text: src/log-conform.h
  - The log-conformation method for some viscoelastic constitutive models
  The log-conformation method for some 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 where is the deformation tensor and is the solvent viscosity of the viscoelastic fluid.
  The polymeric stress represents memory effects due to the polymers. Several constitutive rheological models are available in the literature where the polymeric stress is typically a function of the conformation tensor such as where is the relaxation parameter and is the polymeric viscosity.
  The conformation tensor is related to the deformation of the polymer chains. is governed by the equation where denotes the material derivative and is the relaxation function.
  In the case of an Oldroyd-B viscoelastic fluid, , and the above equations can be combined to avoid the use of
  Comminal et al. (2015) gathered the functions and for different constitutive models. In the present library we have implemented the Oldroyd-B model and the related FENE-P model for which
  Parameters
  The primary parameters are the retardation or relaxation time and the polymeric viscosity . The solvent viscosity is defined in the Navier-Stokes solver.
  (const) scalar λ = unity; (const) scalar mup = unity;
  Constitutive models other than Oldroyd-B (the default) are defined through the two functions and .
  void (* f_s) (double, double *, double *) = NULL; void (* f_r) (double, double *, double *) = NULL;
  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, . 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 rather than the viscoelastic stress tensor to circumvent the instability.
  The constitutive equation for the log of the conformation tensor is where and are tensors that result from the decomposition of the transpose of the tensor gradient of the velocity
  The antisymmetric tensor requires only the memory of a scalar in 2D since, The log-conformation tensor, , is related to the polymeric stress tensor , by the strain function where denotes the trace of the tensor and is an additional property of the viscoelastic fluid.
  We will use the Bell–Collela–Glaz scheme to advect the log-conformation tensor .
  #include "bcg.h"
  Variables
  The main variable will be the stress tensor . The trace of the conformation tensor, , is often necessary for constitutive viscoelastic models other than Oldroyd-B.
  symmetric tensor tau_p[]; #if AXI scalar tau_qq[]; #endif (const) scalar trA = zeroc; event defaults (i = 0) { if (is_constant (a.x)) a = new face vector; if (f_s || f_r) trA = new scalar; foreach() { foreach_dimension() tau_p.x.x[] = 0.; tau_p.x.y[] = 0.; #if AXI tau_qq[] = 0; #endif }
  Boundary conditions
  By default we set a zero Neumann boundary condition for all the components except if the bottom is an axis of symmetry.
  for (scalar s in {tau_p}) { s.v.x.i = -1; // just a scalar, not the component of a vector foreach_dimension() { s[left] = neumann(0); s[right] = neumann(0); } } #if AXI scalar s = tau_p.x.y; s[bottom] = dirichlet (0.); #endif } event init (i = 0) { #if AXI boundary((scalar *){tau_p, tau_qq}); #else boundary((scalar *){tau_p}); #endif }
  Numerical Scheme
  The first step is to implement a routine to calculate the eigenvalues and eigenvectors of the conformation tensor .
  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; static void diagonalization_2D (pseudo_v * Lambda, pseudo_t * R, pseudo_t * A) {
  The eigenvalues are saved in vector computed from the trace and the determinant of the symmetric conformation tensor .
  if (sq(A->x.y) < 1e-15) { R->x.x = R->y.y = 1.; R->y.x = R->x.y = 0.; Λ->x = A->x.x; Λ->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, are saved by columns in tensor .
  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 *) Λ; 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, (or more precisely the related variable ) 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:
  - the advection term:
  - the model term (but set in terms of the conformation tensor ). In an Oldroyd-B viscoelastic fluid, the model is
  The implementation below assumes that the values of and are never needed simultaneously. This means that can be used to store (temporarily) the values of (i.e. is just an alias for ).
  event tracer_advection (i++) { tensor Ψ = tau_p; #if AXI scalar Psiqq = tau_qq; #endif
  Computation of and upper convective term
  foreach() { if (λ[] == 0.) { foreach_dimension() Ψ.x.x[] = 0.; Ψ.x.y[] = 0.; #if AXI Psiqq[] = 0.; #endif } else { // lambda[] != 0.
  We assume that the stress tensor depends on the conformation tensor as follows In most of the viscoelastic models, and are nonlinear parameters that depend on the trace of the conformation tensor, .
  double η = 1., ν = 1.; if (f_s) f_s (trA[], &ν, &η); double fa = (mup[] != 0 ? λ[]/(mup[]*η) : 0.); pseudo_t A; A.x.y = fa*tau_p.x.y[]/ν; foreach_dimension() A.x.x = (fa*tau_p.x.x[] + 1.)/ν;
  In the axisymmetric case, . Therefore .
  #if AXI double Aqq = (1. + fa*tau_qq[])/ν; Psiqq[] = log (Aqq); #endif
  The conformation tensor is diagonalized through the eigenvector tensor and the eigenvalues diagonal tensor, .
  pseudo_v Λ; pseudo_t R; diagonalization_2D (&Λ, &R, &A);
  is easily obtained after diagonalization, .
  Ψ.x.y[] = R.x.x*R.y.x*log(Λ.x) + R.y.y*R.x.y*log(Λ.y); foreach_dimension() Ψ.x.x[] = sq(R.x.x)*log(Λ.x) + sq(R.x.y)*log(Λ.y);
  We now compute the upper convective term .
  The diagonalization will be applied to the velocity gradient to obtain the antisymmetric tensor and the traceless, symmetric tensor, . If the conformation tensor is , and .
  pseudo_t B; double OM = 0.; if (fabs(Λ.x - Λ.y) <= 1e-20) { B.x.y = (u.y[1,0] - u.y[-1,0] + u.x[0,1] - u.x[0,-1])/(4.*Δ); foreach_dimension() B.x.x = (u.x[1,0] - u.x[-1,0])/(2.*Δ); } 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.*Δ); 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.*Δ); } double ω = (Λ.y*M.x.y + Λ.x*M.y.x)/(Λ.y - Λ.x); OM = (R.x.x*R.y.y - R.x.y*R.y.x)*ω; 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 in time, adding the upper convective contribution.
  double s = - Ψ.x.y[]; Ψ.x.y[] += dt*(2.*B.x.y + OM*(Ψ.y.y[] - Ψ.x.x[])); foreach_dimension() { s *= -1; Ψ.x.x[] += dt*2.*(B.x.x + s*OM); }
  In the axisymmetric case, the governing equation for only involves that component, with . Therefore step (a) for is
  #if AXI Psiqq[] += dt*2.*u.y[]/y; #endif } }
  Advection of
  We proceed with step (b), the advection of the log of the conformation tensor , but first we apply boundary conditions.
  #if AXI boundary ({Ψ.x.x, Ψ.x.y, Ψ.y.y, Psiqq}); advection ({Ψ.x.x, Ψ.x.y, Ψ.y.y, Psiqq}, uf, dt); #else boundary ({Ψ.x.x, Ψ.x.y, Ψ.y.y}); advection ({Ψ.x.x, Ψ.x.y, Ψ.y.y}, uf, dt); #endif
  Model term
  foreach() { if (λ[] == 0.) {
  If the stress tensor for the polymeric part reduces to that of a Newtonian fluid with the rate-of-strain tensor. Note that is in this case independent of time.
  foreach_dimension() tau_p.x.x[] = mup[]*(u.x[1,0] - u.x[-1,0])/Δ; // 2*mu*dxu; tau_p.x.y[] = mup[]*(u.y[1,0] - u.y[-1,0] + u.x[0,1] - u.x[0,-1])/(2.*Δ); // mu*(dxv+dyu) #if AXI tau_qq[] = 2.*mup[]*u.y[]/y; #endif } else { // lambda != 0.
  It is time to undo the log-conformation, again by diagonalization, to recover the conformation tensor and to perform step (c).
  pseudo_t A = {{Ψ.x.x[], Ψ.x.y[]}, {Ψ.y.x[], Ψ.y.y[]}}, R; pseudo_v Λ; diagonalization_2D (&Λ, &R, &A); Λ.x = exp(Λ.x), Λ.y = exp(Λ.y); A.x.y = R.x.x*R.y.x*Λ.x + R.y.y*R.x.y*Λ.y; foreach_dimension() A.x.x = sq(R.x.x)*Λ.x + sq(R.x.y)*Λ.y; #if AXI double Aqq = exp(Psiqq[]); #endif
  We perform now step (c) by integrating to obtain . This step is analytic,
  double η = 1., ν = 1.; if (f_r) { #if 0 // Set to one if the midstep trace is to be used. scalar t = trA; t[] = A.x.x + A.y.y; #if AXI t[] += Aqq; #endif #endif f_r (trA[], &ν, &η); } double fa = exp(-ν*η*dt/λ[]); #if AXI Aqq = (1. - fa)/ν + fa*exp(Psiqq[]); Psiqq[] = log (Aqq); #endif A.x.y *= fa; foreach_dimension() A.x.x = (1. - fa)/ν + A.x.x*fa;
  The trace at time is also needed for some models.
  if (f_s || f_r) { scalar t = trA; t[] = A.x.x + A.y.y; #if AXI t[] += Aqq; #endif }
  Then the stress tensor is computed from according to the constitutive model, .
  ν = 1; η = 1.; if (f_s) f_s (trA[], &ν, &η); fa = mup[]/λ[]*η; tau_p.x.y[] = fa*ν*A.x.y; #if AXI tau_qq[] = fa*(ν*Aqq - 1.); #endif foreach_dimension() tau_p.x.x[] = fa*(ν*A.x.x - 1.); } } #if AXI boundary((scalar *){tau_p, tau_qq}); #else boundary((scalar *){tau_p}); #endif }
  Divergence of the viscoelastic stress tensor
  The viscoelastic stress tensor 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 -component in Cartesian coordinates has the following terms: . 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])* α.x[]/(sq(fm.x[])*Δ); } #if AXI foreach_face(y) if (y > 0.) av.y[] -= (tau_qq[] + tau_qq[0,-1])*α.y[]/sq(y)/2.; #endif }