MAE 123 : Mechanical Engineering Laboratory II - Fluids
Laboratory 5: Turbulent Free-Jet Dispersion
Dr. J. M. Meyers | Dr. D. G. Fletcher | Dr. Y. Dubief
ME 123:
Mechanical
Engineering
Lab II: Fluids
Image:
Technical
University
of Denmark
Laboratory 5: Turbulent Free-Jet Dispersion
1
Self Similarity
• A similarity solution is one in which the number of independent variables is reduced by at least
one, usually by a coordinate transformation
• Normally, coordinates are collapsed into dimensionless groups that scale the velocities
• One can take advantage of self-similar solutions to achieve useful simplifications in solving
problems or comparing data from other experiments.
Blasius solution for the flat plate boundary layer
• Solid line is theory
• Data points represent experiments at
different Reynolds numbers
ME 123: Mechanical Engineering Lab II: Fluids
Laboratory 5: Turbulent Free-Jet Dispersion
2
Free Shear Flow
Free shear flows are flows unaffected by walls and develop and spread in an open ambient fluid
Pipe flow (wall bounded flow)
Free jet flow (free shear flow)
Velocity gradients in 𝑦 are created by some upstream mechanism, that are smoothed out by
viscous diffusion in the presence of convective deceleration
Because of the lack of wall confinement, the static pressure field is constant unlike in wall
confined flows:
𝜕𝑃
𝜕𝑃
Pipe Flow
=0
≠0
𝜕𝑦
𝜕𝑥
Free Shear Flow
ME 123: Mechanical Engineering Lab II: Fluids
Laboratory 5: Turbulent Free-Jet Dispersion
𝜕𝑃
=0
𝜕𝑦
𝜕𝑃
=0
𝜕𝑥
3
Free Shear Flow
Shear flow satisfies the flat plate equations (this is for
2D plane flow but arguments hold for axisymmetric
as well)
𝜕𝑢 𝜕𝑢
+
=0
𝜕𝑥 𝜕𝑦
𝜕𝑢
𝜕𝑢
𝜕2𝑢
𝑢
+𝑣
≈𝜈 2
𝜕𝑥
𝜕𝑦
𝜕𝑦
Continuity
Steady state 𝑥-momentum
• Figure above shows two parallel uniform streams of two different velocities meeting at 𝑥 = 0
• As we move downstream the initial discontinuity imposed is smoothed out by viscosity into as Sshaped free-shear layer
• The simplest application would be for 𝑈2 = 0 which would represent a jet emerging from a slot
(recall we are thinking in terms of 2D plane flow for simplicity!) into ambient air at rest
ME 123: Mechanical Engineering Lab II: Fluids
Laboratory 5: Turbulent Free-Jet Dispersion
4
Free Shear Flow
• We can generalize this into two different fluids
with properties (𝜌1 , 𝜇1 ) and (𝜌2 , 𝜇2 )
• A self-similarity variable for each stream can be
defined as:
𝑈
𝜂𝑗 = 𝑦
2𝑥𝜈𝑗
𝑓′𝑗 =
𝑢
𝑈𝑗
ME 123: Mechanical Engineering Lab II: Fluids
Laboratory 5: Turbulent Free-Jet Dispersion
5
Free Shear Flow
Free turbulence just refers to high Reynolds number shear flow in an open fluid
environment unconfined by rigid boundaries
Some common types include:
a) A mixing layer between two streams of different velocity
b) A jet issuing into a still stream
c) A wake behind a body
ME 123: Mechanical Engineering Lab II: Fluids
Laboratory 5: Turbulent Free-Jet Dispersion
6
Turbulent Flow
Reynolds decomposition refers to separation of the flow variable (like velocity ) into the mean (timeaveraged) component, 𝑢 𝑥 , and the fluctuating component, 𝑢′(𝑥, 𝑡).
𝑢 𝑥, 𝑡 = 𝑢 𝑥 + 𝑢′(𝑥, 𝑡)
absolute mean
velocity
velocity velocity fluctuation
The RANS (Reynolds Average Navier-Stokes) equations are primarily used to describe turbulent flows.
These equations can be used with approximations based on knowledge of the properties of flow
turbulence to give approximate time-averaged solutions to the Navier-Stokes equations.
How does one measure mean and fluctuating
velocities?
𝑢′(𝑥, 𝑡)
Recall how in the pipe flow labs manometer
fluctuations were higher toward the wall…
this is where higher turbulence is present.
ME 123: Mechanical Engineering Lab II: Fluids
Laboratory 5: Turbulent Free-Jet Dispersion
𝑢 𝑥
7
Turbulent Free Jet
•
•
•
•
•
This figure shows the details of turbulent jet formation in still ambient fluid similar fluids (𝑘 = 1)
(1) Near the orifice exit the jet issues at a nearly flat, fully developed, turbulent velocity 𝑈𝑒𝑥𝑖𝑡
(2) Mixing layers form at exit lip and grow toward the end of the inviscid potential core (about 1
orifice diameter, 𝐷, downstream)
(3) After the core extinguishes the flow develops into a distinctive Gaussian-shaped “jet”
(4) Finally, after a distance of about 20 × 𝐷 downstream the velocity profile reaches and
maintains a self-preserving (self-similar) shape
(1)
ME 123: Mechanical Engineering Lab II: Fluids
Laboratory 5: Turbulent Free-Jet Dispersion
(3)
(4)
(2)
8
Turbulent Free Jet
For an axisymmetric jet the self-similar shape takes on the following form:
𝑢
𝑈𝑚𝑎𝑥
≈𝑓
𝑟
𝑏
The jet width, 𝑏, above may also have the notation of 𝛿 later in your notes or lab handout.
It is the asymptotic self-similar form of free turbulent flows that we wish to study.
(3)
(1)
ME 123: Mechanical Engineering Lab II: Fluids
Laboratory 5: Turbulent Free-Jet Dispersion
(2)
(4)
9
Turbulent Free Jet
Mass Flow
•
It is important to note that mass flow is not conserved in each velocity profile as flow is entrained
and entering the system. THIS IS A SIGNIFICANT DIFFERENCE BETWEEN FREE SHEAR AND CONFINED
PIPE FLOW
2𝜋
∞
𝑚(𝑥) =
𝜌𝑢 𝑥 𝑟𝑑𝑟𝑑𝜃 ≠ const
0
−∞
(3)
(1)
ME 123: Mechanical Engineering Lab II: Fluids
Laboratory 5: Turbulent Free-Jet Dispersion
(2)
(4)
10
Turbulent Free Jet
•
Momentum
Since there is no pressure gradient the jet momentum , 𝐽, must remain constant at each cross
section. For an axisymmetric jet:
2𝜋
𝑀(𝑥) =
0
•
∞
𝜌𝑢2 (𝑥, 𝑟, 𝜃)𝑟𝑑𝑟𝑑𝜃 = const = const × 𝜌𝑏 2 𝑈𝑚𝑎𝑥 2
−∞
In the self-similar region the centerline velocity should only depend on jet momentum, density, and
distance… NOT viscosity (as there are no rigid boundaries… i.e. walls)
1/2
𝑀
𝑈𝑚𝑎𝑥 = const ×
𝑥 −1/2
𝜌
(3)
(1)
ME 123: Mechanical Engineering Lab II: Fluids
Laboratory 5: Turbulent Free-Jet Dispersion
(2)
(4)
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Turbulent Free Jet
Kinetic Energy
•
The flux of mean kinetic energy is not conserved
2𝜋
𝐾(𝑥) =
0
∞
𝑢2 (𝑥, 𝑟, 𝜃)
𝜌𝑢(𝑥, 𝑟, 𝜃)
𝑟𝑑𝑟𝑑𝜃 ≠ const
2
−∞
(3)
(1)
ME 123: Mechanical Engineering Lab II: Fluids
Laboratory 5: Turbulent Free-Jet Dispersion
(2)
(4)
12
Theoretical Models of Self-Similar Solutions
Görtler Theory
𝑢
𝑈𝑚𝑎𝑥
𝜂2
≈ 1+
4
−2
𝜂 ≈ 15.2
𝑦
𝑥
Eq. 6-152
Plane Jet Solution Variation:
𝑢
𝑈𝑚𝑎𝑥
≈ sech2 10.4
𝑦
𝑥
Eq. 6-153
Average velocity
These are analyzing the time averaged mean velocity:
𝑢 𝑥, 𝑡 = 𝑢 𝑥 + 𝑢′(𝑥, 𝑡)
absolute mean
velocity
velocity velocity fluctuation
The time-averaged velocity, 𝑢 𝑥 , approaches a self
similar state wll before the turbulent velocity
fluctuation contribution, 𝑢′(𝑥, 𝑡)
How could you measure the level of turbulence?
ME 123: Mechanical Engineering Lab II: Fluids
Laboratory 5: Turbulent Free-Jet Dispersion
Turbulent velocity
13
Free Jet Experiment
•
•
Density must be constant over the whole flow domain
From the Venturi lab, we had one known quantity, which was the total pressure of the working
fluid was the atmospheric pressure since laboratory air was drawn into the wind tunnel:
𝑃𝑡𝑜𝑡𝑎𝑙 = 1 atm for Venturi lab
•
•
•
Static pressure values (measured by hydrostatic devices, manometers) decreased as velocity
increased and vica versa as dictated by the Bernoulli relation
These principles hold for a free jet experiment but the working conditions are different.
For this experiment, air enters the pipe, is compressed, and then is ejected as a free jet into the lab
ME 123: Mechanical Engineering Lab II: Fluids
Laboratory 5: Turbulent Free-Jet Dispersion
14
Free Jet Experiment
•
Now we are working at the exit of a compressor rather than at the entrance like we did with the
wind tunnel
•
What does the compressor do to the working fluid?
• Increase Pressure
•
The jet flowing into still air entrains some of the surrounding air and causes it to have forward
momentum, but the entire flow is subjected to a constant pressure boundary condition imposed
by the medium it enters…
•
Will this fluid (air) leaving the jet have a pressure greater or less than ambient?
• Static pressure is equal to the atmospheric pressure
• Total pressure will increase due to velocity and is not equal to atmospheric in this lab
ME 123: Mechanical Engineering Lab II: Fluids
Laboratory 5: Turbulent Free-Jet Dispersion
15
Free Jet Experiment
•
The Pitot probe for this lab is designed specifically for incompressible flow applications to
measure total pressure
•
We will be using a different Pitot probe but the same manometer setup as the last lab.
•
By now you should feel confident with the hydrostatic force balance on the manometer fluid and
how to extract velocity
•
The Pitot probe has only one opening at the tip. The jet is flowing in the ambient air where the
density is taken to be constant
•
As with the Venturi experiment the total pressure is nearly constant… but is the total pressure the
same as it was for the wind tunnel? Why or why not?
• Please address this in your report.
ME 123: Mechanical Engineering Lab II: Fluids
Laboratory 5: Turbulent Free-Jet Dispersion
16
Free Jet Experiment
•
•
•
•
•
A diagram of such a flow is shown below where a jet exits from a slot in a plane wall and draws
fluid along because of what fluid property?
If we measure the mass flow in the axial direction away from the jet, we find that this is not
constant. How can we show this?
However, we find that the total momentum is conserved in the axial direction. How can we show
this?
Kinetic energy will also not be conserved. How can we show this?
How would a 2-D jet (planar) vs. an axisymmetric jet differ? Or are they similar? Why?
ME 123: Mechanical Engineering Lab II: Fluids
Laboratory 5: Turbulent Free-Jet Dispersion
17
Free Jet Experiment
•
•
•
•
We will measure the Pitot pressure as a function of radial position at a number of axial locations
POTENTIALLY for D = 30 mm: x/D = 1, 3, 5, 7 , 15, and 22
We are looking for a velocity profile development and for a self-similar region
The velocity profile comes from the entrainment of the air
Conservation of momentum suggests that:
∞
𝑀=
𝑑
𝑑𝑥
𝜌𝑢2 𝑑𝐴 = const = const × 𝜌𝑏 2 𝑈𝑚𝑎𝑥 2
−∞
•
∞
𝑢2 𝑥, 𝑦 𝑑𝑦 = 0
−∞
In the self-similar region, we can express velocity as:
𝜂=
𝑢 𝑥, 𝑦 = 𝑢0 𝑥, 𝑦 𝑓(𝜂)
Where:
𝑦
𝛿(𝑥)
𝑑𝑦 = 𝛿 𝑥 𝑑𝜂
ME 123: Mechanical Engineering Lab II: Fluids
Laboratory 5: Turbulent Free-Jet Dispersion
18
Free Jet Experiment
•
In the self similar region, this shape factor is constant:
𝑑
𝑈0 𝑥, 𝑦 2 𝛿 𝑥
𝑑𝑥
∞
𝑓(𝜂)𝑑𝜂 = 0
−∞
𝑑
𝑈0 𝑥, 𝑦 2 𝛿 𝑥 = 0
𝑑𝑥
•
So the integral is also constant, so we realize that:
•
Your task is to measure the parameter defining the jet width, 𝛿 𝑥 , which we can define as the
distance where the local value of velocity is 5% of the centerline
Then with the centerline velocity value, give the region of the jet where this relation holds
(identify the measurement locations where this relation is valid)
Keep in mind what the Pitot probe is measuring!
Interpret your measurements and your conclusions considering propagation of error.
•
•
•
ME 123: Mechanical Engineering Lab II: Fluids
Laboratory 5: Turbulent Free-Jet Dispersion
19
Free Jet Experiment
The blower section with attached orifice plate at the end of
the pipe flow facility will act as your experimental apparatus
A Pitot probe is attached to a 3-axis translating support with
built-in measurement scales to facilitate velocity profile
measurements
𝒛
At each measurement point in your profile you need only to
record total pressure and atmospheric pressure heights from
the manometer.
Velocity, as you are very familiar in calculating, will come
from the known relation:
𝑃𝑡 =
1 2
𝜌𝑢 + 𝑃𝑠
2
𝑢=
2
𝑃𝑡 −𝑃𝑠
𝜌𝑎𝑖𝑟
What is the static pressure in this case?
𝒚
𝒙
Why is the total pressure not the atmospheric pressure?
ME 123: Mechanical Engineering Lab II: Fluids
Laboratory 5: Turbulent Free-Jet Dispersion
20
Free Jet Experiment
For this lab you will measure radial velocity profiles of a jet
exiting a 30 mm orifice at a sufficient number of chosen axial
locations
𝒛
These surveys must be capable of addressing
• evidence of entrainment
• mass flow evolution in the x-direction
(numerically integrate slide 10)
• evidence of the conservation of momentum
(numerically integrate slide 11)
• a study of the evolution of kinetic energy
(numerically integrate slide 12)
You are also required to study, within the range of
measurement by the apparatus, any evidence of self-similar
behavior
𝒚
An appropriate uncertainty and sensitivity analysis of your
results is also required
ME 123: Mechanical Engineering Lab II: Fluids
Laboratory 5: Turbulent Free-Jet Dispersion
𝒙
21