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COMPUTER HYDROMECHANICS, 2024
IX International Scientific & Practical Conference "Computer Hydromechanics"
HYDRODYNAMICS AND ACOUSTICS
2021 ◊ Volume 2 (92) ◊ Issue 1 ◊ p. 13-29
I. O. Brovchenko*, V. S. Maderich*, S. I. Kovalets *
* Institute of Mathematical Machines and Systems of NAS of Ukraine, Kyiv, Ukraine
The method of random walks for inhomogeneous flows and diffusion
Gidrodin. akust. 2021, 2(1):13-29
TEXT LANGUAGE: Ukrainian
ABSTRACT
The physical and mathematical description of the transfer of passive impurities by the velocity field in the fluid flow (an advection) is important for modeling the spread of various pollutants on the surfaces of water areas. In some cases, one cannot neglect the influence of molecular diffusion on the advection process. This article is devoted to developing the methods of numerical modeling of the transfer of passive particles in the fields of inhomogeneous flows in the presence of diffusion. The center of mass of an ensemble of particles or the mathematical expectation of the position of each particle was found not to coincide with the streamlines constructed by the velocity vector field when solving the equation of transport and diffusion by the method of random walks. It is shown that with a non-zero second derivative of the velocity along the spatial coordinate, the deviation from the streamline has the second order of smallness. Expressions for correction terms in the equation of motion of a passive particle are derived. The high-precision trajectory integration methods in random walk methods can lead to significant errors associated with particle dispersion during movement. The schemes of a high order of accuracy, without the allowance for the correction terms, introduce a greater error than the first-order schemes. The equation for the moments of the distribution of the particle's position is derived, which relates the dispersion and mathematical expectation to the inhomogeneous field of velocity and diffusion coefficient. They can be used for constructing high-precision random walk methods. The proposed approach allows building the numerical schemes of the three-dimensional transfer taking into account the influence of particle dispersion on the movement of the center of mass, as well as the influence of the inhomogeneity of the flow field on the dispersion of the particle coordinate distribution. The analytical statements and numerical calculations on examples of two-dimensional problems illustrate the validity of the derived equations.
KEY WORDS
advection, diffusion, stochastic methods, random walk