P M V Subbarao

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Modeling of Turbulent Flows

• P M V Subbarao
• Professor
• Mechanical Engineering Department
• I I T Delhi

Development of Deterministic Methods to Solve
Stochastic Problem....
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Simplified Reynolds Averaged Navier Stokes
equations
4 equations 5 unknowns ? We need one
more ???
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Modeling of Turbulent Viscosity
MVM Mean velocity models TKEM Turbulent kinetic
energy equation models
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MVM Eddy-viscosity models

• Eddy-viscosity models
• Compute the Reynolds-stresses from explicit
  expressions of the mean strain rate and a
  eddy-viscosity.
• Boussinesq eddy-viscosity approximation

The k term is a normal stress and is typically
treated together with the pressure term.
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Algebraic MVM

• Prandtl was the first to present a working
  algebraic turbulence model that is applied to
  wakes, jets and boundary layer flows.
• The model is based on mixing length hypothesis
  deduced from experiments and is analogous, to
  some extent, to the mean free path in kinetic gas
  theory.

Turbulent transport
Molecular transport
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Pradntls Hypothesis of Turbulent Flows

• In a laminar flow the random motion is at the
  molecular level only.
• Macro instruments cannot detect this randomness.
• Macro Engineering devices feel it as molecular
  viscosity.
• Turbulent flow is due to random movement of fluid
  parcels/bundles.
• Even Macro instruments detect this randomness.
• Macro Engineering devices feel it as enhanced
  viscosity.!

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Prandtl Mixing Length Hypothesis
The fluid particle A with the mass dm located at
the position , ylm and has the longitudinal
velocity component U?U is fluctuating. This
particle is moving downward with the lateral
velocity v? and the fluctuation momentum
dI?ydm?v?. It arrives at the layer which has a
lower velocity U. According to the Prandtl
hypothesis, this macroscopic momentum exchange
most likely gives rise to a positive fluctuation
u? gt0.
Y
y
X
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Definition of Mixing Length

• Particles A B experience a velocity difference
  which can be approximated as

The distance between the two layers lm is called
mixing length. Since ?U has the same order of
magnitude as u?, Prandtl arrived at
By virtue of the Prandtl hypothesis, the
longitudinal fluctuation component u? was
brought about by the impact of the lateral
component v? , it seems reasonable to assume that
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Prandtl Mixing Length Model

• Thus, the component of the Reynolds stress tensor
  becomes

• The turbulent shear stress component becomes

• This is the Prandtl mixing length hypothesis.
• Prandtl deduced that the eddy viscosity can be
  expressed as

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Estimation of Mixing Length

• To find an algebraic expression for the mixing
  length lm, several empirical correlations were
  suggested in literature.
• The mixing length lm does not have a universally
  valid character and changes from case to case.
• Therefore it is not appropriate for
  three-dimensional flow applications.
• However, it is successfully applied to boundary
  layer flow, fully developed duct flow and
  particularly to free turbulent flows.
• Prandtl and many others started with analysis of
  the two-dimensional boundary layer infected by
  disturbance.
• For wall flows, the main source of infection is
  wall.
• The wall roughness contains many cavities and
  troughs, which infect the flow and introduce
  disturbances.

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Quantification of Infection by seeing the Effect

• Develop simple experimental test rigs.
• Measure wall shear stress.
• Define wall friction velocity using the wall
  shear stress by the relation

Define non-dimensional boundary layer coordinates.
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Approximation of velocity distribution for a
fully turbulent 2D Boundary Layer
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Approximation of velocity distribution for a
fully turbulent 2D Boundary Layer
For a fully developed turbulent flow, the
constants are experimentally found to be ?0.41
and C5.0.
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Measures for Mixing Length

• Outside the viscous sublayer marked as the
  logarithmic layer, the mixing length is
  approximated by a simple linear function.

• Accounting for viscous damping, the mixing length
  for the viscous sublayer is modeled by
  introducing a damping function D.
• As a result, the mixing length in viscous
  sublayer

The damping function D proposed by van Driest
with the constant A 26 for a boundary layer at
zero-pressure gradient.
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Effect of Mean Pressure Gradient on Mixing Length

• Based on experimental evaluation of a large
  number of velocity profiles, Kays and Moffat
  developed an empirical correlation for that
  accounts for different pressure gradients.

With
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Van Driest damping function
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Distribution of Mixing length in near-wall region
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Mixing length in far wall Region
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Conclusions on Algebraic Models

• Few other algebraic models are
• Cebeci-Smith Model
• Baldwin-Lomax Algebraic Model
• Mahendra R. Doshl And William N. Gill (2004)
• Gives good results for simple flows, flat plate,
  jets and simple shear layers
• Typically the algebraic models are fast and
  robust
• Needs to be calibrated for each flow type, they
  are not very general
• They are not well suited for computing flow
  separation
• Typically they need information about boundary
  layer properties, and are difficult to
  incorporate in modern flow solvers.

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Steady Turbulent flow
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A Segment of Reconstructed Turbulent Flame in SI
Engines
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Large Scales Parents Vortices
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Creation of Large Eddies an I.C. Engines

• There are two types of structural turbulence that
  are recognizable in an engine tumbling and
  swirl.
• Both are created during the intake stroke.
• Tumble is defined as the in-cylinder flow that is
  rotating around an axis perpendicular with the
  cylinder axis.

Swirl is defined as the charge that rotates
concentrically about the axis of the cylinder.
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Instantaneous Energy Cascade in Turbulent
Boundary Layer.
A state of universal equilibrium is reached when
the rate of energy received from larger eddies is
nearly equal to the rate of energy of when the
smallest eddies dissipate into heat.
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One-Equation Model by Prandtl

• A one-equation model is an enhanced version of
  the algebraic models.
• This model utilizes one turbulent transport
  equation originally developed by Prandtl.
• Based on purely dimensional arguments, Prandtl
  proposed a relationship between the dissipation
  and the kinetic energy that reads

• where the turbulence length scale lt is set
  proportional to the mixing length, lm, the
  boundary layer thickness ? or a wake or a jet
  width.
• The velocity scale is set proportional to the
  turbulent kinetic energy as suggested
  independently.
• Thus, the expression for the turbulent viscosity
  becomes

with the constant C? to be determined from the
experiment.
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Transport equation for turbulent kinetic Energy
x-momentum equation for incompressible steady
turbulent flow
Reynolds averaged x-momentum equation for
incompressible steady turbulent flow
subtract the second equation from the second
equation to get
Multiply above equation with u? and take Reynolds
averaging
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Reynolds Equations for Normal Reynolds Stresses
Similarly
Define turbulent kinetic energy as
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Turbulent Kinetic Energy Conservation Equation
The Cartesian index notation is
Boundary conditions
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One and Two Equation Turbulence Models

• The derivation is again based on the Boussinesq
  approximation
• The mixing velocity is determined by the
  turbulent turbulent kinetic energy
• The length scale is determined from another
  transport equation

with
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Second equation
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Dissipation of turbulent kinetic energy

• The equation is derived by the following
  operation on the Navier-Stokes equation

The resulting equation have the following form
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The k-e model

• Eddy viscosity
• Transport equation for turbulent kinetic energy
• Transport equation for dissipation of turbulent
  kinetic energy
• Constants for the model

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Dealing with Infected flows

• The RANS equations are derived by an averaging or
  filtering process from the Navier-Stokes
  equations.
• The averaging process results in more unknown
  that equations, the turbulent closure problem
• Additional equations are derived by performing
  operation on the Navier-Stokes equations
• Non of the model are complete, all model needs
  some kind of modeling.
• Special care may be need when integrating the
  model all the way to the wall, low-Reynolds
  number models and wall damping terms.