Advertisement

Toward Reduction of Conservation Equations in Curvilinear Coordinate Systems into a Set of ODEs Using the Method of Characteristics

  • Gunawan NugrohoEmail author
  • Totok R. Biyanto
Chapter

Abstract

This work examined the conservation equations in curvilinear coordinates using the method of characteristics. The method is commonly applied to solve first-order partial differential equations. When applying this method to the conservation laws, the main difficulty is mass and momentum equations which are simultaneous and nonlinear PDEs. The method utilizes the separation of order in order to solve the problem. The resulting nonlinear ODEs in the main equation and characteristic variables were performed by the implementation of the system of Riccati and polynomial equation. The system of polynomial and Riccati equation is handled by the proposed method to solve each polynomial and Riccati equation. Both results were then equated to define each ODE solution. The procedure was then repeated sequentially to reach the final solutions for velocities, pressure, and temperature.

Keywords

Method of characteristics Conservation equations Reduction of PDEs Analytical solutions Curvilinear coordinates 

References

  1. Ahmadi K, Mogensen K, Norman R (2011) Limitation of current method-of-characteristics (MOC) methods using shock-jump approximations to predict MMPs for complex gas/oil displacements. SPE J 16(04):743–750CrossRefGoogle Scholar
  2. Cinnella P (2008) Transonic flows of dense gases over finite wings. Phys Fluids 20(4):046103CrossRefzbMATHGoogle Scholar
  3. Debnath L (1997) Nonlinear partial differential equations for scientists and engineers. Birkhauset, BostonCrossRefzbMATHGoogle Scholar
  4. Deglaire P, Agren O, Bernhoff H, Leijon M (2008) Conformal mapping and efficient boundary element method without boundary elements for fast vortex particle simulations. Eur J Mech B/Fluids 27:150–176MathSciNetCrossRefzbMATHGoogle Scholar
  5. Deng X, Liu H, Jiang Z, Baldock TE (2016) Swash flow properties with bottom resistance based on the method of characteristics. Coast Eng 114:25–34CrossRefGoogle Scholar
  6. Desantes JM, Serrano JR, Arnau F, Piqueras P (2012) Derivation of the method of characteristics for the fluid dynamics solution of flow advection along porous wall channel. Appl Math Model 36:3134–3152MathSciNetCrossRefzbMATHGoogle Scholar
  7. Eklund M, Alamaniotis M, Hernandez H, Jevremovic T (2015) Method of chraracteristics—a review with applications to science and nuclear engineering computation. Prog Nucl Energy 85:548–567CrossRefGoogle Scholar
  8. Galindo J, Tiseira A, Fajardo P, Navarro R (2013) Analysis of the influence of different real flow effects on computational fluid dynamics boundary conditions based on the method of characteristics. Math Comput Model 57:1957–1964CrossRefGoogle Scholar
  9. Kolev NI (2011) Numerical methods for multi-phase flow in curvilinear coordinate systems. In: Kolev NI (ed) Multiphase flow dynamics 1 fundamentals. Springer, Berlin, HeidelbergGoogle Scholar
  10. Munusamy S, Narasimhan S, Kaisare NS (2013) Order reduction and control of hyperbolic, countercurrent distributed parameter systems using method of characteristics. Chem Eng Sci.  https://doi.org/10.1016/j.ces.2013.12.029 Google Scholar
  11. Nugroho G, Soehartanto T, Biyanto TR (2015) The existence of polynomial solution of the nonlinear dynamical systems. Open Access Journal of Information Systems (OAJIS)Google Scholar
  12. Oosthuizen PH, Naylor D (1999) An introduction to convective heat transfer analysis. McGraw-Hill, SingaporezbMATHGoogle Scholar
  13. Sim W-G, Park J-H (1997) Transient analysis for compressible fluid flow in transmission line by method of characteristics. KSME Int J 11(2):173–185CrossRefGoogle Scholar
  14. Tumilowicz E, Chan CL, Li P, Xu B (2014) An enthalpy formulation for thermocline with encapsulated PCM thermal storage and benchmark solution using the method of characteristics. Int J Heat Mass Transf 79:362–377CrossRefGoogle Scholar
  15. Van TD, Tsuji M, Son NDT (2000) The characteristic method and its generalizations for first-order nonlinear partial differential equations. Chapman & Hall/CRCGoogle Scholar
  16. Woyczynski WA (1998) Burgers-KPZ turbulence. Springer, BerlinCrossRefzbMATHGoogle Scholar

Copyright information

© Springer International Publishing AG 2018

Authors and Affiliations

  1. 1.Department of Engineering PhysicsInstitut Teknologi Sepuluh NopemberSurabayaIndonesia

Personalised recommendations