RIVER & STREAMS HYDRAULICS

Transcription

1 RIVER & STREAMS HYRAULICS Benoit Csman-Roisin artmot College River flow is 3 and nsteady (trblent). Bt: lengt of river >> widt & dept As a reslt, te downstream velocity, aligned wit te cannel, dominates te flow. Te 1 assmption may be made, wit (x,t) = water dept (x,t) = water velocity wit x = downstream distance and t = time. 1

2 Te presence of two flow variables [(x,t) and (x,t)] necessitates two pysical statements. Tese are: 1. Conservation of mass (wat goes in, goes ot). Momentm bdget (wit 3 forces: pressre, gravity and friction) Additional assmption: incompressibility density = constant = 1 kg/m 3 (freswater) Bdget for a stretc dx of river A = Area W = Widt = max dept P = Wetted perimeter S = Slope = sin

3 Conservation of mass: Amont stored in stretc dx = wat goes in wat goes ot. Adx t dt t A x A xdx dx, dt Ten constant t A A A t x x A were A() is a known fnction of te water dept (cannel profile given) Tis eqation is attribted to Leonardo da Vinci ( ). Case of a rectanglar cannel wit constant widt: A = W, wit W = constant P = + W + A t x t x A 3

4 Momentm Bdget: d dt Momentm inside stretc Momentm entering at x Momentm exiting at x dx Psing pressre force in rear Braking pressre force aead ownslope gravitational force Braking frictional force along bottom Pressre force F p pda p( w( dz p( ydrostatic (gage) pressre g( w( cannel widt at level z ( z ) F p g( w( dz dfp [ g( w( ] z d g w( dz ga gw( dz For later: F p x dfp d ga x x 4

5 Gravitational force ( mg)sin ( Adx) gs gasdx Frictional force bottom stress bottom area ( Pdx) ( C wit bottom stress C b and P wetted perimeter b ) Pdx Ptting it altogeter: [ Adx] at tdt [ Adx] dt at t A at x F p at x gasdx A C P dx F at xdx p at xdx Momentm in and ot Pressre force, rear and front ownslope gravity Bottom friction or, in differential form: t x F p A A gas C P x and after some simplifications and se of volme conservation: inertia g gs C t x x gravity P A friction 5

6 For convenience, we define te ydralic radis: A cross - sectional area R P wetted perimeter so tat te momentm eqation becomes: g gs C t x x For a broad flat cannel, wic is a good approximation for most rivers: R W W R W W g gs C t x x Tis eqation is attribted to Ademar de Saint-Venant ( ). Togeter, tis momentm eqation and te mass-conservation eqation form a x nonlinear system for te flow variables and. 1. Uniform frictional flow: and t x Only te momentm eqation remains and it becomes: gs C gs C Te balance is between te forward force of gravity and te retarding force of bottom friction. Te formla is de to Antoine de Cezy ( ). Te Cezy formla specifies one relation between te velocity and te water dept. How can tese qantities be determined separately? 6

7 Answer: We need to know te volmetric flow (discarge) of te river! Wen Q A W CQ gsw gsq CW is given, ten Note tat bot water dept and velocity increase wit te discarge. Tis explains wy te water level rises and te crrent increases simltaneosly wen te river discarge rises. Note: As te discarge increases, te water dept ( Q /3 ) increases faster tan te velocity ( Q 1/3 ). Manning s formla River data sow tat te drag coefficient C is not a constant bt depends on dept. If we se te logaritmic velocity profile of wall trblence, we obtain: and te Cezy formla becomes C [ln( / z ) 1] gs gs ln 1 C z Using abndant data, Robert Manning ( ) determined tat a power of was adeqate, wit te /3 power giving te best fit, and e wrote: 1 R n /3 S 1/ in wic te coefficient n is now called te Manning Coefficient. Note tat tis expression is not dimensionally correct. So, care mst be taken to se metric nits. 7

8 Examples of Manning Coefficients: Smoot cement canal n =.1 Clark Fork at St. Regis, Montana n =.8 Clark Fork above Missola, Montana n =.3 Middle Fork Flatead River near Essex, Montana n =.41 Sot Fork Clearwater River near Grangeville, Idao n =.51 Rock Creek near arby, Montana n =.75 ttp://wwwrcamnl.wr.sgs.gov/sws/fieldmetods/indirects/nvales/index.tm Nmerical vales of te Manning Coefficient: CHANNEL TYPE n Artificial cannels finised cement.1 nfinised cement.14 brick work.15 rbble masonry.5 smoot dirt. gravel.5 wit weeds.3 cobbles.35 Natral cannels montain streams.45 clean and straigt.3 clean and winding.4 wit weeds and stones.45 most rivers.35 wit deep pools.4 irreglar sides.45 dense side growt.8 Flood plains farmland.35 small brses.15 wit trees.15 8

9 Te enigma of Roman water engineers Roman engineers ad no conception of time at te scale of te minte and second. So, tey ad no concept of water velocity and dealt only wit water depts. Segovia, Spain So, ow were tey able to bild properly designed aqedcts and sewage draining passages? Answer: Te Romans were lcky becase velocity is directly related to water dept, and water dept cold ten be sed as te only variable. It also elped tat water dept appens to be te more sensitive fnction of discarge among te two variables. 9

10 A nice exercise: Sbject te Cezy soltion to small, time-dependent flctations, to find tat it is stable only as long as Fr S 4 C If tis condition is not met, waves grow to finite amplitde. Tese are so-called roll waves. Steady frictionless flow: Now, take (steadiness) and C (no friction) t Te pair of governing eqations become: Mass : Momentm : If we define te bottom elevation b( x) above a reference datm (sea level) ten d dx ( ) db S dx d d g gs dx dx Q constant W and te momentm eqation can be cast as : d g gb dx 1

11 Terefore, te expression B g gb is conserved along te flow. Tis is relation is de to aniel Bernolli (17-178), and is known as te Bernolli principle. Te essence of te Bernolli principle is conservation of energy : kinetic energy, g( b ) potential energy. 11