Jordanian-German Winter Academy February 5th-11th 2006 Eddy

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Eddy Viscosity Model
Jordanian-German Winter Academy February 5th-11th
2006
Participant Name Eng. Tareq Salameh Mechanical
Engineering Department University of jordan
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Reynolds Averaged Navier-Stokes (RANS)

• The averaging procedure introduces additional
  unknown terms containing products of the
  fluctuating quantities, which act like additional
  stresses in the fluid. These terms, called
  turbulent or Reynolds stresses, are difficult
  to determine directly and so become further
  unknowns.

Reynolds stress
and the Reynolds flux
These terms arise from the non-linear convective
term in the un-averaged equations.
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Reynolds Averaged Navier-Stokes (RANS) (Cont.)

• The Reynolds (turbulent) stresses need to be
  modeled by additional equations of known
  quantities in order to achieve closure.
• Closure implies that there is a sufficient
  number of equations for all the unknowns,
  including the Reynolds-Stress tensor resulting
  from the averaging procedure.

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Reynolds Averaged Navier-Stokes (RANS) (Cont.)

• The equations used to close the system define
  the type of turbulence model Reynolds Averaged
  Navier Stokes (RANS) Equations.
• Type of turbulence model based on RANS
  (classical type)
• Eddy viscosity model (I)
• Reynolds stress model (II)
• Another type
• Large Eddy Simulation model
• Detached Eddy Simulation model

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Eddy viscosity model

• One proposal suggests that turbulence consists
  of small eddies which are continuously forming
  and dissipating.
• Definition of eddy viscosity model.
• The eddy viscosity hypothesis assumes that the
  Reynolds stresses can be related to the mean
  velocity gradients and Eddy (turbulent) Viscosity
  by the gradient diffusion hypothesis, in a manner
  analogous to the relationship between the stress
  and strain tensors in laminar Newtonian flow.

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Eddy viscosity model

• Here, ?t is the Eddy Viscosity or Turbulent
  Viscosity. This has to be prescribed.

Analogous to the eddy viscosity hypothesis is the
eddy diffusivity hypothesis, which states that
the Reynolds fluxes of a scalar are linearly
related to the mean scalar gradient
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Eddy viscosity model

• Here, ?t is the Eddy Diffusivity, and this has
  to be prescribed. The Eddy Diffusivity can be
  written

Where Prt is the turbulent Prandtl number. Eddy
diffusivities are then prescribed using the
turbulent Prandtl number.
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Eddy viscosity model

• The above equations can only express the
  turbulent fluctuation terms of functions of the
  mean variables if the turbulent viscosity, ?t ,
  is known. Both the k-? and k-? two-equation
  turbulence models provide this variable.
• Subject to these hypotheses, the Reynolds
  averaged momentum and scalar transport equations
  become

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Eddy viscosity model

• where B is the sum of the body forces, µeff is
  the Effective Viscosity, and Geff is the
  Effective Diffusivity, defined by,

and
is a modified pressure, defined by
where
is the bulk viscosity.
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The Zero Equation Model

• Very simple eddy viscosity models compute a
  global value for µt from the product of turbulent
  mean velocity scale, Ut, and a a turbulence
  geometric length scale , lt using an empirical
  formula, as proposed by Prandtl and Kolmogorov

Where
f?0.01
is a proportionality constant
Because no additional transport equations are
solved, these models are termed zero equation
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The Zero Equation Model (Cont.)

• The velocity scale is taken to be the maximum
  velocity in the fluid domain.
• The length scale is derived using the
  formula

where VD is the fluid domain volume
This model has little physical foundation and is
not recommended.
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Two Equation Turbulence Models

• Two-equation turbulence models (k-? and k-? )
  are very widely used and much more sophisticated
  than the zero equation models .
• Both the velocity and length scale are solved
  using separate transport equations (hence the
  term two-equation)
• The turbulence velocity scale and the
  turbulent length scale are computed from the
  turbulent kinetic energy and turbulent kinetic
  energy and its dissipation rate respectively.

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The k-? model

• k is the turbulence kinetic energy and is
  defined as the variance of the fluctuations in
  velocity.
• ? is the turbulence eddy dissipation (the rate
  at which the velocity fluctuations dissipate)
• The k-? model introduces two new variables
  into the system of equations.
• The continuity equation is then

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The k-? model (Cont.)

• and the momentum equation becomes

where
B is the sum of body forces.
µeff is the effective viscosity accounting for
turbulence.
is the modified pressure given by
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The k-? model (Cont.)

• The k-? model, like the zero equation model,
  is based on the eddy viscosity concept, so that

where µt is the turbulence viscosity.
The model k- ? assumes that the turbulence
viscosity µt is linked to the turbulence kinetic
energy and dissipation via the relation
Where C?0.09
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The k-? model (Cont.)
The values of k and ? come directly from the
differential transport equations for the
turbulence kinetic energy and turbulence
dissipation rate
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The k-? model (Cont.)

• Where Ce11.44 , Ce21.92 , ?k1.0 and ??1.3
  are constants. Pk is the turbulence production
  due to viscous and buoyancy forces, which is
  modeled using

For incompressible flow,
is small
and the second term on the right side of equation
does not contribute significantly to the
production.
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The k-? model (Cont.)

• For compressible flow,

is only large in regions with
high velocity divergence, such as at shocks.
The term 3µt in the second term is based on the
frozen stress assumption . This prevents the
values of k and ? becoming too large through
shocks.
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The k-? model (Cont.)

• there are applications for which these models
  may not be suitable

Flows with boundary layer separation. Flows
with sudden changes in the mean strain
rate. Flows in rotating fluids. Flows over
curved surfaces.
A Reynolds Stress model may be more appropriate
for flows with sudden changes in strain rate or
rotating flows, while the SST model may be more
appropriate for separated flows.
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The k-? model (Cont.)
Turbulent Recirculation Flow
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Buoyancy Turbulence

• If the full buoyancy model is being used, the
  buoyancy production term Pkb is modeled as

and if the Boussinesq buoyancy model is being
used, it is
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The RNG k-? Model

• The RNG k-? model is based on renormalisation
  group analysis of the Navier-Stokes equations.

• The transport equations for turbulence generation
  and dissipation are the same as those for the
  standard k-? model, but the model constants
  differ, and the constant C?1 is replaced by the
  function C?1RNG.

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The RNG k-? Model (Cont.)

• The transport equation for turbulence
  dissipation becomes

Where,
Where ??RNG0.7179, C?2RNG1.68,
?RNG0.012, C?RNG0.085

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The k-? Model

• One of the advantages of the k-? formulation
  is the near wall treatment for low-Reynolds
  number computations.

The model does not involve the complex non-linear
damping functions required for the k-? model and
is therefore more accurate and more robust.
The k-? models assumes that the turbulence
viscosity is linked to the turbulence kinetic
energy and turbulent frequency via the relation
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The Wilcox k-? Model

• The starting point of the present formulation
  is the k-? model developed by Wilcox1.

It solves two transport equations, one for the
turbulent kinetic energy, k, and one for the
turbulent frequency,?.
The stress tensor is computed from the
eddy-viscosity concept.
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The Wilcox k-? Model (Cont.)

• k-equation

?-equation
In addition to the independent variables, the
density, ?, and the velocity vector, U, are
treated as known quantities from the
Navier-Stokes method. Pk is the production rate
of turbulence, which is calculated as in the k-?
model
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The Wilcox k-? Model (Cont.)

• The model constants are given by

The unknown Reynolds stress tensor,?, is
calculated from
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The Wilcox k-? Model (Cont.)

• In order to avoid the build-up of turbulent
  kinetic energy in stagnation regions, a limiter
  to the production term is introduced into the
  equations according to Menter 2 

With clim 10 for ? based models. This limiter
does not affect the shear layer performance of
the model, but has consistently avoided the
stagnation point build-up in aerodynamic
simulations.
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The Baseline (BSL) k-? Model

• The main problem with the Wilcox model is its
  well known strong sensitivity to freestream
  conditions Menter 3.

Depending on the value specified for ? at the
inlet, a significant variation in the results of
the model can be obtained.
This is undesirable and in order to solve the
problem, a blending between the k-? model near
the surface and the k-? model in the outer region
was developed by Menter 2.
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The Baseline (BSL) k-? Model (Cont.)

• It consists of a transformation of the k-?
  model to a k-? formulation and a subsequent
  addition of the corresponding equations.

The Wilcox model is thereby multiplied by a
blending function F1 and the transformed k-?
model by a function 1-F1. F1 is equal to one
near the surface and switches over to zero inside
the boundary layer. At the boundary layer edge
and outside the boundary layer, the standard k-?
model is therefore recovered.
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The Baseline (BSL) k-? Model (Cont.)

• Wilcox model

Transformed k-? model
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The Baseline (BSL) k-? Model (Cont.)

• Now the equations of the Wilcox model are
  multiplied by function F1, the transformed k-?
  equations by a function 1-F1 and the
  corresponding k- and ?- equations are added to
  give the BSL model

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The Baseline (BSL) k-? Model (Cont.)

• The coefficients of the new model are a
  linear combination of the corresponding
  coefficients of the underlying models

coefficients are listed again for completeness
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The Shear Stress Transport (SST) k-? Based Model

• The k-? based SST model accounts for the
  transport of the turbulent shear stress and gives
  highly accurate predictions of the onset and the
  amount of flow separation under adverse pressure
  gradients.

The BSL model combines the advantages of the
Wilcox and the k-e model, but still fails to
properly predict the onset and amount of flow
separation from smooth surfaces
The main reason is that both models do not
account for the transport of the turbulent shear
stress.
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The Shear Stress Transport (SST) k-? Based Model
(Cont.)

• This results in an over prediction of the
  eddy-viscosity.

The proper transport behavior can be obtained by
a limiter to the formulation of the
eddy-viscosity
Where
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The Shear Stress Transport (SST) k-? Based Model
(Cont.)

• Again F2 is a blending function similar to
  F1, which restricts the limiter to the wall
  boundary layer, as the underlying assumptions are
  not correct for free shear flows. S is an
  invariant measure of the strain rate.

The blending functions are critical to the
success of the method.
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The Shear Stress Transport (SST) k-? Based Model
(Cont.)

• Their formulation is based on the distance to
  the nearest surface and on the flow variables.

With
Where y is the distance to the nearest wall and ?
is the kinematic viscosity and
With
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The (k-?)1E Eddy Viscosity Transport Model

• A very simple one-equation model has been
  developed by Menter 4 5.

It is derived directly from the k-e model and is
therefore named the (k-?)1E model
Where
is the kinematic eddy viscosity,
is the
turbulent kinematic eddy viscosity and s is a
model constant.
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The (k-?)1E Eddy Viscosity Transport Model (Cont.)

• The model contains a destruction term, which
  accounts for the structure of turbulence and is
  based on the von Karman length scale

Where S is the shear strain rate tensor. The eddy
viscosity is computed from
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The (k-?)1E Eddy Viscosity Transport Model (Cont.)

• In order to prevent the a singularity of the
  formulation as the von Karman length scale goes
  to zero, the destruction term is reformulated as
  follows

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The (k-?)1E Eddy Viscosity Transport Model (Cont.)

• The coefficients are

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The (k-?)1E Eddy Viscosity Transport Model (Cont.)

• Low Reynolds Number Formulation

Low Reynolds formulation of the model is obtained
by including damping functions. Near wall damping
functions have been developed to allow
integration to the surface
Where D2 is required to compute the
eddy-viscosity which goes into the momentum
equations
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Development of maximum deficit velocity in
axisymmetric wake
mixing length model
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Reference

• 1 Wilcox, D.C..Multiscale model for turbulent
  flows.In AIAA 24th Aerospace Sciences Meeting.
  American Institute of Aeronautics and
  Astronautics, 1986.
• 2 Menter, F.R..Two-equation eddy-viscosity
  turbulence models for engineering
  applications.AIAA-Journal., 32(8), 1994.
• 3Menter, F.R., Multiscale model for turbulent
  flows.In 24th Fluid Dynamics Conference. American
  Institute of Aeronautics and Astronautics, 1993.
• 4 Menter, F. R.,Eddy Viscosity Transport
  Equations and their Relation to the k-e
  Model.NASA Technical Memorandum 108854, November
  1994.
• 5 Menter, F. R.,Eddy Viscosity Transport
  Equations and their Relation to the k-e
  Model.ASME J. Fluids Engineering, vol. 119, pp.
  876-884, 1997.

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Thanks For YourListening
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