src-local/log-conform-viscoelastic.h

See raw file

Log-Conformation Method with Tensor Implementation

Overview

Title: log-conform-viscoelastic.h
Version: 10.5
Description: Tensor-based implementation of the log-conformation method for viscoelastic fluids

Key Features

Conformation tensor A exists across domain and relaxes according to λ
Stress acts according to elastic modulus G
Uses native tensor data structures for better code organization
Supports both 2D and axisymmetric configurations

Author Information

Name: Vatsal Sanjay
Email: [email protected]
Institution: Physics of Fluids
Last Updated: Nov 23, 2024

Implementation Notes

Based on http://basilisk.fr/src/log-conform.h with key improvements:
- Uses G-λ formulation for better physical interpretation
- Fixes surface tension coupling bug where [σ_p] = 0 & [σ_s] = γκ
- Ensures [σ_s+σ_p] = γκ for correct interface behavior

Important Limitations

3D Compatibility

Currently limited to 2D and axisymmetric cases only
3D support is blocked by Basilisk core limitations:
- Boundary conditions for symmetric tensors are not implemented in Basilisk core
- See basilisk/src/grid/cartesian-common.h lines 230-251
- Comment in source: “fixme: boundary conditions don’t work!”

Alternative for 3D

For 3D simulations, use log-conform-viscoelastic-scalar-3D.h
Scalar version uses individual components instead of tensors
Provides full 3D functionality without boundary condition limitations

Technical Notes

Variable Naming

conform_p, conform_qq: Represent the Conformation tensor
Tensor implementation provides more natural mathematical representation
Axisymmetric components handled separately when needed

Mathematical Framework

The implementation follows the standard log-conformation approach: 1. Uses tensor mathematics for clean formulation 2. Handles both planar and axisymmetric geometries 3. Provides natural extension to various constitutive models

Note:

In this code, conform_p, conform_qq are in fact the Conformation tensor.

The log-conformation method for viscoelastic constitutive models

Introduction

Viscoelastic fluids exhibit both viscous and elastic behaviour when subjected to deformation. Therefore these materials are governed by the Navier–Stokes equations enriched with an extra elastic stress \(Tij\) \[ \rho\left[\partial_t\mathbf{u}+\nabla\cdot(\mathbf{u}\otimes\mathbf{u})\right] = - \nabla p + \nabla\cdot(2\mu_s\mathbf{D}) + \nabla\cdot\mathbf{T} + \rho\mathbf{a} \] where \(\mathbf{D}=[\nabla\mathbf{u} + (\nabla\mathbf{u})^T]/2\) is the deformation tensor and \(\mu_s\) is the solvent viscosity of the viscoelastic fluid.

The polymeric stress \(\mathbf{T}\) represents memory effects due to the polymers. Several constitutive rheological models are available in the literature where the polymeric stress \(\mathbf{T}\) is typically a function \(\mathbf{f_s}(\cdot)\) of the conformation tensor \(\mathbf{A}\) such as \[ \mathbf{T} = G_p \mathbf{f_s}(\mathbf{A}) \] where \(G_p\) is the elastic modulus and \(\mathbf{f_s}(\cdot)\) is the relaxation function.

The conformation tensor \(\mathbf{A}\) is related to the deformation of the polymer chains. \(\mathbf{A}\) is governed by the equation \[ D_t \mathbf{A} - \mathbf{A} \cdot \nabla \mathbf{u} - \nabla \mathbf{u}^{T} \cdot \mathbf{A} = -\frac{\mathbf{f_r}(\mathbf{A})}{\lambda} \] where \(D_t\) denotes the material derivative and \(\mathbf{f_r}(\cdot)\) is the relaxation function. Here, \(\lambda\) is the relaxation time.

In the case of an Oldroyd-B viscoelastic fluid, \(\mathbf{f}_s (\mathbf{A}) = \mathbf{f}_r (\mathbf{A}) = \mathbf{A} -\mathbf{I}\), and the above equations can be combined to avoid the use of \(\mathbf{A}\) \[ \mathbf{T} + \lambda (D_t \mathbf{T} - \mathbf{T} \cdot \nabla \mathbf{u} - \nabla \mathbf{u}^{T} \cdot \mathbf{T}) = 2 G_p\lambda \mathbf{D} \]

Comminal et al. (2015) gathered the functions \(\mathbf{f}_s (\mathbf{A})\) and \(\mathbf{f}_r (\mathbf{A})\) for different constitutive models.

Parameters

The primary parameters are the relaxation time \(\lambda\) and the elastic modulus \(G_p\). The solvent viscosity \(\mu_s\) is defined in the Navier-Stokes solver.

Gp and lambda are defined in two-phaseVE.h.

The log conformation approach

The numerical resolution of viscoelastic fluid problems often faces the High-Weissenberg Number Problem. This is a numerical instability appearing when strongly elastic flows create regions of high stress and fine features. This instability poses practical limits to the values of the relaxation time of the viscoelastic fluid, \(\lambda\). Fattal & Kupferman (2004, 2005) identified the exponential nature of the solution as the origin of the instability. They proposed to use the logarithm of the conformation tensor \(\Psi = \log \, \mathbf{A}\) rather than the viscoelastic stress tensor to circumvent the instability.

The constitutive equation for the log of the conformation tensor is \[ D_t \Psi = (\Omega \cdot \Psi -\Psi \cdot \Omega) + 2 \mathbf{B} + \frac{e^{-\Psi} \mathbf{f}_r (e^{\Psi})}{\lambda} \] where \(\Omega\) and \(\mathbf{B}\) are tensors that result from the decomposition of the transpose of the tensor gradient of the velocity \[ (\nabla \mathbf{u})^T = \Omega + \mathbf{B} + N \mathbf{A}^{-1} \]

The antisymmetric tensor \(\Omega\) requires only the memory of a scalar in 2D since, \[ \Omega = \left( \begin{array}{cc} 0 & \Omega_{12} \\ -\Omega_{12} & 0 \end{array} \right) \]

For 3D, \(\Omega\) is a skew-symmetric tensor given by

\[ \Omega = \left( \begin{array}{ccc} 0 & \Omega_{12} & \Omega_{13} \\ -\Omega_{12} & 0 & \Omega_{23} \\ -\Omega_{13} & -\Omega_{23} & 0 \end{array} \right) \]

The log-conformation tensor, \(\Psi\), is related to the polymeric stress tensor \(\mathbf{T}\), by the strain function \(\mathbf{f}_s (\mathbf{A})\) \[ \Psi = \log \, \mathbf{A} \quad \mathrm{and} \quad \mathbf{T} = \frac{G_p}{\lambda} \mathbf{f}_s (\mathbf{A}) \] where \(Tr\) denotes the trace of the tensor and \(L\) is an additional property of the viscoelastic fluid.

We will use the Bell–Collela–Glaz scheme to advect the log-conformation tensor \(\Psi\).

TODO: - Perhaps, instead of the Bell–Collela–Glaz scheme, we can use the conservative form of the advection equation and transport the log-conformation tensor with the VoF color function, similar to http://basilisk.fr/src/navier-stokes/conserving.h

#include "bcg.h"

#if dimension ==
#error "This implementation does not support 3D due to missing tensor boundary conditions in Basilisk (see cartesian-common.h line ~246). Use log-conform-viscoelastic-scalar-3D.h for 3D simulations."
#endif

(const) scalar Gp = unity; // elastic modulus
(const) scalar lambda = unity; // relaxation time

symmetric tensor conform_p[], tau_p[];
#if AXI
scalar conform_qq[], tau_qq[];
#endif

event defaults (i = 0) {
  if (is_constant (a.x))
    a = new face vector;

  foreach() {
    foreach_dimension(){
      tau_p.x.x[] = 0.;
      conform_p.x.x[] = 1.;
    }
    tau_p.x.y[] = 0.;
    conform_p.x.y[] = 0.;
#if AXI
    tau_qq[] = 0;
    conform_qq[] = 1.;
#endif
  }

  for (scalar s in {tau_p}) {
    s.v.x.i = -1; // just a scalar, not the component of a vector
    foreach_dimension(){
      if (s.boundary[left] != periodic_bc) {
        s[left] = neumann(0);
          s[right] = neumann(0);
      }
    }
  }

  for (scalar s in {conform_p}) {
    s.v.x.i = -1; // just a scalar, not the component of a vector
    foreach_dimension(){
      if (s.boundary[left] != periodic_bc) {
        s[left] = neumann(0);
          s[right] = neumann(0);
      }
    }
  }

#if AXI
  scalar s1 = tau_p.x.y;
  s1[bottom] = dirichlet (0.);
#endif

#if AXI
  scalar s2 = conform_p.x.y;
  s2[bottom] = dirichlet (0.);
#endif
}

Useful functions in 2D

The first step is to implement a routine to calculate the eigenvalues and eigenvectors of the conformation tensor \(\mathbf{A}\).

These structs ressemble Basilisk vectors and tensors but are just arrays not related to the grid.

typedef struct { double x, y;}   pseudo_v;
typedef struct { pseudo_v x, y;} pseudo_t;

// Function to initialize pseudo_v
static inline void init_pseudo_v(pseudo_v *v, double value) {
    v->x = value;
    v->y = value;
}

// Function to initialize pseudo_t
static inline void init_pseudo_t(pseudo_t *t, double value) {
    init_pseudo_v(&t->x, value);
    init_pseudo_v(&t->y, value);
}

static void diagonalization_2D (pseudo_v * Lambda, pseudo_t * R, pseudo_t * A)
{

The eigenvalues are saved in vector \(\Lambda\) computed from the trace and the determinant of the symmetric conformation tensor \(\mathbf{A}\).

  if (sq(A->x.y) < 1e-15) {
    R->x.x = R->y.y = 1.;
    R->y.x = R->x.y = 0.;
    Lambda->x = A->x.x; Lambda->y = A->y.y;
    return;
  }

  double T = A->x.x + A->y.y; // Trace of the tensor
  double D = A->x.x*A->y.y - sq(A->x.y); // Determinant

The eigenvectors, \(\mathbf{v}_i\) are saved by columns in tensor \(\mathbf{R} = (\mathbf{v}_1|\mathbf{v}_2)\).

  R->x.x = R->x.y = A->x.y;
  R->y.x = R->y.y = -A->x.x;
  double s = 1.;
  for (int i = 0; i < dimension; i++) {
    double * ev = (double *) Lambda;
    ev[i] = T/2 + s*sqrt(sq(T)/4. - D);
    s *= -1;
    double * Rx = (double *) &R->x;
    double * Ry = (double *) &R->y;
    Ry[i] += ev[i];
    double mod = sqrt(sq(Rx[i]) + sq(Ry[i]));
    Rx[i] /= mod;
    Ry[i] /= mod;
  }
}

The stress tensor depends on previous instants and has to be integrated in time. In the log-conformation scheme the advection of the stress tensor is circumvented, instead the conformation tensor, \(\mathbf{A}\) (or more precisely the related variable \(\Psi\)) is advanced in time.

In what follows we will adopt a scheme similar to that of Hao & Pan (2007). We use a split scheme, solving successively

the upper convective term: \[ \partial_t \Psi = 2 \mathbf{B} + (\Omega \cdot \Psi -\Psi \cdot \Omega) \]
the advection term: \[ \partial_t \Psi + \nabla \cdot (\Psi \mathbf{u}) = 0 \]
the model term (but set in terms of the conformation tensor \(\mathbf{A}\)). In an Oldroyd-B viscoelastic fluid, the model is \[ \partial_t \mathbf{A} = -\frac{\mathbf{f}_r (\mathbf{A})}{\lambda} \]

event tracer_advection(i++)
{
    tensor Psi = conform_p;
#if AXI
    scalar Psiqq = conform_qq;
#endif

Computation of \(\Psi = \log \mathbf{A}\) and upper convective term

    foreach() {

We assume that the stress tensor \(\mathbf{\tau}_p\) depends on the conformation tensor \(\mathbf{A}\) as follows \[ \mathbf{\tau}_p = G_p (\mathbf{A}) = G_p (\mathbf{A} - I) \]

      pseudo_t A;
      A.x.y = conform_p.x.y[];

      foreach_dimension()
        A.x.x = conform_p.x.x[];

In the axisymmetric case, \(\Psi_{\theta \theta} = \log A_{\theta \theta}\). Therefore \(\Psi_{\theta \theta} = \log [ ( 1 + \text{fa} \tau_{p_{\theta \theta}})]\).

#if AXI
      double Aqq = conform_qq[];
      Psiqq[] = log (Aqq);
#endif

The conformation tensor is diagonalized through the eigenvector tensor \(\mathbf{R}\) and the eigenvalues diagonal tensor, \(\Lambda\).

      pseudo_v Lambda;
      pseudo_t R;
      diagonalization_2D (&Lambda, &R, &A);

      /*
      Check for negative eigenvalues -- this should never happen. If it does, print the location and value of the offending eigenvalue.
      Please report this bug by opening an issue on the GitHub repository.
      */
      if (Lambda.x <= 0. || Lambda.y <= 0.) {
        fprintf(ferr, "Negative eigenvalue detected: Lambda.x = %g, Lambda.y = %g\n", Lambda.x, Lambda.y);
        fprintf(ferr, "x = %g, y = %g\n", x, y);
        exit(1);
      }

\(\Psi = \log \mathbf{A}\) is easily obtained after diagonalization, \(\Psi = R \cdot \log(\Lambda) \cdot R^T\).

      Psi.x.y[] = R.x.x*R.y.x*log(Lambda.x) + R.y.y*R.x.y*log(Lambda.y);
      foreach_dimension()
        Psi.x.x[] = sq(R.x.x)*log(Lambda.x) + sq(R.x.y)*log(Lambda.y);

We now compute the upper convective term \(2 \mathbf{B} + (\Omega \cdot \Psi -\Psi \cdot \Omega)\).

The diagonalization will be applied to the velocity gradient \((\nabla u)^T\) to obtain the antisymmetric tensor \(\Omega\) and the traceless, symmetric tensor, \(\mathbf{B}\). If the conformation tensor is \(\mathbf{I}\), \(\Omega = 0\) and \(\mathbf{B}= \mathbf{D}\).

Otherwise, compute M = R * (nablaU)^T * R^T, where nablaU is the velocity gradient tensor. Then,

Calculate omega using the off-diagonal elements of M and eigenvalues: omega = (Lambda.yM.x.y + Lambda.xM.y.x)/(Lambda.y - Lambda.x) This represents the rotation rate in the eigenvector basis.
Transform omega back to physical space to get OM: OM = (R.x.xR.y.y - R.x.yR.y.x)*omega This gives us the rotation tensor Omega in the original coordinate system.
Compute B tensor components using M and R: B is related to M and R through:

In 2D: \[ B_{xx} = R_{xx}^2 M_{xx} + R_{xy}^2 M_{yy} \\ B_{xy} = R_{xx}R_{yx} M_{xx} + R_{xy}R_{yy} M_{yy} \\ B_{yx} = B_{xy} \\ B_{yy} = -B_{xx} \]

Where: - R is the eigenvector matrix of the conformation tensor - M is the velocity gradient tensor in the eigenvector basis - The construction ensures B is symmetric and traceless

      pseudo_t B;
      double OM = 0.;
      if (fabs(Lambda.x - Lambda.y) <= 1e-20) {
        B.x.y = (u.y[1,0] - u.y[-1,0] + u.x[0,1] - u.x[0,-1])/(4.*Delta);
        foreach_dimension()
          B.x.x = (u.x[1,0] - u.x[-1,0])/(2.*Delta);
      } else {
        pseudo_t M;
        foreach_dimension() {
          M.x.x = (sq(R.x.x)*(u.x[1] - u.x[-1]) +
          sq(R.y.x)*(u.y[0,1] - u.y[0,-1]) +
          R.x.x*R.y.x*(u.x[0,1] - u.x[0,-1] +
          u.y[1] - u.y[-1]))/(2.*Delta);

          M.x.y = (R.x.x*R.x.y*(u.x[1] - u.x[-1]) +
          R.x.y*R.y.x*(u.y[1] - u.y[-1]) +
          R.x.x*R.y.y*(u.x[0,1] - u.x[0,-1]) +
          R.y.x*R.y.y*(u.y[0,1] - u.y[0,-1]))/(2.*Delta);
        }
        double omega = (Lambda.y*M.x.y + Lambda.x*M.y.x)/(Lambda.y - Lambda.x);
        OM = (R.x.x*R.y.y - R.x.y*R.y.x)*omega;

        B.x.y = M.x.x*R.x.x*R.y.x + M.y.y*R.y.y*R.x.y;
        foreach_dimension()
          B.x.x = M.x.x*sq(R.x.x)+M.y.y*sq(R.x.y);
      }

We now advance \(\Psi\) in time, adding the upper convective contribution.

      double s = - Psi.x.y[];
      Psi.x.y[] += dt*(2.*B.x.y + OM*(Psi.y.y[] - Psi.x.x[]));
      foreach_dimension() {
        s *= -1;
          Psi.x.x[] += dt*2.*(B.x.x + s*OM);
      }

In the axisymmetric case, the governing equation for \(\Psi_{\theta \theta}\) only involves that component, \[ \Psi_{\theta \theta}|_t - 2 L_{\theta \theta} = \frac{\mathbf{f}_r(e^{-\Psi_{\theta \theta}})}{\lambda} \] with \(L_{\theta \theta} = u_y/y\). Therefore step (a) for \(\Psi_{\theta \theta}\) is

#if AXI
      Psiqq[] += dt*2.*u.y[]/max(y, 1e-20);
#endif

}

Advection of \(\Psi\)

We proceed with step (b), the advection of the log of the conformation tensor \(\Psi\).

#if AXI
  advection ({Psi.x.x, Psi.x.y, Psi.y.y, Psiqq}, uf, dt);
#else
  advection ({Psi.x.x, Psi.x.y, Psi.y.y}, uf, dt);
#endif

Convert back to _p

    foreach() {

It is time to undo the log-conformation, again by diagonalization, to recover the conformation tensor \(\mathbf{A}\) and to perform step (c).

      pseudo_t A = {{Psi.x.x[], Psi.x.y[]}, {Psi.y.x[], Psi.y.y[]}}, R;
      pseudo_v Lambda;
      diagonalization_2D (&Lambda, &R, &A);
      Lambda.x = exp(Lambda.x), Lambda.y = exp(Lambda.y);

      A.x.y = R.x.x*R.y.x*Lambda.x + R.y.y*R.x.y*Lambda.y;
      foreach_dimension()
        A.x.x = sq(R.x.x)*Lambda.x + sq(R.x.y)*Lambda.y;
#if AXI
      double Aqq = exp(Psiqq[]);
#endif

We perform now step (c) by integrating \(\mathbf{A}_t = -\mathbf{f}_r (\mathbf{A})/\lambda\) to obtain \(\mathbf{A}^{n+1}\). This step is analytic, \[ \int_{t^n}^{t^{n+1}}\frac{d \mathbf{A}}{\mathbf{I}- \mathbf{A}} = \frac{\Delta t}{\lambda} \]

     double intFactor = (lambda[] != 0. ? (lambda[] == 1e30 ? 1: exp(-dt/lambda[])): 0.);

#if AXI
      Aqq = (1. - intFactor) + intFactor*exp(Psiqq[]);
#endif

      A.x.y *= intFactor;
      foreach_dimension()
        A.x.x = (1. - intFactor) + A.x.x*intFactor;

Then the Conformation tensor \(\mathcal{A}_p^{n+1}\) is restored from \(\mathbf{A}^{n+1}\).

      conform_p.x.y[] = A.x.y;
      tau_p.x.y[] = Gp[]*A.x.y;
#if AXI
      conform_qq[] = Aqq;
      tau_qq[] = Gp[]*(Aqq - 1.);
#endif

      foreach_dimension(){
        conform_p.x.x[] = A.x.x;
        tau_p.x.x[] = Gp[]*(A.x.x - 1.);
      }

  }
}

Divergence of the viscoelastic stress tensor

The viscoelastic stress tensor \(\mathbf{\tau}_p\) is defined at cell centers while the corresponding force (acceleration) will be defined at cell faces. Two terms contribute to each component of the momentum equation. For example the \(x\)-component in Cartesian coordinates has the following terms: \(\partial_x \mathbf{\tau}_{p_{xx}} + \partial_y \mathbf{\tau}_{p_{xy}}\). The first term is easy to compute since it can be calculated directly from center values of cells sharing the face. The other one is harder. It will be computed from vertex values. The vertex values are obtained by averaging centered values. Note that as a result of the vertex averaging cells [] and [-1,0] are not involved in the computation of shear.

event acceleration (i++)
{
  face vector av = a;
  foreach_face()
    if (fm.x[] > 1e-20) {
      double shear = (tau_p.x.y[0,1]*cm[0,1] + tau_p.x.y[-1,1]*cm[-1,1] -
              tau_p.x.y[0,-1]*cm[0,-1] - tau_p.x.y[-1,-1]*cm[-1,-1])/4.;
      av.x[] += (shear + cm[]*tau_p.x.x[] - cm[-1]*tau_p.x.x[-1])*
    alpha.x[]/(sq(fm.x[])*Delta);
    }
#if AXI
  foreach_face(y)
    if (y > 0.)
      av.y[] -= (tau_qq[] + tau_qq[0,-1])*alpha.y[]/sq(y)/2.;
#endif
}