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Symmetric factorization of the conformation tensor in viscoelastic fluid models

Nusret Balci, Becca Thomases, Michael Renardy, Charles R. Doering

arXiv:1006.3488v1math.NAcs.NAmath.APphysics.flu-dyn

TL;DR

Viscoelastic-fluid models are difficult to analyze and simulate, in part because the positive-definite conformation tensors do not form a vector space. The paper evolves their symmetric square roots in Oldroyd-B and FENE-P, reporting improved numerical accuracy and stability over direct tensor evolution without artificial stress diffusion.

  • Problem

    Viscoelastic-fluid models pose numerical accuracy and stability challenges, and the positive-definite conformation tensors do not form a vector space with a natural energy norm.

    Source: §1 · p. 1, §2 · p. 2

  • Method

    The paper derives and evolves the unique symmetric square root of the conformation tensor, choosing its evolution to preserve symmetry.

    Source: §1 · p. 1, §2 · p. 2

  • Results

    The square-root formulation yields significant accuracy and stability gains over direct conformation-tensor evolution without artificial stress diffusion.

    Source: §1 · p. 1

  • Takeaways & Limitations

    In practice, the square-root method can be used at higher Weissenberg numbers and for longer times without artificial numerical stress diffusion than direct conformation-tensor evolution.

    Source: §4 · p. 6

    With fixed resolution and no stress diffusion, accuracy inevitably decreases; whether the Hilbert-space formulation assists rigorous analysis remains open.

    Source: §3.2 · p. 5, §4 · p. 6

Abstract

from arXiv · show

The positive definite symmetric polymer conformation tensor possesses a unique symmetric square root that satisfies a closed evolution equation in the Oldroyd-B and FENE-P models of viscoelastic fluid flow. When expressed in terms of the velocity field and the symmetric square root of the conformation tensor, these models' equations of motion formally constitute an evolution in a Hilbert space with a total energy functional that defines a norm. Moreover, this formulation is easily implemented in direct numerical simulations resulting in significant practical advantages in terms of both accuracy and stability.

1. Introduction

The paper derives a symmetric-square-root evolution for Oldroyd-B and FENE-P by exploiting the matrix square root’s O(n) degeneracy. The formulation reports accuracy and stability gains over direct conformation-tensor evolution without artificial stress diffusion.

1. Introduction

  • Viscoelastic-fluid models remain difficult to analyze and simulate; weak polymer diffusion is often neglected, while numerical schemes face accuracy and stability problems.The introduction also notes the absence of effective a priori estimates for analysis.Source: §1 · p. 1
  • Prior approaches include log-conformation, eigenvalue, and deformation-tensor methods; the paper presents its square-root evolution as distinct from prior work it knows.Source: §1 · p. 1
  • The paper derives an evolution equation for the positive-definite square root by using the O(n) degeneracy of the matrix square root.Source: §1 · p. 1
  • At practically no additional computational cost, the formulation gains numerical stability and accuracy over direct conformation-tensor evolution without artificial stress diffusion.For the studied Oldroyd-B flow, simulations extend far beyond the Weissenberg number at which the associated steady-flow stress becomes infinite.Source: §1 · p. 1

2. Mathematical Framework

The framework replaces the positive-definite conformation tensor with its symmetric square root, restoring a vector-space formulation. Choosing the square-root evolution to preserve symmetry gives the models an energy-based norm, with the FENE-P norm result restricted to a neighborhood of the origin.

2. Mathematical Framework

  • Total mechanical energy combines fluid kinetic and polymer elastic energy and is formally conserved in the infinite-Reynolds- and infinite-Weissenberg-number limits, but excludes the entropic free-energy term.Source: §2 · p. 2
  • The positive-definite conformation-tensor space is not a vector space, so total energy defines neither a natural norm nor a metric there, complicating analysis.Source: §2 · p. 2
  • The reformulation uses the unique symmetric square root b of c, and the polymer energy density depends on the matrix norm of b.Source: §2 · p. 2
  • An antisymmetric matrix a can be chosen from ∇u and symmetric b to keep the evolving square root symmetric; for positive-definite b, that choice is unique.In three dimensions, the coefficients are obtained by solving a system of three linear equations.Source: §2 · pp. 2–3
  • The reformulation returns the dynamics to a vector space; Oldroyd-B energy is proportional to its squared norm, while FENE-P has a natural energy norm near the origin for ∥b∥ < l.The paper says this formulation leads to significant practical improvements in some direct numerical simulations.Source: §2 · p. 3

3. Numerical Experiments

The experiments compare direct conformation-tensor evolution with the symmetric square-root formulation in Stokes Oldroyd-B and FENE-P, testing accuracy and stability without artificial stress diffusion.

3.1. Accuracy

  • At Wi = 5, the square-root method is more accurate away from the extensional stagnation point, while direct c is more accurate very near it.Although the square-root method has larger absolute error at the stagnation point, its relative error there is small because the stress is very large.Source: Figure 2 · p. 4, §3.1 · p. 4
  • For Oldroyd-B, the square-root method has smaller domain-averaged error at every tested resolution, with larger accuracy gains at higher Wi, especially on coarser grids.The comparisons use T = t/Wi = 2.Source: Figure 3 · p. 4, §3.1 · p. 4

3.2. Stability

  • At Wi = 10, direct c fails to produce finite values by t = 20, while square-root simulations run to t = 1500 and beyond without artificial stress diffusion.Fixed-resolution runs still lose accuracy, but qualitatively reproduce previously reported instabilities.Source: Figure 4 · p. 5, §3.2 · p. 5

3.3. FENE-P

  • For FENE-P at Wi = 5, the accuracy improvement is about 65% at N = 256^2 and 31% at N^2 = 512^2.The plotted quantity S is closely related to physical stress and includes the factor that grows as tr(c) approaches the cutoff.Source: §3.3 · p. 6
  • For FENE-P, direct c fails to produce finite values before t = 60 at Wi = 50, while square-root evolution appears to continue indefinitely, with some loss of accuracy.The reported perturbed-initial-data simulation uses l^2 = 225 and is shown at t = 500.Source: §3.3 · p. 6

4. Discussion and Conclusions

The symmetric square-root formulation offers numerical gains through positivity preservation and reduced amplitudes, while expressing the models in an energy-normed vector space remains a theoretical advantage. Its accuracy and stability claims have scope limits, including unresolved high-Weissenberg-number behavior and an unimplemented three-dimensional algorithm.

4. Discussion and Conclusions

  • Taking the square root reduces large amplitudes and may reduce time-stepping stiffness; symmetrization may further reduce stiffness because its components generally have less variance than a nonsymmetric square root.The comparison is with a square root without symmetry.Source: §4 · p. 6
  • The proposed method's comparison with logarithmic-conformation and eigenvalue-evolution methods is left for future study.The authors note that logarithmic methods can be computationally expensive and more complicated to implement.Source: §4 · p. 6
  • The square-root computation preserves positivity and, in practice, runs at higher Wi and for longer without artificial stress diffusion than direct conformation-tensor evolution.The authors report this permits numerical solutions in more situations.Source: §4 · p. 6
  • Under the assumed locality of modal interactions, the first N modes of the square root may provide information about up to the first 2N modes of the conformation tensor.This expectation depends on Galerkin approximation working well for both variables at sufficiently large N; other spatial discretizations require further investigation.Source: §4 · p. 6
  • Implementing the full three-dimensional symmetric square-root algorithm and systematically comparing it with conventional schemes remains future work.The authors mention schemes used to investigate turbulent drag reduction as an example.Source: §4 · p. 6
  • The method still has limitations at sufficiently large Wi, and matching standard conformation-tensor diffusion requires a more complicated nonlinear diffusion term for the square root.A more physically realistic diffusion coefficient is left for future work.Source: §4 · p. 6
  • The Hilbert-space formulation of Oldroyd-B and FENE-P is currently a theoretical advantage; whether it assists rigorous analysis remains an open question.Source: §4 · p. 6

Appendix A. Entries of the antisymmetric matrix for n =

Appendix A identifies the antisymmetric-matrix entries and gives auxiliary quantities used in their expressions. It specifies the T and B quantities directly and refers to earlier equations for the w quantities.

Appendix A. Entries of the antisymmetric matrix for n =

  • The appendix gives expressions for the entries a12, a13, and a23 of the antisymmetric matrix a introduced in (11).Source: Appendix A · p. 7
  • It defines T1 = b22 + b33, T2 = b11 + b33, and T3 = b11 + b22, with B3 = b12, B2 = b13, and B1 = b23.Source: Appendix A · p. 7
  • The quantities w1, w2, and w3 are specified by equations (17)–(19).Source: Appendix A · p. 7
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