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

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 512 3 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 256 3 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 512 3 DNS flow is 1.68 ±.15. The probability distribution of the locally averaged disspation rate ε r , 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 ε r , and r for the conditionally averaged velocity increments δᵣu|εᵣ̄ are quantified, and the direction of their variations conforms with the refined similarity theory. The dimensionless averaged velocity increments (δᵣuⁿ|εᵣ̄)/(εᵣr)n/3 are found to depend on the local Reynolds number Re ε r = ε 1/3 r r 4/3 /ν in a manner consistent with the refined similarity hypotheses. In the inertial subrange, the probability distribution of δ r u /(ε r r ) 1/3 is found to be universal. Because the local Reynolds number of K62, R ε r = ε 1/3 r r 4/3 /ν, spans a finite range at a given scale r as compared to a single value for the local Reynolds number R r = ε −1/3 r 4/3 /ν in Kolmogorov's (1941 a,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 R λ . 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 ε r . The velocity increments conditioned on ε′ r do not follow the refined similarity hypotheses to the same degree as those conditioned on ε r .