Abstract: In order to analyse the instability of turbulent flow from the viewpoint of fractal dynamics, an arbitrary curve with discrete disturbance wave is chosen as the element of fractal structures of turbulence. Meanwhile, a quantized scale and coordinate method is adopted to handle the strongly nonlinear phenomena of flows despite their complex structures and large amplitudes of disturbance. Two of the main difficulties for the complexity of turbulence,i.e. nonlinearity and discreteness, have been approximately solved by the above-mentioned means. The bifurcated solution leading to transition and turbulence is meaningful, which embraces all the effects of nonlinearity and discreteness, including shearing, stretching, rotation, dispersion, pressure-strain interaction and memory. Therefore, the result should deepen our comprehension of turbulence.