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Taylor Green Vortex 2D

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 benchmark case for verifying mesh convergence, CFL stability and processor scalability. The domain is a square of 2Dx2D with periodic boundaries over the 4 sides. The initial solution is obtained 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=true, printmodel=true, 
vtu_export = ["uh","ph","uh_analytic", "ph_analytic"], extra_export=["VelocityError","PressureError"])

at each time-step, a new line on a .csv file will be written with: timestep-VelocityError-PressureError in L2 norm. You can use this file to analyze the error decreasing with mesh/order - refinement.

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)