Correlation, Spectrum, and Scales. The longitudinal correlation coefficient is defines as. = u

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1 Coelation Spectm and Scales Deinition: Coelation Tenso (Two-Point) Conside a tblent low ield as shown in Fie. Hee and ae the components o the velocity vectos and x x is the distance between the two points. The two-point coelation tenso is deines as Q ( x x) i ( x) ( x ) ( x) ( x ) ( x) ( x ) x x In a homoenos tblent low the coelations (and all the statistics) ae independent o the shit o space oiin. That is Fie. Geometic eates o two-point coelations in a tblent low ield. Q ( x x ) Q ( ) Deinition: onitdinal Coelation Coeicient The lonitdinal coelation coeicient is deines as whee () Q () ( x ) ( x ) Q ( x ) ( x ) Note that () is an even nction. That is () ( ) typical lonitdinal coelation coeicient is shown in Fie. Fie. Schematics o lonitdinal coelation coeicient. M639

2 Deinition: ateal Coelation Coeicient The lateal coelation coeicient is deines as Q () ( x ) ( x ) Q () The lateal coelation coeicient is also a symmetic Fnction. That is () ( ) typical lateal coelation coeicient is shown in Fie 3. Deinition: Taylo's Micoscales 8 Fie 3. Schematics o lateal coelation nction and the coespondin Taylo Scale. The Taylo micoscales ae deines as () () whee and ae espectively the Taylo lonitdinal and lateal micoscales. The micoscales may be deined by ittin a paabola to the coelation coeicient cves at. That is () + () +! Deinition: Inteal Scales Macoscales The macoscales o tblence ae deined as () d onitdinal Macoscale () d ateal Macoscale M639

3 Deinition: leian Time Coelation (stationay lows) The leian time coelation is deined as ( τ) ( x t) ( xt + τ) The leian time micoscale τ then is iven by τ () The leian time macoscale (inteal scale) T is deined as T ( τ) dτ Usin the niom low and ozen ield appoximations the scales may be elated. That is UT Uτ (U τ ) ( τ ) t U Deinition: aanian Time Coelation The aanian velocity coelation is deined as ( τ ) v (t)v (t + τ) v whee v is the aanian lctation velocity. The coespondin aanian time micoscale τ and the macoscale T ae iven as and τ () M639 3

4 T ( τ) dτ Deinition: ney Spectm Tenso ney spectm o is deined as the Foie Tansom o the coelation tenso. That is ( k) 8π Q ( x) Q ( x) e ( k) e ik x ik x Deinition: One Dimensional ney Spectm The one dimensional eney spectm is deined as dk dx (k ) l + x π ik (x)e dx + ik x (x) (k)e dk Symmety o (x ) implies that (k) (x)coskxdx π (k ) (x) (k)cos kxdk typical one dimensional eney spectm is shown in Fie. Settin x eqal to zeo we ind (k) dk. Fie. Schematics o onedimensional eney spectm. k M639

5 M639 5 lso x )dk (k k x xample: The lonitdinal coelation may be appoximated as e () The coespondin one dimensional spectm is iven as k ) (k + π

6 stimates o Taylo Micoscales The eney dissipation is iven as i i ε () Fo isotopic tblence it can be shown that It then ollows that ε 5( ) 5 () () Since ε (3) () Usin the macoscopic estimate o the dissipation we ind 3 ε 3 (5) Theeoe 3 / (6) Theeoe << since >> (7) Similaly 5 / (8) M639 6

7 It may also be shown that 5 (9) and η 5 ( ) / / 5 / / () Fom qation (3) it ollows that.6 ε.6 τ () whee the Kolmooov time scale is iven by η τ () υ ε This means that the Taylo micoscale is not a chaacteistic lenth o the dissipation eddies. It howeve povides a sel atiicial lenth scale o estimatin the velocity adients o the small eddies when macoscopic velocity scale is sed o the velocity o the eddies. That is qations () and () imply that i i ε ~ ( ) (3) Othe sel estimates ae lso d i i d ~ ~ ( ) 3 ~ ~ () and ~ / ~ η ~ 3/ ~ 3/ η ~ / ~ / (5) 3 η (6) M639 7