Free Surface Hydrodynamics

Transcription

1 Water Sciece ad Egieerig Free Surface Hydrodyamics y A part of Module : Hydraulics ad Hydrology Water Sciece ad Egieerig Dr. Shreedhar Maskey Seior Lecturer UNESCO-IHE Istitute for Water Educatio S. Maskey (0) About Module Hydraulics ad Hydrology Free surface hydrodyamics (5%) Egieerig hydrology (5%) GIS ad remote sesig (0%) S. Maskey (0)

2 Part-: Fudametal priciples ad equatios Cotiuity ad Mometum priciples Euler ad Navier-Stokes equatios Beroulli equatio ad applicatios Cotiuity priciple Epresses the coservatio of mass i a cotrol volume occupied by a fluid. Obtaied by equatig the chage i fluid mass to the differece betwee the rate of mass IN ad OUT i a give time. Chage i mass i time t is ( ( t)) yz tyz tt tt Chage i mass due to the chage i ρ ad u (i -directio) u u ( ua A ua ) t A t Y Z ua u ua A Rate of fluid mass iflow ad outflow for the cotiuity equatio. X u tyz 4 S. Maskey (0)

3 S. Maskey (0) Cotiuity priciple X Z Y ua u A ua Geeral form of cotiuity equatio: 0 w v u w v u t For icompressible fluid with costat ρ: z y z y t 0 z w y v u For a steady ad icompressible flow i a pipe: A V A V Also valid for a steady cotiuous flow i a ope chael. 5 Cotiuity priciple For usteady flow i a ope chael: 0 t h B 0 t A t t 6

4 Mometum priciple Iertia force: To chage a eistig motio of a body of mass M, it is ecessary to apply a force F to this mass, which causes a acceleratio a = dv/dt, such that dv M V dv F (uit volume) dt dt FTwo types of iertia forces are cosidered This is the Newto's Equatio of Motio. The product M(dV/dt) is the iertia force. Two types of iertia forces are cosidered Due to local accelatio chage i velocity i time Due to covective acceleratio chage i velocity over a distace dv dt dv V d 7 Mometum priciple Iertia force compoets i dimesios (, y, z aes): u t v t w t u u u u v w y z v v v u v w y z w w w u v w y z Local acceleratio term Covective acceleratio terms 8 S. Maskey (0) 4

5 Mometum priciple Applied forces: Gravity force (F g = Mg) Pressure force Z Pressure forces actig o two opposite faces o the yz-plae. p( yz) p p p yz p( yz) p p p X Dyamic yz yz Kiematic Temperature Viscosity Viscosity -t- -µ- ( o C) (N s/m - ν - ) 0 - (m /s) 0-6 Y du dy Coefficiet of viscosity Viscous force (recall Newto s Law of Viscosity) Shear stress Mometum priciple I dimesios Z zy z z y y z ( yz) y( z) z( y) y z y yy ( yz) y z yz ( yz) y y( z) z y( z) z zy zz z( y) z( y) Y yz y X Covetio used to represet shear stresses o various plaes. The first subscript represets the ais ormal to the plae ad the secod subscript represets the ais alog which the force is actig. 0 S. Maskey (0) 5

6 Mometum equatio Z zy Obtaied from balacig iertia ad applied forces yz z z y y Euler equatio eglects viscous X forces u u u u p u v w t y z v v v v p u v w t y z y w w w w p u v w g t y z z Y Covetio used to represet shear stresses o various plaes. The first subscript represets the ais ormal to the plae ad the secod subscript represets the ais alog which the force is actig. Mometum equatio Navier-Stokes equatio also icludes viscous forces du p y z dt y z dv p y yy zy dt y y z dw p z yz zz g dt z y z Y Z yz zy y z z y X Covetio used to represet shear stresses o various plaes. The first subscript represets the ais ormal to the plae ad the secod subscript represets the ais alog which the force is actig. S. Maskey (0) 6

7 Beroulli equatio Ca be derived from Euler equatio Steady flow assumptio is used Has wide applicatio i hydraulics Represets the total eergy ad sigifies that total the eergy remais costat V g V p g g TOTAL ENERGY LINE S f V h h z Costat Eergy lie V g Free surface V S 0 Chael bottom Part-: D Chael Flow Steady-uiform flow Frictio coefficiet ad velocity distributio Specific discharge, critical depth No-uiform gradually varied flow (backwater curve) Usteady flow St-Veat equatio Kiematic wave 4 S. Maskey (0) 7

8 Steady-uiform flow Velocity is costat (i.e. V/t = 0) ad is uiform i space (i.e. i oe dimesio V/ = 0). The water surface lie remais parallel to the chael bed profile. Gravity force to be balaced by the shear force due to the bed resistace. gasi P b dp 0 A b g si grsi P Remember: P is wetted perimeter Hydraulic radius, R = A / P 5 Steady-uiform flow For small slope si z ta b S b grs0 0 For turbulet flows, the shear stress ca be assumed to be related to the average velocity V ad frictio coefficiet C f, as Fially b C f V V g C f RS 0 6 S. Maskey (0) 8

9 Steady-uiform flow Chezy s formula V C CA RS 0 RS A 0 C S 0 P Maig s formula 0 A V R S AR S0 S 0 P 5 Remember: Discharge, = A V Hydraulic radius, R = A / P 7 Steady-uiform flow Normal depth depth based o uiform flow From Maig s formula Remember: A P 5 S 0 A P 5 / / S 0 For a wide rectagular chael (i.e. B >> h) P B R h Rectagular chael Wide rectagular chael 5 / ( Bh ) 5 ( B h ) / S0 h B S 0 Ca you work out similarly the relatioship for the ormal depth usig the Chezy s formula? 8 S. Maskey (0) 9

10 Uiform flow computatio i a compoud chael The flow velocity ad the carryig capacity of the chael are ormally differet for the mai chael ad the over bak flow or the floodplai. The roughess coefficiet may also vary i the mai chael ad the floodplai. A h h A A b b b b h The usualapproach is total Ai R i / i S / b b b That is, the total discharge is the sum of the discharges through each segmet of the cross sectio. 9 Specific eergy The eergy (i water head) due to the depth of water (h) plus its velocity (the velocity head) is the specific eergy. E V h h g ga h c Supecriticalflow rage Differetiatig w.r.t. h E mi de dh ga da dh For a give discharge there eists a mi. specific eergy E mi (at de/dh = 0): de V dh gd V gd With da/dh = B ad D = A/B T. V gh For a rectagular chael 0 S. Maskey (0) 0

11 Critical depth, Froude umber Water depth at E mi is called a critical depth, h c. V gd A D g h c gb q g h c E mi Supecriticalflow rage Froude umber (Fr) represets the ratio of iertia forces to gravity forces, give by V Fr At critical depth (h c ), Fr =. gd Subcritical ad supercritical flows Subcritical flow: Fr < Critical flow: Fr = Supercritical flow: Fr > V < (gd) V = (gd) V > (gd) Subcritical flow h h c Supercritical flow h c h S. Maskey (0)

12 Celerity, subcritical ad supercritical flow The critical velocity (gd) is also the speed, or more precisely, "celerity" of small gravity waves that occur i shallow water i ope chaels as a result of a disturbace to the free surface. c gd c A gravity wave ca be propagated upstream i water of subcritical flow (c > V), but ot i water of supercritical flow (c < V). gh For a rectagular chael Wave patters created by disturbaces: (a) Still water, V = 0; (b) subcritical flow, c > V; (c) critical flow, c = V; ad (d) supercritical flow, c < V. No-uiform, steady flow Gradually varied flow- backwater curve computatio (o-uiform, steady flow) 4 S. Maskey (0)

13 Gradually varied (o-uiform) steady flow Gradually varied (o-uiform) steady flow equatios ca be derived either by () Takig / t terms equal to zero (because steady flow) i the St. Veat equatio, or () From the cosideratio of total eergy (Beroulli equatio) betwee two sectios i a uiform chael. 5 Gradually varied (o-uiform) steady flow Equatio i terms of Froude Number (Fr): 4 A R dh S0 S f B d Fr Fr T OR dh d S 0 de S f dh S f ga P A 4/ 0/ Maig s formula S f = frictio slope, S 0 = chael bed slope, P = wetted perimeter, A = cross-sectio area, h = water depth, = legth alog the chael, = Maig s coefficiet, B T = top width of chael cross sectio, Fr = Froude umber, g = acceleratio due to gravity, = discharge, E = specific eergy. Eercise: Rewrite the formula for S f for a wide rectagular chael. Ca you workout S f usig Chezy s formula? 6 S. Maskey (0)

14 Gradually varied (o-uiform) steady flow Equatio i terms of ormal ad critical depths for a wide-rectagular chael: dh d S 0 h h h h c h C h c B gb S 0 C = Chezy s coefficiet, B = costat bed width, h = ormal depth, h c = critical depth. 7 Gradually varied flow flow profiles M (backwater curve) S f P A 4/ 0/ M (drawdow curve) M y Regio Regio y c Regio 8 S. Maskey (0) 4

15 Solutio of gradually varied steady flow equatio Numerical solutio (direct step solutio) dh S d Fr 0 S f The problem here is S f ad Fr are depedet o h, i.e. the equatio is implicit o h. h OR S0 h Fr Startig at the dowstream boudary, Fr compute the horizotal distace that correspods to a give chage i S 0 S f depth h. S f Startig at the dowstream boudary, compute chage i depth hfor a chose horizotal distace. 9 Solutio of gradually varied steady flow equatio Numerical solutio (improved Euler s method) dh S0 d Fr h S f S S Fr 0 f j/ h j Remember! Whe startig from dowstream towards upstream, make sure that is egative. Compute S f ad Fr with h = h j. h j h j h j S0 S Fr f h j / Compute S f ad Fr with h = h j+/ h j = water depth at sectio j, h j+ = water depth the sectio et to sectio j, h j+/ = water depth betwee sectios j ad j+. 0 S. Maskey (0) 5

16 Solutio of gradually varied steady flow equatio Numerical solutio (Predictor-corrector method) Predictor : Corrector : dh S0 d ( Fr h h S f ) S S Fr Remember! pre 0 f j hj cor j h j S0 S Fr Whe startig from dowstream towards upstream, make sure that t is egative. h j f Compute S f ad Fr with h = h j. h S0 S Fr f pre j h j Repeat Corrector step with a ew value of predictor h util the predictor h is close eough to corrector h. The ew value of predictor is ormally take as the curret value of the corrector h. Hydraulic jump Occurs whe supercritical flow chages ito a subcritical flow. h h 8Fr Hydraulic jump Subcritical Supercritical h V h V Practical applicatio: to dissipate eergy dowstream of dams, weirs, etc. Eeryloss, E E E h h 4h h S. Maskey (0) 6

17 Usteady, o-uiform flow Usteady Flow, St. Veat Equatio Flood Wave Propagatio Usteady o-uiform flow: movig flood wave H, Logitudial view of water surface i a ope chael H, H, at time T at time T X X at time T X 4 S. Maskey (0) 7

18 Usteady o-uiform flow Logitudial Profile: Water surface profile alog a river chael at a give time. Represets variatio i space X Hydrograph: Discharge at various times through a give sectio of a river. Represets variatio i time T Computatio of usteady o-uiform flow cosiders variatio i both space ad time together. 5 Sait Veat equatios for D chael flow Assumptios of Sait-Veat equatio Oe-dimesioal flow Gradually varyig flow i space ad time Hydrostatic pressure prevails, vertical acceleratio is egligible Stable ad relatively small bed slope Logitudial ais of the chael is approimated as a straight lie. Icompressible, homogeeous ad costat viscosity fluid Cosists of Cotiuity ad Mometum equatios Mometum equatio ca be derived from Euler equatio with a added bed shear force. 6 S. Maskey (0) 8

19 Sait Veat equatios Mometum force: V t V V A Gravity force compoet: gas 0 Pressure force: h g A Shear force due to frictio: gas f V g V S f V g Eergy lie Free surface h h V S 0 Chael bottom 7 Sait Veat equatios Equatig mometum ad applied forces: V t V V g h g S 0 0 S f V = /A t A h ga ga S 0 0 S f Local acceleratio term Covective acceleratio term Pressure force term Gravity force term Frictio force term 8 S. Maskey (0) 9

20 Sait Veat equatios Various flood wave approimatios from the Sait- Veat equatio: Cotiuity equatio A 0 t Mometum equatio t A h ga ga( S ( 0 S f ) 0 Kiematic wave Diffusio wave Full dyamic wave 9 Kiematic wave approimatio From cotiuity equatio: A 0 t Kiematic wave approimatio at o eglects ects acceleratio ad pressure terms of the mometum equatio, i.e. Note: If there is lateral iflow q, the cotiuity equatio becomes: A t q S0 S f This assumptio allows to use Maig or Chezy equatio: AR / S / OR CAR / S / 40 S. Maskey (0) 0

21 Kiematic wave approimatio I both Maig ad Chezy equatio, A ca be epressed i terms of as A P S Usig Maig equatio, the coefficiets alpha ad beta are give by / / 0.6 ad 0.6 Eercise: Verify that the give ad are for the Maig equatio ad derive them for the Chezy equatio. 4 Kiematic wave approimatio I cotiuity equatio, replacig A with ad simplifyig. 0 t 0 c t k c k is the kiematic wave celerity. Where c 5 5 V k A 4 S. Maskey (0)

22 Usteady flow ad flood wave propagatio A ote o umerical methods for the solutio D usteady flow 4 Usteady flow ad flood wave propagatio I practice almost always solved usig umerical methods. Fiite Differece Method is oe of the widely used umerical methods. Discretizatio is used i space ad time to approimate the partial differece equatio to a fiite differece (algebraic) equatio. 44 S. Maskey (0)

23 Fiite differece umerical solutio Uderstadig space-time discretisatio of umerical schemes t + t j j j j - = 0 j = 0 j- j j+ kow poits ukow poit iitial coditio boudary coditio 45 Fiite differece umerical solutio Eplicit ad Implicit schemes: I a eplicit scheme the value of the variable at time level + ca be computed directly (or eplicitly) from the values at time level. I a implicit scheme the computatio of the value of a variable at + ivolves oe or more of the values from the same time level (i.e. +). 46 S. Maskey (0)

24 Numerical solutio of kiematic wave (a eample) A eample of a eplicit scheme 0 c t j k j c j j k t t Partial differetial equatio TO Fiite differece equatio j j j j j Cr j ( Cr ) j Where t Cr ck C r is the Courat Number. j- j j+ kow poits ukow poit 47 Numerical solutio of kiematic wave (a eample) A eample of a implicit scheme 0 c t j j k Where j c j j k t Cr j C C t Cr ck Partial differetial equatio TO Fiite differece equatio + 0 j r r C r is the Courat Number. Multiply both sides by - t j j j- j j+ j j kow poits ukow poit 48 S. Maskey (0) 4

25 Fiite differece umerical solutio Stability of a scheme I umerical methods, the choice of the space step () ad time step (t) is importat! A eplicit scheme becomes ustable whe the Courat umber (C r = ct/) is greater tha. Where c is the celerity. A implicit solutio is always stable. However, Cr >> is ot recommeded. d 49 Fiite differece umerical solutio Iitial ad boudary coditios: Numerical schemes are based o the assumptio that variable values are kow at all poits o space at time level zero also called a iitial coditio. A umerical scheme also requires that values at all time levels at the startig poit o space (called upstream boudary coditio) ad/or at the ed poit o space (called the dowstream boudary coditio). 50 S. Maskey (0) 5

26 Commoly used iitial ad boudary coditios Iitial coditio: Global or local water depth ad steady state computatio Upstream boudary coditio: Discharge hydrograph Dowstream boudary coditio: -h relatioship Ed of Presetatio 5 S. Maskey (0) 6