On the stability and diffusive characteristics of Roe-MUSCL and Runge–Kutta schemes for inviscid Taylor–Green vortex

In this study, the stability and diffusive characteristics of second-order Roe-MUSCL in conjunction with three Runge–Kutta-based temporal schemes are investigated for implicit large-eddy simulation (ILES) of the Taylor–Green vortex transitional flow. The excessive dissipation inherent to Roe-MUSCL is mitigated using the Thornber et al. [12] and Bidadi–Rani [8] approaches, referred to as the low-Mach-number and current modifications LM [12] and CM, respectively. The temporal schemes investigated are those of Jameson et al. [3], Shu and Osher [15], and Spiteri and Ruuth [16]. The time evolution of statistics such as turbulent kinetic energy (TKE) and enstrophy are shown for various combinations of modified Roe-MUSCL and temporal schemes at 643 and 1283 grid resolutions. Also, the spectra of TKE, and of the numerical viscosity and dissipation-rate inherent to the modified schemes are presented. In general, it is observed that Shu–Osher method is the most stable among the three time schemes. For the 643 grid case, LM [12] resulted in a stable solution when combined with all three time schemes, whereas CM maintained stability only with Shu–Osher. The dissipation rate spectra reveal that the instability arises due to the insufficient rate of energy transfer from the energy-containing to the inertial-range scales. The instability of Jameson et al. and Spiteri–Ruuth schemes (with CM) is manifested as the larger magnitude of the slope of dissipation rate spectrum at low wavenumbers. For the 1283 grid simulations, temporal evolution of TKE showed that only Shu–Osher with LM remained stable. For the cases where both LM and CM are stable, the former exhibited better accuracy in capturing transition and the subsequent decaying turbulent flow. Estimates of effective Reynolds number from current ILES simulations show good agreement with actual Reynolds numbers in prior direct numerical simulations.