope z = 0, and Manning's n = 0.015. What is the normal-depth Froude number in the steep channel? Use m = 100 and n = 100.

9. A steep channel of slope S_o = 0.035 enters into a milder steep channel of slope S_o = 0.012. Identify the type of water surface profile in the milder steep channel. Using ONLINE CALC, calculate the normal depth in the downstream channel, and the length [to normal depth] of the water surface profile. The channel has Q = 3.2 m³/s, bottom width b = 4 m, side slope z = 2, and Manning's n = 0.015. What are the normal-depth Froude numbers? What is the normal depth in the upstream channel? What would be the length of the water surface profile if Manning's n was instead estimated at 0.013? Use m = 100 and n = 100.

10. A perennial stream has the following properties:   discharge Q = 15 m³/s, bottom width b = 8 m, side slope z = 2, bottom slope S_o = 0.0025, and Manning's n = 0.035. A 2.0-m high diversion dam is planned on the stream to raise the head for an irrigation canal (Fig. 7-23).

    • Calculate the normal depth.

    • Calculate the total length of the M₁ water-surface profile. Use three resolutions: (a) n = 100 and m = 100, (b) n = 200 and m = 200, and (c) n = 400 and m = 400. Comment on the results.

    • Using the higher resolution results, calculate the partial length of the M₁ profile, from the dam location to a point upstream where the normal depth is exceeded by 1%.

    Use ONLINE CHANNEL 01 and ONLINE WSPROFILES 21.

    Fig. 7-23  A diversion dam.

11. An overflow spillway flows into a mild channel, producing a hydraulic jump. The channel is rectangular, with Q = 3.6 m³/s, bottom width b = 5 m, and Manning's n = 0.015. The flow depth at the toe of the spillway is 0.15 m and the approximate slope at the toe of the spillway is 0.1. The slope of the downstream mild channel, which functions as a stilling basin, is 0.00016. Use n = 100 and m = 100. Calculate:

    • the critical depth,

    • the length L_tc from the toe of the spillway to critical depth downstream,

    • the Froude number at the toe of the spillway,

    • the Froude number downstream of the hydraulic jump,

    • the sequent depth y₂ (the normal depth downstream of the hydraulic jump),

    • the length L_j of the jump, assuming L_j = 6.2 y₂

    • the minimum length of the stilling basin L_sb = L_tc + L_j

    What would be the length of the stilling basin if the bottom width were to be increased to 7 m?

    Use ONLINE CHANNEL 02 and ONLINE WSPROFILES 23. In the latter, use n = 100 and m = 100.

12. A 5-m high diversion dam is planned on a channel operating at critical flow. The channel is rectangular, with Q = 100 m³/s, bottom width b = 4.7 m, bottom slope S_o = 0.01, and Manning's n = 0.023. Calculate the length of the C₁ water-surface profile. Assume n = 100 and m = 100.

13. A steep channel, with bottom slope S_o = 0.03, flows into a channel operating at critical flow. The channel is rectangular, with Q = 100 m³/s, bottom width b = 4.7 m, bottom slope S_o = 0.01, and Manning's n = 0.024. Calculate the length of the C₃ water-surface profile. Assume n = 100 and m = 100.

14. A horizontal channel of length L = 500 m is designed to convey Q = 5 m³/s from a reservoir to a free overfall. The bottom width is b = 2 m, and side slope z = 1.5. The channel is lined with gabions and the Manning's n recommended by the manufacturer is n = 0.028.

    • What is the headwater depth (accuracy to 1 cm) required to pass the design discharge?

    • What is the tailwater depth, that is, the critical flow depth at the downstream boundary?

    Use ONLINE_WSPROFILES_32; assume n = 100 and m = 100.

15. A sluice gate is designed to release supercritical flow into a stilling basin, where a hydraulic jump will occur. The basin channel bottom is horizontal, of rectangular cross section. The design discharge is Q = 5 m³/s, the bottom width b = 5 m, and Manning's n = 0.015.

    • What should be the gate opening (accuracy to 1 mm) to ensure that the length of the downstream water surface profile is not greater than 10 m?

    • What is the critical depth?

    Use ONLINE_WSPROFILES_35; assume n = 100 and m = 100. Assume S_o,u/s = 0.05 so that the flow through the sluice gate remains supercritical.

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REFERENCES

•   [Top]   [Equation of Gradually Varied Flow]   [Characteristics of Flow Profiles]   [Limits to Water Surface Profiles]   [Methodologies]   [Direct Step Method Example]   [Questions]   [Problems]

Chow, V. T. 1959. Open-channel Hydraulics. McGraw Hill, New York.

U.S. Army Corps of Engineers. (2014). HEC-RAS: Hydrologic Engineering Center River Analysis System.

USDA Soil Conservation Service. (1971). Classification system for varied flow in prismatic channels. Technical Release No. 47 (TR-47), Washington, D.C.

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