Asymptotic scaling laws for the irrotational motions bordering a turbulent region

Turbulent flows are often bounded by regions of irrotational or non-turbulent flow, where the magnitude of the potential velocity fluctuations can be surprisingly high. This includes virtually all turbulent free-shear flows and also turbulent boundary layers, and is particularly true near the so-called turbulent/non-turbulent interface (TNTI) layer, which separates the regions of turbulent and non-turbulent fluid motion. In the present work, we show that in the non-turbulent region and for distances x(2) sufficiently far from the TNTI layer, the asymptotic variation laws for the variance of the velocity fluctuations < u(i)(2)> (i = 1, 2, 3), Taylor micro-scale lambda and viscous dissipation rate epsilon depend on the shape of the kinetic energy spectrum in the infrared region E(k) similar to k(n). Specifically, by using rapid distortion theory (RDT), we show that for Saffman turbulence (E(k) similar to k(2)), we obtain the asymptotic laws < u(i)(2)> similar to x(2)(-3) (i = 1, 2, 3), lambda similar to x(2) and epsilon similar to x(2)(-5). Additionally, we confirm the classical results obtained by Phillips (Proc. Camb. Phil. Soc., vol. 51, 1955, p. 220) for Batchelor turbulence (E(k) similar to k(4)), with < u(i)(2)> similar to x(2)(-4) (i = 1, 2, 3), lambda similar to x(2) and epsilon similar to x(2)(-6). The new theoretical results are confirmed by direct numerical simulations (DNS) of shear-free turbulence and are shown to be independent of the Reynolds number. Therefore, these results are expected to be valid in other flow configurations, such as in turbulent planar jets or wakes, provided the kinetic energy spectra in the turbulence region can be described by a Batchelor or a Saffman spectrum.

Automated summary

This study investigates the characteristics of velocity fluctuations in turbulent flows using theoretical analysis and numerical simulations, revealing the asymptotic laws for variance of velocity fluctuations, Taylor micro-scale, and viscous dissipation rate at different distances from turbulent/non-turbulent interface. The results are confirmed to be independent of Reynolds number and applicable to other flow configurations with appropriate kinetic energy spectra.