Lecture 04 Open Channel Flow PDF

Open Channel Faculty of Hydrology and Water Resources Engineering Kong Chhuon Institute of Technology of Cambodia  Content ❑Objectives ❑What is Open Channel ❑Types of Open Channel ❑Types of flowing water and control ❑Solution of uniform flow in Open Channel CE 312:...

Open Channel Faculty of Hydrology and Water Resources Engineering Kong Chhuon Institute of Technology of Cambodia  Content ❑Objectives ❑What is Open Channel ❑Types of Open Channel ❑Types of flowing water and control ❑Solution of uniform flow in Open Channel CE 312: Hydraulics 2 Chapter 10: Channel Geometry  Objectives Understand how flow in open channels differs from flow in pipes Learn the different flow regimes in open channels and their characteristics Predict if hydraulic jumps are to occur during flow, and calculate the fraction of energy dissipated during hydraulic jumps Learn how flow rates in open channels are measured using sluice gates and weirs CE 312: Hydraulics 3 Chapter 10: Channel Geometry  Types Of Open Channel Open Channel Natural Channel Artificial Channel Irregular shape Regular shape i.e. : river, hillsides i.e. : drains, culverts, rivulets, tidal estuaries sewer, tunnels CE 312: Hydraulics 4 Chapter 10: Channel Geometry  Types Of Open Channel CE 312: Hydraulics 5 Chapter 10: Channel Geometry  Channel Geometry CE 312: Hydraulics 6 Chapter 10: Channel Geometry  Channel Geometry Channel geometry encompasses all characteristics that define the channel geometry At a point, it includes area, depth, friction, bottom width, top width, side slope, wetted perimeter. For the entire channel reach, it includes length, slope, average A concrete trapezoidal channel CE 312: Hydraulics 7 Chapter 10: Channel Geometry  Channel Geometry 1 + Z2 1  z Cot  = z/1 CE 312: Hydraulics 8 Chapter 10: Channel Geometry  Example CE 312: Hydraulics 9 Chapter 10: Channel Geometry  Classification of Open-Channel Flows In an open channel, Velocity is zero on bottom and sides of channel due to no-slip condition Velocity is maximum at the midplane of the free surface In most cases, velocity also varies in the stream wise direction Therefore, the flow is 3D Nevertheless, 1D approximation is made with good success for many practical problems. CE 312: Hydraulics 10 Chapter 10: Channel Geometry  Classification of Open-Channel Flows Open Open Channel Channel Unsteady Flow Steady Flow ≠0 =0 Uniform Flow Non-Uniform Flow = 0 ≠0 Gradually Rapidly Varied flow Varied Flow i.e : upstream i.e : hydraulic of obstruction jump CE 312: Hydraulics 11 Chapter 10: Channel Geometry  Classification of Open-Channel Flows CE 312: Hydraulics 12 Chapter 10: Channel Geometry CE 312: Hydraulics 13 Chapter 10: Channel Geometry  Classification of Open-Channel Flows Uniform y (depth of water) and v (velocity) remain constant along the channel. Figure 2.0 Figure 2.0 Non - uniform Flow y and v change along the length of the channel CE 312: Hydraulics 14 Chapter 10: Channel Geometry  Classification of Open-Channel Flows Flow in open channels is also classified as being uniform or nonuniform, depending upon the depth y. Uniform flow (UF) encountered in long straight sections where head loss due to friction is balanced by elevation drop. Depth in UF is called normal depth yn CE 312: Hydraulics 15 Chapter 10: Channel Geometry CE 312: Hydraulics 16 Chapter 10: Channel Geometry  Classification of Open-Channel Flows Like pipe flow, OC flow can be laminar, transitional, or turbulent depending upon the value of the Reynolds number Where  = density,  = dynamic viscosity, = kinematic viscosity V = average velocity Rh = Hydraulic Radius = Ac/p Ac = cross-section area P = wetted perimeter Note that Hydraulic Diameter was defined in pipe flows as Dh = 4Ac/p = 4Rh (Dh is not 2Rh, BE Careful!) CE 312: Hydraulics 17 Chapter 10: Channel Geometry  Classification of Open-Channel Flows Flow Classifications 1) Depending on the Reynolds number, Re Laminar Flow (if Re < 500): very slow and shallow flowing water in very smooth open channels. Turbulent Flow (if Re > 1000): ordinary flow in ordinary open channels. Transition Flow (if 500 < Re < 1000) CE 312: Hydraulics 18 Chapter 10: Channel Geometry  Classification of Open-Channel Flows 2) Depending on Froude number, Fr OC flow is also classified by the Froude number Resembles classification of compressible flow with respect to Mach number CE 312: Hydraulics 19 Chapter 10: Channel Geometry  TURBULENT LAMINAR CE 312: Hydraulics 20 Chapter 10: Channel Geometry  Froude Number and Wave Speed Critical depth yc occurs at Fr = 1 At low flow velocities (Fr < 1) Disturbance travels upstream y > yc At high flow velocities (Fr > 1) Disturbance travels downstream y < yc CE 312: Hydraulics 21 Chapter 10: Channel Geometry  Froude Number and Wave Speed Important parameter in study of OC flow is the wave speed c0, which is the speed at which a surface disturbance travels through the liquid. Derivation of c0 for shallow- water Generate wave with plunger Consider control volume (CV) which moves with wave at c0 CE 312: Hydraulics 22 Chapter 10: Channel Geometry  Froude Number and Wave Speed For shallow water, where y Es,min, there are two different depths, or alternating depths, which can occur for a fixed value of Es A small change in Es near the critical point causes a large difference between alternate depths and may cause violent fluctuations in flow level. Operation near this point should be avoided. CE 312: Hydraulics 26 Chapter 10: Channel Geometry  Continuity and Energy Equations 1D steady continuity equation can be expressed as 1D steady energy equation between two stations Head loss hL is expressed as in pipe flow, using the friction factor, and either the hydraulic diameter or radius CE 312: Hydraulics 27 Chapter 10: Channel Geometry  Continuity and Energy Equations The change in elevation head can be written in terms of the bed slope  Introducing the friction slope Sf The energy equation can be written as CE 312: Hydraulics 28 Chapter 10: Channel Geometry  Uniform Flow in Channels Uniform depth occurs when the flow depth (and thus the average flow velocity) remains constant Common in long straight runs Flow depth is called normal depth yn Average flow velocity is called uniform-flow velocity V0 CE 312: Hydraulics 29 Chapter 10: Channel Geometry  Uniform Flow in Channels Uniform depth is maintained as long as the slope, cross-section, and surface roughness of the channel remain unchanged. During uniform flow, the terminal velocity reached, and the head loss equals the elevation drop We can the solve for velocity (or flow rate) Where C is the Chezy coefficient. f is the friction factor determined from the Moody chart or the Colebrook equation CE 312: Hydraulics 30 Chapter 10: Channel Geometry  Best Hydraulic Cross Sections Best hydraulic cross section for an open channel is the one with the minimum wetted perimeter for a specified cross section (or maximum hydraulic radius Rh) Also reflects economy of building structure with smallest perimeter CE 312: Hydraulics 31 Chapter 10: Channel Geometry  Best Hydraulic Cross Sections Example: Rectangular Channel Cross section area, Ac = yb Perimeter, p = b + 2y Solve Ac for b and substitute Taking derivative with respect to To find minimum, set derivative to zero Best rectangular channel has a depth 1/2 of the width CE 312: Hydraulics 32 Chapter 10: Channel Geometry  Best Hydraulic Cross Sections Same analysis can be performed for a trapezoidal channel Similarly, taking the derivative of p with respect to q, shows that the optimum angle is For this angle, the best flow depth is CE 312: Hydraulics 33 Chapter 10: Channel Geometry  Gradually Varied Flow In GVF, y and V vary slowly, and the free surface is stable In contrast to uniform flow, Sf  S0. Now, flow depth reflects the dynamic balance between gravity, shear force, and inertial effects To derive how how the depth varies with x, consider the total head CE 312: Hydraulics 34 Chapter 10: Channel Geometry  Gradually Varied Flow Take the derivative of H Slope dH/dx of the energy line is equal to negative of the friction slope Bed slope has been defined Inserting both S0 and Sf gives CE 312: Hydraulics 35 Chapter 10: Channel Geometry  Gradually Varied Flow Introducing continuity equation, which can be written as Differentiating with respect to x gives Substitute dV/dx back into equation from previous slide, and using definition of the Froude number gives a relationship for the rate of change of depth CE 312: Hydraulics 36 Chapter 10: Channel Geometry  Gradually Varied Flow This result is important. It permits classification of liquid surface profiles as a function of Fr, S0, Sf, and initial conditions. Bed slope S0 is classified as Steep : yn < yc Critical : yn = yc Mild : yn > yc Horizontal : S0 = 0 Adverse : S0 < 0 Initial depth is given a number 1 : y > yn 2 : yn < y < yc 3 : y < yc CE 312: Hydraulics 37 Chapter 10: Channel Geometry  Gradually Varied Flow 12 distinct configurations for surface profiles in GVF. CE 312: Hydraulics 38 Chapter 10: Channel Geometry  Gradually Varied Flow Typical OC system involves several sections of different slopes, with transitions Overall surface profile is made up of individual profiles described on previous slides CE 312: Hydraulics 39 Chapter 10: Channel Geometry  Rapidly Varied Flow and Hydraulic Jump Flow is called rapidly varied flow (RVF) if the flow depth has a large change over a short distance Sluice gates Weirs Waterfalls Abrupt changes in cross section Often characterized by significant 3D and transient effects Backflows Separations CE 312: Hydraulics 40 Chapter 10: Channel Geometry  Rapidly Varied Flow and Hydraulic Jump Consider the CV surrounding the hydraulic jump Assumptions 1. V is constant at sections (1) and (2), and 1 and 2 1 2. P = gy 3. w is negligible relative to the losses that occur during the hydraulic jump 4. Channel is wide and horizontal 5. No external body forces other than gravity CE 312: Hydraulics 41 Chapter 10: Channel Geometry  Rapidly Varied Flow and Hydraulic Jump Continuity equation X momentum equation Substituting and simplifying Quadratic equation for y2/y1 CE 312: Hydraulics 42 Chapter 10: Channel Geometry  Rapidly Varied Flow and Hydraulic Jump Solving the quadratic equation and keeping only the positive root leads to the depth ratio Energy equation for this section can be written as Head loss associated with hydraulic jump CE 312: Hydraulics 43 Chapter 10: Channel Geometry  Rapidly Varied Flow and Hydraulic Jump Often, hydraulic jumps are avoided because they dissipate valuable energy However, in some cases, the energy must be dissipated so that it doesn’t cause damage A measure of performance of a hydraulic jump is its fraction of energy dissipation, or energy dissipation ratio CE 312: Hydraulics 44 Chapter 10: Channel Geometry  Rapidly Varied Flow and Hydraulic Jump Experimental studies indicate that hydraulic jumps can be classified into 5 categories, depending upon the upstream Fr CE 312: Hydraulics 45 Chapter 10: Channel Geometry  Flow Control and Measurement Flow rate in pipes and ducts is controlled by various kinds of valves In OC flows, flow rate is controlled by partially blocking the channel. Weir : liquid flows over device Underflow gate : liquid flows under device These devices can be used to control the flow rate, and to measure it. CE 312: Hydraulics 46 Chapter 10: Channel Geometry  Flow Control and Measurement Underflow Gate Underflow gates are located at Free outflow the bottom of a wall, dam, or open channel Outflow can be either free or drowned In free outflow, downstream flow is supercritical Drowned outflow In the drowned outflow, the liquid jet undergoes a hydraulic jump. Downstream flow is subcritical. CE 312: Hydraulics 47 Chapter 10: Channel Geometry  Flow Control and Measurement Underflow Gate Schematic of flow depth-specific Es remains constant for energy diagram for flow through idealized gates with underflow gates negligible frictional effects Es decreases for real gates Downstream is supercritical for free outflow (2b) Downstream is subcritical for drowned outflow (2c) CE 312: Hydraulics 48 Chapter 10: Channel Geometry  Flow Control and Measurement Overflow Gate Specific energy over a bump at station 2 Es,2 can be manipulated to give This equation has 2 positive solutions, which depend upon upstream flow. CE 312: Hydraulics 49 Chapter 10: Channel Geometry  Flow Control and Measurement Broad-Crested Weir Flow over a sufficiently high obstruction in an open channel is always critical When placed intentionally in an open channel to measure the flow rate, they are called weirs CE 312: Hydraulics 50 Chapter 10: Channel Geometry  Flow Control and Measurement Sharp-Crested V-notch Weirs Vertical plate placed in a channel that forces the liquid to flow through an opening to measure the flow rate Upstream flow is subcritical and becomes critical as it approaches the weir Liquid discharges as a supercritical flow stream that resembles a free jet CE 312: Hydraulics 51 Chapter 10: Channel Geometry  Flow Control and Measurement Sharp-Crested V-notch Weirs Flow rate equations can be derived using energy equation and definition of flow rate, and experimental for determining discharge coefficients Sharp-crested weir V-notch weir where Cwd typically ranges between 0.58 and 0.62 CE 312: Hydraulics 52 Chapter 10: Channel Geometry