Self-similar solutions of the bed deformation

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Abstract

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.

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About the authors

А. G. Petrov

Ishlinsky Institute for Problems in Mechanics of the Russian Academy of Sciences

Author for correspondence.
Email: petrovipmech@gmail.com
Russian Federation, Moscow

I. I. Potapov

Computing Center of the FEB RAS

Email: petrovipmech@gmail.com

Хабаровский федеральный исследовательский центр

Russian Federation, Khabarovsk

A. S. Epikhin

Ivannikov Institute for System Programming of the Russian Academy of Sciences

Email: petrovipmech@gmail.com
Russian Federation, Moscow

References

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Supplementary files

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2. Fig. 1. Evolution of the bottom surface: a – under a pipe on the river bottom [11], b – under the influence of a flat bottom jet according to data from [6], 1 – sinusoidal approximation.

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3. Fig. 2. Graphs of functions Ф(ξ) (a) and Г(ξ) (b).

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4. Fig. 3. Statement of the problem of flow past a cylinder: a – scheme of the computational domain. Гin – input boundary, Гout – output boundary, Гbed – bottom boundary, Гtop – upper boundary; b – shapes of the bottom surface; results of calculation of flow past a cylinder over bottom waves: c – 3D wavelength; g – 6D wavelength.

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5. Fig. 4. Comparison of results: a – calculated points with power approximation; b – calculation [12] (1), experimental data [11] (red dots), proposed model (2).

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Note

Presented by Academician of the RAS R.I. Nigmatullin


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