Numerical Modelling of Momentum and Heat Transfer

Garant předmětu:

Lecturer:

Tutor:

Supervisor:

Department of Process Engineering

Synopsis:

Differential equations of momentum and heat transfer in laminar and turbulent flow. Basic methods of numerical solution of the continuity and transport equations (Method of weighted residuals, method of finite differences, control volumes, method of finite elements.) Methods securing the stability of solution, artificial viscosity.

Requirements:

Transfer phenomena

Syllabus of lectures:

1. Constitutive equations Newtonian and nonnewtonian fluids. Turbulent viscosity.

2. Continuity equation, momentum and heat transport. Special cases of almost incmpressible liquids. Eliptic, parabolic, hyperbolic equations. Different formulations: primitive variables (velocities and pressure), velocity potential, vorticity and stream function.

3. Principles of weighted residual method and application for collocation method, spectral method, finite differences, control volume method, Galerkin method and boundary integral method.

4. Requirements for numerical solution: consistency and order of accuracy (Taylor expansion), stability (spectral analysis) and principle of maximum.

5. Control volume method for transport equations: central schemes, upwind-schemes, QUICK, compact schema operator. Comparison with the exact solution of exponential type (numerical viscosity).

6. The method of control volume: solution of NS equations in primitive variables (velocity and pressure). Staggered grid for velocities and pressures (MAC). Pressure as a constraint and ways of solving pressure linked equations. Methods SIMPLE, SIMPLEC, SIMPLER.

7. Possible ways of solving the problem of stability (the checkerboard pattern) for non-staggered (collocated) grid of control volumes. Discussion article Rhie, Chow.

8. Finite Difference: solving parabolic equation for the vorticity transport and elliptic equations for the stream function. Stability analysis of explicit method of solution of the vorticity equation. Solving the equation for the stream function using methods ADI, SOR and conjugate gradient.

9. Boundary conditions for vorticity and stream function. The problem of singularities in the flow around the corner.

10. A method of finite elements applied to the primitive variables. Green's theorem. Method of penalty function. Shape functions used for approximation of velocities and pressure.

11. Design of basis functions in the triangular and quadrilateral elements. Izoparametrické elements. Numerical integration.

12. Meshless methods.

Syllabus of tutorials:

Basic priciples of CFD, making possible not only using but also writing CFD programs.

Study Objective:

Basic understanding of transport and constitutive equations. Ability to understand basic differences between different numerical methods used in CFD. Correctly select the menus in the CFD programes. Cultivate the ability to read articles in scientific journals.

Study materials:

Zienkiewicz O.C., Taylor R.L.: The finite element method, Volume 3 Fluid dynamics, Butterworth Heinemann, Oxford, 2000

http://users.fs.cvut.cz/rudolf.zitny/NAPen6.ppt (classification of PDE, method of characteristic)

http://users.fs.cvut.cz/rudolf.zitny/NAPen7.ppt (Euler equations, finite differences, Lax Fridrichs...)

http://users.fs.cvut.cz/rudolf.zitny/NAPen10.ppt (Navier Stokes, primitive variables, Fluent...)

http://users.fs.cvut.cz/rudolf.zitny/NAPen13.ppt (FEM and meshless methods)

Note:

Further information:

No time-table has been prepared for this course

The course is a part of the following study plans: