SAI RAJESWARI INSTITUTE OF TECHNOLOGY Lingapuram (V), Proddatur, Y S R District – 516 362, A.P Department of civil Engineering Classification of flows, Types of channels 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 ME33 : Fluid Flow 2 Chapter 13: Open Channel Flow Classification of Open-Channel Flows Open-channel flows are characterized by the presence of a liquid-gas interface called the free surface. Natural flows: rivers, creeks, floods, etc. Human-made systems: fresh-water aqueducts, irrigation, sewers, drainage ditches, etc. ME33 : Fluid Flow 3 Chapter 13: Open Channel Flow 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 streamwise direction Therefore, the flow is 3D Nevertheless, 1D approximation is made with good success for many practical problems. ME33 : Fluid Flow 4 Chapter 13: Open Channel Flow 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 ME33 : Fluid Flow 5 Chapter 13: Open Channel Flow Classification of Open-Channel Flows Obstructions cause the flow depth to vary. Rapidly varied flow (RVF) occurs over a short distance near the obstacle. Gradually varied flow (GVF) occurs over larger distances and usually connects UF and RVF. ME33 : Fluid Flow 6 Chapter 13: Open Channel Flow Classification of Open-Channel Flows The wetted perimeter does not include the free surface. Examples of Rh for common geometries shown in Figure at the left. ME33 : Fluid Flow 7 Chapter 13: Open Channel Flow Froude Number and Wave Speed OC flow is also classified by the Froude number Resembles classification of compressible flow with respect to Mach number ME33 : Fluid Flow 8 Chapter 13: Open Channel Flow Specific Energy For a channel with constant width b, Plot of Es vs. y for constant V and b ME33 : Fluid Flow 9 Chapter 13: Open Channel Flow Specific Energy This plot is very useful Easy to see breakdown of Es into pressure (y) and dynamic (V2/2g) head Es as y 0 Es y for large y Es reaches a minimum called the critical point. There is a minimum Es required to support the given flow rate. Noting that Vc = sqrt(gyc) For a given Es > 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. ME33 : Fluid Flow 10 Chapter 13: Open Channel Flow 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 ME33 : Fluid Flow 11 Chapter 13: Open Channel Flow 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 ME33 : Fluid Flow 12 Chapter 13: Open Channel Flow 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 ME33 : Fluid Flow 13 Chapter 13: Open Channel Flow 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 ME33 : Fluid Flow 14 Chapter 13: Open Channel Flow 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 ME33 : Fluid Flow 15 Chapter 13: Open Channel Flow 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 ME33 : Fluid Flow 16 Chapter 13: Open Channel Flow 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 ME33 : Fluid Flow 17 Chapter 13: Open Channel Flow Gradually Varied Flow 12 distinct configurations for surface profiles in GVF. ME33 : Fluid Flow 18 Chapter 13: Open Channel Flow 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 ME33 : Fluid Flow 19 Chapter 13: Open Channel Flow 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 ME33 : Fluid Flow 20 Chapter 13: Open Channel Flow 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 ME33 : Fluid Flow 21 Chapter 13: Open Channel Flow 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 ME33 : Fluid Flow 22 Chapter 13: Open Channel Flow Rapidly Varied Flow and Hydraulic Jump Experimental studies indicate that hydraulic jumps can be classified into 5 categories, depending upon the upstream Fr ME33 : Fluid Flow 23 Chapter 13: Open Channel Flow