5.1 PASS:
Potential flow around a sphere

Author
 Stéphane Popinet
 Command
 sh axi.sh axi.gfs
 Version
 1.3.0
 Required files
 axi.gfs (view) (download)
axi.sh error.ref order.ref isolines.gfv
 Running time
 48 seconds
The axisymmetric potential flow around a sphere is computed (Figure
70) and compared to the theoretical solution
[23]. A large domain is used together with variable spatial
resolution to minimise the influence of the finite domain size.
Figure 71 and 72 illustrate the convergence of the
solution for the horizontal component of velocity with increased
resolution.
Figure 70: Isolines of the velocity components (x in red, y in blue). 
Figure 71: Evolution of the error as a function of resolution. 
Figure 72: Corresponding convergence order. 
5.1.1 PASS:
Viscous flow past a sphere

Author
 Stéphane Popinet
 Command
 sh viscous.sh
 Version
 1.3.0
 Required files
 viscous.gfs (view) (download)
viscous.sh cp12200 fadlun fadluncp100 fadluncp200 Re12 zhang blanco1995 masliyah1970 isolines.gfv fornberg
 Running time
 29 minutes 10 seconds
When viscosity is added, a recirculation region develops behind the
sphere (Figure 73).
Figure 73: Viscous flow around a sphere at Reynolds
100. Isolines of the velocity components (x in red, y in
blue). The recirculation region is indicated by the green isoline
where the value of the horizontal velocity component vanishes. 
The length of the recirculation depends on the Reynolds
number. Figure 74 plots the results obtained with Gerris
as well as previously published results. Published results agree
with Gerris for Reynolds numbers smaller than 100. The mismatch for
results at Reynolds 200 can be attributed to the coarse mesh used to
resolve the wake in the studies of Fornberg [16] and
Fadlun et al [15].
Figure 74: Relative length of the recirculation region
as a function of the Reynolds number. The results of Gerris are
compared with the results of Masliyah & Epstein
[28], Fornberg [16], Blanco &
Magnaudet [8], Fadlun et al [15] and
Zhang & Zheng [48]. 
The pressure profiles are also in good agreement with those reported
by Fadlun et al (which also agree with those of Fornberg) (Figure
75).
Figure 75: Pressure coefficient over the sphere surface at
Reynolds numbers 100 and 200. 