Time Averaging
Steven A. Jones
BIEN 501
Monday, April 14, 2008
Louisiana Tech University
Ruston, LA 71272
Slide 1
Time Averaging
Major Learning Objectives:
1. Apply time averaging to the momentum
and energy transport equations.
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Slide 2
Time Averaging
Minor Learning Objectives:
1. Define a time average.
2. State reasons for time averaging.
3. Demonstrate how linearity and nonlinearity affect time
averaging.
4. Demonstrate the main rules for time averaging.
5. Compare time averaging to linear filtering.
6. Time average the momentum equation.
7. Describe Reynolds stresses.
8. Time average the energy equation.
9. Describe turbulent energy flux.
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Slide 3
Definition of a Time Average
If we have a variable, such as velocity, we can define
the time average of that variable as:
1
u t  
t

t  t
t
u   d
What is t?
What is t?
How can u(t) be a function of time if it is time-averaged?
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Slide 4
Definition of a Time Average
Time averaging is a special case of a linear (low pass)
filter (moving average).
1 t  t
u t  
W t   u   d

t t
Where W(t) is a weighting window.
You should recognize this form as a convolution (or crosscorrelation) between the weighting function and the
variable of interest.
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Slide 5
What are t and t?
t
In this
example, t
is 10 msec.
0.5
1
True Velocity
Time Averaged Velocity
Hanning Window
0
Velocity (cm/s)
1.5
The definition is a moving average, and t is the time at
which the window is applied.
0
10
20
30
40
50
Time (msec)
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Slide 6
Long Time/Short Time
When we talk about u(t), t is referred to as short time.
When we talk about u t , t is referred to as long time.
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Slide 7
Why Time Average?
1. We may be interested in changes that occur over
longer periods of time.
2. We may want to filter out noise in a signal.
3. Measurements are often filtered. All instruments have
some kind of time constant.
1. Examples:
Do weather patterns suggest global warming?
What is the overall flow rate from a piping system?
What is the average shear stress to which an
endothelial cell is subjected?
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Slide 8
Continuity and Linearity
The equation of continuity is:

    u   0
t

 

    u   0 
   u  0
The time average is:
t
t
 
If density is constant:   0     u     u
t
  u  0
When an equation is linear, the time average for the
equation can be found simply by substituting the time
averaged variable for the time dependent variable. E.g.
incompressible continuity is   u  0 and time-averaged
incompressible continuity is   u  0.
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Slide 9
Consequences of Linearity
The time average of a derivative is the derivative of a
time average.
 ut  1 t t  ut 
  1 t t
 ut 

dt    ut  dt  

t
x
t
x
x  t t
x

The same result holds for time derivatives:
ut  1 t t u 
  1 t t
 ut 

d    u  d  

t
t
t
t
t  t t
t

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Slide 10
Time Averaged Time Average
From linear systems, a signal filtered twice is different
from a signal filtered once, as can readily be seen from
the frequency domain.
P    P  H  
2
P    P  H  
4
But if the slow fluctuations are sufficiently separated in
frequency from the fast fluctuations, the average of the
average is approximately the same as the average.
In particular, for fluctuations in steady flow, the two
averages are the same.
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Slide 11
Sample “Steady Flow” Data
200
50
100 150
Turbulent
0
Velocity (cm/sec)
250
Disturbed
0
20
40
60
80
In each case,
what is the “time
average?”
Time (msec)
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Slide 12
Is This Flow Disturbed?
100
What is the correct
averaging time?
Velocity (cm/sec)
80
60
40
20
0
0
0.2
0.4
0.6
0.8
1
Time (sec)
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Slide 13
Choices
We need to determine what time frame we are interested.
The time frame is determined by the value of t.
1. How does the earth’s rotation affect temperature? (t ~
hours)
2. How does the earth’s tilt affect weather? (t ~ days)
3. How does the earth’s magnetic field affect weather? (t
~ years)
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Slide 14
Consequences of Nonlinearity
The time average of a product is not the product of time
averages.
1
vxv y 
t

t  t
t
v x v y dt
We will often divide a variable into two components, one of
which is constant and one of which is time variant.
v x t   v x  v~x t 
The time average becomes simply
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vx .
Slide 15
Consequences of Nonlinearity
With v x t   v x  v~x t , since v x  v x t   v x  v~x t   v x  v~x t 
it follows that v x  v x  v~x t  so v~x t   0 .
The time average of the product becomes:
1 t t
vx v y 
vx  vx  t    v y  v y  t   dt


t t
1 t t

vx v y  vx  t  v y  vx v y  t   vx  t  v y  t  dt

t t
 vx v y  vx  t  v y  vx v y  t   vx  t  v y  t 
 vx v y  vx  t  v y  t 
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Slide 16
Additional Relationships
For functions f & g and constant, a :
f  g  f  g,

af  a f
Proofs :
1 t  t
1 t  t
1 t  t
f  g    f  g  dt    f  dt   g  dt  f  g
t t
t t
t t
1 t  t
 1 t  t

af  
af dt  a 
f dt   a f
t
t
t
 t

 
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

Slide 17
Additional Relationships
For functions f & g and constant, a :
f  f
Proof :
f

1 t  t  1 t  t
1 t  t

     f  dt  dt  
f dt
t
t
t
t
t
 t

...but f is a constant v alue, so the time average of a constant v alue
over the same period as the original integratio n is that same constant.. .
1 t  t
 f
dt  ...  f 1

t
t
Example :
1 12 
1 1 
 1 
1 

1 









t
dt
dt

dt


dt

1
dt

 0
  
 0

 





0
0
0
 2 
 
  2 
 2 
2 

1

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Slide 18
Additional Relationships
For functions f & g and constant, a :
~
f 0
Proof :
~
From an earlier definition , f  f  f
~
f
 ff
~
 ff
~
 ff
Rearrangin g,
~
f  f  f 0
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Slide 19
Additional Relationships
For functions f & g and constant, a :
2
~
~~
f f  f
f f 0
ff  0
Proofs :

f f
~
ff
~~
ff

2
 f f  f
~
 f f  f 0   0
~2
 f 0
 
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Slide 20
Time Averaged Momentum
Consider the z1 Momentum Equation in the form.
  2 v1  2 v1  2 v1 
v1
v1
v1
v1
P

 v1
 v2
 v3

   2  2  2 
t
z1
z2
z3
z1
z2
z3 
 z1
Let  be constant and let:
v1  v1  v1 ; v2  v2  v2 ; v3  v3  v3 ; P  P  P
Then:
v1 
v1
v1  
v1
v1  
v1
v1 

   v1
  v1
  v2
  v3
    v2
    v3

t 
z1
z1  
z2
z2  
z3
z3 
  2 v1  2 v1  2 v1 
P

 2  2  2 
z1
z2 z3 
 z1
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Slide 21
Time Averaged Momentum
The equation
v1 
v1
v~1  
v1
v~1  
v1
v~1 
~
~
~

   v2
   v3

  v1
 v1
 v2
 v3
t 
z1
z1  
z 2
z 2  
z 3
z 3 
  2 v1  2 v1  2 v1 
P

   2  2  2 
z1
z 2
z 3 
 z1
Looks like the non time-averaged version, except for the
extra terms:
 ~ v~1   ~ v~1   ~ v~1 

 v1
   v2
   v3
z1  
z2  
z3 

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Slide 22
Time Averaged Momentum
Consider these terms, and apply the product rule for
differentiation (in reverse):
~
~ v~ 
~


v

v

v
v~1 1   1 1  v~1 1
v~2
Then:
v~3
z 2
v~
1
z 2
v~
1
z3


z1
 v~ v~ 
2 1
z 2
 v~ v~ 
3 1
z3
 v~1
 v~1
z1
v~
2
z 2
v~
3
z3
v~1
v~1
v~1
~
~
~
v1
 v 2
 v 3
z1
z 2
z 3
 v~3v~1 
v~3
 v~1v~1 
v~1
 v~2 v~1 
v~2
~
~
~

 v1

 v1

 v1
z1
z1
z 2
z 2
z3
z 3
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Slide 23
Time Averaged Momentum
Rearrange:
 v~3v~1 
v~3
 v~1v~1 
v~1
 v~2 v~1 
v~2
~
~
~

 v1

 v1

 v1
z1
z1
z 2
z 2
z3
z3
 v~3v~1   ~ v~1
v~3 
 v~1v~1 
 v~2 v~1 
v~2
~
~




  v1
 v1
 v1
z1
z 2
z3
z1
z 2
z3 

Then the term in parentheses is:
~
~
~
~ v~ v~ 


v

v

v

v
3
1
2
v~1
 v~1
 v~1
 v~1  1  2  3   v~1  ~
v
z1
z2
z3
 z1 z2 z3 
Which is zero by continuity.
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Slide 24
Time Averaged Momentum
Now take the time average:

v1
v
v
v
 v1 1  v2 1  v3 1
t
z1
z 2
z3
  2 v1  2 v1  2 v1   v~1v~1
v~1v~3 
v~1v~2
P


   2  2  2    


z1
z 2
z3   z1
z 2
z3 
 z1
To get:
v1
v1
v1
v1

 v1
 v2
 v3
t
z1
z 2
z3
  2 v1  2 v1  2 v1    v~1v~1
 v~1v~ 2
 v~1v~3 
P


   2  2  2    


z1
z 2
z3   z1
z 2
z3 
 z1
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Slide 25
Time Averaged Momentum
 


 
 v~3v~1
 v~1v~1
 v~2 v~1



z1
z 2
z3
The terms:
Look like the divergence of a second order tensor
defined by:
~~ ~~ ~~
 v1v1

 R     v~2 v~1
v~3v~1

v1v2
v~ v~
2 2
v~3v~ 2
v1v3 

v~2 v~3 
v~3v~ 3 
Consequently, it is customary to write the time averaged
momentum equations in the form:

v
 v  v  P         R
t
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Slide 26
Reynolds Stresses
• Have the form of a stress tensor.
• Act as true stresses on the mean flow.
• Are referred to in the biomedical
engineering literature relating cell damage
and platelet activation to turbulence.
• But are not the stresses directly imposed
on the cells. (Viscous shearing).
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Slide 27
Reynolds Stresses
• If an eddy of fluid suddenly move through
the velocity field:
The fluid would tend to change the local
momentum.
• Thus, the Reynolds Stress is not a shear
upon the fluid itself, only upon the velocity
field.
– THIS IS A VERY IMPORTANT DISTINCTION
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Slide 28
Reynolds Stresses
• Cell Damage
– Since cells such as Red Blood Cells (RBC),
monocytes, and platelets can be affected by
shearing, it is important to determine the
degree of shearing to which a cell is subjected
in a given flow geometry.
– This is particularly important in regions of
turbulence such as downstream of a stenosed
valve or downstream of tight vascular
constrictions.
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Slide 29
Reynolds Stresses – 2D Flow
• If the rate of strain is given as follows:
Time Averaged portion :
Fluctuatin g Portion :
1  v x v y 

S xy  

2  x y 
~
1  v~x v y 

S xy  

2  x y 
• Then we can write the energy extracted
from the mean flow and converted to
turbulent fluctuations due to strain rate as:
~
~
 vx v y S xy
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Slide 30
Reynolds Stresses – 2D Flow
• The energy which is extracted from the
turbulent kinetic energy and converted to
heat through viscous shearing is called
viscous dissipation and is designated by ε
 xy  2 S xy S xy
where  is the kinematic viscosity .
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Slide 31
Reynolds Stresses – 2D Flow
• Then in homogeneous steady flow such
that
 1 ~ ~ 
vx  vx vx   0
x  2

• It follows that over the entire turbulent
region,
 
~
~
 vx v y S xy   xy
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Slide 32
Reynolds Stresses – 2D Flow
• There are several mechanisms that can
damage blood cells. Two of these are
pressure fluctuations and shear stress.
– Pressure fluctuations are generally more
important for larger particles since a net shear
on the particle requires a difference in
pressure along its length.
– Shearing is a more likely mechanism for
damage in cells the size of RBC.
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Slide 33
Reynolds Stresses – 2D Flow
~
~

v
v
• The Reynolds Stresses, x y, are often
used as a measure of the stresses on the
individual cells.
• Even though  v~x v~y are called Reynolds
Shearing Stresses, they do not represent
the shearing stresses on individual cells.
– Rather, they are the stresses on the mean
flow field, as stated before.
Louisiana Tech University
Ruston, LA 71272
Slide 34
Reynolds Stresses – 2D Flow
• It is, however, much easier to measure the
Reynolds Stresses than it is to measure
the viscous dissipation
– Because the Reynolds Stresses occur over a
much larger scale than the viscous stresses.
– Thus, we use this identity to estimate the
viscous dissipation, and thus total stresses on
the flow from the Reynolds Stresses.
1
~
~
 vx v y  S xy  xy
Louisiana Tech University
Ruston, LA 71272
How would you measure the
Reynolds Stresses?
Slide 35
Time Averaged Energy Equation
T


 cˆ     T v     q  0
 t

 T

 cˆ     Tv     q  0
 t

 


  T  T~
~

ˆ
 c
  T T

 t
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Ruston, LA 71272




~
v  v     q  q~  0



Slide 36
Time Averaged Energy Equation
From:



 T

~
~
 cˆ 
   T  T v  v      q  0
 t

The cross terms between time averaged and
fluctuating values again become zero.
So:


 T
~~ 
 cˆ 
   T v  T v     q
 t

 T

~~
 cˆ 
   T v     q   cˆ  T v
 t

Louisiana Tech University
Ruston, LA 71272
Slide 37
Time Averaged Energy Equation
 T

~~
 cˆ 
   T v     q   cˆ  T v
 t

From:
Apply the product rule:
(Incompressible)
 T

~~
 cˆ 
 T   v  v  T     q   cˆ  T v
 t

q (t )
 T

~ ~ Same as in
 cˆ 
 v  T     q   cˆT v
book because:
 t



~
Tv  T  T


~~
~~
~
v  v   T v  T v So Tv  T v  T v
Louisiana Tech University
Ruston, LA 71272
Slide 38
Use of Turbulent Energy Flux
• Solutions to the energy equation depend on
finding empirical and semi-theoretical relations
for the turbulent energy flux.
• Turbulent energy flux will depend on
temperature gradient and the stress tensor.
• Turbulence tends to transport energy,
momentum and mass through mixing. I.e.
turbulence carries things across “mean
streamlines” and distributes them more evenly.
Louisiana Tech University
Ruston, LA 71272
Slide 39