3.3 Properties of Vortex Structures

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Ira Atkins
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1 .0 - Maine Hydodynamics, Sping 005 Lecte Maine Hydodynamics Lecte 8 In Lecte 8, paagaph 3.3 we discss some popeties of otex stctes. In paagaph 3.4 we dedce the Benolli eqation fo ideal, steady flow. 3.3 Popeties of Votex Stctes 3.3. Votex Stctes otex line is a line eeywhee tangent to ω. otex line ω Ω ω Ω otex tbe (filament) is a bndle of otex lines. otex tbe ω otex lines

2 otex ing is a closed otex tbe. sketch and two pictes of the podction of otex ings fom oifices ae shown in Figes,, and 3 below. (Figes,3: Van Dyke, n lbm of Flid Motion 98 p.66, 7) side iew U Γ ω coss section ω U ω Fige : Sketch of otex ing podction

3 3.3. No Net Flx of Voticity Thogh a Closed Sface Calcls identity, fo any ecto : ( ) }{{} =0 ω ω =0 V ω = }{{} ω nˆ ds = 0 Diegence S oticity flx Theoem i.e. The net oticity flx thogh a closed sface is zeo. (a) No net oticity flx thogh a otex tbe: (Voticity Flx) in = (Voticity Flx) ot ( ω nˆ) in δ in =(ω nˆ) ot δ ot (ω nˆ) ot ω n ˆ = 0 (ω nˆ) in (b) Voticity cannot stop anywhee in the flid. It eithe taeses the flid beginning o ending on a bonday o closes on itself (otex ing). ω ω 3

4 3.3.3 Conseation of Voticity Flx 0=Γ 3 = d x = ω nds ˆ = 0 C 3 S 3 C 3 ˆn 3 ˆn C C Γ = d x = ω nˆds = ω nˆds = Γ C S S Theefoe, ciclation is the same in all cicits embacing the same otex tbe. Fo the special case of a otex tbe with small aea: Γ= ω = ω ω ω n application of the eqation aboe is displayed in the fige below: ω ω = ω = ω / 4

5 3.3.4 Votex Stctes ae Mateial Stctes Conside a mateial patch m on a otex tbe at time t. m m By definition, ω nˆ =0 on n Then, Γ m = d x = ω nds ˆ = 0 m m t time t +Δt, m moes, and fo an ideal flid nde the inflence of conseatie body foces, Kelin s theoem states that Γ m =0 So, ω nˆ =0on m still, i.e., m still on the otex tbe. Theefoe, the otex tbe is a mateial tbe fo an ideal flid nde the inflence of conseatie foces. In the same manne it can be shown that a otex line is a mateial line, i.e., it moes with the flid. 5

6 3.3.5 Votex stetching Conside a small otex filament of length L and adis R, whee by definition ω is tangent to the tbe. R Γ = ω = constant (in time) Stokes Theoem Kelins Theoem Bt tbe is mateial with olme = L = πr L = constant in time (continity) Γ ω ω = = = constant Volme L L s a otex stetches, L inceases, and since the olme is constant (fom continity), and R decease, and de to the conseation of the angla momentm, ω inceases. In othe wods, Votex stetching L ω (conseation of angla momentm) and R (continity) 6

7 3.3.6 Smmay on Votex Stctes ω Γ Γ R ω Votex ing length L = πr [L] Coss sectional aea = π [L ] Votex ing olme = L = const [L 3 ] continity Voticity ω = [T ] Ciclation Γ = const Kelin s theoem Γ = ω = const oticity flx thogh Γ U = const [L T ] [L T ] [L T ] 7

8 Continity elates length atios = L = const L as L as L L Kelin s theoem + Continity elate length atios to Γ, ω, U Γ /L L U Γ = const U U Γ as L U Γ /L L ω Γ = const ω ω Γ as L ω Example : Example : ω Γ ω ω Γ L L Γ ω < ω >ω L <L > Γ =Γ ω <ω U <U Gien Fom continity only Fom Kelin s theoem Fom Kelin s theoem + continity Fom Kelin s theoem + continity 8

9 3.4 Benolli Eqation fo Steady ( =0), Ideal(ν = 0), t Rotational flow ( ) p = f Viscos flow: Naie-Stokes Eqations (Vecto Eqations) p = f( ) Ideal flow: Benolli Eqation (Scala eqation) Steady, iniscid Ele eqation (momentm eqation): ( ) p = + gy () ρ Fom Vecto Calcls we hae ( ) = ( ) +( ) + ( )+ ( ) ( ) = + ( ) ( ) = ( ) whee = Fom the peios identity and Eqation () we obtain ( ) ( ) p () ( ) = + gy }{{} ρ }{{} momentm () enegy Theefoe, ( ) ( ) p D + ρ + gy =0= Dt p + ρ + gy steamline pathline i.e., p + ρ + gy = constant on a steamline In geneal, + p + gy = F (Ψ) whee Ψ is a tag fo a paticla steamline. ρ ssmptions: Ideal flid, Steady flow, Rotational in geneal. 0 9

10 3.4. Example: Contaction in Wate o Wind Tnnel Contaction Ratio: γ = R /R >> ( tnnel) γ = O(0) fo wind tnnel ; γ = O(5) fo wate Let U and U denote the aeage elocities at sections and espectiely. ( ) (. Fom continity: Ū ) ( πr = Ū ) πr Ū R = = γ >> U R. Since =0, ω = 0 otex ing. 0

11 ω ω ω R = = << πr πr ω R γ ω/l = constant ( ) ( ) since ω << i.e., Section Section 3. Nea the cente, let U = U ( + ε ) and U = U ( + ε ) whee ε and ε mease the elatie elocity flctations. pply the Benolli eqation along a efeence aeage steamline pply Benolli Eqation to a paticla steamline Fom () and (3) we obtain P + ρū = P + ρū () [ ] [ P + ] ρ U ( + ε ) = P + ρ Ū ( + ε ) ε Ū = ε Ū + O(ε ) ε Ū << ε Ū γ 4

ECE Spring Prof. David R. Jackson ECE Dept. Notes 5

ECE Spring Prof. David R. Jackson ECE Dept. Notes 5 ECE 634 Sping 06 Pof. David R. Jacson ECE Dept. Notes 5 TM x Sface-Wave Soltion Poblem nde consideation: x h ε, µ z A TM x sface wave is popagating in the z diection (no y vaiation). TM x Sface-Wave Soltion