Computational Fluid Dynamics 5 Turbulence

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Computational Fluid Dynamics 5Turbulence

• 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 GAMBIT
• Produce meshes of different densities and of
  varying density
• Solve for laminar flow in a 2D channel
• Solve for laminar flow in a 3D pipe
• 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
• Test the appropriateness of your mesh density
• Test the appropriateness of the extent of your
  domain

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Modelling turbulence in CFD

• Versteeg Malalasekera Ch3

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Turbulent flow
Laminar Flow
Turbulent Flow
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Measuring Turbulence
Small scale
Large scale
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Turbulent spectrum
The scale of the turbulence is set by the size of
the bodies in the flow and by the viscosity of
the fluid
There is turbulent energy at frequencies up to
105 Hz. In a 1m/s flow this is equivalent to
eddies passing which are 10µm in size
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Turbulent PIV Vectors (measured)
Turbulent pipe flow Re 5300 10085 vectors
Hairpin vortex
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Law of the Wall
The slope gives a0.4, and B5.5, so
This is the law of the wall for smooth pipes.
Note that it does not apply to rough pipes or at
a distance from the wall.
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Modelling turbulent flow (1)

• Why not solve the Navier-Stokes equations?
• No analytical solution possible
• In a computer, every small whirl would need to be
  modelled. Even a 10cm3 volume would require
  100,000,000 cells
• Need to simplify
• Crossing of streamlines transfers momentum
  between parts of the flow

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Modelling turbulent flow (2)

• Apparent shear stress - Boussinesq(1877)
• Turbulence provides a shear in the flow in
  addition to viscous shear
• Even in low viscosity fluids, there will be a
  shear
• Propose an apparent viscosity
• In general ?Tgt? , so ordinary viscosity can be
  neglected

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Time average the N-S equations continuity (what
goes in must come out)
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Time average the N-S equations momentum
Similarly, the N-S equations become (Schlichting,
Ch 18)
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Closure

• N-S equations
• 4 unknowns (p, u, v, w)
• 4 equations (3 momentum continuity)
• Time-averaged
• 10 unknowns (3 direct stresses, 3 shear stresses)
• Need more equations

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The k-? model

• Most widely used
• k is turbulent kinetic energy
• ? is dissipation rate of k
• Relies on the Boussinesq approximation
• Turbulent viscosity is calculated from

Where Cµ is a dimensionless constant
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The k-? model

• k, ? are scalar quantities with own transport
  equations
• k is produced by shear in the flow
• equations for k and ? contain 4 further
  constants
• ?k ?? are the Prandtl numbers which relate the
  diffusion of k, ? to µT.
• C1? and C2? govern the rate of production and
  destruction of ?

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Law of the Wall
5.5
Turbulent layer
Laminar sub-layer
Buffer zone
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Gridding

• k-e model assumes isotropic turbulence
• Assumes log law at the wall
• Must not have grid points in laminar sub-layer
• Must have enough grid points to model boundary
  layer
• Min grid point gt 30y

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Other models

• k-?
• modification of k-e for low Re
• RSM (Reynolds stress model)
• each component of the direct and shear stresses
  is modelled indepedently
• 7 additional equations
• LES (Large eddy simulation )
• only the small-scale turbulence is modelled
• large scale turbulence is simulated directly
• DNS (Direct Numerical Simulation)
• exact solution of N-S
• no turbulence model

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

2H
x1
x2
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Examples

• 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