PowerPointPrsentation

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Title of Presentation
Physics of Turbulence and Turbulence Modeling
by Wangda Zuo M.Sc. Student of Computational
Engineering Lehrstuhl für Strömungsmechanik FAU
Erlangen-NürnbergCauerstr. 4, D-91058 Erlangen

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Contents

• What is turbulence?
• Typical numerical methods to predict turbulence
• RANS(Reynolds Averaged Navier-Stokes Equations)
• Reynolds decompositions
• Derivation of Reynolds Averaged equations
• Modeling
• Eddy viscosity model
• Zero equation model
• One equation model
• Two equation model
• Outlook Conclusion

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Laminar, Transitional and Turbulent Flows
Laminar Flow
Transitional Flow
Turbulent Flow
It seemed, however, to be certain, if the eddies
were due to one particular cause, that
integration would show the birth of eddies to
depend on some definite of DU/v. Osborne
Reynolds in 1883
Reynolds Number
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Definition and Motivation

• Definition
• Turbulent fluid motion is an irregular condition
  of flow in which the various quantities show a
  random variation with time and space coordinates,
  so that statistically distinct average values can
  be discerned. (Hinze)

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Properties of Turbulence

• Turbulence Properties
• Irregular, random, chaotic, large spectrum of
  scales, high Re, 3D, dissipative.

Internal Energy
Kinetic Energy
Dissipation
Turbulence Energy Transfer Cascade Process
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Contents

• What is turbulence?
• Typical numerical methods to predict turbulence
• RANS(Reynolds Averaged Navier-Stokes Equations)
• Reynolds decompositions
• Derivation of Reynolds Averaged equations
• Modeling
• Eddy viscosity model
• Zero equation model
• One equation model
• Two equation model
• Outlook Conclusion

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Problems to Predict Turbulent Flow Directly
Example Turbulent Channel Flow ( ReUh?/µ 106 )
Air
Directly solve the N-S Equation
U65km/h
H1m
500 Floating point operations per grid point and
time step
2000 years
Total calculation time for simulation
Direct prediction for high Re is impossible!
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Typical Numerical Methods to


Solve Turbulence I

• DNS Direct Numerical Simulation
• Solves the unsteady Navier-Stokes equations
  directly.
• RANS Model (Reynolds Averaged Navier-Stokes
  Model)
• Motivation Engineers are normally interested in
  knowing just a few quantitative properties of a
  turbulent flow.
• Method Using Reynolds-Averaged form of the
  Navier-Stokes equations with appropriate
  turbulence models.
• LES Large Eddy Simulation
• MotivationThe large scale motions are generally
  much more energetic than the small scales, and
  they are the most effective transporters of the
  conserved properties.
• Method Large Scales -gt Solve, Small Scales
  -gtModel

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Typical Numerical Methods to Solve Turbulence II
Methods
Fields
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Contents

• What is turbulence?
• Typical numerical methods to predict turbulence
• RANS(Reynolds Averaged Navier-Stokes Equations)
• Reynolds decompositions
• Derivation of Reynolds Averaged equations
• Modeling
• Eddy viscosity model
• Zero equation model
• One equation model
• Two equation model
• Outlook Conclusion

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Reynolds Decomposition I

• Time Averaging
• In the statistical approach to turbulent flow,
  every instantaneous variable can be written as

Fluctuation
Average Value
where
U
t
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Reynolds Decomposition II

• Properties of Reynolds Decomposition

Average
Addition
Multiplication
Derivation
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Contents

• What is turbulence?
• Typical numerical methods to predict turbulence
• RANS(Reynolds Averaged Navier-Stokes Equations)
• Reynolds decompositions
• Derivation of Reynolds Averaged equations
• Modeling
• Eddy viscosity model
• Zero equation model
• One equation model
• Two equation model
• Outlook Conclusion

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Navier-Stokes Equations I
In previous presentation, Navier-Stokes equations
for incompressible fluid ( ?const ) were
derived.
Continuity Equation
Momentum Equation
Energy Equation
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Navier-Stokes Equations II

• Momentum Equation for Steady Incompressible
  Newtonian (µconst) Fluid

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Derivation of Reynolds Equations I

• Reynolds Decomposition of Variables

• Average of Continuity Equation

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Derivation of Reynolds Equations II

• Average of Momentum Equations

expand and average in time
Reynolds Shear Stress
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Derivation of Reynolds Equations III

• Reynolds Shear Stresses

symmetrical tensors Introduce 6 new unknowns
the system of equations is not closed!
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Modeling


Eddy Viscosity Model Reynolds Stress Model PDF
Model (probability density function)
accuracy
complexity
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Contents

• What is turbulence?
• Typical numerical methods to predict turbulence
• RANS(Reynolds Averaged Navier-Stokes Equations)
• Reynolds decompositions
• Derivation of Reynolds Averaged equations
• Modeling
• Eddy viscosity model
• Zero equation model
• One equation model
• Two equation model
• Outlook Conclusion

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Eddy Viscosity Model I

• Boussinesq Approximation

Molecular Momentum Transport
v Viscosity (fluid)
Turbulent Momentum Transport
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Eddy Viscosity Model II

• Eddy Viscosity

The eddy viscosity models are not closed unless
uc and lc are specified!
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Eddy Viscosity Model III
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Contents

• What is turbulence?
• Typical numerical methods to predict turbulence
• RANS(Reynolds Averaged Navier-Stokes Equations)
• Reynolds decompositions
• Derivation of Reynolds Averaged equations
• Modeling
• Eddy viscosity model
• Zero equation model
• One equation model
• Two equation model
• Outlook Conclusion

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Zero Equation Model I

• Zero Equation Model
• uc and lc are specified algebraically using
  empirical information from experiments and past
  experience.

The eddy viscosity models are closed!
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Zero Equation Model II

• Mixing length Model in Wall Boundary Layers

where
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Contents

• What is turbulence?
• Typical numerical methods to predict turbulence
• RANS(Reynolds Averaged Navier-Stokes Equations)
• Reynolds decompositions
• Derivation of Reynolds Averaged equations
• Modeling
• Eddy viscosity model
• Zero equation model
• One equation model
• Two equation model
• Outlook Conclusion

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One Equation Model I

• One Equation Model

lcl0 algebraically using experienced
information
Then the eddy viscosity can be written as
Model assumptions are made for k because
transport equation contains new unknowns
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One Equation Model II

• k-Equation (Engery Equation)

derived from basic equations
Convection
Production
Dissipation
Diffusion
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Contents

• What is turbulence?
• Typical numerical methods to predict turbulence
• RANS(Reynolds Averaged Navier-Stokes Equations)
• Reynolds decompositions
• Derivation of Reynolds Averaged equations
• Modeling
• Eddy viscosity model
• Zero equation model
• One equation model
• Two equation model
• Outlook Conclusion

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Two Equation Model I

• Two Equation Model

lc is also solved by transport equation

• k- e Model

Model assumptions are made for e because
e-equation contains new unknowns
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Two Equation Model II

• e Equation (Dissipation Equation)

Production due to the mean flow velocity
gradient Production due to the deformation of
vortices (vortex stretching) Production due to
the mixed effects of the gradients of mean and
fluctuating velocities Production due to mean
velocity gradient Diffusive transport due to
turbulent fluctuations Diffusive transport dur
to the turbulent pressure fluctuations Viscous
destructions Viscous diffusion
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Two Equation Model III

• Standard k-e-Model

k Equation
Convection
Production
Dissipation
Diffusion
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Example of k-e model

• Turbulent Recirculating Flow

b) velocity profile
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Comparison of Models I

• Comparison of lc, uc and vT

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Comparison of Models II

• Example Development of maximum deficit velocity
  in axisymmetric wake

mixing length model
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Contents

• What is turbulence?
• Typical numerical methods to predict turbulence
• RANS(Reynolds Averaged Navier-Stokes Equations)
• Reynolds decompositions
• Derivation of Reynolds Averaged equations
• Modeling
• Eddy viscosity model
• Zero equation model
• One equation model
• Two equation model
• Outlook Conclusion

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Outlook I

• k-e model is standard today.
• k-e model is only valid for isotropic turbulence
• In k-e model it is assumed that turbulence is
  isotropic.
• Isotropic the three fluctuating quantities are
  of the same size.

• In reality, turbulence is not always isotropic!

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Outlook II

• Anisotropy Invariant Map
• All turbulences have to lie within the map.

1-component Isotropic
II aijaji
Two-components
DNS Data
2-components Isotropic
k-e model
Isotropic
Future turbulence model should take anisotropy
of turbulence into account.
III ajkakjaij
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Conclusion

• Definition and properties of turbulence
• Introduction and comparison of DNS, RANS and LES
• RANS Reynolds decompositions and derivation of
  Reynolds-Averaged equations
• Eddy viscosity model Zero/One/Two equation
  model
• Outlook Conclusion

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Thank you for your attention!