The development of pre-set wavelength G$ddot{o}$rtler vortices are studied in the boundary-layer flows on concave surfaces of 1.0 and 2.0 m radius of curvature. The wavelengths of the vortices were pre-set by thin wires of 0.2 mm diameter placed 10 mm upstream and perpendicular to the concave surface leading edge. Velocity contours were obtained from velocity measurements using a single hot-wire anemometer probe. The most amplified or dominant wavelength is found to be 15 mm for free-stream velocity of 2.1 m/s and 3.0 m/s on the concave surface of R = 1 m and 2 m, respectively. The velocity contours in the cross-sectional planes at several streamwise locations show the growth and breakdown of the vortices. Three different regions can be identified based on the growth rate of the vortices. The occurrence of a secondary instability mode is also shown in the form of mushroom-like structures as a consequence of the non-linear growth of the G$ddot{o}$rtler vortices. By pre-setting the vortex wavelength to be much larger and much smaller than the most amplified one, the splitting and merging of G$ddot{o}$rtler vortices can be respectively observed.