Computational Fluid Dynamics 5 Solution Behaviour

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Computational Fluid Dynamics 5Solution Behaviour

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

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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?
• Hint Try x12H, 5H, 10H x25H, 10H, 15H

2H
x1
H
x2
Re 500, ? 10-6, H25mm, u0.01ms-1
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Vectors shear stress x1x25H
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x110H, x210H 15H
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Shear Stressx110H, x210H 15H
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Shear stress for x210H,15H
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Things you can do

• Create simple geometries in Star-Design
• Produce meshes of different densities and of
  varying density (by changing the parameters
  before meshing)
• Solve for laminar flow in a 2D channel
• Present the output in a variety of formats
• Solve for 2D laminar jets
• Solve for 2D flows with wall attachment
• Solve to 1st 2nd order simulations (check this)
• Test the appropriateness of your mesh density
  (check)
• Test the appropriateness of the extent of your
  domain

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Solution behaviour

• Anderson Ch3
• Versteeg Malalasekera Ch2

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Types of equation

• Parabolas, Hyperbolas and Ellipses - reminder
• Same class of curves
• Can be cut from a cone
• General equation
• Used in definition of equation types

Hyperbola
Parabola
Ellipse
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General quasi-linear PDE(not the NS equations)
Define a vector equation from the above
simultaneous equations
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Equation types
Define
The solution of this set of equations has been
reduced to a simple matrix equation that has a
key matrix N.
If the eigenvalues of N are real, the equation
is hyperbolic
complex elliptic zero parabolic
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Why does this matter?

• Each equation type has a different mathematical
  behaviour
• Mathematical behaviour is related to physical
  behaviour
• Physical behaviour should be taken into account
  when posing the problem
• Hyperbolic and parabolic equations lend
  themselves to marching solutions and are
  generally more stable to solve
• Marching can be in space or in time
• Elliptic equations must be solved everywhere at
  once and are generally more difficult to solve

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Classification of NS

• General NS equations are of mixed class

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Example 3D incompressible flow through a
constriction at Re1000
steady elliptic unsteady - parabolic
Not well-posed
well-posed
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Well-posed problem

• If there exists a unique solution which depends
  continuously on the boundary conditions, the
  problem is well-posed
• In the above example the steady-state problem was
  not well-posed, so an unsteady simulation was
  performed which was well-posed

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Examples for this week

• Flow through a 3D pipe at Re 107, 106 105 104
  103
• Can you deduce the friction factors?
• What is the effect of increasing surface
  roughness at 107?
• Force on a cylinder in a steady turbulent flow
  (can be done in 2D)
• What is the drag coefficient?
• Consider the grid design and domain carefully
• Allow the walls of your virtual water/wind tunnel
  to slip