A New Approach to the Use of Non-Primitive Variables in the Mechanics of Continuous Media

A. Savitsky, M. Radkevich, A. Salokhiddinov, O. Ashirova, T. Khankelov, K. Shipilova, M. Abdukadirova, A. Gapirov, R. Razzakov


The problem of an approximate solution to hydrodynamic problems is the consideration of pressure. To exclude it from the equations, the transition to “non-primitive variables” (vortex and velocity vector divergence) is made. In this case, there are difficulties in the algorithmization of new equations for solving the inverse problem of hydrodynamics and a lot of internal iterative calculations. The object of this study includes equations in “non-primitive” variables. The research methods are based on the transformation without simplifications and assumptions of hydrodynamic equations into a form containing “non-primitive” variables and the demonstration of the possibilities of solving the equations. The GAMS programming language was used for approximate solutions for the first time. The aim of this paper is to demonstrate the possibility of solving the full equations in “non-primitive” variables for various conditions. The results showed the possibility of considering the compressibility of the medium when solving the inverse problem of hydrodynamics; the identity of solutions of the proposed system of equations and equations using the potential; and the possibility of using optimizing programming languages for hydrodynamics problems. The scientific novelty of this research consists of solving the full equations of hydrodynamics with the use of “non-primitive” variables but without the use of the current function.


Doi: 10.28991/ESJ-2024-08-02-021

Full Text: PDF


Continuous Medium; Non-Primitive Variable; Vortex; Motion Vector Potential; Compressibility of the Medium.


Kulesh, V. P., Sergeyev, Y. K., & Schreit, E. (1971). Baroclinic quasi-static model of circulation of waters of the Baltic Sea. Vestnik LHU, 18(18), 34–56. (In Russian)

Marchuk, G. I. (1980). Mathematical models of circulation in the ocean. Nauka, Novosibirsk (In Russian).

Calvino, C., Dabrowski, T., & Dias, F. (2023). A study of the wave effects on the current circulation in Galway Bay, using the numerical model COAWST. Coastal Engineering, 180, 104251. doi:10.1016/j.coastaleng.2022.104251.

Balagansky, M. Y., & Zakharov, Y. N. (2003). Iterative schemes for Navier-Stokes equations solving in vorticity-stream formulation. Vychislitelnye Tekhnologii, 8(5), 14–22. (In Russian).

Blohin, N. S., & Soloviev, D. A. (2006). Influence of wind on the dynamics of thermobar development during spring warming of a water body. Vestnik of Moscow University, Series 3: Physics. Astronomy, 3, 59–63. (In Russian).

Ivanov, V.G. (2005). Numerical Solution of Navier-Stokes Equations in the Stream Function-Vortex Variables. National Research Tomsk State University, Tomsk (In Russian).

Brenner, H. (2006). Fluid mechanics revisited. Physica A: Statistical Mechanics and its Applications, 370(2), 190-224. doi:10.1016/j.physa.2006.03.066.

Pastuhov, D. F., & Pastuhov, U. F. (2017). Approximation of the Poisson equation on a rectangle of increased accuracy. Bulletin of Polotsk State University. Series C, Fundamental Sciences, 12, 62–77. (In Russian).

Frick, P.G. (2003). Turbulence: Approaches and Models. Institute for Computer Research, Moscow-Izhevsk. (In Russian).

Obeidat, N. A., & Rawashdeh, M. S. (2023). On theories of natural decomposition method applied to system of nonlinear differential equations in fluid mechanics. Advances in Mechanical Engineering, 15(1), 1-15. doi:10.1177/16878132221149835.

Ivanov K.S. (2008). Numerical solution of the non-stationary Navier-Stokes equations. Kemerovo State University–Computational Technologies, 13(4), 35–49. (In Russian).

Mazo, A. B. (2006). Numerical simulation of viscous flow around a system of bodies on the basis of the Navier-Stokes equations in stream function-vorticity variables. Journal of Engineering Physics and Thermophysics, 79(5), 963–970. doi:10.1007/s10891-006-0192-0.

Liu, W. T., Zhang, A. M., Miao, X. H., Ming, F. R., & Liu, Y. L. (2023). Investigation of hydrodynamics of water impact and tail slamming of high-speed water entry with a novel immersed boundary method. Journal of Fluid Mechanics, 958, A42. doi:10.1017/jfm.2023.120

Popov, I. V., & Timofeeva, Y. E. (2015). Construction of a difference scheme of higher order approximation for the transfer equation using adaptive artificial viscosity. Keldysh Institute of Applied Mathematics, 39, 25-26. (In Russian).

Danaev, N. T., & Amenova, F. S. (2014). Studying convergence of an iterative algorithm for numerically solving the thermal convection problems in the variables “stream function-vorticity.” Journal of Applied and Industrial Mathematics, 8(4), 500–509. doi:10.1134/S1990478914040061.

Kuttykozhaeva, S. N., & Uvalieva, S. K. (2015). Navier-Stokes equations of alternating stream function and velocity vortex. International Journal of Experimental Education, 7, 168-168. (In Russian).

Abdallah, S. (2017). On the Non-primitive Variables Formulations for the Incompressible Euler Equations. Global Journal of Technology and Optimization, 8(1), 111. doi:10.4172/2229-8711.1000e111.

Kochin, N. E. (1937). Vector Calculus and the Beginning of Tensor Calculus. Gostekhteorizdat, Moscow, Russia. (In Russian).

Sazonov, Y. A., Mokhov, M. A., Gryaznova, I. V., Voronova, V. V., Tumanyan, K. A., & Konyushkov, E. I. (2023). Solving Innovative Problems of Thrust Vector Control Based on Euler's Scientific Legacy. Civil Engineering Journal, 9(11), 2868-2895. doi:10.28991/CEJ-2023-09-11-017.

Kochin, N.E., Kibel, I.A., & Roze N.V. (1963). Theoretical Hydromechanics. Fizmatgiz, Moscow, Russia. (In Russian).

Batchelor, J., & Moffat, G. (1984). Contemporary Hydrodynamics, Successes and Problems. Mir Publ., Moscow, Russia. (In Russian).

Brown, C. (2008). Attachment 2: HEC-RAS model development-Attachment 2. Technical Report, Hydrological Center, U.S. Army Corps of Engineers, Wahington, United States.

Lax, P. D., & Richtmyer, R. D. (1956). Survey of the stability of linear finite difference equations. Communications on Pure and Applied Mathematics, 9(2), 267–293. doi:10.1002/cpa.3160090206.

Salokhiddinov, A., Savitsky, A., McKinney, D., & Ashirova, O. (2023). An improved finite-difference scheme for the conservation equations of matter. E3S Web of Conferences, 386, 6002. doi:10.1051/e3sconf/202338606002.

Yarmitskiy, A. G. (1978). About one class of axisymmetric unsteady flows of viscous incompressible fluid. Siberian Department of Russian Academy of Sciences, 59-66. (In Russian).

Vorozhtsov, E. V., & Shapeev, V. P. (2015). Numerical Solution of the Poisson Equation in Polar Coordinates by the Method of Collocations and Least Residuals. Modeling and Analysis of Information Systems, 22(5), 648. doi:10.18255/1818-1015-2015-5-648-664.

Kuhnert, J., & Tiwari, S. (2001). Grid free method for solving the Poisson equation. Fraunhofer-Institut für Techno- und Wirtschaftsmathematik ITWM, 25, 1-12. (In Russian).

Matsumoto, T., & Hanawa, T. (2003). A Fast Algorithm for Solving the Poisson Equation on a Nested Grid. The Astrophysical Journal, 583(1), 296–307. doi:10.1086/345338.

Liu, S., Li, J., Chen, L., Guan, Y., Zhang, C., Gao, F., & Lin, J. (2019). Solving 2D Poisson-type equations using meshless SPH method. Results in Physics, 13, 102260. doi:10.1016/j.rinp.2019.102260.

Ma, Q. W., Zhou, Y., & Yan, S. (2016). A review on approaches to solving Poisson’s equation in projection-based meshless methods for modelling strongly nonlinear water waves. Journal of Ocean Engineering and Marine Energy, 2(3), 279–299. doi:10.1007/s40722-016-0063-5.

Khujaev, I., Khujaev, J., Eshmurodov, M., & Shaimov, K. (2019). Differential-difference method to solve problems of hydrodynamics. Journal of Physics: Conference Series, 1333(3), 32037. doi:10.1088/1742-6596/1333/3/032037.

Salokhiddinov, A. T., Savitsky, A. G., & Ashirova, O. A. (2022). Studies of conservative finite-difference scheme for transfer equations. Journal of Irrigation and Melioration, 1(27), 13–17. (In Russian).

Beliyaev, V. A., & Shapeev, V. P. (2017). Variants of the collocation method and the least disjoint method for solving problems of mathematical physics in trapezoidal domains. Vychislitelnye Tekhnologii, 22(4), 22–42. (In Russian).

Beliyaev, V. A., & Shapeev, V. P (2018). Solution of the Dirichlet problem for the Poisson equation by collocation and least squares method in a region with discretely defined boundary. Vychislitelnye Tekhnologii, 23(3), 15–30. (In Russian).

Serrin, J. (1959). Serrin, J. (1959). Mathematical Principles of Classical Fluid Mechanics. Fluid Dynamics I, 125–263, Springer, Berlin, Germany. doi:10.1007/978-3-642-45914-6_2.

Full Text: PDF

DOI: 10.28991/ESJ-2024-08-02-021


  • There are currently no refbacks.

Copyright (c) 2024 Andre Savitsky, Maria Radkevich, Abdulkhakim Salokhiddinov, Olga Ashirova, Tavbay Khankelov, Kamila Shipilova, Maloxat Abdukadirova, Abdusamin Gapirov, Ruslan Razzakov