Transcription of Navier-Stokes Equations { 2d case
1 navier -StokesEquations{2dcaseNSE(A)Equat ionanalysisEquationanalysisEquationanaly sisEquationanalysisEquationanalysisLamin ar owb etweenplates(A)Flowdowninclinedplane(A)T ips(A) navier -StokesEquations{2dcaseSOE32 11/2 FluidMechanicslecture3 navier -StokesEquations{2dcaseNSE(A)Equat ionanalysisEquationanalysisEquationanaly sisEquationanalysisEquationanalysisLamin ar owb etweenplates(A)Flowdowninclinedplane(A)T ips(A)NSE(A) conservationofmass,momentum. oftenwrittenassetofp de's di erentialform{ uid owatap oint 2dcase,incompressible ow:Continuityequation:@ux@x+@uy@y=0 conservationofmass seenb efore{p otential owNavier-StokesEquations{2dcaseNSE(A)Equ ationanalysisEquationanalysisEquationana lysisEquationanalysisEquationanalysisLam inar owb etweenplates(A)Flowdowninclinedplane(A)T ips(A)Momentumequations:@ux@t+ux@ux@x+uy @ux@y= 1 @p@x+ @2ux@x2+@2ux@y2 +fx@uy@t+ux@uy@x+uy@uy@y= 1 @p@y+ @2uy@x2+@2uy@y2 +fy (xandycmpts) 3variables,ux,uy,p linkedequations needtosimplfybyconsideringdetailsofprobl emNavier-StokesEquations{2dcaseNSE(A)Equ ationanalysisEquationanalysisEquationana lysisEquationanalysisEquationanalysisLam inar owb etweenplates(A)Flowdowninclinedplane(A)T ips(A)TheNSEare Non-linear{termsinvolvingux@ux@x Partialdi erentialequations{ux,pfunctionsofx,y,t 2ndorder{highestorderderivatives@2ux@x2 Coupled{momentumequationinvolvesp,ux,uyT wowaystosolvetheseequations1 Applytosimplecases{simplegeometry,simple conditions{andreduceequationsuntilwecans olvethem2 Usecomputationalmetho ds{CFD(SOE3212/3) navier -StokesEquations{ 2dcaseNSE(A)EquationanalysisEquationanal ysisEquationanalysisEquationanalysisEqua tionanalysisLaminar owb etweenplates(A)Flowdowninclinedplane(A)T ips(A)}}}}}}}}}}}}}}}
2 TheNSEare Non-linear{termsinvolvingux@ux@x Partialdi erentialequations{ux,pfunctionsofx,y,t 2ndorder{highestorderderivatives@2ux@x2 Coupled{momentumequationinvolvesp,ux,uyT wowaystosolvetheseequations1 Applytosimplecases{simplegeometry,simple conditions{andreduceequationsuntilwecans olvethem2 Usecomputationalmetho ds{CFD(SOE3212/3) navier -StokesEquations{ 2dcaseNSE(A)EquationanalysisEquationanal ysisEquationanalysisEquationanalysisEqua tionanalysisLaminar owb etweenplates(A)Flowdowninclinedplane(A)T ips(A)Equationanalysis(A)Considerthevari ousterms:@ux@t+ux@ux@x+uy@ux@y= 1 @p@x+ @2ux@x2+@2ux@y2 +fx@ux@t changeofuxatap ointNavier-StokesEquations{2dcaseNSE(A)E quationanalysisEquationanalysisEquationa nalysisEquationanalysisEquationanalysisL aminar owb etweenplates(A)Flowdowninclinedplane(A)T ips(A)Equationanalysis(A)Considerthevari ousterms:@ux@t+ux@ux@x+uy@ux@y= 1 @p@x+ @2ux@x2+@2ux@y2 +fxux@ux@x+uy@ux@y transp ort/advectionterm howdo es ow(ux;uy)moveux? non-linearNavier-StokesEquations{2dcaseN SE(A)EquationanalysisEquationanalysisEqu ationanalysisEquationanalysisEquationana lysisLaminar owb etweenplates(A)Flowdowninclinedplane(A)T ips(A)Equationanalysis(A)Considerthevari ousterms:@ux@t+ux@ux@x+uy@ux@y= 1 @p@x+ @2ux@x2+@2ux@y2 +fx 1 @p@x pressuregradient{usuallydrives owNavier-StokesEquations{2dcaseNSE(A)Equ ationanalysisEquationanalysisEquationana lysisEquationanalysisEquationanalysisLam inar owb etweenplates(A)Flowdowninclinedplane(A)T ips(A)Equationanalysis(A)Considerthevari ousterms:@ux@t+ux@ux@x+uy@ux@y= 1 @p@x+ @2ux@x2+@2ux@y2 +fx @2ux@x2+@2ux@y2 viscousterm{e ectofviscosity on ow hasadi usivee ectNavier-StokesEquations{2dcaseNSE(A)Eq uationanalysisEquationanalysisEquationan alysisEquationanalysisEquationanalysisLa minar owb etweenplates(A)Flowdowninclinedplane(A)T ips(A)Equationanalysis(A)Considerthevari ousterms.}}}}}}}}}}}}}}
3 @ux@t+ux@ux@x+uy@ux@y= 1 @p@x+ @2ux@x2+@2ux@y2 +fxfx externalb o dyforces{ {2dcaseNSE(A)EquationanalysisEquationana lysisEquationanalysisEquationanalysisEqu ationanalysisLaminar owb etweenplates(A)Flowdowninclinedplane(A)T ips(A)Laminar owb etweenplates(A)Fullydevelop edlaminar owb etweenin niteplatesaty= aWhatdoweexp ectfromthe ow?xy-a+a u=0atwalls Flowsymmetricaroundy=0 FlowparalleltowallsNavier-StokesEquation s{2dcaseNSE(A)EquationanalysisEquationan alysisEquationanalysisEquationanalysisEq uationanalysisLaminar owb etweenplates(A)Flowdowninclinedplane(A)T ips(A)@ux@t+ux@ux@x+uy@ux@y= 1 @p@x+ @2ux@x2+@2ux@y2 @uy@t+ux@uy@x+uy@uy@y= 1 @p@y+ @2uy@x2+@2uy@y2 navier -StokesEquations{2dcaseNSE(A)Equat ionanalysisEquationanalysisEquationanaly sisEquationanalysisEquationanalysisLamin ar owb etweenplates(A)Flowdowninclinedplane(A)T ips(A)@ux@t+ux@ux@x+uy@ux@y= 1 @p@x+ @2ux@x2+@2ux@y2 @uy@t+ux@uy@x+uy@uy@y= 1 @p@y+ @2uy@x2+@2uy@y2 Flowparalleltowalls{weexp ectuy=0;dpdy=0andux=ux(y) navier -StokesEq uations{2dcaseNSE(A)EquationanalysisEqua tionanalysisEquationanalysisEquationanal ysisEquationanalysisLaminar owb etweenplates(A)Flowdowninclinedplane(A)T ips(A)@ux@t+ux@ux@x+uy@ux@y= 1 @p@x+ @2ux@y2 @uy@t+ux@uy@x+uy@uy@y= 1 @p@y+ @2uy@x2+@2uy@y2 Flowparalleltowalls{weexp ectuy=0.}}}}}}}
4 Dpdy=0andux=ux(y) navier -StokesEquations{ 2dcaseNSE(A)EquationanalysisEquationanal ysisEquationanalysisEquationanalysisEqua tionanalysisLaminar owb etweenplates(A)Flowdowninclinedplane(A)T ips(A)@ux@t+ux@ux@x+uy@ux@y= 1 @p@x+ @2ux@y2 @uy@t+ux@uy@x+uy@uy@y= 1 @p@y+ @2uy@x2+@2uy@y2 Flowfullydevelop ed{nochangeinpro @@t=0;@@x=0 navier -StokesEquations{2dcaseNSE(A)Equat ionanalysisEquationanalysisEquationanaly sisEquationanalysisEquationanalysisLamin ar owb etweenplates(A)Flowdowninclinedplane(A)T ips(A)@ux@t+ux@ux@x+uy@ux@y= 1 @p@x+ @2ux@y2 @uy@t+ux@uy@x+uy@uy@y= 1 @p@y+ @2uy@x2+@2uy@y2 Flowfullydevelop ed{nochangeinpro @@t=0;@@x=0 navier -StokesEquations{2dcaseNSE(A)Equat ionanalysisEquationanalysisEquationanaly sisEquationanalysisEquationanalysisLamin ar owb etweenplates(A)Flowdowninclinedplane(A)T ips(A)Somomentumequationb ecomes0= 1 dpdx+ d2uxdy2 Integrateonce:ydpdx= duxdy+C1 Butaty=0,duxdy=0(symmetry),soC1= ux+C2 Butaty= a,ux=0,soC2=12a2dpdxNavier-StokesEquatio ns{2dcaseNSE(A)EquationanalysisEquationa nalysisEquationanalysisEquationanalysisE quationanalysisLaminar owb etweenplates(A)Flowdowninclinedplane(A)T ips(A)Somomentumequationb ecomes0= 1 dpdx+ d2uxdy2 Integrateonce:ydpdx= duxdy+C1 Butaty=0,duxdy=0(symmetry),soC1= ux+C2 Butaty= a,ux=0,soC2=12a2dpdxNavier-StokesEquatio ns{2dcaseNSE(A)EquationanalysisEquationa nalysisEquationanalysisEquationanalysisE quationanalysisLaminar owb etweenplates(A)Flowdowninclinedplane(A)T ips(A)Somomentumequationb ecomes0= 1 dpdx+ d2uxdy2 Integrateonce.}}}}}}}
5 Ydpdx= duxdy+C1 Butaty=0,duxdy=0(symmetry),soC1= ux+C2 Butaty= a,ux=0,soC2=12a2dpdxNavier-StokesEquatio ns{2dcaseNSE(A)EquationanalysisEquationa nalysisEquationanalysisEquationanalysisE quationanalysisLaminar owb etweenplates(A)Flowdowninclinedplane(A)T ips(A)Somomentumequationb ecomes0= 1 dpdx+ d2uxdy2 Integrateonce:ydpdx= duxdy+C1 Butaty=0,duxdy=0(symmetry),soC1= ux+C2 Butaty= a,ux=0,soC2=12a2dpdxNavier-StokesEquatio ns{2dcaseNSE(A)EquationanalysisEquationa nalysisEquationanalysisEquationanalysisE quationanalysisLaminar owb etweenplates(A)Flowdowninclinedplane(A)T ips(A)Somomentumequationb ecomes0= 1 dpdx+ d2uxdy2 Integrateonce:ydpdx= duxdy+C1 Butaty=0,duxdy=0(symmetry),soC1= ux+C2 Butaty= a,ux=0,soC2=12a2dpdxNavier-StokesEquatio ns{2dcaseNSE(A)EquationanalysisEquationa nalysisEquationanalysisEquationanalysisE quationanalysisLaminar owb etweenplates(A)Flowdowninclinedplane(A)T ips(A)Finalsolutionux(y)=12 y2 a2 dpdx{equationofaparab olaAlso,rememb erthat = @ux@ySofromthisweseethatinthiscase =ydpdxNavier-StokesEquations{2dcaseNSE(A )EquationanalysisEquationanalysisEquatio nanalysisEquationanalysisEquationanalysi sLaminar owb etweenplates(A)Flowdowninclinedplane(A)T ips(A)Flowdowninclinedplane(A){Flowofliq uiddowninclinedplanehyx uxTakex-comp onentmomentumequation@ux@t+ux@ux@x+uy@ux @y= 1 @p@x+ @2ux@x2+@2ux@y2 +fxNavier-StokesEquations{2dcaseNSE(A)Eq uationanalysisEquationanalysisEquationan alysisEquationanalysisEquationanalysisLa minar owb etweenplates(A)Flowdowninclinedplane(A)T ips(A)Note.}}}}}}}
6 1 Steady ow2ux(y)only3 Nopressuregradient4fx=gsin Equationb ecomesd2uxdy2= g sin {2dcaseNSE(A)EquationanalysisEquationana lysisEquationanalysisEquationanalysisEqu ationanalysisLaminar owb etweenplates(A)Flowdowninclinedplane(A)T ips(A)Note:1 Steady ow2ux(y)only3 Nopressuregradient4fx=gsin Equationb ecomesd2uxdy2= g sin {2dcaseNSE(A)EquationanalysisEquationana lysisEquationanalysisEquationanalysisEqu ationanalysisLaminar owb etweenplates(A)Flowdowninclinedplane(A)T ips(A)Boundaryconditions: lowersurface{ux(0)=0 upp ersurface{duxdy=0 Solutionux=g sin hy y22 navier -StokesEquations{2dcaseNSE(A)Equat ionanalysisEquationanalysisEquationanaly sisEquationanalysisEquationanalysisLamin ar owb etweenplates(A)Flowdowninclinedplane(A)T ips(A)Boundaryconditions: lowersurface{ux(0)=0 upp ersurface{duxdy=0 Solutionux=g sin hy y22 navier -StokesEquations{2dcaseNSE(A)Equat ionanalysisEquationanalysisEquationanaly sisEquationanalysisEquationanalysisLamin ar owb etweenplates(A)Flowdowninclinedplane(A)T ips(A)Tips(A)MostNSEproblemswillb etime-indep ow,andoneco eeitherpressuredriven(sonoviscousterm)or sheardriven( ,sonopressureterm).}}}}}}}}
7 Thus,mostNSEproblemswillleadtoa2ndorderO DEforavelo citycomp onent(uxoruy)asafunctionofoneco ordinate(xory).Thuswewouldexp ecttointegratetwice,andtoimp osetwob oundaryconditions. Awallb oundaryconditionpro ducesa Afreesurfacepro ducesazerogradientcondition.