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Accurate modeling of boundary-layer transition is an important aspect of developing greener air transport technologies. In that regard, transition models based on auxiliary transport equations offer a robust approach that is easily integrated into the Reynolds-averaged Navier-Stokes (RANS) solvers. Recent workshops under NATO and AIAA have identified the verification of transport-equations-based transition modeling as a critical aspect of reducing the scatter between the predictions of different CFD codes. Follow-on work has highlighted the need for highly dense grids to achieve an asymptotic convergence of transition related flow metrics. The present work examines the role of automatic near-body mesh adaptation capability in the NASA OVERFLOW CFD solver to enable verification studies in an efficient manner, and for establishing best practices for designing grids for the RANS-based transition models. A sensor function relevant to the Langtry-Menter \gamma-Re_{\theta t}\ transition model has been identified and used for error-based mesh adaptation for canonical configurations comprising a flat plate, and the S809 and NLR-7301 airfoils. The efficacy of the mesh adaptation approach is assessed for flow conditions involving multiple transition scenarios such as natural transition, separation-induced transition, and shock-induced transition. The results from this exploratory study indicate that the meshes adapted using the proposed sensor provide solutions that approach the references solutions obtained with uniformly refined hand-crafted meshes, in terms of the chosen metrics, and yield modest yet significant savings in grid count. We also highlight areas for improvement in the grid adaptation methodology within OVERFLOW.