Revision as of 17:17, 13 August 2020

Step 2 Description

The next step in the benchmark is to simulate the Taylor-Green Vortex configuration in 3-D, still for a single-component incompressible flow. For this configuration, a reference solution obtained with a pseudo-spectral solver is available in the scientific literature^[1] and can be used for validation by a direct comparison.

The domain size has been set to ${\displaystyle L=2\pi L_{0}}$ in each direction, where ${\displaystyle L_{0}=1\;\mathrm {m} }$ is an arbitrarily-chosen reference length scale for dimensional codes. As in the previous section, periodic boundary conditions are used in all three spatial directions.

The initial conditions for 3-D TGV are given by the following set of equations:

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

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

${\displaystyle w(x,y,z,0)=0}$

The reference velocity magnitude is set again to ${\displaystyle u_{0}=1\;\mathrm {m/s} }$. The desired Reynolds number is again ${\displaystyle Re={u_{0}L_{0}}/{\nu }=1,600}$, as in the previous section, since this value has been used as well in the reference study employed for comparison^[1]. To obtain the proper value of Re, the kinematic viscosity is set once again to ${\displaystyle \nu =6.25\times 10^{-4}\;\mathrm {m^{2}/s} }$. With this set of parameters, the reference time scale is ${\displaystyle \tau _{\mathrm {ref} }=L_{0}/u_{0}=1\;\mathrm {s} }$. The flow simulation must be carried out for a physical time of ${\displaystyle t=20\;\tau _{\mathrm {ref} }}$.

Regarding spatial discretization, a resolution of at least 512 nodes (or cells) in each direction is recommended. This value leads to an over-resolved simulation during the first few ${\displaystyle \tau _{\mathrm {ref} }}$ but is necessary to capture properly the transition to turbulence that should occur between 10 and 15 ${\displaystyle \tau _{\mathrm {ref} }}$. As a point of comparison, the reference solution from^[1] has been obtained with a spectral code on a ${\displaystyle 512^{3}}$ grid. The timestep should be chosen to ensure that the maximum values of the Courant-Friedrichs-Lewy (CFL) and Fourier (Fo) numbers remain below ${\displaystyle 0.3}$ and ${\displaystyle 0.2}$, respectively. This values are actually quite conservative since the focus of this study is set on accuracy comparisons and not on pure computational performance.

2D Large-Eddy Simulation, injection of a premixed kerosene/air mixture on the left with a high level of turbulence. Some kerosene droplets are added to this premixing.

Aside Suggestion

Check mesh convergence and time integration independence.

Check that ${\displaystyle dKE/dt=\int Sij:Sij=\int Enstrophy}$

References

• van Rees WM, Leonard A, Pullin D, Koumoutsakos P. A comparison of vortex and pseudo-spectral methods for the simulation of periodic vortical flows at high reynolds numbers. J Comput Phys. 2011;230(8):2794-2805.
• Carton de Wiart C, Hillewaert K, Duponcheel M, Winckelmans G. Assessment of a discontinuous galerkin method for the simulation of vortical flows at high reynolds number. Int J Numer Meth Fluids. 2014;74(7):469-493.
• Duponcheel M, Orlandi P, Winckelmans G. Time-reversibility of the euler equations as a benchmark for energy conserving schemes. J Comput Phys. 2008;227(19):8736-8752.
• Bricteux, L, Zeoli, S, Bourgeois, N. Validation and scalability of an open source parallel flow solver. Concurrency Computat: Pract Exper. 2017; 29:e4330. https://doi.org/10.1002/cpe.4330