Marginal turbulent state of viscoelastic fluids: A polymer drag reduction perspective

The laminar-turbulent (LT) transition of dilute polymer solutions is of great interest not only for the complex transition dynamics itself, but also for its potential link to the maximum drag reduction (MDR) phenomenon. We present an in-depth investigation of the edge state (ES), an asymptotic solution on the LT boundary, in viscoelastic channel flow. For given $\text{Re}$ and simulation domain size, mean flow statistics of the ES do not vary with the introduction of polymers, proving that there is a region of turbulent states not susceptible to polymer drag reduction effects. The dynamics of the ES features low-frequency fluctuations and in the longer domains we studied it is nearly periodic with regular bursts of turbulent activities separated by extended quiescent periods. Its flow field is dominated by elongated vortices and streaks, with very weak extensional and rotational flow motions. Polymer stretching is almost exclusively contributed by the mean shear and polymer-turbulence interaction is minimal. Flow structures and the kinematics of the ES match hibernating turbulence, an MDR-like phase intermittently occurring in turbulent dynamics. Its observation now seems to result from recurrent visits to certain parts of the ES. The ES offers explanations for the existence and universality of MDR, the quantitative magnitude of which, however, still remains unsolved.

Figure 1

Time series of bulk-averaged TKE of DNS shooting trajectories pinching an edge solution (Newtonian, with $L_x^{+}\times L_z^{+}=360\times 140$). Different colors are used for different rounds of shooting. Shooting start points are shown with circles. Solid lines denote ${\mathit{X}}_{{\omega }_e^{+}}\left(t\right)$ and dashed lines ${\mathit{X}}_{{\omega }_e^{-}}\left(t\right)$.

Figure 2

The ES time series for $\text{Wi}=28$ and $L_x^{+}\times L_z^{+}=720\times 140$: the blue solid line (left axis) shows the peak value of the instantaneous RSS profile and the green dashed line (right axis) shows the TKE (bulk average).

Figure 3

Time series of the peak value of the instantaneous RSS profile: comparison between regular DNS (top) and the ES (bottom) (both with $\text{Wi}=28$ and $L_x^{+}\times L_z^{+}=720\times 230$).

Figure 4

State-space projections of regular DNS trajectories and the ES with $L_x^{+}\times L_z^{+}=720\times 650$ for $\text{Wi}=55$ DNS, as per [30], and $720\times 230$ for the rest. Two diverging DNS trajectories from the ES are shown in thin blue (solid and dashed) lines.

Figure 5

Spatiotemporal wall shear rate evolution ($\text{Wi}=28$) for (a) the ES with $L_x^{+}\times L_z^{+}=720\times 140$, (b) the ES with $720\times 230$, and (c) DNS with $720\times 230$. The distribution of $\partial v_x{/\partial y|}_w$ is measured along the $x=0$ line and is plotted in $z^{+}-t$ coordinates. The same color scale, from 1 (dark) to 6 (bright), is used for all panels. The time axes are consistent with those in Figs. 2 and 3.

Figure 6

Mean velocity profiles of Newtonian (Newt.) and viscoelastic (VE) ($\text{Wi}=28$) ESs for $L_x^{+}\times L_z^{+}=720\times 140$ (140) and $720\times 230$ (230). The viscous sublayer is denoted by , the PvK logarithmic law by , and the Virk MDR by .)

Figure 7

Flow and polymer conformation statistics for (a) mean velocity (viscous sublayer, ; PvK logarithmic law, ; and Virk MDR, ), (b) Reynolds shear stress [dashed line shows total shear stress (12)], and (c) $\mathrm{tr}\left(\mathbf{\alpha }\right)$. Profiles include the time average of the $\text{Wi}=28$ ES ($L_x^{+}\times L_z^{+}=720\times 140$), marked with ES (VE), and its selected instants (indicated in Fig. 2), as well as time averages of the Newtonian (Newt.) and $\text{Wi}=28$ (VE) DNS ($L_x^{+}\times L_z^{+}=720\times 230$). In (a) and (c), $+$ units are used for time-average profiles and $*$ units for instantaneous ones; in (b) $+$ units are used for all profiles.

Figure 8

Newtonian ES flow statistics at different $\text{Re}$ ($L_x^{+}\times L_z^{+}=720\times 140$) for the (a) mean velocity (viscous sublayer, ; PvK logarithmic law, ; and Virk MDR, ) and (b) RSS.

Figure 9

Isosurfaces of $Q$ (pink solid), $\mathrm{tr}\left(\mathbf{\alpha }\right)$ (light blue meshed), and ${\alpha }_{yy}+{\alpha }_{zz}$ (dark blue translucent) for the three selected instants indicated in Fig. 2; the isosurface levels $[Q_{\text{iso}},\mathrm{tr}{\left(\mathbf{\alpha }\right)}_{\text{iso}},{({\alpha }_{yy}+{\alpha }_{zz})}_{\text{iso}}]$ used in each instant are different: (a) I $[1.68\times {10}^{-3},529.61,2.99]$, (b) II $[1.12\times {10}^{-2},670.98,10.52]$, and (c) III $[6.00\times {10}^{-4},847.56,2.22]$.

Figure 10

Isosurfaces of $Q$ (pink solid), $\mathrm{tr}\left(\mathbf{\alpha }\right)$ (light blue meshed), and ${\alpha }_{yy}+{\alpha }_{zz}$ (dark blue translucent) for typical active and hibernating instants in the regular DNS ($\text{Wi}=28$); the isosurface levels $[Q_{\text{iso}},\mathrm{tr}{\left(\mathbf{\alpha }\right)}_{\text{iso}},{({\alpha }_{yy}+{\alpha }_{zz})}_{\text{iso}}]$ used in each instant are different: (a) active $[5.25\times {10}^{-2},1455.36,1370.65]$ and (b) hibernating $[1.83\times {10}^{-2},514.78,156.16]$.

Figure 11

Distribution of $\widehat{Q}$ at $y^{+}=24.85$ for the three selected instants indicated in Fig. 2: (a) I, (b) II, and (c) III.

Figure 12

Distribution of $\widehat{Q}$ at $y^{+}=24.85$ for typical instants in the regular DNS ($\text{Wi}=28$): (a) active and (b) hibernating.