Transcription of Acknowledgement Computational Fluid Dynamics …
1 1 Computational Fluid Dynamics An IntroductionSimulation in Computer GraphicsUniversity of FreiburgWS 04/05 University of Freiburg -Institute of Computer Science -Computer Graphics LaboratoryThis slide set is based on:nJohn D. Anderson, Jr., Computational Fluid Dynamics The basics with Applications, McGraw-Hill, Inc.,New York, ISBN 0-07-001685-2 AcknowledgementUniversity of Freiburg -Institute of Computer Science -Computer Graphics LaboratoryMotivationD. Enright, S. Marschner, R. Fedkiw, Animation and Rendering of Complex Water Surfaces, Siggraph2002, ACM TOG, vol.
2 21, pp. 736-744, of Freiburg -Institute of Computer Science -Computer Graphics Laboratorynintroductionnpre-requisitesng overning equationsncontinuity equationnmomentum equationnsummarynsolution techniquesnLax-WendroffnMacCormacknnumer ical aspectsnrecent research in graphicsOutline 2 University of Freiburg -Institute of Computer Science -Computer Graphics Laboratorynphysical aspects of Fluid flow are governed bythree principlesnmass is conservednforce = mass acceleration (Newton s second law)nenergy is conserved (not considered in this lecture)
3 Nprinciples can be described with integral equations orpartial differential equationsnin CFD, these governing equations are replaced by discretizedalgebraic formsnalgebraic forms provide quantities at discrete pointsin time and space, no closed-form analytical solutionIntroductionUniversity of Freiburg -Institute of Computer Science -Computer Graphics LaboratoryNumerical Solution -OverviewGoverningEquationsDiscretizatio nSystem DifferenceDiscreteNodal ValuesLax-WendroffMacCormackUniversity of Freiburg -Institute of Computer Science -Computer Graphics Laboratorynx, y, z 3D coordinate systemnt timen (x,y,z,t) densitynv(x,y,z,t) velocitynv(x,y,z,t) = (u(x,y,z,t), v(x,y,z,t), w(x,y,z,t) )
4 TContinuous QuantitiesUniversity of Freiburg -Institute of Computer Science -Computer Graphics LaboratoryProblemv (x,y,z,t) (x,y,z,t)v (x,y,z,t+ t) (x,y,z,t+ t) 3 University of Freiburg -Institute of Computer Science -Computer Graphics Laboratorynintroductionnpre-requisitesng overning equationsncontinuity equationnmomentum equationnsummarynsolution techniquesnLax-WendroffnMacCormacknnumer ical aspectsnrecent research in graphicsOutlineUniversity of Freiburg -Institute of Computer Science -Computer Graphics Laboratoryninfinitesimally small Fluid element moving with the flown(x1,y1,z1 ) position at time t1n(x2,y2,z2 ) position at time t2nv1(x1,y1,z1,t1)
5 = (u(x1,y1,z1,t1),v(x1,y1,z1,t1),w(x1,y1,z 1,t1))Tnv2(x2,y2,z2,t2) = (u(x2,y2,z2,t2),v(x2,y2,z2,t2),w(x2,y2,z 2,t2))Tn 1(x1,y1,z1,t1)n 2(x2,y2,z2,t2)Substantial Derivative of University of Freiburg -Institute of Computer Science -Computer Graphics LaboratorynTaylor series at point 1 t2 t1 Substantial Derivative of ()()()()12112112112112tttzzzyyyxxx + + + += 11212112121121211212 + + + = tttzzzttyyyttxxxtt 11111212 + + + = twzvyuxtt University of Freiburg -Institute of Computer Science -Computer Graphics Laboratorynsubstantial derivative of nsubstantial derivative = local derivative + convective derivativenlocal derivative time rate of change at a fixed locationnconvective derivative time rate of change due to Fluid flownsubst.
6 Derivative = total derivative with respect to timeSubstantial DerivativetzwyvxuDtD + + + TzyxtDtD = + ,,)(v 4 University of Freiburg -Institute of Computer Science -Computer Graphics Laboratoryndivergence of velocity v= time rate of change of the volume V of a moving Fluid element per unit volumeDivergence of v()DtVDVzwyvxu 1= + + = vUniversity of Freiburg -Institute of Computer Science -Computer Graphics Laboratorynintroductionnpre-requisitesng overning equationsncontinuity equationnmomentum equationnsummarynsolution techniquesnLax-WendroffnMacCormacknnumer ical aspectsnrecent research in graphicsOutlineUniversity of Freiburg -Institute of Computer Science -Computer Graphics Laboratorynmass is conservedninfinitesimally small Fluid element moving with the flownfixed mass m, variable volume V.
7 Variable density ntime rate of change of the mass of the moving fluidelement is zero Continuity EquationVm =()0=+= =DtVDDtDVDtVDDtmD 01=+DtVDVDtD 0= +v DtDUniversity of Freiburg -Institute of Computer Science -Computer Graphics LaboratoryContinuity Equation0= +v DtDdivergence of velocity -time rate of change of volumeof a moving Fluid element pervolumesubstantial derivative time rate of change ofdensity of a moving Fluid element 5 University of Freiburg -Institute of Computer Science -Computer Graphics Laboratorynnon-conservation form (considers moving element)
8 Nmanipulationnconservation form (considers element at fixed location)Continuity Equation0= +v DtD()vvvv + = + + = +ttDtD()0= + v tUniversity of Freiburg -Institute of Computer Science -Computer Graphics Laboratorynmotivation for conservation formninfinitesimally small element at a fixed locationnmass flux through elementndifference of mass inflow and outflow = net mass flownnet mass flow = time rate of mass increaseContinuity Equation()0= + v tnet mass flowper volumetime rate of massincrease per volumeUniversity of Freiburg -Institute of Computer Science -Computer Graphics Laboratorynintroductionnpre-requisitesng overning equationsncontinuity equationnmomentum equationnsummarynsolution techniquesnLax-WendroffnMacCormacknnumer ical aspectsnrecent research in graphicsOutlineUniversity of Freiburg -Institute of Computer Science -Computer Graphics Laboratorynconsider a moving Fluid elementnphysical principle.
9 F= m a(Newton s second law)ntwo sources of forcesnbody forcesngravitynsurface forcesnbased on pressure distribution on the surfacenbased on shear and normal stress distribution on the surfacedue to deformation of the Fluid elementnormalstressMomentum Equationpressurefluidelementgravityshear stressfrictionvelocityvelocity 6 University of Freiburg -Institute of Computer Science -Computer Graphics Laboratorynf= body force per unit massngravity: f = m g/ m = g body force on Fluid element f(dxdydz)Body Forcedxdzdyfluid elementUniversity of Freiburg -Institute of Computer Science -Computer Graphics Laboratorynconsider the xcomponentnpressure forceacts orthogonalto surface intothe Fluid elementnnet pressure force in xdirectionPressure Forcedxdydzp dydz( p + p/ x dx)
10 Dydzdzdydxxpdzdydxxppp = + University of Freiburg -Institute of Computer Science -Computer Graphics Laboratorynnormal stress related to the time rate of change of volumenshear stress related to the time rate of change of the shearing deformationn jk stress in kdirection applied to a surface perpendicularto the jaxisn xx normal stress in xdirectionn zx, yx shear stresses in x directions Stress xxxy yxUniversity of Freiburg -Institute of Computer Science -Computer Graphics Laboratorynnormal stress is related to pressure orthogonal to surfacenshear stress is related to friction parallel to surfacenfriction and pressure are related to the velocity gradient nfriction (shear stress) models viscosity (viscous flow)
