Nonhomogeneous elastic turbulence in the two-dimensional Taylor-Couette flow

TL;DR

This study investigates elastic turbulence in a two-dimensional creeping Taylor–Couette flow using the Oldroyd‑B model and log-conformation stabilization in DNS. It demonstrates a supercritical purely elastic instability with critical Weissenberg number $Wi_c\approx5.525$, and reveals a dynamically active elastic boundary layer near the inner wall where nonlinear dynamics concentrate. In the fully developed turbulent-like state, the flow is weakly anisotropic and nonhomogeneous, with elastic and kinetic energy spectra showing distinct scaling laws and a Wi-dependent boundary-layer structure that governs scale interactions. The findings clarify the onset mechanism, quantify boundary-layer confinement, and connect numerical results with experimental and theoretical predictions for wall-bounded elastic turbulence, with implications for mixing at low Reynolds numbers.

Abstract

Elastic turbulence is a spatially and temporally disordered flow state appearing in viscoelastic fluids at vanishing fluid inertia and large elasticity. The resulting flows have broad technological interest, particularly to enhance mixing and heat transfer in microdevices. Although its experimental characterization is now well established in different setups, its theoretical understanding and numerical reproducibility remain challenging, especially in wall-bounded geometries. By means of extensive numerical simulations, we investigate the onset of elastic turbulence and the characteristics of the developed turbulent-like states in the two-dimensional, confined, Taylor-Couette system. We find that the purely elastic instability is supercritical, which clarifies previously contrasting evidences. We then show that the fully nonlinear dynamics are weakly anisotropic and strongly nonhomogeneous. Indeed, they are confined in a dynamically active region adjacent to the inner wall, akin to the elastic boundary layer from previous predictions. Within this region, the statistical and spectral turbulent properties are close to the theoretical expectations and experimental observations.

Figures (15)

• Figure 1: (a) Schematic of the Taylor-Couette viscoelastic setup. The inner wall is fixed, while the outer wall rotates in counter-clockwise direction at peripheral velocity $\Omega R_o$. The radial gap parameter is $\eta = R_i/R_o = 1/4$. (b) Schematic of a typical distribution for the numerical mesh: cells are uniformly distributed in the azimuthal direction and expand in the radial one by a constant ratio of $9.5$.
• Figure 2: Secondary flow strength measured by the order parameter $\Phi_{u_r}$ evaluated at $r=0.351$ for different $Wi$ by ramping up in $Wi$ starting from rest (red markers) or ramping down in $Wi$ starting from fully-developed conditions at $Wi = 50$ (green markers, with a tolerance range of 10%). The dashed line corresponds to the best fit with $(Wi - Wi_c)^{1/2}$. The red crosses ($Wi \leq 5$) denote stable conditions, the empty red circle is within the tolerance range across the critical Weissenberg number $Wi_c = 5.525\pm 0.025$, while the full red markers denote unstable conditions. The three radial velocity fields depicted for $Wi = 5$, 6 and 25 demonstrate the significance of the secondary flow over the domain. The dashed line in $u_r$ snapshots represents the line on which the order parameter $\Phi_{u_r}$ is computed (in this case $r=0.351$). The decay of $\Phi_{u_r}(r=0.351)$ for $Wi>25$ is due to the outward radial shift of the secondary flow extrema (compare $Wi=6$ and $Wi=25$).
• Figure 3: Kinetic energy $E_k$ divided by $Re$ (top) and elastic energy $E_e$ (bottom) versus time, for $Wi = 12.5, 25, 50, 100$. The transient to reach the statistically steady state ($t< t_{stat} = 600$) was removed, the durations of the shown time series correspond to $40 Wi$ in non-dimensional time units.
• Figure 4: Time-averaged elastic energy $\overline{E_e}$ as a function of $Wi$. The green markers for $Wi<1.3\times Wi_c$ match the subcritical steady linear scaling $\overline{E_e}\sim Wi$. The purple markers and the corresponding power-law fit, $\overline{E_{e}}\sim Wi^{0.6}$ for $Wi > 1.3\times Wi_c$, denote the supercritical scaling.
• Figure 5: Time behavior of the rms velocity components $u^{rms}_{r}$ and $u^{rms}_{\phi}$ for $Wi = 12.5$ (a), 25 (b), 50 (c), 100 (d) increasing from top to bottom. The transient to reach the statistically steady state ($t < t_{stat} = 600$) was removed.