Viscous Fluid Flow in a Coaxial Pipe
Abstract
In this paper, we propose a method to solve problems of stability of viscous incompressible fluid in pipes. The method is based on the R-functions with which the boundary function is obtained. Boundary function role is similar to the mesh in the finite element method, but it depends on shape of the region only and does not require any changes to increase accuracy. With the help of the boundary function and Chebyshev polynomials we design and elaborate structures for an approximate solution of the equations that satisfy the boundary conditions. This method is used for calculation of laminar steady flow in a pipe due to the constant pressure gradient and for solution of the linearized eigenvalue problem for perturbations. The pressure is handled by the Poisson equation. An example of flow stability in a rectangular duct is investigated. We obtain the eigenvalues and plots of perturbation velocity. The proposed algorithm does not depend on the shape of duct cross-section and can be used to investigate the stability of flows in arbitrary pipes including internal elements. This algorithm is simpler than the collocation method, tau method or spectral element method. Therefore, it should work faster and can be quickly adapted to new computer architectures, such as CUDA, OpenCL, Xeon Phi.
DOI 10.14258/izvasu(2016)1-10
Downloads
Metrics
References
Гольдштик М.А., Штерн В.Н. Гидродинамическая устойчивость и турбулентность. -Новосибирск, 1977.
Henningson D.S., Schmid P.J. Stability and transition in shear flows -New-York, 2001.
Theofilis V. Advances in global linear instability analysis of non-parallel and threedimensional flows//Progress in Aerospace Sciences. -2003. -№39.
Кравченко В.Ф., Рвачев В.Л. Алгебра логики, атомарные функции и вейвлеты в физических приложениях. -М., 2006.
Рвачёв В.Л. Теория R-функций и некоторые ее приложения. -Киев, 1982.
Shapiro V. Semi-analytic geometry with R-functions//Acta Numerica. -2007. -V. 16.
Tsukanovl., Shapiro V., Zhang S. A Mesh-free Method for Incompressible Fluid Dynamics Problems//Int. J. Numer. Meth. Engng. -2003. -V. 58.
Shapiro V., Tsukanov I. The Architecture of SAGE -A Meshfree System Based on RFM//Engineering with Computers. -2002. -V. 18., №4.
Fougerolle Y., Gribok A., Foufou S., Truchetet F. and others Boolean operations with implicit and parametric representation of primitives using R-functions//Visualization and Computer Graphics, IEEE Transactions. -2005. -V. 11., № 5.
Freytag M., Shapiro V., Tsukanov I. Field modeling with sampled distances//Computer-Aided Design. -2006. -V. 38., №2.
Proskurin A., Sagalakov A. The numerical investigation of the stability of the localized perturbation in Poiseuille flow//Computational technologies. -2013. -V. 18., №3.
Слесаренко А.П. S-функции в обратных задачах аналитической геометрии и моделировании тепловых процессов//ВосточноЕвропейский журнал передовых технологий. -2012. -Т. 1, №4(55).
Fletcher C. Computational techniques for fluid dymamics 2. -Springer, 1991.
Проскурин А.В., Сагалаков А.М. Математическое моделирование течений в трубах с помощью метода функций Рвачева//XI Всероссийский съезд по фундаментальным проблемам теоретической и прикладной механики: сборник трудов (Казань, 20-24 августа 2015 г.). -Казань, 2015.
Proskurin A., Sagalakov A. A method for modelling MHD flows in pipes//Book of abstracts of Russian conference on Magnetohydrodynamics. June 22-25, 2015, Perm, Russia. -Perm, 2015.
Saad J. Numerical methods for large eigenvalue problems. -Manchester, 1992.
Hernandez V., Roman J. E., Vidal V. SLEPc: A scalable and flexible toolkit for the solution of eigenvalue problems//ACM Trans. Math. Software -2005. -V. 31., №2.
Izvestiya of Altai State University is a golden publisher, as we allow self-archiving, but most importantly we are fully transparent about your rights.
Authors may present and discuss their findings ahead of publication: at biological or scientific conferences, on preprint servers, in public databases, and in blogs, wikis, tweets, and other informal communication channels.
Izvestiya of Altai State University allows authors to deposit manuscripts (currently under review or those for intended submission to Izvestiya of Altai State University) in non-commercial, pre-print servers such as ArXiv.
Authors who publish with this journal agree to the following terms:
- Authors retain copyright and grant the journal right of first publication with the work simultaneously licensed under a Creative Commons Attribution License (CC BY 4.0) that allows others to share the work with an acknowledgement of the work's authorship and initial publication in this journal.
- Authors are able to enter into separate, additional contractual arrangements for the non-exclusive distribution of the journal's published version of the work (e.g., post it to an institutional repository or publish it in a book), with an acknowledgement of its initial publication in this journal.
- Authors are permitted and encouraged to post their work online (e.g., in institutional repositories or on their website) prior to and during the submission process, as it can lead to productive exchanges, as well as earlier and greater citation of published work (See The Effect of Open Access).
