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

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.

Figures (7)

• Figure 1: Characteristic Function $\chi$=$\chi(R_\lambda)$
• Figure 2: Probability Density Function of $s = {\partial u_r/\partial r}/{\sqrt{\left\langle \left( \partial u_r/\partial r\right)^2\right\rangle}}$ for $R_\lambda \rightarrow \infty$ in comparison with a Gaussian PDF.
• Figure 3: Probability Density Function of $s = {\partial \vartheta/\partial r}/{\sqrt{\left\langle \left( \partial \vartheta/\partial r\right)^2\right\rangle}}$ for $Pe \rightarrow \infty$ compared with a Gaussian PDF.
• Figure 4: Top a), b), c): PDF of longitudinal velocity derivative for various values of $R_\lambda$. a) Dotted, dash--dotted, and continuous lines represent $R_\lambda = 15, 30$, and $60$, respectively. b) and c) PDFs for $R_\lambda = 255, 416, 514, 1035$, and $1553$. Plot (c) provides a detailed view of the tails shown in (b). Bottom: Experimental data from Ref. Tabeling96.
• Figure 5: Top: Statistical moments of $\Delta u_r$ as a function of the separation distance for $R_\lambda=600$. Bottom: Scaling exponents of the velocity increments at different $R_\lambda$. Solid symbols denote data from the present analysis. The dashed line represents K41 theory K41, the dotted line denotes K62 theory K62, and the continuous line indicates the She--Leveque model She94.