TUBULENT FLUXES OF HEAT, MOISTURE AND MOMENTUM: BASICS OF TURBULENCE

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TUBULENT FLUXES OF HEAT, MOISTURE AND MOMENTUM
BASICS OF TURBULENCE

• What is turbulence? Alfred Blackadar No
  definition of turbulence can be given at this
  time!. Thus, we can only agree on some
  attributes of turbulence (Lumley and Panofsky
  1964)
• 1. Turbulence is stochastic by nature
• Even if the equations are deterministic, they
  are nonlinear
• Results are highly dependent on small
  differences in initial state
• No way to observe the initial state sufficiently

No way to treat the turbulence in a deterministic
way

• 2. Turbulence is three-dimensional
• Although some 2-D cases can be considered (e.g.
  cyclones),
• their ensemble behaviour is different from
  small scale
• turbulence in a large 3-D environment
• 3. In a turbulent environment any 2 particles
  that are free to move tend
• to become increasingly distant from each
  other with time

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• 4. Turbulence is rotational by nature
• Vorticity is an essential attribute of turbulence
• 5. Turbulence is dissipative
• The energy tends to shift from large-scale well
  organized eddies (small wave numbers) towards
  smaller eddies and, finally to molecular motions

Vortices are stretched by turbulence,
their diameter is reduced

• 6. Turbulence is a phenomenon of large Reynolds
  numbers
• Large spatial dimensions
• Small viscosity

The space available for motions is LARGE in
comparison to the dimensions of eddies
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The Reynolds number Turbulent motions are highly
dependent on the sizes of eddies and
characteristic scales of the environment where
there eddies exist. The Reynolds number is the
ratio between two lengths

L is defined by boundary configuration (e.g.
depth of the channel, diameter of the pipeline).
In the atmosphere gravity is essential, and
turbulence is largely controlled by stable lapse
rates. ? (laminar sublayer thickness) is related
to viscosity
? ? / U
(1) where ?
is kinematic viscosity U is the velocity of
the !LARGEST! scale of motions
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The dimensionless ratio between L and ? is the
Reynolds number
Re L/? LU/? (2)
In the pipes (pipelines) L usually is taken
as a diameter, and mean current velocity stands
for U Recr ? 1000


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Osborne Reynolds's father was a priest in the
Anglican church but he had an academic background
having graduated from Cambridge in 1837, being
elected to a fellowship at Queens' College, and
being headmaster of first Belfast Collegiate
School and then Dedham School in Essex. In fact
it was a family with a tradition of the Church
and three generations of Osborne's father's
family had been the rector of Debach-with-Boulge.
Osborne was born in Belfast when his father was
Principal of the Collegiate School there but
began his schooling at Dedham when his father was
headmaster of the school in that Essex town.
After that he received private tutoring to
complete his secondary education. He did not go
straight to university after his secondary
education, however, but rather he took an
apprenticeship with the engineering firm of
Edward Hayes in 1861. Reynolds wrote (actually in
his application for the chair in Manchester in
1868) of his father's influence on him while he
was growing up In my boyhood I had the
advantage of the constant guidance of my father,
also a lover of mechanics, and a man of no mean
attainments in mathematics and its application to
physics. Reynolds, after gaining experience in
the engineering firm, studied mathematics at
Cambridge, graduating in 1867. As an
undergraduate Reynolds had attended some of the
same classes as Rayleigh who was one year ahead
of him. As his father had before him, Reynolds
was elected to a scholarship at Queens' College.
He again took up a post with an engineering firm,
this time the civil engineers John Lawson of
London, spending a year as a practicing civil
engineer. In 1868 Reynolds became the first
professor of engineering in Manchester (and the
second in England). Kargon writes- ... a newly
created professorship of engineering was
advertised at Owens College, Manchester, at ?500
per annum. Reynolds applied for the position
and, despite his youth and inexperience, was
awarded the post.
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We should note in passing that Owens College
would later become the University of Manchester.
Reynolds held this post until he retired in 1905.
His early work was on magnetism and electricity
but he soon concentrated on hydraulics and
hydrodynamics. He also worked on electromagnetic
properties of the sun and of comets, and
considered tidal motions in rivers. After 1873
Reynolds concentrated mainly on fluid dynamics
and it was in this area that his contributions
were of world leading importance. We summarise
these contributions. He studied the change in a
flow along a pipe when it goes from laminar flow
to turbulent flow. In 1886 he formulated a theory
of lubrication. Three years later he produced an
important theoretical model for turbulent flow
and it has become the standard mathematical
framework used in the study of turbulence. An
account of Reynolds' work on hydrodynamic
stability published in 1883 and 1895 is looked at
in 8. The 1883 paper is called An experimental
investigation of the circumstances which
determine whether the motion of water in parallel
channels shall be direct or sinuous and of the
law of resistance in parallel channels. The
'Reynolds number' (as it is now called) used in
modelling fluid flow which is named after him
appears in this work. Reynolds became a Fellow
of the Royal Society in 1877 and, 11 years later,
won their Royal Medal. In 1884 he was awarded an
honorary degree by the University of Glasgow. By
the beginning of the 1900s Reynolds health began
to fail and he retired in 1905. Not only did he
deteriorate physically but also mentally, which
was sad to see in so brilliant a man who was
hardly 60 years old.
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Not only is Reynolds important in terms of his
research, but he is also important for the
applied mathematics course he set up at
Manchester. Anderson writes in 3- Reynolds
was a scholarly man with high standards.
Engineering education was new to English
universities at that time, and Reynolds had
definite ideas about its proper form. He believed
that all engineering students, no matter what
their speciality, should have a common background
based in mathematics, physics, and particularly
the fundamentals of classical mechanics. ...
Despite his intense interest in education, he was
not a great lecturer. His lectures were difficult
to follow, and he frequently wandered among
topics with little or no connection. Lamb, who
knew Reynolds well both as a man and as a fellow
worker in fluid dynamics, wrote- The character
of Reynolds was like his writings, strongly
individual. He was conscious of the value of his
work, but was content to leave it to the mature
judgement of the scientific world. For
advertisement he had no taste, and undue
pretension on the part of others only elicited a
tolerant smile. To his pupils he was most
generous in the opportunities for valuable work
which he put in their way, and in the share of
cooperation. Somewhat reserved in serious or
personal matters and occasionally combative and
tenacious in debate, he was in the ordinary
relations of life the most kindly and genial of
companions.
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Osborne Reynolds Born 23 Aug 1842 in Belfast,
IrelandDied 21 Feb 1912 in Watchet, Somerset,
England
On the dynamical theory of incompressible viscous
fluids and the determination of the criterion.
Royal Society, Phil. Trans., 1895.
Reynolds proceeded to measure the critical
velocity for onset of eddies using three tubes of
different diameter and in each case varying the
water temperature. To a first approximation, the
Reynolds Numbers based on these critical values
of velocity were found to be the same (about
13000) for each of the tubes and for all water
temperatures. He then set out to find the
critical condition for an eddying flow to change
into non-turbulent flow, referring to this as the
inferior limit'. To do this, he allowed water to
flow in a disturbed state from the mains through
a length of pipe and measured the pressure-drop
over a five-foot distance near the outlet.
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Reynolds three tubes
Pressure measurements
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The Reynolds approach to the equations of a
turbulent fluid Reynolds separated each of the
velocity components u, v and w into two parts
(3)
Mean values
Turbulent portion
If we now observe a sequence of velocities of
particles at times that are sufficiently
separated to be considered as uncorrelated with
each other, the mean value of each such sequence
is independent of the other samples and the mean
values of deviations from the mean (u, v, w)
are zero
(4)
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Equation (4) represents the so-called ensemble
averaging. Reynolds approach Averaging of the
equations of motion and the equation of
continuity, i.e. to replace in each equation u,
v and w
by ?u?, ?v? and ?w? and
u, v and w and to get the equations
for the changes in
?u?, ?v? and ?w?
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Continuity equation
Lagrangian derivative
The rate of change following the instantaneous
motion
Time and spatial derivatives at a fixed point
Eulerian derivative
Two forms of the continuity equation

(5)

(6)
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(7)

k stands for a dummy index which implies
summation over the indices corresponding to the
three Cartesian coordinate directions
the derivative of product
(8)
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The Reynolds procedure (substituting from (3) and
averaging the result) gives
(10)

Averaging operators are applied to the
derivatives, i.e. it is the derivative what is
averaged
????? ??uk??????uk?
- Reynolds postulates

• Equation of continuity can be
• applied to the mean motion and
• mean velocity without change

!
(11)
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Flux and general conservation equation Properties
whose amounts are identified with a mass of
fluid

• Specific humidity (the mass of water vapor per
• unit mass of air)
• Kinetic energy per unit mass
• Specific entropy (Cpln?)

These properties are assumed to be conservative,
i.e. they do not change just because of their
motion
Equation of the conservation of property q
(12)
Source strength of the quantity q per unit mass
Lets assume that q is specific humidity. Then
(12) can be expanded to
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Lets assume that q is specific humidity. Then
(12) can be expanded to

(13)
Continuity equation multiplied by q term-by-term

(14)
?
(15)
The rate of convergence of a vector
whose components are (?ukq)
The rate of internal production of q per unit
volume at the same fixed point
The rate of change of the amount of q per unit
volume measured at a fixed position

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A is a small area on the surface ? to one
of the 3 axis. ukA is a volume of a cylinder
with a height of uk?(unit time). ?uk
is the mass transported through a unit area of
the surface in one unit of time. ?uk?q
is the amount of property transported per unit
area and per unit time across a
surface ? to xk.
This is flux of q in xk.direction
Lets average (15), substituting q?q?q,
uk?uk?uk
0
?(?q?q)(?uk?uk)?
?q??uk? ?q??uk??q??uk? ?q??uk?
(16)
The averaged conservation equation
(17)
The flux due to mean state
The eddy flux
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The closure problem (K-theory)

For ?q? we use Taylor series expansion
We need to estimate the eddy flux of property q
at ref. level zref
(18)

Parcel moves down through zref with a vertical
velocity -w. It has been last mixed with its
environment at distance l1 from zref where it
took its mean value. Deviation from the mean at
reference level and the corresponding
contribution to the flux
(19)
Averaging over all parcels
Size of the largest energy containing eddy
(mixing Prandtl length)
(20)
Kinematic exchange coefficients (eddy diffusivity)
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Surface stress What is stress? Stress is the
force acting across the boundary surface and
proportional to the area of surface, across which
it acts.
Motion of the fluid, when velocity is
expanded into Taylor series. ?ij is the unit
symmetric tensor.
To get the rate of pure deformation, we have to
subtract out the divergence
Stress
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The motion equations
General form of the Navier-stokes equations
(without Coriolis force)
(21)
(22)
Averaging
Eddy diffusivity 10-5m2s-1
Using K-theory
(23)
Molecular viscosity 10-5m2s-1
(24)
Vertical momentum fluxes
Vertical fluxes of heat and moisture
(25)
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latent heat flux (evaporation)
sensible heat flux
momentum flux
measurements
parameterization