# Fluid Lab manual

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A Manual for the

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

2

TABLE OF CONTENTS

Item

Page

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

3

REPORT WRITING

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

specified otherwise.

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

Reports:

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

past tense.

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-

Bibliography

Calibration Curves

Original Data Sheet

(Sample Calculation)

Appendix

Title Page

Discussion & Conclusion

(Interpretation)

Results (Tables

and Graphs)

Procedure (Drawings

and Instructions)

Theory

(Textbook Style)

Object

(Past Tense)

Table of Contents

Each page numbered

Experiment Number

Experiment Title

Your Name

Due Date

Partners’ Names

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

nothing.

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.

4

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

section.

Appendix

(1) Original data sheet.

(2) Show how data were used by a sample

calculation.

(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.

Graphs

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.

0.2

•

•

•

•

•

•

•

•

Border is drawn about the entire graph.

Axis labels defined with symbols and

units.

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.

3

flow rate Q in m /s

Features of note

Q

th

0.15

Q

ac

0.1

0.05

0

0

0.2

0.4

0.6

0.8

1

head loss ∆ h in m

FIGURE R.2. Theoretical and actual volume flow rate

through a venturi meter as a function of head loss.

5

CLEANLINESS AND SAFETY

Cleanliness

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

equipment.

Safety

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

communication opportunities.

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

properly.

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.

6

EXPERIMENT 1

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.

W2

W1

Part I: Density Measurement.

Equipment

Graduated cylinder or beaker

Liquid whose properties are to be

measured

Hydrometer cylinder

Scale

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

Equipment

Surface tension meter

Beaker

Test fluid

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

Figure 1.2.

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

found as

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

calculated.

Questions

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

following equation

7

balance rod

platinum

iridium ring

FIGURE 1.2. A schematic of the

surface tension meter.

clamp

torsion wire

test liquid

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

8

EXPERIMENT 2

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

4

4

ρsg πR3 - ρg πR3 - 6πµVR = 0

3

3

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

Experiment 1.)

Questions

1. Should the terminal velocity of two

different size spheres be the same?

2. Does a larger sphere have a higher

terminal velocity?

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?

ρVD

< 1

µ

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

liquid.

Equipment

Hydrometer cylinder

Scale

Stopwatch

Several small spheres with weight and

diameter to be measured

Test liquid

d

V

FIGURE 2.1. Terminal velocity measurement (V =

d/time).

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

9

EXPERIMENT 3

CENTER OF PRESSURE ON A SUBMERGED

PLANE SURFACE

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

submerged surfaces.

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.

yF

F

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

Equipment

Center of Pressure Apparatus

Weights

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

F=

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

yF =

Ix x

+ yc

y cA

(3.2)

in which Ixx is the second area moment of the

plane about its centroid. The experimental

WL

(y + y F )

(3.3)

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.

Questions

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?

10

L

level

y

Ri

pivot point

(point of contact)

torus

weight

hanger

zeroing weight

yF

Ro

h

F

FIGURE 3.2. A schematic of the center of pressure apparatus.

11

w

EXPERIMENT 4

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.

Manometry

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

hydrostatic equation.

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,

p2

p1

z2

2

1

z1

h

pA

pA

m

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.

Questions

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

type?

2. Manometer 5 is a combined U/tube/inclined

manometer. What is the advantage of this

type?

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

possible.

4. What advantages or disadvantages does

the gage have over the manometers?

12

5. Is a low value of the standard deviation

expected? Why?

6. What does a low standard deviation

imply?

7. In your opinion, which device gives the

most accurate reading. What led you to this

conclusion?

High pressure tank

Low pressure tank

Bleed valves

Gage

U-tube manometers

Well-type

manometer

U-tube/inclined

manometer

FIGURE 4.2. A schematic of the apparatus used in this experiment.

13

EXPERIMENT 5

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

Equipment

Jet Impact Apparatus

Object plates

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

below:

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

in.

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

equation is:

F=

ρQ2

A

(flat plate)

For a hemisphere,

F=

2ρQ2

A

(hemisphere)

For a cone whose included half angle is α,

F=

ρQ2

(1 + cos α)

A

(cone)

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.

14

balancing

weight

pivot

lever arm with

flat plate attached

flat plate

water

jet

nozzle

drain

flow control

valve

weigh tank

tank pivot

plug

weight hanger

sump tank

motor

pump

FIGURE 5.1. A schematic of the jet impact apparatus.

15

EXPERIMENT 6

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

experiment.

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:

Re =

Critical Reynolds Number Measurement

Equipment

Critical Reynolds Number Determination

Apparatus

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

VD

ν

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

liquid.

The hydraulic diameter is calculated from

its definition:

D=

4 x Area

Wetted Perimeter

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.

16

Questions

1. Can a similar procedure be followed for

gases?

2. Is the Reynolds number obtained at

transition dependent on tube size or shape?

3. Can this method work for opaque liquids?

dye reservoir

drilled partitions

B

on/off valve

rotameter

transparent tube

inlet to

tank

overflow

to drain

A

to drain

FIGURE 6.1. The critical Reynolds number determination apparatus.

17

EXPERIMENT 7

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

this experiment.

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.

h

1

2

and substituting from the hydrostatic equation, it

can be shown after simplification that the

volume flow rate through the venturi meter is

given by

Qth = A2

√

2g ∆ h

1 - (D 24/D 14)

(7.1)

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

into account.

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

Cv =

Qac

Qth

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

Re =

V2D2

ν

(7.2)

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

(after simplification):

p 1 - p2

= ∆h

ρg

By combining the continuity equation,

Q = A1 V 1 = A2 V 2

with the Bernoulli equation,

p 1 V 12 p2 V 22

+

= +

ρ

2

ρ

2

The Orifice Meter and

Nozzle-Type Meter

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

18

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.

rotor supported

on bearings

(not shown)

to receiver

h

turbine rotor

rotational speed

proportional to

flow rate

flow

straighteners

2

1

FIGURE 7.4. A schematic of a turbine-type flow

meter.

FIGURE 7.2. Cross sectional view of the orifice

meter.

h

1

2

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

Equation 7.2.

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

meter.

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.

outlet

freely

suspended

float

tapered, graduated

transparent tube

inlet

FIGURE 7.5. A schematic of the rotameter and its

operation.

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

19

2.

3.

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

readings.

Experimental Procedure

Equipment

Fluid Meters Apparatus

Stopwatch

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

Cp =

g∆H

V2/2

(7.3)

All velocities are based on actual flow rate and

pipe diameter.

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

venturi meter.

6. Plot Co vs Re on a log-log grid for the orifice

meter.

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.

Questions

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

to use?

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

and accuracy?

20

manometer

orifice meter

volumetric

measuring

tank

venturi meter

rotameter

return

sump tank

turbine-type meter

motor

pump

valve

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.)

21

EXPERIMENT 8

PIPE FLOW

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

f=

Equipment

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

rate.

2g ∆ h

V 2 (L/D)

It is customary to graph the friction factor as a

function of the Reynolds number:

Re =

VD

ν

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

K=

∆h

V2/2g

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

open.

22

rotameter

tank

valve

motor

static pressure tap

pump

FIGURE 8.1. Schematic of the pipe friction apparatus.

23

EXPERIMENT 9

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

experiment.

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.

Wake

FIGURE 9.1. Streamlines of flow about a circular

cylinder.

separation

point

0

Stagnation

Streamline

Freestream

Velocity V

p

30

60

90

120

150

180

separation

point

(a) Polar Coordinate Graph

(b) Linear Graph

FIGURE 9.2. Pressure distribution around a circular cylinder placed in a uniform flow. 24

Pressure Measurement

Equipment

A Wind Tunnel

A Right Circular Cylinder with Pressure

Taps

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

technique).

The drag force can be calculated by

integrating the flow-direction-component of each

pressure over the area of the cylinder:

π

Df = 2RL

∫ ∆p cosθdθ

0

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

CD =

Df

ρV2A/2

The Reynolds number is

Re =

ρVD

µ

inlet flow

straighteners

nozzle

diffuser

fan

test section

FIGURE 9.3. A schematic of the wind tunnel used in this experiment.

25

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.

60

90

120

30

0

static pressure

taps attach to

manometers

150

180

FIGURE 9.4. Schematic of the experimental

apparatus used in this experiment.

26

EXPERIMENT 10

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.

Equipment

Subsonic Wind Tunnel

Objects

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

energy):

CD =

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

Re =

VD

ν

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

appropriate.

Questions

1. How does the mounting piece affect the

readings?

2. How do you plan to correct for its effect, if

necessary?

object

uniform flow

mounting stand

drag force

measurement

lift force

measurement

FIGURE 10.1. Schematic of an object mounted in

the test section of the wind tunnel.

Df

ρV2A/2

27

EXPERIMENT 11

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

the wing.

Experiment I

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.

lift

c

drag

c

uniform flow

stagnation

point

Cp

stagnation

point

drag force

measurement

negative pressure

gradient on upper

surface

pressure

coefficient

mounting stand

lift force

measurement

positive pressure

on lower surface

FIGURE 11.2. Schematic of lift and drag

measurement in a test section.

chord, c

FIGURE 11.1. Streamlines of flow about a wing

and the resultant pressure distribution.

Lift and Drag Measurements for a Wing

Equipment

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 %

errors.

28

Experiment II

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.

Analysis

Lift and drag data are usually expressed in

dimensionless terms using lift coefficient and drag

coefficient. The lift coefficient is defined as

CL =

Lf

2A/2

ρV

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

CD =

Df

ρV2A/2

in which Df is the drag force.

29

EXPERIMENT 12

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.

h0 =

In terms of flow rate, the velocities are written as

V0 =

V=

Flow Through a Sluice Gate

Equipment

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

downstream depths.

sluice gate

hand crank

V 2 V 02

+h

2g 2g

Q Q

=

A bh0

Q

bh

where b is the channel width at the gate.

Substituting into the Bernoulli Equation and

simplifying gives

h0 =

Q2 1

- 1 +h

2gb2 h 2 h 0 2

Dividing by h 0,

1=

Q2

1 - 1 + h

2gb 2 h 0 h 2 h 0 2 h 0

Rearranging further,

2

patm

direction of

movement

patm

ho

2

1 - h = Q 2 1 - h 2

h0

h 0 2gb 2 h h 0

h

FIGURE 12.1. Schematic of flow under a sluice

gate.

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

p V2

+

+ h0 =

+

+h

ρg 2g

ρg 2g

Pressures at the free surface are both equal to

atmospheric pressure, so they cancel. Rearranging

gives

Multiplying both sides by h2/h02, and continuing

to simplify, we finally obtain

h 2/h02

Q2

=

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

30

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

tank.

Questions

1. For the required report, derive the sluice

gate equation in detail.

2. What if it was assumed that V 0

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

2

TABLE OF CONTENTS

Item

Page

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

3

REPORT WRITING

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

specified otherwise.

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

Reports:

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

past tense.

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-

Bibliography

Calibration Curves

Original Data Sheet

(Sample Calculation)

Appendix

Title Page

Discussion & Conclusion

(Interpretation)

Results (Tables

and Graphs)

Procedure (Drawings

and Instructions)

Theory

(Textbook Style)

Object

(Past Tense)

Table of Contents

Each page numbered

Experiment Number

Experiment Title

Your Name

Due Date

Partners’ Names

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

nothing.

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.

4

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

section.

Appendix

(1) Original data sheet.

(2) Show how data were used by a sample

calculation.

(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.

Graphs

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.

0.2

•

•

•

•

•

•

•

•

Border is drawn about the entire graph.

Axis labels defined with symbols and

units.

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.

3

flow rate Q in m /s

Features of note

Q

th

0.15

Q

ac

0.1

0.05

0

0

0.2

0.4

0.6

0.8

1

head loss ∆ h in m

FIGURE R.2. Theoretical and actual volume flow rate

through a venturi meter as a function of head loss.

5

CLEANLINESS AND SAFETY

Cleanliness

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

equipment.

Safety

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

communication opportunities.

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

properly.

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.

6

EXPERIMENT 1

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.

W2

W1

Part I: Density Measurement.

Equipment

Graduated cylinder or beaker

Liquid whose properties are to be

measured

Hydrometer cylinder

Scale

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

Equipment

Surface tension meter

Beaker

Test fluid

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

Figure 1.2.

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

found as

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

calculated.

Questions

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

following equation

7

balance rod

platinum

iridium ring

FIGURE 1.2. A schematic of the

surface tension meter.

clamp

torsion wire

test liquid

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

8

EXPERIMENT 2

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

4

4

ρsg πR3 - ρg πR3 - 6πµVR = 0

3

3

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

Experiment 1.)

Questions

1. Should the terminal velocity of two

different size spheres be the same?

2. Does a larger sphere have a higher

terminal velocity?

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?

ρVD

< 1

µ

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

liquid.

Equipment

Hydrometer cylinder

Scale

Stopwatch

Several small spheres with weight and

diameter to be measured

Test liquid

d

V

FIGURE 2.1. Terminal velocity measurement (V =

d/time).

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

9

EXPERIMENT 3

CENTER OF PRESSURE ON A SUBMERGED

PLANE SURFACE

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

submerged surfaces.

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.

yF

F

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

Equipment

Center of Pressure Apparatus

Weights

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

F=

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

yF =

Ix x

+ yc

y cA

(3.2)

in which Ixx is the second area moment of the

plane about its centroid. The experimental

WL

(y + y F )

(3.3)

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.

Questions

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?

10

L

level

y

Ri

pivot point

(point of contact)

torus

weight

hanger

zeroing weight

yF

Ro

h

F

FIGURE 3.2. A schematic of the center of pressure apparatus.

11

w

EXPERIMENT 4

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.

Manometry

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

hydrostatic equation.

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,

p2

p1

z2

2

1

z1

h

pA

pA

m

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.

Questions

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

type?

2. Manometer 5 is a combined U/tube/inclined

manometer. What is the advantage of this

type?

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

possible.

4. What advantages or disadvantages does

the gage have over the manometers?

12

5. Is a low value of the standard deviation

expected? Why?

6. What does a low standard deviation

imply?

7. In your opinion, which device gives the

most accurate reading. What led you to this

conclusion?

High pressure tank

Low pressure tank

Bleed valves

Gage

U-tube manometers

Well-type

manometer

U-tube/inclined

manometer

FIGURE 4.2. A schematic of the apparatus used in this experiment.

13

EXPERIMENT 5

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

Equipment

Jet Impact Apparatus

Object plates

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

below:

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

in.

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

equation is:

F=

ρQ2

A

(flat plate)

For a hemisphere,

F=

2ρQ2

A

(hemisphere)

For a cone whose included half angle is α,

F=

ρQ2

(1 + cos α)

A

(cone)

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.

14

balancing

weight

pivot

lever arm with

flat plate attached

flat plate

water

jet

nozzle

drain

flow control

valve

weigh tank

tank pivot

plug

weight hanger

sump tank

motor

pump

FIGURE 5.1. A schematic of the jet impact apparatus.

15

EXPERIMENT 6

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

experiment.

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:

Re =

Critical Reynolds Number Measurement

Equipment

Critical Reynolds Number Determination

Apparatus

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

VD

ν

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

liquid.

The hydraulic diameter is calculated from

its definition:

D=

4 x Area

Wetted Perimeter

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.

16

Questions

1. Can a similar procedure be followed for

gases?

2. Is the Reynolds number obtained at

transition dependent on tube size or shape?

3. Can this method work for opaque liquids?

dye reservoir

drilled partitions

B

on/off valve

rotameter

transparent tube

inlet to

tank

overflow

to drain

A

to drain

FIGURE 6.1. The critical Reynolds number determination apparatus.

17

EXPERIMENT 7

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

this experiment.

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.

h

1

2

and substituting from the hydrostatic equation, it

can be shown after simplification that the

volume flow rate through the venturi meter is

given by

Qth = A2

√

2g ∆ h

1 - (D 24/D 14)

(7.1)

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

into account.

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

Cv =

Qac

Qth

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

Re =

V2D2

ν

(7.2)

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

(after simplification):

p 1 - p2

= ∆h

ρg

By combining the continuity equation,

Q = A1 V 1 = A2 V 2

with the Bernoulli equation,

p 1 V 12 p2 V 22

+

= +

ρ

2

ρ

2

The Orifice Meter and

Nozzle-Type Meter

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

18

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.

rotor supported

on bearings

(not shown)

to receiver

h

turbine rotor

rotational speed

proportional to

flow rate

flow

straighteners

2

1

FIGURE 7.4. A schematic of a turbine-type flow

meter.

FIGURE 7.2. Cross sectional view of the orifice

meter.

h

1

2

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

Equation 7.2.

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

meter.

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.

outlet

freely

suspended

float

tapered, graduated

transparent tube

inlet

FIGURE 7.5. A schematic of the rotameter and its

operation.

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

19

2.

3.

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

readings.

Experimental Procedure

Equipment

Fluid Meters Apparatus

Stopwatch

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

Cp =

g∆H

V2/2

(7.3)

All velocities are based on actual flow rate and

pipe diameter.

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

venturi meter.

6. Plot Co vs Re on a log-log grid for the orifice

meter.

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.

Questions

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

to use?

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

and accuracy?

20

manometer

orifice meter

volumetric

measuring

tank

venturi meter

rotameter

return

sump tank

turbine-type meter

motor

pump

valve

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.)

21

EXPERIMENT 8

PIPE FLOW

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

f=

Equipment

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

rate.

2g ∆ h

V 2 (L/D)

It is customary to graph the friction factor as a

function of the Reynolds number:

Re =

VD

ν

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

K=

∆h

V2/2g

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

open.

22

rotameter

tank

valve

motor

static pressure tap

pump

FIGURE 8.1. Schematic of the pipe friction apparatus.

23

EXPERIMENT 9

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

experiment.

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.

Wake

FIGURE 9.1. Streamlines of flow about a circular

cylinder.

separation

point

0

Stagnation

Streamline

Freestream

Velocity V

p

30

60

90

120

150

180

separation

point

(a) Polar Coordinate Graph

(b) Linear Graph

FIGURE 9.2. Pressure distribution around a circular cylinder placed in a uniform flow. 24

Pressure Measurement

Equipment

A Wind Tunnel

A Right Circular Cylinder with Pressure

Taps

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

technique).

The drag force can be calculated by

integrating the flow-direction-component of each

pressure over the area of the cylinder:

π

Df = 2RL

∫ ∆p cosθdθ

0

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

CD =

Df

ρV2A/2

The Reynolds number is

Re =

ρVD

µ

inlet flow

straighteners

nozzle

diffuser

fan

test section

FIGURE 9.3. A schematic of the wind tunnel used in this experiment.

25

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.

60

90

120

30

0

static pressure

taps attach to

manometers

150

180

FIGURE 9.4. Schematic of the experimental

apparatus used in this experiment.

26

EXPERIMENT 10

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.

Equipment

Subsonic Wind Tunnel

Objects

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

energy):

CD =

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

Re =

VD

ν

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

appropriate.

Questions

1. How does the mounting piece affect the

readings?

2. How do you plan to correct for its effect, if

necessary?

object

uniform flow

mounting stand

drag force

measurement

lift force

measurement

FIGURE 10.1. Schematic of an object mounted in

the test section of the wind tunnel.

Df

ρV2A/2

27

EXPERIMENT 11

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

the wing.

Experiment I

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.

lift

c

drag

c

uniform flow

stagnation

point

Cp

stagnation

point

drag force

measurement

negative pressure

gradient on upper

surface

pressure

coefficient

mounting stand

lift force

measurement

positive pressure

on lower surface

FIGURE 11.2. Schematic of lift and drag

measurement in a test section.

chord, c

FIGURE 11.1. Streamlines of flow about a wing

and the resultant pressure distribution.

Lift and Drag Measurements for a Wing

Equipment

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 %

errors.

28

Experiment II

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.

Analysis

Lift and drag data are usually expressed in

dimensionless terms using lift coefficient and drag

coefficient. The lift coefficient is defined as

CL =

Lf

2A/2

ρV

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

CD =

Df

ρV2A/2

in which Df is the drag force.

29

EXPERIMENT 12

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.

h0 =

In terms of flow rate, the velocities are written as

V0 =

V=

Flow Through a Sluice Gate

Equipment

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

downstream depths.

sluice gate

hand crank

V 2 V 02

+h

2g 2g

Q Q

=

A bh0

Q

bh

where b is the channel width at the gate.

Substituting into the Bernoulli Equation and

simplifying gives

h0 =

Q2 1

- 1 +h

2gb2 h 2 h 0 2

Dividing by h 0,

1=

Q2

1 - 1 + h

2gb 2 h 0 h 2 h 0 2 h 0

Rearranging further,

2

patm

direction of

movement

patm

ho

2

1 - h = Q 2 1 - h 2

h0

h 0 2gb 2 h h 0

h

FIGURE 12.1. Schematic of flow under a sluice

gate.

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

p V2

+

+ h0 =

+

+h

ρg 2g

ρg 2g

Pressures at the free surface are both equal to

atmospheric pressure, so they cancel. Rearranging

gives

Multiplying both sides by h2/h02, and continuing

to simplify, we finally obtain

h 2/h02

Q2

=

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

30

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

tank.

Questions

1. For the required report, derive the sluice

gate equation in detail.

2. What if it was assumed that V 0