Lecture of : the Reynolds equations of turbulent motions

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Lecture of the Reynolds equations of turbulent
motions
JORDANIAN GERMAN WINTER ACCADMEY
Prepared by Eng. Mohammad Hamasha Jordan
University of Science Technology
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• Most of the research on turbulent flow analysis
  is the
• past century has used the concept of time
  averaging.
• Applying time averaging to the basic equations
  of
• motion yield Reynolds equations.
• Reynolds equations involve both mean and
• fluctuating quantities.
• Reynolds equations attempt to model fluctuating
  
• terms by relating them to the mean properties
  or
• gradients.
• Reynolds equations form the basis of the most
• engineering analyses of turbulent flow.

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• assume
• Fluid is in a randomly unsteady turbulent state.
  
• Worked with the time-averaged or mean equations
• of motion.
• So any variable is resolve into mean value
  plus
• fluctuating value

• Where T is large compared to relevant period of
• fluctuation.

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• Itself may vary slowly with time as the
• following figure

fig 1
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• Lets assume the turbulent flow is incompressible
  flow with constant transport properties with
  significant fluctuations in velocity, pressure
  and temperature.
• The variables will be formed as following

equ.. 1

• From the basic integral Equation

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• incompressible continuity equation

equ...2

• Substitute in u, v, w from equ 1 and take the
  time average of entire equation

equ .3

• This is Reynolds-averaged basic differential
  equation for turbulent mean
• continuity.

• Subtract equ 3 from equ 2 but do not take time
  average, this gives

equ4
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• equations 3 and 4 does not valid for fluid with
  density fluctuating
• ( compressible fluid ).
• Now use non linear Navier-Stokes equation

equ5

• Convective- acceleration term

equ.6
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• substitute equ 2 in to equ 5

equ..7

• the momentum equation is complicated by new term
  involving
• the turbulent inertia tensor .

• This new term is never negligible in any
  turbulent flow and is the
• source of our analytic difficulties.

• time-averaging procedure has introduced nine new
  variables (the tensor components) which can
  be defined only through (unavailable) knowledge
  of the detailed turbulent structure.

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• The components of are related not only
  to fluid physical
• properties but also to local flow conditions
  (velocity,
• geometry, surface roughness, and upstream
  history).

• there is no further physical laws are available
  to resolve this
• dilemma.

• Some empirical approaches have been quite
  successful,
• though rather thinly formulated from
  nonrigorous postulates.

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• A slight amount of illumination is thrown upon
  Eq. 7 if it is
• rearranged to display the turbulent inertia
  terms as if they
• were stresses, which of course they are not.
  Thus we write

equ..8

• This is Reynolds-averaged basic differential
  equation for turbulent mean
• momentum.

equ..9
Turbulent
Laminar

• mathematically, then, the turbulent inertia terms
  behave as if the
• total stress on the system were composed of
  the Newtonian
• Viscous Stresses plus an additional or
  apparent turbulent-
• stress tensor .

• is called turbulent shear.

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• Now consider the energy equation (first law of
  thermodynamics)
• for incompressible flow with constant
  properties

equ..10

• Taking the time average, we obtain the
  mean-energy equation

equ..11

• This is Reynolds-averaged basic differential
  equation for turbulent mean
• thermal energy.

equ..12
equ..13
Turbulent
Laminar
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• By analogy with our rearrangement of the momentum
  equation, we
• have collected conduction and turbulent
  convection terms into a
• sort of total-heat-flux vector qi which
  includes molecular flux plus
• the turbulent flux .
• The total-dissipation term is obviously
  complex in the general
• case. In two-dimensional turbulent-boundary-la
  yer flow (the most
• common situation), the dissipation reduces
  approximately to

equ..14

• Reynolds equations can not be achieve without
  additional relation or
• empirical modeling ideas

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The Turbulence Kinetic-Energy Equations
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• Many attempts have been made to add "turbulence
  conservation
• relations to the time-averaged continuity,
  momentum, and energy
• equations.
• the most obvious single addition would be a
  relation for the
• turbulence kinetic energy K of the
  fluctuations, defined by

equ.15
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• A conservation relation for K can be derived by
  forming the mechanical energy
• equation, i.e., the clot product of u and the
  ith momentum equation. then, we
• subtract the instantaneous mechanical energy
  equation from its time-averaged
• value. The result is the turbulence
  kinetic-energy relation for an incompressible
• fluid

equ.16
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The Reynolds stress equation
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II
III
I
IV
V
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• Here the roman numerals denote (I) rate of change
  of Reynolds stress,
• (II) generation of stress, (III) dissipation,
  (IV) pressure strain effects,
• and (V) diffusion of Reynolds stress.

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Thank you