Experimental Lab 3: Shear Flows

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Feb 22, 2014 (3 years and 1 month ago)

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Jet Flows


Axisymmetric Jet


Case of an axisymmetric jet of fluid entering an otherwise
motionless fluid. The factors determining the motion are
the momentum flux, the distance from the orifice, and
turbulent momentum transfer. We are at a distance from

the orifice large enough so that all traces of it are lost
because of the eddies.


In this situation:

z = nR

Where z is the distance along the axis and R is the radius.


Outside of this cone the motion is irrotational and
becomes unsteady near the cone.

S
ince the momentum

flux is proportional

to the square of the velocity, w, and
the cross
-
sectional area


(


R
2
w)w at various stations
and z = nR

For an incompressible jet:



R
2
w
2

=



R
0
2

w
0
2

= constant

Since z = nR




z
2
w
2

=



z
0
2

w
0
2

or

w


z
-
1

The Reynold
s number is equal to wR/



or (z
-
1
)(z/n)/


=捯湳瑡ct

This predicted Reynolds number constancy predicts
that if the jet is turbulent near the orifice it remain
turbulent to infinity.


Next assume the inflow velocity, U, i
s horizontal and
proportional to w, and there fore to z
-
1
, and R
-
1
.


A flux of volume along the axis must equal the
horizontal flux inwards through a distance dz.

D(wR
2
)


2

創Rz


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ASEN 2002 Introduction to Thermodynamics and Aerodynamics Fall 2000


ASEN 2002, Fall 2000

Experiment 3: Shear Flows




Assigned : November 13, 2000

Report Due: December 6, 2000


This experim
ent consists of an axisymetric jet issuing from a circular nozzle
into stagnant ambient air. There are 4 sets of measurements, which will be
completed by any given group in one lab session. Each set of data will be
taken

by one pair of students (a

differe
nt pair for each set) and the data will
be shared among each lab group. Your care and attention in performing
these experiments will be of importance for both you and your fellow
students’ ability to obtain reliable data.




Axisymmetric Turbulent Jet


T
urbulent jets have a wide range of applications in combustion chambers,
nozzles, etc. In this experiment you will consider an incompressible jet.
Note that generally, jet flows become turbulent within a distance along the
jet axis equal to about 10 times

the exit diameter of the jet, d
0
. The region
between the exit plane and the onset of the turbulent region is called the
potential core
. In the potential core the velocity is flat and gradually attains
a bell
-
curve shape as the flow develops. A schemati
c of the flow is shown
below. Photos 1 through 3 ( from Van Dyke, A album of fluid motions)
show the forms such a jet can take.

In free turbulent shear flows like the jet flow in this experiment, because
there are no solid wa
lls, kinematic viscosity is not important and mixing
occurs mostly through
turbulent eddy

viscosity. The half spread of the jet

,
and the centerline velocity U
0

strongly depend on x. In fact for the
axisymmetric turbulent jet,




= 0.11x U
0

~ 1/x


It is also possible to use


and U
0

as the length and velocity scales and obtain
self
-
similar velocity distributions. A good empirical fit to the self
-
similar
velocity profile is given by



u/U
0

= e
-
0.692(

^2)

,


= y/



u is the mean velocity at a given point




Experimental Procedure


This experiment involves the measurement of the mean velocity profile of
the jet at a given jet exit velocity at
four different x locations
. The basic
goals of this experiment are:



To determine the rate of jet spread and the centerline velocity decay
rate



Obtain self
-
similar profiles



Observe the differences between laminar and turbulent signals



Also observe the
intermittent

nature of turbulence towards the outer
edges of the jet.


T
he following set of measurements will be taken using a
hot wire

anemometer
:


Set 1:
Measure and record the jet nozzle diameter. Run the jet at 85% of its
maximum allowable velocity at the exit plane. Record the jet exit velocity
by placing the hot wire
probe at a location as close as possible to the jet exit
plane. Place the traversing mechanism at a x location so that the distance to
the probe from the jet exit plane is 5d
0
. Obtain the velocity as a function of
y with

y = 0.5 inches. Record the posi
tion of the probe and the
corresponding voltage, which will be converted into velocity units by using
the calibration curve.


Set 2:
Repeat the above measurement at x = 25 inches. Use the aligner to
make sure that your traverse goes through the center
of the jet.


Set 3:
Repeat the above at x = 30 inches


Set 4:
Repeat the above at x = 35 inches










Data Analysis


1.

Plot u versus y at the four positions along the axis of
the jet (different x
-
locations)

2.

From each of these plots obtain


and U
0
.

3.

Plot



and U
0
as a function of x and compare these

variations with the theoretical distributions plotted on
the same graph.

4.

For each profile obtain u/U
0

versus y/


and plot all of
these distributions on the same graph. How do they
compare? When does the jet

become self
-
similar?
Where are the largest deviations from self similarity?

5.

Obtain mass flow rate and the momentum flux of the
jet from the formulas given below. What do these
quantities indicate? Are they constant as a function of
the distance, x, or
do they change (increase/decrease)?
What does the change in mass flow rate indicate about
“entrainment”?


Mass flow rate,






y
Uydy
m
0
2
/




and the momentum flux, M







y
ydy
U
M
0
2
2
/



Calculate these quantities at each x
-
station and plot t
hem as
a function of x. The calculation will require a numerical
integration method; you can use EXCEL or MATLAB or
MATHEMATICA.




Unsteady Jet Flows


The experimental set
-
up is equipped with a device that
obtains a jet pulsating at different frequencies
. The effect of
the unsteadyness on the spread of the jet and on its mixing
with the surrounding fluid is of interest to burners and
combustion chambers. In other words
-
can we increase the
rate of spread and thereby the mixing of the jet with the
surroun
ding fluid if the jet is pulsating and at what
frequencies? The Strouhal number (St) becomes an
important parameter. This is given by


St = d F/U
j


In this equation, d, is the jet diameter and F is the
frequency. Can you define a relevant frequency (in
Hertz)
for this set up? Conversely, the mixing could be dominated
by the mixing caused by the unsteady bursts of momentum.


In this part of the experiment, you will qualitatively
observe how the rate of spread of the jet is affected by the
pulsations of t
he nozzle flow. It is instructive to traverse
the probe along the jet axis and observe on the oscilloscope
the radial distance at which all the unsteady (turbulent)
signal will die out. Can you observe how this distance is
affected by the frequency of th
e pulsations? Does the
unsteadiness increase or decrease the rate of spread?


Other Possibilities for Exploration


There are many additional interesting aspects of jet flow
that also could be studied. A number of questions are listed
below.




How does a
single puff of momentum differ from a
continuous jet?



How does the boundary layer develop when a jet strikes
a flat plate?



What happens when a jet strikes a free surface?



If a jet is warmer/colder than its’ surroundings


How
will that change things?



What
controls the distance to the point where the
axisymmetric jet starts to spread?



What happens when a jet interacts with a cylinder
mounted on springs and free to vibrate?





Distance to Spreading of an Axisymmetric Jet









The initial f
low of an axisymmetric jet near the nozzle
opening is initially quite smooth until the turbulence
becomes established. This distance, L
0
, compared with
measurements of central flow speed, U
0
, can provide an
estimate of the time scale for the development o
f
turbulence. You assume that the following variables
are important and use the Buckingham theorem to
identify the key dimensionless groups.


L
0

= distance from the nozzle to jet spreading


U
0

= jet center flow speed


D = jet diameter


T = time scale for
establishing turbulence



e

= eddy viscosity


T = f ( D, U,

e
, L
0
)



Jet Nozzle

D

U
0

L
0

T = f ( D, U,

e
, L
0
)


N=5 K=2 N
-
K=3 Choose

e
, L
0

for the k
-
set



1
= T (

e
)
A

(L
0
)
B


1
= T (L
2
T
-
1
)
A

(L)
B

time: 1
-
A=0 A=1

length: 2A+B=0 B=
-
2


1
= T (

e
)
1

(L
0
)
-
2


1
= T

e
/ (L
0
)
2




= U(

e
)
A

(L
0
)
B


2
= LT
-
1

(L
2
T
-
1
)
A

(L)
B

time:
-
1
-
A=0 A=
-
1

length: 1+2A+B=0 B=1


2
= U(

e
)
-
1
(L
0
)


2
= U(L
0
)/(

e
)



3

= D(

e
)
A

(L
0
)
B


3
= L (L
2
T
-
1
)
A

(L)
B

time: A=0

length: 1+B=0 B=
-
1


3

= D/(L
0
)



1


2
= TU
0
/L
0


Note that U
0

and L
0
can be measure
d and T estimated

Assuming T ~ L
0
/U
0
thus

e

can be computed.



An Experiment to Explore These Results


1.

With the hot wire measure the entry flow near the
nozzle.

2.

Starting near the nozzle perform a series of profiles
until the x
-
axis point of initial sp
reading is identified.

3.

Estimate T from L
0
/ U
0

measurements

4.

Estimate

e

from L
0
2
/T

5.

Estimate R
e
from U
0
D/

e


Puffs









The speed of motion of a puff injected into a quiet medium
is assumed to be dependent upon the initial injection speed
of

the nozzle, U
0
, the volume injected, R
0
3

the position, x,
along the axis, and the eddy viscosity,

e
. Develop
expressions for the speed U as a function of U
0
, R
0
3

and x,
and also the Reynolds number, R
e
, as a function of U
0
, R
0
,

e

and x.


U = f(U
0
, R
0
3
,

x )


R
e

= f(U
0
, R
0
3
,

e
, x)


In general, U = f(U
0
, R
0
3
,

e
, x)


Use the Buckingham theorem to develop these relationships










R
0

U
0

x