TO QUESTION OF APPLICATION OF HYDRODYNAMIC ANALOGY FOR PROCEDURE OF CALCULATION OF PARAMETERS OF TRANSPORT STREAMS

A. V. Sokhatsky; A. V. Trofimov; A. I. Kuzmenko

doi:10.32836/2521-6643-2021-1-61.2

A. V. Sokhatsky Graduate degree and rank: Professor, Doctor of Technical Sciences, Job: University of Customs and Finance, Dnipro, Position: Head at the Department of Transport Technologies and International Logistics, Country: Ukraine, City: Dnipro, Adress: 2/4 Vernadskogo str., Dnipro, Ukraine, 49000. https://orcid.org/0000-0002-3593-6517
A. V. Trofimov Graduate degree and rank: Associate Professor, PhD of Physical and Mathematical Sciences, Job: University of Customs and Finance, Ukraine, Dnipro, Position: Associate Professor of the Department of Transport Technologies and International Logistics Country: Ukraine City: Dnipro, Adress: 2/4 Vernadskogo str., Dnipro, Ukraine, 49000. https://orcid.org/0000-0001-8401-4570
A. I. Kuzmenko Graduate degree and rank: Associate Professor, PhD of Technical Sciences, Job: University of Customs and Finance, Ukraine, Dnipro, Position: Associate Professor of the Department of Transport Technologies and International Logistics Country: Ukraine City: Dnipro, Adress: 2/4 Vernadskogo str., Dnipro, Ukraine, 49000. https://orcid.org/0000-0001-7278-3647

DOI: https://doi.org/10.32836/2521-6643-2021-1-61.2

Keywords: traffic flow; macroscopic models; numerical simulation; equations of Navier – Stokes.

Abstract

A mathematical modeling of traffic flow for today is rather intricate and actual problem. The most perfect mathematical models of traffic flow are described by equations of mathematical physics.

In the theory of traffic flow there are various approaches to classification of their mathematical models. One of widespread is classification on macroscopic models and microscopic models. The whole group of transport vehicles, that is described by the corresponding parameters of motion, is examined in macroscopic models. Microscopic models are based on conception of safe distance to the leader. The most known models are: model of optimal speed, following by a leader model, Treibe’s model of clever driver.

Historically one of the first macroscopic models is a model of Lighthill – Whitham – Richards (LWR). In it the stream of motor-car transport vehicles is considered as an unidimensional stream of compressible liquid. In a model LWR is accepted, that an unambiguous interconnection exists between speed and fluid flow density, and the laws of maintenance of mass are satisfied.

Other macroscopic models based on analogues of a traffic flow to the hydrodynamic features of compressible liquid flow exist. It is a model of Tanaka, Whitham. Payne et al. The model of Helbing Eeler Navier Stokes is proposed in 1995. In this model to the system of Payne’s equations the third equation that represents the law of conservation of energy for variation of speed is added. In the second equation (law of momentum conservation) an additional component is considered that allows to take into account variation of speed. It should be noted that for the system of Navier – Stokes equations it is not known how to formulate the initial Cauchy boundary value problem in order the global solution is unique for all times.

The main difference between the hydrodynamic models of traffic flows from the corresponding hydrodynamic analogs is the formation of the right-hand side of the equations. This refers to the correct notation, as a rule, of hyperbolic systems of equations and their diffusion analogs.

The difficulties arising in the description of the traffic flow are similar to those that arise in the description of the turbulent motion of a liquid.

The aim of work is a construction of mathematical model, numerical method, algorithm for obtaining numerical solutions and creation of software for studying the dynamics of traffic flows.

The task of modeling of traffic flows of motor-car transport vehicles is examined in the article. For description of physical process the system of equations of Navier – Stokes is used. Methodology, numerical algorithm, and software, is worked out. For numerical integration of the systems of differential equations, finite volume method is used. The developed methodology was tested. Based on the results of numerical calculations, a fundamental traffic flow diagram has been constructed.

References

Lighthill M. J., Whitham G. B. On kinematic waves. II. Theory of traffic flow on long crowded roads // Proc. R. Soc. London, Se. A. 1955. V. 229. P. 281–345.

Richards P. I. Shock Waves on the Highway // Oper. Res. 1956. V. 4. P. 42–51.

Whitham Dzh. Lyneinыe y nelyneinыe volnы. M.: Myr, 1977.

Kheit F. Matematycheskaia teoryia transportnіkh potokov. M.: Myr, 1966.

Driu A. Teoryia transportnіkh potokov y upravlenye ymy. “Transport”, 1972 h., str. 1-424

Garbaruk A. V., Strelets M. H., Travin A. K., Shur M. L. Sovremennyie podhodyi k modelirovaniyu turbulentnosti . SPb. Izd-vo Politehn. un-ta, 2016. 234 s.

Sohatskiy A. V. Teoretichni osnovi stvorennya aerodinamichnih komponuvan perspektivnih shvidkisnih transportnih aparativ: dis. doktora tehnichnih nauk: 05.07.01. Dnipropetrovsk. 2010. 364 s.

Vvedenye v matematycheskoe modelyrovanye transportnykh potokov / Hasnykov A. V. y dr. / рod red. A. V. Hasnykova. M.: MFTY, 2010. 362 s.