9.1 PASS:
Circular droplet in equilibrium

Author
 Stéphane Popinet
 Command
 sh spurious.sh spurious.gfs 4e10
 Version
 1.1.2
 Required files
 spurious.gfs (view) (download)
spurious.sh convergence.ref kconvergence.ref
 Running time
 6 minutes 46 seconds
A circular droplet of diameter D=0.8 is initialised centered on
the topleft corner of the unit box. Surface tension is imposed on
the interface. The exact solution is given by Laplace’s law: uniform
zero velocity and a pressure jump accross the interface exactly
balancing the surface tension force.
The initial condition – while close to the exact solution – does
not guarantee the exact balance between the numerical
discretisations of surface tension and pressure gradient. However,
these small initial perturbations generate small capillary waves
which are progressively (on a timescale of order D^{2}/µ) damped
by viscosity so that the exact (to roundoff error) balance is
eventually obtained.
The convergence is obtained for a wide range of Laplace numbers
La=σρ D/µ^{2}, as illustrated on Figure 106.
Correspondingly, convergence of the curvature to a constant value is
also obtained at all Laplace numbers as illustrated on Figure
107.
Figure 108 illustrates the convergence of the error on
the droplet shape as a function of resolution for a Laplace number
of 12000. Both the shape error and the relative error on the
equilibrium curvature value illustrated on Figure
109 show close to secondorder convergence.
Figure 106: Evolution of the amplitude of the capillary currents
max(u)(D/σ)^{1/2} as a function of
nondimensional time τ=tµ/D^{2} for the range of Laplace
numbers indicated in the legend. 
Figure 107: Evolution of the standard deviation of the
value of the curvature along the interface as a function of
nondimensional time τ=tµ/D^{2} for the range of Laplace
numbers indicated in the legend. 
Figure 108: Convergence of the error on the equilibrium shape of the
droplet with resolution. The diameter is given in number of grid
points. 
Figure 109: Convergence of the relative error on the
equilibrium curvature value with resolution. The diameter is given
in number of grid points. 
9.1.1 PASS:
Axisymmetric spherical droplet in equilibrium

Author
 Stéphane Popinet
 Command
 sh ../spurious.sh axi.gfs 5e8
 Version
 1.3.1
 Required files
 axi.gfs (view) (download)
convergence.ref kconvergence.ref
 Running time
 17 minutes 22 seconds
The same test case but using the axisymmetric solver. The results
are comparable.
Figure 110: Evolution of the amplitude of the capillary currents
max(u)(D/σ)^{1/2} as a function of
nondimensional time τ=tµ/D^{2} for the range of Laplace
numbers indicated in the legend. 
Figure 111: Evolution of the standard deviation of the
value of the curvature along the interface as a function of
nondimensional time τ=tµ/D^{2} for the range of Laplace
numbers indicated in the legend. 
Figure 112: Convergence of the error on the equilibrium shape of the
droplet with resolution. The diameter is given in number of grid
points. 
Figure 113: Convergence of the relative error on the
equilibrium curvature value with resolution. The diameter is given
in number of grid points. 