Towards a full-scale version of Yakhot's model of strong turbulence

TL;DR

This work extends Yakhot's model of strong turbulence to dissipative scales by leveraging an empirical link between derivatives of even-order structure functions and the next higher odd order, enabling a closed description when combined with Kolmogorov's four-fifth law. It introduces a Reynolds-number–dependent crossover length scale $\rho$ and, together with a large-scale extension, yields full-scale, closed-form expressions for the second- and third-order structure functions that agree with experimental data from dissipative to system scales, without free parameters aside from $\rho$. The study also identifies a residual-based correction for even orders and reveals an asymmetry between even and odd residuals, pointing to limitations in fully closing the odd-order dynamics alone. Overall, the approach provides a scale-spanning, low-order turbulence description driven by large-scale flow properties and a single dissipation-related length scale, with potential implications for multiscale turbulence modeling.

Abstract

We present first elements of an extension of Yakhot's model of strong turbulence towards small scales. The analysis is based on an empirically observed relation for even order structure functions which extends from the inertial into the dissipation range. With this relation and Kolmogorov's four-fifth law, models for structure functions of orders two and three can be derived that replicate expected small scale limits and describe the transition from dissipative to inertial range scaling regimes correctly. An additional length scale parameter is introduced by the extension. It marks the crossover point of inertial and dissipation range and can be expressed as a function of the Reynolds number. In combination with a recently proposed large-scale extension of Yakhot's model, we ultimately obtain full-scale models for structure functions of second and third order. These expressions are closed-form, do not contain free parameters and are in good agreement with experimental data from the smallest dissipative scales up to the system scale.

Figures (7)

• Figure 1: Functions $d_n(r)$ as defined in eq. (\ref{['impliedDofR']}) implied from experimental data of structure functions of order two (dashed lines) and four (dotted lines) in linear (left) and log--linear (right) scale.
• Figure 2: The residuals $R_n(r)$ as defined in eq. (\ref{['resDefinition']}) for orders $n=2$ (dashed lines) and $4$ (dotted lines) compensated by the structure functions $S_{n+1}$ (left) and additionally scaled by order $n$ (right). The straight line in the right graph shows a fit according to eq. (\ref{['rnDef']}).
• Figure 3: Compensated and scaled residuals $R_n(r)$ for orders $n=3$ (dashed lines) and $n=5$ (dotted lines) with power law fits (solid lines). Left graph: Residuals scaled with order $n$ in comparison to the power law fit of even--order residuals according to eq. (\ref{['rnDef']}). Right graph: Results for alternative scaling with $1/(n-1)$ in comparison to a power--function fit of second order (the fitted pre--factor does not coincide with the value of $\tau$ found for even order).
• Figure 4: Second order structure function determined from experimental data (straight line) in comparison to the small (dashed line) and the large scale models (dotted line) as given by equations (\ref{['smallScaleS2Formula']}) and (\ref{['ExtS2Solution']}), respectively.
• Figure 5: The second order structure function (solid line) in comparison with large--scale (dotted line) and full--scale (dashed line) models in linear (left) and logarithmic (right) scale. The scale parameter $\rho$ of the model (\ref{['fullScaleS2']}) was set to a value of $\rho = 5.7 \cdot 10^{-4}$ according to eq. (\ref{['fullRho']}).