Tollmien-Schlichting route to elastoinertial turbulence in channel flow

Direct simulations of two-dimensional channel flow of a viscoelastic fluid have revealed the existence of a family of Tollmien-Schlichting (TS) attractors that is nonlinearly self-sustained by viscoelasticity [Shekar et al., J. Fluid Mech. 897, A3 (2020)]. Here, we describe the evolution of this branch in parameter space and its connections to the Newtonian TS attractor and to elastoinertial turbulence (EIT). At Reynolds number $\mathrm{Re}=3000$, there is a solution branch with TS-wave structure but which is not connected to the Newtonian solution branch. At fixed Weissenberg number, $\mathrm{Wi}$, and increasing Reynolds number from 3000 to 10 000, this attractor goes from displaying a striation of weak polymer stretch localized at the critical layer to an extended sheet of very large polymer stretch. We show that this transition is directly tied to the strength of the TS critical layer fluctuations and can be attributed to a coil-stretch transition when the local Weissenberg number at the hyperbolic stagnation point of the Kelvin cat's eye structure of the TS wave exceeds $\frac{1}{2}$. At $\mathrm{Re}=10000$, unlike 3000, the Newtonian TS attractor evolves continuously into the EIT state as $\mathrm{Wi}$ is increased from zero to about 13. We describe how the structure of the flow and stress fields changes, highlighting in particular a “sheet-shedding” process by which the individual sheets associated with the critical layer structure break up to form the layered multisheet structure characteristic of EIT.

Figure 1

(a) Time-averaged $L_2$ norm of ${\widehat{\alpha }}_{xx}$ vs $\mathrm{Wi}$ of the three solution branches identified in [15] at $\mathrm{Re}=3000,L_x=5$. Snapshots corresponding to (b) 2D EIT at $\mathrm{Wi}=13$ (point A on the bifurcation diagram), (c) NNTSA at $\mathrm{Wi}=3$ (point B), and (d) VNTSA at $\mathrm{Wi}=10$ (point C). Shown are white contour lines of wall normal velocity, $\widehat{v}$, superimposed on color contours of $xx$ component of polymer conformation tensor (${\widehat{\alpha }}_{xx}$). Here $\widehat{}$ denotes deviations from laminar base state. For point B, we also show the streamlines (blue) in a reference frame moving at the wave speed and the hyperbolic stagnation points (blue dots).

Figure 2

(a) $L_2$ norm of ${\widehat{\alpha }}_{xx}$ (blue) and corresponding mean local stretch rate at the hyperbolic stagnation point (red) along the VNTSA branch at $\mathrm{Wi}=10$. Red-dashed line corresponds to ${\mathrm{Wi}}_{\mathrm{loc}}=\lambda {\sigma }_{\max }=1/2$. Panels (b) and (c) are snapshots of the fluctuation structure of the solution branch at $\text{Re}=6000$ and 8000, respectively. Shown are contour lines of $\widehat{v}$ superimposed on color contours of ${\widehat{\alpha }}_{xx}$. Here $\widehat{}$ denotes deviations from laminar base state. On (c), the hyperbolic stagnation points in the traveling frame are indicated as blue dots and the streamlines attached to them as dashed curves with arrows indicating the direction of flow.

Figure 3

(a) $\left|\right|{\widehat{\alpha }}_{xx}{\left|\right|}_2$ vs $\mathrm{Wi}$ at $\mathrm{Re}=10000$ for simulations in the full space and the shift-reflect subspace. Here, “A” indicates asymmetry in the attractor in the full space. The upper and lower blue symbols at $\mathrm{Wi}=11$ and 12 indicate averages over the two metastable states intermittently visited by the dynamics at these $\mathrm{Wi}$ values. (b)–(i) are snapshots of the fluctuation structure at the indicated $\mathrm{Wi}$.

Figure 4

(a)–(f) Snapshots of the fluctuation structure at $\mathrm{Re}=10000$, $\mathrm{Wi}=4$. Snapshots are taken from $t=4400$ to 4410 every 2 time units, respectively. Green lines correspond to $\widehat{v}=0$, i.e., where wall normal velocity changes sign. Bottom half of the domain shown.

Figure 5

(a)–(j) Snapshots of the fluctuation structure at $\mathrm{Wi}=11$. Snapshots are taken from $t=5230$ to 5320 every 10 time units, respectively. Same format as Fig. 4.

Figure 6

Power spectral density (PSD) of wall normal velocity at $y=\pm 0.8$ at the indicated $\mathrm{Wi}$.

Figure 7

(a) Mean streamwise velocity profiles, (b) Reynolds shear stress profiles, and (c) polymer shear stress profiles of full space solutions. All profiles are shown as a function of distance from the wall $1-\left|y\right|$.

Figure 8

(a) Instantaneous $\left|\right|{\widehat{\alpha }}_{xx}{\left|\right|}_2$ vs time at $\mathrm{Re}=10000$ in the shift-reflect symmetric subspace. Panels (b) and (c) are snapshots of the fluctuation structure of the TS (instant indicated by the black dot) and EIT (red dot) metastable states, respectively.

Figure 9

Dynamics of $x$ in the van der Pol–type system described by Eqs. (6) and (7) for $\epsilon =0.01$ at the indicated values of control parameter $a$.

Figure 10

Snapshots of the attractor at $\mathrm{Re}=10000$ in a box of size $L_x=31$ at the indicated $\mathrm{Wi}$. Contour plots follow the same format as previous figures. Note the compressed scale in $x$.

Figure 11

Snapshots of the attractor at $\mathrm{Re}=10000$ with $b=100000$ at various $\mathrm{Wi}$. At $\mathrm{Wi}=6$, the dynamics are strongly intermittent; (b) and (c), respectively, are snapshots from the low-amplitude (TS) and large amplitude (EIT) intervals. Format is the same as previous plots.