A Nonlinear System of Differential Equations in Supercritical Flow Spread Problem and Its Solution Technique

Abstract

A nonlinear system of differential equations in the problem of free flowing of supercritical flow is considered and a method of its solution is proposed. The analytical method is based on the introduction of the velocity hodograph plane and the obtaining of analytical solutions for the system of partial differential equations. It is pointed out that apart from being purely analytical, the potential flow model has a great practical demand due to its use as a base for the further research of the flow resistance forces. The proposed model can be developed by taking into account flow resistance and gradient, the bottom of the diverting channel flow. The theoretical results are complemented by numerical experiments and compared with experimental data.

Nonlinear Differential Equations of Flow Motion Considering Resistance Forces

“…This paper represents the experimental data obtained by the Department of Hydraulic Engineering and Melioration of the Don State Agrarian University. This work is the development of analytical methods for solving problems of technical mechanics of liquids and gases [1,17,19], using a technique for solving nonlinear problems similar to those proposed in [24][25][26][27][28][29][30].…”

“…The research object is a stationary potential 2D planar open high-velocity (supercritical) water flow [1]. The subject of this research is a mathematical model of a freely spreading high-velocity flow, with the resistance forces from the channel taken into account.…”

“…When calculating high-velocity flows in long conduits, the effect of resistance forces should not be neglected, the presence of which causes a vortex flow structure and leads to a reduction in the total hydrodynamic pressure downstream. In addition, in many cases, the slope of the bottom is of great significance [1][2][3][4]. The predominant approach in calculating uniform currents is based on a 1D presentation of an open flow.…”

Dependence of the Analytical Approximate Solution to the Van der Pol Equation on the Perturbation of a Moving Singular Point in the Complex Domain

“…At the moment, there are two options for solving nonlinear differential equations with moving singular points. The first option is related to the solvability in quadratures, which is allowed only in special cases [17][18][19][20][21][22][23]. The second option is associated with the author's analytical approximate solution method, successfully tested on a number of classes of nonlinear differential equations [15,[24][25][26][27][28].…”

The Maximum Domain for an Analytical Approximate Solution to a Nonlinear Differential Equation in the Neighborhood of a Moving Singular Point

“…At present, there are two alternative solutions to such equations. Solution One deals with special solutions to such equations using special substitution of variables [5][6][7][8][9][10][11][12][13][14][15][16][17][18][19][20][21][22][23]. Solution One, dealing with nonlinear differential equations, does not solve the problem as a whole.…”