CEE 3310 Open Channel Flow,, Nov. 18,

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1 CEE 3310 Open Channel Flow,, Nov. 18, Review Drag & Lit Laminar vs Turbulent Boundary Layer Turbulent boundary layers stay attached to bodies longer Narrower wake! Lower pressure drag! C D = F D 1 ρ 2 A where A is the appropriate area (i.e., surace, cross sectional, wetted) C D = (Re) may have to iterate to get a converged solution 8.2 Open-Channel Flow Pipe/duct low closed, ull, gas or liquid. Pressure riction balance. Open-channel low ree-surace (river, lume, partially illed pipe,...). Gravity riction balance. Open-channel low is oten turbulent with complex geometries (rivers, estuaries, streams!). Hence we will simpliy the geometry (oten an excellent approximation)! We will assume straight channels with simple geometries (prismatic channels) and steady state low (in time). Free-Surace at constant pressure atmospheric. This helps analysis! But ree-surace position is unknown! This hinders analysis! There are entire books written on open-channel low (French, Henderson, Chow,...), we will only discuss the basics. For more details keep an eye open or CEE 3320 (may get a 4000 number) that will cover the details o unsteady open channel/river low, sediment transport, and undamental coastal engineering/water wave theory.

2 CEE 3310 Open Channel Flow,, Nov. 18, D Flow Assumption Consider the energy equation. At surace P 1 = P 2 = P atm, thereore: 1 + z 1 = 2 + z 2 + h From our pipe low work we should immediately suspect that it is a reasonably good approximation to consider: h L D h where L = x 2 x 1 and D h = 4A/P as previously deined. Note P is the wetted perimeter hence only the distance along the wetted sides o the channel. Now, we can relate V 1 to rom conservation o mass since Q 1 = Q 2 = V 1 A 1 A 2. We deine an appropriate local Reynolds number as Re Dh = UD h /ν. I Re Dh > 10 5 the low is turbulent and most are (classic exception is sheet low o paved suraces).

3 CEE 3310 Open Channel Flow,, Nov. 18, Flow Classiication by Variation o Depth with Distance Along Channel Uniorm Flow constant depth dy dx = 0. Varied Flow non-constant depth dy dx 0. Gradually Varied Flow dy dx 1. Rapidly Varied Flow dy dx 1. A picture: 8.5 Uniorm Flow In uniorm low y 1 = y 2, V 1 = = V, thereore the energy equation becomes: z 1 z 2 = S 0 L = h Flow is essentially ully developed, thereore we can apply Darcy-Weissbach relations. h = L D h In open channel low it is more common to work with the: hydraulic radius = A P = D h 4 = R h

4 CEE 3310 Open Channel Flow,, Nov. 18, where again (P) is the length o the wetted perimeter. Combining the above two equations we have: S 0 L = L D h V = ( ) 1/2 8g (R h S 0 ) 1/2 8.6 Chézy Formulas Chézy deined the coeicient C = ( ) 1/2 8g (now called the Chézy coeicient) and ound that it varies by a actor o 3. Thereore V = C (R h S 0 ) 1/2 and Q = CA (R h S 0 ) 1/2 Manning did ield tests and ound C = ( 8g ) 1/2 α R1/6 h n where n is known as Manning s n and is a roughness coeicient and α is a dimensional constant that varies with systems o units (this is not a homogeneous equation, remember?!). For SI α = 1, or BG α = It is let as an exercise or the student to ind the units and veriy the conversion. 8.7 Manning s Equation Substituting Manning s result into the Chézy ormulas we have the celebrated Manning s equation: V α n R2/3 h S1/2 0 and Q α n AR2/3 h S1/2 0 n varies by a actor o 15 and is tabulated in your text and more extensively elsewhere.

5 CEE 3310 Open Channel Flow,, Nov. 18, Example Fall Creek Flow Let s consider Fall Creek where the USGS operates a gaging station. Looking over yesterday s record we see a local maximum low depth occurred on 20:45 EST o d = I we estimate S , n = rom table 10.1 in text, assuming somewhere between clean and straight and sluggish with deep pools and a width o b = 50, d = 0.85 we can ask the question, what is Q? R h = A P = Q = α n AR2/3 h S1/2 0 bd 2d + b d i b d thereore Q = α n bd5/3 S 1/2 0 = t1/3 /s (50 t)(0.85 t) 5/ Q = 51 CFS Actual value rom calibrated low gage was 53 CFS. Just using ball park estimates we were within 4% amazingly good (best I have ever seen)! Hard to be more accurate as with luck we pretty much hit it but in reality one would need a survey o the river slope and the wetted perimeter. We could get these more accurately rom a USGS topographic map but to be truly accurate we would send a survey team out to measure directly in the ield.