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


Slide 2

Time Averaging

Major Learning Objectives:

1. Apply time averaging to the momentum and energy transport equations.


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.


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?


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.


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


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


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?


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


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


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.


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?”


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?


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)


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


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


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


Slide 18


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


000

2

0 0dtdtdtdtdtt

fdtt

f

f

dtft

dtdtftt

f

ff

agf

tt

t

tt

t

tt

t

tt

t


Slide 19


0~

g,Rearrangin

~

~

~

~ ,definitionearlier an From

:Proof

0~


fff

ff

ff

fff

fff

f

agf


Slide 20


0 ~~~

00~~

:Proofs

0~~

0~


2

2

2

fff

fffff

fffff

fffffff

agf


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:


Slide 22


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


Slide 23


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


Slide 24


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.


Slide 25


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


Slide 26


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


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).


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


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.


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


Slide 31


• 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


Slide 32


• Then in homogeneous steady flow such that

• It follows that over the entire turbulent region,

0~~2

1

xxx vvx

v

xyxyyxvv S~~


Slide 33


• 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.


Slide 34


• 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 ~~


Slide 35


• 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?


Slide 36

Time Averaged Energy Equation

0ˆ

qvTt

Tc

0ˆ

qvTt

Tc

0~~~~

ˆ

qqvvTTt

TTc


Slide 37


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


Slide 38


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


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.

louisiana tech university ruston, la 71272 slide 1 time averaging steven a. jones bien 501 monday,...

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