Chapter 4
Mathematical Modeling of Wave Motions
of Fluids
Valentin A. Gushchin , Vasilii G. Kondakov , and Irina A. Smirnova
Abstract The study of wave movements of liquids is of interest both theoretically and practically. This can include flows with a free surface and flows with
internal waves. For correct mathematical modeling of such flows, the finite-difference
schemes of methods must have such properties as follows: high order of approximation, minimal scheme dissipation and dispersion, performance in a wide range
of the Reynolds and the Froude numbers, and that is especially important the property of monotonicity. This chapter presents two approaches: splitting method for
incompressible fluid flow (SMIF) method and compact accurately boundary adjusting
high-resolution technique (CABARET) method, of course, whose finite-difference
schemes have the properties listed above. A number of test tasks are considered and
compared with theoretical, experimental data and calculations of other authors.
4.1 Introduction
In this work, two methods for problems of motion of fluids with a free surface and
problems of internal waves’ destruction in steadily stratified medium are considered.
First method is known as SMIF [1, 2] or splitting on physical factor method for
incompressible fluid flows. SMIF-based schemes have second order of approximation
by both spatial and time steps. Schemes based on SMIF use mesh with spaced
variables: when the velocity components normal to sides are set on the faces of cell,
V. A. Gushchin (B) · I. A. Smirnova
Institute for Computer Aided Design of the RAS, 19/18, Vtoraya Brestskaya ul., Moscow 123056,
Russian Federation
e-mail: gushchin@icad.org.ru
I. A. Smirnova
e-mail: o-ira@yandex.ru
V. G. Kondakov
Nuclear Safety Institute of the RAS, 52, Bol’shaya Tul’skaya ul., Moscow 115191, Russian
Federation
e-mail: kondakov@ibrae.ac.ru
© The Author(s), under exclusive license to Springer Nature Singapore Pte Ltd. 2021
L. C. Jain et al. (eds.), Applied Mathematics and Computational Mechanics for Smart
Applications, Smart Innovation, Systems and Technologies 217,
https://doi.org/10.1007/978-981-33-4826-4_4
35
Mathematical Modeling of Wave Motions
of Fluids
Valentin A. Gushchin , Vasilii G. Kondakov , and Irina A. Smirnova
Abstract The study of wave movements of liquids is of interest both theoretically and practically. This can include flows with a free surface and flows with
internal waves. For correct mathematical modeling of such flows, the finite-difference
schemes of methods must have such properties as follows: high order of approximation, minimal scheme dissipation and dispersion, performance in a wide range
of the Reynolds and the Froude numbers, and that is especially important the property of monotonicity. This chapter presents two approaches: splitting method for
incompressible fluid flow (SMIF) method and compact accurately boundary adjusting
high-resolution technique (CABARET) method, of course, whose finite-difference
schemes have the properties listed above. A number of test tasks are considered and
compared with theoretical, experimental data and calculations of other authors.
4.1 Introduction
In this work, two methods for problems of motion of fluids with a free surface and
problems of internal waves’ destruction in steadily stratified medium are considered.
First method is known as SMIF [1, 2] or splitting on physical factor method for
incompressible fluid flows. SMIF-based schemes have second order of approximation
by both spatial and time steps. Schemes based on SMIF use mesh with spaced
variables: when the velocity components normal to sides are set on the faces of cell,
V. A. Gushchin (B) · I. A. Smirnova
Institute for Computer Aided Design of the RAS, 19/18, Vtoraya Brestskaya ul., Moscow 123056,
Russian Federation
e-mail: gushchin@icad.org.ru
I. A. Smirnova
e-mail: o-ira@yandex.ru
V. G. Kondakov
Nuclear Safety Institute of the RAS, 52, Bol’shaya Tul’skaya ul., Moscow 115191, Russian
Federation
e-mail: kondakov@ibrae.ac.ru
© The Author(s), under exclusive license to Springer Nature Singapore Pte Ltd. 2021
L. C. Jain et al. (eds.), Applied Mathematics and Computational Mechanics for Smart
Applications, Smart Innovation, Systems and Technologies 217,
https://doi.org/10.1007/978-981-33-4826-4_4
35