ENGR 691 – 73: Introduction to Free-Surface Hydraulics in Open Channels

1. ENGR 691 – 73: Introduction to Free-Surface Hydraulics in Open Channels Lecture 04: Nonuniform Flow Yan Ding, Ph.D. Research Assistant Professor, National Center for Computational Hydroscience and Engineering (NCCHE), The University of Mississippi, Old Chemistry 335, University, MS 38677 Phone: 915-8969; Email: ding@ncche.olemiss.edu Course Notes by: Mustafa S. Altinakar and Yan Ding

2. Outline • Transition Between Subcritical and Critical Flow • Introduction to Hydraulic Jump • Gradually Varied Flow (Governing Equations) • Forms of water surface (Channels on Mild Slope, Critical Slope, Steep Slope, Adverse Slope, Horizontal Slope) • Control Points • Computation of Water Surface (Method of successive Approximations; Method of Direct Integration; Method of Graphical Integration) • Rapidly Varied Flow (Weirs; Spillways; Hydraulic Drop; Underflow Gates; Hydraulic Jump) • Transitions (Channel with variable Bed Floor; Channel of variable Width; Oblique Jump) • Lateral Inflow

3. Transitions between subcritical and critical flow Transition from subcritical to supercritical flow Transition from supercritical to subcritical flow When the flow changes from subcritical to supercritical the water surface lowers gradually from a higher depth to a lower depth by passing through critical depth. In the region where the flow changes from subcritical to critical flow, a gradually varied flow takes place. When the flow changes from supercritical to subcritical the water surface rapidly increases from a supercritical depth to subcritical depth. This sudden increase is called a rapidly varied flow. The rapidly varied flow may be preceded by a gradually varied flow region where the flow depth rises but stays below critical depth.

4. Introduction to Hydraulic Jump conjugate depths or sequent depths alternate depths Specific Momentum Specific Energy Equation of continuity • Things to remember: • Conjugate depths or sequent depths (on Specific Momentum Curve) • Alternate depths (on Specific Energy Curve) Momentum equation with

5. Introduction to Hydraulic Jump By combining momentum equation and continuity equation, on gets: or where and

6. Gradually Varied Flow Equation ref. line Consider the steady non uniform flow in a channel. We wish to develop an equation for the variation of the water surface h(x), i.e. longitudinal water surface profile. For this, we will consider the equation of energy: and the equation of continuity: Differentiate the energy equation with respect to x to get: Assuming that the head loss can be expressed using Chezy equation, we have:

7. Gradually Varied Flow Equation ref. line Consider the steady non uniform flow in a channel. We wish to develop an equation for the variation of the water surface h(x), i.e. longitudinal water surface profile. For this, we will consider the equation of energy: and the equation of continuity: Differentiate the energy equation with respect to x to get: Assuming that the head loss can be expressed using Chezy equation, we have:

8. Gradually Varied Flow Equation Note that for a prismatic channel the flow area is only a function of the flow depth, A = f(h) : We can, therefore, write: Substitute this expression back into the previous equation to get: By rearranging the terms, we obtain a differential equation describing the variation of flow depth with distance, i.e. the equation for longitudinal water surface profile: It is important to note that when: i.e. Chezy equation for uniform flow the water surface profile equation reduces to

9. Gradually Varied Flow Equation The flow depth remains constant and is equal to normal depth (uniform flow) The flow depth increases in the direction of flow The flow depth decreases in the direction of flow Consider again the equation for longitudinal water surface profile: For the denominator becomes zero and we have: We can, therefore conclude that, at critical flow (Fr = 1 and h = hc), the water surface profile is perpendicular to bed. The normal is equal to critical depth, hn = hc, when: Uniform flow

10. Review of the Notion of Critical Flow Consider the flow cases below (for all cases channel cross section characteristics are the same): Critical slope is the bed slope when normal depth, hn, is equal to critical depth, hc. When flow is critical, we have: Since the flow is also uniform, Chezy equation holds: Equating two expressions, we have: note that we have changed So to Sc. The expression for critical discharge is obtained as: If Manning-Strickler is used:

11. Gradually Varied Flow Equation in Terms of Conveyance The equation for gradually varied flow can also be written using the notion of conveyance: when the flow is uniform, in either case we can write: Remember the definition of conveyance: when using Manning Strickler or when using Chezy Consider the term in the denominator of gradually varied flow equation: Now consider the term in the nominator of gradually varied flow equation: The gradually varied flow equation can therefore be written as:

12. Special forms of Gradually Varied Flow Equation: Wide Channel Let us now consider a wide rectangular channel. The Chezy equation can be written as: The critical depth in a rectangular channel is given by: Using these expressions and assuming that the Chezy coefficient C does not depend on depth h, the gradually varied flow equation can be written as: This equation is known as equation of Bresse named after the French scientist J.A.C. BRESSE (1822-1883), who developed it first. If we use Manning-Strickler, we have: In this case equation of Bresse becomes:

13. Gradually Varied Flow: Forms of Water Surface Before we present all possible gradually varied flow profiles, let us take a look at the general properties of such curves: • The water surface profile approaches asymptotically to uniform depth hn. • The water surface profile is orthogonal to the critical depth line, when h = hc. Water surface profiles are classified according to the bed slope. Channel on Mild slope type profile For each profile type several possibilities are distinguished. These are called branches. Channel on Steep slope type profile Channel on Critical slope type profile Channel on Horizontal slope type profile Channel on Adverse slope type profile In studying gradually varied water surface profiles we should also keep in mind that: • In subcritical flow (Fr < 1), the perturbations travel both upstream and downstream. The water surface profiles for subcritical flow are controlled by a downstreamcontrol section. • In supercritical flow (Fr > 1), the perturbations travel only downstream. The water surface profiles for supercritical flow are controlled by an upstreamcontrol section.

14. Gradually Varied Flow: Forms of Water Surface Convention for numbering branches: • When the water surface profile is higher than both the normal depth and the critical depth, the branch is numbered as type 1, • the water surface profile is between the normal and critical depths, the branch is numbered as type 2, • the water surface profile is lower than both the normal depth and the critical depth, the branch is numbered as type 3,

15. Gradually Varied Flow: Forms of Water Surface Channel on Mild slope and M-type profiles Branch M1 Branch M2 Branch M3 Towards upstream the profile approaches asymptotically normal depth, towards downstream the curve tends to become horizontal. Towards upstream the profile approaches asymptotically normal depth, towards downstream the curve decreasingly tends to critical depth. Towards downstream the profile approaches increasingly to critical depth. • Encountered: • Upstream of a weir or a dam • Upstream of a pier • Upstream of certain bed slope changes points • Encountered: • Upstream of an increase in bed slope • Upstream of a free drop structure • Encountered: • When a supercritical flow enters a mild channel • After a change in slope from steep to mild

16. Gradually Varied Flow: Forms of Water Surface Channel on Steep slope and S-type profiles Branch S1 Branch S2 Branch S3 Towards upstream the profile approaches asymptotically normal depth, towards downstream the curve tends to become horizontal. Towards upstream the profile approaches asymptotically normal depth, towards downstream the curve decreasingly tends to critical depth. Towards downstream the profile approaches increasingly to critical depth. • Encountered: • Upstream of a weir or a dam • Upstream of a pier • Upstream of certain bed slope changes points • Encountered: • Upstream of an increase in bed slope • Upstream of a free drop structure • Encountered: • When a supercritical flow enters a mild channel • After a change in slope from steep to mild

17. Gradually Varied Flow: Forms of Water Surface Channel on Critical slope and C-type profiles Branch C1 Branch C2 Branch C3 The water surface profile is horizontal, when Chezy equation is used. There is no physically possible C2 profile. The water surface profile is horizontal, when Chezy equation is used. • Encountered: • Upstream of a dam/weir • At certain bed slope change locations • Encountered: • When a supercritical flow enters a mild channel • After a change in slope from steep to mild

18. Gradually Varied Flow: Forms of Water Surface Channel on Horizontal slope H-type profiles Branch H1 Branch H2 Branch H3 Normal depth becomes infinite and is meaningless. Consequently, H1 profile is not possible. Similar to M2 profile Similar to M3 profile • Encountered: • Upstream of a free drop structure • Encountered: • When a supercritical flow enters a horizontal channel

19. Gradually Varied Flow: Forms of Water Surface Channel on Adverse slope H-type profiles Branch A1 Branch A2 Branch A3 Normal depth becomes infinite and is meaningless. Consequently, A1 profile is not possible. Similar to H2 profile (parabolic) Similar to H3 profile (parabolic) • Encountered: • Upstream of a certain bed slope change location • Encountered: • When a supercritical flow enters a channel with adverse slope

20. Gradually Varied Flow: Notion of Control Section Note that the passage from subcritical flow to supercritical flow occurs with a smooth surface. On the other hand, when the flow passes from supercritical flow to subcritical flow, a sudden increase in the water depth is observed. On the figure this is indicated by HJ, which means hydraulic jump. We will study hydraulic jump in more detail later. Photograph from Ohio University's Fluid Mechanics Laboratory. Athens, Ohio USA http://www.lmnoeng.com/Channels/HydraulicJump.htm Control point, as the name implies, is the point that controls the water surface profile. At a control point we can generally write an expression between discharge and depth. Thus, it can be used as boundary condition for calculating the water surface profile.

21. Critical Depth as Control Section and Other Uses of Critical Depth In open channel flow locally a critical flow situation may exist for certain situations, such as slope change from mild to steep, free fall (drop structure), and excessive contraction, etc. Changing from a mild slope to a steep slope (passage from subcritical flow to supercritical flow). Subcritical flow at a free overfall. In fact, the critical depth takes place about 3 to 4 times hc upstream of the brink (due to curvature of streamlines). The depth at the brink is approximately equal to: The cases of critical flow due to excessive contraction and a high positive step will be studied later. In open channel flow critical section is a valuable tool because, knowing the geometry of the section, one can write the relationship between flow depth and discharge. Due to this property, critical condition is sometimes forced at a point in the channel. Then the discharge can be obtained by measuring the flow depth.

22. Computation of Gradually Varied Flow Several methods are available for computing gradually varied water surface profiles: The most obvious is to solve the differential equation of gradually varied flow, equation of Bresse, using a numerical method, such as 4th order Runge-Kutta method. This method is called method of direct integration. Equation of Bresse using Manning-Strickler equation: Equation of Bresse using Chezy equation: 4th order Runge-Kutta method formula can be written as: where: Coordinate along the channel length. The origin can be arbitrarily placed at any location. Flow depth at location x. All flow parameters at this location are known. Flow depth at location x+Dx. This is the unknown flow depth we are calculating. Computations should start from a point where all flow parameters are known (such as a control point) and proceed upstream if the flow is subcritical and downstream if the flow is supercritical.

23. Computation of Gradually Varied Flow Several methods are available for computing gradually varied water surface profiles: The second possibility is to use directly the energy equation to compute the water surface profile by employing an iterative procedure. This approach is called method of successive approximations. This method can be applied in two ways: 2.1 The open channel reach under study is divided into sub-reaches at known intervals starting from a control point where all the hydraulic parameters are known. Based on the depth at the known point the depth at the next station is computed. This method is called method of reaches (Stand Step Method in Open-Channel Flow, MH Chaudhry). 2.2 A control point where all the hydraulic parameters are known is identified. The depth at that station, h, is known. We choose another depth h+Dh, and compute where this depth will be along the channel. This method is called method of depth variation (Direct-Step Method, MH Chaudhry). In this course, we will study only the method of reaches. Please refer to the textbook and other references for more information on other methods that can be used for computation of water surfaces.

24. Computation of Gradually Varied Flow: Method of Reaches Consider the gradually varied flow shown in the figure. We have divided the reach under study into smaller sub reaches of length Dx. We also define the cross sections i, i+1, i+2, ….. etc. We will assume that the geometric properties of the channel (A, P, B, Rh, Dh) at each cross section can be calculated by knowing the depth. We will also assume that the depth at cross section i is known. We would like to calculate the depth at cross section i+1. Let us write the equation of energy Bernoulli equation) between two cross sections i and i+1 :

25. Computation of Gradually Varied Flow Therefore, when using the method of reaches, we will be solving this ordinary differential equation: The basic equation we are using is: Since depth hi , invert elevation zi and the discharge Q are known, we can calculate the left side of the equation, i.e. the total energy head, Hi directly. Let us now assume a depth hi+1 . Since the invert elevation zi and the discharge Q are known, we can also calculate the total energy head, Hi+1 directly. Now the question is weather the assumed that is the correct depth. This can be easily done. If the assumed depth hi+1 is correct, then, the difference between the total heads Hi and Hi+1 should be equal to DxSe. The energy gradient can be calculated using either the equation of Chezy or Manning Strickler: Manning Strickler equation: Chezy equation:

26. Computation of Gradually Varied Flow Since the hydraulic parameters are varying from cross section i to i+1, we may want to use the average value of the energy gradient: Note also that : We should therefore check that: or is satisfied. If the above equation is not satisfied, a new value should be assumed for hi+1 and the computations must be carried out again. All these calculation can easily be carried out on a spread sheet. If there are singular losses between the two cross sections i and i+1, this should also be taken into account. Then the equation becomes: Again considering average values we can write: Consequently:

27. Computation of Gradually Varied Flow A trapezoidal channel having a bottom width of b = 7.0m and side slopes of m = 1.5, conveys a discharge of Q = 28m3/s. The channel has a constant bed slope of So = 0.001. The Manning friction coefficient for the channel is n = 0.025m-1/3s. The channel terminates by a sudden drop of the bed. Determine the type of water surface profile to be expected. Calculate the water surface profile for a reach length of 3200m. The computation of gradually varied flow equations can be easily carried out on a spreadsheet using Goal Seek function

28. Rapidly Varied Flow at Channel Transitions A transition is a change in the channel geometry over a relatively short distance. The change can be contraction or expansion of the section, or a change in the section cross section geometry (say from rectangular to trapezoidal), or an abrupt rise or drop of the channel bed. In designing transition, the attention must be paid to create minimum amount of disturbance to the flow. https://www.fhwa.dot.gov/engineering/hydraulics/pubs/06086/hec14ch06.cfm#fig096 The figure shows, typical designs for channel transition from a rectangular cross section to a trapezoidal cross section.

29. Use of Specific Energy to study Rapidly Varied Flow at Channel Transitions Curve plotted for a constant Q Subcritical flow Specific Energy Alternate depths Supercritical flow Specific energy curve is an extremely useful tool for analyzing various flow situations. In the following slides we will learn how the specific energy curve can be used to analyze various flow situations in channel transitions (flow over a positive or negative step, flow through a contraction or expansion).

30. Rapidly Varied Flow at Channel Transitions In the following pages we will study in detail the rapid change of water surface at four types of channel transitions under both subcritical and supercritical conditions: Subcritical flow over a positive step Supercritical flow over a positive step Subcritical flow over a negative step Supercritical flow over a negative step Subcritical flow through a contraction Supercritical flow through a contraction Subcritical flow through an expansion Supercritical flow through an expansion Side view Side view Top view Top view

31. Rapidly Varied Flow: Subcritical Flow over a Positive Step Assume that the head loss due to the step is negligible (the energy grade line remains parallel to the bed).

32. Rapidly Varied Flow: Supercritical Flow over a Positive Step Assume that the head loss due to the step is negligible (the energy grade line remains parallel to the bed).

33. Rapidly Varied Flow: Subcritical Flow over a Negative Step Assume that the head loss due to the step is negligible (the energy grade line remains parallel to the bed).

34. Rapidly Varied Flow: Supercritical Flow over a Negative Step Assume that the head loss due to the step is negligible (the energy grade line remains parallel to the bed).

35. Rapidly Varied Flow: Subcritical Flow through a Contraction Top view Side view Assume that the head loss due to contraction is negligible (the energy grade line remains parallel to the bed).

36. Rapidly Varied Flow: Supercritical Flow through a Contraction Top view Side view Assume that the head loss due to contraction is negligible (the energy grade line remains parallel to the bed).

37. Rapidly Varied Flow: Subcritical Flow through an expansion Top view Side view Assume that the head loss due to contraction is negligible (the energy grade line remains parallel to the bed).

38. Rapidly Varied Flow: Supercritical Flow through an Expansion Top view Side view Assume that the head loss due to contraction is negligible (the energy grade line remains parallel to the bed).

39. Rapidly Varied Flow: Special Case of Choked Flow due to a High Positive Step By subtracting Dz from Hs1, we cannot fall back onto the specific energy curve At one point on the step the flow goes through the critical depth of the cross section on the step. final energy line initial energy line final water surface initial water surface If the step is too high, subtracting Dz from Hs1, we cannot fall back onto the specific energy curve. The flow is said to be choked. The step is too high. The water accumulates upstream of the step until it can pass over it by going through critical flow over the step. Same equations hold. However, now h1 is also an unknown. Condition of critical flow over the step provides the third equation needed for the analysis.

40. Rapidly Varied Flow: Special Case of Choked Flow due to too much Contraction Top view With the available energy Hs1, we cannot cut the specific energy curve of the contracted section At one point on the contracted section the flow goes through the critical depth of that cross section. final energy line initial energy line final water surface initial water surface Side view If the step is contracted too much, with the specific energy Hs1 we cannot cut the specific energy curve of the contracted section. The flow is said to be choked. The section is contracted too much. The water accumulates upstream of the contraction until it can pass a discharge of Q to the downstream by going through critical flow of the contracted section. Same equations hold. However, now h1 is also an unknown. Condition of critical flow at the contracted section provides the third equation needed for the analysis.

41. Energy Losses For Subritical Flow in Open Channel Transitions Head losses at contractions and expansions can be calculated using the following expressions: U1 U2 U1 U2 Contraction Expansion Taken from USACE (1994) U.S. Army Corps of Engineers, 1994. “Hydraulic Design of Flood Control Channels,” Engineering and Design Manual, EM 1110-2-1601, July 1991, Change 1 (June 1994).

42. Rapidly Varied Flow: Hydraulic Jump Hydraulic jump is a natural phenomenon that occurs when supercritical flow is forced to become subcritical. The passage from supercritical flow to subcritical takes place with a sudden rise of the flow depth accompanied by a very turbulent motion that may entrain air into the flow. To derive the equation governing hydraulic jump in a channel (see figure above), we will make use of momentum and continuity equations simultaneously. Consider a control volume, which comprises the hydraulic jump. The upstream cross section of the control volume is in supercritical flow and the downstream section is in subcritical flow. Forces acting on this control volume are the weight of the fluid, W, the upstream and downstream pressure forces, FP1 and FP2 respectively, and the friction force, Ff. The momentum equation can be written as: Assuming a rectangular channel, we have:

43. Rapidly Varied Flow: Hydraulic Jump Using these expressions and neglecting the component of weight and friction forces, the momentum equation becomes: Note that the left and right hand side of the equation represent the specific momentum, which is defined as: Let us now make use of equation of continuity to write the momentum equation as: Divide both sides by (h2 – h1) to get: Only the positive root of the above quadratic equation is physically meaningful: Written in dimensionless form, the above equation becomes: or This equation is called the equation of Bélangerin honor of the French scientist who developed it for the first time. where and

44. Rapidly Varied Flow: Hydraulic Jump Note that for a hydraulic jump on larger slopes, the weight of the fluid cannot be neglected. In this case, the equation of Bélanger for hydraulic jump becomes: where as given by Rajaratnam. a is in degrees Hydraulic jumps are classified according to the approach flow Froude number. https://www.fhwa.dot.gov/engineering/hydraulics/pubs/06086/hec14ch06.cfm#fig096

45. Rapidly Varied Flow: Hydraulic Jump Photos from (Dr. H. Chanson): http://www.uq.edu.au/~e2hchans/undular.html Energy loss across the hydraulic jump: Length of the hydraulic jump: Classification of hydraulic jumps: Undular jump Weak jump Oscillating jump jump type generally preferred in engineering applications Steady jump Strong jump

46. Use of Hydraulic Jump in Hydraulic Engineering Hydraulic jump is used for dissipating the energy of high speed flow which may harm the environment if released in an uncontrolled way. The hydraulic jump, should take place in a area where the bottom is protected (for example by a concrete slab or large size rocks). If the jump takes place on erodible material the formation of the erosion hole may endanger even the foundations of the structure. In real engineering projects measures are taken to ensure that the hydraulic jump takes place in the area with a protected bottom. This is achieved by creating a stilling basin with the use of chute blocks, baffle piers, and end sill, etc.

47. Examples of the Use of Hydraulic Jump in Hydraulic Engineering http://www.engineering.uiowa.edu/~cfd/gallery/images/hyd8.jpg

48. Design of Stilling Basins USBR Type I Stilling Basin USBR Type II Stilling Basin USBR Type III Stilling Basin USBR Type IV Stilling Basin SAF Stilling Basin Pillari’s Stilling Basin

49. Books on Design of Stilling Basins Hydraulic Design of Stilling Basins and Energy Dissipators by A. J. Peterka, U.S. Department of the Interior, Bureau of Reclamation Energy Dissipators and Hydraulic Jump by Willi H. Hager

50. Positioning of a Hydraulic Jump Draw the downstream subcritical flow profile starting from a control section at the downstream. Draw the upstream supercritical flow profile starting from a control section at the upstream. Draw the conjugate depth curve for the upstream supercritical flow profile. For a hydraulic jump with zero length the jump is a vertical water surface between A’ and Z’. If we wish to take into account the length of the jump for each point on the conjugate depth curve, draw a line parallel to the bed. The length of the line should be equal to the length of the jump, i.e. 3 to 5 times the height difference between the conjugate depth and the water depth. The tips of these lines are joined to obtain a translated conjugate depth curve which takes into account the length of the jump. The intersection of the downstream profile with the translated conjugate depth gives the downstream end of the jump. Thus, the jump takes place between A and Z.