MECHANICS of FLUIDS LABORATORY
William S. Janna
Department of Mechanical Engineering
Memphis State University
©1997 William S. Janna
All Rights Reserved.
No part of this manual may be reproduced, stored in a retrieval system, or transcribed in any form or by any means—electronic, magnetic, mechanical, photocopying, recording, or otherwise—
without the prior written consent of William S. Janna
TABLE OF CONTENTS
Report Writing.................................................................................................................4 Cleanliness and Safety ....................................................................................................6 Experiment 1
Density and Surface Tension.....................................................7 Experiment 2
Viscosity.........................................................................................9 Experiment 3
Center of Pressure on a Submerged Plane Surface.............10 Experiment 4
Measurement of Differential Pressure..................................12 Experiment 5
Impact of a Jet of Water ............................................................14 Experiment 6
Critical Reynolds Number in Pipe Flow...............................16 Experiment 7
Fluid Meters................................................................................18 Experiment 8
Pipe Flow .....................................................................................22 Experiment 9
Pressure Distribution About a Circular Cylinder................24 Experiment 10
Drag Force Determination .......................................................27 Experiment 11
Analysis of an Airfoil................................................................28 Experiment 12
Open Channel Flow—Sluice Gate .........................................30 Experiment 13
Open Channel Flow Over a Weir ..........................................32 Experiment 14
Open Channel Flow—Hydraulic Jump ................................34 Experiment 15
Open Channel Flow Over a Hump........................................36 Experiment 16
Measurement of Velocity and Calibration of
a Meter for Compressible Flow.............................39 Experiment 17
Measurement of Fan Horsepower .........................................44 Experiment 18
Measurement of Pump Performance....................................46 Appendix .........................................................................................................................50
All reports in the Fluid Mechanics
Laboratory require a formal laboratory report
unless specified otherwise. The report should be
written in such a way that anyone can duplicate
the performed experiment and find the same
results as the originator. The reports should be
simple and clearly written. Reports are due one
week after the experiment was performed, unless
The report should communicate several ideas
to the reader. First the report should be neatly
done. The experimenter is in effect trying to
convince the reader that the experiment was
performed in a straightforward manner with
great care and with full attention to detail. A
poorly written report might instead lead the
reader to think that just as little care went into
performing the experiment. Second, the report
should be well organized. The reader should be
able to easily follow each step discussed in the
text. Third, the report should contain accurate
results. This will require checking and rechecking
the calculations until accuracy can be guaranteed.
Fourth, the report should be free of spelling and
grammatical errors. The following format, shown
in Figure R.1, is to be used for formal Laboratory
Title Page–The title page should show the title
and number of the experiment, the date the
experiment was performed, experimenter's
name and experimenter's partners' names.
Table of Contents –Each page of the report must
be numbered for this section.
Object –The object is a clear concise statement
explaining the purpose of the experiment.
This is one of the most important parts of the
laboratory report because everything
included in the report must somehow relate to
the stated object. The object can be as short as
one sentence and it is usually written in the
Theory –The theory section should contain a
complete analytical development of all
important equations pertinent to the
experiment, and how these equations are used
in the reduction of data. The theory section
should be written textbook-style.
Procedure – The procedure section should contain
a schematic drawing of the experimental
setup including all equipment used in a parts
list with manufacturer serial numbers, if any.
Show the function of each part when
necessary for clarity. Outline exactly step-
Original Data Sheet
Discussion & Conclusion
Table of Contents
Each page numbered
FIGURE R.1. Format for formal reports.
by-step how the experiment was performed in
case someone desires to duplicate it. If it
cannot be duplicated, the experiment shows
Results – The results section should contain a
formal analysis of the data with tables,
graphs, etc. Any presentation of data which
serves the purpose of clearly showing the
outcome of the experiment is sufficient.
Discussion and Conclusion – This section should
give an interpretation of the results
explaining how the object of the experiment
was accomplished. If any analytical
expression is to be verified, calculate % error†
and account for the sources. Discuss this
experiment with respect to its faults as well
† % error–An analysis expressing how favorably the
empirical data approximate theoretical information.
There are many ways to find % error, but one method is
introduced here for consistency. Take the difference
between the empirical and theoretical results and divide
by the theoretical result. Multiplying by 100% gives the
% error. You may compose your own error analysis as
long as your method is clearly defined.
as its strong points. Suggest extensions of the
experiment and improvements. Also
recommend any changes necessary to better
accomplish the object.
Each experiment write-up contains a
number of questions. These are to be answered
or discussed in the Discussion and Conclusions
(1) Original data sheet.
(2) Show how data were used by a sample
(3) Calibration curves of instrument which
were used in the performance of the
experiment. Include manufacturer of the
instrument, model and serial numbers.
Calibration curves will usually be supplied
by the instructor.
(4) Bibliography listing all references used.
Short Form Report Format
Often the experiment requires not a formal
report but an informal report. An informal report
includes the Title Page, Object, Procedure,
Results, and Conclusions. Other portions may be
added at the discretion of the instructor or the
writer. Another alternative report form consists
of a Title Page, an Introduction (made up of
shortened versions of Object, Theory, and
Procedure) Results, and Conclusion and
Discussion. This form might be used when a
detailed theory section would be too long.
In many instances, it is necessary to compose a
plot in order to graphically present the results.
Graphs must be drawn neatly following a specific
format. Figure R.2 shows an acceptable graph
prepared using a computer. There are many
computer programs that have graphing
capabilities. Nevertheless an acceptably drawn
graph has several features of note. These features
are summarized next to Figure R.2.
Border is drawn about the entire graph.
Axis labels defined with symbols and
Grid drawn using major axis divisions.
Each line is identified using a legend.
Data points are identified with a
symbol: “ ´” on the Qac line to denote
data points obtained by experiment.
The line representing the theoretical
results has no data points represented.
Nothing is drawn freehand.
Title is descriptive, rather than
something like Q vs ∆h.
flow rate Q in m /s
Features of note
head loss ∆ h in m
FIGURE R.2. Theoretical and actual volume flow rate
through a venturi meter as a function of head loss.
CLEANLINESS AND SAFETY
There are “housekeeping” rules that the user
of the laboratory should be aware of and abide
by. Equipment in the lab is delicate and each
piece is used extensively for 2 or 3 weeks per
semester. During the remaining time, each
apparatus just sits there, literally collecting dust.
University housekeeping staff are not required to
clean and maintain the equipment. Instead, there
are college technicians who will work on the
equipment when it needs repair, and when they
are notified that a piece of equipment needs
attention. It is important, however, that the
equipment stay clean, so that dust will not
accumulate too badly.
The Fluid Mechanics Laboratory contains
equipment that uses water or air as the working
fluid. In some cases, performing an experiment
will inevitably allow water to get on the
equipment and/or the floor. If no one cleaned up
their working area after performing an
experiment, the lab would not be a comfortable or
safe place to work in. No student appreciates
walking up to and working with a piece of
equipment that another student or group of
students has left in a mess.
Consequently, students are required to clean
up their area at the conclusion of the performance
of an experiment. Cleanup will include removal
of spilled water (or any liquid), and wiping the
table top on which the equipment is mounted (if
appropriate). The lab should always be as clean
or cleaner than it was when you entered. Cleaning
the lab is your responsibility as a user of the
equipment. This is an act of courtesy that students
who follow you will appreciate, and that you
will appreciate when you work with the
The layout of the equipment and storage
cabinets in the Fluid Mechanics Lab involves
resolving a variety of conflicting problems. These
include traffic flow, emergency facilities,
environmental safeguards, exit door locations,
etc. The goal is to implement safety requirements
without impeding egress, but still allowing
adequate work space and necessary informal
Distance between adjacent pieces of
equipment is determined by locations of floor
drains, and by the need to allow enough space
around the apparatus of interest. Immediate
access to the Safety Cabinet is also considered.
Emergency facilities such as showers, eye wash
fountains, spill kits, fire blankets and the like
are not found in the lab. We do not work with
hazardous materials and such safety facilities
are not necessary. However, waste materials are
generated and they should be disposed of
Every effort has been made to create a
positive, clean, safety conscious atmosphere.
Students are encouraged to handle equipment
safely and to be aware of, and avoid being
victims of, hazardous situations.
FLUID PROPERTIES: DENSITY AND SURFACE TENSION
There are several properties simple
Newtonian fluids have. They are basic
properties which cannot be calculated for every
fluid, and therefore they must be measured.
These properties are important in making
calculations regarding fluid systems. Measuring
fluid properties, density and viscosity, is the
object of this experiment.
Part I: Density Measurement.
Graduated cylinder or beaker
Liquid whose properties are to be
The density of the test fluid is to be found by
weighing a known volume of the liquid using the
graduated cylinder or beaker and the scale. The
beaker is weighed empty. The beaker is then
filled to a certain volume according to the
graduations on it and weighed again. The
difference in weight divided by the volume gives
the weight per unit volume of the liquid. By
appropriate conversion, the liquid density is
calculated. The mass per unit volume, or the
density, is thus measured in a direct way.
A second method of finding density involves
measuring buoyant force exerted on a submerged
object. The difference between the weight of an
object in air and the weight of the object in liquid
is known as the buoyant force (see Figure 1.1).
Part II: Surface Tension Measurement
Surface tension meter
Surface tension is defined as the energy
required to pull molecules of liquid from beneath
the surface to the surface to form a new area. It is
therefore an energy per unit area (F⋅L/L2 = F/L).
A surface tension meter is used to measure this
energy per unit area and give its value directly. A
schematic of the surface tension meter is given in
The platinum-iridium ring is attached to a
balance rod (lever arm) which in turn is attached
FIGURE 1.1. Measuring the buoyant force on an
object with a hanging weight.
Referring to Figure 1.1, the buoyant force B is
B = W1 - W2
The buoyant force is equal to the difference
between the weight of the object in air and the
weight of the object while submerged. Dividing
this difference by the volume displaced gives the
weight per unit volume from which density can be
1. Are the results of all the density
measurements in agreement?
2. How does the buoyant force vary with
depth of the submerged object? Why?
to a stainless steel torsion wire. One end of this
wire is fixed and the other is rotated. As the wire
is placed under torsion, the rod lifts the ring
slowly out of the liquid. The proper technique is
to lower the test fluid container as the ring is
lifted so that the ring remains horizontal. The
force required to break the ring free from the
liquid surface is related to the surface tension of
the liquid. As the ring breaks free, the gage at
the front of the meter reads directly in the units
indicated (dynes/cm) for the given ring. This
reading is called the apparent surface tension and
must be corrected for the ring used in order to
obtain the actual surface tension for the liquid.
The correction factor F can be calculated with the
FIGURE 1.2. A schematic of the
surface tension meter.
F = 0.725 + √ 0.000 403 3(σa/ρ) + 0.045 34 - 1.679(r/R)
where F is the correction factor, σa is the
apparent surface tension read from the dial
(dyne/cm), ρ is the density of the liquid (g/cm3),
and (r/R) for the ring is found on the ring
container. The actual surface tension for the
liquid is given by
σ = Fσa
FLUID PROPERTIES: VISCOSITY
One of the properties of homogeneous liquids
is their resistance to motion. A measure of this
resistance is known as viscosity. It can be
measured in different, standardized methods or
tests. In this experiment, viscosity will be
measured with a falling sphere viscometer.
The Falling Sphere Viscometer
When an object falls through a fluid medium,
the object reaches a constant final speed or
terminal velocity. If this terminal velocity is
sufficiently low, then the various forces acting on
the object can be described with exact expressions.
The forces acting on a sphere, for example, that is
falling at terminal velocity through a liquid are:
Weight - Buoyancy - Drag = 0
ρsg πR3 - ρg πR3 - 6πµVR = 0
where ρs and ρ are density of the sphere and
liquid respectively, V is the sphere’s terminal
velocity, R is the radius of the sphere and µ is
the viscosity of the liquid. In solving the
preceding equation, the viscosity of the liquid can
be determined. The above expression for drag is
valid only if the following equation is valid:
average the results. With the terminal velocity
of this and of other spheres measured and known,
the absolute and kinematic viscosity of the liquid
can be calculated. The temperature of the test
liquid should also be recorded. Use at least three
different spheres. (Note that if the density of
the liquid is unknown, it can be obtained from any
group who has completed or is taking data on
1. Should the terminal velocity of two
different size spheres be the same?
2. Does a larger sphere have a higher
3. Should the viscosity found for two different
size spheres be the same? Why or why not?
4. If different size spheres give different
results for the viscosity, what are the error
sources? Calculate the % error and account
for all known error sources.
5. What are the shortcomings of this method?
6. Why should temperature be recorded.
7. Can this method be used for gases?
8. Can this method be used for opaque liquids?
9. Can this method be used for something like
peanut butter, or grease or flour dough?
Why or why not?
where D is the sphere diameter. Once the
viscosity of the liquid is found, the above ratio
should be calculated to be certain that the
mathematical model gives an accurate
description of a sphere falling through the
Several small spheres with weight and
diameter to be measured
FIGURE 2.1. Terminal velocity measurement (V =
Drop a sphere into the cylinder liquid and
record the time it takes for the sphere to fall a
certain measured distance. The distance divided
by the measured time gives the terminal velocity
of the sphere. Repeat the measurement and
CENTER OF PRESSURE ON A SUBMERGED
Submerged surfaces are found in many
engineering applications. Dams, weirs and water
gates are familiar examples of submerged
surfaces used to control the flow of water. From
the design viewpoint, it is important to have a
working knowledge of the forces that act on
A plane surface located beneath the surface
of a liquid is subjected to a pressure due to the
height of liquid above it, as shown in Figure 3.1.
Increasing pressure varies linearly with
increasing depth resulting in a pressure
distribution that acts on the submerged surface.
The analysis of this situation involves
determining a force which is equivalent to the
pressure, and finding the location of this force.
FIGURE 3.1. Pressure distribution on a submerged
plane surface and the equivalent force.
verification of these equations for force and
distance is the subject of this experiment.
Center of Pressure Measurement
Center of Pressure Apparatus
Figure 3.2 gives a schematic of the apparatus
used in this experiment. The torus and balance
arm are placed on top of the tank. Note that the
pivot point for the balance arm is the point of
contact between the rod and the top of the tank.
The zeroing weight is adjusted to level the
balance arm. Water is then added to a
predetermined depth. Weights are placed on the
weight hanger to re-level the balance arm. The
amount of needed weight and depth of water are
then recorded. The procedure is then repeated for
four other depths. (Remember to record the
distance from the pivot point to the free surface
for each case.)
From the depth measurement, the equivalent
force and its location are calculated using
Equations 3.1 and 3.2. Summing moments about the
pivot allows for a comparison between the
theoretical and actual force exerted. Referring to
Figure 3.2, we have
For this case, it can be shown that the
equivalent force is:
F = ρgycA
in which ρ is the liquid density, yc is the distance
from the free surface of the liquid to the centroid
of the plane, and A is the area of the plane in
contact with liquid. Further, the location of this
force yF below the free surface is
in which Ixx is the second area moment of the
plane about its centroid. The experimental
(y + y F )
where y is the distance from the pivot point to
the free surface, yF is the distance from the free
surface to the line of action of the force F, and L is
the distance from the pivot point to the line of
action of the weight W. Note that both curved
surfaces of the torus are circular with centers at
the pivot point. For the report, compare the force
obtained with Equation 3.1 to that obtained with
Equation 3.3. When using Equation 3.3, it will be
necessary to use Equation 3.2 for yF.
1. In summing moments, why isn't the buoyant
force taken into account?
2. Why isn’t the weight of the torus and the
balance arm taken into account?
(point of contact)
FIGURE 3.2. A schematic of the center of pressure apparatus.
MEASUREMENT OF DIFFERENTIAL PRESSURE
Pressure can be measured in several ways.
Bourdon tube gages, manometers, and transducers
are a few of the devices available. Each of these
instruments actually measures a difference in
pressure; that is, measures a difference between
the desired reading and some reference pressure,
usually atmospheric. The measurement of
differential pressure with manometers is the
subject of this experiment.
A manometer is a device used to measure a
pressure difference and display the reading in
terms of height of a column of liquid. The height
is related to the pressure difference by the
Figure 4.1 shows a U-tube manometer
connected to two pressure vessels. The manometer
reading is ∆h and the manometer fluid has
density ρm. One pressure vessel contains a fluid of
density ρ1 while the other vessel contains a fluid
of density ρ2. The pressure difference can be found
by applying the hydrostatic equation to each
limb of the manometer. For the left leg,
FIGURE 4.1. A U-tube manometer connected to
two pressure vessels.
p1 + ρ1gz1 = pA
Likewise for the right leg,
p2 + ρ2gz2 + ρmg∆h = pA
Equating these expressions and solving for the
pressure difference gives
p1 - p2 = ρ2gz2 + ρ1gz1 + ρmg∆h
If the fluids above the manometer liquid are both
gases, then ρ1 and ρ2 are small compared to ρµ.
The above equation then becomes
p1 - p 2 = ρ m g ∆ h
Figure 4.2 is a schematic of the apparatus
used in this experiment. It consists of three U-tube
manometers, a well-type manometer, a Utube/inclined manometer and a differential pressure gage. There are two tanks (actually, two
capped pieces of pipe) to which each manometer
and the gage are connected. The tanks have bleed
valves attached and the tanks are connected
with plastic tubing to a squeeze bulb. The bulb
lines also contain valves. With both bleed valves
closed and with both bulb line valves open, the
bulb is squeezed to pump air from the low pressure
tank to the high pressure tank. The bulb is
squeezed until any of the manometers reaches its
maximum reading. Now both valves are closed
and the liquid levels are allowed to settle in
each manometer. The ∆h readings are all
recorded. Next, one or both bleed valves are
opened slightly to release some air into or out of a
tank. The liquid levels are again allowed to
settle and the ∆h readings are recorded. The
procedure is to be repeated until 5 different sets of
readings are obtained. For each set of readings,
convert all readings into psi or Pa units, calculate
the average value and the standard deviation.
Before beginning, be sure to zero each manometer
and the gage.
1. Manometers 1, 2 and 3 are U-tube types and
each contains a different liquid. Manometer
4 is a well-type manometer. Is there an
advantage to using this one over a U-tube
2. Manometer 5 is a combined U/tube/inclined
manometer. What is the advantage of this
3. Note that some of the manometers use a
liquid which has a specific gravity
different from 1.00, yet the reading is in
inches of water. Explain how this is
4. What advantages or disadvantages does
the gage have over the manometers?
5. Is a low value of the standard deviation
6. What does a low standard deviation
7. In your opinion, which device gives the
most accurate reading. What led you to this
High pressure tank
Low pressure tank
FIGURE 4.2. A schematic of the apparatus used in this experiment.
IMPACT OF A JET OF WATER
A jet of fluid striking a stationary object
exerts a force on that object. This force can be
measured when the object is connected to a spring
balance or scale. The force can then be related to
the velocity of the jet of fluid and in turn to the
rate of flow. The force developed by a jet stream
of water is the subject of this experiment.
Impact of a Jet of Liquid
Jet Impact Apparatus
Figure 5.1 is a schematic of the device used in
this experiment. The device consists of a tank
within a tank. The interior tank is supported on a
pivot and has a lever arm attached to it. As
water enters this inner tank, the lever arm will
reach a balance point. At this time, a stopwatch
is started and a weight is placed on the weight
hanger (e.g., 10 lbf). When enough water has
entered the tank (10 lbf), the lever arm will
again balance. The stopwatch is stopped. The
elapsed time divided into the weight of water
collected gives the weight or mass flow rate of
water through the system (lbf/sec, for example).
The outer tank acts as a support for the table
top as well as a sump tank. Water is pumped from
the outer tank to the apparatus resting on the
table top. As shown in Figure 5.1, the impact
apparatus contains a nozzle that produces a high
velocity jet of water. The jet is aimed at an object
(such as a flat plate or hemisphere). The force
exerted on the plate causes the balance arm to
which the plate is attached to deflect. A weight
is moved on the arm until the arm balances. A
summation of moments about the pivot point of
the arm allows for calculating the force exerted
by the jet.
Water is fed through the nozzle by means of
a centrifugal pump. The nozzle emits the water in
a jet stream whose diameter is constant. After the
water strikes the object, the water is channeled to
the weighing tank inside to obtain the weight or
mass flow rate.
The variables involved in this experiment
are listed and their measurements are described
1. Mass rate of flow–measured with the
weighing tank inside the sump tank. The
volume flow rate is obtained by dividing
mass flow rate by density: Q = m/ρ.
2. Velocity of jet–obtained by dividing volume
flow rate by jet area: V = Q/A. The jet is
cylindrical in shape with a diameter of 0.375
3. Resultant force—found experimentally by
summation of moments about the pivot point
of the balance arm. The theoretical resultant
force is found by use of an equation derived by
applying the momentum equation to a control
volume about the plate.
Impact Force Analysis
The total force exerted by the jet equals the
rate of momentum loss experienced by the jet after
it impacts the object. For a flat plate, the force
For a hemisphere,
For a cone whose included half angle is α,
(1 + cos α)
For your report, derive the appropriate
equation for each object you use. Compose a graph
with volume flow rate on the horizontal axis,
and on the vertical axis, plot the actual and
theoretical force. Use care in choosing the
increments for each axis.
lever arm with
flat plate attached
FIGURE 5.1. A schematic of the jet impact apparatus.
CRITICAL REYNOLDS NUMBER IN PIPE FLOW
The Reynolds number is a dimensionless ratio
of inertia forces to viscous forces and is used in
identifying certain characteristics of fluid flow.
The Reynolds number is extremely important in
modeling pipe flow. It can be used to determine
the type of flow occurring: laminar or turbulent.
Under laminar conditions the velocity
distribution of the fluid within the pipe is
essentially parabolic and can be derived from the
equation of motion. When turbulent flow exists,
the velocity profile is “flatter” than in the
laminar case because the mixing effect which is
characteristic of turbulent flow helps to more
evenly distribute the kinetic energy of the fluid
over most of the cross section.
In most engineering texts, a Reynolds number
of 2 100 is usually accepted as the value at
transition; that is, the value of the Reynolds
number between laminar and turbulent flow
regimes. This is done for the sake of convenience.
In this experiment, however, we will see that
transition exists over a range of Reynolds numbers
and not at an individual point.
The Reynolds number that exists anywhere in
the transition region is called the critical
Reynolds number. Finding the critical Reynolds
number for the transition range that exists in pipe
flow is the subject of this experiment.
dye will flow downstream in a threadlike
pattern for very low flow rates. Once steady state
is achieved, the rotameter valve is opened
slightly to increase the water flow rate. The
valve at B is opened further if necessary to allow
more dye to enter the tube. This procedure of
increasing flow rate of water and of dye (if
necessary) is repeated throughout the
Establish laminar flow in one of the tubes.
Then slowly increase the flow rate and observe
what happens to the dye. Its pattern may
change, yet the flow might still appear to be
laminar. This is the beginning of transition.
Continue increasing the flow rate and again
observe the behavior of the dye. Eventually, the
dye will mix with the water in a way that will
be recognized as turbulent flow. This point is the
end of transition. Transition thus will exist over a
range of flow rates. Record the flow rates at key
points in the experiment. Also record the
temperature of the water.
The object of this procedure is to determine
the range of Reynolds numbers over which
transition occurs. Given the tube size, the
Reynolds number can be calculated with:
Critical Reynolds Number Measurement
Critical Reynolds Number Determination
Figure 6.1 is a schematic of the apparatus
used in this experiment. The constant head tank
provides a controllable, constant flow through
the transparent tube. The flow valve in the tube
itself is an on/off valve, not used to control the
flow rate. Instead, the flow rate through the tube
is varied with the rotameter valve at A. The
head tank is filled with water and the overflow
tube maintains a constant head of water. The
liquid is then allowed to flow through one of the
transparent tubes at a very low flow rate. The
valve at B controls the flow of dye; it is opened
and dye is then injected into the pipe with the
water. The dye injector tube is not to be placed in
the pipe entrance as it could affect the results.
Establish laminar flow by starting with a very
low flow rate of water and of dye. The injected
where V (= Q/A) is the average velocity of
liquid in the pipe, D is the hydraulic diameter of
the pipe, and ν is the kinematic viscosity of the
The hydraulic diameter is calculated from
4 x Area
For a circular pipe flowing full, the hydraulic
diameter equals the inside diameter of the pipe.
For a square section, the hydraulic diameter will
equal the length of one side (show that this is
the case). The experiment is to be performed for
both round tubes and the square tube. With good
technique and great care, it is possible for the
transition Reynolds number to encompass the
traditionally accepted value of 2 100.
1. Can a similar procedure be followed for
2. Is the Reynolds number obtained at
transition dependent on tube size or shape?
3. Can this method work for opaque liquids?
FIGURE 6.1. The critical Reynolds number determination apparatus.
FLUID METERS IN INCOMPRESSIBLE FLOW
There are many different meters used in pipe
flow: the turbine type meter, the rotameter, the
orifice meter, the venturi meter, the elbow meter
and the nozzle meter are only a few. Each meter
works by its ability to alter a certain physical
characteristic of the flowing fluid and then
allows this alteration to be measured. The
measured alteration is then related to the flow
rate. A procedure of analyzing meters to
determine their useful features is the subject of
The Venturi Meter
The venturi meter is constructed as shown in
Figure 7.1. It contains a constriction known as the
throat. When fluid flows through the
constriction, it must experience an increase in
velocity over the upstream value. The velocity
increase is accompanied by a decrease in static
pressure at the throat. The difference between
upstream and throat static pressures is then
measured and related to the flow rate. The
greater the flow rate, the greater the pressure
drop ∆p. So the pressure difference ∆h (= ∆p/ρg)
can be found as a function of the flow rate.
and substituting from the hydrostatic equation, it
can be shown after simplification that the
volume flow rate through the venturi meter is
Qth = A2
2g ∆ h
1 - (D 24/D 14)
The preceding equation represents the theoretical
volume flow rate through the venturi meter.
Notice that is was derived from the Bernoulli
equation which does not take frictional effects
In the venturi meter, there exists small
pressure losses due to viscous (or frictional)
effects. Thus for any pressure difference, the
actual flow rate will be somewhat less than the
theoretical value obtained with Equation 7.1
above. For any ∆h, it is possible to define a
coefficient of discharge Cv as
For each and every measured actual flow rate
through the venturi meter, it is possible to
calculate a theoretical volume flow rate, a
Reynolds number, and a discharge coefficient.
The Reynolds number is given by
where V 2 is the velocity at the throat of the
meter (= Q ac/A 2).
FIGURE 7.1. A schematic of the Venturi meter.
Using the hydrostatic equation applied to
the air-over-liquid manometer of Figure 7.1, the
pressure drop and the head loss are related by
p 1 - p2
By combining the continuity equation,
Q = A1 V 1 = A2 V 2
with the Bernoulli equation,
p 1 V 12 p2 V 22
The Orifice Meter and
The orifice and nozzle-type meters consist of
a throttling device (an orifice plate or bushing,
respectively) placed into the flow. (See Figures
7.2 and 7.3). The throttling device creates a
measurable pressure difference from its upstream
to its downstream side. The measured pressure
difference is then related to the flow rate. Like
the venturi meter, the pressure difference varies
with flow rate. Applying Bernoulli’s equation to
points 1 and 2 of either meter (Figure 7.2 or Figure
7.3) yields the same theoretical equation as that
for the venturi meter, namely, Equation 7.1. For
any pressure difference, there will be two
associated flow rates for these meters: the
theoretical flow rate (Equation 7.1), and the
actual flow rate (measured in the laboratory).
The ratio of actual to theoretical flow rate leads
to the definition of a discharge coefficient: Co for
the orifice meter and Cn for the nozzle.
FIGURE 7.4. A schematic of a turbine-type flow
FIGURE 7.2. Cross sectional view of the orifice
FIGURE 7.3. Cross sectional view of the nozzletype meter, and a typical nozzle. For each and every measured actual flow
rate through the orifice or nozzle-type meters, it
is possible to calculate a theoretical volume flow
rate, a Reynolds number and a discharge
coefficient. The Reynolds number is given by
The Turbine-Type Meter
The turbine-type flow meter consists of a
section of pipe into which a small “turbine” has
been placed. As the fluid travels through the
pipe, the turbine spins at an angular velocity
that is proportional to the flow rate. After a
certain number of revolutions, a magnetic pickup
sends an electrical pulse to a preamplifier which
in turn sends the pulse to a digital totalizer. The
totalizer totals the pulses and translates them
into a digital readout which gives the total
volume of liquid that travels through the pipe
and/or the instantaneous volume flow rate.
Figure 7.4 is a schematic of the turbine type flow
The Rotameter (Variable Area Meter)
The variable area meter consists of a tapered
metering tube and a float which is free to move
inside. The tube is mounted vertically with the
inlet at the bottom. Fluid entering the bottom
raises the float until the forces of buoyancy, drag
and gravity are balanced. As the float rises the
annular flow area around the float increases.
Flow rate is indicated by the float position read
against the graduated scale which is etched on
the metering tube. The reading is made usually at
the widest part of the float. Figure 7.5 is a sketch
of a rotameter.
FIGURE 7.5. A schematic of the rotameter and its
Rotameters are usually manufactured with
one of three types of graduated scales:
1. % of maximum flow–a factor to convert scale
reading to flow rate is given or determined for
the meter. A variety of fluids can be used
with the meter and the only variable
encountered in using it is the scale factor. The
scale factor will vary from fluid to fluid.
Diameter-ratio type–the ratio of cross
sectional diameter of the tube to the
diameter of the float is etched at various
locations on the tube itself. Such a scale
requires a calibration curve to use the meter.
Direct reading–the scale reading shows the
actual flow rate for a specific fluid in the
units indicated on the meter itself. If this
type of meter is used for another kind of fluid,
then a scale factor must be applied to the
Fluid Meters Apparatus
The fluid meters apparatus is shown
schematically in Figure 7.6. It consists of a
centrifugal pump, which draws water from a
sump tank, and delivers the water to the circuit
containing the flow meters. For nine valve
positions (the valve downstream of the pump),
record the pressure differences in each
manometer. For each valve position, measure the
actual flow rate by diverting the flow to the
volumetric measuring tank and recording the time
required to fill the tank to a predetermined
volume. Use the readings on the side of the tank
itself. For the rotameter, record the position of
the float and/or the reading of flow rate given
directly on the meter. For the turbine meter,
record the flow reading on the output device.
Note that the venturi meter has two
manometers attached to it. The “inner”
manometer is used to calibrate the meter; that is,
to obtain ∆h readings used in Equation 7.1. The
“outer” manometer is placed such that it reads
the overall pressure drop in the line due to the
presence of the meter and its attachment fittings.
We refer to this pressure loss as ∆H (distinctly
different from ∆h). This loss is also a function of
flow rate. The manometers on the turbine-type
and variable area meters also give the incurred
loss for each respective meter. Thus readings of
∆H vs Qac are obtainable. In order to use these
parameters to give dimensionless ratios, pressure
coefficient and Reynolds number are used. The
Reynolds number is given in Equation 7.2. The
pressure coefficient is defined as
All velocities are based on actual flow rate and
The amount of work associated with the
laboratory report is great; therefore an informal
group report is required rather than individual
reports. The write-up should consist of an
Introduction (to include a procedure and a
derivation of Equation 7.1), a Discussion and
Conclusions section, and the following graphs:
1. On the same set of axes, plot Qac vs ∆h and
Q th vs ∆ h with flow rate on the vertical
axis for the venturi meter.
2. On the same set of axes, plot Qac vs ∆h and
Q th vs ∆ h with flow rate on the vertical
axis for the orifice meter.
3. Plot Qac vs Qth for the turbine type meter.
4. Plot Qac vs Qth for the rotameter.
5. Plot Cv vs Re on a log-log grid for the
6. Plot Co vs Re on a log-log grid for the orifice
7. Plot ∆H vs Qac for all meters on the same set
of axes with flow rate on the vertical axis.
8. Plot Cp vs Re for all meters on the same set
of axes (log-log grid) with C p vertical axis.
1. Referring to Figure 7.2, recall that
Bernoulli's equation was applied to points 1
and 2 where the pressure difference
measurement is made. The theoretical
equation, however, refers to the throat area
for point 2 (the orifice hole diameter)
which is not where the pressure
measurement was made. Explain this
discrepancy and how it is accounted for in
the equation formulation.
2. Which meter in your opinion is the best one
3. Which meter incurs the smallest pressure
loss? Is this necessarily the one that should
always be used?
4. Which is the most accurate meter?
5. What is the difference between precision
FIGURE 7.6. A schematic of the Fluid Meters Apparatus. (Orifice and Venturi meters: upstream diameter is 1.025 inches; throat diameter is 0.625 inches.)
Experiments in pipe flow where the presence
of frictional forces must be taken into account are
useful aids in studying the behavior of traveling
fluids. Fluids are usually transported through
pipes from location to location by pumps. The
frictional losses within the pipes cause pressure
drops. These pressure drops must be known to
determine pump requirements. Thus a study of
pressure losses due to friction has a useful
application. The study of pressure losses in pipe
flow is the subject of this experiment.
With the pump on, record the assigned
pressure drops and the actual volume flow
rate from the rotameter.
Using the valve closest to the pump, change
the volume flow rate and again record the
pressure drops and the new flow rate value.
Repeat this procedure until 9 different
volume flow rates and corresponding pressure
drop data have been recorded.
With pressure loss data in terms of ∆h, the
friction factor can be calculated with
Pipe Flow Test Rig
Figure 8.1 is a schematic of the pipe flow test
rig. The rig contains a sump tank which is used as
a water reservoir from which a centrifugal pump
discharges water to the pipe circuit. The circuit
itself consists of four different diameter lines and
a return line all made of drawn copper tubing. The
circuit contains valves for directing and
regulating the flow to make up various series and
parallel piping combinations. The circuit has
provision for measuring pressure loss through the
use of static pressure taps (manometer board not
shown in schematic). Finally, because the circuit
also contains a rotameter, the measured pressure
losses can be obtained as a function of flow rate.
As functions of the flow rate, measure the
pressure losses in inches of water for (as specified
by the instructor):
1. 1 in. copper tube
2. 3/4-in. copper tube
3. 1/2-in copper tube
4. 3/8 in copper tube
5. 1 in. 90 T-joint
6. 1 in. 90 elbow (ell)
7. 1 in. gate valve
8. 3/ 4-in gate valve
The instructor will specify which of the
pressure loss measurements are to be taken.
Open and close the appropriate valves on the
apparatus to obtain the desired flow path.
Use the valve closest to the pump on its
downstream side to vary the volume flow
2g ∆ h
V 2 (L/D)
It is customary to graph the friction factor as a
function of the Reynolds number:
The f vs Re graph, called a Moody Diagram is
traditionally drawn on a log-log grid. The graph
also contains a third variable known as the
roughness coefficient ε/D. For this experiment
the roughness factor ε is that for drawn tubing.
Where fittings are concerned, the loss
incurred by the fluid is expressed in terms of a loss
coefficient K. The loss coefficient for any fitting
can be calculated with
where ∆h is the pressure (or head) loss across the
fitting. Values of K as a function of Qac are to be
obtained in this experiment.
For the report, calculate friction factor f and
graph it as a function of Reynolds number Re for
items 1 through 4 above as appropriate. Compare
to a Moody diagram. Also calculate the loss
coefficient for items 5 through 8 above as
appropriate, and determine if the loss coefficient
K varies with flow rate or Reynolds number.
Compare your K values to published ones.
Note that gate valves can have a number of
open positions. For purposes of comparison it is
often convenient to use full, half or one-quarter
static pressure tap
FIGURE 8.1. Schematic of the pipe friction apparatus.
PRESSURE DISTRIBUTION ABOUT A CIRCULAR CYLINDER
In many engineering applications, it may be
necessary to examine the phenomena occurring
when an object is inserted into a flow of fluid. The
wings of an airplane in flight, for example, may
be analyzed by considering the wings stationary
with air moving past them. Certain forces are
exerted on the wing by the flowing fluid that
tend to lift the wing (called the lift force) and to
push the wing in the direction of the flow (drag
force). Objects other than wings that are
symmetrical with respect to the fluid approach
direction, such as a circular cylinder, will
experience no lift, only drag.
Drag and lift forces are caused by the
pressure differences exerted on the stationary
object by the flowing fluid. Skin friction between
the fluid and the object contributes to the drag
force but in many cases can be neglected. The
measurement of the pressure distribution existing
around a stationary cylinder in an air stream to
find the drag force is the object of this
Consider a circular cylinder immersed in a
uniform flow. The streamlines about the cylinder
are shown in Figure 9.1. The fluid exerts pressure
on the front half of the cylinder in an amount
that is greater than that exerted on the rear
half. The difference in pressure multiplied by the
projected frontal area of the cylinder gives the
drag force due to pressure (also known as form
drag). Because this drag is due primarily to a
pressure difference, measurement of the pressure
distribution about the cylinder allows for finding
the drag force experimentally. A typical pressure
distribution is given in Figure 9.2. Shown in
Figure 9.2a is the cylinder with lines and
arrowheads. The length of the line at any point
on the cylinder surface is proportional to the
pressure at that point. The direction of the
arrowhead indicates that the pressure at the
respective point is greater than the free stream
pressure (pointing toward the center of the
cylinder) or less than the free stream pressure
(pointing away). Note the existence of a
separation point and a separation region (or
wake). The pressure in the back flow region is
nearly the same as the pressure at the point of
separation. The general result is a net drag force
equal to the sum of the forces due to pressure
acting on the front half (+) and on the rear half
(-) of the cylinder. To find the drag force, it is
necessary to sum the components of pressure at
each point in the flow direction. Figure 9.2b is a
graph of the same data as that in Figure 9.2a
except that 9.2b is on a linear grid.
FIGURE 9.1. Streamlines of flow about a circular
(a) Polar Coordinate Graph
(b) Linear Graph
FIGURE 9.2. Pressure distribution around a circular cylinder placed in a uniform flow. 24
A Wind Tunnel
A Right Circular Cylinder with Pressure
Figure 9.3 is a schematic of a wind tunnel. It
consists of a nozzle, a test section, a diffuser and a
fan. Flow enters the nozzle and passes through
flow straighteners and screens. The flow is
directed through a test section whose walls are
made of a transparent material, usually
Plexiglas or glass. An object is placed in the test
section for observation. Downstream of the test
section is the diffuser followed by the fan. In the
tunnel that is used in this experiment, the test
section is rectangular and the fan housing is
circular. Thus one function of the diffuser is to
gradually lead the flow from a rectangular
section to a circular one.
Figure 9.4 is a schematic of the side view of
the circular cylinder. The cylinder is placed in
the test section of the wind tunnel which is
operated at a preselected velocity. The pressure
tap labeled as #1 is placed at 0° directly facing
the approach flow. The pressure taps are
attached to a manometer board. Only the first 18
taps are connected because the expected profile is
symmetric about the 0° line. The manometers will
provide readings of pressure at 10° intervals
about half the cylinder. For two different
approach velocities, measure and record the
pressure distribution about the circular cylinder.
Plot the pressure distribution on polar coordinate
graph paper for both cases. Also graph pressure
difference (pressure at the point of interest minus
the free stream pressure) as a function of angle θ
on linear graph paper. Next, graph ∆p cosθ vs θ
(horizontal axis) on linear paper and determine
the area under the curve by any convenient
method (counting squares or a numerical
The drag force can be calculated by
integrating the flow-direction-component of each
pressure over the area of the cylinder:
Df = 2RL
∫ ∆p cosθdθ
The above expression states that the drag force is
twice the cylinder radius (2R) times the cylinder
length (L) times the area under the curve of ∆p
cosθ vs θ.
Drag data are usually expressed as drag
coefficient CD vs Reynolds number Re. The drag
coefficient is defined as
The Reynolds number is
FIGURE 9.3. A schematic of the wind tunnel used in this experiment.
where V is the free stream velocity (upstream of
the cylinder), A is the projected frontal area of
the cylinder (2RL), D is the cylinder diameter, ρ
is the air density and µ is the air viscosity.
Compare the results to those found in texts.
taps attach to
FIGURE 9.4. Schematic of the experimental
apparatus used in this experiment.
DRAG FORCE DETERMINATION
An object placed in a uniform flow is acted
upon by various forces. The resultant of these
forces can be resolved into two force components,
parallel and perpendicular to the main flow
direction. The component acting parallel to the
flow is known as the drag force. It is a function of
a skin friction effect and an adverse pressure
gradient. The component perpendicular to the
flow direction is the lift force and is caused by a
pressure distribution which results in a lower
pressure acting over the top surface of the object
than at the bottom. If the object is symmetric
with respect to the flow direction, then the lift
force will be zero and only a drag force will exist.
Measurement of the drag force acting on an object
immersed in the uniform flow of a fluid is the
subject of this experiment.
Subsonic Wind Tunnel
A description of a subsonic wind tunnel is
given in Experiment 9 and is shown schematically
in Figure 9.3. The fan at the end of the tunnel
draws in air at the inlet. An object is mounted on a
stand that is pre calibrated to read lift and drag
forces exerted by the fluid on the object. A
schematic of the test section is shown in Figure
10.1. The velocity of the flow at the test section is
also pre calibrated. The air velocity past the
object can be controlled by changing the angle of
the inlet vanes located within the fan housing.
Thus air velocity, lift force and drag force are
read directly from the tunnel instrumentation.
There are a number of objects that are
available for use in the wind tunnel. These
include a disk, a smooth surfaced sphere, a rough
surface sphere, a hemisphere facing upstream,
and a hemisphere facing downstream. For
whichever is assigned, measure drag on the object
as a function of velocity.
Data on drag vs velocity are usually graphed
in dimensionless terms. The drag force Df is
customarily expressed in terms of the drag
coefficient CD (a ratio of drag force to kinetic
in which ρ is the fluid density, V is the free
stream velocity, and A is the projected frontal
area of the object. Traditionally, the drag
coefficient is graphed as a function of the
Reynolds number, which is defined as
where D is a characteristic length of the object
and ν is the kinematic viscosity of the fluid. For
each object assigned, graph drag coefficient vs
Reynolds number and compare your results to
those published in texts. Use log-log paper if
1. How does the mounting piece affect the
2. How do you plan to correct for its effect, if
FIGURE 10.1. Schematic of an object mounted in
the test section of the wind tunnel.
ANALYSIS OF AN AIRFOIL
A wing placed in the uniform flow of an
airstream will experience lift and drag forces.
Each of these forces is due to a pressure
difference. The lift force is due to the pressure
difference that exists between the lower and
upper surfaces. This phenomena is illustrated in
Figure 11.1. As indicated the airfoil is immersed
in a uniform flow. If pressure could be measured at
selected locations on the surface of the wing and
the results graphed, the profile in Figure 11.1
would result. Each pressure measurement is
represented by a line with an arrowhead. The
length of each line is proportional to the
magnitude of the pressure at the point. The
direction of the arrow (toward the horizontal
axis or away from it) represents whether the
pressure at the point is less than or greater than
the free stream pressure measured far upstream of
Mount the wing with pressure taps in the
tunnel and attach the tube ends to manometers.
Select a wind speed and record the pressure
distribution for a selected angle of attack (as
assigned by the instructor). Plot pressure vs chord
length as in Figure 11.1, showing the vertical
component of each pressure acting on the upper
surface and on the lower surface. Determine
where separation occurs for each case.
Mount the second wing on the lift and drag
balance (Figure 11.2). For the same wind speed
and angle of attack, measure lift and drag exerted
on the wing.
gradient on upper
on lower surface
FIGURE 11.2. Schematic of lift and drag
measurement in a test section.
FIGURE 11.1. Streamlines of flow about a wing
and the resultant pressure distribution.
Lift and Drag Measurements for a Wing
Wind Tunnel (See Figure 9.3)
Wing with Pressure Taps
Wing for Attachment to Lift & Drag
Instruments (See Figure 11.2)
The wing with pressure taps provided
pressure at selected points on the surface of the
wing. Use the data obtained and sum the
horizontal component of each pressure to obtain
the drag force. Compare to the results obtained
with the other wing. Use the data obtained and
sum the vertical component of each pressure to
obtain the lift force. Compare the results
obtained with the other wing. Calculate %
For a number of wings, lift and drag data
vary only slightly with Reynolds number and
therefore if lift and drag coefficients are graphed
as a function of Reynolds number, the results are
not that meaningful. A more significant
representation of the results is given in what is
known as a polar diagram for the wing. A polar
diagram is a graph on a linear grid of lift
coefficient (vertical axis) as a function of drag
coefficient. Each data point on the graph
corresponds to a different angle of attack, all
measured at one velocity (Reynolds number).
Referring to Figure 11.2 (which is the
experimental setup here), the angle of attack α is
measured from a line parallel to the chord c to a
line that is parallel to the free stream velocity.
If so instructed, obtain lift force, drag force and
angle of attack data using a pre selected velocity.
Allow the angle of attack to vary from a negative
angle to the stall point and beyond. Obtain data
at no less than 9 angles of attack. Use the data to
produce a polar diagram.
Lift and drag data are usually expressed in
dimensionless terms using lift coefficient and drag
coefficient. The lift coefficient is defined as
where Lf is the lift force, ρ is the fluid density, V
is the free stream velocity far upstream of the
wing, and A is the area of the wing when seen
from a top view perpendicular to the chord
length c. The drag coefficient is defined as
in which Df is the drag force.
OPEN CHANNEL FLOW—SLUICE GATE
Liquid motion in a duct where a surface of the
fluid is exposed to the atmosphere is called open
channel flow. In the laboratory, open channel
flow experiments can be used to simulate flow in a
river, in a spillway, in a drainage canal or in a
sewer. Such modeled flows can include flow over
bumps or through dams, flow through a venturi
flume or under a partially raised gate (a sluice
gate). The last example, flow under a sluice gate,
is the subject of this experiment.
In terms of flow rate, the velocities are written as
Flow Through a Sluice Gate
Open Channel Flow Apparatus
Sluice Gate Model
Figure 12.1 shows a schematic of the side
view of the sluice gate. Flow upstream of the gate
has a depth h o while downstream the depth is h.
The objective of the analysis is to formulate an
equation to relate the volume flow rate through
(or under) the gate to the upstream and
V 2 V 02
where b is the channel width at the gate.
Substituting into the Bernoulli Equation and
- 1 +h
2gb2 h 2 h 0 2
Dividing by h 0,
1 - 1 + h
2gb 2 h 0 h 2 h 0 2 h 0
1 - h = Q 2 1 - h 2
h 0 2gb 2 h h 0
FIGURE 12.1. Schematic of flow under a sluice
The flow rate through the gate is maintained at
nearly a constant value. For various raised
positions of the sluice gate, different liquid
heights h o and h will result. Applying the
Bernoulli equation to flow about the gate gives
p 0 V 02
+ h0 =
Pressures at the free surface are both equal to
atmospheric pressure, so they cancel. Rearranging
Multiplying both sides by h2/h02, and continuing
to simplify, we finally obtain
1 + h/h 0 2gb 2 h 0 3
here Q is the theoretical volume flow rate. The
right hand side of this equation is recognized as
1/2 of the upstream Froude number. So by
measuring the depth of liquid before and after
the sluice gate, the theoretical flow rate can be
calculated with the above equation. The
theoretical flow rate can then be compared to the
actual flow rate obtained by measurements using
the orifice meters.
For 9 different raised positions of the sluice
gate, measure the upstream and downstream
depths and calculate the actual flow rate. In
addition, calculate the upstream Froude number
for each case and determine its value for
maximum flow conditions. Graph h/h0 (vertical
axis) versus (Q 2/b 2h 03g). Determine h/h0
corresponding to maximum flow. Note that h/h0
varies from 0 to 1.
Figure 12.2 is a sketch of the open channel
flow apparatus. It consists of a sump tank with a
pump/motor combination on each side. Each pump
draws in water from the sump tank and
discharges it through the discharge line to
calibrated orifice meters and then to the head
tank. Each orifice meter is connected to its own
manometer. Use of the calibration curve
(provided by the instructor) allows for finding
the actual flow rate into the channel. The head
tank and flow channel have sides made of
Plexiglas. Water flows downstream in the
channel past the object of interest (in this case a
sluice gate) and then is routed back to the sump
1. For the required report, derive the sluice
gate equation in detail.
2. What if it was assumed that V 0