2The Navier-Stokes and Euler Equations(12)

math

2TheNavier–StokesandEulerEquations–FluidandGasDynamics

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case,thesocalledp-system,withoutexternalforce:

ρt+(ρu)x=0 2 (ρu)t+ρu+p(ρ)x=0

Thisisanonlinearhyperbolicsystem,degenerateatthevacuumstateρ=0.ForanextensivestudyoftheRiemannproblemwereferto[7]andforthepioneeringproofofglobalweaksolutions,usingentropywavesandcompensatedcompactness,to[8].

Finally,weremarkthattheincompressibleinviscidSaint–VenantsystemofhydrologyhasthemathematicalstructureofanisotropiccompressibleEulersystemwithquadraticpressurelawin1or,resp.,2dimensions,wherethespatialground uctuationsplaytheroleofanexternalforce eld.

CommentsontheFigs.2.1–2.4AnimportantassumptioninthederivationoftheSaint–VenantsystemfromtheEuleror,resp.,Navier–Stokesequations–apartfromtheshallowwatercondition–isasmallnessassumptiononthevariationofthebottomtopography,i.e.gradZhastobesmall.Clearly,thisrestrictstheapplicabilityofthemodel,inparticularitsuseforwaterfallmodelling.Recently,anextensionoftheSaint–Venantsystemwaspresentedin[1],whicheliminatesallassumptionsonthebottomtopography.TherethecurvatureoftheriverbottomistakenintoaccountexplicitelyinthederivationfromthehydrostaticEulersystem(assumingasmall uidvelocityinorthogonaldirectiontothe uidbottom).WeremarkthatextensionsoftheSaint–Venantmodelstogranular ows(likedebrisavalanches)existintheliterature,seealso[1].

CommentsontheFigs.2.5–2.6Turbulent ows14arecharcacterizedbyseem-inglychaotic,randomchangesofvelocities,withvorticesappearingonavarietyofscales,occurringatsuf cientlylargeReynoldsnumber15.Non-turbulent owsarecalledlaminar,representedbystreamline ow,wheredifferentlayersofthe uidarenotdisturbedbyscaleinteraction.Simulationsofturbulent owsarehighlycomplicatedandexpensivesincesmallandlargescalesinthesolutionsoftheNavier–Stokesequationshavetoberesolvedcontemporarily.Varioussimpli-fyingattempts(‘turbulencemodeling’)exist,typicallybasedontime-averagingtheNavier–Stokesequationsandusing(moreorless)empiricalclosurecon-ditionsforthecorrelationsofvelocity uctuations.The owsdepictedintheFigs.2.5and2.6arehighlyturbulent,withapparentmicro-scales.

Weremarkthattheturbulentpartsofthe owsdepictedintheFigs.2.1–2.6aretwo-phase ows,duetotheairbubblesentrainedclosetothefreewater-airsurfaceinteractingwiththeturbulentwater ow.

Fig.2.5.TurbulentFlow,CascadadeAguaAzul,Chiapas,Mexico

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