3. Learning Objectives
• Make the student understand the role of C in FD, its applicability,
potential and limitations
• Give a basic foundation in numerical analysis, by teaching the relevance
of accuracy and stability
• Give a working idea of the various choices of numerical methods and
discretization schemes by applying them to simple model equations. In
doing this, always remind them of the connection with the big picture.
• Make the student knowledgeable about the various terminologies in
practical CFD (Grids, BCs, Approximations, Schemes etc)
• Ingrain the basics of good CFD practice (be aware of the
applicability/feasibility of a particular model, its limitations, choose the
right boundary conditions, ascertain grid/time independence,
verification/validation)
• By the end of the class, the student should be in a position to set up
simple aerodynamic problems and analyze them
4. Contents
• Introduction (1.5)
• Classification of PDE, Model equations (1.5)
• Finite difference methods:
Spatial discretization (2.5)
Temporal discretization (1.5)
Convergence, Consistency, Stability (1)
• Grids/Boundary conditions (1)
• Euler equations (0.5)
• DNS/LES (1)
• RANS Equations and Turbulence modeling (1)
• Case studies & Best practices in CFD (1.5)
• Hands-on CFD/Lab sessions (8)
(.) – Approximate number of lectures
6. What is CFD/FD ?
• CFD is a branch of Fluid dynamics
• So what really is Engineering Fluid Dynamics in the first place? Lets
look at some examples:
We are interested in the forces (pressure , viscous stress
etc.) acting on surfaces (Example: In an airplane, we are interested
in the lift, drag, power, pressure distribution etc)
We would like to determine the velocity field (Example: In a
race car, we are interested in the local flow streamlines, so that we
can design for less drag)
We are interested in knowing the temperature distribution
(Example: Heat transfer in the vicinity of a computer chip)
• Roughly put, in Engineering fluid dynamics, we would like to
determine certain flow properties in a certain region of interest, so
that the information can be used to predict the behaviour of
systems, to design more efficient systems etc..
7. Fluid Dynamics• Theoretical
Most important branch of fluid dynamics. Crucial in
understanding concepts (Example: L = ρUΓ), Usually good in
predicting trends (Example: δ ~ Re-1/2
)
Can obtain a lot of information using simplifying
assumptions, sometimes enough for detailed design (Example: the
SR-71 Blackbird was designed completely using theoretical ideas)
However, doesn’t always provide sufficient information
• Experimental
Only way to obtain reliable data in many situations.
However, costly, difficult to achieve exact conditions, difficult to
isolate effects, sometimes difficult to assess error, sometimes not
repeatable
• Computational (CFD)
Becoming important as computers are getting faster and
cheaper. Potential to provide tremendous amount of data at a
fraction of the cost of experiments. But sometimes unreliable
because of numerical/modeling/human errors. Sometimes more
expensive than experiments
Very important to validate with theory/experiments
8. Words of wisdom
(To be taken with a huge helping of salt :)
• Theoretical Fluid dynamics: Most important. Everyone HAS to learn
it.
• Experimental Fluid dynamics: Important. Usually, everyone believes
it except the person that conducted the experiment.
• Computational Fluid dynamics: Also important. Usually, no one
believes it except the person that performed the calculations.
• A good engineer understands the pro’s and con’s of all three
methods, and should be in a position to assess which one is best
under the circumstances
• More importantly, should not be prejudiced against any of the three
approaches
13. Sample Application -3
[Simulation to aid theoretical understanding]
Merger of co-rotating
vortices due to
Elliptical instability
(Movie)
Courtesy: CERFACS
14. Procedures in CFD
• Identification of right approximation (Viscous/Inviscid,
Laminar/Turbulent, Incompressible / compressible, Single-
phase/multi-phase)
• Identification of right solution method (Finite Element /
Difference/Volume, Structured/Unstructured mesh, Order of accuracy)
• Pre-processing (Generate computational grid, assign boundary
conditions, set initial conditions, compile code, prepare input
parameters)
• Solution (Run the code, monitor the solution)
• Post-processing (Collect and organize data, analyze results)
• Verification (Do the results make sense? Are the trends right? Does it
agree with previous calculations on similar configurations?)
• Validation (Does the result (or an aspect of the result)) agree with
theory/experiment?)
• At every step, good understanding of theoretical fluid dynamics is
essential!!!
15. Example: Flow over a pitching
airfoil
• Problem: Predict the loads acting on an airfoil pitching in a wind
tunnel under the following conditions: α =10o
+ 10o
sin(w t), Re =
3.8x106
, M = 0.3, w = 0.06
• Identification of right approximation : Viscous, Turbulent,
compressible, Single-phase
• Identification of right solution method (Finite Volume, Structured
mesh, second order accurate)
18. Example: Flow over a pitching
airfoil
• Post processing: Flow visualization (movie)
19. Example: Flow over a pitching
airfoil
• Post processing: Loads comparison
20. Governing Equations of fluid
dynamics
• Assumptions: Continuum flow, Newtonian fluid
• Lets restrict ourselves to single phase, single species, perfect gases
(this way, incompressible flow is a special case)
• Ignore body forces
• Unknowns: Density (ρ), Velocity (u,v,w), Pressure (p)
• Dynamics of fluids is then given by
Conservation of Mass (Continuity equation) [Law of
common sense]
Conservation of Momentum (Navier-Stokes equations)
[Newton’s second law]
Conservation of Energy (Energy equation) [First law of
thermodynamics]
• 5 equations to determine 5 unknowns.
• All of fluid dynamics is contained in these equations
21. Governing equations
• How to derive these equations?
Integral form
Differential form
• Reynolds transport theorem:
Rate of change of “stuff” inside a control volume = Net flux of “stuff”
entering/leaving the boundaries + generation of “stuff” – destruction
of “stuff”
• In addition, need some more info (such as stress-strain relation,
temperature-heat flux relation etc.)
The “stuff” U is nothing but mass,
momentum and energy
