Comparison between temporal and spatial direct numerical simulations for bypass transition flows

Temporally developing direct numerical simulations (T-DNS) are performed for free-stream turbulence induced bypass transition flow over a flat-plate under zero-pressure gradient, and results are validated using experimental data and spatially developing DNS (S-DNS) results. The temporal simulations predict the growth of near-wall Klebanoff modes in the pre-transition regime and their subsequent breakdown due to the sinuous-like instability, where the increase in free-stream turbulence intensity and length scale enhanced the instability. A formulation for the domain translation velocity is developed using the momentum integral boundary layer equation, to estimate the translation of the domain along the plate. The formulation was found to be robust for a range of free-stream turbulence intensities, and both the boundary layer growth and free-stream decay compared well with spatial simulations, confirming that the same translation velocity can be used for the entire domain. The optimal domain size for a temporal simulation is dictated by the length scale of the turbulent spots and errors due to streamwise periodicity, and is estimated to be $\sim 8$ to $10{\delta }_{\text{onset}}$, where ${\delta }_{\text{onset}}$ is the boundary layer thickness at the transition onset. The T-DNS prediction of the integral boundary layer parameters, mean flow, second and higher-order statistics, stress budgets, and free-stream decay were comparable to those of S-DNS, even though they were obtained on ten times smaller streamwise domain and numerical grid. However, T-DNS are limited for bypass transtion flows for which leading-edge effects are unimportant, and may have limitations for the accurate predictions of the near-wall Klebanoff modes that extend up to breakdown. Nonetheless, temporal approach is identified to be a viable inexpensive alternative to the spatial approach for canonical test cases for transition flow physics analysis.