What about water... What about open channel flow... Continuity Equation. HECRAS Basic Principles of Water Surface Profile Computations

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1 What about water... HECRA Basic Principles o Water urace Proile Computations b G. Parodi WR ITC The Netherlands Incompressible luid must increase or decrease its velocit and depth to adjust to the channel shape High tensile strength allows it to be drawn smoothl along while accelerating No shear strength does not decelerate smoothl, results in standing waves, good canoeing, air entrainment, etc What about open channel low... A ree surace Liuid surace is open to the atmosphere Boundar is not ixed b the phsical boundaries o a closed conduit ince the low is incompressible, the product o the velocit and cross sectional area is a constant. Thereore, the low must increase or decrease its velocit and depth to adjust to the shape o a channel. Continuit Euation (conservation o mass) VA constant Discharge is expressed as Q VA Q is low V is velocit A is cross section area

2 Open Channel Flow - Controls Water goes downhill. What does that mean to us? Deinition: A control is an eature o a channel or which a uniue depth - discharge relationship occurs. Weirs/pillwas Abrupt changes in slope or width Friction - over a distance Water lowing in an open channel tpicall gains energ (kinetic) as it lows rom a higher elevation to a lower elevation It loses energ with riction and obstructions. The gravitational orces that are pushing the low along are in balance with the rictional orces exerted b the wetted perimeter that are retarding the low. Uniorm or Normal Flow Over a long stretch o a river... Average velocit: is a unction (slope and the resistance to drag along the boundaries) Gravit Friction W F

3 What does that mean? Assume a Hpothetical Channel Long prismatic (same section over a long distance) channel No change in slope, section, discharge Uniorm low occurs when the gravitational orces are exactl oset b the resistance orces V V /g EGL Uniorm low occurs when: Mean velocit is constant rom section to section Depth o low is constant rom section to section Area o low is constant rom section to section Hdrostatic pressure distribution Y W Channel bottom is parallel to water surace is parallel to energ grade line. The streamlines are parallel. o Thereore: It can onl occur in ver long, straight, prismatic channels where the terminal velocit o the low is achieved Normal Depth Applies to Dierent Tpes o lopes Yn Yc n stands or normal s stands or critical I the low is a unction o the slope and boundar riction, how can we account or it? Yc Yn

4 Newton's second law F ma 0 Manning s euation I velocit is constant, then acceleration is zero. o use a simple orce balance. τ PL W sinθ 0 0 where τ KV 0 rearrange euation KV PL γalsinθ Θsmall sinθ γ V K 0 R 0 Gravit Friction Antoine Chez (8th centur) V C R o Q Q n coe A area slope P wetted R.49 A R n low( cs) A P perimeter One o the most widel used to account or riction losses Manning s Euation - n units Manning s n.49 Q A R n R A L P L L L / T ( L )( L ) n L length (eet, meters, inches, etc) T time (seconds, minutes, hours, etc) A bulk term, a unction o grain size, roughness, irregularities, etc Values been suggested since turn o centur (King 98) Manning's n values or small, natural streams (top width <0m) Lowland treams Minimum Normal Maximum (a) Clean, straight, no deep pools (b) ame as (a), but more stones and weeds ( c) Clean, winding, some pools and shoals (d) ame as (c ), but some weeds and stones (e) ame as (c ), at lower stages, with less eective and sections () ame as (d) but more stones (g) luggish reaches, weed, deep pools (h) Ver weed reaches, deep pools and loodwas with heav stand o timber and brush Mountain streams (no vegetation in channel, banks steep, trees and brush submerged at high stages (a) treambed consists o gravel, cobbles and ew boulders (b) Bed is cobbles with large boulders (Chow, 959) 4

5 Guidance Available: seek bibliograph and the WEB Manning s n in teep Channel NRC - Fasken, 96 n0.08 n0.050 n0.08 n0.0 treams ma appear to be super critical but are just ast subcritical Jarret s En (ACE J. o Hd Eng, Vol. 0()) ( R hdraulic radius in eet) n0.04 n0.060 n0.5 n R n0.06 n0.00 n0.080 n0.50 What can we do with Manning s Euation? o wh make things an more complicated? Can compute man o the parameters that we are interested in: Velocit Trial and error or depth, width, area, etc.49 V R n.49 Q A R n 5

6 How sensitive is the euation? W00 d n0.05 Q700 cs How oten do we see normal depth in real channels? n0.0 to 0.04 % to 7% d4.5 to 5.5 t 6% to 7% w 90 to 0 t % 0.00 to % to % teep slope: normal depth below critical treams go towards normal depth but seldom get there All 40% to 70% Mild slope: normal depth above critical Limitations o a Normal Depth Computation: Open Channel Flow is tpicall varied Constant ection - natural channel? Constant Roughness - overbank low? Constant lope No Obstructions - bridges, weirs, etc 6

7 Both man made and natural channels upercritical ubcritical Critical In natural graduall varied low channels: Tim McCabe, IA NRC ubcritical Critical Hdraulic Jump ubcritical Velocit and depth changes rom section to section. However, the energ and mass is conserved. upercritical Hdraulic Jump ubcritical Lnn Betts, IA NRC Can use the energ and continuit euations to step rom the water surace elevation at one section to the water surace at another section that is a given distance upstream (subcritical) or downstream (supercritical) HEC-RA uses the one dimensional energ euation with energ losses due to riction evaluated with Manning s euation to compute water surace proiles. This is accomplished with an iterative computational procedure called the tandard tep Method. 7

8 How about that energ euation? First law o thermodnamics Remember - it s an open channel and a hdrostatic pressure distribution Bernoulli s Euation Y Y (V /g) + P /w + Z (V /g) + P /w + Z + h e (V /g) + P /w + Z (V /g) + P /w + Z + h e Kinetic energ + pressure energ + potential energ is conserved Pressure head can be represented b water depth, measured verticall (can be a problem i ver steep - streamlines converge or diverge rapidl) Energ Euation tandard tep Method (V /g) h e (V /g) h e Y (V /g) Y (V /g) Z Datum (V /g) + Y + Z (V /g) + Y + Z + h e Y Z Z Datum Y Z Energ Losses h e L + C αv g α V g W W + ( αv αv ) + g h e tart at a known point. How man unknowns? Trial and error 8

9 HEC-RA - Computation Procedure Energ Loss - important stu. Assume water surace elevation at upstream/ downstream cross-section.. Based on the assumed water surace elevation, determine the corresponding total conveance and velocit head. With values rom step, compute and solve euation or he. 4. With values rom steps and, solve energ euation or W. 5. Compare the computed value o W with value assumed in step ; repeat steps through 5 until the values agree to within 0.0 eet, or the user-deined tolerance. Loss coeicients Used: Manning s n values or riction loss ver signiicant to accurac o computed proile calibrate whenever data is available Contraction and expansion coeicients or X-ections due to losses associated with changes in X-ection areas and velocities contraction when velocit increases downstream expansion when velocit decreases downstream Bridge and culvert contraction & expansion loss coeicients same as or X-ections but usuall larger values Friction loss is evaluated as the product o the riction slope and the discharge weighted reach length Friction loss is evaluated as the product o the riction slope and the discharge weighted reach length h e L L L lob Q + C lob Q + L lob αv g ch + Q α Q ch ch + L + Q V g rob rob Q rob h e L + C.49 Q AR n Q, K αv g αv g K ( Q ) K Channel Conveance 9

10 Friction lopes in HEC-RA Average Conveance (HEC- RA deault) - best results or all proile tpes (M, M, etc.) Average Friction lope - best Q K + Q + K + results or M proiles Flow Classiication teep slope: normal depth below critical Geometric Mean Friction lope - used in UG/FHWA WPRO model Harmonic Mean Friction lope - best results or M proiles + Mild slope: normal depth above critical Friction lopes in HEC-RA HEC-RA has option to allow the program to select best riction slope euation to use based on proile tpe. Friction lopes in HEC-RA HEC-RA has option to allow the program to select best riction slope euation to use based on proile tpe. 0

11 HEC-RA HEC- stle The deault method o conveance subdivision is b breaks in Manning s n values. An optional method is as HEC- does it - subdivides the overbank areas at each individual ground point. Note: can get biggest dierences when have big changes in overbanks Other losses include: Contraction losses Expansion losses he L + C αv g V g α C contraction or expansion coeicient Contraction and Expansion Energ Loss Coeicients he L Note : WP uses the upstream section or the whole reach below it while HEC-RA averages between the two X-sections. + C αv g V g α Note : WP onl uses L in older versions and has added C to its latest version using the LO card.

12 Expansion and Contraction Coeicients peciic Energ Expansion Contraction No transition loss 0 0 Gradual transitions Tpical bridge sections Abrupt transitions Notes: maximum values are. Losses due to expansion are usuall much greater than contraction. Losses rom short abrupt transitions are larger than those rom gradual changes. Deinition: available energ o the low with respect to the bottom o the channel rather than in respect to a datum. Assumption that total energ is the same across the section. Thereore we are assuming -D. b Note: Hdraulic depth () is cross section area divided b top width E V V E Q Q V + b b + g g peciic Energ peciic Energ ( E ) const. E const. g The peciic Energ euation can be used to produce a curve. Q: What use is it? A: It is useul when interpreting certain aspect o open channel low ( E ) const. E g const. For an pair o E and, we have two possible depths that have the same speciic energ. One is supercritical, one is subcritical. The curve has a single depth at a minimum speciic energ. Y Y Minimum speciic energ or Y Note: Angle is 45 degrees or small slopes Y or E

13 peciic Energ Froude Number E + g de d g g 0 Q: What is that minimum? A: Critical low!! Minimum speciic energ E de d + g g Froude g Froude V g Ratio o stream velocit (inertia orce) to wave velocit (gravit orce) Froude Number Ratio o stream velocit (inertia orce) to wave velocit (gravit orce) inertia Fr gravit Fr >, supercritical low Fr <, subcritical low Critical Flow Froude Minimum speciic energ Transition mall changes in energ (roughness, shape, etc) cause big changes in depth Occurs at overall/spillwa ubcritical upercritical : subcritical, deep, slow low, disturbances onl propagate upstream : supercritical, ast, shallow low, disturbance can not propagate upstream ubcritical upercritical Note: Critical depth is independent o roughness and slope

14 Critical Depth Determination HEC-RA computes critical depth at a x-section under 5 dierent situations: In HEC-RA, we have a choice or the calculations upercritical low regime has been speciied. Calculation o critical depth reuested b user. Critical depth is determined at all boundar x-sections. Froude number check indicates critical depth needs to be determined to veri low regime associated with balanced elevation. Program could not balance the energ euation within the speciied tolerance beore reaching the maximum number o iterations. till need to examine transitions closel How about hdraulic jumps? General hape O Proile Water surace jumps up Tpical below dams or obstructions Ver high-energ loss/dissipation in the turbulence o the jump dc dn luice gate H d ra ulic J um p M dn u p e r c r it ic a l F lo w F lo w u b c r it ic a l F lo w M 4

15 A rapidl varing low situation Going rom subcritical to supercritical low, or viceversa is considered a rapidl varing low situation. Energ euation is or graduall varied low (would need to uanti internal energ losses) Can use empirical euations Can use momentum euation Momentum Euation Derived rom Newton s second law, Fma Appl F ma to the bod o water enclosed b the upstream and downstream x-sections. Dierence in pressure + weight o water - external riction mass x acceleration P P + Wx F Q ρ Vx Momentum Euation ( V / g ) + Y + Z ( V / g) + Y + Z + hm The momentum and energ euations ma be written similarl. Note that the loss term in the energ euation represents internal energ losses while the loss in the momentum euation (hm) represents losses due to external orces. HECRA can use the momentum euation or: Hdraulic jumps Hdraulic drops Low low hdraulics at bridges tream junctions. In uniorm low, the internal and external losses are identical. In graduall varied low, the are close. ince the transition is short, the external energ losses (due to riction) are assumed to be zero 5

16 Classiications o Open tead versus Unstead Unstead Examples Natural streams are alwas unstead - when can the unstead component not be ignored? Dam breach Estuaries Bas Flood wave others... HEC-RA is a -Dimensional Model Velocit Distribution - It s -D Flow in one direction Because o ree surace and riction, velocit is not uniorml distributed It is a simpliication o a chaotic sstem Can not relect a super elevation in a bend Can not relect secondar currents V Vx i+ Vj+ Vzk but HEC-RA is -D 6

17 Velocit Distribution econdar circulation pattern at a river bed cross section Actual Max V is at approx. 0.5D (rom the top). Actual Average V is at approx.. 0.6D (rom the top). ingle cell theor (Thomson, 876; Hawthorne, 95; Quick, 974) Outward shoaling low across point bar Outward shoaling low across point bar Current theor o bend low with skew induced outer bank cells (He and Thorne, 975) Other assumptions in HEC-RA HEC-RA is a ixed bed model The cross section is static HEC-6 allows or changes in the bed. HEC-RA can not b itsel relect upstream watershed changes. HEC-RA is a simpliication o the natural sstem 7