Librairies

numericalMethods

Efficient time step management is crucial for modeling biofilm growth, as the characteristic time scales can vary by several orders of magnitude. This library implements an adaptive time step management system based on time truncation error estimates (see Horgue et al for more details).

The time step is computed to maintain a relative time truncation error (estimated using Taylor series) below a user-defined value \(\epsilon_{error}\). The time-step computations for the available time schemes in the code are as follows:

backward Euler: \(\Delta t = \sqrt{ \frac{ 2 \epsilon_{error} x_{max} }{ \left( \partial^2 x / \partial t^2 \right)_{max} } }\)
2nd order backward scheme: \(\Delta t = \sqrt[3]{ \frac{ 4 \epsilon_{error} x_{max} }{ \left( \partial^3 x / \partial t^3 \right)_{max} } }\)
Crank-Nicolson: \(\Delta t = \sqrt[3]{ \frac{ 12 \epsilon_{error} x_{max} }{ \left( \partial^3 x / \partial t^3 \right)_{max} } }\)

where \(x_{max}\) is the value of the variable at the cell where the maximal derivative has been computed.

A third-order time scheme had to be implemented in the toolbox (EulerD3dt3Scheme, using he current and three previous time-step values) to allow adaptive time step management for the available time schemes.

The timestep manager is configured within the file system/controlDict

runTimeModifiable true;

adjustTimeStep    yes;

maxDeltaT         3600;

truncationError   0.001;

and the time scheme is specified in the file system/fvSchemes

ddtSchemes
{
  default         backward;
}

parabolicVelocity

In many configurations of relatively slow flows, it is interesting to impose a parabolic velocity profile directly, rather than giving it the space it needs to establish itself. To do this, we use a dedicated library.

The option to impose a parabolic velocity profile is loaded within the file system/controlDict (when coupling biofilm growth with the fluid)

libs ("libParabolicVelocity.so");

and the properties are specified in the file 0/U

inlet
  {
    type            parabolicVelocity;
    maxValue        1.-3;
    n               (1 0 0);
    y               (0 1 0);
  }