Chapter 4. Gradually-varied Flow Gradually-varied flow (GVF): A steady non-uniform flow in a prismatic channel with gradual changes in its water-surface elevation. Examples of GVF: (a) Drawdown produced by sudden drop in the channel (b) Backwater produced by increased in bed elevation

Two basic assumptions are involved in the analysis of GVF: 1. The pressure distribution at any section is hydrostatic. 2. The resistance to flow at any depth can be assumed using uniform-flow equation, such as the Manning's equation, with the condition that the slope term to be used in the equation is the energy slope and not the bed slope. Thus, if in a GVF the depth of flow at any section is y, the energy slope Sf is: where R is the hydraulic radius of the section at depth y. Differential Equation of GVF The total energy H of a gradually-varied flow in a channel of small slope is: (4.1) where the specific energy

Figure 4.2 Schematic sketch of GVF Since the water surface varies in the longitudinal x-direction, the depth of the flow and the total energy are functions of x. Differentiating Eq. (4.1) with respect to x, (4.2) Energy slope, Bottom slope, water-surface slope relative to the channel bottom

Since Eq. (4.2) can now be rewritten as Rearranging, Dynamic equation of GVF (4.3)

Other forms of Eq. (4.3) (a) If K = conveyance at any depth y and Ko = conveyance corresponding to the normal depth yo, then for GVF for uniform flow (4.4) Similarly, if Z = section factor at depth y and Zc = section factor at the critical depth yc, and Hence (4.5)

Using Eqs. (4.4) and (4.5), Eq. (4.3) can be written as (4.6) This equation is useful in developing direct integration techniques. (b) If Qn represents the normal discharge at a depth y and Qc denotes the critical discharge at the same depth y, Using these definitions, Eq. (4.3) can be rewritten as (4.7) and

(c) Another form of Eq. (4.2) is (4.8) This equation is called the differential-energy equation of GVF to distinguish it from the GVF differential equations of Eqns. 4.3, 4.6 and 4.7. This energy equation is very useful in developing numerical techniques for the GVF profile computation. Classification of Flow Profiles In a given channel, yo and yc are two fixed depths if Q, n and So are fixed. Also, there are three possible relations between yo and yc as: (i) yo > yc (ii) yo < yc (iii) yo = yc Further, there are two cases where yo does not exist, i.e. when the channel bed is horizontal when the channel has an adverse slope So is -ve.

Based on the above, the channels are classified into five categories as indicated in Table 4.1 Table 4.1. Types of GVF profiles

All curves in region 1 have positive slopes - backwater curves All curves in region 2 have negative slopes - drawdown curves Figure 4.3. Various GVF profiles

Example 4.1 A rectangular channel with a bottom width of 4 m and bottom slope of 0.0008 has a discharge of 1.5 m3/s. In a gradually-varied flow in this channel, the depth at a certain location is found to be 0.30 m. Assuming n = 0.016, determine the type of GVF profile. Solution: Find the normal depth yo and the critical depth yc The normal depth yo The critical depth yc Manning equation The channel is a mild slope channel The profile is of the M2 type

Examples of Occurrence of Type M Flow Profiles M1 profile is a subcritical-flow condition. Obstructions to flow such as weirs, dams, control structures and natural features such as bends, produce M1 backwater curves. These extends to several kilometres upstream before merging with the normal depth. M2 profiles occur at a sudden drop in the bed of the channel, at constriction type of transitions and at the canal outlet of pools. M3 profiles occur when there is a supercritical flow enters a mild-slope channel. Examples are the flow leading from a spillway or a sluice gate to a mild slope. M3 normally followed by a small stretch of rapidly-varied flow and the downstream is generally terminated by a hydraulic jump. Compared to M1 and M2 profiles, M3 profiles are relatively short length.

Examples of Occurrence of Type S Flow Profiles S1 profile is produced when the flow from a steep channel is terminated by a deep pool created by an obstruction, such as a weir or dam. At the beginning of the curve, the flow changes from the normal depth (supercritical flow) to subcritical flow through a hydraulic jump. The profiles extend downstream with a positive water-surface slope to reach a horizontal asymptote at the pool elevation. S2 profiles occur at the entrance region of a steep channel leading from a reservoir and at a break of grade from mild slope to steep slope. Generally S2 profile are of short length.

S3 profile is produced when the flow exited from a sluice gate with a steep slope on its downstream. S3 curve is also produced when a flow exists from a steeper slope to a less steep slope.

Examples of Occurrence of Type C Flow Profiles C1 and C3 profiles are very rare and are highly unstable. Examples of Occurrence of Type H Flow Profiles A horizontal channel can be considered as the lower limit reached by a mild slope as its bed slope becomes flatter. It is obvious that there is no region 1 for a horizontal channel as yo = . The H2 and H3 profiles are similar to M2 and M3 profiles respectively. However, the H2 curve has a horizontal asymptote. Adverse slopes are rather rare and A2 and A3 curves are similar to H2 and H3 curves respectively. These profiles are of very short length.

Example 4.2 A triangular channel has side slope 1 horizontal : 1 vertical and bed slope is 0.001. Estimate and categorize this channel whether mild, steep or critical when the flow rate is given as much as 0.2 m3/s through this channel. Assume that Manning’s roughness coefficient n = 0.015. Give various of depths that categorize the flow profile in type 1, 2 and 3. Solution: The normal depth yo The critical depth yc Manning equation The channel is a mild slope channel Curve M1 y > 0.536 m Curve M2 0.536 m > y > 0.382 m Curve M3 y < 0.382 m

Control section A control section is defined as a section in which a fixed relationship exists between the discharge and depth of flow. Weirs, spillways, sluice gates are some typical examples of structures which give rise to control sections. The critical depth is also a control point. Any GVF profile will have at least one control section.

Calculations of Flow Profile Profile of gradually varied flow can be determined using dynamic equation of gradually varied flow. Equations below are differential equations for gradually varied flow and they represent depth of water y at a certain distance x. Calculations are carried out to: i. determine the length L if the depth y1 and y2 known. ii. determine either one of water depth (y1 or y2), if the length of L and either one of depth are known. Methods to determine flow profile: 1. Direct Integration* 2. Numerical Integration* 3. Multiple Intergration a. Direct Step Method* b. Standard Step Method 4. Graphical Integration 5. Numerical/computer methods * Methods that will applied for this subject

(4.9) Direct Integration From previous equation Replacing u = y/yo du /dy = 1 / yo dy = yo du Substituting in the above equation: Intergrate above equation, hence it become : (4.9) Substituting = F(u,N) Replacing with

Then, Applying the above equation between two section (x1, y1) and (x2, y2) in one channel: Values of F(u,N) can be obtained from table provided in appendix I. If (u,N) is replaced with (v,J), then the table can also be used to find the value of F(v, J).

Example 4.3 A very wide river have depth of 3.0 m and slopes of 0.0005. Estimate length of flow profile produced by a weir that caused water surface increased as much as 1.50 m at the upstream of weir (n = 0.035). Solution: 1. Determine yo and yc. Given yo = 3.0 m, y = 4.5 m Critical depth , yc = (q2/g )1/3 = (3.987/ 9.81)1/3 = 1.175 m

2. Determine value of N & M For a very wide rectangular channel, N = 10/3 dan M = 3 3. Calculate J, J = N / (N – M+1) = 10/3 = 2.5 (10/3 – 3 + 1) 4. Calculate u1, u2 and v1, v2 u1 = y1/yo = 4.5 / 3 = 1.5 u2 = y2/yo = 3.03 / 3 = 1.01 v1 = u1N/J = (1.5)3.33/2.5 = 1.72 v2 = u2N/J = (1.01)3.33/2.5 = 1.01

5. Obtain F (u, N) and F ( v, J) from table F(u1, N) = F (1.5 , 3.33) = 0.189 F(u2, N) = F (1.01 , 3.33) = 1.220 F(v1, J) = F (1.72 , 2.50) = 0.333 F(v2, J) = F (1.01 , 2.50) = 1.867

5. Obtain F (u, N) and F ( v, J) from table F(u1, N) = F (1.5 , 3.33) = 0.189 F(u2, N) = F (1.01 , 3.33) = 1.220 F(v1, J) = F (1.72 , 2.50) = 0.333 F(v2, J) = F (1.01 , 2.50) = 1.867

5. Obtain F (u, N) and F ( v, J) from table F(u1, N) = F (1.5 , 3.33) = 0.189 F(u2, N) = F (1.01 , 3.33) = 1.220 F(v1, J) = F (1.72 , 2.50) = 0.333 F(v2, J) = F (1.01 , 2.50) = 1.867

5. Obtain F (u, N) and F ( v, J) from table F(u1, N) = F (1.5 , 3.33) = 0.189 F(u2, N) = F (1.01 , 3.33) = 1.220 F(v1, J) = F (1.72 , 2.50) = 0.261 F(v2, J) = F (1.01 , 2.50) = 1.867

6. Finding length L, from equation. L = - 8711 m ; L = 8711m from back of weir.

Numerical Integration Equations used are: i) For any shape of channel ii) For rectangular channel

iii) For wide rentangular channel equation (using Chezy, C) (using Manning)

Example 4.4 A very wide channel(rectangular) have depth 3.0 m and slope 0.0005. Determined type of flow profile and estimate length of gradualy varied flow profile produced by a weir that elevated the upstream flow as much s 1.50 m ( assumed n = 0.035). Take N = 4 steps/ section. Solution: 1. Calculate yo and yc ; Given yo = 3.0 m, y = 4.5 m Critical depth, yc = (q2/g )1/3 = (3.987/ 9.81)1/3 = 1.175 m y > yo > yc; GVF profile is M1

Equation used for wide rectangular channel is: yc = 1.175 m; yo = 3.0 m; so = 0.0005 Table 1: Calculation for flow profile y (m) dy (m) ŷ(m) 1 – (yc/y)3 1– (yo/y)10/3 dx (m) 4.5 – 4.13 0.37 4.32 0.980 0.703 1031.0 4.13 – 3.76 3.95 0.974 0.600 1201.3 3.76 – 3.39 3.58 0.9645 0.445 1604.7 3.39 – 3.02 3.21 0.951 0.202 3483.9 L = ∑∆x = 7321.5 Stop calculation at, y = 3.00 m

Length of flow profile M1 is 7823.1 m from back of weir.

Direct Step Method Using concept of energy continuity horizontal B hL Slope of energy line, i A v1² 2g v2² Datum So z2 z1 C D dx y1 y2 S Length AC = BD, Energy equation , E1 + z1 = E2 + z2 + hL Gradually varied flow with respect to length of dx

Hence, equation used for calculating gradually varied flow is: If Hence, equation used for calculating gradually varied flow is: Remember! 1. End calculation at y = ( 1 ± 0.01 ) yo 2. Accuraccy of calculation, depending on the number of numerical, N choosen.

Example 4.5 A 100 m wide and 3.0 m deep channel has 0.0005 slope. Determine type of flow profile and estimate length of gradually varied low profile produced by a weir that elevated the upstream flow as much as 1.50 m ( assumed n = 0.035). Solution Assumed the river as a very wide channel, flow/width are: q = 3.987 m3/s/m Critical depth, = 1.175 m

y > yo > yc ; GVF profile is M1 Take N = 4 steps / section Stop calculation at y = 3.01 m R = y, q = 3.987 m3/s/m , n = 0.035 , so = 0.0005 Table 2:Calculation of flow profile by using direct step method 1 y (m) 2 R (m) 3 v (m/s) 4 v2/2g (m) 5 E (m) 6 ∆E (m) 7 ix 10-4 8 avg ī x 10-4 9 (so-avg ī)x10-4 10 ∆x(m) 11 L (m) 4.5 0.886 0.0400 4.540 _ 1.294 4.13 0.965 0.0475 4.178 0.362 1.722 1.508 3.492 1036.7 1036 3.76 1.060 0.0573 3.817 0.361 2.354 2.038 2.962 1218.8 2255 3.39 1.176 0.0704 3.460 0.357 3.327 2.841 2.159 1653.5 3908 3.03 1.316 0.0883 3.118 0.342 4.839 4.083 0.917 3729.6 7638

Hence the length of flow profile is 7525 m from weir. y2 y1 ∆x4 ∆x3 ∆x2 ∆x1 L = ∑dx = 7525 m