Fluid Mechanics: A few conceptsFluid Mechanics: A few concepts Aline Cotel University of Michigan...
Transcript of Fluid Mechanics: A few conceptsFluid Mechanics: A few concepts Aline Cotel University of Michigan...
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FluidMechanics:Afewconcepts
AlineCotelUniversityofMichigan
COESSING2019
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SchemaFcofprocessesthatcontributetovariabilityintheoceansurfaceboundarylayer
[R.Weller,WHOI]
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Outline
1. Diffusionversusturbulence–Mixingexample2. Laminarandturbulentflows3. DensitystraFficaFon4. EffectofrotaFon
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en.wikipedia.org
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MolecularDiffusion
• DefiniFon-BrownianmoFon,i.e.randommovementofthediffusingparFcles.
• DiffusionversussFrring• DiffusiveFmescale?Takeaguessassumingthatthediffusivitycoefficientforthedyeinwateris10-9m2/s.
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EDDIESINTHEOYASHIOCURRENTG.K.VallisPrincetonUniversityThephotographfromNASAshowseddiesintheOyashioCurrent,offtheKamchatkaPeninsulaintheBeringSea,inMarch,1992.Thecurrent,partofthethewesternboundarycurrentoftheNorthPacificsubpolargyre,isbaroclinicallyunstableandsea-iceprovidesflowvisualizaFon.Snowcover(white)coversthethickericeflowsjusttotherightofthecentreoftheimage.
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LaminarandTurbulentflows
• Examplebelow–velocityprofileinastraightpipe.Whichoneistheturbulentcase?
• Determinedbythevalue/rangeoftheReynoldsnumber
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Reynoldsnumber
Re=UL/ν• Uistheflowvelocity(m/s),Ltherelevantlengthscale(m)
andνthekinemaFcviscosity(m2/s).ν=μ/ρ,whereμisthedynamicsviscosityandρthedensity.
• DeterminesthetransiFonfromlaminartoturbulentflows• Importantindeterminingturbulenteddyspectruminterms
ofsize,velocityandrotaFonperiod.ImportantinpredicFngmixinginindustrialandenvironmentalflows.
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Laminar
TransiFonal
Turbulent
Fullyturbulent
Reynolds’experiment
Re
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What’sthemostefficientconfiguraFon?
• Mixing2fluidsthruinjecFon.• ChoiceofconfiguraFon:– OpFon1:(a)isbejer– OpFon2:(b)isbejer– OpFon3:Thereisnodifference
(a) (b)
Cross-secFonofpipe
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DefiniFonofturbulence• Atleast2ways:
1. ThroughitsproperFes–ThisgivesintuiFon2. MathemaFcaldefiniFon–Necessaryfortheoryand
modeling
Isotropicturbulence–Sc=25Jetinambientfluid,V=2cm/s,15cmhorizontalfieldofview
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ProperFesofturbulence• FlowsthataredescribedbytheNavier-StokesequaFonsand
correspondingboundarycondiFons.ConFnuumapproximaFonandNewtonianfluid.
• Turbulentflowsarehighlyirregular.ChaoFc,randombutnotcompletely.Expectedtohavesomestructures:eddies!
• Turbulentflowsareunstable.TheyaretheresultsofflowinstabiliFes.
• Turbulentflowsarestronglynonlinear.Thevelocityfieldcanbeassociatedwithrandomwaves.
• TurbulentflowsarehighlyvorFcal,i.e.characterizedbyalargeamountofvorFcity.
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ProperFescont’d• Turbulentflowsarecharacterizedbyhighratesofdiffusionof
mass,momentum,heat,chemicalspecies…
• TurbulentflowsarehighlydissipaFve.Mechanicalenergyistransformedintointernal energy.Needasupplyofenergytosustainit.
Largeeddiesinaturbulentboundarylayer
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Kolmogorov’senergycascadetheory(1941)
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Navier-StokesequaFons
• Oneofthelastunresolvedproblemsofclassicalphysics.
• Describeallthecomplexityoffluidflows.
€
D! u Dt
+ 2" Ω ×! u = −
1ρo
∇p +ρρo
! g +! F
! 9!
!Now!the!forces!can!be!!
o gravitational:!!
o Viscous!stresses!and!pressure:!!
!The!viscous!stresses!are!due!to!velocity!gradients!in!the!fluid,!and!the!gradients!of!the!stresses!are!what!cause!a!force!on!the!fluid.!!
The!net!force!in!the!xLdirection!is!then:!
!!So!that!!!!Same!in!all!directions:!
!!And!!!!Where!!!!!!
! This!is!the!viscous!stress!tensor!acting!on!a!fluid!element.!And!finally,!!
!!
This!is!it!!It’s!valid!for!any!fluid!and!any!general!motion.!!!For!a!Newtonian!fluid,!the!viscous!stresses!are!proportional!to!the!strain!rate!and!viscosity:!!
dFx,surf = [∂∂x(σ xx ) +
∂∂y(σ yx ) +
∂∂z(σ zx )]dxdydz
(dF
d∀)surf = −∇p + (dF
d∀)viscous
dFgrav = ρgdxdydz
σ ij=
− p + τ xx τ yx τ zxτ xy − p + τ yy τ zyτ xz τ yz − p + τ zz
⎛
⎝
⎜⎜⎜
⎞
⎠
⎟⎟⎟
dFxd∀
= −∂p∂x
+∂∂x(σ xx ) +
∂∂y(σ yx ) +
∂∂z(σ zx )
(dF
d∀)viscous = ∇.τ ij
τ ij=
τ xx τ yx τ zxτ xy τ yy τ zyτ xz τ yz τ zz
⎛
⎝
⎜⎜⎜
⎞
⎠
⎟⎟⎟
ρ dV
dt= −∇p +∇.τ ij + ρg
τ xx = 2µ∂u∂x
τ yy = 2µ∂v∂y
τ zz = 2µ∂w∂z
Gravity:pressuregradientandbuoyancyFricFon:windstressCoriolisForce
RotaFon
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GeneralCirculaFon
TemporalandspaFalscalesofvariousphysicaloceanprocessesTempo
ral(Time)Scale
HorizontalSpaFalScale
Image credit: Exploring the World Ocean, inside cover
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Otherdimensionlessparameters
• Richardsonnumber:
• Rossbynumber:wherefistheCoriolisfrequency,Ω:planetrotaFonΘ:laFtude
Ri = (Δρ ρ)gδU 2
Ro = ULf
f = 2Ωsinθ
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Oceanmeasurements
Lagrangian->followtheparFclesintheflowasafuncFonofFme.Instrumentsarefollowingtheoceancurrents.Eulerian->moFonofparFclesarebothafuncFonofFmeandspace.MooredADCPsforexample.
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THEOCEANISALSOSTRATIFIEDANDTHEEARTHISROTATING…
Afewmorethings….
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Mostoftheoceanisverycold:~4°CSurfacewaterscanreach~28°CWaterhasahighheatcapacity:Itcanabsorbandreleasealotofenergywithoutchangingtemperaturemuch.
Temperatureofseawater
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en.wikipedia.org
Salinityisameasureofthegramsofdissolvedsaltsperkilogramofseawater
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HowdoTemperature,SalinityandPressureinfluencedensity(ρ)?
Seawater�sdensityisafuncFonofT,PandS
AsTemperature↑
ρ↓
AsSalinity↑
ρ↑
AsPressure↑
ρ↑(note:seawaterisonlyalijlecompressible…6%change)
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Densityofseawater-whydoesitmajer?
• VerFcalvariaFonofdensitydeterminesstaFcstability,i.e.,howstronglystraFfiedthewatercolumnis.• DensityvariaFonsdetermineaveryimportantpartofthehorizontallyvaryingpressureandthustheoceancirculaFon.
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Annualmeanofseasurfacetemperature,SST
T(z=0)
madefromabout5millionhydrocasts
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Salinity
Salinityattheoceansurface
LowSalinityalongtheequatorduetoexcessrainattheequator
HighSalinityat30°N&30°SduetoexcessevaporaFon
Isolatedregionsoflowsalinitynearcoasts.Couldbe:upwelling,orriverinput
TheAtlanFcissalFerthanthePacific
NetevaporaFonintheAtlanFc,precipitaFoninthePacific=freshwaterflux
TheMediterraneanisverysalty(alotofevaporaFon).
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StraFficaFonconFnued
InternalwavesformaFonoffaslope->Importantforsedimenttransportintotheinterioroftheocean+slopestability
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NoCorioliseffectattheequator
MoreSpin
NoSpinalignedwiththeaxisofrotaFon
MoreSpin
AndweareonarotaFngplanet…
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3rulesoftheCoriolisEffect1) Anyobjectmovinghorizontallyonearth�ssurfacehasits
trajectorydeflected
2)Thefasteranobjectmoves,thegreateritstendencytodeflect
3)Thetendencytodeflectisgreatestatthepolesanddecreasestozeroattheequator.
We don�t notice it on land because other forces overwhelm its influence
(fromHowtheOceanWorks,byMarkDenny)ww2010.atmos.uiuc.edu
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ImpactofRotaFon
• TendstokeepwatercolumnsverFcallyaligned
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Next…
Wewilllookatsomeofthesephysicalprocessesinthelaboratory!!