Is the velocity spectrum of elastic turbulence universal?
In plain words
The chaotic flow of polymer solutions has an energy spectrum that falls off steeply toward small eddies. Whether its slope is the same in all flows is unknown.
Precise statement
In elastic turbulence at $\mathrm{Re} \to 0$ the velocity spectrum is $E(k) \sim k^{-\alpha}$; theory requires $\alpha > 3$ for a smooth velocity field (Fouxon and Lebedev 2003), experiments in curvilinear flows report $\alpha$ of about 3.3 to 3.6 (approximately), and channel DNS give $\alpha \sim 4$. Determine whether $\alpha$ has a universal value as $\mathrm{Wi} \to \infty$ and, if so, compute it.
What would settle it
A theory predicting $\alpha$, confirmed by spectra over at least a decade in $k$ in two different geometries at matched Wi.
Status in the literature
Unverified note
2024 channel simulations report $k^{-4}$.