Asymptotic-analysis-inspired boundary conditions aiming at eliminating polymer diffusive instability

TL;DR

This work identifies polymer diffusive instability (PDI) as a boundary-condition–driven artifact in simulations with artificial conformation diffusion (ACD) and demonstrates how to suppress it using asymptotic analysis of the near-wall diffusive layer. By reducing the asymptotic problem through four regimes (I–IV) and exploiting the parameter-free regime IV, the authors derive a new conformation boundary condition: a Neumann condition for the streamwise stretching component $c_{11}$ and Dirichlet conditions for the remaining components. They validate these conditions across Oldroyd-B and FENE-P models, showing elimination of PDI without sacrificing other instabilities, and confirm stability in direct numerical simulations. The resulting boundary conditions offer a robust, implementable route to reliable ACD-based polymer-flow simulations, enabling accurate exploration of transition routes to elastic turbulence and elasto-inertial turbulence.

Abstract

The recent discovery of polymer diffusive instability (PDI) by Beneitez et al. (Phys. Rev. Fluids, 2023, 8: L101901) poses challenges in implementing artificial conformation diffusion (ACD) in transition simulations of viscoelastic wall-shear flows. In this paper, we demonstrate that the unstable PDI is primarily induced by the conformation boundary conditions additionally introduced in the ACD equation system, which could be eliminated if a new set of conformation conditions is adopted. To address this issue, we begin with an asymptotic analysis of the PDI within the near-wall thin diffusive layer, which simplifies the complexity of the instability system by reducing the number of the controlling parameters from five to zero. Then, based on this simplified model, we construct a stable asymptotic solution that minimises the perturbations in the wall sublayer. From the near-wall behaviour of this solution, we derive a new set of conformation boundary conditions, prescribing a Neumann-type condition for its streamwise stretching component, $c_{11}$, and Dirichlet-type conditions for all the other conformation components. These boundary conditions are subsequently validated within the original ACD instability system, incorporating both the Oldroyd-B and FENE-P constitutive models. Finally, we perform direct numerical simulations based on the traditional and the new conformation conditions, demonstrating the effectiveness of the latter in eliminating the unstable PDI. Importantly, this improvement does not affect the calculations of other types of instabilities. Therefore, this work offers a promising approach for achieving reliable polymer-flow simulations with ACD, ensuring both numerical stability and accuracy.

Figures (17)

• Figure 1: A summary of the reduction process from the instability system (\ref{['eq:instability']}) to regime IV. Inbox panels (a), (b) and (c) illustrate the asymptotic structures for regimes I, II/III and IV, respectively. Layer (i) denotes the main layer; layer (ii) denotes the diffusive layer; layers (ii-1) and (ii-2) denote the upper and lower diffusive layer, respectively; layers (iii) and (iv) denote the bulk sublayer and the wall sublayer, respectively. The elongated chains represent the polymer states in the high and low shear rates.
• Figure 2: Contours of the rescaled growth rate $c_{1i}$ of the regime-I PDI for various wall boundary conditions for PPF with $\beta=0.9$, where the circles mark the critical point for the instability onset ($Wi_c$,$\bar{k}_c$). (a): BC1; (b): BC2; (c): BC3.
• Figure 3: Neutral curves of regime-I PDI in the $\bar{k}$-$\lambda Wi$ plane for BC1 with representative $\beta$ values, where lines with and without are for the PCF ($\lambda=1$) and PPF ($\lambda=2$) flows, respectively. The fitting curve $\bar{k}\sim Wi^{-3/2}$ indicate the scaling law of the lower-branch neutral curve.
• Figure 4: The properties of the most unstable regime-I PDI mode for different $\beta$ values under BC1. (a) and (b): Rescaled wavenumber $\bar{k}=k\epsilon^{1/2}$ for PPF ($\lambda=2$) and PCF ($\lambda=1$), respectively; (c): rescaled growth rate $c_{1i}$. The light dashed lines in (a,b) indicate the neutral curves extracted from figure \ref{['fig:neutral_curve']}.
• Figure 5: Dependence of the rescaled growth rate $\tilde{c}_{1i}$ on the rescaled wavenumber $\tilde{k}$ with $\beta=0.7$, 0.8, 0.9, 0.95 and 0.98 for regime-II PDI mode for a PPF. The inset in each panel shows the curves obtained after applying an appropriate rescaling. (a): BC12; (b): BC3.