Dynamical model for turbulence. III. Numerical results

Abstract

We present the numerical solutions of the equations for turbulence developed in papers I and II. (1) The model predicts the Kolmogorov law and Ko=5/3, in accord with recent data; (2) in the inertial-conductive regime, the model predicts the Corrsin spectrum for the temperature variance and the Batchelor constant Ba=σt Ko, where σt=0.72 is the turbulent Prandtl number; (3) the predicted energy spectrum in the dissipation region is in agreement with recent laboratory measurements; (4) in the inertial-convective region, the temperature variance spectrum is closer to the spectrum (−11/3) obtained by LES when the velocity field is rapidly stirred at all scales than(−17/3), which holds when the velocity field is frozen in time and has a Gaussian statistics; (5) for freely decaying turbulence, the power law spectra for energy and temperature variance, as well as the velocity and temperature integral scales, agree with the most recent LES data; (6) after a few evolutionary times, the skewness S reaches S=0.5, in accord with a variety of data; (7) for shear-driven flows, the Reynolds stress spectrum E12(k) has an inertial regime with a power −7/3, in accord with recent data; (8) for two shear-driven flows, plane strain and axisymmetric contraction, turbulent kinetic energy, Reynolds stress tensor, and dissipation rate εij versus time compare very well with DNS data; (9) the slow and rapid parts of the pressure–strain correlation tensor compare with DNS data better than with the three most widely used phenomenological models. The rapid parts are also in excellent agreement with the DNS data; (10) for homogeneous shear, turbulent kinetic energy and Reynolds stress tensor versus time match quite closely LES data. We recall that the model does not contain any free parameters.

Physical Regimes and Dimensional Structure of Rotating Turbulence

“…It was stated that [7] "the decoupling was unexpected especially considering the strong coupling between vertical and horizontal fluctuations," and that [6] "most striking is the large growth rate in L 11,3 which attains values between 5-10 larger than L 33,3 . "We show that both DNS-LES results can be reproduced and understood on the basis of a new hierarchy of regimes which we construct using a recent model previously tested on a variety of other data [8,9]. For large V, we show that there exist two quite different regimes separated by the new number N K͞nV (K is the turbulent kinetic energy and n is the viscosity).…”

“…We show that both DNS-LES results can be reproduced and understood on the basis of a new hierarchy of regimes which we construct using a recent model previously tested on a variety of other data [8,9]. For large V, we show that there exist two quite different regimes separated by the new number N K͞nV (K is the turbulent kinetic energy and n is the viscosity).…”

“…Thus, while DIA contains no adjustable parameters, it neglects a large set of diagrams. For the problems encountered in extending DIA to anisotropic, inhomogeneous flows, see [13].The model presented in [8,9] employs the RNG technique to compute the UV part but departs substantially from the one-loop, DIA model. Since the series of IR diagrams cannot be summed, a physical model was suggested based on the assumption that the nonlinear transfer of energy is mostly a local process.…”

“…We also compute the power law exponents for energy and length scales for freely decaying rotating turbulence and show that the results reproduce LES data [7,10]. The nonlinear interactions, though weakened, are the main cause of 2D state which is not of the Proudman-Taylor type since the latter requires negligible nonlinearities.Stationary, homogeneous, rotating turbulence.-Before applying the model of [8,9] to rotating turbulence, it is necessary to give a brief sketch of its physical content. Generally speaking, the interaction of an eddy of wave number k is contributed by two processes: the interaction with all the smaller eddies with wave numbers larger than k (the ultraviolet part, UV) and with the larger eddies with wave numbers smaller than k (infrared part, IR).…”

“…Stationary, homogeneous, rotating turbulence.-Before applying the model of [8,9] to rotating turbulence, it is necessary to give a brief sketch of its physical content. Generally speaking, the interaction of an eddy of wave number k is contributed by two processes: the interaction with all the smaller eddies with wave numbers larger than k (the ultraviolet part, UV) and with the larger eddies with wave numbers smaller than k (infrared part, IR).…”