Modelling turbulent flow 1

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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 nodes
• 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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Reynolds stresses
Object to include the random fluctuations in the
Navier-Stokes equations for the mean flow.
Method represent all quantities by the mean plus
fluctuation.
and so on
(T and ? must also be considered for compressible
flow)
Putting these into the Navier-Stokes equations
and separating out the time averaged and variable
terms leads to a modified set of equations
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Reynolds stresses-continuity
Continuity - what goes in must come out!
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Reynolds stresses - Navier Stokes
Similarly, the N-S equations become (Schlichting,
Ch 18)
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Reynolds stresses

• Compared to the laminar Navier-Stokes equation,
  one new term has been added. The other terms
  have been averaged to remove the time dependency.
• The terms on the left are the forcing terms,
  gravity, pressure, viscosity and turbulence
• The terms on the right are the response terms

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Reynolds stresses in 2D

• No z,w terms
• Steady, turbulent flow in x direction
• Ignore gravity

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Reynolds stresses in 2D
In 2D, the turbulent N-S equation therefore
reduces to
Note that there are now two shear stress terms.
Re-writing
In turbulent flow, therefore the shear stress is
given by
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Reynolds and Boussinesq
Boussinesq proposed an additive turbulent shear
stress
So the additive term is equivalent to the
Reynolds stress.
However, we need to know values for
in order to use this
Are the Reynolds stresses related to the flow
velocity?
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Prandtls Mixing Length

• Analogous to the kinetic theory of gases
• Used because it works

Suppose lumps of fluid move randomly from one
shear layer to another, a distance l apart. This
carries momentum and the velocity difference must
therefore be related to the turbulence
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Prandtls Mixing Length
Turbulence is even in all directions (homogeneous)
So the Reynolds shear stress must be proportional
to the square of the mixing length times the
velocity gradient
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Prandtls Mixing Length
Returning to the equation for the shear stress
This gives a direct relationship between
turbulent viscosity and velocity gradient in
the flow
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Prantdls Mixing Length

• We still need a value for the mixing length, l.
• In free turbulence, l will be constant.
• In wall generated turbulence, l will vary as the
  distance from the wall. (lay)
• For a smooth wall y0, l0
• For a rough wall y0, lk (the surface roughness)

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The Universal Law of The Wall
First define the friction velocity, V, which is
characteristic of the fluctuating flow
Assuming that the shear stress remains constant
throughout, then V const (typically V4 u)
Using the relation from above, lay, gives the
differential equation
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The Universal Law of The Wall
Integrating gives
This is a dimensional logarithm. To make it
dimensionless, bring in the properties of the
fluid, the viscosity, ?, the density, ?, and the
new pseudo-property V. Then noting that the
kinematic viscosity ? ?/ ?
To find the values of A and B, experiments must
be performed.
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Law of the Wall
5.5
Turbulent layer
Laminar sub-layer
Buffer zone
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Mixing length measurement in pipes
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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.