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Competition in aeronautical propulsion imposes new challenges in terms of maintenance costs and product availability. Within the HP turbine design process, aero-thermal engineers are given only a short time in which to propose adequate solutions for HPT blade cooling air supply systems. The industrialization of the SAVIRAT chain at SNECMA MOTEURS offers opportunities for applying Navier-Stokes calculations which are now compatible with the quality and time constraints of a project design phase. One clear way to improve the life time of HPT disks is to remove hooks between the forward sealing plate and the HPT disk itself. In addition to its main mechanical function, this device also optimizes the relative speed of the HPT blade cooling air before it enters the slot bottoms. This document details the contribution made by Navier- Stokes three-dimensional (3D) analyses during the HPT disk design phase, in the event of a simple contact, without hooks, between the forward flange and the disk rim. A fluid and solid wall 3D mesh represents the HPT cooling air supply system, from the forward sealing plate exit holes to the entry of the cooling air in the blade root. Calculations were carried out under adiabatic conditions. Models were created for several configurations. These led the authors to draw the following conclusions: where a 'no hook' device is used, analysis reveals a vortex phenomenon due to bailing; in the case of a conventional solution with hooks, the vortex phenomenon disappears and the pressure distribution in the slot bottoms is homogenous; and fitting blade roots with axial air deflectors restores air supply quality in slot bottoms. Some applications may have a significant impact on the optimization of components in terms of efficiency or maximizing life time. Secondary flows of turbine rotors are one of the most interesting fields of application. (1 table, 4 refs.).