From Bifurcations to State-Variable Statistics in Isotropic Turbulence: Internal Structure, Intermittency, and Kolmogorov Scaling via Non-Observable Quasi-PDFs

TL;DR

This paper develops a theoretical framework linking bifurcation modes, non-observability, and intermittency in isotropic turbulence, analytically deriving internal structure functions and Kolmogorov scaling laws.

Contribution

It introduces a novel analytical approach using quasi-PDFs to connect bifurcation modes and turbulence statistics, explaining intermittency and Kolmogorov scaling.

Findings

Analytically derived velocity and temperature structure functions and PDFs.

Reproduced Kolmogorov scaling law as a consequence of non-observability.

Demonstrated that bifurcation mode amplitudes' third moment scales as R_lambda^(-3).

Abstract

This article investigates the intrinsic link between skewness and statistical intermittency in velocity and temperature increments within homogeneous isotropic turbulence. The theoretical framework builds upon the author's previously established closure schemes for the von Karman-Howarth and Corrsin equations. A transition Taylor-scale Reynolds number is first estimated via a formal bifurcation analysis of the closed von Karman-Howarth equation. A central thesis of this work is that while the nonlinearity of the Navier-Stokes equations is fundamentally responsible for intermittency, it is insufficient on its own to recover the Kolmogorov scaling law. We demonstrate that the non-observability of bifurcation modes constitutes the missing conceptual link: the concomitant effect of nonlinearity and non-observability not only determines the Kolmogorov scaling and drives an intermittency that grows monotonically with the Taylor-scale Reynolds number, but also enables the analytical determination of the internal structure functions of velocity and temperature differences, along with their corresponding PDFs and statistics. By invoking Fisher's principle (1922) for statistical description, we show that the entire statistics of increments can be analytically derived through a decomposition into bifurcation modes governed by quasi-probability distribution functions (quasi-PDFs). These provide the formal mathematical basis to also represent local energy backscatter. Notably, the analysis recovers the Kolmogorov law -- specifically the scaling of the velocity standard deviation ratio as R_lambda^(1/2) -- as a consequence of non-observability. Our analysis reveals that bifurcation modes exhibit amplitudes whose third statistical moment scales as R_lambda^(-3). The results show excellent agreement with benchmark numerical and experimental data in the literature.