Report

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: [email protected] Course Notes by: Mustafa S. Altinakar and Yan Ding Lecture 03. Engr 691-73 Introduction to Free-Surface Hydraulics in Open Channels 1 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 Lecture 03. Engr 691-73 Introduction to Free-Surface Hydraulics in Open Channels 2 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., B M Extensive Property: directly proportional to the system size or the amount of M B b 1 material in the system, e.g., M M sys sys sys Lecture 03. sys Engr 691-73 Introduction to Free-Surface Hydraulics in Open Channels 3 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: ( m I m R ) t ( m R m O ) t dt ( m I ) t ( m O ) t dt ( m I ) t 1 A1 ds1 ( m O ) t dt 2 A2 ds 2 1 A1 Then, ds1 dt 2 A2 ds 2 dt Mass flowrate = 1 A1V1 2 A2V 2 Volume flowrate Q AV Q A1V1 A2V 2 If density variation is negligible Lecture 03. Engr 691-73 Introduction to Free-Surface Hydraulics in Open Channels 4 The 1-D Continuity Equation for Open Channel Flows Lecture 03. Engr 691-73 Introduction to Free-Surface Hydraulics in Open Channels 5 Review: Derivation of Reynolds Transport Theorem D B out D B in Time t System Control surface System Control surface m Quantity of property B present in the fluid system at time t m Quantity of property B present in the fluid system at time t+Dt Bt B t Dt Time t+Dt Quantity of property B present in the control volume at time t Bt B t Dt 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 D B out Quantity of property B which left the control volume through the control surface during the time interval Dt DB m Total change in the quantity of property B in the fluid system during the time interval Dt D B in DB Lecture 03. Total change in the quantity of property B in the control volume during the time interval Dt Engr 691-73 Introduction to Free-Surface Hydraulics in Open Channels 6 Review: Derivation of Reynolds Transport Theorem System Control surface D B out D B in Bt Bt m Time t Time t+Dt At time t, the control volume and the fluid system coincide Bt Bt At time t+Dt, the quantity of B present in the system is B t D t B t D t D B out D B in m m Total change in the quantity of property B in the fluid system during the time interval Dt DB Total change in the quantity of property B in the control volume during the time interval Dt D B B t Dt B t Divide both sides of equation by Dt At the limit Dt 0 DB m Dt DB B m Dt t Dt B Bt Dt t DB D B out DB out D B in Dt DB Dt B t Dt B t DB Combining the first three relationships the change in B in the fluid system is written as m m D B m m B t D t D B out D B in B t out D B in Dt in Dt 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). Lecture 03. Engr 691-73 Introduction to Free-Surface Hydraulics in Open Channels 7 Review: Derivation of Reynolds Transport Theorem System Control surface D B out D B in Bt Bt m Time t The interpretation of different terms : Recalling the definition of extensive property: B Time t+Dt DB Dt rate of change of B in the fluid system b dm B m D B out D B in t Dt rate of change net efflux of of B in the B through the control volume control surface b d B m b d B and s M b d cv DB DB out in b d b d Using this definition we can write: Dt t Dt s cv 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 D b d b d Dt t s cv D Lecture 03. DB DB out in Dt cv t b d D B out D B in Dt Engr 691-73 Introduction to Free-Surface Hydraulics in Open Channels 8 Review: Derivation of Reynolds Transport Theorem System Control surface D B out V B m t B V t D B in n Time t n Time t+Dt (V n ) C S . o u t 0 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: D B out D t b V dA D t b V n dA cs . out cs . out Considering that d A n dA where n is the vector normal to the surface element dA (V n ) C S .in 0 D B in D t b V dA D t b V n dA cs .in cs .in We have, therefore D B out D B in Dt Finally D Dt Lecture 03. b V n dA ( C S . out b d s b V n dA ) C S . in cv C S . out b d b t CS V n dA b V n dA b V n dA C S . in b V n dA CS Reynolds transport theorem in its general form Engr 691-73 Introduction to Free-Surface Hydraulics in Open Channels 9 Review: Conservation of Mass Recall : D Reynolds Transport Theorem Dt b d sv t b d b V n dA cv CS B M sys Extensive property : System Mass Reynolds Transport Theorem for mass Intensive property : D Dt d SV CV t sv = System Volume cv = Control Volume cs = Control Surface B b M sys M sys 1 M sys V n dA d CS M sys Const . 0 Equation for Conservation of Mass Continuity Equation 0 CV Equation for Conservation of Mass Continuity Equation CV If the flow is steady or of uniform density (i.e. time derivative of density is equal to zero) Lecture 03. 0 t t V n dA d d CS V n dA CS V n dA CS Engr 691-73 Introduction to Free-Surface Hydraulics in Open Channels 10 Review: Conservation of (Linear) Momentum Recall : Reynolds Transport Theorem D Dt Extensive property : System Momentum b SV CV B M sys Reynolds Transport Theorem for Momentum d t V D CS d sv t V d V cv D M sys Dt Equation for Conservation of Momentum Momentum Equation M sys V M sys V n dA Momentum of the fluid system dM V dV sys F M sys a M sys dt dt F demonstration F CV If the flow is steady (i.e. time derivatives are equal to zero) B M sys V CS M sys 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. V n dA Intensive property : b V Dt b d b sv = System Volume cv = Control Volume cs = Control Surface F t V V d V V n dA CS V n dA CS Lecture 03. Engr 691-73 Introduction to Free-Surface Hydraulics in Open Channels 11 Review: Conservation of Energy Recall : D Reynolds Transport Theorem Dt b B b M sys t b d b V n dA d sv cv B E E u Extensive property : System Total Energy Intensive property : E M sys Eu Ep M sys D Reynolds Transport Theorem for Energy Dt M sys e CS sv = System Volume cv = Control Volume cs = Control Surface 2 V E p E k M sys u M sys g z M sys 2 2 V Ek e u e p e k e u gz M sys 2 d sv t e d e V n dA cv CS E sys Recall from physics: The 1st principle of thermodynamics. DE sys Dt DQ Dt DW Dt Rate of change of Total Work accomplished by the system Rate of change of Total Energy of the System Lecture 03. Rate of change of Net Heat Efflux (heat entering and/or leaving the system) Engr 691-73 Introduction to Free-Surface Hydraulics in Open Channels 12 Review: Conservation of Energy Using the 1st equation of thermodynamics we can now write: dQ Equation for Conservation of Energy Energy Equation dQ dt dQ dt dW dt dW dt t e u ep ep d cv cv e u dt dW dt t e d e V n dA cv CS e p e p V n dA CS 2 V u gz t 2 d 2 V u gz 2 CS V n dA 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: 1.Rate of change of work done on the fluid contained in the system, dW/dt, and 2.Rate of change of heat energy in the fluid system, dQ/dt. Lecture 03. Engr 691-73 Introduction to Free-Surface Hydraulics in Open Channels 13 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) W Positive when fluid does work on machine and negative when work is done on the fluid by a machine Wp dt Lecture 03. dW s dt p V dA CS Wt Pump is a mechanical device which does work on a fluid system to increase its energy (negative sign) dW Wf Work done by the flow of fluid, i.e. pressure forces Wf Ws Ws p (V n ) dA CS Turbine is a mechanical device on which the fluid system does work and looses some of its energy (positive sign) dW f dW dt dt dW p dt Engr 691-73 Introduction to Free-Surface Hydraulics in Open Channels dW t dt p V n dA CS 14 Review: Conservation of Energy Rate of change of heat energy, dQ/dt dQ If heat is added to the system: 0 dt If heat is extracted from the system: dQ 0 dt 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: Ideal Fluid is defined as a nonviscous fluid. In reality all fluids are viscous. Ideal fluid is a simplification of the reality. Case of Real Fluid: 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. Lecture 03. Engr 691-73 Introduction to Free-Surface Hydraulics in Open Channels 15 Review: Conservation of Energy dQ dW p dt dt dQ dW p dt dt dW t dt dW t dt 2 2 V V t u gz 2 d u gz 2 CS CV CS 2 2 V V V n dA u gz d u gz t 2 2 CS dW p dW t dt dt dt By combining all together : dQ p V n dA CS CV CV p V n dA 2 V u gz t 2 d 2 V u gz 2 CS V n dA Equation for Conservation of Energy Energy Equation If the flow is steady (i.e. time derivatives are equal to zero) dW p dt dt dQ dW p dt dt dW t dt cv 2 V u gz t 2 dt CS dW t p V n dA CS d p V n dA 2 V p u gz 2 CS V n dA V n dA 2 2 u p gz V Specific enthalpy (enthalpy per unit mass) Lecture 03. CS dQ V n dA Engr 691-73 Introduction to Free-Surface Hydraulics in Open Channels h u p 16 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. Lecture 03. Engr 691-73 Introduction to Free-Surface Hydraulics in Open Channels 17 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: u 2 2g u p zP u pt Const Velocity head, i.e. energy per unit weight of fluid Pressure head Elevation of point P, i.e. potential energy 2 2g p zP 2 2g p p zP zP Lecture 03. p* pt H Total mechanical energy head or simply “total head” Piezometric 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. Engr 691-73 Introduction to Free-Surface Hydraulics in Open Channels 18 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: p h cos b with For dz tan dx 6 Sf or we have cos 1 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 U 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 U Lecture 03. thus p h b 2 h z pt h z pt 2g e 2 2g H Const H Const 3 1 u dA where e A AU QU 1 S f 0 .1 2 u A 3 dA is the “kinetic energy correction coefficient”, which accounts for the non-uniform velocity distribution. Engr 691-73 Introduction to Free-Surface Hydraulics in Open Channels 19 Energy Correction Coefficient Refer to Open Channel Flow (M.H. Chaudhry) on Page 12 Lecture 03. Engr 691-73 Introduction to Free-Surface Hydraulics in Open Channels 20 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: e U1 2 2g h1 z1 e Total energy at U2 2 2g h2 z 2 Total energy at h dx Total head loss between and Referring to the figure let us write the above equation in a more explicit form: 2 U2 U 1 U 1 o dP h dh z dz e h z e d e dx dx 2g 2 g 2 g g t g dA U 2 where hr 1 o dP g dA 1 U g t Lecture 03. dx dx 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 Engr 691-73 Introduction to Free-Surface Hydraulics in Open Channels 21 Equation of Energy for Open Channel Flow Simplifying the previous equation, we have 2 U 1 U 1 o dP d e h z dx dx 2g g t g dA or 2 U 1 U d e h z dx h r 2 g g t Head difference between and Total head loss between and h dx 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/dx and the bed slope Sf = -dz/dx., it can also be written as follows: 1 U U U h S f Se g t g x x 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. Lecture 03. Engr 691-73 Introduction to Free-Surface Hydraulics in Open Channels 22 Concept of Specific Energy Let us consider the energy equation for a steady flow: hf 2 V1 / 2 g 2 V2 / 2 g Q z1 h1 U1 2 2g U2 z 2 h2 2 2g hr h1 Total Energy h2 z1 H z h z2 U 2 2g ref. line Specific Energy: H s h L U 2 2g Specific energy is the total mechanic energy with respect to the local invert elevation of the channel. Note that since U Q/A we can also write Hs e 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. We can thus study the variation of Lecture 03. Q / A 2 h e 2g Hs Q Q 2 2 gA 2 h Q 2 2 gA 2 h 2 2 gA 2 h h as a function of Hs for Q = constant Specific Energy Curve h as a function of Q for Hs = constant Specific Discharge Curve Engr 691-73 Introduction to Free-Surface Hydraulics in Open Channels 23 Specific Energy Curve We wish to plot the Specific Energy Curve (i.e. h as a function of Hs for constant Q) : One immediately sees that the curve has two asymptotes: Hs h Q 2 2 gA 2 for h 0 we have A 0 therefore Hs for h we have A therefore Hs h 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). Lecture 03. Engr 691-73 Introduction to Free-Surface Hydraulics in Open Channels 24 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. Hs h Specific Energy U 2 h 2g Q 2 2 gA 2 B dH s 1 dh 2 Q gA dA 3 0 dh B Therefore : dH s dh 1 Q gA Let us work on this equation to see what it means: because 2 3 dA B dh Q B0 2 gA 1 Q 2 g A 2 B A 1 U g 2 3 B 1 1 1 Dh Fr U 1 gD h This shows that the critical flow condition (h = hc and Hs is minimum) is reached when Froude number is equal to one. Lecture 03. Engr 691-73 Introduction to Free-Surface Hydraulics in Open Channels 25 How to plot the specific energy curve for a cross section h 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 Hs h U 2 2g h1 hc h1 hc Alternate depths 2 U c / 2g hc To plot the specific energy curve: 1. Select a discharge Q 2. Assume an h value 3. Calculate A knowing h 4. Calculate U = Q/A 5. Calculate U2/2g 6. Calculate Hs = h + U2/2g 7. Repeat steps 2 to 6 by assuming other h values. 2 h2 U 2 / 2g Supercritical flow h 2 hc Es H s 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 Hs h Lecture 03. U 2 2g h E s or H s Q 2 2 gA 2 Engr 691-73 Introduction to Free-Surface Hydraulics in Open Channels 26 Specific Discharge Curve for a Cross Section h Specific Energy Curve plotted for E s Const . or H s Const . Hs h Hs h q q max H s U 2 2g 2 Q h 2g 2 2 gA h 2 gA Q 2 2 2 2 g H s h Q A hc U For a rectangular section A h B 2 2 h 2 gB 2 Q H s Instead of plotting h vs Hs for a constant discharge Q (or q), i.e. specific energy curve, one can also plot h vs Q (or q) for a constant Hs. This will be called specific discharge curve. Lecture 03. H s h 2g h qh Engr 691-73 Introduction to Free-Surface Hydraulics in Open Channels 2 2 h Q 2 Q 2 B 2 2 q 2 2 g H s h 27 Critical Depth and Specific Discharge Curve We can easily see that for h0 we have Q 0 for h Hs we have Q 0 Discharge curve has a maximum, Qmax, for dQ d dh dh dA Since We can write dQ gB 2 H s h D h dh For a rectangular channel For a triangular channel For a parabolic channel Lecture 03. 2 g H s Dh h Dh h Dh 2h 2 3 h 2 A 2 g H s h B and dh 0 The expression is zero if 2 g H s h dA / dh Ag 2 g H s Dh hc 2 H sc h c 2 hc 0 A B 2 H sc hc D hc 0 2 H sc hc hc 0 2 H sc h c h 2 0 2 0 3 Engr 691-73 Introduction to Free-Surface Hydraulics in Open Channels hc 2 hc 4 hc 3 5 3 4 H sc H sc H sc 28 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. 2 H sc hc D hc For critical flow in a channel: Recall that: Q max Ac 2 g H sc hc Q max Ac Uc The average velocity corresponding to the critical depth is : g D hc g D hc or Uc 2 2g D hc 2 For critical flow in a channel, the velocity head is equal to half of the hydraulic depth One can also write: Uc Critical velocity is given by Flow regimes can be classified according to Fr Lecture 03. Uc g D hc Uc 1 g D hc g D hc c Fr 1 Subcritical flow Fr 1 Critical flow Fr 1 Supercritical flow Fr c 1 Propagation velocity of small perturbations in still water of depth h Engr 691-73 Introduction to Free-Surface Hydraulics in Open Channels 29 Critical Depth for the Special Case of Rectangular Channel Dh h In a rectangular channel, we have Recall that the critical depth, hc, in a rectangular channel is given by: h c Using the definition of unit discharge: One obtains: hc 2 q 2 g hc 3 H sc H sc or hc hc 2 Uc 2 2g q Uh 2 2 2 hc q 2 3 This is valid for a rectangular channel g The maximum unit discharge, q, which may exist in a rectangular channel is: q 3 g hc 2 g H sc 3 3 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. Lecture 03. Engr 691-73 Introduction to Free-Surface Hydraulics in Open Channels 30 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 Curve for a Trapezoidal Channel Specific energy is defined as: B m b b mhh h A m 0.100 0.147 0.194 0.218 0.242 0.265 0.289 0.301 0.317 0.331 0.336 0.349 0.389 0.436 0.469 0.535 0.602 0.668 0.734 0.867 1.000 m2 0.315 0.474 0.640 0.726 0.813 0.901 0.992 1.038 1.103 1.159 1.178 1.231 1.395 1.592 1.737 2.036 2.348 2.674 3.012 3.730 4.500 Lecture 03. V = Q/A m/s 6.349 4.218 3.125 2.757 2.461 2.219 2.016 1.927 1.814 1.726 1.698 1.624 1.434 1.256 1.152 0.982 0.852 0.748 0.664 0.536 0.444 V2/2g m 2.055 0.907 0.498 0.387 0.309 0.251 0.207 0.189 0.168 0.152 0.147 0.134 0.105 0.080 0.068 0.049 0.037 0.029 0.022 0.015 0.010 Hs h 2 h 2g Q 2 2 gA 2 h 1 b 2mh Hs B m 2.155 1.054 0.692 0.605 0.550 0.516 0.496 0.490 0.485 0.483 0.483 0.484 0.494 0.516 0.537 0.584 0.639 0.697 0.757 0.882 1.010 m 3.300 3.442 3.583 3.654 3.725 3.796 3.867 3.902 3.952 3.994 4.008 4.048 4.168 4.307 4.407 4.606 4.805 5.004 5.203 5.602 6.000 V gDh Fr Dh = A/B m 0.095 0.138 0.179 0.199 0.218 0.237 0.257 0.266 0.279 0.290 0.294 0.304 0.335 0.370 0.394 0.442 0.489 0.534 0.579 0.666 0.750 Fr (-) 6.56 3.63 2.36 1.98 1.68 1.45 1.27 1.19 1.10 1.02 1.00 0.94 0.79 0.66 0.59 0.47 0.39 0.33 0.28 0.21 0.16 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: 1.200 1.000 0.800 h (m) Channel and flow data b= 3m m= 1.5 (-) 3 Q= 2 m /s V 0.600 0.400 0.200 0.000 0.000 0.500 1.000 Engr 691-73 Introduction to Free-Surface Hydraulics in Open Channels 1.500 2.000 2.500 Hs (m) 31 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. Channel and flow data b= 3.000 m m= 1.500 (-) Hs = 0.600 m hc = 0.421 m Dh = Fr = B h 1 m b 2mh m 0.100 0.164 0.228 0.260 0.292 0.325 0.357 0.373 0.395 0.414 0.421 0.424 0.435 0.448 0.457 0.475 0.492 0.510 0.528 0.564 0.600 B m 3.300 3.492 3.685 3.781 3.877 3.974 4.070 4.118 4.185 4.243 4.262 4.273 4.305 4.343 4.370 4.424 4.477 4.531 4.585 4.692 4.800 Lecture 03. Hs h b 0.358 m 1.00 (-) h Specific energy is defined as: b mhh A 2 m 0.315 0.533 0.763 0.883 1.006 1.132 1.261 1.326 1.419 1.501 1.528 1.543 1.589 1.644 1.683 1.761 1.841 1.922 2.004 2.170 2.340 V gDh Fr Q 3 m /s 0.987 1.558 2.061 2.279 2.470 2.631 2.755 2.801 2.846 2.864 2.865 2.865 2.859 2.842 2.822 2.764 2.674 2.548 2.376 1.820 0.000 V = Q/A m/s 3.132 2.924 2.700 2.581 2.456 2.325 2.185 2.112 2.005 1.909 1.875 1.856 1.799 1.729 1.677 1.569 1.453 1.326 1.186 0.839 0.000 V2/2g Hs m 0.500 0.436 0.372 0.340 0.308 0.275 0.243 0.227 0.205 0.186 0.179 0.176 0.165 0.152 0.143 0.125 0.108 0.090 0.072 0.036 0.000 m 0.600 0.600 0.600 0.600 0.600 0.600 0.600 0.600 0.600 0.600 0.600 0.600 0.600 0.600 0.600 0.600 0.600 0.600 0.600 0.600 0.600 Dh = A/B m 0.095 0.153 0.207 0.233 0.259 0.285 0.310 0.322 0.339 0.354 0.358 0.361 0.369 0.378 0.385 0.398 0.411 0.424 0.437 0.462 0.488 Fr (-) 3.24 2.39 1.89 1.71 1.54 1.39 1.25 1.19 1.10 1.02 1.00 0.99 0.95 0.90 0.86 0.79 0.72 0.65 0.57 0.39 0.00 Q Hs h 2 2 gA Q A 2 V 2 h 2g Q 2 2 gA 2 2 g H s h 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: 0.7 0.6 0.5 h (m) Specific Discharge Curve for a Trapezoidal Channel 0.4 0.3 0.2 0.1 0.0 0.0 1.0 Engr 691-73 Introduction to Free-Surface Hydraulics in Open Channels 2.0 Q (m3/s) 3.0 4.0 32 Hydraulic Jump • Refer to Open Channel Flow (M.H. Chaudhry) on Page 43 Lecture 03. Engr 691-73 Introduction to Free-Surface Hydraulics in Open Channels 33 Homework Open-Channel Flow, 2nd Edition, by M.H. Chaurdhry • Problems 2.11, 2.12, 2.19, and 2.24 Lecture 03. Engr 691-73 Introduction to Free-Surface Hydraulics in Open Channels 34 Critical Flows in Different Types of Channels • Refer to Chapter 3, Open Channel Flow (M.H. Chaudhry) on Pages 55-85 Lecture 03. Engr 691-73 Introduction to Free-Surface Hydraulics in Open Channels 35 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 Lecture 03. Engr 691-73 Introduction to Free-Surface Hydraulics in Open Channels 36 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. 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 developed the formula that bears his name from Ganguillet-Kutter formula based on the data by Henry Basin. He is well known for his uniform flow formula that he established using his own data and data from literature. http://chezy.sdsu.edu/ Lecture 03. Engr 691-73 Introduction to Free-Surface Hydraulics in Open Channels 37 Concept of Uniform Flow Consider a channel defined by the following characteristics : Cross section shape Bed slope (S = tg) 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. h=? 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). Lecture 03. Engr 691-73 Introduction to Free-Surface Hydraulics in Open Channels 38 Concept of Uniform Flow Consider a prismatic open channel (the section does not change along the flow direction) with S o Consider a free-surface flow of constant depth in this channel i.e. the free surface is parallel to the bed Equilibrium of all forces in the flow direction (no acceleration) L W sin F f W sin Ff o (P L) with W ( A L) gh W with Lecture 03. sin tan gh Ff tan S o ( AL ) tan o ( P L ) small and A tan P o A P Rh o Rh S o Engr 691-73 Introduction to Free-Surface Hydraulics in Open Channels 39 Concept of Uniform Flow with remember also u* 2 u* g Rh S o u * 8 V f by rearranging terms, we get 2 f 2 u* So f 1 8 V V 2 f 8 V 2 g Rh S o 2 4 Rh 2 g 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. Lecture 03. Engr 691-73 Introduction to Free-Surface Hydraulics in Open Channels 40 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. Lecture 03. Engr 691-73 Introduction to Free-Surface Hydraulics in Open Channels 41 Uniform Flow Calculation Using Darcy-Weisbach Equation So f The Darcy-Weissbach equation for open channel flow is given as: This equation can be rewritten as: U 2 1 f since Q UA 1 Q UA 2 4 Rh 2 g S o 4 R h 2 g We can also write U U 8g f 8g f Rh S o Rh S o A 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: k / R bf h 2 . 0 log s f a Re f f 1 Reynolds number is computed as: Lecture 03. with Re 12 a f 15 and 0 bf 6 U 4 Rh Engr 691-73 Introduction to Free-Surface Hydraulics in Open Channels 42 Uniform Flow Calculation Using Darcy-Weisbach Equation Uniform flow problems can be solved by solving the following two equations simultaneously: Q UA 8g f Rh S o A START Read Q, cross section data, ks, and So Estimate hn and Calculate A, P, Rh, and ks/Rh k / R bf 1 h 2 . 0 log s f Re f a f U = Q/A and Re = 4URh/ 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. 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. Lecture 03. Qc Use Colebrook and White equation to calculate f’ f Rh S o A is Q = Qc ? Use Colebrook and White equation to calculate f’ is f = f’ ? 8g no yes yes Output the results: hn, A, P, Rh, ks/Rh Re, f, Q, U no take f = f’ Engr 691-73 Introduction to Free-Surface Hydraulics in Open Channels END 43 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: u u* z 8 .5 ln k s u 1 u* z 5 . 75 log u* ks u z 8 .5 ln( 10 ) log k s 1 8 .5 30 z 5 . 75 log u* ks u u 2 . 3026 u* z 8 .5 log k s It is customary to measure and use point velocities at 0.2h and 0.8h. z 0 .2 h u ( z 0 . 8 ) u 0 .2 ( 30 ) ( 0 . 2 h ) 6h u 0 .2 5 . 75 u * log 5 . 75 u * log k k s s z 0 .8 h u ( z 0 . 2 ) u 0 .8 ( 30 ) ( 0 . 8 h ) 24 h u 0 .8 5 . 75 u * log 5 . 75 u * log ks ks Eliminating u* from these two equations, one obtains: 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: Lecture 03. h 0 . 78 1 . 38 log k 1 s h 6 . 25 5 . 75 ln u* k s U h 2 .2 2 . 03 log f k s 1 Engr 691-73 Introduction to Free-Surface Hydraulics in Open Channels with u 0 .8 u 0 .2 U 1 . 78 0 . 95 1 u* 1 f 1 . 78 0 . 95 8 1 44 Uniform Flow Calculation Using Darcy-Weisbach Equation The table given below is not exhaustive. Consult other references for a more detailed table. Types of Wall ks (mm) Glass, copper, brass < 0.001 Lead 0.025 Steel pipe, old 0.03 to 0.1 Steel pipe, new 0.4 Wrought iron, new 0.25 Wrought iron, old 1.0 to 1.5 Wrought iron, coated 0.1 Concrete, smooth 0.3 to 0.8 Concrete, rough <3.0 Wood 0.9 to 9 Stone, worked rough 8 to 15 Lecture 03. 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) 0 . 95 Large trapezoidal cross section 0 . 80 Triangular (equilateral) cross section 1 . 25 Using these corrections, in the formulate replace Rh by Rh. 1.0 to 2.5 Riveted Steel Rock The values given in the table are for circular industrial pipes. However, they are generally assumed to be valid also for openc channel flows. 90 to 600 Engr 691-73 Introduction to Free-Surface Hydraulics in Open Channels 45 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 /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 . Lecture 03. Engr 691-73 Introduction to Free-Surface Hydraulics in Open Channels 46 Uniform Flow Calculation Using Darcy-Weisbach Equation Without going into details, the derivation of the velocity profile for the inner region leads to: u u* 1 ln z ' C 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: u f o , , , k s , z ' 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. Lecture 03. Engr 691-73 Introduction to Free-Surface Hydraulics in Open Channels 47 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. u (r) Reynolds number Re UD u* Shear Reynolds Number Re * roughness o D U (r ) ks Re < 2000 Shear velocity Laminar flow 2000 < Re < 3000 Transition flow Re > 3000 Turbulent flow 1st level of classification o Hydraulic smooth Hydraulic transition Hydraulic rough u* k s 5 u* k s 5 u* k s u* k s 70 70 2nd level of classification Lecture 03. Engr 691-73 Introduction to Free-Surface Hydraulics in Open Channels 48 Friction Coefficient Formulae for Different Flow Regimes Laminar Flow f 64 Re Turbulent Smooth Flow 1 2 log Re f f 0 .8 Turbulent Rough Flow Colebrook-White Formula k / D 2 .5 s 2 . 0 log f 3 . 7 Re f 1 Swamee and Jain Formula f 0 . 25 log Lecture 03. 5 . 74 10 0 .9 Re 3 .7 D ks 2 hf f L V 2 Darcy-Weisbach equation for head loss D 2g Engr 691-73 Introduction to Free-Surface Hydraulics in Open Channels 49 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 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. Willi H. Hager (2003) :”Hydraulicians in Europe, 1800-2000”, IAHR Monograph. Published by IAHR, Delft, The Netherlands Lecture 03. Lewis F. Moody Professor of Hydraulic Engineering, Princeton University 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”. * http://biosystems.okstate.edu/darcy/DarcyWeisbach/Darcy-WeisbachHistory.htm Engr 691-73 Introduction to Free-Surface Hydraulics in Open Channels 50 Moody-Stanton Diagram for Friction Coefficient in Pipes (and in open channel) Moody-Stanton Diagram for Industrial Pipes Lecture 03. Engr 691-73 Introduction to Free-Surface Hydraulics in Open Channels 51 Colebrook and White Equation for Friction Coefficient We know that: u* k s 5 Turbulent smooth flow 5 In between u* k s 70 u* k s 70 Turbulent transition flow Turbulent rough flow 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: k / R bf h 2 . 0 log s f Re f a f 1 Reynolds number is computed as: with Re and 0 bf 6 a f 12 and b f 3 .4 U 4 Rh For wide channels it is recommended to take: ks 0 Consider the following limiting cases: Re Lecture 03. 12 a f 15 bf 2 . 0 log f f Re Turbulent smooth flow k / R h 2 . 0 log s f a f Turbulent rough flow 1 1 Engr 691-73 Introduction to Free-Surface Hydraulics in Open Channels 52 Chezy and Manning-Strickler Equations for Computing Uniform Flow Chezy Equation start with Darcy-Weissbach eqn rearrange Manning-Strickler Equation So f 1 S o R h f 8g 1 V 2 4 Rh 2 g V 8g C f define the Chezy coefficient to obtain the Chezy equation V C 1 n 2 using S o Rh Rh V K Rh Strickler defined which means 2 V Manning defined K 2/3 2/3 So So 1/ 2 1/ 2 1 n Q VA Q Manning eqn for discharge A n Rh 2/3 So 1/ 2 This is the average velocity for uniform flow in a channel using Q VA we get discharge Q K A Rh Strickler eqn for discharge Q VA CA S o Rh Manning eqn for traditional unit system V 1 . 49 n Rh 2/3 2/3 So So 1/ 2 1/ 2 Relationship between the two friction coefficients C Rh 1/ 6 n Lecture 03. Engr 691-73 Introduction to Free-Surface Hydraulics in Open Channels 53 Chezy and Manning Coefficients ATTENTION ! DIMENSIONAL COEFFICIENTS Chezy Equation C m 1/ 2 s 1 Manning-Strickler Equation n m Tables are available for various surfaces Lecture 03. 1 / 3 s Tables are available for various surfaces Engr 691-73 Introduction to Free-Surface Hydraulics in Open Channels 54 Chezy and Manning Coefficients Chezy Equation Chezy equation is valid only for turbulent rough flows. It should not be used for laminar flows or turbulent smooth flows. Note that one can write the following relationship between the Chezy Coefficient, C, and the friction factor, f: C 8g 1 f Therefore one can use two-point velocity measurements to calculate the Chezy coefficient (Graf & Altinakar 1998, Pp77-78): C 8g 1 f 1 . 78 g 0 . 95 1 Tables of Chezy coefficients for different types of channel materials are given in various textbooks and reference books: Lecture 03. Engr 691-73 Introduction to Free-Surface Hydraulics in Open Channels 55 Manning Coefficients for Various Types of Channels n (m-1/3s) n (m-1/3s) Taken from http://harris.centreconnect.org/Table%20A-1.htm Lecture 03. Engr 691-73 Introduction to Free-Surface Hydraulics in Open Channels 56 Manning Coefficient for Natural Channels Hydraulic computations in natural stream require an evaluation of the roughness characteristics of the channel. In the absence of a satisfactory quantitative procedure this evaluation remains chiefly an art. The ability to evaluate roughness coefficients must be developed through experience. One means of gaining this experience is by examining and becoming acquainted with the appearance of some typical channels whose roughness coefficients are known. The USGS web site http://wwwrcamnl.wr.usgs.gov/sws/fieldmethods/Indirects/nvalues/index.htm displays photos, characteristics and Manning-Strickler coefficients for a wide range of channel conditions. It would be a good idea to study these photos. Familiarity with the appearance, geometry, and roughness characteristics of these channels will improve your ability to select roughness coefficients for channels that you will encounter in your professional life. n (m-1/3s) Stream Photo 1 0.024 Columbia River at Vernita, Washington 0.028 Clark Fork at St. Regis, Montana 0.030 Clark Fork above Missoula, Montana 0.032 Salt River below Stewart Mountain Dam, Arizona Lecture 03. Engr 691-73 Introduction to Free-Surface Hydraulics in Open Channels Photo 2 57 Manning-Strickler Coefficient for Natural Streams: Wenatchee River at Plain, Washington Photo 169: Downstream from above section 1, Wenatchee River at Plain, Washington •Station name Photo 173: Downstream from section 2, Wenatchee River at Plain, Washington Wenatchee River at Plain, Washington •Station number 12-4570 •Gage location Lat 47°45'50'', long 120°39'30'', in lot 8, sec. 12, T. 26 N., R. 17 E., on left bank at Plain, 0.25 mile downstream from Beaver Creek, 7.5 miles downstream from Nason Creek, and 12 miles north of Leavensworth. Section 1 is about 1,360 ft upstream from gage. •Drainage area 591 sq mi. •Date of flood May 29, 1948 •Gage height 12.43 ft at gage: 16.50 ft at section 1 •Peak discharge 22,700 cfs A (ft2) B (ft) h (ft) Rh (ft) U (ft/s) Lenght (ft) between sections Fall (ft) between sections 1 2,480 224 11.1 10.86 9.15 ....... ..... •Computed roughness coefficient Manning n = 0.037 2 2,470 228 10.8 10.58 9.19 311 0.75 •Description of channel 3 2,440 237 10.3 10.05 9.30 325 .75 Bed is boulders; d50 = 162 mm, d85 = 320 mm. Bank are lined with trees and bushes. Sect. Lecture 03. Engr 691-73 Introduction to Free-Surface Hydraulics in Open Channels 58 Grain Size Distribution in Natural Channel Bed Measurement of grain size distribution of sediments by sieving: Weight Percent Cumulative Weight %Retained 0.01 0.6 0.02 100 0 0.063 0.5 0.02 99.98 0.02 0.125 0.6 0.02 99.96 0.04 0.25 1.2 0.04 99.94 0.06 0.354 1.8 0.06 99.9 0.1 0.5 4.7 0.16 99.84 0.16 0.707 17.5 0.59 99.68 0.32 1 172.2 5.81 99.09 0.91 1.41 2570.1 86.74 93.28 6.72 2 152.7 5.15 6.54 93.46 2.83 41.2 1.39 1.39 98.61 4 0 0 0 100 2963.1 100 Weight of Size Fraction (g) Sieve Shaker AS 400 control by Retsch Weight of Size Fraction (g) Cumulative Weight %Retained http://www.retsch.com/279.0.html?&L=0 Cumulative Weight %Passed Grain Size (mm) d (mm) Lecture 03. Engr 691-73 Introduction to Free-Surface Hydraulics in Open Channels 59 Manning Coefficients for Channel with Mobile Bed Made of Granular Material Manning-Strickler Coefficient for Mobile Bed Made of Granular Material n d 50 1/ 6 where 21 . 1 n d 90 d 50 Diameter of the holes of a sieve which would pass 50% of a sediment sample taken from the bed of the channel 1/ 6 where 26 . 0 d 90 Diameter of the holes of a sieve which would pass 90% of a sediment sample taken from the bed of the channel Other expressions that are used in practice: According to “River Mechanics” by Pierre Julien: n d 50 1/ 6 n 16 . 1 n 0 . 062 d 50 Lecture 03. d 75 1/ 6 n 21 . 7 1/ 6 n 0 . 046 d 75 d 90 1/ 6 26 . 3 1/ 6 n 0 . 038 d 90 1/ 6 Engr 691-73 Introduction to Free-Surface Hydraulics in Open Channels 60 Dependence of Manning Coefficient on Relative Depth! C g 8 f Rh n 1/ 6 g U gR h S o At this point, we are only discussing the “grain roughness”. In natural channels, the friction is also caused by bed forms. This will be discussed in detail later. The figure compares the Manning-Strickler and logarithmic law relationships with measured data as a function of the relative depth (flow depth divided by the characteristic height of the sediment grains, such as ds = d50). h ds Lecture 03. Engr 691-73 Introduction to Free-Surface Hydraulics in Open Channels h d 50 61 Composite Roughness Consider an open channel cross section with different roughness types. One way to handle the situation of composite roughness is to divide the flow area into parts, each of which is influenced by a single type of roughness. A1 A3 A2 Each part will have its own flow area (A1, A2, … AN), its own perimeter (P1, P2, … PN), thus its own hydraulic radius (Rh1, Ah2, … AhN), and its own roughness (n1, n2, … nN). We will assume that the velocity through each individual flow area is the same and is equal to the average value through the cross section (U1 = U2 = … = UN = U = Q/A). P3 P1 P2 N Note that the individual flow areas sum up to give the total flow area: A1 A 2 .... A N A i A 1 Assuming that the Manning-Strickler equation is valid for each individual flow area, we can write: 1 A1 U n1 P1 2/3 Sf 1/ 2 1 A2 n 2 P2 2/3 Sf 1/ 2 1 AN .... n N PN After various simplifying assumptions, Einstein and Horton has suggested to use: Lecture 03. n eq 2/3 Sf 1/ 2 1 A n eq P N 3/2 Pi n i 1 P 2/3 Sf 1/ 2 2/3 Engr 691-73 Introduction to Free-Surface Hydraulics in Open Channels 62 Composite Roughness References for Einstein Horton equation are: A1 A3 A2 P3 P1 Horton R. E., “Separate roughness coefficients for channel bottom and sides.” Engineering News-Record, vol iii, No. 22 (30 November 1933), pp 652-653 Einstein.. H. A.. “Der hvdraulische oder Profil-Radius.” Schweizerische Bauzeitung ,“vol 103~ No. 8 (24 February 1934), pp 89-91. P2 In fact, in addition to Einstein-Horton equation, there are other expressions proposed*: N n eq Los Angeles District equation Colebatch equation n eq n i U. S . Army, Office, Chief of Engineers, Hydraulic Design of Flood Control Channels. EM 1110-2-1601 (unpublished Engineer Manual draft ) Ai 1 A N 3/2 n i Ai 1 A 2/3 Colebatch, G. T., “Model tests on Liawenee Canal roughness coefficients.” Transactions of the Institution, Journal of the Institution of Engineers, vol 13, No. 2, Australia (February 1941), pp 27-32. In general, Einstein-Horton equation gives a more conservative estimation. It is preferred for design purposes. (*) Hydraulic design criteria, SHEETS 610-1 to 610-7, TRAPEZOIDAL CHANNELS (http://chl.erdc.usace.army.mil/Media/2/8/3/600.pdf) Lecture 03. Engr 691-73 Introduction to Free-Surface Hydraulics in Open Channels 63 Concept of Conveyance Let us study the concept of conveyance. Consider a prismatic open channel with a fixed bed. The uniform flow discharge can be calculated using Manning-Strickler Equation: Q AU A n Rh 2/3 So 1/ 2 The parameters characterizing the cross section of the channel are: A f (h ) R h f (h ) n Flow area Hydraulic radius Manning coefficient Together they form a term that can be interpreted as a measure of the ability of the cross section to convey flow; it is therefore called conveyance. Conveyance is a function of depth: The Manning-Strickler equation , therefore reduces to: Lecture 03. Q AU K ( h ) S o K (h) A n Rh 2/3 1/ 2 Engr 691-73 Introduction to Free-Surface Hydraulics in Open Channels 64 Best Hydraulic Section (Most Efficient Cross-Section) What is best hydraulic section ? A section that gives the largest flow area for the smallest perimeter ! For a rectangular channel A Bz P P B 2z A 2z z Let us keep A as constant. P is then only a function of z. Let us vary z to minimize the perimeter dP dz A z 2 A 20 z 2 Bz 2 z 2 B 2 z 2 To obtain best rectangular hydraulic section the depth must equal half of the width. r r h B Lecture 03. Engr 691-73 Introduction to Free-Surface Hydraulics in Open Channels 65 Compound Channel Method : Solve for each flow region separately and add the discharges. Am A1 P1 So n1 nm Pm Channel slope is the same for main channel and the flood plain R h1 A1 R hm Am P1 Pm V1 Vm 1 n1 1 nm R h1 2/3 R hm So 2/3 1/ 2 So 1/ 2 Q 1 A1 V1 Q m Am V m A1 n1 Am nm R h1 2/3 R hm So 2/3 1/ 2 So 1/ 2 Q T Q m Q1 Lecture 03. Engr 691-73 Introduction to Free-Surface Hydraulics in Open Channels 66 Example: Computation of Uniform Flow The triangular drainage ditch shown in the figure has a side slope of m = 2: a) Find the critical depth, hc, for a discharge of Q = 0.35 m3/s and the corresponding minimum specific energy. b) Calculate the discharge if the flow depth is h = 0.6m. The channel has a Manning coefficient of n = 0.025m-1/3/s and a bed slope of So = 0.001. Froude number for a triangular channel is given by: V Fr gD h A gD h A mh For a triangular channel, we have: Q 2 mh g 2 2 2 h m h 2 When flow is 2 Fr critical, we have Fr = 1: Q 5 h 2 2Q 5 2 Dh 1 Q A gD h 2 2 Q 1 2 A gD h Q g 2 A Dh 2 h 2 2 hc 2 gm 5 2 0 . 35 2 9 . 81 2 2 0 . 35 2 0 . 362 m Corresponding minimum specific energy is: H s min h c Vc 2 2g hc Q 2 hc 2 Ac 2 g Q 2 mh 2 g 2 2 hc c Q 2 2 4 0 . 362 m hc 2 g 2 0 . 362 2 9 . 81 2 4 Q Uniform flow discharge is calculated using Manning-Strickler equation: h 0 .6 m Q A n Rh Lecture 03. A mh 2 0 . 6 0 . 72 m 2 2/3 S 1/ 2 0 . 72 0 . 268 2 2/3 0 . 001 1/ 2 2 Rh mh 2 1 m 2 n 2 0 .6 2 1 2 A Rh 0 . 453 m 2/3 S 1/ 2 0 . 268 m 2 0 . 379 m / s 3 0 . 025 Engr 691-73 Introduction to Free-Surface Hydraulics in Open Channels 67 Equation of Energy for Open Channel Flow Solved Problem 14.4 A trapezoidal channel having a bottom width of b = 2.0 m and side slopes of m = 1.5 carries a uniform flow with a depth of h = 0. 557m. The channel has a bed slope of S = 0.005 and the coefficient of Manning is n = 0.030 m -1/3s. a) What is the discharge of the uniform flow? b) What is the regime of flow? For a trapezoidal channel the geometric relationships can be calculated as follows: B b 2 mh 2 2 1 . 5 0 . 557 3 . 671 m P b 2h 2 2 2 0 . 557 1 1 .5 2 4 . 008 m A b mh h 2 1 . 5 0 . 557 0 . 557 1 . 579 m R The uniform flow discharge can be calculated using ManningStrickler equation 1 m h 2 A / P 1 . 579 / 4 . 008 0 . 394 m Q A n Rh 2/3 S 1/ 2 1 . 579 ( 0 . 394 ) 2/3 ( 0 . 005 ) 1/ 2 2 . 001 m / s 3 0 . 030 To determine the regime of the flow we need calculate Froude number Fr V g Dh Hydraulic depth for a trapezoidal channel: Q Fr A Lecture 03. g Dh 2 .0 1 . 579 9 . 81 0 . 43 0 . 617 Q A g Dh D h A / B 1 . 579 / 3 . 671 0 . 43 m Fr 0 . 617 1 The uniform flow is subcritical Engr 691-73 Introduction to Free-Surface Hydraulics in Open Channels 68 Example: Computation of Critical Slope A rectangular laboratory channel has a width of B = 2.0m. The Manning coefficient of for this channel is n = 0.020 m-1/3/s. What should be the bed slope to achieve a critical uniform flow in this channel for a discharge of Q = 3.0m3/s ? Hint: critical uniform flow is achieved when uniform flow depth for a given discharge is equal to the critical depth for that same discharge. Fr Froude number for a rectangular channel is given by: V gh Bh Q 2 2 2 B h gh Q 2 2 B gh 3 1 h Q 3 2 hc 2 B g Uniform flow discharge is given by the Manning-Strickler equation. Qn Bh 2 / 3 From which we obtain an equation S for the slope S, Bh B 2 h If the uniform flow is going to be critical, its depth should be hc = 0.612m, thus we get. Lecture 03. A Q Fr When flow is critical Froude number is equal to 1: Q gh Bh 3 Q gh Bh gh 1 gh Q 3 2 2 B g Q A n 2 Q 3 2 2 9 . 81 2 Rh 2/3 2 S 1/ 2 0 . 612 m Bh Bh n B 2h 2/3 S 1/ 2 2 Qn Bh 2 / 3 3 0 . 020 2 0 . 612 2 / 3 S 0 . 00874 Bh B 2 h 2 0 . 612 2 2 0 . 612 Engr 691-73 Introduction to Free-Surface Hydraulics in Open Channels 69 ANGLE OF REPOSE A pile of sediment at resting at the angle of repose jr represents a threshold condition; any slight disturbance causes a failure. Here the pile of sediment is under water. Consider the indicated grain. The net downslope gravitational force acting on the grain (gravitational force – buoyancy force) is 3 Fc 3 4 D D s g sin j r g sin j r 3 3 2 2 4 F gt 3 D Rg sin j r 3 2 4 , The net normal force is F gn s R j r r 3 3 D Fc c Rg cos j r 3 2 Force balance requires that Fgt D Rg cos j r 3 2 4 The net Coulomb resistive force to motion is 4 1 Fgn F gt Fc 0 or thus: tan j r c which is how c is measured (note that it is dimensionless). For natural sediments, jr ~ 30 ~ 40 and c ~ 0.58 ~ 0.84. 1D Sediment Transport Morphodynamics with applications to Rivers and Turbidity Currents, Gary Parker, http://cee.uiuc.edu/people/parkerg/morphodynamics_e-book.htm Lecture 03. Engr 691-73 Introduction to Free-Surface Hydraulics in Open Channels 70 Hjulström curve Hjulstrom has provided a chart for the initiation of motion and sedimentation as a function of average velocity. This brings in the notion of critical velocity, or erosion velocity, Ucr=UE. This chart shows that the velocity for eroding the bed is greater than the velocity for sedimentation, i.e. Ucr=UE > UD. This indicates that, once the particle is eroded it may stay in suspension even at lower velocities. Henning Filip Hjulström (October 6, 1902–March 26, 1982) was a Swedish geographer. Hjulström was professor of geography at Uppsala University from 1944, and in 1949, when the subject of geography was split, he became professor of Physical Geography. Lecture 03. Engr 691-73 Introduction to Free-Surface Hydraulics in Open Channels 71