src/compressible/two-phase.h

    A compressible two-phase flow solver

    This solves the two-phase compressible Navier-Stokes equations including the total energy equation. \displaystyle \frac{\partial (f \rho_i)}{\partial t } + \nabla \cdot (f \rho_i \mathbf{u}) = 0 \displaystyle \frac{\partial (\rho_i \mathbf{u})}{\partial t } + \nabla \cdot ( \rho_i \mathbf{u} \mathbf{u}) = -\nabla p + \nabla \cdot \tau_i' \displaystyle \frac{\partial [\rho_i (e_i + \mathbf{u}^2/2)]}{\partial t } + \nabla \cdot [ \rho_i \mathbf{u} (e_i + \mathbf{u}^2/2)] = -\nabla \cdot (\mathbf{u} p_i) + \nabla \cdot \left( \tau'_i \mathbf{u} \right) an advection equation for the volume fraction f \displaystyle \frac{\partial f}{\partial t} + \mathbf{u} \cdot \nabla f = 0 and an Equation Of State (EOS) that needs to be defined.

    The method is described in detail in Fuster & Popinet, 2018 and relies on a time splitting technique were we first solve the advection step using the VOF method for f, f_i \rho_i, f_i \rho_i \mathbf{u} f_i \rho_i e_{tot,i} and then perform the projection step using the all-Mach solver.

    #include "all-mach.h"

    We transport VOF tracers without the one-dimensional compressive term.

    #define NO_1D_COMPRESSION 1
#include "vof.h"

    The primary fields are: \displaystyle \begin{aligned} \text{frho1} & = f \rho_1, \\ \text{frho2} & = (1-f) \rho_2, \\ \text{q} & = f \rho_1 \mathbf{u} + (1-f) \rho_2 \mathbf{u}, \\ \text{fE1} & = f \rho_1 (e_1 + \mathbf{u}^2/2), \\ \text{fE2} & = (1-f) \rho_2 (e_2 + \mathbf{u}^2/2) \end{aligned} 

    scalar f[], * interfaces = {f};
scalar frho1[], frho2[], fE1[], fE2[];

    The timestep can be limited by a CFL based on the speed of sound.

    double CFLac = HUGE;

    The dynamic viscosities for each phase, as well as the volumetric viscosity coefficients.

    double mu1 = 0., mu2 = 0.;
double lambdav1 = 0., lambdav2 = 0. ;

    These functions are provided by the Equation Of State. See for example the Mie–Gruneisen Equation of State.

    extern double sound_speed          (Point point);
extern double average_pressure     (Point point);
extern double bulk_compressibility (Point point);
extern double internal_energy      (Point point, double fc);

    By default the Harmonic mean is used to compute the phase-averaged dynamic viscosity.

    #ifndef mu
# define mu(f) (mu1*mu2/(clamp(f,0,1)*(mu2 - mu1) + mu1))
#endif

    The volumetric viscosity uses arithmetic average by default.

    // fixme: Include the term depending on $\nabla \cdot u$. It is tricky because this
// quantity can be very different upon the phase
#ifndef lambdav
# define lambdav(f)  (clamp(f,0,1)*(lambdav1 - lambdav2) + lambdav2)
// # define lambdav(f)  (clamp(f,0.,1.)*(lambdav1 - lambdav2 - 2./3*(mu1 - mu2)) + lambdav2 - 2./3*mu2)
#endif

    Auxilliary fields

    Auxilliary fields need to be allocated. The quantity \rho c^2, the average density rhov and its inverse alphav.

    scalar rhoc2v[];
face vector alphav[];
scalar rhov[];
const face vector lambdav0[] = {0,0,0};
(const) face vector lambdav = lambdav0;

    Time step restriction based on the speed of sound

    event stability (i++)
{
  if (CFLac < HUGE)
    foreach (reduction (min:dtmax)) {
      double c = sound_speed (point);
      if (CFLac*Delta < c*dtmax)
	dtmax = CFLac*Delta/c;
    }
}

#if TREE

    Energy refinement function

    The energy is refined from the refined pressures, momentum and densities, using the equation of state.

    void fE_refine (Point point, scalar fE)
{
  foreach_child() {
    double Ek = 0.;
    foreach_dimension()
      Ek += sq(q.x[]);
    Ek /= 2.*(frho1[] + frho2[]);
    fE[] = fE.inverse ?
      (internal_energy (point, 0.) + Ek)*(1. - f[]) :
      (internal_energy (point, 1.) + Ek)*f[];
  }
}
#endif // TREE

    Initialisation

    We set the default values.

    event defaults (i = 0)
{
  alpha = alphav;
  rho = rhov;

    If the viscosity is non-zero, we need to allocate the face-centered viscosity field.

      if (mu1 || mu2) {
    mu = new face vector;
    lambdav = new face vector;
  }

  rhoc2 = rhoc2v;  
  foreach () {
    frho1[] = 1., fE1[] = 1.;
    frho2[] = 0., fE2[] = 0.;
    p[] = 1.;
    f[] = 1.;
  }

    We also initialize the list of tracers to be advected with the VOF function f (or its complementary function).

      f.tracers = list_copy ({frho1, frho2, fE1, fE2});
  for (scalar s in {frho2, fE2})
    s.inverse = true;

    We set limiting.

      for (scalar s in {frho1, frho2, fE1, fE2, q}) {
    s.gradient = minmod2;
#if TREE

    On trees, we ensure that limiting is also applied to prolongation and refinement.

        s.prolongation = s.refine = refine_linear;
#endif
  }

    We add the interface and the density to the default display.

      display ("draw_vof (c = 'f');");
  display ("squares (color = 'rhov', spread = -1);");
}

event init (i = 0)
{
  trash ({uf});
  foreach_face()
    uf.x[] = fm.x[]*(q.x[]/(frho1[] + frho2[]) + q.x[-1]/(frho1[-1] + frho2[-1]))/2.;

    We update the fluid properties.

      event ("properties");

    We set the initial timestep (this is useful only when restoring from a previous run).

      dtmax = DT;
  event ("stability");

    For the associated tracers we use the gradient defined by f.gradient.

      if (f.gradient)
    for (scalar s in {frho1, frho2, fE1, fE2, q})
      s.gradient = f.gradient;  
}

    VOF advection of momentum

    We overload the vof() event to transport consistently the volume fraction and the momentum of each phase.

    static scalar * interfaces1 = NULL;

event vof (i++)
{

    We split the total momentum q into its two components q1 and q2 associated with f and 1 - f respectively.

      vector q1 = q, q2[];
  foreach()
    foreach_dimension() {
      double u = q.x[]/(frho1[] + frho2[]);
      q1.x[] = frho1[]*u;
      q2.x[] = frho2[]*u;
    }

    Momentum q2 is associated with 1 - f, so we set the inverse attribute to true. We use the same limiting for q1 and q2.

      foreach_dimension() {
    q2.x.inverse = true;
    q2.x.gradient = q1.x.gradient;
  }

#if TREE

    The refinement function is modified by vof_advection(). To be able to restore it, we store its value.

      void (* refine) (Point, scalar) = q1.x.refine;
#endif

    We associate the transport of q1 and q2 with f and transport all fields consistently using the VOF scheme.

      scalar * tracers = f.tracers;
  f.tracers = list_concat (tracers, (scalar *){q1, q2});
  vof_advection ({f}, i);
  free (f.tracers);
  f.tracers = tracers;

    We recover the total momentum.

      foreach() {
    foreach_dimension()
      q.x[] = q1.x[] + q2.x[];

    We avoid negative densities and energies which may have been caused by round-off during VOF advection.

        if (f[] <= 0.){
      frho1[] = 0.;
      fE1[] = 0.;
    }
    if (f[] >= 1.){
      frho2[] = 0.;
      fE2[] = 0.;
    }
  }
  
#if TREE

    We restore the refinement function for the total momentum.

      for (scalar s in {q}) {
    s.refine = s.prolongation = refine;
    s.dirty = true;
  }

#if 0

    This is switched off by default for now as the standard refinement in /src/vof.h#vof_concentration_refine seems to work fine.

      for (scalar s in {fE1,fE2}) {
    s.refine = s.prolongation = fE_refine;
    s.dirty = true;
  }
#endif
#endif // TREE

    We set the list of interfaces to NULL so that the default vof() event does nothing (otherwise we would transport f twice).

      interfaces1 = interfaces, interfaces = NULL;  
}

    We set the list of interfaces back to its default value.

    event tracer_advection (i++) {
  interfaces = interfaces1;
}

    Pressure and density

    During the projection step we compute the provisional pressure from the EOS using the values after advection.

    event properties (i++)
{
  foreach() {
    rhov[] = frho1[] + frho2[];
    ps[] = average_pressure (point);

    We also compute \rho c^2.

        rhoc2v[] = bulk_compressibility (point);
  }
  
  foreach_face() {

    If viscosity is present we obtain the averaged viscosity at the cell faces.

        if (mu1 || mu2) {
      face vector muv = mu, lambdavv = lambdav;
      double ff = (f[] + f[-1])/2.;
      muv.x[] = fm.x[]*mu(ff);
      lambdavv.x[] = lambdav(ff);
    }

    We also compute the averaged density at the cell faces.

        alphav.x[] = 2.*fm.x[]/(rhov[] + rhov[-1]);

  }

    The all-Mach solver needs rho*cm.

      foreach()
    rhov[] *= cm[];
  
  /* I still miss a source term in the divergence related to viscous
   * dissipation. Usually it is small and a little bit complicated to
   * calculate. In the current form of allmach it cannot be added
   * because it does not allow user-defined divergence sources. */
}

    Update of the total energy

    After projection we update the values of the total energy adding the term missing from the projection step.

    For a Newtonian fluid, the stress tensor comprises the pressure term, the viscous uncompressible term and the viscous compressible term,

    \displaystyle \tau = -p \mathbf{I} + \mu (\nabla \mathbf{u} + \nabla \mathbf{u}^T) + \lambda_v (\nabla \cdot \mathbf{u}) \mathbf{I}

    where \mathbf{I} is the unity tensor and \lambda_v = \mu_v - 2/3 \mu. The volumetric viscosity \mu_v is negligible in most cases.

    event end_timestep (i++)
{
#if 0

    We first compute the divergence of the velocity at cell centers using the cell face velocity. Note that the face vector uf incorporates the metric factor.

      scalar divU[];
  foreach () {
    divU[] = 0.;
    foreach_dimension ()
      divU[] += (uf.x[1] - uf.x[])/(Delta*cm[]);
  }

    We compute explicitly the contribution of the compressible viscous term to the momentum equation and the total energy equation, \displaystyle \partial_t \mathbf{q} = \nabla \cdot [\lambda_v(\nabla \cdot \mathbf{u}) \mathbf{I}] \displaystyle \partial_t [\rho (e + \mathbf{u}^2/2)] = \nabla \cdot [\lambda_v (\nabla \cdot \mathbf{u}) \mathbf{u}]

    The axysimmetric case allows some simplifications. Since \displaystyle \mathbf{S} = \lambda_v(\nabla \cdot \mathbf{u}) \mathbf{I} = \left( \begin{array}{ccc} S_{xx} & 0 & 0 \ \ 0 & S_{yy} & 0 \ \ 0 & 0 & S_{\theta \theta} \end{array} \right) with S = S_{xx} = S_{yy} = S_{\theta \theta}= \lambda_v \nabla \cdot \mathbf{u}. Hence, the radial component of \nabla \cdot \mathbf{S} is simply \displaystyle (\nabla \cdot \mathbf{S}) \cdot \mathbf{e}_y = \frac{1}{y} \frac{\partial(y S_{yy}) }{\partial y} - \frac{S_{\theta \theta}}{y} = \frac{\partial S }{\partial y} Also u_\theta = 0.

      foreach() {
    double momentum = 0., energy = 0.;
    foreach_dimension() {
      double right = lambdav.x[1]*(divU[1]  + divU[])/2.;
      double left = lambdav.x[]*(divU[-1] + divU[])/2.;
      momentum += right - left;
      energy += uf.x[1]*right - uf.x[]*left;
    }
    foreach_dimension ()
      q.x[] += dt*momentum/Delta;
    energy *= dt/(Delta*cm[]);
    double fc = clamp(f[],0,1);
    fE1[] += energy*fc;
    fE2[] += energy*(1. - fc);
  }
#endif

    The contribution of the pressure to the energy of each phase is lacking.

      {
    face vector upf[];
    foreach_face()
      upf.x[] = - uf.x[]*(p[] + p[-1])/2.;
 
    foreach () {
      double energy = 0.; 
      foreach_dimension()
	energy += upf.x[1] - upf.x[];
      energy *= dt/(Delta*cm[]);
      double fc = clamp(f[],0,1);
      fE1[] += energy*fc;
      fE2[] += energy*(1. - fc);
    }
  }

    This is the contribution of the incompressible viscous term.

      if (mu1 || mu2) {

    The velocity \mathbf{u} is first obtained from the updated momentum \mathbf{q}.

        vector u = q;
    foreach()
      foreach_dimension() 
        u.x[] = q.x[]/(frho1[] + frho2[]);

    We add the incompressible contribution of the \nabla \cdot (u_i \tau'_i) term. Note that the compressible contribution, which is typically small, is neglected.

        // fixme: add the compressible contribution (careful, $\nabla \cdot
    // u$ can be very different upon the phase and also very different
    // from the value defined for the mixture with an averaged velocity
    face vector ueijk[];
    foreach_dimension() {
      foreach_face(x)
	ueijk.x[] = 2.*uf.x[]*(u.x[] - u.x[-1])/Delta;
#if dimension > 1
      foreach_face(y)
	ueijk.y[] = uf.y[]*(u.x[] - u.x[0,-1] + 
			    (u.y[1,-1] + u.y[1,0])/4. -
			    (u.y[-1,-1] + u.y[-1,0])/4.)/Delta;
#endif
#if dimension > 2
      foreach_face(z)
	ueijk.z[] = uf.z[]*(u.x[] - u.x[0,0,-1] + 
			    (u.z[1,0,-1] + u.z[1,0,0])/4. -
			    (u.z[-1,0,-1] + u.z[-1,0,0])/4.)/Delta;
#endif
      foreach () {
	double energy = 0.; 
	foreach_dimension()
	  energy += ueijk.x[1] - ueijk.x[];
	energy *= dt/(Delta*cm[]);
	double fc = clamp(f[],0,1);
	fE1[] += mu1*energy*fc;
	fE2[] += mu2*energy*(1. - fc);
      }   

      // fixme: Formally, we need to include a correction term due to
      // the normal viscous stress jump (to be done)
    }

    Finally we recover momentum.

        foreach()
      foreach_dimension() 
        q.x[] *= frho1[] + frho2[];
  }
}

    At the end of the simulation we clean the tracer list.

    event cleanup (i = end) {
  free (f.tracers);
  f.tracers = NULL;
}

    References

    [fuster2018]

    Daniel Fuster and Stéphane Popinet. An all-Mach method for the simulation of bubble dynamics problems in the presence of surface tension. Journal of Computational Physics, 374:752–768, December 2018. [ DOI | http | .pdf ]

    See also

    Usage

    Tests