src-local/log-conform-viscoelastic-scalar-2D.h
Log-Conformation Method for 2D Viscoelastic Fluids
Overview
- Title: log-conform-viscoelastic-scalar-2D.h
- Version: 2.5
- Description: 2D and axisymmetric scalar implementation of viscoelastic fluid dynamics using the log-conformation method
Key Features
- Conformation tensor A exists across domain and relaxes according to λ
- Stress acts according to elastic modulus G
- Supports both 2D and axisymmetric configurations
- Scalar implementation approach for better performance
- Compatible with log-conform-viscoelastic.h
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
Version History
v1.0 (Oct 18, 2024)
- Initial implementation with 2D+axi support
- Scalar-based implementation for efficiency
v2.0 (Nov 3, 2024)
- Major documentation improvements
- Made code an axisymmetric mirror of log-conform-viscoelastic-scalar-3D.h
- Added negative eigenvalue detection with location reporting
- Added initialization functions for pseudo_v and pseudo_t
v2.1 (Nov 14, 2024)
- Added support for infinite Deborah number cases
v2.5 (Nov 23, 2024)
- Enhanced documentation clarity and completeness
Future Work
Tensor Formulation
- Convert to consistent tensor formulation for:
- Improved readability and maintainability
- Better computational efficiency
- Reduced potential for bugs
- Prerequisites for axi compatibility in 3D version
- Related issues:
- https://github.com/comphy-lab/Viscoelastic3D/issues/11
- https://github.com/comphy-lab/Viscoelastic3D/issues/5
Code Improvements
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"
(const) scalar Gp = unity; // elastic modulus
(const) scalar lambda = unity; // relaxation time
scalar A11[], A12[], A22[]; // conformation tensor
scalar T11[], T12[], T22[]; // stress tensor
#if AXI
scalar AThTh[], T_ThTh[];
#endif
event defaults (i = 0) {
if (is_constant (a.x))
a = new face vector;
/*
initialize A and T
*/
for (scalar s in {A11, A22}) {
foreach () {
s[] = 1.;
}
}
for (scalar s in {T11, T12, T22, A12}) {
foreach(){
s[] = 0.;
}
}
#if AXI
foreach(){
T_ThTh[] = 0;
AThTh[] = 1.;
}
#endif
for (scalar s in {T11, T12, T22}) {
if (s.boundary[left] != periodic_bc) {
s[left] = neumann(0);
s[right] = neumann(0);
}
}
for (scalar s in {A11, A12, A22}) {
if (s.boundary[left] != periodic_bc) {
s[left] = neumann(0);
s[right] = neumann(0);
}
}
#if AXI
T12[bottom] = dirichlet (0.);
A12[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); // DeterminantThe 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++)
{
scalar Psi11 = A11;
scalar Psi12 = A12;
scalar Psi22 = A22;
#if AXI
scalar Psiqq = AThTh;
#endifComputation 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.x = A11[]; A.y.y = A22[];
A.x.y = A12[];
#if AXI
double Aqq = AThTh[];
Psiqq[] = log (Aqq);
#endifThe conformation tensor is diagonalized through the eigenvector tensor \(\mathbf{R}\) and the eigenvalues diagonal tensor, \(\Lambda\).
pseudo_v Lambda;
init_pseudo_v(&Lambda, 0.0);
pseudo_t R;
init_pseudo_t(&R, 0.0);
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, \(\Psi = R \cdot \log(\Lambda) \cdot R^T\).
Psi12[] = R.x.x*R.y.x*log(Lambda.x) + R.y.y*R.x.y*log(Lambda.y);
Psi11[] = sq(R.x.x)*log(Lambda.x) + sq(R.x.y)*log(Lambda.y);
Psi22[] = sq(R.y.y)*log(Lambda.y) + sq(R.y.x)*log(Lambda.x);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;
init_pseudo_t(&B, 0.0);
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;
init_pseudo_t(&M, 0.0);
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 = -Psi12[];
Psi12[] += dt * (2. * B.x.y + OM * (Psi22[] - Psi11[]));
s *= -1;
Psi11[] += dt * 2. * (B.x.x + s * OM);
s *= -1;
Psi22[] += dt * 2. * (B.y.y + 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 ({Psi11, Psi12, Psi22, Psiqq}, uf, dt);
#else
advection ({Psi11, Psi12, Psi22}, uf, dt);
#endifConvert back to Aij
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 = {{Psi11[], Psi12[]}, {Psi12[], Psi22[]}}, R;
init_pseudo_t(&R, 0.0);
pseudo_v Lambda;
init_pseudo_v(&Lambda, 0.0);
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[]);
#endifWe 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 \(\mathbf{A}^{n+1}\).
A12[] = A.x.y;
T12[] = Gp[]*A.x.y;
#if AXI
AThTh[] = Aqq;
T_ThTh[] = Gp[]*(Aqq - 1.);
#endif
A11[] = A.x.x;
T11[] = Gp[]*(A.x.x - 1.);
A22[] = A.y.y;
T22[] = Gp[]*(A.y.y - 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(x){
if (fm.x[] > 1e-20) {
double shearX = (T12[0,1]*cm[0,1] + T12[-1,1]*cm[-1,1] -
T12[0,-1]*cm[0,-1] - T12[-1,-1]*cm[-1,-1])/4.;
av.x[] += (shearX + cm[]*T11[] - cm[-1]*T11[-1])*
alpha.x[]/(sq(fm.x[])*Delta);
}
}
foreach_face(y){
if (fm.y[] > 1e-20) {
double shearY = (T12[1,0]*cm[1,0] + T12[1,-1]*cm[1,-1] -
T12[-1,0]*cm[-1,0] - T12[-1,-1]*cm[-1,-1])/4.;
av.y[] += (shearY + cm[]*T22[] - cm[0,-1]*T22[0,-1])*
alpha.y[]/(sq(fm.y[])*Delta);
}
}
#if AXI
foreach_face(y)
if (y > 1e-20)
av.y[] -= (T_ThTh[] + T_ThTh[0,-1])*alpha.y[]/sq(y)/2.;
#endif
}