Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Numerical simulation of dilute polymeric fluids with memory effects in the turbulent flow regime

Jonas Beddrich, Stephan B. Lunowa, Barbara Wohlmuth

Abstract

We address the numerical challenge of solving the Hookean-type time-fractional Navier--Stokes--Fokker--Planck equation, a history-dependent system of PDEs defined on the Cartesian product of two $d$-dimensional spaces in the turbulent regime. Due to its high dimensionality, the non-locality with respect to time, and the resolution required to resolve turbulent flow, this problem is highly demanding. To overcome these challenges, we employ the Hermite spectral method for the configuration space of the Fokker--Planck equation, reducing the problem to a purely macroscopic model. Considering scenarios for available analytical solutions, we prove the existence of an optimal choice of the Hermite scaling parameter. With this choice, the macroscopic system is equivalent to solving the coupled micro-macro system. We apply second-order time integration and extrapolation of the coupling terms, achieving, for the first time, convergence rates for the fully coupled time-fractional system independent of the order of the time-fractional derivative. Our efficient implementation of the numerical scheme allows turbulent simulations of dilute polymeric fluids with memory effects in two and three dimensions. Numerical simulations show that memory effects weaken the drag-reducing effect of added polymer molecules in the turbulent flow regime.

Numerical simulation of dilute polymeric fluids with memory effects in the turbulent flow regime

Abstract

We address the numerical challenge of solving the Hookean-type time-fractional Navier--Stokes--Fokker--Planck equation, a history-dependent system of PDEs defined on the Cartesian product of two -dimensional spaces in the turbulent regime. Due to its high dimensionality, the non-locality with respect to time, and the resolution required to resolve turbulent flow, this problem is highly demanding. To overcome these challenges, we employ the Hermite spectral method for the configuration space of the Fokker--Planck equation, reducing the problem to a purely macroscopic model. Considering scenarios for available analytical solutions, we prove the existence of an optimal choice of the Hermite scaling parameter. With this choice, the macroscopic system is equivalent to solving the coupled micro-macro system. We apply second-order time integration and extrapolation of the coupling terms, achieving, for the first time, convergence rates for the fully coupled time-fractional system independent of the order of the time-fractional derivative. Our efficient implementation of the numerical scheme allows turbulent simulations of dilute polymeric fluids with memory effects in two and three dimensions. Numerical simulations show that memory effects weaken the drag-reducing effect of added polymer molecules in the turbulent flow regime.

Paper Structure

This paper contains 19 sections, 2 theorems, 85 equations, 10 figures.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Hermite spectral method
  4. Kernel compression method
  5. Derivation of a macroscopic formulation
  6. Extra-stress tensor
  7. Fokker--Planck equation
  8. The choice of the scaling parameter
  9. Macroscopic system
  10. Numerical method
  11. Projection method and time integration
  12. Finite element approximation
  13. Numerical results
  14. Convergence studies
  15. Flow around a cylinder
  16. ...and 4 more sections

Key Result

Lemma 1

Let $\boldsymbol{\mathbf{D}} \in \mathbb{R}^{3 \times 3}$ be a trace-free, symmetric matrix, $\mathrm{De} \in \mathbb{R}$, such that is symmetric and positive definite. Then

Figures (10)

  • Figure 1: Hermite polynomials (left) and Hermite functions with scaling parameter $a = 1 / \sqrt{2}$ (right) up to order 5.
  • Figure 2: Error decay of the numerical solution (a) in comparison to an analytical solution for the TFFP equation with $\alpha = 0.5$ and (b) in comparison to a numerical reference solution obtained at a finer resolution for the fully coupled time-fractional NSFP system.
  • Figure 3: Influence of the fractional order on the magnitude of the polymer-induced force $\|\nabla \cdot \boldsymbol{\mathbf{\uptau}}\|_{L^2(\Omega)}$.
  • Figure 4: Comparison of the velocity magnitude of dilute polymeric fluid flow for $\alpha = 0.5, 0.8, 1$ at $t=2,3,4$.
  • Figure 5: Visualization of the $\omega$-method for the pure solvent fluid at $t=20$.
  • ...and 5 more figures

Theorems & Definitions (5)

  • Lemma 1
  • proof
  • Lemma 2
  • proof
  • proof