Follow the curvature of viscoelastic stress: Insights into the steady arrowhead structure

Abstract

Focusing on simulated dilute polymer solutions, this letter investigates the interactions between flow structures and organized polymer stress sheets for the steady arrowhead coherent structure in a two-dimensional periodic channel flow. Formulating the problem in a frame of reference moving with the arrowhead velocity, streamlines, which are also pathlines in this frame, enables the identification of two distinct topological regions linked to two stagnation points. The streamlines help connecting the spatial distribution of polymer stress within the sheets and the dynamics of polymers transported by the flow. Using stresslines, lines parallel to the eigenvectors of polymer stress, a novel formulation of the viscoelastic stress term in the momentum transport equation proposes a more intuitive interpretation of the relation between the curvature of the stresslines, and the variation of stress along these lines, with the local flow topology. An approximation of this formulation is shown to explain the pressure jump observed in the arrowhead structure as a function of the local curvature of the polymer stress sheet.

Figures (4)

• Figure 1: Contour plot of the relative polymer extension $C_{kk}/L^2$ (a) and pressure field $p$ (b). Streamwise (c) and wall-normal (d) velocity profiles evaluated at the location of the pressure minimum ($x=0$) (blue) and at $x=\pm\pi$ (orange). Because of the solution symmetry, only the upper half of the channel is shown. In the chosen moving frame of reference, the flow goes from right to left. $Re=1000$, $Wi=50$, $\beta=0.9$, $L=90$ and $Sc=500$.
• Figure 2: Streamlines in the frame of the steady arrowhead overlaid on the contour of the relative polymer extension for the entire computational domain (a) and in the region of the arrowhead (b). The separating streamline (thicker white line) shows the transition between the two topological regions: external flow (a) and the recirculation zone (b). Its end-points on the centerline (white dots) correspond to stagnation points (zero relative velocity); the left stagnation point is denoted SP1 in the following, the other is denoted SP2. The blue dot indicates the position of the maximum of polymer extension, and the blue line is the corresponding streamline. $Re=1000$, $Wi=50$, $\beta=0.9$, $L=90$ and $Sc=500$.
• Figure 3: Illustration of the flow and polymer topology around the two stagnation points. The background contour represents the normalized polymer extension on top of which are shown the streamlines of relative velocity (white), the dividing streamline (thicker white line) and the stresslines of principal stress $\boldsymbol{\mathcal{T}}_{1}$ (green). The complementary stresslines $\boldsymbol{\mathcal{T}}_{2}$ are not shown but are everywhere normal to the green lines. The blue dot indicates the position of the overall maximum polymer extension and the two white dots on the centerline represent the two stagnation points. $Re=1000$, $Wi=50$, $\beta=0.9$, $L=90$ and $Sc=500$.
• Figure 4: (a): Scaled polymer body force $\mathbf{f}$ (arrows) on top of the contour of the normalized polymer extension $C_{kk}/L^2$. (b): Scaled resultant forcing $\mathbf{f}-\boldsymbol{\nabla}p$ (arrows) on top of the contour of its magnitude. In (a-b): green and blue lines are the stresslines generated from the eigenvectors $\mathbf{e}^{(1)}$ and $\mathbf{e}^{(2)}$, respectively. Note the green stressline passing through the point of maximum of extension $C_{kk}/L^2$ (blue dot), referred to as $\mathcal{S}^{(1)}_\mathrm{max}$ in (c). Vector fields are represented by white arrows (arrows for which the magnitude of the vector field is less than $4\%$ of the maximum are not shown for clarity). (c): Comparison between the pressure jump $\Delta P_\perp=p_\mathrm{II} - p_\mathrm{I}$ (green line) across a stress polymer sheet centered on $\mathcal{S}^{(1)}$ and the polymer body force component integrated over the thickness of the sheet $\int_{-\delta}^{+\delta}f_2~\mathrm{d}s_2$ (red line). The evolutions of both quantities are calculated along the stressline $\mathcal{S}^{(1)}_\mathrm{max}$ and ploted as a function of the horizontal location $x$ for direct correspondence with (a-b). The thickness of the sheet is $2\delta$, $\delta=0.02$, a choice explained in the text.$Re=1000$, $Wi=50$, $\beta=0.9$, $L=90$ and $Sc=500$.