Nonasymptotic elastoinertial turbulence for asymptotic drag reduction

Polymer-induced drag reduction is bounded by an asymptotic limit of maximum drag reduction (MDR). For decades, researchers have presumed that MDR reflects the convergence to an ultimate flow state that is not further changed by polymers. Our simulation shows that, as drag reduction converges to its invariant limit, the underlying dynamics continues to evolve with no sign of convergence. The stage of asymptotic drag reduction is not represented by any single flow state, but encompasses states with varying dynamical patterns, all of which are partially sustained by polymer elasticity.

Figure 1

(a) Friction factor (arrows mark Newtonian turbulence, MDR [1], and laminar values) for 2D EIT and 3D DNS (data provided in SM [23]; the “Large” domain is $2L_x^{+}\times L_y^{+}\times 2L_z^{+}$); (b) mean velocity profiles of 3D DNS (with reference lines: (dot-dashed) viscous sublayer $U_{\text{m}}^{+}=y^{+}$; (dashed) Newtonian log law $U_{\text{m}}^{+}=2.5ln{y}^{+}+5.5$ [24]; (solid) MDR $U_{\text{m}}^{+}=11.7ln{y}^{+}-17.0$ [1]; instants I and II are marked in Figs. 2 and 4). Error bars use block average [25].

Figure 2

Dynamical trajectories of 3D DNS at (a) $\mathrm{Wi}=40$ (solid; color maps to time) and (b) $\mathrm{Wi}=400$ (solid; purple). The edge state [30] and 2D EIT solutions are both at $\mathrm{Wi}=40$; active turbulence is represented by 3D DNS solution at $\mathrm{Wi}=30$. $A^{*}\equiv y^{*}(\partial U_{\text{m}}^{*}/\partial y^{*})$ is the logarithmic slope function and $A_{y^{*}=25}^{*}$ measures the local steepness of the $U_{\text{m}}^{*}\left(y^{*}\right)$ profile at $y^{*}=25$ [3]. $|\langle v_x^{\prime *}{v}_y^{\prime *}\rangle {|}_{\text{max}}$ is the maximum instantaneous Reynolds shear stress value (after $xz$ average) [22, 30].

Figure 3

Flow structures of 3D DNS at representative instants as marked in Figs. 2, 4, and 5. Colors map to $\mathrm{tr}\left(\alpha \right)/b$. Isosurfaces show vortex configuration using the $Q$ criterion [36, 37] ($Q=0.004$; bottom half channel only).

Figure 4

Time series of instantaneous friction factor $C_{\text{f}}^{\text{ins}}$ and TKE elastic conversion rate $-{\langle {\epsilon }_{\text{p}}^k\rangle }_{\mathcal{V}}$ (scaled by a factor $s$ for clarity) of 3D DNS: (a) $\mathrm{Wi}=40$ ($s=20$) showing types A and B cycles, (b) $\mathrm{Wi}=64$ ($s=12$) showing type B cycles, and (c) $\mathrm{Wi}=400$ ($s=4$) showing type C cycles. Arrows mark $C_{\text{f}}$ of Newtonian turbulence and MDR.

Figure 5

(a) Enlargement of type C cycle [Fig. 4] time series, with Newtonian shooting trajectories showing the dynamics after removing polymer stress from instants III and IV; (b) time autocorrelation function of $C_{\text{f}}^{\prime ,\text{ins}}$ (3D DNS).