Governing Equations for Incompressible Turbulent Flow

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Governing Equations for Incompressible Turbulent
Flow

• RANS equations,
• Reynolds Transport Equations,

Needs to be modeled
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• Equation for Triple Velocity correlation can be
  obtained by multiplying equation for by
  and multiply equation for by
  then add and average to get,

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Transport Equation for Triple Velocity
Correlations,
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• Models
• Davidov (Dav), 1959. Differential model
• Approximate Dijk , for distributions not far from
  normal.

Neglect viscous terms, terms proportional to
?. Approximate
Final Model
2. Daly and Harlow (DL), 1970. Algebraic
model. Assume a gradient diffusion form, Final
Model
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3. Mellor and Herring (MH), 1971. Algebraic
model. Assume a gradient diffusion form, Final
Model
, where l is a length scale.
4. Hanjalic and Launder (HL), 1971. Algebraic
model. Neglected convection and viscous terms as
well as terms proportional to mean flow
gradients. Applied same approximations as Davidov
for Dijk and pressure correlations. Final Model
5. Lumely, 1978.
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6. Cormack, 1978. Algebraic model. Express Cijk
in terms of tke and stress anisotropy gradients.
Where
7. Nagano and Tagawa (NT), 1990. Structural
model for triple velocity correlations. Needs a
model for all other correlations are
obtained from algebraic equations.
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8. Magnaudet (MG), 1993. Algebraic model This
model expresses Cijk in terms of Reynolds
stresses gradients and dissipation
gradients. Final Model
9. Younis, Gatski and Speziale, 1999.
10. Other models There are other structural and
differential models that are less popular than
the above ones.
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• Notes
• All algebraic models neglect convection of Cijk.
• All models neglect terms with mean velocity
  gradients.
• All models neglect viscous terms, which might be
  very important close to the wall.
• Ideas
• Try to incorporate mean flow gradients and
  viscous terms in a consistent way.
• Try to validate the assumption DCijk/Dt0.