The current research aims to study how changing the endwall shape of the inner cylinder affects the appearance of Ekman cells and the onset of Taylor vortices. A two-dimensional numerical method simulates the flow between two concentric cylinders, with the inner cylinder rotating while the outer cylinder and the upper and lower endwalls remain at rest. The method of solving the Navier–Stokes equations for incompressible viscous flows is provided by the Ansys Fluent software, which is based on the finite volumes method. The geometric parameters, such as the radius ratio (η) and the height ratio (Г), are fixed at η = 0.9 and Г = 9.85, respectively. The working fluid is an aqueous solution (μ = 48 mPa.s, ρ = 1050 kg/m³). The study focuses on modeling four configurations (A, B, C, and D): configuration (A) includes two flat plate endwall, configuration (B) features a hemispherical upper endwall and a flat lower plate endwall, configuration (C) consists of a flat upper endwall and a hemispherical lower endwall, and configuration (D) comprises two hemispherical endwalls. The obtained results indicate that the appearance of Ekman cells and the transition to Taylor vortex flow are specifically delayed for configuration (D). Additionally, the friction coefficient is affected by the modification of the endwalls. Indeed, configuration (D) contributes to a reduction of the skin friction coefficient by approximately 43% at the appearance of Ekman cells, and by 20% near the appearance of Taylor vortices compared to configuration (A).