Two-dimensional dynamics of elasto-inertial turbulence and its role in polymer drag reduction

The goal of the present study is threefold: (i) to demonstrate the two-dimensional nature of the elasto-inertial instability in elasto-inertial turbulence (EIT), (ii) to identify the role of the bidimensional instability in three-dimensional EIT flows, and (iii) to establish the role of the small elastic scales in the mechanism of self-sustained EIT. Direct numerical simulations of viscoelastic fluid flows are performed in both two- and three-dimensional straight periodic channels using the Peterlin finitely extensible nonlinear elastic model (FENE-P). The Reynolds number is set to ${\mathrm{Re}}_{\tau }=85$, which is subcritical for two-dimensional flows but beyond the transition for three-dimensional ones. The polymer properties selected correspond to those of typical dilute polymer solutions, and two moderate Weissenberg numbers, ${\mathrm{Wi}}_{\tau }=40,100$, are considered. The simulation results show that sustained turbulence can be observed in two-dimensional subcritical flows, confirming the existence of a bidimensional elasto-inertial instability. The same type of instability is also observed in three-dimensional simulations where both Newtonian and elasto-inertial turbulent structures coexist. Depending on the $\mathrm{Wi}$ number, one type of structure can dominate and drive the flow. For large $\mathrm{Wi}$ values, the elasto-inertial instability tends to prevail over the Newtonian turbulence. This statement is supported by (i) the absence of typical Newtonian near-wall vortices and (ii) strong similarities between two- and three-dimensional flows when considering larger $\mathrm{Wi}$ numbers. The role of small elastic scales is investigated by introducing global artificial diffusion (GAD) in the hyperbolic transport equation for polymers. The aim is to measure how the flow reacts when the smallest elastic scales are progressively filtered out. The study results show that the introduction of large polymer diffusion in the system strongly damps a significant part of the elastic scales that are necessary to feed turbulence, eventually leading to flow laminarization. A sufficiently high Schmidt number (weakly diffusive polymers) is necessary to allow self-sustained turbulence to settle. Although EIT can withstand a low amount of diffusion and remains in a nonlaminar chaotic state, adding a finite amount of GAD in the system can have an impact on the dynamics and lead to important quantitative changes, even for Schmidt numbers as large as ${10}^2$. The use of GAD should therefore be avoided in viscoelastic flow simulations.

Figure 1

Instantaneous contours of polymer elongation $\mathrm{tr}\left(\mathbf{C}\right)/L^2$ in two-dimensional simulations. Left: ${\mathrm{Wi}}_{\tau }=40$; right: ${\mathrm{Wi}}_{\tau }=100$.

Figure 2

Visualization of flow structures in three-dimensional flows. Contours of polymer elongation $\mathrm{tr}\left(\mathbf{C}\right)/L^2$ in vertical planes, contours of streamwise shear fluctuations in a near-wall horizontal plane, and turbulent structures identified by large positive (white) and negative (black) $Q$-criterion values (only shown in the bottom half of the channel). Left: ${\mathrm{Wi}}_{\tau }=40$; right: ${\mathrm{Wi}}_{\tau }=100$.

Figure 3

Streamwise spectra of turbulent kinetic energy $k$ averaged over the computational domain and over time. Left: Two- and three-dimensional flows at $\mathrm{Sc}=\infty$ for different Weissenberg numbers: (dashed gray) 3D, Newtonian; (blue) 3D, ${\mathrm{Wi}}_{\tau }=40$; (red) 3D, ${\mathrm{Wi}}_{\tau }=100$; and (green) 2D, ${\mathrm{Wi}}_{\tau }=40$. Right: Two-dimensional flow at ${\mathrm{Wi}}_{\tau }=40$ for different Schmidt numbers: (red) $\mathrm{Sc}=9$; (orange) $\mathrm{Sc}=16$; (blue) $\mathrm{Sc}=25$; (green) $\mathrm{Sc}=100$; and (black) $\mathrm{Sc}=\infty$.

Figure 4

Mean streamwise velocity profile (left) and mean polymer elongation profile (right) for a two-dimensional flow at ${\mathrm{Wi}}_{\tau }=40$ and different Schmidt numbers: (gray) $\mathrm{Sc}=0.5$ (laminar); (red) $\mathrm{Sc}=9$; (orange) $\mathrm{Sc}=16$; (blue) $\mathrm{Sc}=25$; (green) $\mathrm{Sc}=100$; and (black) $\mathrm{Sc}=\infty$.