Spectral Reconstruction for Under-Resolved Turbulence Measurements Using a Variational Cutoff Dissipation Model

TL;DR

This work introduces a bounded-dissipation-range spectral model derived from a variational cascade-dynamics principle, yielding a hard cutoff at the Kolmogorov wavenumber and a Ginzburg-Landau domain-wall solution for the dissipation tail. By modeling the spectrum as the inertial-range form times a transmission function with a switchable dissipation boundary, the method reconstructs the full energy spectrum from under-resolved measurements without extra flow-specific calibration. Validation against high-Reynolds-number data shows accurate spectral shapes and superior TKE recovery, achieving over 98% reconstruction for spectra truncated at $k\eta=0.15$, outperforming traditional Pao and Pope models. The approach offers a robust, parameter-universal tool for turbulence diagnostics in industrial, aeroacoustic, and atmospheric sensing where bandwidth limitations hinder direct dissipation-range resolution.

Abstract

This technical note addresses the challenge of accurate turbulence characterization using robust, bandwidth-limited sensors which fail to resolve the high-wavenumber dissipation range. To correct the resulting underestimation of turbulent kinetic energy (TKE), a novel analytical spectral model is derived from a variational principle governing cascade resistance, yielding a Ginzburg-Landau domain wall solution. Unlike classical asymptotic decay formulations such as the Pao or Pope models, the proposed formulation features bounded spectral support with a hard energetic cutoff at the Kolmogorov wavenumber ($k_η$) and requires no adjustable parameters beyond the Kolmogorov constant ($C_K$). Validation against high-Reynolds-number experimental data confirms that the model accurately captures the spectral rolloff and achieves superior TKE recovery, restoring over 98\% of the variance from spectra truncated as early as $kη=0.15$, thereby offering a robust tool for industrial and aeroacoustic flow diagnostics.

Figures (3)

• Figure 1: Schematic representation of the phenomenological Ginzburg-Landau analogy applied to cascade dynamics. (a) The thermodynamic potential $V(\Gamma) = \frac{1}{2}(1-\Gamma^2)^2$. The cascade dynamics follow a trajectory (solid line red arrow) from the unstable fixed point of the inertial range ($\Gamma=0$) toward the stable attractor of the dissipation limit ($\Gamma=1$), enforcing the asymptotic boundary conditions. (b) The resulting heteroclinic orbit solution $\Gamma(Z) = \tanh(Z)$ in conformal coordinate space. The finite slope of the transition illustrates the effect of the gradient term $(d\Gamma/dZ)^2$ in Eq. (9), which represents the spectral "stiffness" of the energy transfer and prevents discontinuous jumps in the cascade flux.
• Figure 2: Compensated spectrum $E(k)\varepsilon^{-2/3}k^{5/3}C_{\kappa}^{-1}$ vs $k\eta$. Comparison of novel model (Eq. (\ref{['eq:final_spectrum']})) (solid), Pao (dashed), Pope (dash-dot), and experimental data (symbols) from Mishra Ref. mishra:tel-04908316
• Figure 3: Compensated spectrum $E(k)\varepsilon^{-2/3}k^{5/3}C_{\kappa}^{-1}$ vs $k\eta$. Comparison of novel model (Eq. (\ref{['eq:final_spectrum']})) (solid), Pao (dashed), Pope (dash-dot), and experimental data (symbols) from Saddoughi and Veeravalli Ref. Saddoughi1994