Implicit algorithm for solving equations of motion of incompressible fluid

Aksenov A.A., Zhluktov S.V., Pokhilko V.I., Sorokin K.E.
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A large number of methods have been developed to solve the Navier – Stokes equations in the case of incompressible flows, the most popular of which are methods with velocity correction by the SIMPLE algorithm and its analogue — the method of splitting by physical variables. These methods, developed more than 40 years ago, were used to solve rather simple problems — simulating both stationary flows and non-stationary flows, in which the boundaries of the calculation domain were stationary. At present, the problems of computational fluid dynamics have become significantly more complicated. CFD problems are involving the motion of bodies in the computational domain, the motion of contact boundaries, cavitation and tasks with dynamic local adaptation of the computational mesh. In this case the computational mesh changes resulting in violation of the velocity divergence condition on it. Since divergent velocities are used not only for Navier – Stokes equations, but also for all other equations of the mathematical model of fluid motion — turbulence, mass transfer and energy conservation models, violation of this condition leads to numerical errors and, often, to undivergence of the computational algorithm.

This article presents an implicit method of splitting by physical variables that uses divergent velocities from a given time step to solve the incompressible Navier – Stokes equations. The method is developed to simulate flows in the case of movable and contact boundaries treated in the Euler paradigm. The method allows to perform computations with the integration step exceeding the explicit time step by orders of magnitude (Courant – Friedrichs – Levy number $CFL\gg1$). This article presents a variant of the method for incompressible flows. A variant of the method that allows to calculate the motion of liquid and gas at any Mach numbers will be published shortly. The method for fully compressible flows is implemented in the software package FlowVision.

Numerical simulating classical fluid flow around circular cylinder at low Reynolds numbers ($50 < Re < 140$), when laminar flow is unsteady and the Karman vortex street is formed, are presented in the article. Good agreement of calculations with the experimental data published in the classical works of Van Dyke and Taneda is demonstrated.

Keywords: hydrodynamics, gas dynamics, Navier – Stokes equations, velocity-pressure split algorithm, implicit numerical method
Citation in English: Aksenov A.A., Zhluktov S.V., Pokhilko V.I., Sorokin K.E. Implicit algorithm for solving equations of motion of incompressible fluid // Computer Research and Modeling, 2023, vol. 15, no. 4, pp. 1009-1023
Citation in English: Aksenov A.A., Zhluktov S.V., Pokhilko V.I., Sorokin K.E. Implicit algorithm for solving equations of motion of incompressible fluid // Computer Research and Modeling, 2023, vol. 15, no. 4, pp. 1009-1023
DOI: 10.20537/2076-7633-2023-15-4-1009-1023
URL: http://crm-en.ics.org.ru/journal/article/3371/
Creative Commons License This work is licensed under a Creative Commons Attribution-NoDerivs 3.0 Unported License.

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