Revision as of 05:42, 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~\cite{Taylor1937}.

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~\cite{Taylor1937}:

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

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

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

Notes

1. ↑ A. Abdelsamie and G. Fru and F. Dietzsch and G. Janiga and D. Thévenin , Towards direct numerical simulations of low-Mach number turbulent reacting and two-phase flows using immersed boundaries, Comput. Fluids, 2016 (131-5, pp123--141)
2. ↑ S. Laizet and E. Lamballais , High-order compact schemes for incompressible flows: A simple and efficient method with quasi-spectral accuracy, J. Comput. Phys., 2009 (228, pp5989--6015)