5.00 credits
30.0 h + 30.0 h
Q2
Teacher(s)
Winckelmans Grégoire;
Language
English
Main themes
- Reminder of the conservation equations in fluid mechanics; Reminder of the differents types of PDEs and of their classification.
- Finite differences et numerical schemes for ODEs and discretized PDEs : consistency, stability, convergence, explicit and implicit schemes.
- Case of 2-D and of 3-D flows, steady and unsteady.
- Incompressible flows : formulation in velocity-pressure and formulation in vorticity-velocity (streamfunction) .
- Compressible flows, including capture of discontinuities.
- Structured grids, also with mapping from physical to computational space. Introduction to finite volumes approaches, and to unstructured grids.
- Lagrangian vortex element method (VEM) eventually combined with the boundary element method (BEM)
Learning outcomes
At the end of this learning unit, the student is able to : | |
| 1 |
In consideration of the reference table AA of the program "Masters degree in Mechanical Engineering", this course contributes to the development, to the acquisition and to the evaluation of the following experiences of learning:
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Content
Reminder of the different types of partial differential equations (PDE) : hyperbolic, parabolic, elliptic. Systems of PDEs. Method of characteristics for hyperbolic cases and applications in simple compressible flows.
Discretisation using explicit finite differences, centered and decentered: obtention by Taylor series, truncation error and order. Definition of fundamental operators and obtention of finite difference stencils using operators. Implicit finite differences and compact schemes.
Model convection equation in 1-D: discretisation of the convective term using centered finite differences, explicit and implicit, modal analysis and modified wavenumber: phase error (= numerical dispersion); decentered finite differences (upwinding) and amplitude error (= numerical diffusion).
Model diffusion equation in 1-D: discretisation of the diffusion term using centered finite differences, explicit and implicit, modal analysis and modified wavenumber: amplitude error.
Temporal integration schemes for discretized problems: numerical integration of ODE and system of ODEs; reminder of basic schemes and new schemes, stability analysis : explicit Euler, implicit Euler, Crank-Nicolson (= trapezoid rule), multi substeps schemes (Runge-Kutta), multi step schemes (Leap Frog, Adams-Bashforth, Adams-Moulton), predictor-corrector schemes, Hyman scheme, 3BDF scheme.
Equations of convection and/or diffusion : mesh Reynolds number, Fourier number and CFL (Courant-Friedrichs-Lewy) number, linear and non linear cases, integration schemes and stability, decentered finite differences for the convection (upwinding), ADI schemes for multi-dimensional problems.
Numerical methods for incompressible flows : velocity-presure formulation : discretisation (MAC mesh), imposition of boundary conditions, method of artificial evolution for steady flows, methods for unsteady flows, stability, Brinkman penalization method for case with an immersed body. Vorticity-velocity formulation: discretisation, obtention of the velocoty field from the vorticity field, obtention of an approximate boundary condition on the vorticity, method or artifical evolution for steady flows, methods for unsteady flows (also introduction to the method of vortex particles).
Hyperbolic systems in conservative form : model non-linear equation (Burgers), Euler equations for compressible flows and boundary conditions (based on the characteristics) ; explicit integation schemes (Lax, Lax-Wendroff, Richtmeyer, MacCormack), implicit integration schemes; numerical capture of discontinuities. Transformation of a computational structured domain (block) in a physical domain, and obtention of equations in a conservative form in the computational domain ; multi-blocks approach. Delta form of the discretized equations for multi-dimensionanl domains and generalized ADI schemes (Beam-Warming).
Introduction to the method of finite volumes for unstructured grids: treatment of the convective and diffusive fluxes.
Discretisation using explicit finite differences, centered and decentered: obtention by Taylor series, truncation error and order. Definition of fundamental operators and obtention of finite difference stencils using operators. Implicit finite differences and compact schemes.
Model convection equation in 1-D: discretisation of the convective term using centered finite differences, explicit and implicit, modal analysis and modified wavenumber: phase error (= numerical dispersion); decentered finite differences (upwinding) and amplitude error (= numerical diffusion).
Model diffusion equation in 1-D: discretisation of the diffusion term using centered finite differences, explicit and implicit, modal analysis and modified wavenumber: amplitude error.
Temporal integration schemes for discretized problems: numerical integration of ODE and system of ODEs; reminder of basic schemes and new schemes, stability analysis : explicit Euler, implicit Euler, Crank-Nicolson (= trapezoid rule), multi substeps schemes (Runge-Kutta), multi step schemes (Leap Frog, Adams-Bashforth, Adams-Moulton), predictor-corrector schemes, Hyman scheme, 3BDF scheme.
Equations of convection and/or diffusion : mesh Reynolds number, Fourier number and CFL (Courant-Friedrichs-Lewy) number, linear and non linear cases, integration schemes and stability, decentered finite differences for the convection (upwinding), ADI schemes for multi-dimensional problems.
Numerical methods for incompressible flows : velocity-presure formulation : discretisation (MAC mesh), imposition of boundary conditions, method of artificial evolution for steady flows, methods for unsteady flows, stability, Brinkman penalization method for case with an immersed body. Vorticity-velocity formulation: discretisation, obtention of the velocoty field from the vorticity field, obtention of an approximate boundary condition on the vorticity, method or artifical evolution for steady flows, methods for unsteady flows (also introduction to the method of vortex particles).
Hyperbolic systems in conservative form : model non-linear equation (Burgers), Euler equations for compressible flows and boundary conditions (based on the characteristics) ; explicit integation schemes (Lax, Lax-Wendroff, Richtmeyer, MacCormack), implicit integration schemes; numerical capture of discontinuities. Transformation of a computational structured domain (block) in a physical domain, and obtention of equations in a conservative form in the computational domain ; multi-blocks approach. Delta form of the discretized equations for multi-dimensionanl domains and generalized ADI schemes (Beam-Warming).
Introduction to the method of finite volumes for unstructured grids: treatment of the convective and diffusive fluxes.
Teaching methods
Lectures : there are typically 13 lectures in class, each of 2 hours.
Sessions of exercices are also organised in class, each of 2 hours, to further develop concepts covered during the lectures and to do some applications.
The students must perform two homeworks which require to be program in C. These homeworks are mandatory and they must be done during the quadrimester, each with a start date and a deadline date for the report, which is graded. Depending on the amplitude of the work/effort expected, these homeworks are done alone or in teams of two students.
Sessions of exercices are also organised in class, each of 2 hours, to further develop concepts covered during the lectures and to do some applications.
The students must perform two homeworks which require to be program in C. These homeworks are mandatory and they must be done during the quadrimester, each with a start date and a deadline date for the report, which is graded. Depending on the amplitude of the work/effort expected, these homeworks are done alone or in teams of two students.
Evaluation methods
The homeworks are essential to this course on numerical methods in fluid mechanics. They correspond to work that is mandatory and that must be performed during the quadrimester; each within a well-defined time period and with a given deadline for the report, that is graded.
It is not possible to do, or even re-do, any of the homeworks outside of the time period that was defined for it within the quadrimester.
The final exam is a written exam, with questions that can cover all parts of the course (lectures, exercice sessions, homeworks).
The calculation of the final grade obtained by the student for the course is a weighted sum of the grade obtained for the homeworks (for 60 %) and of the grade obrained for the final written exam (for 40 %).
It is not possible to do, or even re-do, any of the homeworks outside of the time period that was defined for it within the quadrimester.
The final exam is a written exam, with questions that can cover all parts of the course (lectures, exercice sessions, homeworks).
The calculation of the final grade obtained by the student for the course is a weighted sum of the grade obtained for the homeworks (for 60 %) and of the grade obrained for the final written exam (for 40 %).
Online resources
Bibliography
- R.W. Hamming, « Numerical Methods for Scientists and Engineers », second ed., Dover, 1986.
- J.H. Ferziger, « Numerical Methods for Engineering Applications », Wiley, 1981.
- J. H. Ferziger and M. Peric, « Computational Methods for Fluid Dynamics », Springer, 1996.
- R. Peyret and T.D. Taylor, « Computational Methods for Fluid Flow », Springer, 1986.
- C.A. J. Fletcher, « Computational Techniques for Fluid Dynamics 1, Fundamental and General Techniques », second ed., Springer 1991.
- C.A. J. Fletcher, « Computational Techniques for Fluid Dynamics 2, Specific Techniques for Different Flow Categories » second ed., Springer, 1991.
- K. Srinivas and C.A.J Fletcher, « Computational Techniques for Fluid Dynamics, A Solutions Manual », Springer, 1991.
- D.A. Anderson, J.C. Tannehill, R.H. Pletcher, « Computational Fluid Mechanics and Heat Transfer », Hemisphere Publishing, 1984.
- D. Drikakis and W. Rider, « High-Resolution Methods for Incompressible and Low-Speed Flows », Springer, 2005.
Teaching materials
- Notes et documentation (e;g. slides) du titulaire
Faculty or entity
MECA
