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In this work, the laminar-to-turbulent transition phenomenon around the two- and three-dimensional ellipsoid at different Reynolds numbers is numerically investigated. In the present paper, Reynolds Averaged Navier Stokes (RANS) equations with the Spalart-Allmaras, SST k-ω, and SST-Trans models are used for numerical simulations. The possibility of laminar-to-turbulent boundary layer transition is summarized in phase diagrams in terms of skin friction coefficient and Reynolds number. The numerical results show that SST-Trans method can detect different aspects of flow such as adverse pressure gradient and laminar-to-turbulent transition onset. Our numerical results indicate that the laminar-to-turbulent transition location on the 6:1 prolate spheroid is in a good agreement with the experimental data at high Reynolds numbers.