Interaction of a Vortex Pair with a Polymeric Fluid Layer

TL;DR

The paper investigates how a vortex pair interacts with a finite, nonuniform polymer layer modeled by the FENE-P equations, uncovering that polymer stresses can both dissipate vorticity and generate new coherent structures via gradients in elastic stresses. By solving a 2-D reduction of the viscoelastic flow with a localized polymer concentration and performing parametric scans over initial concentration $oldsymbol{ abla} abla$, relaxation time $oldsymbol{ abla} abla$, layer thickness, and maximum extension $L_{max}$, the authors show that a secondary and tertiary vortex can form, with a transient bump in the total kinetic energy $E$ co-occurring with tertiary-vortex formation. Enstrophy $oldsymbol{ abla}oldsymbol{ abla}$ is consistently amplified in the presence of polymers, reflecting strong velocity gradients at the polymer–fluid interface, while the primary vortex may completely dissipate in some parameter regimes. These findings connect polymer-induced drag reduction to a mechanism of vorticity injection via polymer torques and provide new insight into vortex–polymer interactions in nonuniform polymer layers, highlighting the importance of time-scale matching through the Weissenberg number $Wi = u^* rac{\lambda}{l_f}$.

Abstract

The interaction of vortical structures with boundaries has been extensively studied in Newtonian fluids, where conditions such as no slip walls, free surfaces, or contaminated surfaces dictate whether vortices rebound, dissipate, or generate secondary structures. In this work, we investigate a related but fundamentally different problem: the interaction of a vortex pair with a finite, non uniform layer of polymeric fluid. Numerical simulations employing the finitely extensible nonlinear elastic Peterlin model are used to examine the effects of polymer concentration, relaxation time, polymer layer thickness, and maximum polymer extension on the evolution of kinetic energy and enstrophy. The results show that, while the polymeric fluid dissipates vortical motion, vortex polymer layer interactions can also generate new coherent structures. In particular, the formation of secondary and tertiary vortices coincides with transient increases in kinetic energy, a behavior absent in the Newtonian case. Unlike classical vortex boundary interactions, where the primary vortex survives, we find that under certain conditions it completely dissipates upon interaction with the polymer layer. These findings emphasize that fluids with non-uniform polymer concentrations, act not only as dissipative agents but also as sources of vorticity, extending the traditional view of polymer induced drag reduction and providing new insight into vortex polymer interactions.

Figures (11)

• Figure 1: Schematic (not to scale) of the $x-z$ plane at the center of the computational domain ( $y=0$ ) showing the problem setup. The body force $F_0$ acts to the right, as indicated by the red arrow, and only within the rectangular region whose dimensions are given by $l_f$ and $R$. The center of this region is $x_0,y_0,z_0$, the midpoint of the computational domain. The dimensions of the computational domain are $L_x$ and $L_z$ in the $x$ and $z$ directions, respectively. A polymer layer of thickness $t_l$ is positioned downstream of the body force, and $h$ is the distance between the force center and the initial location of the center of the polymer layer. The $y$-axis points outward, normal to the plane of the page. The center of the force is located at $x=L_x/2$ and $z=L_z/2$.
• Figure 2: Evolution of a vortex pair interacting with a passive scalar (no polymer). Velocity vectors are color coded to indicate their magnitude (cm/s). The visualizations are taken from a zoomed-in region of the $x-z$ plane at the center ($y=0$). The core of the primary vortex is indicated.
• Figure 3: Evolution of a vortex pair interacting with a polymer layer for the case of $\gamma_0=100PPM$, $\lambda=1s$, $t_l=\pi/5cm$, and $L_{max}=100$. Velocity vectors are color coded to indicate their magnitude (cm/s). The visualizations are taken from a zoomed-in region of the $x-z$ plane at the center ($y=0$). The core of the primary vortex as well as the formation of a secondary vortex is indicated.
• Figure 4: Evolution of a vortex pair interacting with a polymer layer for base case where $\gamma_0=1000PPM$, $\lambda=1s$, $t_l=\pi/5cm$, and $L_{max}=100$. Velocity vectors are color coded to indicate their magnitude (cm/s). Right panels display the corresponding evolution of the kinetic energy, E. Red dots indicate the specific time instants shown corresponding to the images on the left. The visualizations are taken from a zoomed-in region of the $x-z$ plane at the center ($y=0$). The core of the primary vortex as well as the formation of a secondary vortex is indicated. The blue arrow indicates the direction of translation of the tertiary vortex. In panel (a), R1 denotes the forcing phase, R2 denotes the period from force removal to the local minimum in kinetic energy, and R3 denotes the subsequent bump and decay of the kinetic energy.
• Figure 5: Evolution of the vorticity magnitude for the case shown in Fig \ref{['fig:vortex']}. Color bar represents the vorticity magnitude in $s^{-1}$. The scalar field identifying the polymer layer is displayed with partial transparency and overlaid on the vorticity magnitude. Arrows point to the centers of the primary, secondary and tertiary vortices, as well as the shear layers in (b).