CE Supplementary Notes for Open Channel Flow

Transcription

1 CE upplemetary Notes or Ope Chael Flow 10.5 Gradually Varyig Flow Flow quatities, depths, ad bed slopes are ot always costat. Chages are cotrolled by two competig actors: gravitatioal eergy added rom water lowig dowhill, ad eergy lost rom bottom rictio ad turbulece. I chages are rapid eough that we ca igore bottom rictio (weirs, hydraulic jumps, etc.), the we ca use the rapidly varyig low techiques that were developed i Chapter 10. I chages are slower, the we must take ito accout both gravity ad bottom rictio. The easiest way to do this is to look at the exteded Beroulli equatio icludig dissipatio betwee two poits: E + z = E + z + h 1 1 E = E + ( z z ) + h 1 1 E = E + ( ) l 1 0 l l (10.5.1) where E1, E, are the speciic eergies at the two locatios, = h / l is the rictio slope, ad 0 is the bed slope. For uiorm low, 1 E l = E ad thus the bed slope ad rictio slope are idetical. For gradually varyig low, they will be dieret ad thus the speciic eergies ad water levels will chage. It ote useul to trasorm (10.5.1) to a dieretial equatio valid over a short distace E = x ( 0 ) (10.5.) This gives the usurprisig result that i gravity is stroger tha rictio ( 0 > ), the low will gai eergy. Coversely, i rictio wis over gravity, the low will lose eergy. Like may topics i ope chael low, it is useul to look at speciic eergy curves. For the example show i Figure , the iitial low is supercritical. I the chael is o costat cross-sectio, the low must stay o the origial curve. For supercritical low with icreasig eergy, the low depth decreases, while or decreasig eergy, the low depth icreases. This is exactly the same as beore whe we had o rictio but did have bed elevatio chages. I the iitial low were 1

2 subcritical istead o supercritical, a similar procedure would be ollowed, but with the opposite result. For subcritical low with icreasig eergy, the low depth icreases, while or decreasig eergy the low depth decreases. Figure peciic eergy ad depth chages or gradually varyig low I the chael cotracts, as show i Figure 10.5., the low must ow ollow the speciic eergy curve o the right. I E > E1, the low depth will the be greater tha i the chael width was costat, ad is oud at the itersectio o the vertical lie with the curve. I E < E1 as show i Figure , there is a problem. There is o itersectio betwee eergy E ad the ew speciic eergy curve. This idicates that it will ot be possible to pass the required amout o low past this poit. Very likely a hydraulic jump will develop beore this situatio occurs, but we will look at this type o behavior later. For subcritical low, similar behaviors exist but with opposite sese: the eect o the chael arrowig is to cause a decrease i the water level.

3 Figure peciic eergy ad depth chages or gradually varyig low i a cotractio. Kowledge o this geeral behavior is useul, but quatitative solutio o gradually-varyig proiles requires some more work. First, we eed to deie the rictio slope,. The rictio slope is the rate o eergy dissipatio rom the low, ad it is ot immediately clear how to calculate it util we realize that or uiorm low = 0. For uiorm low we already kow how to calculate 0 usig Maig s equatio, ad gradually varyig low. is calculated exactly the same or V = κ R 4/3 h (10.5.3) tadard ad Direct tep Methods Now that we kow this, we ca compute eergy chages usig (10.5.). The diiculty ow becomes how to compute chages i depth alog a chael. A very simple but powerul method or doig this is the stadard step method that may be used to compute chages i low depth by had or both prismatic (chael cross-sectio does ot chage) ad varyig chaels. Here, we rewrite (10.5.) as 3

4 ( ) E = E E = x (10.5.4) 1 0 mea The stadard step method is the, give iitial depth y 1 ad velocity V 1 ad thus speciic eergy E 1 : 1. Choose a step legth, x. Calculate the rictio slope,, at poit 1 3. Calculate E at x+ x usig (10.5.4) but oly usig 0 ad rom poit 1 4. Give E = E1+ E, calculate dowstream (or upstream i x is egative) depth y at locatio x+ x 5. Give y, calculate V 6. Calculate the rictio slope,, at poit 7. Recalculate E at x+ x usig (10.5.4) with mea values o 0 ad 8. Does E = E1+ E? I yes, the you are doe with this x, ad move o to the ext oe. I they are ot close eough, go to step 4. There is oe major subtlety that is ot immediately apparet: which directio to itegrate upstream or dowstream? At irst, it seems obvious that you should start upstream ad itegrate dowstream because this is the directio water lows. For supercritical lows, this is true: you start at a upstream locatio ad march dowstream usig this method ad everythig works well. However, it does ot work this way or subcritical lows. Why is this? Remember that or supercritical lows, all disturbaces are swept dowstream. Dowstream thus has o eect o upstream. However, or subcritical lows, disturbaces move both dowstream ad upstream ad thus dowstream has a huge impact o upstream coditios. Because o this subcritical lows are usually itegrated rom a dowstream cotrol or equilibrium poit to a upstream locatio. The direct step method is somewhat similar to the stadard step method, except that y is speciied rather tha x. The direct step method has the sigiicat advatage that o iteratio is ote required, but it is more diicult to cotrol the itegratio step legth, x. However, it works best or prismatic chaels (costat cross-sectio). For o-prismatic chaels, the direct step method would become iterative just like the stadard step method, ad the chael crosssectio is geerally kow oly at a ew locatios i ay case. For these reasos, the direct step method is ot used or atural chaels. It is summarized by 4

5 1. Choose a depth chage, y so that y = y1+ y. Give y, calculate V ad E 3. Calculate the rictio slope,, at poits 1 ad, ad average 4. Calculate x usig (10.5.4). Move o to the ext sectio o chael. The diiculty with this method is that it ca be diicult to cotrol how ar the step x goes, particularly close to the ormal depth. You may also choose the wrog sig or y ad ed up goig i the wrog directio. till, this method ca work well. For both the direct step ad stadard step methods, you should always have velocity chages o less tha 0% betwee steps. I you are gettig velocity chages larger tha this, choose a smaller step. Costat Chael Width: Numerical Itegratio I the chael has costat width, there are methods that are easy to use, do ot require iteratio, ad allow us to use stadard umerical methods. ome aalytical work is required irst. We start with (10.5.) E = x ( 0 ) (10.5.) Usig the chai rule (there are some subtleties to the way we do it here that mea we ca oly use this method or costat cross-sectios), we write E E y = x y x (10.5.5) Now, kowig E = y+ V /g, E V = 1+ y y g Q = 1 ga 3 QB = 1 ga = 1 Fr A y 3 (10.5.6) 5

6 where i the process we ow have a good way to deie Froude umber or o-rectagular chaels, Fr = Q B / ga where Ay ( ) is the cross-sectioal area ad By ( ) is the surace width. 3 The dieretial equatio or slowly varyig chael low the becomes y 0 = x 1 Fr (10.5.7) that ca be solved usig stadard methods. Agai, itegrate dowstream or supercritical low ad upstream or subcritical low. Classiicatio o Gradually Varyig Flow As was also see i the speciic eergy curve o Figure , equatio (10.5.7) shows that icreasig or decreasig depths depeds ot oly o whether eergy is gaied or lost, but o whether there is supercritical or subcritical low. I act, it is possible to classiy graduallyvaryig low ito more ormal divisios based o the relative positios o three depths: the depth o ormal low, y ; the depth or critical low, y c, ad the actual depth, y. The irst classiicatio is whether the depth o ormal (uiorm) low is greater tha, less tha, or equal to the critical depth. This is equivalet to aswerig whether uiorm low is subcritical (M or mild slope), supercritical ( or steep slope), or critical (C). Additioal criteria o horizotal slope (H) or adverse slope (A) are also sometimes used. The secod compoet o the matrix is the relative depths y, y c, ad y. I y > max( yc, y), it is regio 1. I y is i betwee y c ad y, it is regio, ad i y< mi( y, y ), the low is i regio 3. Figure shows all o these c regios ad how the curves relate. I a low is i a regio it will always have the same upstream ad dowstream behavior with respect to critical ad ormal depths. The shape o the curves may also be ierred rom (10.5.7). Lookig at these curves also helps to show why we should itegrate subcritical low goig upstream. I we look at curves M1 ad M, at the let side o the igure both have depths approachig y, ad the diereces i depth are very small. However, i we itegrate dowstream, curve M1 icreases depth (maybe ito a lake), while M decreases towards the critical depth (probably a weir). From very similar begiigs there are very dieret eds. However, i we itegrate upstream, both curves very quickly ad stably approach the ormal depth. This is ot a proo o aythig, but demostrates the perils o itegratig subcritical lows i the dowstream directio. 6

7 Oe diiculty with Figure is that there are may curves ad it ca be diicult to uderstad all o them. There is a simpler way to thik o this that uites all o the curves ito oe priciple: subcritical lows approach the ormal depth headig upstream, ad supercritical lows approach the ormal depth headig dowstream. (We must thik o adverse ad zero slopes here as havig a iiite ormal depth.) Figure Gradually varyig low classiicatios. 7

8 Thought Experimet: What sorts o coditios would create each o these types o lows? Cotrol Poits ometimes computed curves try to pass through the critical poit, where they break dow ad predict a iiite slope. I reality, oe o two thigs will happe: there will either be a weir low, or a hydraulic jump. Both o these are called cotrol poits, where the low passes through the critical depth this is because you kow the low properties at this locatio ad ca start to itegrate water levels away rom it. May times cotrol poits are ixed, as with a weir or a spillway. Here, we ca itegrate the subcritical low startig a short distace upstream o the weir, ad the supercritical low startig a short distace dowstream. We do ot start immediately at the weir, as there will be a iiite slope to the water surace. The hydraulic jump is more o a issue, as we must match the supercritical low water levels, itegratig dowstream, with the subcritical low levels headig upstream. These ca ever meet, so how do we recocile the two? The aswer is to use the hydraulic jump cojugate depths (equatio i Youg luids text), either or the upstream or dowstream low, as see i Figure The upstream ad dowstream depths will be i mometum equilibrium at the locatio where the cojugate depth or the upstream low matches the actual depth or the dowstream low, ad a hydraulic jump will orm aroud here. (Note that a hydraulic jump is ot just a poit but stretches out over several water depths. However, our methods caot describe this.) Figure Hydraulic jump locatio or a M1 curve. This situatio occurs ote just dowstream o a spillway or a udershot weir. Thought experimet: What will happe to the locatio o the hydraulic jump i the dowstream water level icreases? 8

9 Also, some times the low over a broad-crested weir may be subcritical (i the low rate Q is too small) ad sometimes it will be supercritical, so it is always ecessary to check assumptios about cotrol poits. 9