fluid mechanics: a few conceptsfluid mechanics: a few concepts aline cotel university of michigan...

FluidMechanics:Afewconcepts

AlineCotelUniversityofMichigan

COESSING2019

SchemaFcofprocessesthatcontributetovariabilityintheoceansurfaceboundarylayer

[R.Weller,WHOI]

Outline

1.  Diffusionversusturbulence–Mixingexample2.  Laminarandturbulentflows3.  DensitystraFficaFon4.  EffectofrotaFon

en.wikipedia.org

MolecularDiffusion

•  DefiniFon-BrownianmoFon,i.e.randommovementofthediffusingparFcles.

•  DiffusionversussFrring•  DiffusiveFmescale?Takeaguessassumingthatthediffusivitycoefficientforthedyeinwateris10-9m2/s.

EDDIESINTHEOYASHIOCURRENTG.K.VallisPrincetonUniversityThephotographfromNASAshowseddiesintheOyashioCurrent,offtheKamchatkaPeninsulaintheBeringSea,inMarch,1992.Thecurrent,partofthethewesternboundarycurrentoftheNorthPacificsubpolargyre,isbaroclinicallyunstableandsea-iceprovidesflowvisualizaFon.Snowcover(white)coversthethickericeflowsjusttotherightofthecentreoftheimage.

LaminarandTurbulentflows

•  Examplebelow–velocityprofileinastraightpipe.Whichoneistheturbulentcase?

•  Determinedbythevalue/rangeoftheReynoldsnumber

Reynoldsnumber

Re=UL/ν•  Uistheflowvelocity(m/s),Ltherelevantlengthscale(m)

andνthekinemaFcviscosity(m2/s).ν=μ/ρ,whereμisthedynamicsviscosityandρthedensity.

•  DeterminesthetransiFonfromlaminartoturbulentflows•  Importantindeterminingturbulenteddyspectruminterms

ofsize,velocityandrotaFonperiod.ImportantinpredicFngmixinginindustrialandenvironmentalflows.

Laminar

TransiFonal

Turbulent

Fullyturbulent

Reynolds’experiment

Re

What’sthemostefficientconfiguraFon?

•  Mixing2fluidsthruinjecFon.•  ChoiceofconfiguraFon:– OpFon1:(a)isbejer– OpFon2:(b)isbejer– OpFon3:Thereisnodifference

(a) (b)

Cross-secFonofpipe

DefiniFonofturbulence•  Atleast2ways:

1.  ThroughitsproperFes–ThisgivesintuiFon2.  MathemaFcaldefiniFon–Necessaryfortheoryand

modeling

Isotropicturbulence–Sc=25Jetinambientfluid,V=2cm/s,15cmhorizontalfieldofview

ProperFesofturbulence•  FlowsthataredescribedbytheNavier-StokesequaFonsand

correspondingboundarycondiFons.ConFnuumapproximaFonandNewtonianfluid.

•  Turbulentflowsarehighlyirregular.ChaoFc,randombutnotcompletely.Expectedtohavesomestructures:eddies!

•  Turbulentflowsareunstable.TheyaretheresultsofflowinstabiliFes.

•  Turbulentflowsarestronglynonlinear.Thevelocityfieldcanbeassociatedwithrandomwaves.

•  TurbulentflowsarehighlyvorFcal,i.e.characterizedbyalargeamountofvorFcity.

ProperFescont’d•  Turbulentflowsarecharacterizedbyhighratesofdiffusionof

mass,momentum,heat,chemicalspecies…

•  TurbulentflowsarehighlydissipaFve.Mechanicalenergyistransformedintointernal energy.Needasupplyofenergytosustainit.

Largeeddiesinaturbulentboundarylayer

Kolmogorov’senergycascadetheory(1941)

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

GeneralCirculaFon

TemporalandspaFalscalesofvariousphysicaloceanprocessesTempo

ral(Time)Scale

HorizontalSpaFalScale

Image credit: Exploring the World Ocean, inside cover

Otherdimensionlessparameters

•  Richardsonnumber:

•  Rossbynumber:wherefistheCoriolisfrequency,Ω:planetrotaFonΘ:laFtude

Ri = (Δρ ρ)gδU 2

Ro = ULf

f = 2Ωsinθ

Oceanmeasurements

Lagrangian->followtheparFclesintheflowasafuncFonofFme.Instrumentsarefollowingtheoceancurrents.Eulerian->moFonofparFclesarebothafuncFonofFmeandspace.MooredADCPsforexample.

THEOCEANISALSOSTRATIFIEDANDTHEEARTHISROTATING…

Afewmorethings….

Mostoftheoceanisverycold:~4°CSurfacewaterscanreach~28°CWaterhasahighheatcapacity:Itcanabsorbandreleasealotofenergywithoutchangingtemperaturemuch.

Temperatureofseawater

en.wikipedia.org

Salinityisameasureofthegramsofdissolvedsaltsperkilogramofseawater

HowdoTemperature,SalinityandPressureinfluencedensity(ρ)?

Seawater�sdensityisafuncFonofT,PandS

AsTemperature↑

ρ↓

AsSalinity↑

ρ↑

AsPressure↑

ρ↑(note:seawaterisonlyalijlecompressible…6%change)

Densityofseawater-whydoesitmajer?

• VerFcalvariaFonofdensitydeterminesstaFcstability,i.e.,howstronglystraFfiedthewatercolumnis.• DensityvariaFonsdetermineaveryimportantpartofthehorizontallyvaryingpressureandthustheoceancirculaFon.

Annualmeanofseasurfacetemperature,SST

T(z=0)

madefromabout5millionhydrocasts

Salinity

Salinityattheoceansurface

LowSalinityalongtheequatorduetoexcessrainattheequator

HighSalinityat30°N&30°SduetoexcessevaporaFon

Isolatedregionsoflowsalinitynearcoasts.Couldbe:upwelling,orriverinput

TheAtlanFcissalFerthanthePacific

NetevaporaFonintheAtlanFc,precipitaFoninthePacific=freshwaterflux

TheMediterraneanisverysalty(alotofevaporaFon).

StraFficaFonconFnued

InternalwavesformaFonoffaslope->Importantforsedimenttransportintotheinterioroftheocean+slopestability

NoCorioliseffectattheequator

MoreSpin

NoSpinalignedwiththeaxisofrotaFon

MoreSpin

AndweareonarotaFngplanet…

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

ImpactofRotaFon

•  TendstokeepwatercolumnsverFcallyaligned

Next…

Wewilllookatsomeofthesephysicalprocessesinthelaboratory!!