Computational Fluid Dynamics 5 Lecture 2

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Computational Fluid Dynamics 5Lecture 2

• Professor William J Easson
• School of Engineering and Electronics
• The University of Edinburgh

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Last weeks examples

• Create new working directory
• Create a simple geometry in GAMBIT and mesh
• Solve for laminar flow in the channel
• Present the output in a variety of formats
• Model 1 is incompressible, laminar flow through a
  channel
• Reynolds numbers must be ltlt Recrit Velocities ltlt
  speed of sound if gas
• low velocity and/or channel width
• YOU must calculate appropriate numbers
• Garbage in garbage out

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Last weeks examples (cont)

• The main objective of the exercise with the flow
  between planes is to familiarise you with the
  software
• Further numerical experiments before next week
• 3D Laminar flow through a circular pipe
• How does the point of fully developed flow vary
  with velocity?
• 2D Laminar jet into chamber
• What is the rate of expansion of the jet?
• Attempt some of the GAMBIT tutorials

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Laminar Jet

• How fast does the jet spread?
• How large should the domain be?
• Is a special grid required?

5
Discretising equations

• What are we solving?

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Components of the N-S equations

• Need to know
• values of each variable (eg u) at each point
• values of the first derivative
• values of cross-derivatives
• values of second derivatives
• ..and more

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Forward approximation to value of the 1st
derivative of u in space
u
x
i-1
i1
i
dx
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Rearward approximation to value of the 1st
derivative of u in space
u
x
i-1
i1
i
dx
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Central approximation to value of the 1st
derivative of u in space
u
x
i-1
i1
i
dx
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Approximations to values of the 1st derivative of
u in space
u
forward
rearward
central
x
i-1
i1
i
dx
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1st 2nd Order Finite Difference
1st order forward difference
1st order rearward difference
2nd order central difference
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Discretising equations(Anderson)
The value of the variable, u, at the grid point
i1,j can be approximated by a Taylor expansion
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1st 2nd Order Finite Difference
From the previous equation, we can find
expressions for the derivatives
1st order forward difference
2nd order central difference
2nd order central difference
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Practical consequences of discretisation

• Errors arise from spacing of grid needs to be
  small enough to represent the key aspects of the
  flow
• Errors arise from the order of the equations
• 1st order should generally not be used
• Only 2nd order solutions are acceptable for
  journal publication

15
Testing solution

• Start with a coarse grid
• Solve the problem
• Double the grid density
• Compare with the first solution
• If the values have not changed significantly, it
  is likely that the solution is grid-independent
• If the values have changed significantly,
  continue until they stop changing

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Week 2 - example

• Flow over a backward-facing step
• Flow expands and leaves a recirculating vortex
  behind the step
• Solve to 2nd order and maintain laminar flow
• How long does the domain have to be to ensure
  that the solution is valid
• Upstream?
• Downstream?