Self-similar solutions of the bed deformation

In this paper, a conclusion about the self-similar behavior of the bed surface evolution is made. It is based on the analysis of experimental and numerical studies of the bed surface evolution under the mechanical impact of liquid flow. The bottom wave has a form close to one period of a sinusoidal function with a time-varying wavelength and constant steepness. A method of constructing the automodel dependence of the bed surface on time and spatial coordinate in analytical form is proposed. It was shown that it is enough to select five bottom surfaces with given wavelengths from a series of shapes. Next, the mean values of shear stresses are calculated for them, and the rates of change of wavelengths are found. Then a degree of approximation of the wavelength dependence of its rate of change is determined, and, finally, the exact solution of the corresponding differential equation is obtained. Comparison with experimental data and numerical solutions shows that the solution error does not exceed a few percent and that computational time is reduced by 25–30 times.