Small Scale Intermittency and the Renormalization Group

Long time fate of two-dimensional incompressible high Reynolds number Navier–Stokes turbulence: A quantitative comparison between theory and simulation

“…10 (multimedia view). Thus, our numerical observation shows quite good agreement with the theoretical prediction obtained by KMRS theory [16][17][18]. The set of fitting parameters for Eq.…”

“…As there exists strong velocity shear between the vortex layer, such initial conditions are inherently Kelvin-Helmholtz unstable and hence leads to turbulence. Our results are compared with the statistical mechanical model of patch vortices [16][17][18] (KMRS theory) from small to large initial circulation values and a systematic deviation from sinh-Poisson model is demonstrated for increasing values of C + & C − or in other words, for increasing area of occupancy of nonzero circulation. As is well known, maximum number of scales k max resolvable at a grid size say N grid is k max ∼ N grid /3 [4].…”

“…From a statistical mechanics point of view, by considering the finite size vortices which takes into account the incompressibility effect (KMRS theory), it has been shown [16][17][18] that vorticity and stream function obeys the general relation for a system with zero total circulation, or in the other words, the most probable state is expressed as,…”

“…In Case A above, the vorticity packing fraction was 62.5%, which is now reduced to 50.0% and the remaining 50.0% of the domain contains zero vorticity. Circulation due to positive vortex strips (C + ) are, C + = 16 16 π 2 , and for minus vortex strips (C − ) = − 16 16 π 2 . Here we point out that the total initial circulation is C = C + + C − = ωdxdy = 0, where as circulation for positive and negative strips are reduced.…”

“…To properly account for "incompressibility" in a statistical mechanical model of 2-dimensional Navier-Stokes turbulence, Kuz'min [16], Miller [17] and later Roberts & Sommeria [18], rather independently proposed that "patch vortices"' or vortices with finite size, if used as "building blocks", should be able to account for incompressibility effects. For example, for small values of total initial circulations of a given type (i.e, + or −) viz C ± = ω ± dxdy, where ω ± (x, y, t) is the local vortic-ity function of + and − type quantifying the + and − vortices at a given time t, the regions of zero circulation dominate and thus statistical mechanical predictions of point vortex model should suffice.…”