GitHub

Taylor Green Vortex

TGx

It solves the Taylor Green Vortex case.

For the 2D case there is an analytical solution for velocity and pressure:

$u_x= U_{a}-V_{s} \cos \bigg (\frac{\pi}{D}(x-U_{a} t)\bigg ) \sin \bigg (\frac{\pi}{D}(y-V_{a} t)\bigg ) e^{-\frac{2 v \pi^{2}}{D^{2}} t}$

$u_y= V_{a}+V_{s} \sin \bigg (\frac{\pi}{D}(x-U_{a} t)\bigg ) \cos \bigg (\frac{\pi}{D}(y-V_{a} t)\bigg ) e^{-\frac{2 \nu \pi^{2}}{D^{2}} t}$

$p=-\frac{V_{s}^{2}}{4}\bigg ((\cos(2 \frac{\pi}{D}(x-U_{a} t) )+\cos (2 \frac{\pi}{D}(y-V_{a} t))\bigg ) e^{-\frac{4 v \pi^{2}}{D^{2}} t}$

$\omega=\frac{2 V_{s} \pi}{D} \cos \bigg (\frac{\pi}{D}(x-U_{a} t)\bigg ) \cos \bigg (\frac{\pi}{D}(y-V_{a} t)\bigg ) e^{-\frac{4 \nu \pi^{2}}{D^{2}} t}$

Because of an analytical solution it is used as a benchamark case for verifing mesh convergence, CFL stability and processor scalability. The domain is a squqare of 2Dx2D with periodic boundaries over the 4 sides. The initial solution is retrived from the analytical solution. The pressure is fixed in the centre of the domain equal to the analytical solution. The parameters set by the user are overwritten by the following standard values:

  • $Vs = 1\;m/s$ swirling velocity
  • $Ua = 0.2\;m/s$ translational velocity in $x$ direction
  • $Va = 0.3\;m/s$ translational velocity in $y$ direction
  • $D = 0.5\;m$ vortex dimensions
  • $\nu = 0.001\; m^2/s$
using PartitionedArrays
using SegregatedVMSSolver
using SegregatedVMSSolver.ParametersDef
using SegregatedVMSSolver.SolverOptions


t0 =0.0
dt = 0.1
tF = 0.5

Re = 1000
D = 2
rank_partition = (2,2)

backend = with_debug

sprob = StabilizedProblem(VMS(1))
timep = TimeParameters(t0=t0,dt=dt,tF=tF)

physicalp = PhysicalParameters(Re=Re)
solverp = SolverParameters()
exportp = ExportParameters(printinitial=false,printmodel=false)

We create a mesh of 32x32 elements

meshp= MeshParameters(rank_partition,D;N=32,L=0.5)
simparams = SimulationParameters(timep,physicalp,solverp,exportp)

bc_tgv = Periodic(meshp,physicalp ) 
mcase = TaylorGreen(bc_tgv, meshp,simparams,sprob)

SegregatedVMSSolver.solve(mcase,backend)

It is also possible to use natural boundary conditions:

bc_tgv = Natural(meshp,physicalp) 
mcase = TaylorGreen(bc_tgv, meshp,simparams,sprob)
SegregatedVMSSolver.solve(mcase,backend)