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Lec8MECH222F-EulerianandLagrangianFrameshandout

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MECH 222
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Ref.: WikimediaMECH 222F – EULERIAN AND LAGRANGIAN
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LECTURE OVERVIEW• EULERIAN FRAME OF REFERENCE• LAGRANGIAN FRAME OF REFERENCE• EULERIAN AND LAGRANGIAN DERIVATIVES
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EULERIAN FRAMEYou are already familiar with the Eulerianframe of reference, but you call it something else:----Q: When we say that the wind speed measured by an anemometer is 10 m/s, is that an Eulerian measurement?LabframeInertialframeYE3
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LAGRANGIAN FRAME•e.g. ping pong ball floating in a river࠵ౡ࠵?= 0࠵ౡ࠵?=1࠵ౡ࠵?=2The Lagrangianframe of reference is the frame of reference that follows the fluid as it moves:Ref. WalmartRef. Wikipedia
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EULERIAN VERSUS LAGRANGIANQ: Can you give an example when you want to know what is happening in an Eulerian frame?Q: Can you give an example when you are interested in a Lagrangian frame?•In general, it is only possible to relate Eulerian and Lagrangian frames for infinitesimal motions. •For finite motions you have to integrate …·massflowratethroughpipe·windforcesonabuilding-dispersionofpollution
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RELATING EULERIAN AND LAGRANGIAN FRAMESConsider a particle that moves with the fluid. Let’s start by considering a steady1D flow in the ࠵౥࠵?direction. Then, ࠵౉࠵?࠵౥࠵?,࠵౦࠵?,࠵౧࠵?,࠵ౡ࠵?=࠵ౢ࠵?(࠵౥࠵?):࠵౥࠵?࠵౥࠵?1࠵ౢ࠵?(࠵౥࠵?1)࠵౥࠵?2=࠵౥࠵?1+࠵ౢ࠵?࠵౥࠵?1࠵౑࠵?࠵ౡ࠵?࠵ౢ࠵?(࠵౥࠵?2)࠵౑࠵?࠵ౢ࠵?࠵౑࠵?࠵ౡ࠵?=࠵ౢ࠵?࠵౥࠵?2− ࠵ౢ࠵?(࠵౥࠵?1)࠵౑࠵?࠵ౡ࠵?=࠵༕࠵?࠵ౢ࠵?࠵༕࠵?࠵౥࠵?×࠵ౢ࠵?࠵౥࠵?1࠵౑࠵?࠵ౡ࠵?࠵౑࠵?࠵ౡ࠵?=࠵ౢ࠵?࠵༕࠵?࠵ౢ࠵?࠵༕࠵?࠵౥࠵?࠵ౢ࠵?࠵౥࠵?2=࠵ౢ࠵?࠵౥࠵?1+࠵༕࠵?࠵༕࠵?࠵༕࠵?࠵༕࠵?࠵౥࠵?2− ࠵౥࠵?1= ࠵ౢ࠵?࠵౥࠵?1+࠵༕࠵?࠵༕࠵?࠵༕࠵?࠵༕࠵?×࠵ౢ࠵?࠵౥࠵?1࠵౑࠵?࠵ౡ࠵?IimIst=0t=dt3m/s1-iit-definitionofderivativefluisisconnectedatspeed4(x,/LagranginEulerianderivativespace(followfluid)steadyIthefluidspeedCunsteady)fluidacceleratesconstant
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Now, consider a particle that moves with the fluid but the fluid velocity is the same everywhere but varies with time, ࠵౉࠵?࠵౥࠵?,࠵౦࠵?,࠵౧࠵?,࠵ౡ࠵?=࠵ౢ࠵?(࠵ౡ࠵?):࠵౥࠵?࠵ౢ࠵?(࠵ౡ࠵?1)࠵ౢ࠵?(࠵ౡ࠵?2)࠵౑࠵?࠵ౢ࠵?࠵౑࠵?࠵ౡ࠵?=࠵ౢ࠵?࠵ౡ࠵?2− ࠵ౢ࠵?(࠵ౡ࠵?1)࠵౑࠵?࠵ౡ࠵?=࠵༕࠵?࠵ౢ࠵?࠵༕࠵?࠵ౡ࠵?So, combining this result with the result on the previous slide, if a flow is 1D and unsteady, i.e. ࠵ౢ࠵?=࠵ౢ࠵?࠵౥࠵?,࠵ౡ࠵?, then:࠵౑࠵?࠵ౢ࠵?࠵౑࠵?࠵ౡ࠵?=࠵༕࠵?࠵ౢ࠵?࠵༕࠵?࠵ౡ࠵?+࠵ౢ࠵?࠵༕࠵?࠵ౢ࠵?࠵༕࠵?࠵౥࠵?Generalizing to 3D (steps are the same), i.e. ࠵ౢ࠵?=࠵ౢ࠵?࠵౥࠵?,࠵౦࠵?,࠵౧࠵?,࠵ౡ࠵?, then:࠵౑࠵?࠵ౢ࠵?࠵౑࠵?࠵ౡ࠵?=࠵ష࠵?࠵ౢ࠵?࠵ష࠵?࠵ౡ࠵?=࠵༕࠵?࠵ౢ࠵?࠵༕࠵?࠵ౡ࠵?+࠵ౢ࠵?࠵༕࠵?࠵ౢ࠵?࠵༕࠵?࠵౥࠵?+࠵ౣ࠵?࠵༕࠵?࠵ౢ࠵?࠵༕࠵?࠵౦࠵?+࠵౤࠵?࠵༕࠵?࠵ౢ࠵?࠵༕࠵?࠵౧࠵?RELATING EULERIAN AND LAGRANGIAN FRAMESLagrangianEuleriandevitativenotationforvagranginIIEulerian
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DIFFERENTIAL EQUATION FORM OF RTTWe can also derive the same result using the chain rule:࠵౑࠵?࠵ౢ࠵?࠵౑࠵?࠵ౡ࠵?=࠵ష࠵?࠵ౢ࠵?࠵ష࠵?࠵ౡ࠵?=࠵༕࠵?࠵ౢ࠵?࠵༕࠵?࠵ౡ࠵?+࠵ౢ࠵?࠵༕࠵?࠵ౢ࠵?࠵༕࠵?࠵౥࠵?+࠵ౣ࠵?࠵༕࠵?࠵ౢ࠵?࠵༕࠵?࠵౦࠵?+࠵౤࠵?࠵༕࠵?࠵ౢ࠵?࠵༕࠵?࠵౧࠵?࠵౑࠵?࠵ౢ࠵?(࠵౥࠵?,࠵౦࠵?,࠵౧࠵?,࠵ౡ࠵?)࠵౑࠵?࠵ౡ࠵?=࠵༕࠵?࠵ౢ࠵?࠵༕࠵?࠵౥࠵?࠵༕࠵?࠵౥࠵?࠵༕࠵?࠵ౡ࠵?+࠵༕࠵?࠵ౢ࠵?࠵༕࠵?࠵౦࠵?࠵༕࠵?࠵౦࠵?࠵༕࠵?࠵ౡ࠵?+࠵༕࠵?࠵ౢ࠵?࠵༕࠵?࠵౧࠵?࠵༕࠵?࠵౧࠵?࠵༕࠵?࠵ౡ࠵?+࠵༕࠵?࠵ౢ࠵?࠵༕࠵?࠵ౡ࠵?࠵༕࠵?࠵ౡ࠵?࠵༕࠵?࠵ౡ࠵?=࠵ౢ࠵?࠵༕࠵?࠵༕࠵?࠵༕࠵?࠵༕࠵?+࠵ౣ࠵?࠵༕࠵?࠵༕࠵?࠵༕࠵?࠵౦࠵?+࠵౤࠵?࠵༕࠵?࠵༕࠵?࠵༕࠵?࠵౧࠵?+࠵༕࠵?࠵༕࠵?࠵༕࠵?࠵༕࠵?࠵౑࠵?࠵ౣ࠵?࠵౑࠵?࠵ౡ࠵?=࠵ష࠵?࠵ౣ࠵?࠵ష࠵?࠵ౡ࠵?=࠵༕࠵?࠵ౣ࠵?࠵༕࠵?࠵ౡ࠵?+࠵ౢ࠵?࠵༕࠵?࠵ౣ࠵?࠵༕࠵?࠵౥࠵?+࠵ౣ࠵?࠵༕࠵?࠵ౣ࠵?࠵༕࠵?࠵౦࠵?+࠵౤࠵?࠵༕࠵?࠵ౣ࠵?࠵༕࠵?࠵౧࠵?࠵౑࠵?࠵౤࠵?࠵౑࠵?࠵ౡ࠵?=࠵ష࠵?࠵౤࠵?࠵ష࠵?࠵ౡ࠵?=࠵༕࠵?࠵౤࠵?࠵༕࠵?࠵ౡ࠵?+࠵ౢ࠵?࠵༕࠵?࠵౤࠵?࠵༕࠵?࠵౥࠵?+࠵ౣ࠵?࠵༕࠵?࠵౤࠵?࠵༕࠵?࠵౦࠵?+࠵౤࠵?࠵༕࠵?࠵౤࠵?࠵༕࠵?࠵౧࠵?In general, for any quantity ࠵ీ࠵?convected with the fluid: ࠵ష࠵?࠵ీ࠵?࠵ష࠵?࠵༕࠵?=࠵༕࠵?࠵ీ࠵?࠵༕࠵?࠵༕࠵?+࠵౉࠵? � ∇ ࠵ీ࠵?⟺࠵ష࠵?࠵౉࠵?࠵ష࠵?࠵༕࠵?=࠵༕࠵?࠵౉࠵?࠵༕࠵?࠵༕࠵?+࠵౉࠵? � ∇࠵౉࠵?rateofchangeinfixedspaceh**00000
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EXAMPLEA velocity field is given by ࠵౉࠵?࠵౥࠵?,࠵౦࠵?,࠵౧࠵?,࠵ౡ࠵?=3࠵౥࠵?࠵ౡ࠵?,2࠵౥࠵? − 3࠵౦࠵?+࠵౧࠵? − ࠵ౡ࠵?, 7࠵౥࠵?2࠵౦࠵?࠵౧࠵?3What is the acceleration of a fluid particle at ࠵౥࠵?,࠵౦࠵?,࠵౧࠵?,࠵ౡ࠵?= (1,2,3,4)?YVWInx-direction:d=B=+kw+wu+wbu-wibydz=3x+3ut(t)+12x-3y+2-490]+[7xyz37%)=3x+9x+2=3(1)+9(1)(4)=147my2voruberIny-direction:a*****iszI-+(3xt)(2)+(2x-3y+z-H(-3)+7xyz3(1)--I+6xt-6x+9y32+3t+7x2yz3=-+6(1)(4)-6(1)+9(2)-3(3)+3(4)+7(1)-()(35=416m/st·Allm
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EXAMPLEA velocity field is given by ࠵౉࠵?࠵౥࠵?,࠵౦࠵?,࠵౧࠵?,࠵ౡ࠵?=3࠵౥࠵?࠵ౡ࠵?,2࠵౥࠵? − 3࠵౦࠵?+࠵౧࠵? − ࠵ౡ࠵?, 7࠵౥࠵?2࠵౦࠵?࠵౧࠵?3and the corresponding density variation is ࠵༌࠵?࠵౥࠵?,࠵౦࠵?,࠵౧࠵?,࠵ౡ࠵?=5࠵౥࠵?+2࠵౥࠵?࠵౦࠵? − 3࠵౥࠵?࠵౧࠵?࠵ౡ࠵?+2࠵ౡ࠵?.What can you say about the density of a fluid particle at ࠵౥࠵?,࠵౦࠵?,࠵౧࠵?,࠵ౡ࠵?= (1,2,3,4)?WWzu*=10=1+4+v6p+w6p--dtet5tSchdydz=-3xz+2+3xt(5+2y-327)+(2x-by+z-t)(2x)+7xyz3(3xt)=-3(1)(3)+2+3/)(4)(5+2(2)-3(3)(4))+(1-3(2)+3-4][2]+7(z)(3))-3(41)=-4877=-4877-==(- 3xz+2)+3xt(5+zy-3zt)+(2x-3y+z-t)(2x)+7xyz3)-3xt)=-3(1)(3)+2+3(1)(4)(5+2(2)-3(3)(4)+(2(1)-3(2)+3-4)(2)+7(2)(3)(-3(4))=-4877
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RECAP•What is an Eulerian frame of reference?•What is a Lagrangian frame of reference?For anything, ࠵ీ࠵?, convected with the fluid: ࠵ష࠵?࠵ీ࠵?࠵ష࠵?࠵༕࠵?=࠵༕࠵?࠵ీ࠵?࠵༕࠵?࠵༕࠵?+࠵౉࠵? � ∇ ࠵ీ࠵?
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