Louisiana Tech University Ruston, LA 71272 Slide 1 Time Averaging Steven A. Jones BIEN 501 Monday,...
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Transcript of Louisiana Tech University Ruston, LA 71272 Slide 1 Time Averaging Steven A. Jones BIEN 501 Monday,...
Louisiana Tech UniversityRuston, LA 71272
Slide 1
Time Averaging
Steven A. Jones
BIEN 501
Monday, April 14, 2008
Louisiana Tech UniversityRuston, LA 71272
Slide 2
Time Averaging
Major Learning Objectives:
1. Apply time averaging to the momentum and energy transport equations.
Louisiana Tech UniversityRuston, LA 71272
Slide 3
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.
Louisiana Tech UniversityRuston, LA 71272
Slide 4
Definition of a Time Average
If we have a variable, such as velocity, we can define the time average of that variable as:
dut
tutt
t
1
What is t?
What is t?
How can u(t) be a function of time if it is time-averaged?
Louisiana Tech UniversityRuston, LA 71272
Slide 5
Definition of a Time Average
Time averaging is a special case of a linear (low pass) filter (moving average).
dutWt
tutt
t
1
Where W(t) is a weighting window.
You should recognize this form as a convolution (or cross-correlation) between the weighting function and the variable of interest.
Louisiana Tech UniversityRuston, LA 71272
Slide 6
00.
51
1.5
0 10 20 30 40 50
Time (msec)
Ve
locit
y (
cm
/s)
True VelocityTime Averaged VelocityHanning Window
What are t and t?
The definition is a moving average, and t is the time at which the window is applied.
In this example, t is 10 msec.
t
Louisiana Tech UniversityRuston, LA 71272
Slide 7
Long Time/Short Time
When we talk about u(t), t is referred to as short time.
When we talk about , t is referred to as long time. tu
Louisiana Tech UniversityRuston, LA 71272
Slide 8
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?
Louisiana Tech UniversityRuston, LA 71272
Slide 9
Continuity and Linearity
00
uu tt
The equation of continuity is:
The time average is:
0
ut
0
0
u
uu t
If density is constant:
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 and time-averaged incompressible continuity is .
0 u0 u
Louisiana Tech UniversityRuston, LA 71272
Slide 10
Consequences of Linearity
The time average of a derivative is the derivative of a time average.
x
tdtt
txdt
x
t
tx
t tt
t
tt
t
uu
uu 11
The same result holds for time derivatives:
t
td
ttd
ttt
t tt
t
tt
t
uu
uu 11
Louisiana Tech UniversityRuston, LA 71272
Slide 11
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.
4
2
HPP
HPP
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.
Louisiana Tech UniversityRuston, LA 71272
Slide 12
050
100
150
200
250
0 20 40 60 80
Time (msec)
Velo
cit
y (
cm
/sec)
Sample “Steady Flow” Data
Disturbed
Turbulent
In each case, what is the “time average?”
Louisiana Tech UniversityRuston, LA 71272
Slide 13
Is This Flow Disturbed?
0
20
40
60
80
100
0 0.2 0.4 0.6 0.8 1
Time (sec)
Vel
oci
ty (
cm/s
ec) What is the correct
averaging time?
Louisiana Tech UniversityRuston, LA 71272
Slide 14
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)
Louisiana Tech UniversityRuston, LA 71272
Slide 15
Consequences of Nonlinearity
tt
t yxyx dtvvt
vv1
We will often divide a variable into two components, one of which is constant and one of which is time variant.
tvvtv xxx~
The time average of a product is not the product of time averages.
The time average becomes simply . xv
Louisiana Tech UniversityRuston, LA 71272
Slide 16
Consequences of Nonlinearity
tvvtvvtvv xxxxxx~~ With , since
it follows that .
tvvtv xxx~
1
1
t t
x y x x y yt
t t
x y x y x y x yt
x y x y x y x y
x y x y
v v v v t v v t dtt
v v v t v v v t v t v t dtt
v v v t v v v t v t v t
v v v t v t
The time average of the product becomes:
0~~ tvtvvv xxxx so
Louisiana Tech UniversityRuston, LA 71272
Slide 17
Additional Relationships
fadtft
adtaft
af
gfdtgt
dtft
dtgft
gf
faafgfgf
agf
tt
t
tt
t
tt
t
tt
t
tt
t
11
111
:Proofs
,
: constant, and & functionsFor
Louisiana Tech UniversityRuston, LA 71272
Slide 18
Additional Relationships
2
11
1
22
11
2
1111
:Example
1...1
.constant.. same that isn integratio original theas period same over the
alueconstant v a of average time theso alue,constant v a is but ...
111
:Proof
: constant, and & functionsFor
000
2
0 0dtdtdtdtdtt
fdtt
f
f
dtft
dtdtftt
f
ff
agf
tt
t
tt
t
tt
t
tt
t
Louisiana Tech UniversityRuston, LA 71272
Slide 19
Additional Relationships
0~
g,Rearrangin
~
~
~
~ ,definitionearlier an From
:Proof
0~
: constant, and & functionsFor
fff
ff
ff
fff
fff
f
agf
Louisiana Tech UniversityRuston, LA 71272
Slide 20
Additional Relationships
0 ~~~
00~~
:Proofs
0~~
0~
: constant, and & functionsFor
2
2
2
fff
fffff
fffff
fffffff
agf
Louisiana Tech UniversityRuston, LA 71272
Slide 21
Time Averaged Momentum
23
12
22
12
21
12
13
13
2
12
1
11
1
z
v
z
v
z
v
z
P
z
vv
z
vv
z
vv
t
v
Consider the z1 Momentum Equation in the form.
Let be constant and let:
1 1 1 2 2 2 3 3 3; ; ;v v v v v v v v v P P P
1 1 1 1 1 1 11 1 2 2 3 3
1 1 2 2 3 3
2 2 21 1 1
2 2 21 1 2 3
v v v v v v vv v v v v v
t z z z z z z
v v vP
z z z z
Then:
Louisiana Tech UniversityRuston, LA 71272
Slide 22
Time Averaged Momentum
The equation
Looks like the non time-averaged version, except for the extra terms:
23
12
22
12
21
12
1
3
13
3
13
2
12
2
12
1
11
1
11
1~
~~
~~
~
z
v
z
v
z
v
z
P
z
vv
z
vv
z
vv
z
vv
z
vv
z
vv
t
v
3
13
2
12
1
11
~~
~~
~~
z
vv
z
vv
z
vv
Louisiana Tech UniversityRuston, LA 71272
Slide 23
Time Averaged Momentum
3
31
3
13
3
13
2
21
2
12
2
12
1
11
1
11
2
11
~~
~~~~
~~
~~~~
~~
~~~~
z
vv
z
vv
z
vv
z
vv
z
vv
z
vv
z
vv
z
vv
z
vv
Consider these terms, and apply the product rule for differentiation (in reverse):
Then:
3
31
3
13
2
21
2
12
1
11
1
11
3
13
2
12
1
11
~~
~~~~
~~~~
~~
~~
~~
~~
z
vv
z
vv
z
vv
z
vv
z
vv
z
vv
z
vv
z
vv
z
vv
Louisiana Tech UniversityRuston, LA 71272
Slide 24
Time Averaged Momentum
3
31
2
21
1
11
3
13
2
12
1
11
3
31
3
13
2
21
2
12
1
11
1
11
~~
~~
~~
~~~~~~
~~
~~~~
~~~~
~~
z
vv
z
vv
z
vv
z
vv
z
vv
z
vv
z
vv
z
vv
z
vv
z
vv
z
vv
z
vv
Rearrange:
Then the term in parentheses is:
v~~~~~
~~
~~
~~
~1
3
3
2
2
1
11
3
31
2
21
1
11
vz
v
z
v
z
vv
z
vv
z
vv
z
vv
Which is zero by continuity.
Louisiana Tech UniversityRuston, LA 71272
Slide 25
Time Averaged Momentum
Now take the time average:
3
31
2
21
1
1123
12
22
12
21
12
1
3
13
2
12
1
11
1
~~~~~~
z
vv
z
vv
z
vv
z
v
z
v
z
v
z
P
z
vv
z
vv
z
vv
t
v
To get:
3
31
2
21
1
1123
12
22
12
21
12
1
3
13
2
12
1
11
1
~~~~~~
z
vv
z
vv
z
vv
z
v
z
v
z
v
z
P
z
vv
z
vv
z
vv
t
v
Louisiana Tech UniversityRuston, LA 71272
Slide 26
Time Averaged Momentum
3
13
2
12
1
11~~~~~~
z
vv
z
vv
z
vv
The terms:
Look like the divergence of a second order tensor defined by:
332313
322212
312111
~~~~~~
~~~~~~
~~~~~~
vvvvvv
vvvvvv
vvvvvv
R
Consequently, it is customary to write the time averaged momentum equations in the form:
RPt
vvv
Louisiana Tech UniversityRuston, LA 71272
Slide 27
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).
Louisiana Tech UniversityRuston, LA 71272
Slide 28
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
Louisiana Tech UniversityRuston, LA 71272
Slide 29
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.
Louisiana Tech UniversityRuston, LA 71272
Slide 30
Reynolds Stresses – 2D Flow
• If the rate of strain is given as follows:
• Then we can write the energy extracted from the mean flow and converted to turbulent fluctuations due to strain rate as:
xyyxvv S ~~
y
v
x
v
y
v
x
v
yxxy
yxxy
~~
2
1 :Portion gFluctuatin
2
1 :portion Averaged Time
S
S
Louisiana Tech UniversityRuston, LA 71272
Slide 31
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
. viscositykinematic theis where
2
xyxyxy SS
Louisiana Tech UniversityRuston, LA 71272
Slide 32
Reynolds Stresses – 2D Flow
• Then in homogeneous steady flow such that
• It follows that over the entire turbulent region,
0~~2
1
xxx vvx
v
xyxyyxvv S~~
Louisiana Tech UniversityRuston, LA 71272
Slide 33
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.
Louisiana Tech UniversityRuston, LA 71272
Slide 34
Reynolds Stresses – 2D Flow
• The Reynolds Stresses, , are often used as a measure of the stresses on the individual cells.
• Even though 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.
yxvv ~~
yxvv ~~
Louisiana Tech UniversityRuston, LA 71272
Slide 35
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.
xyxyyxvv 1~~ S How would you measure the Reynolds Stresses?
Louisiana Tech UniversityRuston, LA 71272
Slide 36
Time Averaged Energy Equation
0ˆ
qvTt
Tc
0ˆ
qvTt
Tc
0~~~~
ˆ
qqvvTTt
TTc
Louisiana Tech UniversityRuston, LA 71272
Slide 37
Time Averaged Energy Equation
0~~ˆ
qvvTTt
TcFrom:
The cross terms between time averaged and fluctuating values again become zero.
qvTv
~~ˆ T
t
TcSo:
vTqv ~~ˆˆ
cTt
Tc
Louisiana Tech UniversityRuston, LA 71272
Slide 38
Time Averaged Energy Equation
From:
Apply the product rule:
vqv ~~ˆˆ TcTt
Tc
vqvv ~~ˆˆ TcTTt
Tc
vqv ~~ˆˆ TcTt
Tc
(Incompressible)
vvv ~~TTT
Same as in book because:
vvvvv ~~~~TTTTT So
)(tq
Louisiana Tech UniversityRuston, LA 71272
Slide 39
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.