ME 412 Numerical Methods in Thermal Science

1. ME 412 Numerical Methods in Thermal Science S. P.Vanka Professor of Mechanical Engineering, UIUC, Illinois, USA

2. IN MEMORIUM This lecture is dedicated to my mother who passed away recently (Nov. 29th, 2011) at the age of 89 years

3. What this course is about This course teaches you the basics of developing and applying computational methods for solving problems of fluid flow and heat transfer It covers both fundamental theory and application of the techniques to develop practical fluid flow software. In addition, you will be also using one of the commercial fluid flow software to solve an industrially relevant flow problem

4. Course Objectives • To expose students to fundamentals of computational fluid dynamics and heat transfer. • To make students confident of developing as well as using software for computational fluid dynamics. • To make students familiar with simulation of complex fluid flows with complex boundary shapes and boundary conditions.

5. Course Administration • Instructor: S. P. Vanka; 3011 MEL, 4-8388, spvanka@uiuc.edu • Office hours: M, W 3-5 pm • Lectures: 400 Engineering Hall, MWF 1-1:150pm. • Text Book: No specific book, but the book by Ferziger and Peric is worth purchasing and reading. • Class notes will be distributed as appropriate.

6. Course Organization • Project based approach. • Students will develop / use projects to learn and apply CFD. • Each project can take two to three weeks depending on complexity. • There will be a total of 6 projects that graduate students need to complete. Undergraduate students will not do the final project. One student per project. Help will be given during the projects. • Grade will be assigned based on successful completion of the projects.

7. My Pledge • I will try to make the course interesting, beneficial and intellectually challenging • I will explain the material to the best of my ability and give you a clear perspective of the fundamentals as well as the applications • I will make myself available to you for answering questions and helping in the completion of projects

8. My expectations from you • Regular attendance in the class • Punctual submission of assignments • Attention in the class • Benefit from frequent interactions • Ask any questions whenever needed

9. Tentative List of Projects 1. Solution of two-dimensional unsteady heat conduction equation; 2. Solution of two-dimensional steady heat conduction equation; 3. Solution of scalar transport equation (with convective terms); 4. Boundary layer flow over a flat plate; 5. Two-dimensional unsteady recirculating flow; 6. Application of a commercial CFD code.

10. Grading • Grading will be based on the projects and one presentation. • 5 projects will be worth 75 points; 25 points for final project presentation and report. • Final letter grade will be based on points scored on a curve. • Each project should be accompanied by a report and code/results from commercial code. • Help on projects will be provided during office hours.

11. Grading • To balance course work loads and previous academic experiences, for undergraduate students, the projects will be based on use of a commercial code (Fluent/Ansys). Graduate students should write their own codes, and will be graded separately. • Undergraduate students need not complete the final project if they are enrolled for 3 credit hours. The last project is worth 1 credit hr.

12. Fluid Mechanics (Mechanics of Fluids) Statics and Dynamics • Statics: useful in estimating forces and stresses on earthen dams, tankers, sluice gates, etc. Relatively easy to compute. Based on pressure variation with depth. • Dynamics: may be the most complex branch of science. Nonlinear interactions lead to complex phenomena such as three-dimensionality, chaos and turbulence.

13. Fluid Flow is Omnipresent • Pumping of blood by the heart • Circulation of blood and nutrients in the body • Convective mass exchange in various organs • Water flow in piping systems and rivers, canals, seas • Atmospheric circulations, weather patterns • Combustion of gases, vehicle propulsion • Space and aeronautics • Chemical processing, heat transfer, • Etc.

15. Example Demonstrations • Light an incense stick and observe the rising incense smoke. Observe the transition from a smooth flow to a unsteady and eventually turbulent flow • Watch the flow of water in a river with some rocks. Observe the complex patterns behind the rock, and the stagnation of flow ahead of the rock • Light a match stick and observe the flame • Mix a colored dye in a beaker of water and observe the propagation of the dye and the small structures

16. Computational Fluid Dynamics (CFD) • Complimentary technique to experimental fluid dynamics and theoretical fluid dynamics • Has become attractive because of rapid development of computing technology • Several advantages over real-life experiments Cost, Speed, Feasibility, Accessibility, Convenience

17. CFD-Issues • Several important issues Accuracy Scale resolution Representation of all physical processes through mathematical models Code validation and uncertainty Is it only ‘colorful fluid dynamics’ ?

18. Governing Equations of Fluid Flow

19. Complexities of Fluid Flows Multi-dimensional Steady / Transient Compressible / incompressible Chemically reacting Newtonian / Non-Newtonian May contain external forces (magnetic , electric fields, surface tension, buoyancy, etc.)

20. Computational Methodologies • Finite-Difference Methods (x) • Finite Volume Methods (x) • Finite Element Methods (x) • Spectral Methods (x) • Spectral Element Methods (x) • Vortex Methods • Boundary Element Methods • Lattice Boltzmann Methods (x)

21. Finite Difference Method • Each derivative in the differential equation is first expressed in the form of a relation (stencil) between discrete values of the variable on a grid (rectangular, curvilinear, triangular, etc.) • The discrete equations for each derivative are derived using Taylor series expansions of the continuous variable

22. Finite Difference Method • The accuracy of the solution depends on how the expansion is truncated • The error is determined by the leading term that has been truncated (first order, second-order, fourth order schemes are commonly used) • Stability and convergence are important requirements of the finite difference method

23. Finite Volume Method • Finite Volume Method is based on conservation principles. Each relevant dependent variable is conserved over discrete volumes. For example, mass, momentum per unit mass, energy, species, are conserved quantities, which are expressed as “balances” over discrete control volumes

24. Finite Volume Method • The finite volume method can be seen as directly related to how the basic governing equations were initially derived. The discrete relations also assume some polynomial variation of variables between the locations where they are computed • The quantities computed by the finite volume method are “cell averages” not point values

25. Finite Difference / FVM • Both finite difference method and finite volume method satisfy the governing equation at each discrete location / control volume to the level of solution accuracy • The error in the discrete solution from the continuous solution is a combination of discretisation error and solution error

26. Evolution of Computing Power Since the 1950s computing power has increased dramatically. In 1968, I used the IBM 1620 computer, which probably was slower than a hand held calculator of current time. Input was typically through punched cards and output was in paper. Most computers were at data centers, and one had to pay for computing power in real dollars. The availability of personal computers and the graphics based use made the computers more user friendly and powerful. Simultaneously, the “chip” in the computer has continually become more powerful

27. Evolution of Computing Power Advances in processor speed are correlated well with Moore’s law by which processor speed doubles every 18 months. This has happened until now, but can be challenged in future due to heating considerations Another advance in computing speed has been the development of parallel computers. Consider networking hundreds of small computers to make one big calculation. Each small computer performs calculations on limited data and then exchanges results with other processors. The end result is an increase in computational speed equivalent to number of processor times the single processor speed

28. Evolution of Computing Power Personal Computers can perform calculations at a few GFLOPS (Giga Floating Point Operations) Networked PCs can increase the performance ten fold or more NCSA has clusters that can deliver teraflop speeds We are now expecting Peta Flops from a recently funded NSF center (Blue Waters) Simultaneously we now have significant increases in RAM (random access memory)

29. Advances in Scientific Visualization In my graduate school days, I had to make plots by hand by joining the dots by a “French curve”, trace them on transparent paper by “India ink”, and then copy into a thesis. There were no color printers, CDs, USB devices or WYSIWYG screens. In current days, you have LCD screens on your computers, beautiful animation software, and inexpensive color printers, storage devices, etc. THE FUTURE IS YET TO COME !!!

30. Selection of GRID Several types of grids can be used: Cartesian (x-y-z) grids Triangular / quadrilateral/tetrahedral elements Curvilinear grids Grid-less (meshless) methods do not need a grid Vortex methods also do not need a grid Accuracy depends on the type of grid (grid quality) and the number of mesh points/elements Mesh generation time is usually quite large for a complex industrial problem

31. Selection of Numerical Parameters The numerical parameters are: Steady or time marching algorithm Time of integration and time step Numerical relaxation parameters Number of iterations, parameters in the solvers Turbulence constants, and constants in the combustion model

32. Some Examples

33. Natural Convection in Enclosures Velocity vectors and temperature contours for natural convection in a square cavity, (a) Ra = 104. and (b). Ra = 105

34. Natural Convection in Enclosures (b) (a) Velocity vectors and temperature contours for natural convection in a square cavity with a cylinder, (a). Pr = 0.71, Ra = 105. and (b). Pr = 0.71, Ra = 106

35. Three Dimensional Natural Convection Natural convection in a "green house" with bottom wall maintained at T = 1.0 and all other walls at T = 0. Vector color corresponds to fluid temperature.

36. Rayleigh-Taylor instabilities (Re=1024) Density contours

37. Buoyancy-Induced Mixing in Tilted Channel

38. Mixing in a Curved Duct

39. Mixing in a Curved Duct

40. Turbulent Flow in a Circular Pipe, Re (tau) = 400 40

41. Instantaneous Fields in a Square Duct 41

42. Lagrangian Particle Tracking St = 5.0 St = 0.3 St = 0.1

43. Wavy Channel Flow • Rush et al. (1999) observed mixing and heat transfer characteristics of developing flow in serpentine and wavy channels

44. Wavy Channel Flow • Stone and Vanka (1999) numerically simulated developing flow and heat transfer in a wavy channel.

45. Some Good Websites http://www.cfd-online.com/ http://www.cfd-online.com/Wiki/Codes http://www.fluent.com/ http://www.lanl.gov/orgs/t/t3/codes.shtml# Google: CFD