Hysteresis in the dissipation in turbulent flows

Abstract

We present evidence of the hysteretic nature of dissipation in unsteady turbulent flows. Wind tunnel experiments and direct numerical simulations in oscillating flows show that, at fixed mean Reynolds number, the dissipation constant is larger for decelerating flows. Consequently, a periodic behavior of the flow produces a hysteresis cycle, whose area scales with a parameter combining the Strouhal number and the relative amplitude of the forcing. This phenomenon can be explained and quantified through the influence of the unsteady term in the Karman-Howarth equation, with implications for a wide range of out-of-equilibrium systems.

Figures (4)

• Figure 1: (a) Example of the triple decomposition of an experimental velocity signal into its mean, periodic, and turbulent components. (b) The resulting phase-averaged hysteresis loop in the ($\langle Re_{\lambda} \rangle_{\phi}$, $\langle C_{\epsilon} \rangle_{\phi}$) plane for a representative case, showing the distinct decelerating (blue) and accelerating (red) paths. The shaded area in magenta represents the area of the hysteresis, $\Delta A_H$. $\varepsilon$ and the Taylor scale $\lambda$ were estimated using standard spectral techniques mora2020estimating.
• Figure 2: Gallery of phase-averaged hysteresis loops in the ($\langle Re_{\lambda} \rangle_{\phi}$, $\langle C_{\epsilon} \rangle_{\phi}$) plane for various forcing conditions, demonstrating the robustness of the phenomenon in both (a) experiments and (b) simulations.
• Figure 3: Normalized hysteresis area, $\Delta \langle C_{\epsilon}(\phi) \rangle_\phi$, as a function of the dimensionless forcing parameter $X = St_f \cdot (|\widetilde{U}|/\overline{U})$. The data includes all experimental (HWA) and numerical (DNS) configurations.
• Figure 4: (a) Experimental result of the phase-averaged profiles of the unsteady function, $\langle F(r) \rangle_{\phi}$, versus the non-dimensional lag, $r/\lambda_{\phi}$, for a representative case. (b) DNS result, for a representative case. The curves in (a) and (b) are colored by phase, where phases $\phi \in [0, \pi)$ correspond to the decelerating part of the cycle (cool colors) and $\phi \in [\pi, 2\pi)$ to the accelerating part (warm colors). (c) Scaling of the peak-to-peak amplitude of the unsteady function evaluated at the Taylor scale, $|\langle F(\lambda) \rangle_{\phi}|$, with the dimensionless forcing parameter, $X$. The scaling includes all experimental and numerical data.