Introducing Ideal Flow
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1 D f f f p D p D p D f T k p D e The Continit eqation The Naier Stokes eqations The iscos Flo Energ Eqation These form a closed set hen to thermodnamic relations are specified Introdcing Ideal Flo
2 Getting to Ideal Flo st ne Assmption: Constant densit flo Implies: The Continit eqation The Naier Stokes eqations Energ Eqation decoples nd ne Assmption:??
3 Constant Densit Flo D f f f p D p D p D The Continit eqation The Naier Stokes eqations
4 4 Constant Densit Flo f T k p D e The iscos Flo Energ Eqation T k e D Sbtract Momentm, se continit and take iscosit as constant This eqation of thermal energ controls ho thermal energ is condcted into the flid and generated b iscos action are redistribted b the flo This redistribtion hoeer has no impact back on the elocit or pressre fields of the flo, hich are determined entirel b the continit and momentm eqns
5 Constant Densit Iniscid Flo? 5
6 Constant Densit Irrotational Flo? 6
7 A Eropean Starling in an 9-m/s Airflo Irrotational 0 Rotational Sbmitted b: Chris Johnston Taken from: 7
8 Goerning Eqations of Ideal Flo Constant densit irrotational flo Continit 0 or 0 Laplace s eqation Momentm t g p C t Condition of irrotationalit 0 Decopled eqations Sole them separatel An soltion to these eqations are actall soltions to the N-S eqations 8
9 Understanding Ideal Flo Grait When can o ignore it? Bondar Conditions Principle of Sperposition 4 elocit Potential What is it? connection ith circlation Ho does it behae? In a flo? At infinit? 5 Forces on a General Bod in Arbitrar Motion 9
10 Grait When can o ignore it? Depth Depth same P P Stationar sbmarine in stagnant ocean Moing sbmarine in ocean
11 Bondar Conditions? Stationar srface Flo elocit =r n Moing srface Flo elocit =r n Srface elocit U s
12 Initial Conditions Initial conditions are not needed becase the instantaneos state of an ideal flo is entirel determined b the instantaneos state of the its bondar conditions See proof, Karamcheti section 99
13 The Principle of Sperposition Laplace s eqation is linear, ie different soltions to Laplace s eqation can be added together to make ne soltions What abot bondar conditions? What does this sa abot soling comple flo problems?
14 4a The Potential What is it? Eplicit eqation for the potential B A A,B 4
15 Can the circlation be non-ero? Stokes theorem S Ω nds ds C c nds Open Srface S ith Perimeter C ==0, so The circlation is ero: Wheneer Stokes Theorem ma be applied 5
16 Topolog of Potential Flo A B D bod X D bod A Y B D bod X Toroid A X Loops A and B are redcible the ma be contracted to a point ithot passing ot of the flid domain Hence X and Y are irredcible For the D bod loop A is reconcilable ith loop B A ma be made coincident ith B b moing and deforming them ithot passing ot of the flid domain Hence for the D bod X and Y are reconcilable bt A and X are irreconcilable A space hich contains no irredcible loops is termed simpl connected A space containing irredcible loops is called dobl or mltipl connected depending on the nmber of irredcible irreconcilable loops that ma be dran 6
17 Can the circlation be non-ero? Stokes theorem S Ω nds ds C c nds Open Srface S ith Perimeter C ==0, so The circlation is ero: Wheneer Stokes Theorem ma be applied 7
18 4b Behaior Of The elocit Potential And Related Qantities Within A Flo Within a flo - the elocit potential ma not reach a maimm or minimm, the elocit components ma not reach a maimm or minimm, the elocit magnitde ma not reach a maimm, 4 the pressre ma not reach a minimm 5 the elocit potential ma onl be determined p to an additie constant 6 the flo throgh a mltipl connected region is onl niqel determined if the circlations is specified See proofs, Karamcheti section 98 8
19 4c Behaior Of The elocit Potential And Related Qantities At Infinit As r r = φ/r ~ /r for a D rigid bod /r for a D dilating bod /r for a D rigid bod /r for a D dilating bod θ = /r φ/θ ~ /r for an D bod also /r for a D bod ith no circlation /r for a D bod ith circlation r r See proofs, Karamcheti section 96, 97 9
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