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ME 412 Numerical Methods in Thermal Science


ME 412 Numerical Methods in Thermal Science S. P.Vanka Professor of Mechanical Engineering, UIUC, Illinois, USA Natural Convection in Enclosures Velocity vectors and ... – PowerPoint PPT presentation

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Title: ME 412 Numerical Methods in Thermal Science

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

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

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

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
  • To make students familiar with simulation of
    complex fluid flows with complex boundary shapes
    and boundary conditions.

Course Administration
  • Instructor S. P. Vanka 3011 MEL, 4-8388,
  • Office hours M, W 3-5 pm
  • Lectures 400 Engineering Hall, MWF 1-1150pm.
  • Text Book No specific book, but the book by
    Ferziger and Peric is worth purchasing and
  • Class notes will be distributed as appropriate.

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.

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

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

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.

  • Grading will be based on the projects and one
  • 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

  • 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
  • Undergraduate students need not complete the
    final project if they are enrolled for 3 credit
    hours. The last project is worth 1 credit hr.

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

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,
  • Atmospheric circulations, weather patterns
  • Combustion of gases, vehicle propulsion
  • Space and aeronautics
  • Chemical processing, heat transfer,
  • Etc.

(No Transcript)
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
  • 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

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,

  • Several important issues
  • Accuracy
  • Scale resolution
  • Representation of all physical processes through
    mathematical models
  • Code validation and uncertainty
  • Is it only colorful fluid dynamics ?

Governing Equations of Fluid Flow
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.)

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)

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,
  • The discrete equations for each derivative are
    derived using Taylor series expansions of the
    continuous variable

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

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

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

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

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
Evolution of Computing Power
Advances in processor speed are correlated well
with Moores 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
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
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,
Selection of GRID
Several types of grids can be used Cartesian
(x-y-z) grids Triangular / quadrilateral/tetrahed
ral 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
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
Some Examples
Natural Convection in Enclosures
Velocity vectors and temperature contours for
natural convection in a square cavity, (a) Ra
104. and (b). Ra 105
Natural Convection in Enclosures
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
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
  • Rayleigh-Taylor instabilities (Re1024)

Density contours
Buoyancy-Induced Mixing in Tilted Channel
Mixing in a Curved Duct
Mixing in a Curved Duct
Turbulent Flow in a Circular Pipe, Re (tau) 400
Instantaneous Fields in a Square Duct
Lagrangian Particle Tracking
St 5.0
St 0.3
St 0.1
Wavy Channel Flow
  • Rush et al. (1999) observed mixing and heat
    transfer characteristics of developing flow in
    serpentine and wavy channels

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

Some Good Websites
http// http//
om/Wiki/Codes http// http// Google CFD