A Manual for the
removable glass cover
manifold with injectors
dye water flow
grid on surface
manifold with injectors
grid on surface
William S. Janna
Department of Mechanical Engineering
The University of Memphis
TABLE OF CONTENTS
Course Learning Outcomes, Cleanliness and Safety................................................4
Code of Student Conduct ...............................................................................................5
Statistical Treatment of Experimental.........................................................................6
Experiment 1 Density and Surface Tension...................................................18
Experiment 2 Viscosity.......................................................................................20
Experiment 3 Center of Pressure on a Submerged Plane Surface.............21
Experiment 4 Impact of a Jet of Water ............................................................23
Experiment 5 Critical Reynolds Number in Pipe Flow...............................26
Experiment 6 Fluid Meters................................................................................28
Experiment 7 Pipe Flow .....................................................................................32
Experiment 8 Pressure Distribution About a Circular Cylinder................34
Experiment 9 Drag Force Determination .......................................................37
Experiment 10 Analysis of an Airfoil................................................................38
Experiment 11 Open Channel Flow—Sluice Gate .........................................40
Experiment 12 Open Channel Flow Over a Weir ..........................................42
Experiment 13 Open Channel Flow—Hydraulic Jump ................................44
Experiment 14 Measurement of Pump Performance....................................46
Experiment 15 Measurement of Velocity and Calibration of
a Meter for Compressible Flow.............................50
Experiment 16 Measurement of Fan Horsepower .........................................55
Experiment 17 External Laminar Flows Over Immersed Bodies................57
Experiment 18 Series-Parallel Pump Performance ........................................59
Experiment 19 Design of Experiments: Calibration of an Elbow Meter.....63
Experiment 20 Design of Experiments: Measurement of Force on a
Conical Object ...........................................................65
Course Learning Outcomes
The Fluid Mechanics Laboratory experiments are
set up so that experiments can be performed to
complement the theoretical information taught
in the fluid mechanics lecture course. Thus topical
areas have been identified and labeled as Course
Learning Outcomes (CLOs). The CLOs in the
MECH 3335 Laboratory are as follows:
TABLE 1. Course Learning Outcomes
1. Identify safe operating practices and
requirements for laboratory experiments
2. Measure fluid properties
3. Measure hydrostatic forces on a submerged
4. Use flow meters to measure flow rate in a
5. Measure pressure loss due to friction for pipe
6. Measure drag/lift forces on objects in a flow,
or measure flow rate over a weir
7. Design and conduct an experiment, as well as
analyze and interpret data
8. Function effectively as a member of a team
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 heavily.
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,
unused equipment stored in the lab, etc. The goal
is to implement safety requirements without
impeding egress, but still allowing adequate work
space and necessary informal communication
Distance between adjacent pieces of
equipment is determined by locations of water
supply valves, floor drains, electrical outlets,
and by the need to allow enough space around the
apparatus of interest. Immediate access to the
Safety Cabinet and the Fire Extinguisher is also
considered. We do not work with hazardous
materials and safety facilities such as showers,
eye wash fountains, spill kits, fire blankets, etc.,
are not necessary.
Safety Procedures. There are five exit doors in
this lab, two of which lead to other labs. One
exit has a double door and leads directly to the
hallway on the first floor of the Engineering
Building. Another exit is a single door that also
leads to the hallway. The fifth exit leads
directly outside to the parking lot. In case of fire,
the doors to the hallway should be closed, and
the lab should be exited to the parking lot.
There is a safety cabinet attached to the
wall of the lab adjacent to the double doors. In
case of personal injury, the appropriate item
should be taken from the supply cabinet and used
in the recommended fashion. If the injury is
serious enough to require professional medical
attention, the student(s) should contact the Civil
Engineering Department in EN 104, Extension
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.
THE CODE OF STUDENT CONDUCT
Taken from The University of Memphis
1998–1999 Student Handbook
Institution Policy Statement
The University of Memphis students are citizens
of the state, local, and national governments, and
of the academic community. They are, therefore,
expected to conduct themselves as law abiding
members of each community at all times.
Admission to the University carries with it
special privileges and imposes special
responsibilities apart from those rights and
duties enjoyed by non-students. In recognition of
this special relationship that exists between the
institution and the academic community which it
seeks to serve, the Tennessee Board of Regents
has, as a matter of public record, instructed “the
presidents of the universities and colleges under
its jurisdiction to take such action as may be
necessary to maintain campus conditions…and to
preserve the integrity of the institution and its
The following regulations (known as the Code
of Student Conduct) have been developed by a
committee made up of faculty, students, and staff
utilizing input from all facets of the University
Community in order to provide a secure and
stimulating atmosphere in which individual and
academic pursuits may flourish. Students are,
however, subject to all national, state and local
laws and ordinances. If a student’s violation of
such laws or ordinances also adversely affects the
University’s pursuit of its educational objectives,
the University may enforce its own regulations
regardless of any proceeding instituted by other
authorities. Additionally, violations of any
section of the Code may subject a student to
disciplinary measures by the University whether
or not such conduct is simultaneously violative of
state, local or national laws.
The term “academic misconduct” includes, but
is not limited to, all acts of cheating and
The term “cheating” includes, but is not limited
a. use of any unauthorized assistance in taking
quizzes, tests, or examinations;
b. dependence upon the aid of sources beyond
those authorized by the instructor in writing
papers, preparing reports, solving problems,
or carrying out other assignments;
c. the acquisition, without permission, of tests
or other academic material before such
material is revealed or distributed by the
d. the misrepresentation of papers, reports,
assignments or other materials as the product
of a student’s sole independent effort, for the
purpose of affecting the student’s grade,
credit, or status in the University;
e. failing to abide by the instructions of the
proctor concerning test-taking procedures;
examples include, but are not limited to,
talking, laughing, failure to take a seat
assignment, failing to adhere to starting and
stopping times, or other disruptive activity;
f. influencing, or attempting to influence, any
University official, faculty member,
graduate student or employee possessing
academic grading and/or evaluation
authority or responsibility for maintenance of
academic records, through the use of bribery,
threats, or any other means or coercion in
order to affect a student’s grade or
g. any forgery, alteration, unauthorized
possession, or misuse of University documents
pertaining to academic records, including, but
not limited to, late or retroactive change of
course application forms (otherwise known as
“drop slips”) and late or retroactive
withdrawal application forms. Alteration or
misuse of University documents pertaining to
academic records by means of computer
resources or other equipment is also included
within this definition of “cheating.”
The term “plagiarism” includes, but is not limited
to, the use, by paraphrase or direct quotation, of
the published or unpublished work of another
person without full or clear acknowledgment. It
also includes the unacknowledged use of
materials prepared by another person or agency
engaged in the selling of term papers or other
Academic misconduct (acts of cheating and of
plagiarism) will not be tolerated. The Student
Handbook is quite specific regarding the course of
action to be taken by an instructor in cases where
academic misconduct may be an issue.
Statistical Treatment of Experimental Data
This laboratory course concerns making
measurements in various fluid situations and
geometries, and relating results of those
measurements to derived equations. The objective
is to determine how well the derived equations
describe the physical phenomena we are
modeling. In doing so, we will need to make
physical measurements, and it is essential that
we learn how to practice good techniques in
making scientific observations and in obtaining
measurements. We are making quantitative
estimates of physical phenomena under
There are certain primary desirable
characteristics involved when making these
physical measurements. We wish that our
measurements would be:
a ) Observer-independent,
b) Consistent, and
So when reporting a measurements, we will be
stating a number. Furthermore, we will have to
add a dimension because a physical value
without a unit has no significance. In reporting
measurements, a question arises as to how should
we report data; i.e., how many significant digits
should we include? Which physical quantity is
associated with the measurement, and how
precise should it or could it be? It is prudent to
scrutinize the claimed or implied accuracy of a
In the course of performing an experiment, we
first would develop a set of questions or a
hypothesis, or put forth the theory. We then
identify the system variables to be measured or
controlled. The apparatus would have to be
developed and the equipment set up in a
particular way. An experimental protocol, or
procedure, is established and data are taken.
Several features of this process are
important. We want accuracy in our
measurements, but increased accuracy generally
corresponds to an increase in cost. We want the
experiments to be reproducible, and we seek to
minimize errors. Of course we want to address all
safety issues and regulations.
After we run the experiment, and obtain data,
we would analyze the results, draw conclusions,
and report the results.
Comments on Performing Experiments
• Keep in mind the fundamental state of
questions or hypotheses.
• Make sure the experiment design will answer
the right questions.
• Use estimation as a reality check, but do not
let it affect objectivity.
• Consider all possible safety issues.
• Design for repeatability and the appropriate
level of accuracy.
Error & Uncertainty—Definitions
The fluid mechanics laboratory is designed to
provide the students with experiments that
verify the descriptive equations we derive to
model physical phenomena. The laboratory
experience involves making measurements of
depth, area, and flow rate among other things. In
the following paragraphs, we will examine our
measurement methods and define terms that
apply. These terms include error, uncertainty,
accuracy, and precision.
Error. The error E is the difference between a
TRUE value, x, and a MEASURED value, xi:
E x xi= − (1)
There is no error-free measurement. All
measurements contain some error. How error is
defined and used is important. The significance of
a measurement cannot be judged unless the
associated error has been reliably estimated. In
Equation 1, because the true value of x is unknown,
the error E is unknown as well. This is always the
The best we can hope for is to obtain the
estimate of a likely error, which is called an
uncertainty. For multiple measurements of the
same quantity, a mean value, x, (also called a
nominal value) can be calculated. Hence, the
E x x= −
However, because x is unknown, E is still
Uncertainty. The uncertainty, ∆x, is an estimate
of E as a possible range of errors:
∆x E≈ (2)
For example, suppose we measure a velocity and
report the result as
V = 110 m/s ± 5 m/s
The value of ± 5 m/s is defined as the uncertainty.
Alternatively, suppose we report the results as
V = 110 m/s ± 4.5%
The value of ± 4.5% is defined as the relative
uncertainty. It is common to hear someone speak
of “experimental errors,” when the correct
terminology should be “uncertainty.” Both terms
are used in everyday language, but it should be
remembered that the uncertainty is defined as an
estimate of errors.
Accuracy. Accuracy is a measure (or an estimate)
of the maximum deviation of measured values, xi,
from the TRUE value, x:
accuracy estimate of x xi= −max (3)
Again, because the true value x is unknown, then
the value of the maximum deviation is unknown.
The accuracy, then, is only an estimate of the
worst error. It is usually expressed as a
percentage; e.g., “accurate to within 5%.”
Accuracy and Precision. As mentioned, accuracy is
a measure (or an estimate) of the maximum
deviation of measured values from the true value.
So a question like:
“Are the measured values accurate?”
can be reformulated as
“Are the measured values close to the true
Accuracy was defined in Equation 3 as
accuracy estimate of x xi= −max (3)
Precision, on the other hand, is a measure (or an
estimate) of the consistency (or repeatability).
Thus it is the maximum deviation of a reading
(measurement), xi, from its mean value, x :
precision estimate of x xi= −max
Note the difference between accuracy and
Regarding the definition of precision, there is
no true value identified, only the mean value (or
average) of a number of repeated measurements of
the same quantity. Precision is a characteristic of
the measurement. In everyday language we often
conclude that “accuracy” and “precision” are the
same, but in error analysis there is a difference.
So a question like:
“Are the measured values precise?”
can be reformulated as
“Are the measured values close to each
As an illustration of the concepts of accuracy and
precision, consider the dart board shown in the
accompanying figures. Let us assume that the blue
darts show the measurements taken, and that the
bullseye represents the value to be measured.
When all measurements are clustered about the
bullseye, then we have very accurate and,
therefore, precise results (Figure 1a).
When all measurements are clustered
together but not near the bullseye, then we have
very precise but not accurate results (Figure 1b).
When all measurements are not clustered
together and not near the bullseye, but their
nominal value or average is the bullseye, then we
have accurate (on average) but not precise results
When all measurements are not clustered
together and not near the bullseye, and their
average is the not at the bullseye, then we have
neither accurate nor precise results (Figure 1d).
We conclude that accuracy refers to the
correctness of the measurements, while precision
refers to their consistency.
Classification of Errors
Random error. A random error is one that arises
from a random source. Suppose for example that a
measurement is made many thousands of times
using different instruments and/or observers
and/or samples. We would expect to have random
errors affecting the measurement in either
direction (±) roughly the same number of times.
Such errors can occur in any scenario:
• Electrical noise in a circuit generally produces
a voltage error that may be positive or
negative by a small amount.
FIGURE 1a. Accurate and Precise
FIGURE 1b. Precise but not Accurate.
FIGURE 1c. Precise but not Accurate.
FIGURE 1d. Neither Precise nor Accurate.
• By counting the total number of pennies in a
large container, one may occasionally pick up
two and count only one (or vice versa).
The question arises as to how can we reduce
random errors? There are no random error free
measurements. So random errors cannot be
eliminated, but their magnitude can be reduced.
On average, random errors tend to cancel out.
Systematic Error. A systematic error is one that is
consistent; that is, it happens systematically.
Typically, human components of measurement
systems are often responsible for systematic
errors. For example, systematic errors are common
in reading of a pressure indicated by an inclined
Consider an experiment involving dropping a
ball from a given height. We wish to measure the
time it takes for the ball to move from where it is
dropped to when it hits the ground. We might
repeat this experiment several times. However,
the person using the stopwatch may consistently
have a tendency to wait until the ball bounces
before the watch is stopped. As a result, the time
measurement might be systematically too long.
Systematic measurements can be anticipated
and/or measured, and then corrected. This can be
done even after the measurements are made.
The question arises as to how can we reduce
systematic errors? This can be done in several
1. Calibrate the instruments being used by
checking with a known standard. The
standard can be what is referred to as:
a) a primary standard obtained from the
“National Institute of standards and
technology” (NIST— formerly the National
Bureau of Standards); or
b) a secondary standard (with a higher
accuracy instrument); or
c) A known input source.
2. Make several measurements of a certain
quantity under varying test conditions, such
as different observers and/or samples and/or
3. Check the apparatus.
4. Check the effects of external conditions
5. Check the coherence of results.
A repeatability test using the same instrument is
one way of gaining confidence, but a far more
reliable way is to use an entirely different
method to measure the desired quantity.
Determining Uncertainty. When we state a
measurement that we have taken, we should also
state an estimate of the error, or the uncertainty.
As a rule of thumb, we use a 95% relative
uncertainty, or stated otherwise, we use a 95%
Suppose for example, that we report the
height of a desk to be 38 inches ± 1 inch. This
suggests that we are 95% sure that the desk is
between 37 and 39 inches tall.
When reporting relative uncertainty, we
generally restrict the result to having one or two
significant figures. When reporting uncertainty in
a measurement using units, we use the same
number of significant figures as the measured
value. Examples are shown in Table 1:
TABLE 1. Examples of relative and absolute
Relative uncertainty Uncertainty in units
3.45 cm ± 8.5% 5.23 cm ± 0.143 cm
6.4 N ± 2.0% 2.5 m/s ± 0.082 m/s
2.3 psi ± 0.1900% 9.25 in ± 0.2 in
9.2 m/s ± 8.598% 3.2 N ± 0.1873 N
The previous tables shows uncertainty in
measurements, but to determine uncertainty is
usually difficult. However, because we are using
a 95% confidence interval, we can obtain an
estimage. The estimate of uncertainty depends on
the measurement type: single sample
measurements, measurements of dependent
variables, or multi variable measurements.
Single-sample measurements. Single-sample
measurements are those in which the
uncertainties cannot be reduced by repetition. As
long as the test conditions are the same (i.e., same
sample, same instrument and same observer), the
measurements (for fixed variables) are single-
sample measurements, regardless of how many
times the reading is repeated.
Single-sample uncertainty. It is often simple to
identify the uncertainty of an individual
measurement. It is necessary to consider the limit
of the “scale readability,” and the limit
associated with applying the measurement tool
to the case of interest.
Measurement Of Function Of More Than One
Independent Variables. In many cases, several
different quantities are measured in order to
calculate another quantity—a dependent
variable. For example, the measurement of the
surface area of a rectangle is calculated using
both its measured length and its measured width.
Such a situation involves a propagation of
Consider some measuring device that has as
its smallest scale division δx. The smallest scale
division limits our ability to measure something
with any more accuracy than δx/2. The ruler of
Figure 2a, as an example, has 1/4 inch as its
smallest scale division. The diameter of the
circle is between 4 and 4 1/4 inches. So we would
correctly report that
D = 41/8 ± 1/8 in.
This is the correct reported measurement for
Figure 2a and Figure 2b, even though the circles
are of different diameters. We can “guesstimate”
the correct measurement, but we cannot report
something more accurately than our measuring
apparatus will display. This does not mean that
the two circles have the same diameter, merely
that we cannot measure the diameters with a
greater accuracy than the ruler we use will allow.
0 1 2 3 4 5 6
0 1 2 3 4 5 6
FIGURE 2. A ruler used to measure the diameter
of a circle.
The ruler depicted in the figure could be any
arbitrary instrument with finite resolution. The
uncertainty due to the resolution of any
instrument is one half of the smallest increment
displayed. This is the most likely single sample
uncertainty. It is also the most optimistic because
reporting this values assumes that all other
sources of uncertainty have been removed.
Multi-Sample Measurements. Multi-sample
measurements involve a significant number of
data points collected from enough experiments so
that the reliability of the results can be assured
by a statistical analysis.
In other words, the measurement of a
significant number of data points of the same
quantity (for fixed system variables) under
varying test conditions (i.e., different samples
and/or different instruments) will allow the
uncertainties to be reduced by the sheer number of
Uncertainty In Measurement of a Function of
Independent Variables. The concern in this
measurement is in the propagation of
uncertainties. In most experiments, several
quantities are measured in order to calculate a
desired quantity. For example, to estimate the
gravitational constant by dropping a ball from a
known height, the approximate equation would
Now suppose we measured: L = 50.00 ± 0.01 m and
t = 3.1 ± 0.5 s. Based on the equation, we have:
2 2 50 00
10 42 2
We now wish to estimate the uncertainty ∆g in
our calculation of g. Obviously, the uncertainty
∆g will depend on the uncertainties in the
measurements of L and t. Let us examine the
“worst cases.” These may be calculated as:
2 49 99
2 50 01
The confidence interval around g then is:
7 7 14 82 2
. .m/s m/s≤ ≤g (6)
Now it is unlikely for all single-sample
uncertainties in a system to simultaneously be the
worst possible. Some average or “norm” of the
uncertainties must instead be used in estimating a
combined uncertainty for the calculation of g.
Uncertainty In Multi-Sample Measurements.
When a set of readings is taken in which the
values vary slightly from each other, the
experimenter is usually concerned with the mean
of all readings. If each reading is denoted by xi
and there are n readings, then the arithmetic
mean value is given by:
Deviation. The deviation of each reading is
d x xi i= − (8)
The arithmetic mean deviation is defined as:
= ∑ =
Note that the arithmetic mean deviation is zero:
Standard Deviation. The standard deviation is
( )x x
Due to random errors, experimental data is
dispersed in what is referred to as a bell
distribution, known also as a Gaussian or Normal
Distribution, and depicted in Figure 3.
FIGURE 3. Gaussian or Normal Distribution.
The Gaussian or Normal Distribution is what
we use to describe the distribution followed by
random errors. A graph of this distribution is
often referred to as the “bell” curve as it looks
like the outline of a bell. The peak of the
distribution occurs at the mean of the random
variable, and the standard deviation is a common
measure for how “fat” this bell curve is. Equation
10 is called the Probability Density Function for
any continuous random variable x.
f x e
The mean and the standard deviation are all
the information necessary to completely describe
any normally-distributed random variable.
Integrating under the curve of Figure 3 over
various limits gives some interesting results.
• Integrating under the curve of the normal
distribution from negative to positive
infinity, the area is 1.0 (i.e., 100 %). Thus the
probability for a reading to fall in the range
of ±∞ is 100%.
• Integrating over a range within ± σ from the
mean value, the resulting value is 0.6826. The
probability for a reading to fall in the range
of ± σ is about 68%.
• Integrating over a range within ± 2σ from the
mean value, the resulting value is 0.954. The
probability for a reading to fall in the range
of ± 2σ is about 95%.
• Integrating over a range within ± 3σ from the
mean value, the resulting value is 0.997. The
probability for a reading to fall in the range
of ± 3σ is about 99%.
TABLE 2. Probability for Gaussian Distribution
(tabulated in any statistics book)
Probability ± value of the mean
Estimating Uncertainty. We can now use the
probability function to help in determining the
accuracy of data obtained in an experiment. We
use the uncertainty level of 95%, which means
that we have a 95% confidence interval. In other
words, if we state that the uncertainty is ∆x, we
suggest that we are 95% sure that any reading xi
will be within the range of ± ∆x of the mean.
Thus, the probability of a sample chosen at
random of being within the range ± 2σ of the
mean is about 95%. Uncertainty then is defined as
twice the standard deviation:
∆x ≈ 2σ
Example 1. The manufacturer of a particular
alloy claims a modulus of elasticity of 40 ± 2 kPa.
How is that to be interpreted?
Solution: The general rule of thumb is that ± 2
kPa would represent a 95% confidence interval.
That is, if we randomly select many samples of
this manufacturer’s alloy we should find that
95% of the samples meet the stated limit of 40 ± 2
Now it is possible that we can find a sample
that has a modulus of elasticity of 37 kPa;
however, it means that it is very unlikely.
Example 2 If we assume that variations in the
product follow a normal distribution, and that
the modulus of elasticity is within the range 40 ±
2 kPa, then what is the standard deviation, σ?
Solution: The uncertainty ≈ 95% of confidence
interval ≈ 2σ. Thus
± 2 kPa = ± 2σ
σ = 1 kPa
Example 3. Assuming that the modulus of
elasticity is 40 ± 2 kPa, estimate the probability
of finding a sample from this population with a
modulus of elasticity less than or equal to 37 kPa.
Solution: With σ = 1 kPa, we are seeking the
value of the integral under the bell shaped curve,
over the range of -∞ to – 3σ. Thus, the probability
that the modulus of elasticity is less than 37 kPa
P(E < 37 kPa) =
100 - 99.7
Statistically Based Rejection of “Bad” Data–
Occasionally, when a sample of n
measurements of a variable is obtained, there
may be one or more that appear to differ
markedly from the others. If some extraneous
influence or mistake in experimental technique
can be identified, these “bad data” or “wild
points” can simply be discarded. More difficult is
the common situation in which no explanation is
readily available. In such situations, the
experimenter may be tempted to discard the
values on the basis that something must surely
have gone wrong. However, this temptation must
be resisted, since such data may be significant
either in terms of the phenomena being studied or
in detecting flaws in the experimental technique.
On the other hand, one does not want an erroneous
value to bias the results. In this case, a statistical
criterion must be used to identify points that can
be considered for rejection. There is no other
justifiable method to “throw away” data points.
One method that has gained wide acceptance
is Chauvenet’s criterion; this technique defines
an acceptable scatter, in a statistical sense,
around the mean value from a given sample of n
measurements. The criterion states that all data
points should be retained that fall within a band
around the mean that corresponds to a
probability of 1-1/(2n). In other words, data
points can be considered for rejection only if the
probability of obtaining their deviation from the
mean is less than 1/(2n). This is illustrated in
1 - 1/(2n)
FIGURE 4. Rejection of “bad” data.
The probability 1-1/(2n) for retention of data
distributed about the mean can be related to a
maximum deviation dmax away from the mean by
using a Gaussian probability table. For the given
probability, the non dimensional maximum
deviation τmax can be determined from the table,
|(xi – –x )|max
and sx is the precision index of the sample.
All measurements that deviate from the
mean by more than dmax/sx can be rejected. A new
mean value and a new precision index can then be
calculated from the remaining measurements. No
further application of the criterion to the sample
Using Chauvenet’s criterion, we say that the
values xi which are outside of the range
x C± σ (11)
are clearly errors and should be discarded for the
analysis. Such values are called outliers. The
constant C may be obtained from Table 3. Note
that Chauvenet’s criterion may be applied only
once to a given sample of readings.
The methodology for identifying and
discarding outlier(s) is a follows:
1. After running an experiment, sort the
outcomes from lowest to highest value. The
suspect outliers will then be at the top and/or
the bottom of the list.
2. Calculate the mean value and the standard
3. Using Chauvenet’s criterion, discard outliers.
4. Recalculate the mean value and the standard
deviation of the smaller sample and stop. Do
not repeat the process; Chauvenet’s criterion
may be applied only once.
TABLE 3. Constants to use in Chauvenet’s
criterion, Equation 11.
Example 4. Consider an experiment in which we
measure the mass of ten individual “identical”
objects. The scale readings (in grams) are as
shown in Table 4.
By visual examination of the results, we
might conclude that the 4.85 g reading is too high
compared to the others, and so it represents an
error in the measurement. We might tend to
disregard it. However, what if the reading was
2.50 or 2.51 g? We use Chauvenet’s criterion to
determine if any of the readings can be discarded.
TABLE 4. Data obtained in a series of
Number, n reading in g
We apply the methodology described earlier.
The results of the calculations are shown in Table
1. Values in the table are already sorted.
Column 1 shows the reading number, and
there are 10 readings of mass, as indicated in
2. We calculate the mean and standard
deviation. The data in column 2 are added to
obtain a total of 26.8. Dividing this value by
10 readings gives 2.68, which is the mean
value of all the readings:
m– = 2.68 g
In column 3, we show the square of the
difference between each reading and the
mean value. Thus in row 1, we calculate
(x– – x1)2 = (2.68 – 2.41)2 = 0.0729
We repeat this calculation for every data
point. We then add these to obtain the value
5.235 shown in the second to last row of
column 3. This value is then divided by (n –1)
= 9 data points, and the square root is taken.
The result is 0.763, which is the standard
deviation, as defined earlier in Equation 9:
( )x x
= 0.763 (9)
3. Next, we apply Chauvenet’s criterion; for 10
data points, n = 10 and Table 3 reads C = 1.96.
We calculate Cσ = 1.96(0.763) = 1.50. The
range of “acceptable” values then is 2.68 ±
m– – Cσ ≤ mi ≤ m– + Cσ
1.18 g ≤ m– ≤ 4.18 g
Any values outside the range of 1.18 and 4.18
are outliers and should be discarded.
4. Thus for the data of the example, the 4.85
value is an outlier and may be discarded. All
other points are valid. The last two columns
show the results of calculations made
without data point #10. The mean becomes
2.44, and the standard deviation is 0.019
(compare to 2.68, and 0.763, respectively).
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, all
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
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
†% 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.
experiment with respect to its faults as well
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.
FEATURES OF NOTE
• 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.
0 0.2 0.4 0.6 0.8 1
∆ hhead loss in m
FIGURE R.2. Theoretical and actual volume flow rate
through a venturi meter as a function of head loss.
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 surface tension, is
the object of this experiment.
Part I: Density Measurement.
Graduated cylinder or beaker
Liquid whose properties are to be
Method 1. 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.
Method 2. 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. 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
Method 3. A third method of making a density
measurement involves the use of a calibrated
hydrometer cylinder. The cylinder is submerged
in the liquid and the density is read directly on
the calibrated portion of the cylinder itself.
Measure density using the methods assigned by
the instructor. Compare results of all
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?
3. In your opinion, which method yielded the
“most accurate” results?
4. Are the results precise?
5. What is the mean of the values you
6. What is the standard deviation of the
7. Using Chauvenent’s rule, can any of the
measurements be discarded?
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
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
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
Measure the surface tension of the liquid
assigned. Each member of your group should make
a measurement to become familiar with the
apparatus. Are all measurements in agreement?
FIGURE 1.2. A schematic of the
surface tension meter.
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
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:
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
Cylinder filled with test liquid
Several small spheres with weight and
diameter to be measured
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
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
FIGURE 2.1. Terminal velocity measurement (V =
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. What are the shortcomings of this method?
5. Why should temperature be recorded?
6. Can this method be used for gases?
7. Can this method be used for opaque liquids?
8. Can this method be used for something like
peanut butter, or grease or flour dough?
Why or why not?
9. Perform an error analysis for one of the data
points. That is, determine the error
associated with all the measurements, and
provide an error band about the mean value.
CENTER OF PRESSURE ON A SUBMERGED
Submerged surfaces are found in many
engineering applications. Dams, weirs and water
gates are familiar examples of submerged plane
surfaces. It is important to have a working
knowledge of the forces that act on submerged
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.
Pressure increases 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 line of
action of this force.
FIGURE 3.1. Pressure distribution on a submerged
plane surface and the equivalent force.
For this case, it can be shown that the
equivalent force is:
F = ρgycA (3.1)
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
+ yc (3.2)
in which Ixx is the second area moment of the
plane about its centroid. The experimental
verification of these equations for force and
distance is the subject of this experiment.
Figure 3.2a is a sketch of an apparatus that
we use to illustrate the concepts behind this
experiment. The apparatus consists of one-fourth
of a torus, consisting of a solid piece of material.
The torus is attached to a lever arm, which is
free to rotate (within limits) about a pivot point.
The torus has inside and outside radii, Ri and Ro
respectively, and it is constructed such that the
center of these radii is at the pivot point of the
lever arm. The torus is now submerged in a liquid,
and there will exist an unbalanced force F that is
exerted on the plane of dimensions h x w. In order
to bring the torus and lever arm back to their
balanced position, a weight Wmust be added to
the weight hanger. The force and its line of
action can be found with Equations 3.1 and 3.2.
Consider next the apparatus sketched in
Figure 3.2b. It is quite similar to that in Figure
3.2a, in that it consists of a torus attached to a
lever arm. In this case, however, the torus is
hollow, and can be filled with liquid. If the
depth of the liquid is equal to that in Figure 3.2a,
(as measured from the bottom of the torus), then
the forces in both cases will be equal in magnitude
but opposite in direction. Moreover, the distance
from the free surface of the liquid to the line of
action of both forces will also be equal. Thus,
there is an equivalence between the two devices.
Center of Pressure Measurement
Center of Pressure Apparatus
The torus and balance arm are located on a pivot
rod. Note that the pivot point for the balance
arm is the point of contact between the rod and
the torus. Place the weight hanger on the
apparatus, and add water into the trim tank (not
shown in the figure) to bring the submerged plane
back to the vertical position.
Start by adding 20 g to the weight hanger.
Then pour water into the torus until the
submerged plan is brought back to the vertical
position. Record the weight and the liquid
depth. Repeat this procedure for 4 more weights.
(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 3.2b, we have
(y + yF)
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. Recalling that both
curved surfaces of the torus are circular with
centers at the pivot point, we see that the forces
acting on the curved surfaces have a zero moment
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 in Figure 3.2a?
2. Why isn’t the weight of the torus and the
balance arm taken into account?
FIGURE 3.2b. A schematic of the center of pressure apparatus.
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 4.1 is a schematic of the device used in
this experiment. The device consists of a catch
basin within a sump tank. A pump moves water
from the sump tank to the impact apparatus,
after which the water drains to the catch basin.
The plug is used to allow water to accumulate in
the catch basin. On the side of the sump tank is a
sight glass (not shown in Figure 4.1) showing the
water depth in the catch basin.
When flow rate is to be measured, water is
allowed to accumulate in the catch basin, and a
stopwatch is used to measure the time required
for the water volume to reach a pre-determined
volume, using the sight glass as an indicator. In
other words, we use the stopwatch to measure the
time required for a certain volume of water to
accumulate in the catch basin.
The sump tank acts as a support for the table
top which supports the impact apparatus. As
shown in Figure 4.1, 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 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 catch basin to obtain the volume flow rate.
The variables involved in this experiment
are listed and their measurements are described
1. Volume rate of flow–measured with the
catch basin (to obtain volume) and a
stopwatch (to obtain time). The volume flow
rate is obtained by dividing volume by time:
Q = V/t.
2. Velocity of jet–obtained by dividing volume
flow rate by jet area: V = Q/A. The jet is
cylindrical in shape.
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) (cone)
These equations are easily derivable from the
momentum equation applied to a control volume
about the object.
balancing weight lever arm with
flat plate attached
FIGURE 4.1. A schematic of the jet impact apparatus.
I. Figure 4.2 shows a sketch of the lever arm
in the impact experiment. The impact object
should be in place and the thumbscrew on
the spring should be used to zero the lever
arm. This is done without any water flow.
(Units of the scales in the figures are
II. The pump is now turned on and a water jet
hits the impact object, which will deflect
the lever arm causing it to rotate slightly
counterclockwise. The balancing weight is
moved from the zero position to the
position required to re-balance the lever
arm (in this case identified as “3” in Figure
4.3). The spring is left untouched. Only the
balancing weight is moved in order to re-
balance the lever arm.
III. During the time that the water jet impacts
the object, the time required to calculate
volume flow rate is measured.
40 1 2 3 5
FIGURE 4.2. Lever arm in zero position without
any water flow.
40 1 2 3 5
FIGURE 4.3. Lever arm in zero position when the
water jet is on.
SYMBOL FORCE DISTANCE
Fs spring force ds
Fw balancing weight dw
Fo impact object do
F exerted by water jet do
Analysis (Actual Force as Measured)
Summing moments about point O in Figure 4.2
gives the following equation for the lever arm:
Fsds + Fodo + Fwdw1 = 0 (4.1)
Summing moments about point O gives the
following equation for the lever arm in Figure 4.3:
Fsds + Fodo – Fdo + Fwdw2 = 0 (4.2)
Now we compare Equations 4.1 and 4.2. We can
identify parameters that appear in both
equations that are constants. These are Fsds and
Fodo. We rearrange Equation 4.1 to solve for the
sum of these force-distance products:
Fsds + Fodo = – Fwdw1 (4.3)
Likewise, Equation 4.2 gives
Fsds + Fodo = + Fdo – Fwdw2 (4.4)
Subtracting Equation 4.4 from 4.3, we get
0 = – Fwdw1 – Fdo + Fwdw2
The force we are seeking is that exerted by the
water jet F; rearranging gives
Fdo = – Fwdw1 + Fwdw2 = Fw(dw2 – dw1)
Fw(dw2 – dw1)
Thus, the force exerted by the water equals the
weight of what we have called the balancing
weight times a ratio of distances. The distance
(dw2 – dw1) is just the difference in readings of the
position of the balancing weight. The distance do
is the distance from the pivot to the location of
the impact object.
For your report, derive the appropriate
equation for each object you are assigned to 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.
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.
Critical Reynolds Number Measurement
Critical Reynolds Number Determination
Figure 5.1 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
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:
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 5.1. 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 6.1. 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.
FIGURE 6.1. A schematic of the Venturi meter.
Using the hydrostatic equation applied to
the air-over-liquid manometer of Figure 6.1, the
pressure drop and the head loss are related by
p1 - p2
By combining the continuity equation,
Q = A1V1 = A2V2
with the Bernoulli equation,
and substituting from the hydrostatic equation, it
can be shown after simplification that the
volume flow rate through the venturi meter is
Qth = A2
1 - (D2
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 6.1
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 V2 is the velocity at the throat of the
meter (= Qac/A2).
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
6.2 and 6.3). 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 6.2 or Figure
6.3) yields the same theoretical equation as that
for the venturi meter, namely, Equation 6.1. For
any pressure difference, there will be two
associated flow rates for these meters: the
theoretical flow rate (Equation 6.1), 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 6.2. Cross sectional view of the orifice
FIGURE 6.3. Cross sectional view of the nozzle-
type 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 6.4 is a schematic of the turbine type flow
FIGURE 6.4. A schematic of a 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 6.5 is a sketch
of a rotameter.
FIGURE 6.5. 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.
2. 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.
3. 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 6.6. 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 6.1. 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 6.2. 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 6.1), a Discussion and
Conclusions section, and the following graphs:
1. On the same set of axes, plot Qac vs ∆h and
Qth 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
Qth 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 Cp vertical axis.
1. Referring to Figure 6.2, 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
Air Over Liquid Manometry
Each corresponding pair of pressure taps on
the apparatus is attached to an air over liquid
(water, in this case), inverted U-tube manometer.
Use of the manometers can lead to some
difficulties that may need attention.
Figure 6.7 is a sketch of one manometer. The
left and right limbs are attached to pressure taps,
denoted as p1 and p2. Accordingly, when the
system is operated, the liquid will rise in each
limb and reach an equilibrium point. The pressure
difference will appear as a difference in height
of the water columns. That is, the pressure
difference is given by:
p1 – p2 = ρg∆h
where ρ is that of the liquid, and ∆h is read
directly on the manometer.
In some cases, the liquid levels are at places
beyond where we would like them to be. To
alleviate this problem, the air release valve
may be opened (slowly) to let air out or in. When
this occurs, the two levels will still have the
same ∆h reading, but located at a different place
on the manometer.
Sometimes, air bubbles will appear within
the liquid. The apparatus used has water with a
small amount of liquid soap dissolved to reduce
the surface tension of the water. However, if the
presence of bubbles persists, the pump should be
cycled on and off several times, and this should
solve the problem.
FIGURE 6.7. Air over liquid manometer.
rotameter sump tank
FIGURE 6.6. 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.
Pipe Flow Test Rig
Figure 7.1 is a schematic of the pipe flow test
rig. The rig contains a sump tank which is used as
a water reservoir from which one or two
centrifugal pumps discharge water to the pipe
circuit. The circuit itself consists of six different
diameter lines and a return line all made of PVC
pipe. The circuit contains ball valves for
directing and regulating the flow, and can be used
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 and pressure taps
are not shown in the schematic).
The six lines are 1/2, 3/4, 1, 11/4, 11/2, and 2
inch schedule 80 pipe. The topmost line is a return
line, and it is made of 2 inch pipe as well. The
apparatus contains two flow meters. The 1/2 line
contains a flow meter which is used only for that
pipe. The other flow meter in the return line is for
all other flows through the system. Because the
circuit contains flow meters, the measured
pressure losses can be obtained as a function of
As functions of the flow rate, measure the
pressure losses in inches of water for whatever
combination of flows and minor losses specified by
• 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(s) on its
downstream side to vary the volume flow
• With the pump on, record the assigned
pressure drops and the actual volume flow
rate from the flow meter.
• Using the valve closest to the pump, change
the volume flow rate and again record the
pressure drops and the new flow rate.
• 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
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 smooth walled
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 the flow rate
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
the assigned pipe(s). Compare to a Moody
diagram. Also calculate the loss coefficient for
the fitting(s) assigned, and determine if the loss
coefficient K varies with flow rate or Reynolds
number. Compare your K values to published ones.
open ball valve
closed ball valve
FIGURE 7.1. Schematic of the pipe friction apparatus (not to scale).
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 8.1. 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 8.2. Shown in
Figure 8.2a 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 8.2b is a
graph of the same data as that in Figure 8.2a
except that 8.2b is on a linear grid.
FIGURE 8.1. Streamlines of flow about a circular
0 30 60 90 120 150 180
(a) Polar Coordinate Graph (b) Linear Graph
FIGURE 8.2. Pressure distribution around a circular cylinder placed in a uniform flow.
A Wind Tunnel
A Right Circular Cylinder with Pressure
Figure 8.3 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 8.4 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 ∫
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 8.3. 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 8.4. 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 8 and is shown schematically
in Figure 8.3. 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
9.1. 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 rotational
speed of the fan. Thus air velocity, lift force and
drag force are read directly from the tunnel
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
uniform flow mounting stand
FIGURE 9.1. 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 10.1. 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 10.1
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
gradient on upper
on lower surface
FIGURE 10.1. Streamlines of flow about a wing
and the resultant pressure distribution.
Lift and Drag Measurements for a Wing
Wind Tunnel (See Figure 8.3)
Wing with Pressure Taps
Wing for Attachment to Lift & Drag
Instruments (See Figure 10.2)
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 10.2 (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.
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.
uniform flow mounting stand
FIGURE 10.2. Schematic of lift and drag measurement in a test section.
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.
Flow Through a Sluice Gate
Open Channel Flow Apparatus
Sluice Gate Model
Figure 11.1 is a sketch of the flow pattern
under a sluice gate. Upstream of the gate, the
velocity is V0, and the liquid height is h0. The
gate is a distance h1 above the bottom of the
channel, and downstream, the liquid height is
h2. The channel width is b.
The objective of this experiment is to make
measurements for a number of gate positions and
flow rates, and to determine whether the
equations we derive for a sluice gate are accurate
in their description of the resulting flows.
FIGURE 11.1. Schematic of flow under a sluice
The continuity equation applied about the
sluice gate is
V0h0b = V2h2b (11.1)
Under real conditions, h2 is somewhat less than
h1. We therefore introduce a contraction
coefficient Cc defined as
Substituting into Equation 11.1, canceling the
channel width b, and solving for V0, we get
V0 = V2
The Bernoulli Equation applied about the gate is
+ h0 =
+ h2 (11.3)
Substituting for V0 from Equation 11.2, we obtain
+ h0 =
Rearranging and solving for V2, we have
h0 – Cch1
1 – Cc
Factoring h0 and noting the relationship between
the numerator and denominator, the preceding
1 + Cch1/h0
The flow rate is the product of area and velocity.
At section 2, the flow rate may be written as
Q = V2h2b = V2Cch1b
Substituting from Equation 11.4 yields
Q = Cch1b
1 + Cch1/h0
Introducing a discharge coefficient Cs, we write
Q = Csb √2gh0 (11.6)
Comparison with Equation 11.5 gives
√1 + Cch1/h0
We see that the contraction coefficient and the
discharge coefficient depend only on the
upstream height and the gate height.
Set up the open channel flow apparatus
(Figure 11.2) to obtain flow under a sluice gate.
For nine (if possible) different flow rate/gate
position combinations, record upstream height,
gate height, downstream height, and volume
Calculate the contraction coefficient, the
discharge coefficient (Equation 11.7), and the
expected volume flow rate (Equation 11.6).
Compare the calculated flow rate (theoretical)
with the measured (actual) value.
Include in Your Report
• Detailed derivation of the equations
• Graph of contraction coefficient as a function
of the ratio h1/h0
• Graph of discharge coefficient as a function of
the ratio h1/h0
• Graph of flow rates (actual and theoretical)
as a function of the ratio h1/h0
• Discussion of your results
FIGURE 11.2. Schematic of the open channel flow apparatus.
OPEN CHANNEL FLOW OVER A WEIR
Flow meters used in pipes introduce an
obstruction into the flow which results in a
measurable pressure drop that in turn is related to
the volume flow rate. In an open channel, flow
rate can be measured similarly by introducing an
obstruction into the flow. A simple obstruction,
called a weir, consists of a vertical plate
extending the entire width of the channel. The
plate may have an opening, usually rectangular,
trapezoidal, or triangular. Other configurations
exist and all are about equally effective. The use
of a weir to measure flow rate in an open channel
is the subject of this experiment.
Flow Over a Weir
Open Channel Flow Apparatus (See
The open channel flow apparatus allows for
the insertion of a weir and measurement of liquid
depths. The channel is fed by two centrifugal
pumps. Each pump has a discharge line which
contains an turbine meter with digital readout,
which provide the means of determining the
actual flow rate into the channel.
Figure 12.1 is a sketch of the side and
upstream view of a 90 degree (included angle) V-
notch weir. Analysis of this weir is presented
here for illustrative purposes. Note that
upstream depth measurements are made from the
lowest point of the weir over which liquid flows.
This is the case for the analysis of all
conventional weirs. A coordinate system is
imposed whose origin is at the intersection of the
free surface and a vertical line extending upward
from the vertex of the V-notch. We select an
element that is dy thick and extends the entire
width of the flow cross section. The velocity of
the liquid through this element is found by
applying Bernoulli's equation:
+ gh =
+ g(h - y)
Note that in pipe flow, pressure remained in the
equation when analyzing any of the differential
pressure meters (orifice or venturi meters). In open
channel flows, the pressure terms represents
atmospheric pressure and cancel from the
Bernoulli equation. The liquid height is
therefore the only measurement required here.
From the above equation, assuming Vo negligible:
V = √2gy (12.1)
Equation 12.1 is the starting point in the analysis
of all weirs. The incremental flow rate of liquid
through layer dy is:
dQ = 2Vxdy = √2gy(2x)dy
From the geometry of the V-notch and with
respect to the coordinate axes, we have y = h - x.
FIGURE 12.1. Side and upstream views of a 90° V-notch weir.
Q = ∫
(2√2g)y1/2(h - y)d y
√2g h5/2 =Ch5/2 (12.2)
where C is a constant. The above equation
represents the ideal or theoretical flow rate of
liquid over the V-notch weir. The actual
discharge rate is somewhat less due to frictional
and other dissipative effects. As with pipe
meters, we introduce a discharge coefficient
The equation that relates the actual volume flow
rate to the upstream height then is
Qac = C'Ch5/2
It is convenient to combine the effects of the
constant C and the coefficient C’ into a single
coefficient Cvn for the V-notch weir. Thus we
reformulate the previous two equations to obtain:
Qac = Cvnh5/2 (12.4)
Each type of weir will have its own coefficient.
Calibrate each of the weirs assigned by the
instructor for 7 different upstream height
measurements. Derive an appropriate equation
for each weir used (similar to Equation 12.4)
above. Determine the coefficient applicable for
each weir tested. List the assumptions made in
each derivation. Discuss the validity of each
assumption, pointing out where they break down.
Graph upstream height vs actual and theoretical
volume flow rates. Plot the coefficient of
discharge (as defined in Equation 12.3) as a
function of the upstream Froude number.
FIGURE 12.2. Other types of weirs–semicircular, contracted and suppressed, respectively.