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
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
Spectral
energy
Vortices are stretched
by turbulence, their
diameter is reduced
Energy
shift

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
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
Re =

Characteristic scale of the dimensions of the
space available for all scales of motion
Length which measures the thickness of the laminar
(viscous) sub-layer (a layer where turbulence is not
maintained (due to its thickness), even if it is present in
the other parts of the space considered
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
where
 is kinematic viscosity
U is the velocity of the !LARGEST! scale of motions
(1)
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
U
L
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 Ј500 per annum. Reynolds applied for the position and, despite his
youth and inexperience, was awarded the post.
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.
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.
Osborne Reynolds
Born: 23 Aug 1842 in Belfast, Ireland
Died: 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.
Reynolds’ three
tubes
Pressure
measurements
The Reynolds approach to the equations of a turbulent fluid
Reynolds separated each of the velocity components u, v and w into
two parts:
u
Mean values
 u
v  v
w  w
 u
 v
 w
(3)
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:
u  u , u  0, (u v )  u v , u u  0,...
(4)
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
and to get the equations for the changes in
u, v and w
u’, v’ and w’
Continuity equation
d
dt


,
x t
The rate of change following
the instantaneous motion
Time and spatial
derivatives at a
fixed point
Lagrangian
derivative
Eulerian
derivative
Two forms of the continuity equation
d
u
v
w

 
0
dt
x
y
z




u
v
w
u
v
w

 
0
t
x
y
z
x
y
z
(5)
(6)
d
u
v
w

 
0
dt
x
y
z




u
v
w
u
v
w

 
0
t
x
y
z
x
y
z
uk


 uk

0
t
xk
xk
(7)
k stands for a dummy index which implies summation over
the indices corresponding to the three Cartesian
coordinate directions
the derivative
of product
 uk

0
t
xk
(8)
 uk

0
t
xk
u  u  u
The Reynolds procedure (substituting from (3) and averaging
the result) gives:

t

uk
xk
0
(10)
Averaging operators are applied to the derivatives, i.e. it
is the derivative what is averaged

 
t
ukuk

  uk
xk
 0 (11)
- Reynolds postulates
Equation of continuity can be
applied to the mean motion and
mean velocity without change
!
Flux and general conservation equation
Properties whose amounts are identified with a mass of fluid:
q
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:
dq
 Sq
dt
(12)
Source strength of the
quantity q per unit mass
Let’s assume that q is specific humidity. Then (12) can be
expanded to
+
q
q
  uk
 Sq
t
xk

uk
q
q
0
t
xk
 q 
 S q
t
The rate of change
of the amount of q
per unit volume
measured
at a fixed position
Continuity
equation multiplied
by q term-by-term
uk q 

xk
The rate of internal
production of q per
unit volume at the
same fixed point
(13)
(14)
?
(15)
The rate of
convergence
of a vector whose
components are
(ukq)
A
ukA
xk
V
uk
is a small area on the surface  to one of the 3 axis.
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.
A
This is flux of q in xk.direction
Let’s average (15), substituting: q=q+q’, uk=uk+uk’:
=0
(q+q’)(uk+uk’) =
=quk+ q’uk+quk’ +q’uk’
The averaged conservation equation:
  q
t
  Sq
The flux due to mean state
 

uk
xk
q   

The eddy flux
uk' q '
xk
(16)

(17)
The closure problem (K-theory)
Joseph Boussinesq
(1842-1929)
For q we use Taylor series expansion:
q
q( z )  q( zref )  ( z  zref )
z
Ludwig Prandtl
1875-1953
Ernst Schmidt
1892-1975
We need to estimate the eddy flux of
(18) property q at ref. level zref
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:
q1  q( z1 )  q( zref )  l1
q
z
q
q2  q( z2 )  q( zref )   l2
Averaging over all parcels:
 wq    K
q
,
z
, w1q1   w1 l1
z
q
z
, w2 q2   w2 l2
q
(19)
z
Size of the largest energy containing eddy (mixing Prandtl length)
K ~ L2T 1
(20)
Kinematic exchange coefficients (eddy diffusivity)
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.
3 components of the force x 3 surfaces = 9 stress components
ui (xk )  ui (0) 
ui
xk
xk
Stress is the tensor of rank 2
normal stresses
Motion of the fluid, when
velocity is expanded
into Taylor series. ij is the
unit symmetric tensor.
 ui
Scalar product: ½ displacement vector●  x
k

u k
xi



tangential stresses
 ii  ij  ik 


   ji  jj  jk 
 ki  kj  kk 


To get the rate of pure
deformation, we have to
subtract out the divergence:
 ui uk  2 ui

 

xk 
 xk xi  3 xk

 ik   
ui (xk )

ui (0)
1  u u 
  i  k xk 
2  xk xi 
1  ui uk 

xk

2  xk xi 
Stress
The motion equations
General form of the Navier-stokes equations (without Coriolis force):
 ui uk  2 u j





  3 x xk 

x

x
 k

i 
j
ui
u
1 p

 uk i   g 3i 

t
xk
 xi
xk
Averaging:

  ui  u k

   

x
xi

k

 ui
 ui
1  p
1 
 uk
  g 3i 

t
xk
 xi
 xk
Using K-theory:
total stress
  ui  uk
 ki    


x
xi
 k

   uk ui


 zx  
  K 
  uu   u v
 u w
1  p

 f v  


dt
 x
y
z
 x
  u v  vv
d v
 vw
1  p

 f u  


dt
 y
y
z
 x
  u w  vw
d w
 ww
1  p

 g  


dt
 z
y
z
 x
d u
d

 
dt


d q
 
dt


   uk ui

u 
x
uq
x


 v 
y
 vq
y


 w 
z
 wq
z




















(21)



(22)
Eddy
diffusivity
10-5m2s-1
u
(23)
z
Molecular
viscosity
10-5m2s-1
(24)
Vertical
momentum
fluxes
Vertical fluxes of
heat and moisture
(25)
momentum
flux
measurements
sensible
heat
flux
latent
heat flux
(evaporation)
parameterization