Investigation of a Smooth Local Correlation-based Transition Model in a Discrete-Adjoint Aerodynamic Shape Optimization Algorithm

Compressibility corrections to extend a smooth local correlation-based transition model to transonic flows

“…This formulation is equivalent to that presented by Piotrowski and Zingg [36] which was non-dimensionalised following the procedure for the Navier-Stokes equations (41), where velocity and distance are normalised by the sound speed and reference length, l, respectively. The new crossflow source term formulation was validated using the TU Braunschweig Sickle Wing transition test case [50], with the results presented by Piotrowski [40]. Simulations of transonic swept wings revealed that both crossflow source-term approaches, D scf and F onset,scf , produce a transition front that is dependent on the initialisation of the flow field.…”

“…This section presents the compressibility correction developed for the Tollmien-Schlichting instability empirical correlation, ψ, and the framework created for applying the crossflow compressibility correction developed by Malik et al [32], ψ scf . The SA-sLM2015 transition model equations are presented in Appendix A, while a full description of the model, including validation with two-and three-dimensional subsonic transition test cases, can be found in Piotrowski and Zingg [36] and Piotrowski [40]. All equations are presented in non-dimensional form to be consistent with previous work [36].…”

“…Machine-zero residual convergence for the total residual is achieved for the cases presented unless stated otherwise. More details on the solution strategy and the iterative and grid convergence of the smooth transition model with the compressibility corrections can be found in previous work [40,52].…”

“…The SA-sLM2015 smooth transition model [36] is a local two-equation model which couples the LM2009 and LM2015 empirical correlations with the Spalart-Allmaras (SA) one-equation turbulence model [37], resulting in a three-equation coupled system. Smooth approximations were introduced to replace discontinuous and stiff functions in the transition model source terms, which improves the iterative convergence of the model and provides continuous gradients that facilitate integration in gradient-based optimisation algorithms [36,38]. The model provides a less expensive and simpler alternative to non-local and higher-fidelity approaches described above.…”

Airfoil Optimization using Design-by-Morphing

“…For the optimization of airfoil shapes using CFD-based solvers, the evaluation of the objective functions (aerodynamic properties) typically falls into two categories: the full Reynolds-averaged Navier-Stokes (RANS) based approach and the interacted viscous/inviscid zonal approach. The RANS-based approach is computationally expensive and demands the optimizer to be highly efficient, and, to accommodate the large number of design variables as often seen in the aerodynamic designs, gradient-based optimizers coupled with adjoint methods for computing the derivatives are deemed the most feasible [41][42][43]. On the other hand, the viscous/inviscid zonal approach, which combines separate solutions for the inviscid external flow and the viscous shear layer flow iteratively to form a continuous profile, is faster and less expensive.…”