introduction to fs hydrodynamics

8/12/2019 Introduction to FS Hydrodynamics

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IntroductionIntroduction toto

FREE SURFACEFREE SURFACEHYDRODYNAMICSHYDRODYNAMICS

(module 2)(module 2)

Prof Arthur E MynettProf Arthur E Mynett

Professor of Hydraulic EngineeringProfessor of Hydraulic Engineering

in addition to the lecture notes by Maskey et al.in addition to the lecture notes by Maskey et al.



basic principles:basic principles:

(i)(i) physicalphysical conceptsconcepts

(ii)(ii) mathematicalmathematical formulationsformulations

Introduction toIntroduction to





learning objectives:learning objectives:

(re)fresh knowledge(re)fresh knowledge

•• physicalphysical conservation principles:conservation principles:

– – MASS / MOMENTUM / ENERGYMASS / MOMENTUM / ENERGY

•• mathematicalmathematical formulations:formulations:

– – CONTINUITY equationCONTINUITY equation

– – MOMENTUM equations (F = ma)MOMENTUM equations (F = ma)

(Euler, Navier (Euler, Navier --Stokes)Stokes)

– – ENERGY equation (Bernouilli)ENERGY equation (Bernouilli)





•• physical conservation principles:physical conservation principles:

– – REFERENCEREFERENCE COCO--ORDINATEORDINATE SYSTEMSYSTEM

– – EULER / LAGRANGE descriptionEULER / LAGRANGE description

– –

laminar / turbulent flows (500 <laminar / turbulent flows (500 <

ReRe

< 700)< 700)

•• mathematical formulations:mathematical formulations:

– – TOTAL (MATERIAL)TOTAL (MATERIAL) DERIVATIVEDERIVATIVE

– – FROUDEFROUDE number,number, REYNOLDSREYNOLDS number number

(Euler, Navier (Euler, Navier --Stokes)Stokes) – – SPECIFICSPECIFIC ENERGY,ENERGY, CRITICALCRITICAL DEPTHDEPTH

(Bernouilli)(Bernouilli)



basic principlesbasic principles

•• conservation of MASSconservation of MASS

– – continuity principle (control volume)continuity principle (control volume)

– – for for inincompressible flow (compressible flow (hydrohydrodynamics):dynamics):

“what comes in“what comes in – – must go out” must go out”

•• conservation of MOMENTUM / ENERGYconservation of MOMENTUM / ENERGY

– – essentially derived fromessentially derived from SAMESAME (!)(!) equation:equation:

– – Newton’s second LawNewton’s second Law F = maF = ma



example:example: hydraulic jumphydraulic jump

•• given bc’s upstream hgiven bc’s upstream h11, v, v11

•• => what are downstream=> what are downstream hh22, v, v22 ??

_______ _______

__ __ hh11 ____ ____

-->> vv11

___________________________________ ___________________________________



example:example: hydraulic jumphydraulic jump

•• conservation of MASSconservation of MASS – – continuity principlecontinuity principle

hh11 vv11 = h= h22 vv22

(“what comes in(“what comes in – – must go out”)must go out”)

•• conservation of MOMENTUM (mv)conservation of MOMENTUM (mv)

– – F = maF = ma can also be written ascan also be written as Fdt = d(mv)Fdt = d(mv)

•• Fdt = ½Fdt = ½ g (hg (h1122

– – hh2222

)dt)dt•• d(mv) = (mv)d(mv) = (mv)outout – – (mv)(mv)inin

= (= (vv22dthdth22)v)v22 – – ((vv11dthdth11)v)v11



excercise:excercise: impinging jet impinging jet

•• given bc’s upstream (diameter dgiven bc’s upstream (diameter d11, velocity v, velocity v11))

•• => what is horizontal Force on wall=> what is horizontal Force on wall FFxx ??

FF

xx dt =dt =

--

((vv

11 dt) (¼dt) (¼

dd

11

22

) v) v

11

___d ___d11 ____ ____

_________=>v _________=>v11 <= F<= F

xx

|

|

|

|

|

||

|

|

|



basic principles (ctd)basic principles (ctd)



– – for for inincompressible flow (compressible flow (hydrohydrodynamics):dynamics):


•• conservation of MOMENTUM / ENERGYconservation of MOMENTUM / ENERGY





BernouilliBernouilli EquationEquation



– – for incompressible flow (hydrodynamics):for incompressible flow (hydrodynamics):


•• conservation of MOMENTUM /conservation of MOMENTUM / ENERGYENERGY





BernoulliBernoulli EquationEquation

•• basic assumptionsbasic assumptions – – steady steady flow conditions (d/dt=0)flow conditions (d/dt=0)

– – validvalid along a streamlinealong a streamline

(bottom, free surface, …)(bottom, free surface, …)

•• practical (hydraulics) formulationpractical (hydraulics) formulation

z + p/z + p/ g + vg + v22 /2g = H (constant) [L] /2g = H (constant) [L]

•• z = position head [L]z = position head [L]

•• p/p/ g = pressure head [L]g = pressure head [L]

•• (z+p/(z+p/ g) = piezometric head [L]g) = piezometric head [L]

•• vv22 /2g = velocity head [L] /2g = velocity head [L]

•• H = energy head [L]H = energy head [L]



example:example: flow over weir flow over weir

•• given bc’s upstream h1, v1given bc’s upstream h1, v1

•• => what are weir conditions=> what are weir conditions h2, v2 ?h2, v2 ?

__ __ h1h1 ____ ____ __ h2, v2 __ __ h2, v2 __

-->> v1v1 _____________ _____________

/ /

_________________/________________________ _________________/________________________




•• specific energy (in terms of h, q)specific energy (in terms of h, q)(NB appropriate choice of reference frame !!)(NB appropriate choice of reference frame !!)

h + vh + v

22

/2g = H (constant) /2g = H (constant)

q = vhq = vh

leads toleads to

h + qh + q22 /2gh /2gh22 = H= H (3(3rdrd order eq in h)order eq in h)




•• critical depth hcritical depth hcc and velocity vand velocity vcc

(NB choice of appropriate reference frame !!)(NB choice of appropriate reference frame !!)

hhcc = 2/3 H= 2/3 H

vvcc22 /2g = 1/3 H /2g = 1/3 H

vizviz

vvcc

22 /2g = h /2g = hcc /2 /2

=> v=> vcc22 / (gh / (ghcc) = Fr ) = Fr 22 = 1= 1



SpecificSpecific Energy (h, q)Energy (h, q)

•• (ii) super critical flow over weir (ii) super critical flow over weir (NB choice of appropriate reference frame !!)(NB choice of appropriate reference frame !!)

h + vh + v

22

/2g = H /2g = H

q = vhq = vh

leads toleads to

h + qh + q22 /2gh /2gh22 = H= H (3(3rdrd order eq in h)order eq in h)



Energy LOSSESEnergy LOSSES – – Carnot’s RuleCarnot’s Rule

•• sudden expansionsudden expansion

(continuity + momentum):(continuity + momentum):

H = HH = H11 – – HH22

= (v= (v11 – – vv22))22 / 2g / 2g



•• contraction (coefficientcontraction (coefficient ~ …)~ …)

•• expansion (=> energy lossexpansion (=> energy loss – – Carnot)Carnot)

•• pipe flow (=> frictionpipe flow (=> friction – – head loss)head loss)

•• velocity head (from continuity)velocity head (from continuity)

•• piezometric head (from Bernouilli)piezometric head (from Bernouilli)

example:example: flow throughflow through culvert culvert



momentum principle:momentum principle: F =F = mmaa

•• aa = F/m= F/m

•• Dv/Dt = F/mDv/Dt = F/m

•• D/Dt{D/Dt{vvii(t,x,y,z)(t,x,y,z)} = F} = Fii /m /m

u u / / tt ++ u u / / xx dx/dt +dx/dt + u/u/ y y dy/dt +dy/dt + u/u/ z z dz/dt = f dz/dt = f xx

v/v/ tt ++ v/v/ xx dx/dt +dx/dt + v/v/ y y dy/dt +dy/dt + v/v/ z z dz/dt = f dz/dt = f yy

w w / / tt ++ w w / / xx dx/dt +dx/dt + w/w/ y y dy/dt +dy/dt + w/w/ z z dz/dt = f dz/dt = f zz




•• aa = F/m= F/m


•• D/Dt{vD/Dt{vii(t,x,y,z)} = F(t,x,y,z)} = Fii /m /m

u u / / t +t + u u / / xx dx/dtdx/dt ++ u/u/ y y dy/dtdy/dt ++ u/u/ z z dz/dtdz/dt = f = f xx

v/v/ t +t + v/v/ xx dx/dtdx/dt ++ v/v/ y y dy/dtdy/dt ++ v/v/ z z dz/dtdz/dt = f = f yy

w w / / t +t + w w / / xx dx/dtdx/dt ++ w/w/ y y dy/dtdy/dt ++ w/w/ z z dz/dtdz/dt = f = f zz




•• aa = F/m= F/m


•• D/Dt{vD/Dt{vii(t,x,y,z)} = F(t,x,y,z)} = Fii /m /m

u u / / t +t + uu u u / / x +x + vv u/u/ y y ++ ww u/u/ z z = f = f xx

v/v/ t +t + uu v/v/ x +x + vv v/v/ y y ++ ww v/v/ z z = f = f yy

w w / / t +t + uu w w / / x +x + vv w/w/ y y ++ ww w/w/ z z = f = f zz

D/Dt = TOTAL (material) DERIVATIVED/Dt = TOTAL (material) DERIVATIVE



momentum principle:momentum principle: F = mF = maa

•• a =a = F/mF/m

•• Dv/Dt =Dv/Dt = F/mF/m

•• D/Dt{vD/Dt{vii(t,x,y,z)} =(t,x,y,z)} = FFii /m /m

u u / / t + ut + u u u / / x + vx + v u/u/ y y + w+ w u/u/ z z == - - 1/ 1/ p/ p/ x x

v/v/ t + ut + u v/v/ x + vx + v v/v/ y y + w+ w v/v/ z z == - - 1/ 1/ p/ p/ y y

w w / / t + ut + u w w / / x + vx + v w/w/ y y + w+ w w/w/ z z == - - 1/ 1/ p/ p/ z z – –g g

EULEREULER EQUATIONSEQUATIONS




•• a =a = F/mF/m



Du/Dt Du/Dt == - - 1/ 1/ p/ p/ x x

Dv/Dt Dv/Dt == - - 1/ 1/ p/ p/ y y

Dw/Dt Dw/Dt == - - 1/ 1/ p/ p/ z z – –g g

EULEREULER EQUATIONSEQUATIONS




•• a =a = F/mF/m



Du/Dt Du/Dt == - - 1/ 1/ ( ( p/ p/ x x ++ xx xx / / x x ++ yx yx / / y y ++ zx zx / / z )z )

Dv/Dt Dv/Dt == - - 1/ 1/ ( ( p/ p/ y y ++ xy xy / / x x ++ yy yy / / y y ++ zy zy / / z z ))

Dw/Dt Dw/Dt == - - 1/ 1/ ( ( p/ p/ z +z + xz xz / / x x ++ yz yz / / y y ++ zz zz / / z )z ) – –g g

NAVIERNAVIER--STOKESSTOKES EQUATIONSEQUATIONS





•• physical conservation principles:physical conservation principles:

– – REFERENCEREFERENCE COCO--ORDINATEORDINATE SYSTEMSYSTEM

– – EULER / LAGRANGE descriptionEULER / LAGRANGE description

– – Laminar / turbulent flows (500 <Laminar / turbulent flows (500 < ReRe < 700)< 700)

•• mathematical formulations:mathematical formulations:

– – TOTAL (MATERIAL)TOTAL (MATERIAL) DERIVATIVEDERIVATIVE

– – FROUDEFROUDE number,number, REYNOLDSREYNOLDS number number

(Euler, Navier (Euler, Navier --Stokes)Stokes) – – SPECIFICSPECIFIC ENERGY,ENERGY, CRITICALCRITICAL DEPTHDEPTH

(Bernouilli)(Bernouilli)





•• physicalphysical conservation principles:conservation principles:

– – MASS / MOMENTUM / ENERGYMASS / MOMENTUM / ENERGY

•• mathematicalmathematical formulations:formulations:

– – CONTINUITY equationCONTINUITY equation

– – MOMENTUM equationsMOMENTUM equations (F = ma)(F = ma)

(Euler, Navier (Euler, Navier --Stokes)Stokes)

– – ENERGY equation (Bernouilli)ENERGY equation (Bernouilli)



basic principles:basic principles:

(i)(i) physicalphysical conceptsconcepts

(ii)(ii) mathematicalmathematical formulationsformulations

Introduction toIntroduction to





IntroductionIntroduction toto



Dr Shreedhar MaskeyDr Shreedhar Maskey

Dr Luigia BrandimarteDr Luigia Brandimarte

Prof Dano RoelvinkProf Dano Roelvink

introduction to fs hydrodynamics

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