1 / 34

Semi-Lagrangian Approximation in the Time-Dependent Navier-Stokes Equations

Explore the Semi-Lagrangian approximation methods for solving time-dependent Navier-Stokes equations using Finite Element Method, with emphasis on advection derivative accuracy. Learn about interpolation stability and conservation laws in computational fluid dynamics.

perrye
Download Presentation

Semi-Lagrangian Approximation in the Time-Dependent Navier-Stokes Equations

An Image/Link below is provided (as is) to download presentation Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author. Content is provided to you AS IS for your information and personal use only. Download presentation by click this link. While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server. During download, if you can't get a presentation, the file might be deleted by the publisher.

E N D

Presentation Transcript


  1. Semi-Lagrangian Approximation in the Time-Dependent Navier-Stokes Equations Vladimir V. Shaydurov Institute of Computational Modeling of Siberian Branch of Russian Academy of Sciences, Krasnoyarsk Beihang University, Beijing shaidurov04@mail.ru in cooperation with G. Shchepanovskaya and M. Yakubovich

  2. Contents • Convection-diffusion equations: • Modified method of characteristics. • Conservation law of mass: • Approximation in norm. • Finite element method: • Approximation in norm.

  3. Pironneau O. (1982): Method of characteristics The main feature of several semi-Lagrangian approaches consists in approximation ofadvection members as one “slant”(substantial or Lagrangian) derivative in the direction of vector

  4. Pointwise approach in convection-diffusion equation The equation with this right-hand side is self-adjoint.

  5. Approximation of slant derivative Apply finite element method at the time level and use appropriate quadrature formulas for the lumping effect Two ways for approximation of slant derivative 1. Approximation along vector 2. Approximation along characteristics (trajectory)

  6. Approximation of substantial derivative along trajectory Solution smoothness usually is better along trajectory Asymptotically both way have the same first order of approximation

  7. Finite element formulation at time level Intermediate finite element formulation Final formulation

  8. Interpolation-1 Stability in norm: Chen H., Lin Q., Shaidurov V.V., Zhou J. (2011), …

  9. Interpolation-1 Stability in norm and conservation law: impact of four neighboring points into the weight of

  10. Interpolation-2

  11. Connection between interpolations

  12. Improving by higher order differences

  13. Solving two problems with the first and second order of accuracy

  14. Navier-Stokes equations. Computational geometric domain

  15. Navier-Stokes equations In the cylinder we write 4 equations in unknowns

  16. Notation

  17. Notation

  18. Notation

  19. Initial and boundary conditions

  20. Boundary conditions at outlet supersonic and rigid boundary

  21. Boundary conditions at subsonic part of computational boundary a wake

  22. Direct approximation of Curvilinear hexahedron V: Trajectories:

  23. Due to Gauss-Ostrogradskii Theorem: Approximation of curvilinear quadrangle Q:

  24. Gauss-Ostrogradskii Theorem in the case of boundary conditions:

  25. Discrete approach

  26. Matrix of finite element formulation at time layer

  27. Supersonic flow around wedge M=4, Re=2000 angle of the wedge β ≈ 53.1º, angle of attack  = 0º Density and longitudinal velocity at t = 8 Density and longitudinal velocity at t = 20

  28. Density and longitudinal velocity at t = 50

  29. Supersonic flow around wedge for nonzero angle of attack M=4, Re=2000 angle of the wedge β= 53.1º, angle of attack  = 5º Density and longitudinal velocity at t = 6 Density and longitudinal velocity at t = 8

  30. Density and longitudinal velocity at t = 10 Density and longitudinal velocity at t = 20

  31. Density M=4, Re=2000 Angle of the wedgeβ ≈ 53.1° Longitudinal velocity

  32. Conclusion • Conservation of full energy (kinetic + inner) • Approximation of advection derivatives in the frame of finite element method without artificial tricks • The absence of Courant-Friedrichs-Lewy restriction on the relation between temporal and spatial meshsizes • Discretization matrices at each time level have better properties • The better smooth properties and the better approximation along trajectories

  33. Thanks for your attention!

More Related