4.1 PASS:
Estimation of the numerical viscosity

Author
 Stéphane Popinet
 Command
 sh reynolds.sh reynolds.gfs 1
 Version
 0.6.4
 Required files
 reynolds.gfs (view) (download)
reynolds.sh div5.ref div6.ref div7.ref reynolds.ref
 Running time
 3 minutes 26 seconds
The velocity field is initialised with an exact stationary solution of
the Euler equations in a periodic 2D domain. An exact Euler solver
would not change this field, however any numerical solver will
introduce numerical dissipation which will slowly dissipate the
kinetic energy of the initial solution. By monitoring the evolution of
the kinetic energy, the dissipative properties of the numerical scheme
can be measured (see [37] for details).
Figures 45 and figure 46 illustrate the evolution
of the divergence of the velocity field with time. This is a check of
the stability of the approximate projection and should remain bounded.
Figures 47 and 48 illustrates the evolution of
the kinetic energy and the corresponding equivalent Reynolds number as
a function of resolution. The higher the Reynolds number, the less
dissipative the scheme.
Figure 45: Evolution of the maximum divergence. 
Figure 46: Evolution of the L2 norm of the divergence. 
Figure 47: Evolution of the kinetic energy. 
Figure 48: Equivalent Reynolds number as a function of resolution. 
Figure 49: Accuracy of the solution as a function of the level of refinement. 
4.1.1 PASS:
Estimation of the numerical viscosity with refined box

Author
 Stéphane Popinet
 Command
 sh ../reynolds.sh box.gfs 4
 Version
 0.6.4
 Required files
 box.gfs (view) (download)
../reynolds.sh div5.ref div6.ref div7.ref reynolds.ref
 Running time
 19 minutes 45 seconds
Same as the previous test but with a refined box in the middle and four
modes of the exact Euler solution.
Figure 50: Evolution of the maximum divergence. 
Figure 51: Evolution of the L2 norm of the divergence. 
Figure 52: Evolution of the kinetic energy. 
Figure 53: Equivalent Reynolds number as a function of resolution. 
Figure 54: Accuracy of the solution as a function of the level of refinement. 
4.1.2 PASS:
Numerical viscosity for vorticity/streamfunction formulation

Author
 Stéphane Popinet
 Command
 sh ../reynolds.sh stream.gfs 1
 Version
 110610
 Required files
 stream.gfs (view) (download)
div5.ref div6.ref div7.ref reynolds.ref
 Running time
 1 minutes 27 seconds
Same as the previous test but using a vorticity/streamfunction formulation.
Figure 55: Evolution of the maximum divergence. 
Figure 56: Evolution of the L2 norm of the divergence. 
Figure 57: Evolution of the kinetic energy. 
Figure 58: Equivalent Reynolds number as a function of resolution. 
Figure 59: Accuracy of the solution as a function of the level of refinement. 
4.1.3 PASS:
Numerical viscosity for the skewsymmetric scheme

Author
 Daniel Fuster
 Command
 sh skew.sh skew.gfs 1
 Version
 110723
 Required files
 skew.gfs (view) (download)
skew.sh div5.ref div6.ref div7.ref reynolds.ref
 Running time
 1 minutes 17 seconds
Same as the previous test but using the skewsymmetric module.
Figure 60: Evolution of the maximum divergence. 
Figure 61: Evolution of the L2 norm of the divergence. 
Figure 62: Evolution of the kinetic energy. 
Figure 63: Accuracy of the solution as a function of the level of refinement. 
4.1.4 PASS:
Numerical viscosity for the skewsymmetric scheme with refined box

Author
 Daniel Fuster
 Command
 sh skew.sh skewbox.gfs 1
 Version
 120229
 Required files
 skewbox.gfs (view) (download)
skew.sh error.ref
 Running time
 12 minutes 28 seconds
The skewsymmetric module with AMR.
Figure 64: Evolution of the kinetic energy. 
Figure 65: Accuracy of the solution as a function of the level of refinement. 