A steady and unsteady numerical research is carried out to explore some effects of a specific non-axisymmetric tip clearance layout on the overall performance and stability of an axial compressor stage. For a 4-stage low-speed research compressor (LSRC) in Shanghai Jiao Tong University (SJTU), one-eighth annulus of the inlet guide vane and the first stage rotor was modeled for this study. After the validation for the uniform tip clearance case, a specific non-axisymmetric tip clearance layout is chosen from several random cases generated by the Gaussian Probabilistic Density Function method. Unsteady time-averaged results at the near stall condition show that the chosen non-axisymmetric layout can improve the isentropic efficiency by 1.3% and extend the stall margin by 4%. Detailed analyses on flow fields are carried out to interpret the performance improvement. Due to the circumferential layout of clearance sizes, the inlet mass flow and incidence are redistributed in both the radial and circumferential directions. It leads to blade loading and tip leakage flow varying with the tip clearance size. The quantification of blockage manifests that the blockage arising from the tip leakage flow is significantly alleviated in the non-axisymmetric layout, which leads to improvements in overall performance and stall margin. Transient flow fields at the rotor tip are also analyzed at the near stall condition. For the non-axisymmetric layout, low-momentum regions originating from larger clearance sizes oscillate and develop downstream in one blade passage period.