Revision as of 07:05, 27 July 2020

Step 1 Description

The 2-D TGV test case has been already well documented in the scientific literature, e.g. ^[1][2]. Therefore, only a brief description is given here. This configuration is used as verification step, since an analytic solution exists^[3].

An incompressible flow with constant density and viscosity is simulated.

The fluid is contained in a square box of dimension ${\displaystyle [0;L]^{2}}$ with periodic boundary conditions in both directions. The value ${\displaystyle L=2\pi L_{0}}$ with ${\displaystyle L_{0}=1\;\mathrm {m} }$ has been arbitrarily chosen in the present benchmark for dimensional codes. The numerical value for the kinematic viscosity is chosen as ${\displaystyle \nu =6.25\times 10^{-4}\;\mathrm {m^{2}/s} }$ to get suitable conditions, as discussed below. The initial velocity components are prescribed as follows:

${\displaystyle u(x,y,0)=+u_{0}\times \sin(x/L_{0})\times \cos(y/L_{0})\,,}$

${\displaystyle v(x,y,0)=-u_{0}\times \cos(x/L_{0})\times \sin(y/L_{0})\,.}$

with ${\displaystyle u_{0}=1\;\mathrm {m/s} }$. The flow is thus initially composed of four vortices, one in each quarter of the box. The vortex turnover time is defined as ${\displaystyle \tau _{\mathrm {ref} }=L_{0}/u_{0}=1\;\mathrm {s} }$. For non-dimensional codes, the relevant parameter for this simulation is the vortex Reynolds number, ${\displaystyle Re=u_{0}L_{0}/\nu =1,600}$ in the present setup. It should be noted that the theoretical solution is only stable for sufficiently small Reynolds numbers, explaining why the particular value ${\displaystyle Re=1,600}$ has been chosen here, as done in many previous studies using the TGV configuration.

A major interest of this test-case is that an analytical solution has been derived for such conditions^[3]:

${\displaystyle u(x,y,t)=u(x,y,0)\times \mathrm {e} ^{-2\nu t/L_{0}^{2}}=u(x,y,0)\times \mathrm {e} ^{-2\,(t/\tau _{\mathrm {ref} })/\mathrm {Re} }\,,}$

${\displaystyle v(x,y,t)=v(x,y,0)\times \mathrm {e} ^{-2\nu t/L_{0}^{2}}=v(x,y,0)\times \mathrm {e} ^{-2\,(t/\tau _{\mathrm {ref} })/\mathrm {Re} }\,.}$

All physical conditions are now completely defined and only numerical parameters remain to be set. Based on previously published studies, the recommended parameters for time and space discretization are ${\displaystyle \Delta t=\tau _{\mathrm {ref} }/2000=5\times 10^{-4}\;\mathrm {s} }$ and ${\displaystyle N=64}$ grid points in each direction. The flow must be simulated for a physical time of ${\displaystyle t=10\;\tau _{\mathrm {ref} }=10\;\mathrm {s} }$.

Aside suggestions

This step should clearly not be a big issue for any code. However, it is a good way to verify the orders of convergence (both in space and time), to check any stability issues, to measure the influence of the convergence criteria of iterative solvers (if any), etc...

Notes

1. ↑ A. Abdelsamie, G. Fru, F. Dietzsch, G. Janiga, D. Thévenin, Towards direct numerical simulations of low-Mach number turbulent reacting and two-phase flows using immersed boundaries, Comput. Fluids 131(5):123--141, 2016, Bibtex
   Author : A. Abdelsamie, G. Fru, F. Dietzsch, G. Janiga, D. Thévenin
   Title : Towards direct numerical simulations of low-Mach number turbulent reacting and two-phase flows using immersed boundaries
   In : Comput. Fluids -
   Address :
   Date : 2016
   
2. ↑ S. Laizet, E. Lamballais, High-order compact schemes for incompressible flows: A simple and efficient method with quasi-spectral accuracy, J. Comput. Phys. 228:5989--6015, 2009, Bibtex
   Author : S. Laizet, E. Lamballais
   Title : High-order compact schemes for incompressible flows: A simple and efficient method with quasi-spectral accuracy
   In : J. Comput. Phys. -
   Address :
   Date : 2009
   
3. ↑ ^{3.0 3.1} G.I. Taylor, A.E. Green, Mechanism of the production of small eddies from large ones, Proc. Royal Soc. A 158:499--521, 1937, Bibtex
   Author : G.I. Taylor, A.E. Green
   Title : Mechanism of the production of small eddies from large ones
   In : Proc. Royal Soc. A -
   Address :
   Date : 1937