Turbulence Decay Corrections for Transitional Flow Calculations

Transitional flow calculations are becoming increasingly common due to the emergence of applications operating at low Reynolds numbers and the appearance of mathematical models focused on modeling transition. From a physical perspective, transition is a complex phenomenon, non-linear and unsteady, in which flow disturbances from the freestream cause the laminar flow in the boundary layer to become unstable and transition to turbulent flow. As a result, the exact location for the transition region is dependent on the characteristics of the freestream flow and its disturbances. From the numerical standpoint, this sensitivity to the freestream conditions is obtained from inlet boundary conditions for turbulence. In calculations using the Reynolds-Averaged Navier-Stokes (RANS) equations, when transition models are not employed, transition is handled by the underlying turbulence model. This leads to transition occurring at too low Reynolds numbers, which originates turbulent flow close to the leading edge of a body, regardless of the specified turbulence quantities at the inlet. However, when transition modelling is desired, the inlet turbulence quantities have a strong influence on the transition location as shown by Eça et al. (2016). Despite being physically expected, this influence causes difficulties, since the specification of these values becomes a challenge as little information about turbulence is known in order to determine both variables (in the case of two-equation models). Additionally, common two-equation eddy-viscosity models such as the k - E and k - w turbulence models predict a very strong decay of the turbulence variables in the freestream, which is related to the value of the eddy-viscosity at the inlet according to Spalart and Rumsey (2007). Thus it is common to observe that calculations with transition modelling are accompanied by very high values of v1/v in order to maintain a ’reasonable’ decay of the turbulence intensity. This means that for practical applications, one must not only know the correct value for the turbulence intensity, but also has to estimate the eddy-viscosity value that will result in the correct evolution of the turbulence intensity along the flow. This situation severely hinders the predictive capability of transition models (Li et al. (2019)), and often results in awkward values for the eddy-viscosity at the inlet, which may become physically questionable. In this paper, we explore an alternative technique to control the decay of turbulence kinetic energy in the freestream that modifies dissipation in the k and ! transport equations. This modification is calibrated for the flow around a flat-plate and then subsequently tested on the flow around the NACA 0012 airfoil. The mathematical formulation is described in section 2. The test cases and numerical settings are described in sections 3 and 4 while the results are presented and discussed in section 5. Section 6summarizes the conclusions of this work.