Axisymmetric turbulent boundary layers that develop around streamwise oriented long cylinder-like objects can be found in many applications, such as towed array sonars or marine seismic streamers. In many of these applications, turbulent fluctuations within the boundary layer flow can have a negative impact compared with laminar flow conditions. The aim of the present work is to design a surface modification that influences the turbulent boundary layer around a cylinder in axial flow in order to reduce turbulent fluctuations. To design the surface we consider recent findings regarding the turbulence damping effects of groove-like surface structures and combine these insights with the effect of convex transverse curvature on turbulence. We use large-eddy simulations to investigate the flow around a cylinder of modified design and around a reference circular cylinder. Both flows have a radius-based Reynolds number of Re_a≈1.23⋅10⁴. The modified design leads to a 20 % decrease in the average wall shear stress and results in local reductions in the turbulent intensities, Reynolds stress, the temporal velocity spectrum, and the turbulent dissipation rate. The analysis within the anisotropy-invariant space reveals a tendency towards flow relaminarization. However, the new design has no effect on turbulent pressure fluctuations. We provide suggestions on how to further improve the surface design to achieve even greater flow stabilization.