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Lecture 4.Engr Introduction to Free-Surface Hydraulics in Open Channels1 ENGR 691 – 73: Introduction to Free-Surface Hydraulics in Open Channels Lecture 04: Nonuniform Flow Course Notes by: Mustafa S. Altinakar and Yan Ding Yan Ding, Ph.D. Research Assistant Professor, National Center for Computational Hydroscience and Engineering (NCCHE), The University of Mississippi, Old Chemistry 335, University, MS Phone: ;

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Lecture 4.Engr Introduction to Free-Surface Hydraulics in Open Channels2 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

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Lecture 4.Engr Introduction to Free-Surface Hydraulics in Open Channels3 Transition from subcritical to supercritical 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. Transition from supercritical to subcritical flow 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. Transitions between subcritical and critical flow

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Lecture 4.Engr Introduction to Free-Surface Hydraulics in Open Channels4 Specific Energy conjugate depths or sequent depths Specific Momentum Equation of continuity Momentum equation with alternate depths Things to remember: Conjugate depths or sequent depths (on Specific Momentum Curve) Alternate depths (on Specific Energy Curve) Things to remember: Conjugate depths or sequent depths (on Specific Momentum Curve) Alternate depths (on Specific Energy Curve) Introduction to Hydraulic Jump

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Lecture 4.Engr Introduction to Free-Surface Hydraulics in Open Channels5 By combining momentum equation and continuity equation, on gets: or whereand Introduction to Hydraulic Jump

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Lecture 4.Engr Introduction to Free-Surface Hydraulics in Open Channels6 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: Gradually Varied Flow Equation

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Lecture 4.Engr Introduction to Free-Surface Hydraulics in Open Channels7 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: Gradually Varied Flow Equation

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Lecture 4.Engr Introduction to Free-Surface Hydraulics in Open Channels8 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: the water surface profile equation reduces to i.e. Chezy equation for uniform flow Gradually Varied Flow Equation

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Lecture 4.Engr Introduction to Free-Surface Hydraulics in Open Channels9 Consider again the equation for longitudinal water surface profile: Forthe denominator becomes zero and we have: We can, therefore conclude that, at critical flow (Fr = 1 and h = h c ), the water surface profile is perpendicular to bed. The normal is equal to critical depth, h n = h c, when: Uniform flow 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 Gradually Varied Flow Equation

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Lecture 4.Engr Introduction to Free-Surface Hydraulics in Open Channels10 Consider the flow cases below (for all cases channel cross section characteristics are the same): Critical slope is the bed slope when normal depth, h n, is equal to critical depth, h c. 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 S o to S c. The expression for critical discharge is obtained as: If Manning-Strickler is used: Review of the Notion of Critical Flow

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Lecture 4.Engr Introduction to Free-Surface Hydraulics in Open Channels11 The gradually varied flow equation can therefore be written as: The equation for gradually varied flow can also be written using the notion of conveyance: Remember the definition of conveyance: when using Manning Strickler when using Chezy Consider the term in the denominator of gradually varied flow equation: when the flow is uniform, in either case we can write: or Now consider the term in the nominator of gradually varied flow equation: Gradually Varied Flow Equation in Terms of Conveyance

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Lecture 4.Engr Introduction to Free-Surface Hydraulics in Open Channels12 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 If we use Manning-Strickler, we have: In this case equation of Bresse becomes: named after the French scientist J.A.C. BRESSE ( ), who developed it first. Special forms of Gradually Varied Flow Equation: Wide Channel

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Lecture 4.Engr Introduction to Free-Surface Hydraulics in Open Channels13 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 h n. The water surface profile is orthogonal to the critical depth line, when h = h c. Water surface profiles are classified according to the bed slope. Channel on M ild slope Channel on S teep slope Channel on C ritical slope Channel on H orizontal slope Channel on A dverse slope type profile For each profile type several possibilities are distinguished. These are called branches. 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 downstream control section. In supercritical flow (Fr > 1), the perturbations travel only downstream. The water surface profiles for supercritical flow are controlled by an upstream control section. Gradually Varied Flow: Forms of Water Surface

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Lecture 4.Engr Introduction to Free-Surface Hydraulics in Open Channels14 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, Gradually Varied Flow: Forms of Water Surface

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Lecture 4.Engr Introduction to Free-Surface Hydraulics in Open Channels15 Channel on M ild slope M-type profiles and Branch M1Branch M2Branch M3 Towards upstream the profile approaches asymptotically normal depth, towards downstream the curve tends to become horizontal. Encountered: Upstream of a weir or a dam Upstream of a pier Upstream of certain bed slope changes points Towards upstream the profile approaches asymptotically normal depth, towards downstream the curve decreasingly tends to critical depth. Encountered: Upstream of an increase in bed slope Upstream of a free drop structure Towards downstream the profile approaches increasingly to critical depth. Encountered: When a supercritical flow enters a mild channel After a change in slope from steep to mild Gradually Varied Flow: Forms of Water Surface

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Lecture 4.Engr Introduction to Free-Surface Hydraulics in Open Channels16 Channel on S teep slope S-type profiles and Branch S1Branch S2Branch S3 Towards upstream the profile approaches asymptotically normal depth, towards downstream the curve tends to become horizontal. Encountered: Upstream of a weir or a dam Upstream of a pier Upstream of certain bed slope changes points Towards upstream the profile approaches asymptotically normal depth, towards downstream the curve decreasingly tends to critical depth. Encountered: Upstream of an increase in bed slope Upstream of a free drop structure Towards downstream the profile approaches increasingly to critical depth. Encountered: When a supercritical flow enters a mild channel After a change in slope from steep to mild Gradually Varied Flow: Forms of Water Surface

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Lecture 4.Engr Introduction to Free-Surface Hydraulics in Open Channels17 Channel on C ritical slope C-type profiles and Branch C1Branch C2Branch C3 The water surface profile is horizontal, when Chezy equation is used. Encountered: Upstream of a dam/weir At certain bed slope change locations There is no physically possible C2 profile. The water surface profile is horizontal, when Chezy equation is used. Encountered: When a supercritical flow enters a mild channel After a change in slope from steep to mild Gradually Varied Flow: Forms of Water Surface

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Lecture 4.Engr Introduction to Free-Surface Hydraulics in Open Channels18 Channel on H orizontal slope H-type profiles Branch H1Branch H2Branch H3 Normal depth becomes infinite and is meaningless. Consequently, H1 profile is not possible. Similar to M2 profile Encountered: Upstream of a free drop structure Similar to M3 profile Encountered: When a supercritical flow enters a horizontal channel Gradually Varied Flow: Forms of Water Surface

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Lecture 4.Engr Introduction to Free-Surface Hydraulics in Open Channels19 Channel on A dverse slope H-type profiles Branch A1Branch A2Branch A3 Normal depth becomes infinite and is meaningless. Consequently, A1 profile is not possible. Similar to H2 profile (parabolic) Encountered: Upstream of a certain bed slope change location Similar to H3 profile (parabolic) Encountered: When a supercritical flow enters a channel with adverse slope Gradually Varied Flow: Forms of Water Surface

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Lecture 4.Engr Introduction to Free-Surface Hydraulics in Open Channels20 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 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. Gradually Varied Flow: Notion of Control Section

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Lecture 4.Engr Introduction to Free-Surface Hydraulics in Open Channels21 Changing from a mild slope to a steep slope (passage from subcritical flow to supercritical flow). Subcritical flow at a free overfall. 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. 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. In fact, the critical depth takes place about 3 to 4 times h c 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. Critical Depth as Control Section and Other Uses of Critical Depth

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Lecture 4.Engr Introduction to Free-Surface Hydraulics in Open Channels22 Several methods are available for computing gradually varied water surface profiles: 1.The most obvious is to solve the differential equation of gradually varied flow, equation of Bresse, using a numerical method, such as 4 th order Runge-Kutta method. This method is called method of direct integration. Equation of Bresse using Chezy equation: Equation of Bresse using Manning- Strickler equation: 4 th 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+ x. 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. Computation of Gradually Varied Flow

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Lecture 4.Engr Introduction to Free-Surface Hydraulics in Open Channels23 Several methods are available for computing gradually varied water surface profiles: 2.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: 2.1The 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). 2.2A control point where all the hydraulic parameters are known is identified. The depth at that station, h, is known. We choose another depth h+ h, 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. Computation of Gradually Varied Flow

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Lecture 4.Engr Introduction to Free-Surface Hydraulics in Open Channels24 Consider the gradually varied flow shown in the figure. We have divided the reach under study into smaller sub reaches of length x. 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, R h, D h ) 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 : Computation of Gradually Varied Flow: Method of Reaches

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Lecture 4.Engr Introduction to Free-Surface Hydraulics in Open Channels25 Therefore, when using the method of reaches, we will be solving this ordinary differential equation: The basic equation we are using is: Since depth h i, invert elevation z i and the discharge Q are known, we can calculate the left side of the equation, i.e. the total energy head, H i directly. Let us now assume a depth h i+1. Since the invert elevation z i and the discharge Q are known, we can also calculate the total energy head, H i+1 directly. Now the question is weather the assumed that is the correct depth. This can be easily done. If the assumed depth h i+1 is correct, then, the difference between the total heads H i and H i+1 should be equal to x S e. The energy gradient can be calculated using either the equation of Chezy or Manning Strickler: Chezy equation: Manning Strickler equation: Computation of Gradually Varied Flow

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Lecture 4.Engr Introduction to Free-Surface Hydraulics in Open Channels26 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:oris satisfied. If the above equation is not satisfied, a new value should be assumed for h i+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: Computation of Gradually Varied Flow

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Lecture 4.Engr Introduction to Free-Surface Hydraulics in Open Channels27 The computation of gradually varied flow equations can be easily carried out on a spreadsheet using Goal Seek function A trapezoidal channel having a bottom width of b = 7.0m and side slopes of m = 1.5, conveys a discharge of Q = 28m 3 /s. The channel has a constant bed slope of S o = The Manning friction coefficient for the channel is n = 0.025m -1/3 s. The channel terminates by a sudden drop of the bed. 1.Determine the type of water surface profile to be expected. 2.Calculate the water surface profile for a reach length of 3200m. Computation of Gradually Varied Flow

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Lecture 4.Engr Introduction to Free-Surface Hydraulics in Open Channels28 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. Rapidly Varied Flow at Channel Transitions

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Lecture 4.Engr Introduction to Free-Surface Hydraulics in Open Channels29 Alternate depths Specific Energy Curve plotted for a constant Q Subcritical flow 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). Use of Specific Energy to study Rapidly Varied Flow at Channel Transitions

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Lecture 4.Engr Introduction to Free-Surface Hydraulics in Open Channels30 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: 1.Subcritical flow over a positive step 2.Supercritical flow over a positive step 3.Subcritical flow over a negative step 4.Supercritical flow over a negative step 5.Subcritical flow through a contraction 6.Supercritical flow through a contraction 7.Subcritical flow through an expansion 8.Supercritical flow through an expansion Side view Top view Rapidly Varied Flow at Channel Transitions

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Lecture 4.Engr Introduction to Free-Surface Hydraulics in Open Channels31 Assume that the head loss due to the step is negligible (the energy grade line remains parallel to the bed). Rapidly Varied Flow: Subcritical Flow over a Positive Step

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Lecture 4.Engr Introduction to Free-Surface Hydraulics in Open Channels32 Assume that the head loss due to the step is negligible (the energy grade line remains parallel to the bed). Rapidly Varied Flow: Supercritical Flow over a Positive Step

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Lecture 4.Engr Introduction to Free-Surface Hydraulics in Open Channels33 Assume that the head loss due to the step is negligible (the energy grade line remains parallel to the bed). Rapidly Varied Flow: Subcritical Flow over a Negative Step

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Lecture 4.Engr Introduction to Free-Surface Hydraulics in Open Channels34 Assume that the head loss due to the step is negligible (the energy grade line remains parallel to the bed). Rapidly Varied Flow: Supercritical Flow over a Negative Step

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Lecture 4.Engr Introduction to Free-Surface Hydraulics in Open Channels35 Assume that the head loss due to contraction is negligible (the energy grade line remains parallel to the bed). Top view Side view Rapidly Varied Flow: Subcritical Flow through a Contraction

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Lecture 4.Engr Introduction to Free-Surface Hydraulics in Open Channels36 Assume that the head loss due to contraction is negligible (the energy grade line remains parallel to the bed). Top view Side view Rapidly Varied Flow: Supercritical Flow through a Contraction

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Lecture 4.Engr Introduction to Free-Surface Hydraulics in Open Channels37 Assume that the head loss due to contraction is negligible (the energy grade line remains parallel to the bed). Top view Side view Rapidly Varied Flow: Subcritical Flow through an expansion

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Lecture 4.Engr Introduction to Free-Surface Hydraulics in Open Channels38 Rapidly Varied Flow: Supercritical Flow through an Expansion Assume that the head loss due to contraction is negligible (the energy grade line remains parallel to the bed). Top view Side view

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Lecture 4.Engr Introduction to Free-Surface Hydraulics in Open Channels39 Rapidly Varied Flow: Special Case of Choked Flow due to a High Positive Step If the step is too high, subtracting z from H s1, 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 h 1 is also an unknown. Condition of critical flow over the step provides the third equation needed for the analysis. By subtracting z from H s1, we cannot fall back onto the specific energy curve initial water surface final water surface At one point on the step the flow goes through the critical depth of the cross section on the step. initial energy line final energy line

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Lecture 4.Engr Introduction to Free-Surface Hydraulics in Open Channels40 Rapidly Varied Flow: Special Case of Choked Flow due to too much Contraction Top view Side view initial water surface final water surface At one point on the contracted section the flow goes through the critical depth of that cross section. With the available energy H s1, we cannot cut the specific energy curve of the contracted section If the step is contracted too much, with the specific energy H s1 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 h 1 is also an unknown. Condition of critical flow at the contracted section provides the third equation needed for the analysis. initial energy line final energy line

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Lecture 4.Engr Introduction to Free-Surface Hydraulics in Open Channels41 Energy Losses For Subritical Flow in Open Channel Transitions Taken from USACE (1994) U.S. Army Corps of Engineers, “Hydraulic Design of Flood Control Channels,” Engineering and Design Manual, EM , July 1991, Change 1 (June 1994). Head losses at contractions and expansions can be calculated using the following expressions: U1U1 U2U2 U1U1 U2U2 ContractionExpansion

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Lecture 4.Engr Introduction to Free-Surface Hydraulics in Open Channels42 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, F P1 and F P2 respectively, and the friction force, F f. The momentum equation can be written as: Assuming a rectangular channel, we have:

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Lecture 4.Engr Introduction to Free-Surface Hydraulics in Open Channels43 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 (h 2 – h 1 ) to get: Only the positive root of the above quadratic equation is physically meaningful: Written in dimensionless form, the above equation becomes: or whereand This equation is called the equation of Bélanger in honor of the French scientist who developed it for the first time.

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Lecture 4.Engr Introduction to Free-Surface Hydraulics in Open Channels44 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: whereas given by Rajaratnam. https://www.fhwa.dot.gov/engineering/hydraulics/pubs/06086/hec14ch06.cfm#fig096 Hydraulic jumps are classified according to the approach flow Froude number. is in degrees

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Lecture 4.Engr Introduction to Free-Surface Hydraulics in Open Channels45 Rapidly Varied Flow: Hydraulic Jump Energy loss across the hydraulic jump: Length of the hydraulic jump: Classification of hydraulic jumps: Undular jump Weak jump Oscillating jump Strong jump Steady jump jump type generally preferred in engineering applications Photos from (Dr. H. Chanson):

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Lecture 4.Engr Introduction to Free-Surface Hydraulics in Open Channels46 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.

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Lecture 4.Engr Introduction to Free-Surface Hydraulics in Open Channels47 Examples of the Use of Hydraulic Jump in Hydraulic Engineering

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Lecture 4.Engr Introduction to Free-Surface Hydraulics in Open Channels48 Design of Stilling Basins USBR Type I Stilling BasinUSBR Type II Stilling Basin USBR Type IV Stilling Basin USBR Type III Stilling Basin SAF Stilling BasinPillari’s Stilling Basin

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Lecture 4.Engr Introduction to Free-Surface Hydraulics in Open Channels49 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

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Lecture 4.Engr Introduction to Free-Surface Hydraulics in Open Channels50 Positioning of a Hydraulic Jump Draw the upstream supercritical flow profile starting from a control section at the upstream. Draw the downstream subcritical flow profile starting from a control section at the downstream. 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.

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Lecture 4.Engr Introduction to Free-Surface Hydraulics in Open Channels51 Oblique Hydraulic Jump Consider again the specific energy curve. Alternate depths Specific Energy Curve plotted for a constant Q Subcritical flow Supercritical flow It can be seen that when the flow is supercritical, a small variation in depth (say h) causes a large variation in kinetic energy and, thus the specific energy ( E s or H s ). Therefore, in supercritical flow, a transition, such as a change in width or a change in direction, will provoke an abrupt variation of flow depth and stationary, stable gravity waves will appear on the free surface.

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Lecture 4.Engr Introduction to Free-Surface Hydraulics in Open Channels52 Oblique Hydraulic Jump Referring to the figure on the left, consider the case of a supercritical deflected by a side wall making an angle with the approach channel. A standing wave front forms making an angle with the approach channel direction. This is called an oblique (hydraulic) jump. Note that this is somewhat different than a classical hydraulic jump due to the fact that the flow is still subcritical downstream of the wave front: and The continuity equation in the direction normal to the wave front gives: Neglecting the bottom friction, the momentum equation in the direction normal to the wave front gives: Froude numbers using the velocity components normal to the oblique wave front are defined as: and

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Lecture 4.Engr Introduction to Free-Surface Hydraulics in Open Channels53 Oblique Hydraulic Jump Note that in the direction tangent to the wave front no momentum change takes place. The equation of momentum in tangential direction becomes: This clearly shows that: Combining continuity and momentum equations, one obtains the equation for change of depth across an oblique jump: In which the Froude number normal to the wave front is defined as: Geometric considerations allow us to write:

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Lecture 4.Engr Introduction to Free-Surface Hydraulics in Open Channels54 The angle of the wave front can be expressed as: Note that, for small variations of depth, thus for gradual transitions, one gets: Combining two equations for change of depth across an oblique jump, we can write: Using equation of continuity and geometric relationships, the equation for the oblique jump can also be written as: These derivations were originally carried out by Ippen (1949). He also experimentally verified the relationship above which gives the relationship between and for contracting channels (only). Oblique Hydraulic Jump

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Lecture 4.Engr Introduction to Free-Surface Hydraulics in Open Channels55 Oblique Hydraulic Jump The relation ship between q and is plotted on the left. Following observations can be made: For all Froude numbers there exists a maximum value for the angle of deflection, max. For all values smaller than max, two values of are possible. However, since the analysis is made for the case the flow remains supercritical after the jump, i.e. Fr 2 > 1, we should consider only the values on the left side (solid lines). It is important to note that, any perturbation created by one wall will be reflected by the other wall and so on. To study this behavior, we will consider two cases: Asymmetrically converging channel, and Symmetrically converging channel.

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Lecture 4.Engr Introduction to Free-Surface Hydraulics in Open Channels56 Reflection of Oblique Jumps in an Asymmetrical Channel Contraction Consider the channel on the left. The left wall is deflected into the flow by an angle , while the right wall remains straight. … and so on. If the contracting channel is sufficiently long, the wave reflection continues until the flow finally becomes subcritical.

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Lecture 4.Engr Introduction to Free-Surface Hydraulics in Open Channels57 Reflection of Oblique Jumps in an Asymmetrical Channel Contraction Consider the channel on the left. The left wall is deflected into the flow by an angle , while the right wall remains straight. … and so on. If the contracting channel is sufficiently long, the wave reflection continues until the flow finally becomes subcritical.

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Lecture 4.Engr Introduction to Free-Surface Hydraulics in Open Channels58 Reflection of Oblique Jumps in an Symmetrical Channel Contraction Causon D. M., C. G. Mingham and D. M. Ingram (1999), Advances in Calculation Methods for Supercritical Flow in Spillway Channels, ASCE, Journal of Hydraulic Engineering, Vol. 125, No. 10, pp View of Supercritical Flow in Curved Transition: (a)Experimental Visualization of Standing Wave Patterns, after Ippen and Dawson (1951); (b)Computer Visualization Based on 2D, Shock-Capturing, Numerical-Model Predictions

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Lecture 4.Engr Introduction to Free-Surface Hydraulics in Open Channels59 Designing a Symmetrical Channel Contraction for Supercritical Flow A good channel contraction design for supercritical flow should reduce or eliminate the undesirable cross wave pattern. This can be achieved by choosing a linear contraction length L T, thus by choosing a contraction angle ’, such that the positive waves emanating from points A and A’, due to converging walls, arrive directly at points D and D’, where negative waves are generated due to diverging walls. Such a design is shown in the figure on the left. The choice of the angle ’ depends on the approach Froude number, Fr 1, and the contraction ratio B 3 /B 1. Based on continuity equation, and assuming that the flow remains supercritical in the contracted section, i.e. Fr 3 > 1, we can write: From geometric considerations, we also have the relationship: The angle ’ to satisfy these two equations is calculated using an iterative procedure.

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Lecture 4.Engr Introduction to Free-Surface Hydraulics in Open Channels60 Gradually Varied Flow with Lateral Inflow We will now consider the case of gradually varied flow with lateral inflow. In the most general case, the discharge added subtracted laterally affects both to the mass and the momentum of the flow. The continuity equation with lateral flow becomes: represents lateral discharge which can be positive if a discharge is added, or negative if a discharge subtracted. We can also write: Consider the gradually varied flow with lateral inflow as shown on the left. Equation of the momentum states that the sum of all forces is equal to the change in momentum: Let us now analyze all the terms in this equation one by one.

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Lecture 4.Engr Introduction to Free-Surface Hydraulics in Open Channels61 The net hydrostatic pressure force will be: z P is the distance from the free surface to the centroid of the flow area A: The weight of the water prism between two sections that are dx apart is: The rightmost side assumes that is small. The friction force can be written as:since The change in the momentum can be written as: Note that Note also that the lateral flow is entering or leaving the channel at an angle and with velocity U ℓ. Let us insert above expression the equation of momentum, and simplify: Gradually Varied Flow with Lateral Inflow

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Lecture 4.Engr Introduction to Free-Surface Hydraulics in Open Channels62 This is the equation of free surface for a steady gradually varied flow with lateral inflow, which is also called a steady spatially varied flow. Recalling that:and Simplifying also second order terms AdA and dQdA, the spatially varied flow equation can be written as: It can be verified that, this equation reduces to gradually varied flow formula when lateral flow is zero, q ℓ = 0. Spatially varied flow equation can also be solved using the same methods for solving gradually varied flow equation. Gradually Varied Flow with Lateral Inflow

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Lecture 4.Engr Introduction to Free-Surface Hydraulics in Open Channels63 Example of Structures for Spatially Varied Flow: Side Channel Spillway Side channel spillway of Hoover Dam in Nevada. Side channel spillway of Hoover Dam in Nevada as seen from the reservoir side.

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Lecture 4.Engr Introduction to Free-Surface Hydraulics in Open Channels64 Example of Structures for Spatially Varied Flow: Side Channel Spillway Side channel spillway of Hoover Dam in Nevada, looking downstream.

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Lecture 4.Engr Introduction to Free-Surface Hydraulics in Open Channels65 Quiz No 1(5 minutes) A steep channel is connected to a mild channel as shown in figure. Both channels have a rectangular cross section. The following data is given: The flow in steep channel is steady and uniform. Determine in which channel, steep or mild, the hydraulic jump will take place.

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Lecture 4.Engr Introduction to Free-Surface Hydraulics in Open Channels66 Quiz No 1(5 minutes): Solution A steep channel is connected to a mild channel as shown in figure. Both channels have a rectangular cross section. The following data is given: The flow in steep channel is steady and uniform. Determine in which channel, steep or mild, the hydraulic jump will take place. Solution: Assume that the steady uniform flow continues all the way down to the point where the slope becomes mild. Let us see if there is a jump at that point what would be the conjugate depth. A jump taking place at the point of slope change will be too strong. It can jump higher than the normal depth in the channel. Therefore, the flow continues into the mild channel without a jump and creates an M3 type profile up to a depth whose conjugate depth is equal to the uniform flow depth of the mild channel.

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Lecture 4.Engr Introduction to Free-Surface Hydraulics in Open Channels67 Quiz No 2 (5 minutes) Consider the channel on the left with the following data: Determine if the flow is choked due to the positive step. What will be the flow depth over the step?

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Lecture 4.Engr Introduction to Free-Surface Hydraulics in Open Channels68 Quiz No 2 (5 minutes): Solution Consider the channel on the left with the following data: Thus the flow is choked. The flow will go through the critical depth over the step, i.e. Solution: Assuming no singular energy losses due to the step, the energy grade line remains at the same level. Over the step, the energy is reduced by an amount z. Determine if the flow is choked due to the positive step. What will be the flow depth over the step?

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