OC FLOW: ENERGY CONCEPTS, CHANNEL ANALYSIS [ energy considerations: the background ] When we consider a sample OC flow segment above: We say the slope of the channel bottom is equal to So=(z1-z2)/l Note that the fluid depth (y1, y2) is measured in the vertical direction and x is horizontal In most OC flows, So is very small (So=0.000118 for Mississippi River), thus in such circumstances x and y are usually taken as channel bottom distance and depth normal to the bottom with minimal discrepancy 87-351 Fluid Mechanics

OC FLOW: ENERGY CONCEPTS, CHANNEL ANALYSIS [ energy considerations: the background ] If we assume a uniform velocity profile across any channel section, we can write a one dimensional energy equation as follows - [1] 87-351 Fluid Mechanics

OC FLOW: ENERGY CONCEPTS, CHANNEL ANALYSIS [ energy considerations: the background ] - [1] In [1], hL is the head loss due to viscous effects between (1) and (2), we can also write z1-z2=Sol Further inspection of [1] reveals that the hydrostatic pressure assumption allows us to write p1/g as y1 and p2/g as y2 87-351 Fluid Mechanics

OC FLOW: ENERGY CONCEPTS, CHANNEL ANALYSIS [ energy considerations: the background ] Now we can re-write [1] - [2] In [2] we will use an expression for head loss adapted for OC flow, hL = l Sf where Sf is the friction slope (the slope of the total energy line) 87-351 Fluid Mechanics

OC FLOW: ENERGY CONCEPTS, CHANNEL ANALYSIS [ energy considerations: the background ] We recall that the total energy line’s magnitude is comprised of the elevation head, z, the pressure head, p/g, and the velocity head v2/2g  then [2] becomes - [3] 87-351 Fluid Mechanics

OC FLOW: ENERGY CONCEPTS, CHANNEL ANALYSIS [ energy considerations: the background ] When Sf=0 (no head loss), the energy line is horizontal, thus the total energy is free to shift between kinetic and potential energy (in a conservative fashion), we write [3] as - [4] 87-351 Fluid Mechanics

OC FLOW: ENERGY CONCEPTS, CHANNEL ANALYSIS [ energy considerations: specific energy ] Specific energy is the sum of potential and kinetic energy per unit weight (an often useful expression in OC flow) - [5] We can rewrite the energy equation [3] as - [6] If head losses are negligible, then Sf = 0, (Sf – So)l = -Sol = z2 - z1 and thus the sum of the specific energy and the elevation of the channel bottom remains constant, that is, E1 + z1 = E2 + z2 (this is BE) 87-351 Fluid Mechanics

OC FLOW: ENERGY CONCEPTS, CHANNEL ANALYSIS [ energy considerations: specific energy ] Now, let us consider a channel with a rectangular cross-section, that has a width, b, we can write the specific energy in terms of the flow rate per unit width, q = Q/b = Vyb/b = Vy, as - [7] Therefore, for any channel of constant width, q will remain constant, but depth, y, may vary – for us to greater appreciate this we consider the specific energy diagram (a graph of E=E(y) with fixed q) 87-351 Fluid Mechanics

OC FLOW: ENERGY CONCEPTS, CHANNEL ANALYSIS [ energy considerations: specific energy ] - [7] Inspection of [7] reveals that it is a cubic If we solve [7] for the depth, y, at a given specific energy, E, **(E>Emin)**, we can expect three values for y Examination of the specific energy diagram to the right shows us where we may typically find ysub, ysup, and yneg yneg is without physical meaning and is ignored Thus, we are left with two depths, a subcritical flow depth, ysub, and a supercritical flow depth, ysup, these are referred to as alternate depths 87-351 Fluid Mechanics

OC FLOW: ENERGY CONCEPTS, CHANNEL ANALYSIS [ energy considerations: specific energy ] - [7] We note from the figure, that ysup< ysub And since, by definition, q=Vy is constant along the curve, it follows that Vsup>Vsub We also note that Emin occurs at the nose of the curve, the nose of the curve corresponds to critical conditions (Fr=1) To determine the value of Emin, we differentiate [7] and set dE/dy = 0 87-351 Fluid Mechanics

OC FLOW: ENERGY CONCEPTS, CHANNEL ANALYSIS [ energy considerations: specific energy ] At dE/dy=0 we obtain - [8] or - [9] Here, the subscript c denotes critical conditions 87-351 Fluid Mechanics

OC FLOW: ENERGY CONCEPTS, CHANNEL ANALYSIS [ energy considerations: specific energy ] - [9] - [7] Now if we sub this yc ([9]) back into [7] we yield - [10] 87-351 Fluid Mechanics

OC FLOW: ENERGY CONCEPTS, CHANNEL ANALYSIS [ energy considerations: specific energy ] - [9] - [11] (by definition) Now if we combine [9] with [11] we can write - [12] 87-351 Fluid Mechanics

OC FLOW: ENERGY CONCEPTS, CHANNEL ANALYSIS [ energy considerations: specific energy ] Further, then we can also write - [12] So, we can say critical conditions occur at Emin (the nose of the curve) 87-351 Fluid Mechanics

OC FLOW: ENERGY CONCEPTS, CHANNEL ANALYSIS [ energy considerations: the chezy and manning equations ] Chezy and Manning equations were developed many years ago They can describe the flow rate and geometry for a uniform open channel flow Following is a brief derivation of Chezy’s equation followed by the refinement made by Manning, which is still in use today Consider the following CV around a typical uniform flow in an open channel 87-351 Fluid Mechanics

OC FLOW: ENERGY CONCEPTS, CHANNEL ANALYSIS [ energy considerations: the chezy and manning equations ] For steady, uniform flow, we write the x-momentum eqn as - [13] In [13] the RHS of the eqn is zero as V2=V1 (momentum flux across (1) is the same as across (2)) So the flow is driven simply by the force balance on the LHS of [13] - [14] Here in [14], F1 and F2 represent the pressure forces on the volume, but the pressure is hydrostatic, thus if y1=y2, then F1=F2, we are left with a balance between the streamwise component of the weight force and the shear force acting opposite it on the channel’s wetted perimeter 87-351 Fluid Mechanics

OC FLOW: ENERGY CONCEPTS, CHANNEL ANALYSIS [ energy considerations: the chezy and manning equations ] So we write [14] as - [15] Now we recognize that for small angles, sinq~tanq=So, and we recall that Rh=A/P so we write [15] as - [16] 87-351 Fluid Mechanics

OC FLOW: ENERGY CONCEPTS, CHANNEL ANALYSIS [ energy considerations: the chezy and manning equations ] - [16] Now if we invoke the fact that the wall shear stress, tw is proportional to the dynamic pressure, rV2/2, (and independent of viscosity) for large Reynolds number flows (like most OC flows), we can write [16] as - [17] Here K is a constant dependent on pipe roughness 87-351 Fluid Mechanics

OC FLOW: ENERGY CONCEPTS, CHANNEL ANALYSIS [ energy considerations: the chezy and manning equations ] - [16] - [17] Equating [16] and [17] we have - [18] Which we can re-write as - [19] This is the Chezy Equation, where, C, is the Chezy coefficient (which must be determined by experiment and is dimensional) 87-351 Fluid Mechanics

OC FLOW: ENERGY CONCEPTS, CHANNEL ANALYSIS [ energy considerations: the chezy and manning equations ] - [19] Manning ran a series of experiments (on [19]) that refined the relationship between the velocity and hydraulic radius, Rh - [20] [20] is the Manning Equation, where n is the Manning coefficient, a dimensional, tabulated coefficient dependent on channel roughness **NOTA BENA** be sure you have the correct ‘n’ based on the system of units in use 87-351 Fluid Mechanics

OC FLOW: ENERGY CONCEPTS, CHANNEL ANALYSIS [ energy considerations: the chezy and manning equations ] - [21] - [22] [21] and [22] are the popular forms of Manning’s equation, here the k is special unit base coefficient that must be applied due to the dimensional nature of the expression k=1 for SI units and k=1.49 if BG units are used 87-351 Fluid Mechanics