Sreamline Pahline Eulerian Lagrangian
Sagnaion Poin Flow V V V = + = + = + o V xi y j a V V xi y j o
Pahline and Sreakline
Insananeous Sreamlines Pahlines Sreaklines
Maerial Derivaive Acceleraion
D = + V D DT T = + V T D DV V = + V V D
Sreamline Coordinaes
HC Chen //09 Conrol volume and Sysem Sysem a collecion of maer of fixed ideniy which may move, flow and inerac wih is surroundings Sysem may change is size and shape coninuously, bu always conains he same mass. The sysem may inerac wih is surroundings hrough hea ransfer, exerion of pressure, shear forces and elecromagneic forces, ec. Conrol Volume a volume in space a geomery eniy, independen of mass hrough which fluid may flow Typical Conrol Volumes he air in earh amosphere, he waer in a lake, he blood in your vein, he coolan in a refrigeraor, or a single fluid paricle Sysem Approach Sysem Approach Lagrangian approach All he conservaion laws conservaion of mass, momenum, angular momenum, energy, ec. can be applied direcly o he sysem. Free-body diagram and similar analyses can be used for a sysem which consiss of he same mass similar o fluid saics. u i is ofen very difficul o idenify and rack a specific fluid sysem in moion. Polluan in a river? Exhaus gas from an engine? The air affeced by an airplane? Reynolds Transpor Theorem
HC Chen //09 Conrol Volume Approach Conrol Volume Approach Eulerian approach Aerodynamic force around airplanes, auomobiles, buildings, ec. Hydrodynamic forces around ships, submarines, offshore srucures, ec. I is difficul and no necessary o employ sysem approach I is more convenien o use he conrol volume approach by idenifying a specific volume in space surrounding he airplane, ship, or oher objecs of ineres and analyze he fluid wihin, hrough, and around ha volume In general, conrol volume may be a moving, deformable volume alloon, pison in moion, airplane ake-off/landing,. d d SYS d 0 bu d d CV d 0 Reynolds Transpor Theorem
HC Chen //09 Fundamenal Laws Conservaion of geomery of moion and deformaion of maer - ranslaion, roaion, linear and angular deformaions fluid kinemaics Conservaion of Mass - Coninuiy equaion Conservaion of momenum - Newon s second law Conservaion of Angular Momenum Conservaion of Energy - Firs Law of Thermodynamics Enropy, Irreversible flow - Second Law of Thermodynamics Equaion of sae, sress/rae-of-srain relaions, Fourier law, and oher consiuive equaions Lagrangian Descripion Paricle approach Maerial volume, maerial surface, maerial curve Fluids in maerial volume sysem will move, disor, and change size and shape, bu always consiss of he same fluid paricles Pahline r r ro, p p ro, V V r, o ec. MV o pahlines MV Reynolds Transpor Theorem 3
HC Chen //09 Eulerian Descripion Field Approach Abandon he edious and ofen unnecessary ask of racking individual paricles Individual paricles are no labeled and no disinguished from one anoher Focuses aenion on wha happens a a fixed poin or volume as differen paricles goes by r, p p r, V V r, ec. CV Lagrangian vs. Eulerican Lagrangian Approach man-o-man, ag each individual paricle difficul o rack many paricles a a ime may be irrelevan o he problem of ineres Eulerian Approach zone defense, observe fluid paricles in a preseleced, ofen fixed, conrol volume always focus on regions of ineres easy o seup experimen or compuaional domain u he conservaion laws need o be derived from paricle approach!!! Reynolds Transpor Theorem 4
HC Chen //09 Lagrangian and Eulerian Descripions Largrangian D V D DV V V V D Unseady effec Eulerian Convecive effec Lagrangian vs. Eulerian Lagrangian Eulerian Reynolds Transpor Theorem 5
HC Chen //09 Mahemaical formulaion Differenial formulaion Inegral CV formulaion Provide all deails of he flow Obey fundamenal laws a every poin in he fluid domain Solve problems wih minimum inpu informaion boundary condiions Produce differenial equaions ha are ofen difficul o solve Expensive for complex flows May give more informaion han acually needed Does no reveal all flow deails Obey fundamenal laws in an average manner over conrol volume Require more inpu, such as velociy profiles and pressure disribuions a convenien boundaries Simpler mahemaics, inegral equaions for global quaniies Less compuaionally inensive Ofen canno give as much as needed, yield only approximae answers Reynolds Transpor Theorem Kinemaics of Moving Conrol Volume Conrol Mass * Largrangian - sysem approach Conrol Volume * Eulerian - field approach Reynolds ranspor heorem * Conversion from Lagrangian o Eulerian descripion Reynolds Transpor Theorem 6
HC Chen //09 Reynolds Transpor Theorem 7 Reynolds Transpor Theorem V =V V =V CVI II I Fixed conrol surface and sysem boundary a ime Sysem boundary a ime + SYS = CV = CVI + I SYS + = CV-I + II II I CV sys CV sys Conrol volume and Sysem Reynolds Transpor Theorem I II CV CV CV II I CV sys sys sys CV in ou CV I 0 II 0 CV CV 0 sys 0 II I b A V b A V bd A V b A b b A V b A b b lim lim lim lim CV SYS b A V b A V D D
HC Chen //09 Fixed Conrol Volume nˆ S I I V III nˆ ds S II nˆ II V MS+ MS, CV=CV+ Maerial Volume Sysem Maerial Surface Sysem boundary CS = MS = S I + S II MV = CV = I + III MV + = II + III CV + = CV Fixed CV Conrol Volume Analysis Ouflow, V nˆ 0 Inflow, V nˆ 0 Reynolds Transpor Theorem 8
HC Chen //09 D D d rd, rd, rv, nda ˆ d MV CV CS D rd r d rd D, lim,, 0 MV MV MV lim r, d rd, 0 MV MV lim rd, rd, 0 MV MV lim r, r, d 0 MV MV CV lim rd, rd, rd, rd, 0 II III I III D D d rd, rd, r, VndA ˆ d MV CV CS D d rd, rd, lim rd, rd, D d 0 MV CV II I d ˆ I Vd nda across SI d ˆ II Vd nda across SII D d rd, rd, r, VndA ˆ D d MV CV CS Rae of change of F in MV Rae of change of F in CV Convecive Transpor flux of F across CS Reynolds Transpor Theorem 9
HC Chen //09 Moving and Deforming Conrol Volume MV = CV nˆ V r nˆ W CV+ Moving Conrol Volume V V W CV V MV+ CV W V V CV : Absolue fluid velociy wih respec o a fixed coordinae sysem : absolue velociy of he conrol volume : relaive fluid velociy wih respec o a moving conrol volume Reynolds Transpor Theorem 0
HC Chen //09 D D d rd, rd, r, WndA ˆ d MV CV CS Lagrangian descripion following he sysem ; Eulerian descripion following conrol volume Le b hen d b d D D D sys MV CV CS MV d b d b d bw nda ˆ d d d f only; d d d D sys Largrangian d bd bwnda ˆ d CV Eulerian CS V W V CV Conrol Volume Analysis Reynolds Transpor Theorem