CFD Modeling of Turbulent Flows

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CFD Modeling of Turbulent Flows

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Overview

• Properties of turbulence
• Predicting turbulent flows
• DNS
• LES
• RANS Models
• Summary

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Properties of Turbulence

• Most flows encountered in industrial processes
  are turbulent.
• Turbulent flows exhibit three-dimensional,
  unsteady, aperiodic motion.
• Turbulence increases mixing of momentum, heat and
  species.
• Turbulence mixing acts to dissipate momentum and
  the kinetic energy in the flow by viscosity
  acting to reduce velocity gradients.
• Turbulent flows contain coherent structures that
  are deterministic events.

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Role of Numerical Turbulence Modeling

• An understanding of turbulence and the ability to
  predict turbulence for any given application is
  invaluable for the engineer.
• Examples
• Increased turbulence is needed in chemical mixing
  or heat transfer when fluids with dissimilar
  properties are brought together.
• turbulence increases drag due to increased
  frictional forces.
• Historically, experimental measurement of the
  system was the only option available. This makes
  design optimization incredibly tedious.

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Characteristics of the Engineering Turbulence
Model

• Numerical modeling of turbulence can serve to
  improve the engineers ability to analyze
  turbulent flow in design particularly when
  precise measurements cannot be obtained and when
  extensive experimentation is costly and
  time-consuming.
• The ideal turbulence model should introduce
  minimal complexity while capturing the essence of
  the relevant physics.

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Modeling Turbulent Flows

• Turbulent flows can be modeled in a variety of
  ways. With increasing levels of complexity they
  are
• Correlations
• Moody chart, Nusselt number correlations
• Integral equations
• Derive ODEs from the equations of motion
• Reynolds Averaged Navier Stokes or RANS equations
• Average the equations of motion over time
• Requires closure
• Large Eddy Simulation or LES
• Solve Navier-Stokes equations for large scale
  motions of the flow. Model only the small scale
  motions
• DNS
• Navier-Stokes equations solved for all motions in
  the turbulent flow

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Turbulence Modeling Approaches
Zero-Equation Models One-Equation
Models Two-Equation Models Standard k-e
RNG k-e Second-order closure Reynolds-St
ress Model Large-Eddy Simulation Direct
Numerical Simulation
Increase Computational Cost Per Iteration
Include More Physics
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Direct Numerical Simulation (DNS)

• Currently DNS is the most exact approach to
  modeling turbulence since no averaging is done or
  approximations are made
• Since the smallest scales of turbulence are
  modeled, called the Kolmogoroff scale, the size
  of the grid must be scaled accordingly.
• A DNS simulation scales with ReL3 (uL/n) where
  ReL 0.01Re. Turbulent flow past a cylinder
  would require at least (0.01 x 20,000)3 or 8
  million cells

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Direct Numerical Simulation (DNS)

• Given the current processing speed and memory of
  the largest computers, only very modest Reynolds
  number flows with simple geometries are possible.
• Advantages DNS can be used as numerical flow
  visualization and can provide more information
  than experimental measurements DNS can be used
  to understand the mechanisms of turbulent
  production and dissipation.
• Disadvantages Requires supercomputers limited
  to simple geometries.

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Large Eddy Simulation (LES)
DNS
u
DNS
LES
LES
t

• LES is a three dimensional, time dependent and
  computationally expensive simulation, though less
  expensive than DNS.
• LES solves the large scale motions and models the
  small scale motions of the turbulent flow.
• The premise of LES is that the large scale
  motions or eddies contain the larger fraction of
  energy in the flow responsible for the transport
  of conserved properties while the small.

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Large Eddy Simulation (LES)

• The large scale components of the flow field are
  filtered from the small scale components using a
  wavelength criteria related to the size of the
  eddies
• The filter produces the following equation used
  to model the small scale motions
• where
• The inequality is then modeled as
• tij is called the subgrid scale Reynolds Stress.
  Different subgrid scale models are available to
  approximate tij .

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Large Eddy Simulation (LES)

• Mixing plane between two streams with unequal
  scalar concentrations
• Unsteady vortex motions with growing length scale

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Reynolds Averaged Navier-Stokes (RANS) Models

• DNS and LES can produce an overwhelming quantity
  of detailed information about a flow structure.
  Generally, in engineering flows, such levels on
  instantaneous information is not required.
• Typical engineering flows are focused on
  obtaining a few quantitative of the turbulent
  flow. For example, average wall shear stress,
  pressure and velocity field distribution, degree
  of mixedness in a stirred tank, etc.
• The approach would be to model turbulence by
  averaging the unsteadiness of the turbulence.
• This averaging process creates terms that cannot
  be solved analytically but must be modeled.
• This modeling approach has been around for 30
  years and is the basis of most engineering
  turbulence calculations.

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RANS Equations

• Velocity or a scalar quantity can be represented
  as the sum of the mean value and the fluctuation
  about the mean value as
• Using the above relationship for velocity(let f
  u) in the Navier-Stokes equations gives (as
  momentum equation for incompressible flows with
  body forces).
• The Reynolds Stresses cannot be represented
  uniquely in terms of mean quantities and the
  above equation is not closed. Closure involves
  modeling the Reynolds Stresses.

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Closure of RANS equations

• The RANS equations contain more unknowns than
  equations.
• The unknowns are the Reynolds Stress terms.
• Closure Models are
• zero-equation turbulence models
• Mixing length model
• no transport equation used
• one-equation turbulence models
• transport equation modeled for turbulent kinetic
  energy k
• two-equation models
• more complete by modeling transport equation for
  turbulent kinetic energy k and eddy dissipation e
• second-order closure
• Reynolds Stress Model
• does not use Bousinesq approximation as
  first-order closure models

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Modeling Turbulent Stresses in Two-Equation Models

• RANS equations require closure for Reynolds
  stresses and the effect of turbulence can be
  represented as an increased viscosity
• The turbulent viscosity is correlated with
  turbulent kinetic energy k and the dissipation
  rate of turbulent kinetic energy e

Boussinesq Hypothesis eddy viscosity model
Turbulent Viscosity
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Turbulent kinetic energy and dissipation

• Transport equations for turbulent kinetic energy
  and dissipation rate are solved so that turbulent
  viscosity can be computed for RANS equations.

Turbulent Kinetic Energy
Dissipation Rate of Turbulent Kinetic Energy
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Standard k-? Model
Turbulent Kinetic Energy
Dissipation
Convection
Generation
Diffusion
Dissipation Rate
are empirical constants
(equations written for steady, incompressible
flow w/o body forces)
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Deficiencies of the Boussinesq Approximation

• The two-equation models based on the eddy
  viscosity approximation provide excellent
  predictions for many flows of engineering
  interest.
• Applications for which the approximation is weak
  typically are flows with extra rate of strain
  (due to isotropic turbulent viscosity
  assumption). Examples of these are
• flows over boundaries with strong curvature
• flows in ducts with secondary motions
• flows with boundary layer separation
• flows in rotating and stratified fluid
• strongly three dimensional flows
• The RNG k-e model is an improvement over the
  standard k-e for these classes of flow by
  incorporating the influence of additional strains
  rates.
• A higher-order closure approximation can be also
  applied to include wider class of problems
  including those with extra rates of strain.

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RNG k-? Model
Turbulent Kinetic Energy
where
Dissipation Rate
(equations written for steady, incompressible
flow w/o body forces)
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Second-Order Closure models

• The Second-Order Closure Models include the
  effects of streamline curvature, sudden changes
  in strain rate, secondary motions, etc.
• This class of models is more complex and
  computationally intensive than the RANS models
• The Reynolds-Stress Model (RSM) is a second-order
  closure model and gives rise to 6 Reynolds stress
  equations and the dissipation equation

Reynolds Stress Transport Equation
Diffusion
Convection
Pressure-Strain redistribution
Generation
Dissipation
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Reynolds Stress Model
Generation
(computed)
Pressure-Strain Redistribution
(modeled)
Dissipation
(related to e)
Turbulent Diffusion
(modeled)
(equations written for steady, incompressible
flow w/o body forces)
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Near Wall Treatment

• The RANS turbulence models require a special
  treatment of the mean and turbulence quantities
  at wall boundaries and will not predict correct
  near-wall behavior if integrated down to the wall
• Special near-wall treatment is required
• Standard wall functions
• Nonequilibrium wall functions
• Two-layer zonal model

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Comparison RNG k-e vs. Standard k-e
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Summary

• Turbulence modeling comes in varying degrees of
  complexity. Determining the right choice of
  turbulence model depends on the detail of results
  expected.
• DNS and LES are still far from being engineering
  tools but in the near future this will be
  possible.
• Two-equation models are widely used for their
  relatively simple overhead. However, increased
  complexity of the turbulent flow reduces the
  adequacy of the models.
• Improvements to the two-equation models to
  incorporate extra strain rates, and the
  second-order closure RSM model provide the extra
  terms to model complex engineering turbulent
  flows.