Examination of hypotheses in the Kolmogorov refined turbulence theory through high-resolution simulations. Part 1. Velocity field

Abstract

The fundamental hypotheses underlying Kolmogorov-Oboukhov (1962) turbulence theory (K62) are examined directly and quantitutivezy by using high-resolution numerical turbulence fields. With the use of massively parallel Connection Machine-5, we have performed direct Navier-Stokes simulations (DNS) at 5123 resolution with Taylor microscale Reynolds number up to 195. Three very different types of flow are considered: free-decaying turbulence, stationary turbulence forced at a few large scales, and a 2563 large-eddy simulation (LES) flow field. Both the forced DNS and LES flow fields show realistic inertial-subrange dynamics. The Kolmogorov constant for the k−5/3 energy spectrum obtained from the 5123 DNS flow is 1.68 ±.15. The probability distribution of the locally averaged disspation rate εr, over a length scale r is nearly log-normal in the inertial subrange, but significant departures are observed for high-order moments. The intermittency parameter p, appearing in Kolmogorov's third hypothesis for the variance of the logarithmic dissipation, is found to be in the range of 0.20 to 0.28. The scaling exponents over both εr, and r for the conditionally averaged velocity increments $\overline{\delta_ru|\epsilon_r}$ are quantified, and the direction of their variations conforms with the refined similarity theory. The dimensionless averaged velocity increments $(\overline{\delta_ru^n|\epsilon_r})/(\epsilon_rr)^{n/3}$ are found to depend on the local Reynolds number Reεr = ε1/3rr4/3/ν in a manner consistent with the refined similarity hypotheses. In the inertial subrange, the probability distribution of δru/(εrr)1/3 is found to be universal. Because the local Reynolds number of K62, Rεr = ε1/3rr4/3/ν, spans a finite range at a given scale r as compared to a single value for the local Reynolds number Rr = ε−1/3r4/3/ν in Kolmogorov's (1941a,b) original theory (K41), the inertial range in the K62 context can be better realized than that in K41 for a given turbulence field at moderate Taylor microscale (global) Reynolds number Rλ. Consequently universal constants in the second refined similarity hypothesis can be determined quite accurately, showing a faster-than-exponential growth of the constants with order n. Finally, some consideration is given to the use of pseudo-dissipation in the context of the K62 theory where it is found that the probability distribution of locally averaged pseudo-dissipation ε′r deviates more from a log-normal model than the full dissipation εr. The velocity increments conditioned on ε′r do not follow the refined similarity hypotheses to the same degree as those conditioned on εr.