The study of boundary layer transition plays a fundamental role in the field of turbomachinery owing to its strong influence on skin friction and heat transfer. The understanding of the laminar to turbulent transition can help designers to improve the aerodynamic and thermodynamic performances both of the components and of the whole machine. Turbulent transition models are nowadays commonly used tools in both research and design practice. In the context of high-pressure turbines design, it is then of particular interest to understand if such models are able to predict the effect of temperature on bypass transition and, in case of positive answer, the reasons of their behaviour. This becomes even more interesting as the effect of the flow aero-thermal coupling becomes prominent in the analysis of such phenomena, as this effect is typically not accountedfor in the validation of turbulence models. Two state-of-the-art transition models are examined in the present contribution: the γ–Reθ model developed by Langtry and Menter [1] and the k–kl–ω model by Walters and Cokljat [2]. The two models have been chosen also as they use two radically different approaches to describe the transition process: an empirical, correlation-based one for the former model opposed to a phenomenological, based on local transport, for the latter. To isolate the effects of the temperature ratio on the transition, the simulations have been performed keeping the same values of Reynolds and Mach numbers and changing the value of the wall to free stream Temperature Ratio (TR). The results of the two transition models have been compared between them as well as with experimental results obtained as part of a parallel effort. The results show that both models are sensitive to TR and can have qualitative agreement with the observations from experimental data. Most importantly the present results show how a transition modelling based on local transport, rather than empirical correlations should be favoured.
A methodology to generate sparse Galerkin models of chaotic/unsteady fluid flows containing a minimal number of active triadic interactions is proposed. The key idea is to find an appropriate set of basis functions for the projection representing elementary flow structures that interact minimally one with the other, resulting in a triadic interaction coefficient tensor with sparse structure. Interpretable and computationally efficient Galerkin models are obtained, since a reduced number of triadic interactions are computed to evaluate the right-hand side of the model. To find the basis functions, a subspace rotation technique is used, whereby a set of proper orthogonal decomposition (POD) modes is rotated into a POD subspace of larger dimension using coordinates associated with low-energy dissipative scales to alter energy paths and the structure of the triadic interaction coefficient tensor. This rotation is obtained as the solution of a non-convex optimisation problem that maximises the energy captured by the new basis, promotes sparsity and ensures long-term temporal stability of the sparse Galerkin system. We demonstrate the approach on two-dimensional lid-driven cavity flow at $Re = 2 \times 10^4$ where the motion is chaotic and energy interactions are scattered in modal space. We show that the procedure generates Galerkin models with a reduced set of active triadic interactions, distributed in modal space according to established knowledge of scale interactions in two-dimensional flows. This property, however, is observed only if long-term temporal stability is included explicitly in the formulation, indicating that a dynamical constraint is necessary to obtain a physically consistent sparsification.
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