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

1. ENGR 691 – 73: Introduction to Free-Surface Hydraulics in Open Channels Lecture 03: Conservation Laws Energy Equation and Critical Depth Uniform Flow and Normal Depth 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 • Review of Reynolds Transport Theorem, Control Volume, and Conservation Laws • Concept of Energy in Open Channel Flow • Energy equation for Open Channel Flow • Specific Energy Curve and Specific Discharge Curve • Critical Depth and its Computation • Uniform Flow and Normal Depth • Computation of Uniform Flow • Friction Coefficient • Chezy and Manning Coefficients

3. System vis-a-vis Control Volume System: A particular collection of matter, which is identified and viewed as being separated from everything external to the system by an imagined or real closed boundary. Control Volume: A volume in space through whose boundary matter, mass, momentum, energy, and the like can flow. Its boundary is called a control surface. The control volume may be of any useful size (finite and infinitesimal) and shape; the control surface is a closed boundary. Inertial Reference Frame: a frame of reference that describes time homogeneously and space homogeneously, isotropically, and in a time independent manner. All inertial frames are in a state of constant, rectilinear motion with respect to one another; they are not accelerating. Noninertial Reference Frame: a frame of reference that is under acceleration Intensive Property: does not depend on the system size or the amount of material in the system, e.g., Extensive Property: directly proportional to the system size or the amount of material in the system, e.g.,

4. Conservation of Mass: The 1-D Continuity Equation (without free surface) A Fluid System: All the matter (fluid) within control volume (I+R) at time t, but within the control volume (R+O) at time t+dt Control Volume: a volume fixed in space (between Section 1 and 2) From the conservation of system mass: Then, Mass flowrate = Volume flowrate If density variation is negligible

5. The 1-D Continuity Equation for Open Channel Flows

6. Review: Derivation of Reynolds Transport Theorem Time t Time t+Dt System System Control surface Control surface Quantity of property B present in the fluid system at time t Quantity of property B present in the fluid system at time t+Dt Quantity of property B present in the control volume at time t Quantity of property B present in the control volume at time t+Dt Quantity of property B which entered the control volume through the control surface during the time interval Dt Quantity of property B which left the control volume through the control surface during the time interval Dt Total change in the quantity of property B in the fluid system during the time interval Dt Total change in the quantity of property B in the control volume during the time interval Dt

7. System Control surface Review: Derivation of Reynolds Transport Theorem Time t Time t+Dt At time t, the control volume and the fluid system coincide At time t+Dt, the quantity of B present in the system is Total change in the quantity of property B in the fluid system during the time interval Dt Total change in the quantity of property B in the control volume during the time interval Dt Combining the first three relationships the change in B in the fluid system is written as Divide both sides of equation by Dt At the limit Dt  0 The material derivative operator “D/Dt” underlines the fact that the derivative applies to a fluid system moving in the coordinate system (derivative contains both local and convective changes).

8. System Control surface Review: Derivation of Reynolds Transport Theorem Time t Time t+Dt The interpretation of different terms : Recalling the definition of extensive property: and Using this definition we can write: The integral on the right hand side is carried out over the control volume which is invariant in time. Therefore, one can bring the derivative sign inside the integral

9. System Control surface Review: Derivation of Reynolds Transport Theorem Time t Time t+Dt Now let us take a look at the last term on the right hand side more closely. We have already shown that the net efflux of property B through the control surface can be expressed as: Considering that where is the vector normal to the surface element dA We have, therefore Finally Reynolds transport theorem in its general form

10. sv = System Volume cv = Control Volume cs = Control Surface Review: Conservation of Mass Reynolds Transport Theorem Recall : Extensive property : System Mass Intensive property : Reynolds Transport Theorem for mass Equation for Conservation of Mass  Continuity Equation Equation for Conservation of Mass  Continuity Equation If the flow issteady or of uniform density (i.e. time derivative of density is equal to zero)

11. sv = System Volume cv = Control Volume cs = Control Surface Review: Conservation of (Linear) Momentum Reynolds Transport Theorem Recall : Extensive property : System Momentum Intensive property : Reynolds Transport Theorem for Momentum Momentum of the fluid system Recall from physics: The rate change of change of Momentum for a system is equal to the net sum of the external forces acting on the system. demonstration Equation for Conservation of Momentum  Momentum Equation If the flow is steady (i.e. time derivatives are equal to zero)

12. sv = System Volume cv = Control Volume cs = Control Surface Review: Conservation of Energy Reynolds Transport Theorem Recall : Extensive property : System Total Energy Intensive property : Reynolds Transport Theorem for Energy Recall from physics: The 1st principle of thermodynamics. Rate of change of Total Work accomplished by the system Rate of change of Total Energy of the System Rate of change of Net Heat Efflux (heat entering and/or leaving the system)

13. Review: Conservation of Energy Using the 1st equation of thermodynamics we can now write: Equation for Conservation of Energy  Energy Equation • The terms on the left side of the energy equation are written in a very general context. Let us now analyze these two terms in more detail: • Rate of change of work done on the fluid contained in the system, dW/dt, and • Rate of change of heat energy in the fluid system, dQ/dt.

14. Review: Conservation of Energy Rate of change of work, dW/dt In a fluid system the work can be done in two ways: Work due to a mechanical device (shaft) Work done by the flow of fluid, i.e. pressure forces Positive when fluid does work on machine and negative when work is done on the fluid by a machine Pump is a mechanical device which does work on a fluid system to increase its energy (negative sign) Turbine is a mechanical device on which the fluid system does work and looses some of its energy (positive sign)

15. Review: Conservation of Energy Rate of change of heat energy, dQ/dt If heat is added to the system: If heat is extracted from the system: Let us investigate the physical interpretation of the change of variation of the heat energy of the system. We will consider two cases: Case of Ideal Fluid: Case of Real Fluid: Ideal Fluid is defined as a nonviscous fluid. In reality all fluids are viscous. Ideal fluid is a simplification of the reality. Real Fluid is defined as a viscous fluid. In reality all fluids are viscous. In case of an ideal fluid, if the flow process is adiabatic (i.e. no energy is transferred in or out of fluid system) the internal energy of the fluid system remains constant. In case of a real fluid, even if the flow process is adiabatic (i.e. no energy is transferred in or out of fluid system) the internal energy of the fluid system decreases. Ideal fluid does not experience any internal energy loss, since there is no friction. The reason for this is the loss of a portion of the mechanical energy by conversion into heat (due to internal friction and friction with the surroundings). The lost mechanical energy cannot be recovered by the flow, and it is forever lost.

16. Review: Conservation of Energy By combining all together : Equation for Conservation of Energy  Energy Equation If the flow is steady (i.e. time derivatives are equal to zero) Specific enthalpy (enthalpy per unit mass)

17. Concept of Energy in Open Channel Flow We will get back to these notions later. Let us now start discussing the concept of energy for open channel flow. We will first introduce the definition of energy Then we will look into difference energy between two cross sections. This will be used to derive an equation of energy for open channel flow.

18. Equation of Energy for Open Channel Flow Let us consider the open channel flow on the left. The total energy for the fluid element at point P, located at elevation zP, where local velocity is u, can be written as: Velocity head, i.e. energy per unit weight of fluid Pressure head Elevation of point P, i.e. potential energy Total mechanical energy head or simply “total head” Important note:If the pressure distribution over the depth h is hydrostatic, the piezometric head is constant along the direction normal to the bed. Piezometric head

19. Equation of Energy for Open Channel Flow Note that the pressure head at the bottom of the channel, i.e. zP = z, can be written as: with For we have thus If we consider an ideal fluid, inviscid fluid with no friction losses, the velocity is constant over the depth, u(z) = U, we have then If we consider a real fluid, viscous fluid with friction losses, the velocity has a distribution over the depth u(z) = U+f(z); we have then where is the “kinetic energy correction coefficient”, which accounts for the non-uniform velocity distribution.

20. Energy Correction Coefficient Refer to Open Channel Flow (M.H. Chaudhry) on Page 12

21. Equation of Energy for Open Channel Flow Let us now write the equation of energy between two cross sections: Consider the flow of a real fluid in an open channel as shown in the figure. The conservation of energy between cross sections  and  can be written as: Total head loss between  and  Total energy at  Total energy at  Referring to the figure let us write the above equation in a more explicit form: where Head loss (or energy loss per unit weight) due to friction (this is also called linear head loss or regular head loss) Head loss (or energy loss per unit weight) due to acceleration in the flow in x-direction

22. Equation of Energy for Open Channel Flow Simplifying the previous equation, we have or Head difference between  and  Total head loss between  and  The above equations are the energy equations for unsteady non-uniform flow. They express the conservation of energy between two cross sections. Note that the energy slope Se = hr/dxand the bed slope Sf = -dz/dx., it can also be written as follows: So far we have not proposed any method to calculate energy loss due to friction. This point will be developed later in detail and various methods will be discussed. The energy equation for unsteady non uniform flow developed above can be manipulated to obtain the equation of conservation of (linear) momentum for unsteady non uniform flow, which is also called dynamic equation of open channel flow. This is what we propose to do in the next slide.

23. ref. line Concept of Specific Energy Let us consider the energy equation for a steady flow: Total Energy Specific Energy: Specific energy is the total mechanic energy with respect to the local invert elevation of the channel. Note that since we can also write For a given cross section, the flow area, A, is a function of h; therefore, the specific energy is a function of Q and h. h as a function of Hs for Q = constant Specific Energy Curve We can thus study the variation of h as a function of Q for Hs = constant Specific Discharge Curve

24. Specific Energy Curve We wish to plot the Specific Energy Curve (i.e. h as a function of Hsfor constant Q) : for we have therefore One immediately sees that the curve has two asymptotes: for we have therefore In addition, for a given Q, the curve has a minimum value, Hsc. We will see about this minimum later in detail. Some observations impose: • For a given Hs, there are always (except when Hs = Hsc) two depths h1 and h2. They are called alternate depths. • The depth corresponding to the minimum specific energy, Hsc, is called critical depth, hc. • Minimum specific energy, Hsc, increases with increasing discharge, Q. • There are three possible flow regimes: subcritical (h > hc), critical (h = hc), and supercritical (h < hc).

25. Critical Depth and Specific Energy The critical depth hc, can be investigated by taking the derivative of specific energy, Hs, with respect to the depth h, and then equating it to zero (point of minimum); i.e. dHs/dh = 0. Specific Energy because Therefore : Let us work on this equation to see what it means: This shows that the critical flow condition (h = hc and Hs is minimum) is reached when Froude number is equal to one.

26. How to plot the specific energy curve for a cross section It is important to note that for some h = hc, the specific energy curve is at its minimum value. Curve plotted for a constant Q • To plot the specific energy curve: • Select a discharge Q • Assume an h value • Calculate A knowing h • Calculate U= Q/A • Calculate U2/2g • Calculate Hs = h + U2/2g • Repeat steps 2 to 6 by assuming other h values. Subcritical flow Alternate depths Supercritical flow The specific energy, Hs, which is always measured with respect to the channel bed, is composed of pressure energy (h) and kinetic energy (V2/2g). Specific Energy

27. Specific Discharge Curve for a Cross Section Specific Energy Curve plotted for For a rectangular section Instead of plotting hvsHs for a constant discharge Q (or q), i.e. specific energy curve, one can also plot hvsQ (or q) for a constant Hs. This will be called specific discharge curve.

28. Critical Depth and Specific Discharge Curve for we have We can easily see that for we have Discharge curve has a maximum, Qmax, for Since and The expression is zero if We can write For a rectangular channel For a triangular channel For a parabolic channel

29. Critical Depth and its Importance Critical depth, hc, in a channel is the flow depth at which: • The specific energy is minimal, Hsc, for a given discharge, Q, • The discharge is maximal, Qmax, for a given specific energy Hsc. For critical flow in a channel: Recall that: The average velocity corresponding to the critical depth is : or For critical flow in a channel, the velocity head is equal to half of the hydraulic depth One can also write: Propagation velocity of small perturbations in still water of depth h Critical velocity is given by Subcritical flow Flow regimes can be classified according to Fr Critical flow Supercritical flow

30. Critical Depth for the Special Case of Rectangular Channel In a rectangular channel, we have Recall that the critical depth, hc, in a rectangular channel is given by: or Using the definition of unit discharge: One obtains: This is valid for a rectangular channel The maximum unit discharge, q, which may exist in a rectangular channel is: Critical flow is unstable and, generally, it cannot be maintained over a long distance. Critical flow is rather a local phenomenon. For a given cross section shape, the critical depth depends only on discharge. This property is exploited to design flow measuring methods and devices in open channels.

31. Example: Plotting a Specific Energy Curve A trapezoidal channel has a bottom width of b = 3.0m and side slopes of m = 1.5. Calculate and plot the specific energy curve for a discharge of Q = 2.0m3/s. Specific energy is defined as: The calculation of Hs for different h was carried out on an MS Excel spreadsheet as shown on the left. The calculated values are plotted below:

32. Example: Plotting a Specific Discharge Curve Solved Problem 14.2 A trapezoidal channel has a bottom width of b = 3.0m and side slopes of m = 1.5. Calculate and plot the specific discharge curve for a specific energy of Hs = 0.6m. Specific energy is defined as: The calculation of Q for different h was carried out on an MS Excel spreadsheet as shown on the left. The calculated values are plotted below:

33. Hydraulic Jump • Refer to Open Channel Flow (M.H. Chaudhry) on Page 43

34. Homework Open-Channel Flow, 2nd Edition, by M.H. Chaurdhry • Problems 2.11, 2.12, 2.19, and 2.24

35. Critical Flows in Different Types of Channels • Refer to Chapter 3, Open Channel Flow (M.H. Chaudhry) on Pages 55-85

36. Concept of Uniform Flow Now we will introduce an important concept: The Uniform Flow In relation with uniform flow, we will also define: Normal Depth or Uniform Flow Depth

37. Pioneers who Introduced the Concept of Uniform Flow Willi Hager (2003) : “Hydraulicians in Europe, 1800-2000”; IAHR Monograph, IAHR, Delft, Netherlands Antoine de Chézy born at Chalon-sur-Marne, France, on September 1, 1718, died on October 4, 1798 Robert Manning born on Oct 22, 1816 in Normandy, died on Dec 9, 1897 in Dublin Strickler born on July 27, 1887 in Wädensville, died on Feb 1, 1963 in Küsnacht Chézy was given the task to determine the cross section and the related discharge for a proposed canal on the river Yvette, which is close to Paris, but at a higher elevation. Since 1769, he was collecting experimental data from the canal of Courpalet and from the river Seine. His studies and conclusions are contained in a report to Mr. Perronet dated October 21, 1775. The original document, written in French, is titled "Thesis on the velocity of the flow in a given ditch," and it is signed by Mr. Chézy, General Inspector of des Ponts et Chaussées At the age of 30, Robert Manning entered the service of the commissioners of public works to work on the projects of arterial drainage. In 1855 he started his own business and was involved in harbor works in Dundrum. In 1869, he returned to the public service and was promoted chief engineer in 1874. In 1880 he was in charge of the improvement of river Shannon and later he worked on fishery piers. Manning retired in 1881. He developed the formula that bears his name from Ganguillet-Kutter formula based on the data by Henry Basin. Obtained his diploma of mechanical engineering at ETH Zurich in 1916. He submitted a Ph.D. thesis related to turbine design. He was appointed head of section in Federal Water Resources Office, where he was involved with low head power plants. In 1928 he was elected the director of the Swiss Power Transmission Society in Bern. Later he founded a private company and worked on projects in eastern Switzerland. He is well known for his uniform flow formula that he established using his own data and data from literature. http://chezy.sdsu.edu/

38. Q h = ? h = ? Concept of Uniform Flow Consider a channel defined by the following characteristics : Cross section shape Bed slope (S = tga) The roughness of the bed (ks) i.e. the relationships: A = f(h), B = f(h), and P = f(h) Assume also that the channel is sufficiently long. a The question is: what will be the flow depth in the channel for a given discharge ? To answer this question we must consider the equilibrium between the forces driving the flow (gravitational force) and forces resisting the flow (friction due to viscous forces). The flow depth will become constant when an equilibrium is reached between driving and resisting forces (i.e. no net force is acting on the flow).

39. Equilibrium of all forces in the flow direction (no acceleration) with with and Concept of Uniform Flow Consider a prismatic open channel (the section does not change along the flow direction) with So Consider a free-surface flow of constant depth in this channel i.e. the free surface is parallel to the bed small

40. Concept of Uniform Flow with remember also by rearranging terms, we get which is Darcy-Weissbach eqn It tells us that in case of uniform flow the slope of the energy gradient line (right hand side of Darcy-Weissbach eqn) is also parallel to the bed slope, So. In uniform flow in an open channel, the water surface, the bed and the energy line are all parallel to each other.

41. Methods for Computing Uniform Flow • Several methods are available for calculating the uniform flow in an open channel: • Using Darcy-Weissbach equation and friction factor, • Using Chezy equation, and • Using Manning-Strickler equation. • We will now study these three methods is detail.

42. Uniform Flow Calculation Using Darcy-Weisbach Equation The Darcy-Weissbach equation for open channel flow is given as: This equation can be rewritten as: since We can also write In this equation both hydraulic radius and flow area are functions of depth h: A= f(h) and Rh= f(h) The friction coefficient can be computed either using Moody-Stanton diagram or Colebrook and White equation. Colebrook and White equation for friction coefficient in pipes (for all flow regimes) was adapted for open channel flows (valid for all regimes) by Silberman et al. (1963) as follows: with and Reynolds number is computed as:

43. Uniform Flow Calculation Using Darcy-Weisbach Equation Uniform flow problems can be solved by solving the following two equations simultaneously: START Read Q, cross section data, ks, and So Estimate hn and Calculate A, P, Rh, and ks/Rh U = Q/A and Re = 4URh/n Calculate discharge with Estimate f Note that if it is required to solve for the flow depth for a given discharge and cross section geometry, a trial and error procedure, such as the one shown on the right, must be used. Use Colebrook and White equation to calculate f’ is Q = Qc ? no Use Colebrook and White equation to calculate f’ The trial and error procedure has two loops. The outer loop iterates the value of h until we reach the normal depth hn. The criteria for stopping the iteration is that the computed discharge is equal to the given discharge. The inner loop finds the value of f iteratively. The criteria for stopping the iteration is that the computed friction factor is equal to the estimated friction factor. yes is f = f’ ? Output the results: hn, A, P, Rh, ks/Rh Re, f, Q, U yes no take f = f’ END

44. Uniform Flow Calculation Using Darcy-Weisbach Equation In rough channels of large width, Rh = h, the friction coefficient, f , can be obtained making in situ measurements of two point velocities and assuming a logarithmic velocity distribution: It is customary to measure and use point velocities at 0.2h and 0.8h. Eliminating u* from these two equations, one obtains: with The expression for the average velocity for turbulent rough flow in a wide channel (Rh ≈ h) is: The expression for the frcition coefficient for turbulent rough flow in a wide channel (Rh ≈ h) is:

45. Uniform Flow Calculation Using Darcy-Weisbach Equation The table given below is not exhaustive. Consult other references for a more detailed table. The values given in the table are for circular industrial pipes. However, they are generally assumed to be valid also for openc channel flows. Since open channel cross sections are not circular in general, a correction factor must be used to multiply the hydraulic radius. This correction factor takes into account the influence of the shape of the channel. Rectangular cross section (B = 2h) Large trapezoidal cross section Triangular (equilateral) cross section Using these corrections, in the formulate replace Rh by fRh.

46. Uniform Flow Calculation Using Darcy-Weisbach Equation • We begin by dividing the flow into several regions: • The viscous sublayer (~0.00 ≤ z’/h ≤ ~0.05) is where the viscous forces are dominant. The velocity profile varies linearly with the distance from the bed. • The inner region (~0.05 ≤ z’/h ≤ ~0.2) is where the turbulence production is important. The length and velocity scales are n/u* and u*, respectively, • The outer region (z’/h ≥ ~0.6) is where the free surface properties are important. The length and velocity scales are flow depth h and maximum flow velocity Uc, respectively, • The intermediate region (~0.2 ≤ z’/h ≤ ~0.6) is where turbulent energy production and dissipation are approximately equal. Outer region Outer region Intermediate region Inner region Inner region Viscous sublayer • From here on, however, we will assume that there are only two layers: • The inner region will be assumed to include also the viscous sublayer. The inner region, therefore, is defined as: ~0.00 ≤ z’/h ≤ ~0.20 , • The outer region will be assumed to include also the intermediate region. The outer region is, therefore, defined as: ~0.2 ≤ z’/h ≤ 1.00 .

47. Uniform Flow Calculation Using Darcy-Weisbach Equation Without going into details, the derivation of the velocity profile for the inner region leads to: This equation is called law of the wall, or logarithmic velocity profile. It is only valid in the inner region. It is important to remember that it has been derived by assuming that the longitudinal pressure distribution is negligible and the shear stress is constant and equal to the wall shear stress over the entire inner region. The integration constant C needs to be determined experimentally. To summarize, inn the inner region the velocity has the following functional relationship: In the above equation, ks represents Nikuradze’s equivalent sand roughness, which can be interpreted as the characteristics length scale corresponding to the height of the roughness elements. Note: Although it is not correct, for simplicity, sometimes the logarithmic velocity profile is assumed to apply over the entire flow depth.

48. Flow Regimes and Friction Coefficient The conclusion of Nikuradse’s experiments was that there is no unique relationship between friction factor, f , Reynolds number, VD/n, and the relative pipe roughness, ks /D. Different relationships must be used for different flow types. The classification of flow types is done using “Reynolds number” and “shear Reynolds number” as criteria. Shear velocity Reynolds number roughness Shear Reynolds Number Re < 2000 Laminar flow 2000 < Re < 3000 Transition flow Hydraulic smooth Re > 3000 Turbulent flow 1st level of classification Hydraulic transition Hydraulic rough 2nd level of classification

49. Friction Coefficient Formulae for Different Flow Regimes Laminar Flow Turbulent Smooth Flow Turbulent Rough Flow Colebrook-White Formula Swamee and Jain Formula Darcy-Weisbach equation for head loss

50. Developing a Diagram for Friction Coefficient The Darcy-Weisbach equation was not made universally useful until the development of the Moody diagram (Moody, 1944) based on the work of Hunter Rouse. Rouse always felt that Moody was given too much credit for what he himself and others did (http://biosystems.okstate.edu/darcy/DarcyWeisbach/Darcy-WeisbachHistory.htm) Sir Thomas Ernest Stanton December 12, 1865, Atherstone, GB August 30, 1931, Eastbourne, GB Lewis F. Moody Professor of Hydraulic Engineering, Princeton University Received his BSc from Owen’s College , Manchester, in 1891 and worked as assistant to Osborne Reynolds. In 1896 took a position of lecturer in engineering at the University College, Liverpool, together with Henry S. Hele-Shaw. Submitted his Ph.D. Thesis in 1898 and became professor of civil and mechanical engineering at Bristol University in 1899. In 1901, he was appointed superintendent of the newly inaugurated National Physical Laboratory, Teddington, where he stayed until his retirment in 1930. He did research on sterngth of materials, lubrication, heat transmission, and hydrodynamics. His main contribution is his 1914 paper with J.R. Pannell: “Similarity relations of motion in relation to the surface friction of fluids, Philosophical Transactions 214: 199-224”. Stantaon received numerous prizes. He became a fellow of the Royal Society in 1914. He was knighted in 1928. He drowned in the sea near Pevensey. The current form of the Moody-Stanton diagram (or chart) was proposed by Moody in his paper: “Moody, L. F., 1944. Friction factors for pipe flow. Transactions of the ASME, Vol. 66”. Willi H. Hager (2003) :”Hydraulicians in Europe, 1800-2000”, IAHR Monograph. Published by IAHR, Delft, The Netherlands * http://biosystems.okstate.edu/darcy/DarcyWeisbach/Darcy-WeisbachHistory.htm