AI Article Synopsis
- The text discusses the use of Heisenberg's approximation to derive formulas for eddy viscosity and eddy diffusivity based on energy transfer and scalar dynamics.
- It evaluates flux integrals to calculate universal constants like the Batchelor constant (B), Kolmogorov constant (C), and turbulent Prandtl number (σ), showing consistency with existing research.
- The findings reveal relationships between these constants and highlight changes in the Prandtl number and Batchelor constant as space dimensions increase, culminating in an asymptotic relationship for B.
Article Abstract
We use Heisenberg's approximation to derive analytic expressions for eddy viscosity and eddy diffusivity from the transfer integrals of energy and mean-square scalar arising from the Navier-Stokes and passive scalar dynamics. In the same scheme, we evaluate the flux integrals for the transports of energy and mean-square scalar. These procedures allow for the evaluation of relevant amplitude ratios, from which we calculate the universal numbers, namely, Batchelor constant B, Kolmogorov constant C, and turbulent Prandtl number σ, under two different schemes (with and without ε expansion). Our results are comparable with existing theoretical, numerical, and experimental values. As a byproduct, we obtain a relation between C, B, and σ, namely, B=σ C. To compare our results with the experimental values, we calculate Batchelor constant in one dimension (B'). Within the same framework, we also see that with increasing values of space dimension d, the Prandtl number σ increases and approaches unity, while the Kolmogorov constant C and Batchelor constant B approach very close to each other. For large space dimensions, we find the asymptotic B=B(0)d(1/3), and evaluate B(0).
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| http://dx.doi.org/10.1103/PhysRevE.84.036315 | DOI Listing |
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