Modelling Bypass Transition with Low-Reynolds-Number Nonlinear Eddy-Viscosity Closure

Abstract

The ability of two nonlinear, low-Reynolds-number eddy-viscosity models, one cubic and the other quadratic, to predict transitional boundary layers is investigated. The latter model distinguishes itself by its ability to return the correct wall-asymptotic variation of all the Reynolds-stress components. The choice of low-Reynolds-number models is motivated by the fact that transitional flows in turbomachinery can feature both transition and relaminarisation in different parts of the same flow, the latter demanding the inclusion of closure terms that represent the effects of viscosity on turbulence. Four flat-plate boundary layers are considered, each subjected to different combinations of free-stream turbulence and pressure gradient. A fifth flow is that around a VKI turbine blade. The models are first applied on their own. Whilst returning qualitative features of the transition process, the models do not provide an adequate quantitative description. In particular, neither model is able to predict the distinctive rise in turbulence intensity in the boundary layer well upstream of the location at which the skin friction and shear stress rise, taken to signify the onset of transition. It is then shown that the combination of the models with conventional intermittency-factor-based formulations is ineffective. This finally leads to a proposal which involves the introduction of modifications that combine a separate transport equation for the pre-transitional laminar fluctuation energy with an intermittency-type factor, governed by a correlation function. This factor is used to ‘blend’ pre-transitional fluctuation energy and turbulence-energy components, and it also features in the eddy-viscosity relation. The resulting modified model is shown to procure a substantial improvement in the representation of the transition process, including the pre-transitional rise in turbulence intensity and shear stress. To assist the assessment of the physical realism of the pre-transitional model, a large eddy simulation of a transitional flat-plate boundary layer was undertaken, and turbulence/fluctuation-energy budgets derived from the simulation are included.