2.29 Numerical Fluid Mechanics Fall 2011 Lecture 6

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1 REVIEW of Lecture Numercal Flud Mechancs Fall 2011 Lecture 6 Contnuum Hypothess and conservaton laws Macroscopc Propertes Materal covered n class: Dfferental forms of conservaton laws Materal Dervatve (substantal/total dervatve) Conservaton of Mass Dfferental Approach Integral (volume) Approach Use of Gauss Theorem Incompressblty Reynolds Transport Theorem Conservaton of Momentum (Cauchy s Momentum equatons) The Naver-Stokes equatons Consttutve equatons: Newtonan flud Naver-stokes, compressble and ncompressble 2.29 Numercal Flud Mechancs PFJL Lecture 6, 1 1

2 Flud flow modelng: the Naver-Stokes equatons and ther approxmatons Cont d Today s Lecture References : Chapter 1 of J. H. Ferzger and M. Perc, Computatonal Methods for Flud Dynamcs. Sprnger, New York, thrd edton, Chapter 4 of I. M. Cohen and P. K. Kundu. Flud Mechancs. Academc Press, Fourth Edton, 2008 Chapter 4 n F. M. Whte, Flud Mechancs. McGraw-Hll Companes Inc., Sxth Edton For today s lecture, any of the chapters above suffce Note each provde a somewhat dfferent perspectve 2.29 Numercal Flud Mechancs PFJL Lecture 6, 2 2

3 Integral Conservaton Law for a scalar d dv d dv ( v. n ) da q. n da s dv dt CM CV dt fxed CS CS CV fxed Advectve fluxes ("convectve" fluxes) Other transports (dffuson, etc) Sum of sources and snks terms (reactons, et c) CV, fxed ρ,φ s Φ v q Applyng the Gauss Theorem, for any arbtrary CV gves:.(v). q s t For a common dffusve flux model (Fck s law, Fourer s law): q k Conservatve form of the PDE.(v) t.( k) s 2.29 Numercal Flud Mechancs PFJL Lecture 6, 3 3

4 Strong-Conservatve form of the Naver-Stokes Equatons ( v) d Cons. of Momentum: vdv v ( v. n ) da p nda.nda gdv dt CV CS CS CS CV F Applyng the Gauss Theorem gves: p. g dv CV CV, fxed ρ,φ s Φ v For any arbtrary CV gves: v.( vv) p. g t Wth Newtonan flud + ncompressble + constant μ: v Momentum:.(v v) p 2 v g q t Mass:.v 0 Equatons are sad to be n strong conservatve form f all terms have the form of the dvergence of a vector or a tensor. For the th Cartesan component, n the general Newtonan flud case: Wth Newtonan flud only: v u u j 2 u j.( vv ). p e e j e g x e t x j x 3 x j 2.29 Numercal Flud Mechancs PFJL Lecture 6, 4 4

5 Naver-Stokes Equatons: For an Incompressble Flud wth constant vscosty V(x,y,z) Flud Velocty Feld Conservaton of Mass Hydrostatc Pressure: -g z for z postve upward Dynamc Pressure P = P actual -gz Naver-Stokes Equaton Densty Knematc vscosty 2.29 Numercal Flud Mechancs PFJL Lecture 6, 5 5

6 Incompressble Flud Pressure Equaton Naver-Stokes Equaton Conservaton of Mass Dvergence of Naver-Stokes Equaton V)= V= - Dynamc Pressure Posson Equaton More general than Bernoull Vald for unsteady and rotatonal flow 2.29 Numercal Flud Mechancs PFJL Lecture 6, 6 6

7 Incompressve Flud Vortcty Equaton Vortcty Naver-Stokes Equaton curl of Naver-Stokes Equaton 2.29 Numercal Flud Mechancs PFJL Lecture 6, 7 7

8 Invscd Flud Mechancs Euler s Equaton Naver-Stokes Equaton: ncompressble, constant vscosty If also nvscd flud Euler s Equatons 2.29 Numercal Flud Mechancs PFJL Lecture 6, 8 8

9 Invscd Flud Mechancs Bernoull Theorems Theorem 1 Theorem 2 Irrotatonal Flow, ncompressble Steady, Incompressble, nvscd, no shaft work, no heat transfer Flow Potental Defne Naver-Stokes Equaton X Along stream lnes and vortex lnes Introduce P T = Thermodynamc pressure 2.29 Numercal Flud Mechancs PFJL Lecture 6, 9 9

10 Potental Flows Integral Equatons Irrotatonal Flow Flow Potental Conservaton of Mass Laplace Equaton Mostly Potental Flows: Only rotaton occurs at boundares due to vscous terms In 2D: Velocty potental : u, v x y Stream functon : u, v y x Snce Laplace equaton s lnear, t can be solved by superposton of flows, called panel methods What dstngushes one flow from another are the boundary condtons and the geometry: there are no ntrnsc parameters n the Laplace equaton 2.29 Numercal Flud Mechancs PFJL Lecture 6, 10 10

11 Potental Flow Boundary Integral Equatons Green s Theorem n Green s Functon Homogeneous Soluton S V Boundary Integral Equaton Dscretzed Integral Equaton Lnear System of Equatons Panel Methods 2.29 Numercal Flud Mechancs PFJL Lecture 6, 11 11

12 MIT OpenCourseWare Numercal Flud Mechancs Fall 2011 For nformaton about ctng these materals or our Terms of Use, vst:

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