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This paper present a simulation strategy for DNS of turbulent co-current Taylor bubble flow. This is a continuation of the work presented in [2] in which Large Eddy Simulation (LES) of co-current turbulent Taylor bubble flow was presented. It was observed that one of the main challenges is the physically accurate prediction of the behavior of the Taylor bubble skirt, and the related bubble shedding. An underestimation of the turbulent fluctuations in the wake of the Taylor bubble was observed in the LES results. It was suggested that this is related to over-prediction of the loss of void of the Taylor bubble due to bubble shedding induced by an LES mesh resolution which is not sufficient to capture the break-up and bubble formation accurately. To counter this, in the current work we present a DNS approach of co-current turbulent Taylor bubble flow called RK-Basilisk, based on the Basilisk code with local adaptive grid refinement. This strategy allows for very high mesh resolution near the bubble's interface while elsewhere the grid is allowed to be coarser. Each time step, the mesh is adapted based on a void fraction criterion. Basilisk's underlying data structure which is based on an 'octree' allows for much faster solution procedures and, therefore, a much greater number of grid points as compared to the LES simulations which were performed using the more general OpenFOAM code. We compare the results against experimental data of the same setting, as well as the co-current LES OpenFOAM results. The setting of turbulent Taylor bubble flow in co-current conditions allows for both lower order turbulence model development and the validation of more general two-phase modeling strategies. Taylor bubble flow in itself also bears relevance to specific two-phase flow situations. The current work contributes to an advancement in simulation capability for such situations.