# The Taylor-Green Vortex

To assess the attributes of the discontinuous Galerkin spectral element method (DGSEM) for both direct numerical (DNS) and large eddy simulations (LES), the prominent Taylor-Green Vortex has been thoroughly investigated for up to high polynomial degrees N.

Key Facts:

• Re=5000
• up to 125,000 compute cores @ HLRS Nehalem and Jugene (Juelich)
• Polynomial degree N up to 20
• 216 million degrees of freedom per variable

The classical Taylor-Green Vortex problem constitutes the simplest flow for which a turbulent energy cascade can be observed numerically. Starting from an initial analytical solution containing only a single length scale, the flow field undergoes a rapid build-up of a fully turbulent dissipative spectrum because of non-linear interactions of the developing eddies. Due to its simplicity in both initial and boundary conditions, the Taylor-Green Vortex has been studied extensively in literature and serves as a well-established reference and benchmark test problem for direct numerical simulation (DNS) solvers and large eddy simulation (LES) subgrid scale models.

Our rationale for choosing this flow as an initial test case for the DGSEM solver FLEXI was two-fold: Firstly, the Taylor-Green Vortex allows for an easy validation of the code due to the readily available reference data while at the same time being accessible on a simple structured grid. Secondly, the physics of the flow field and the absence of wall boundaries constitute an excellent testbed for first LES implementations.

Our simulations were run at a relatively high Reynolds number of 5000, which causes a broad spectrum of turbulent scales and an associated well-developed energy spectrum with typical turbulent features. Due to the periodic, isotropic nature of the flow, periodic boundary conditions on all sides of the hexahedral domain $(2\pi \times 2\pi \times 2\pi)$ were selected. Since the flow field is essentially incompressible, the flow Mach number was set to 0.1.

Our simulations of the Taylor-Green Vortex were run on the Nehalem (HLRS) and Jugene (Juelich) supercomputers with up to 600³ DOF on 125k processors. The spectra of the kinetic energy distribution were by postprocessing the DGSEM solution by means of parallel Fast Fourier Transform (FFT). Since this operation was very memory consuming, it was also run in parallel on up to 64 processors, requiring only about 1 minute of CPU time. As shown in Figure 1, the resulting data shows a fast convergence of the spectra up to the smallest scales for the high resolution test cases, proving the DNS character of our simulation. In addition, the spectra and their features agree very well with the reference data published e.g. by Brachet, thereby validating our code and its postprocessing toolchain for this type of simulation. Figure 2 depicts the temporal development of the vorticity contours within the computational domain.

Figure 2: Temporal development of the vorticity contours of the Taylor-Green Vortex problem.

Contact:

Andrea Beck, beck@iag.uni-stuttgart.de
Claus-Dieter Munz, munz@iag.uni-stuttgart.de