A Study of Active and Inactive Regions in Turbulence by Photon Correlation Spectroscopy

Abstract

In a turbulent flow, it is the relative velocity V(R) between a pair of particles separated by a distance R, that is important rather than the instantaneous velocity v(t) at a point in the fluid¹. If v₁ and v₂ are the velocities of a pair of particles seeded in the fluid with a separation R, then the velocity difference between them at the same instant of time will be V(R) = v₁ − v₂. By virtue of the Central limit theorem and Gaussian random nature of v₁ and v₂, the probability distribution function for the relative velocity fluctuations V(R) will have the form²

EquationSource% MathType!MTEF!2!1!+- % feaagCart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn % hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr % 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq-Jc9 % vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9pg0FirpepeKkFr0xfr-x % fr-xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamiuamaaBa % aaleaacqaHbpGCaeqaaOGaaiikaiaadAfacaGGOaGaamOuaiaacMca % caGGPaGaeyypa0ZaaSaaaeaaciGGLbGaaiiEaiaacchadaGadaqaam % aalaaabaGaeyOeI0IaamOvamaaCaaaleqabaGaaGOmaaaakiaacIca % caWGsbGaaiykaaqaaiaaisdacaGGOaGaaGymaiabgUcaRiabeg8aYj % aacMcacqaHdpWCdaahaaWcbeqaaiaaikdaaaaaaaGccaGL7bGaayzF % aaaabaWaaOaaaeaacaaIYaGaeqiWdaNaaiOlaiaaikdacaGGOaGaaG % ymaiabgUcaRiabeg8aYjaacMcaaSqabaGccqaHdpWCdaahaaWcbeqa % aiaaikdaaaaaaOGaey41aq7aaSaaaeaacaWGobaabaGaamyyamaaCa % aaleqabaGaaGOmaaaakiabgUcaRiabeg8aYnaaCaaaleqabaGaaGOm % aaaaaaaaaa!6441!]]</EquationSource><EquationSource Format="TEX"><![CDATA[$${P_\rho }(V(R)) = \frac{{\exp \left\{ {\frac{{ - {V^2}(R)}}{{4(1 + \rho ){\sigma ^2}}}} \right\}}}{{\sqrt {2\pi .2(1 + \rho )} {\sigma ^2}}} \times \frac{N}{{{a^2} + {\rho ^2}}}$$

(1.1)

where ρ defines the correlation between v₁ and v, and has a Lorentzian form. In our experiment, we calculated P_ρ(V(R)) from the measurements of intensity autocorrela-tion function g(τ) behind a grid of mesh size 3 mm. using the equation² given below and is displayed in Fig. 1.