Computational Fluid Dynamics 5 Gridding

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Computational Fluid Dynamics 5Gridding
time-dependance

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

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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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Things you can do

• Simulate steady, turbulent flow
• Simulate flow past objects in a domain
• Calculate the drag coefficient using the sum of
  forces on an object in a flow
• Determine whether flow solution is dominated by
  hyperbolic, parabolic or elliptic behaviour
• Utilise time-dependant equations to enhance
  convergence for elliptically-dominated solutions

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Moody Diagram (Chart)
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Gridding

• Anderson and Versteek Malalasekera both weak on
  gridding

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Structured grids (eg hex)

• Advantages
• Equations relatively simple
• Cell shape is easily controlled
• Numerical errors are smaller
• Can be fitted to flow direction
• Can be fitted to gradients in flow
• Optimises memory use
• Disadvantages
• Fitting to complex geometries requires a lot of
  time and skill

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Structured grid
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Structured grid (2)
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Un-structured grids (eg tet)

• Advantages
• Can be fitted to unusual shapes
• One-click creation
• Disadvantages
• Generally larger errors
• Limited control over cell shape skewness

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Unstructured grid
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Exercise 1

• Simulate laminar flow through a straight pipe 8mm
  dia and 50mm long. Increase the cell density next
  to the wall
• Using a structured grid by creating a prism layer
• Using a triangular grid with a growth function
• Note the number of cells created in each case and
  the relative convergence time

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Time-dependant flows

• Anderson Ch4
• Versteeg Malalasekera Ch8

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Time dependant modelling

• Many flows are inherently time-dependant and
  therefore require to be modelled as such
• Some steady-state flow solutions are not
  well-behaved and therefore require to be modelled
  by time-dependent equations to achieve convergence

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Classification of NS

• General NS equations are of mixed class

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Explicit techniques

• Solution at time-step t?t is obtained by
  marching forward from time-step t and obtaining
  gradients for estimating new values from those at
  previous time-step
• Easy to program and quick to solve, but can be
  unstable if ?t too large (conditionally stable)
• ?tmax is proportional to (?x)2, so quickly
  becomes very small and limits usefulness of
  solution

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Implicit techniques

• Solution at time-step t?t is obtained by
  marching forward from time-step t and obtaining
  gradients for estimating new values from those at
  previous and current time-step
• Crank-Nicolson is an early example
• Requires solution of large number of simultaneous
  equations (also extra loops inside iteration
  process)
• Conditionally, or even unconditionally, stable
• Large ?t results in errors not instability
• May be quicker to converge than explicit due to
  improved stability

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Time-dependant example

• Vortex shedding past a cylinder
• 2D example
• Use laminar flow, 16mm cylinder at 7.5mm/s
  (Re120 )
• Create unstructured grid using sizing function
• Use an unsteady solver
• Calculate the shedding frequency and check that
  it matches the strouhal number

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