MAE 123 : Mechanical Engineering Laboratory II -Fluids ...jmmeyers/ME123/Lectures/ME 123 Lecture...

ME 123: Mechanical Engineering Lab II: Fluids

Laboratory 5: Turbulent Free-Jet Dispersion1

MAE 123 : Mechanical Engineering Laboratory II - Fluids

Laboratory 5: Turbulent Free-Jet DispersionDr. J. M. Meyers | Dr. D. G. Fletcher | Dr. Y. Dubief

Image: Technical University of Denmark



• 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.

Self Similarity

Blasius solution for the flat plate boundary layer

• Solid line is theory

• Data points represent experiments at

different Reynolds numbers



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�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: �� 0�� 0

Pipe Flow

Free Shear Flow

�� 0�� 0



Free Shear Flow

�� +

�� 0

�� + � �� ≈ ��

��

Continuity

Steady state �-momentum

Shear flow satisfies the flat plate equations (this is for

2D plane flow but arguments hold for axisymmetric

as well)

• Figure above shows two parallel uniform streams of two different velocities meeting at � � 0• As we move downstream the initial discontinuity imposed is smoothed out by viscosity into as S-

shaped free-shear layer

• The simplest application would be for �� 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



Free Shear Flow

�� 2� �

• We can generalize this into two different fluids

with properties (��, ��) and (��, ��)

• A self-similarity variable for each stream can be

defined as:

�′� � ��



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



Reynolds decomposition refers to separation of the flow variable (like velocity ) into the mean (time-

averaged) component, � � , and the fluctuating component, ′(�, �). �, � � � � + ′(�, �)

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.

absolute

velocity

mean

velocity

velocity

fluctuation

Turbulent Flow

� �

′(�, �)

How does one measure mean and fluctuating

velocities?

Recall how in the pipe flow labs manometer

fluctuations were higher toward the wall…

this is where higher turbulence is present.



• This figure shows the details of turbulent jet formation in still ambient fluid similar fluids (� � 1)

• (1) Near the orifice exit the jet issues at a nearly flat, fully developed, turbulent velocity �� !"• (2) Mixing layers form at exit lip and grow toward the end of the inviscid potential core (about 1

orifice diameter, #, downstream)

• (3) After the core extinguishes the flow develops into a distinctive Gaussian-shaped “jet”

• (4) Finally, after a distance of about 20 × # downstream the velocity profile reaches and

maintains a self-preserving (self-similar) shape

Turbulent Free Jet

(1)(2)

(3) (4)



Turbulent Free Jet

For an axisymmetric jet the self-similar shape takes on the following form:

It is the asymptotic self-similar form of free turbulent flows that we wish to study.

��%&

≈ � '(

The jet width, (, above may also have the notation of ) later in your notes or lab handout.

(1) (2)

(3) (4)



Turbulent Free Jet

• 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

Mass Flow

(1) (2)

(3) (4)

*+ (�) � , , �� '-'-./

0/

�1

2� const



• Since there is no pressure gradient the jet momentum , 8, must remain constant at each cross

section. For an axisymmetric jet:

Turbulent Free Jet

(1) (2)

(3) (4)

9(�) � , , ��(�, ', .)'-'-./

0/

�1

2� const � const × �(��%& �

Momentum

• 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)

�%& � const × 9�

�/��0�/�



Turbulent Free Jet

• The flux of mean kinetic energy is not conserved

Kinetic Energy

(1) (2)

(3) (4)

;(�) � , , ��(�, ', .) ��(�, ', .)

2 '-'-./

0/

�1

2� const



The time-averaged velocity, � � , approaches a self

similar state wll before the turbulent velocity

fluctuation contribution, ′(�, �)

�, � � � � + ′(�, �)absolute

velocity

mean

velocity

velocity

fluctuation

Average velocity

Turbulent velocity

��%&

≈ sech� 10.4 �� Eq. 6-153

��%&

≈ 1 + ��4

0�Eq. 6-152� ≈ 15.2 ��

Theoretical Models of Self-Similar Solutions

Görtler Theory

Plane Jet Solution Variation:

These are analyzing the time averaged mean velocity:

How could you measure the level of turbulence?



• 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:

�"A"&B � 1atm 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

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

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.

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?

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:

--�, � �, � -�

/

0/� 0

• In the self-similar region, we can express velocity as:

�, � � 2 �, � �(�)� � �

)(�)Where:

-� � ) � -�

9 � , ��-E/

0/� const � const × �(��%& �

Free Jet Experiment



• In the self similar region, this shape factor is constant:--� �2 �, � �) � , �(�)-�

/

0/� 0

• So the integral is also constant, so we realize that:--� �2 �, � �) � � 0

• Your task is to measure the parameter defining the jet width, ) � , 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.

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

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:

�" � 12�� + �F � 2 �"−�F�&!H

What is the static pressure in this case?

Why is the total pressure not the atmospheric pressure?

Free Jet Experiment

I

JK



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

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


• a study of the evolution of kinetic energy


You are also required to study, within the range of

measurement by the apparatus, any evidence of self-similar

behavior

An appropriate uncertainty and sensitivity analysis of your

results is also required

Free Jet Experiment

I

JK

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