The document discusses the Reynolds number, a critical concept in fluid mechanics that distinguishes laminar and turbulent flow. It highlights the historical contributions of Osborne Reynolds and presents his key experiments, defining the Reynolds number as the ratio of inertial to viscous forces. Additionally, it addresses the practical applications of the Reynolds number in various fluid dynamics scenarios, including calculations for pipe flow, modeling swimming organisms, and wind tunnel testing.

1. Fundamentals of Soil Mechanics
Reynolds Equation
Presented To Dr. Mohamed El-Taher
Group D
‫عبدالسالم‬ ‫أسماء‬–‫عادل‬ ‫إيمان‬–‫عادل‬ ‫أميرة‬–‫جمال‬ ‫ريهام‬–‫محمود‬ ‫فاطمة‬–‫أسماء‬
‫منسي‬–‫محمد‬ ‫أسماء‬

2. . we shall have the pleasure of learning about one of the most
important numbers in the field of fluid mechanics, which
establishes a relation between the forces acting within a fluid -
Reynolds Number. But before we can get to its definition,
. Did You Know?
The concept of mathematically predicting the flow pattern of
fluids was first introduced by George Gabriel Stokes in 1851.
However, it was later researched, developed, and popularized by
Osborne Reynolds in 1883. Thus, it came to be called the
Reynolds Number.

3. the British physicist and engineer “Obsorne Reynolds”
. Born 23 August 1842
Belfast, Ireland
. Died 21 February 1912 (aged 69)
Watchet, Somerset, England
. Attended Queens ‘ College, Cambridge and graduated in 1867
. In 1868 he was appointed professor of engineering at (now
the University of Manchester), becoming in that year one of the
first professors in UK university history to hold the title of
"Professor of Engineering”
. his earliest professional research dealt with such properties as
magnetism, electricity, and heavenly bodies, Reynolds soon began
to concentrate on fluid mechanics. In this area he made a number
of significant contributions , his studies of heat transfer between
solids and fluids brought improvements in boiler and condenser
design , while his work on turbine pumps permitted their rapid
development.
. He also studied wave engineering and tidal motions in rivers.

4. Osborne
Reynolds
Dimensional
analysis
Reynolds
number
𝑅
𝑒 =
𝜌𝑣𝑙
𝜇
• Distinguish
between
laminar flow
and turbulent
flow
• Characteristic
• Length L
• Velocity v
• Density 𝝆
• Viscosity µ

5. Osborne Reynolds:
“The internal motion of water assumes one or other of two broadly distinguishable forms – either the
elements of the fluid follow one another along lines of motion which lead in the most direct manner to their
destination, or they eddy about in sinuous paths the most indirect possible”
Reynolds' experiment on fluid dynamics
in pipes
In which conditions the flow is laminar and in which
turbulent ?
He varied the speed of the flow, the density of the
water (by varying the water temperature), and the
diameter of the pipe. He used colorants in the
water to detect the transition from Laminar to
Turbulent.
He found that when 2000~1900

UD
The flow becomes turbulent

6. Reynolds' observations of the nature of the flow in his experiment:
From these experiments came the dimensionless Reynolds number which is the ratio of inertial
forces to viscous forces.
Inertial forces : resistance of an object to change in its state of motion
( Promotes Turbulent Flow )
Viscous forces : resistance of a liquid to change of form
( promotes laminar flow )
characteristic Length L (SI units : m )
Velocity v ( m/s)
Densityρ(Kg/m^3)
Viscosity µ (Pa. S or N .S/ or Kg/m .s )

7. The Reynolds Number can be expressed in terms of Discharge (Q)

8. For Low Reynolds numbers the flow is Laminar
A laminar flow is characterized by smooth, orderly and slow motions. Streamlines are
parallel and adjacent layers (laminae) of fluid slide past each other with little mixing and
transfer (only at molecular scale) of properties across the layers. A small perturbation
does not increase with time. The flow is regular and predictable.
3 conditions
fluid moves slowly
viscosity is relatively high
flow channel is relatively small
Blood flow through capillaries is laminar flow, as
it satisfies the 3 conditions

9. For high Reynolds numbers the flow is turbulent
Turbulent flows are highly irregular, three-dimensional, rotational, and very
diffusive and dissipative. A small perturbation increases with time.
They cannot be predicted exactly as function of time and space. Only
statistical averaged variables can be predicted.
Examples of turbulence :
- Oceanic and atmospheric layers and ocean
currents
- External flow of air/water over vehicles such as
cars/ships/submarines
- In racing cars, e.g. leading car causes understeer
at fast corners
- Turbulence during air-plane’s flight

10. Turbulent
Flow
Laminar Flow

13. For Open ChannelFor pipe Flow
Reynolds number for
channel flow :
NR = vR/υ
Where R is the hydraulic
radius ( metric units )
For channel flow
NR < 500 – laminar
NR > 2000 – turbulent
Reynolds number for pipe
flow :
NR = vD/υ
where v is the mean velocity ,
D is the diameter of pipe , and
υ is the kynematic viscosity
of fluid
For pipe flow
NR < 2000 – laminar
NR > 4000 – turbulent

14. Solved example….
Question 1: Find the reynolds number if a fluid of viscosity 0.4 Ns/m2 and relative density of 900 Kg/m3 through
a 20 mm pipe with a Velocity of 2.5 m/s?
Solution:
Viscosity of fluid μ = 0.4 Ns/m2,
Density of fluid ρ = 900 Kg/m3,
Diameter of the fluid L = 20 × 10-3 m
The Reynold formula is given by 𝑅𝑒 =
𝜌𝑙𝑣
𝜇
=
900×2.5×20×10−3 ÷0.4 = 112.5
Here we observe that the value of Reynolds number
is less than 2000, so the flow of liquid is laminar.

15. Applications
the Reynolds number, almost a century after its conception, still plays an important part in
the study of fluid mechanics. It has several applications, and it continues to be an
irreplaceable part of modern physics.
1) Reynolds number plays an important part in the calculation of the friction factor
in a few of the equations of fluid mechanics, including the Darcy’s equation.

16. 2) It is used when modeling the movement of organisms swimming through water.
- It is important to be able to calculate the Reynolds number of specific types of flow so that
appropriate models can be used when designing flow systems
- Since the Reynolds number is a measure of how a fluid behaves, you can also substitute
dynamically similar fluids, which are fluids with the same Reynolds number, when testing
models to see how they would behave in a specific environment
- In many real world systems , precise flow analysis can only be performed using scale models

17. FISH BODY
FORM
RELATES
TO
SWIMMING
ABILITY,
HABITAT,
AND NICHE

18. 3) in wind tunnel testing
Atmospheric air is considered to be a fluid. Hence, the Reynolds number can be calculated for it. This makes
it possible to apply it in wind tunnel testing to study the aerodynamic properties of various surfaces.
-If the air in your wind tunnel is not dense enough for what you need to test (i.e. Reynolds number too low),
you can make it less sticky by cooling the air to increase the Reynolds number to that of the dense air you
need.
-The developing turbulent flow across the wing results in loss of serious amounts of lift
( Turbulence causes losses in efficiency in cars and planes )
The more flow separation that appears over the wings of a plane or the
back of a vehicle increases the drag force and reduces the efficiency

19. 4) Pipe systems must be designed accordingly for
the flow they will endure over a given of time
Turbulent fluids can behave unpredictably and
cause large differences over short areas , leading to
reduced life times and even to pipe failures

20. The movement of water through soil
The term seepage flow is used to describe the movement of water through or
under soil structure to wells , drains and reservoirs . Generally, the Reynolds
number in fluid mechanics is less than 1 , and therefore the flow is laminar and
Darcy’s law is valid. Any departure from laminar flow is not serious as long as
the Reynolds number is below 10, the upper limit for the validity of Darcy’s law .
and this depends in the dimension of interstices, which in turn, depends upon
the particle size . In fine grained soils, the dimension of the interstices are very
small and the flow is laminar, while the flow may be turbulent in very coarse-
grained soils.

21. Experiment
EQUIPMENTS:
A stop watch, a graduated cylinder ,and Reynolds apparatus which consists of
water tank having a glass tube leading out of it. The glass tube has a bell mouth at
entrance and a regulating valve at outlet ,a dye container with an arrangement for
injecting a fine filament of dye at the entrance of the glass tube.
PROCEDORE:
I . Fill the water tank with water and allow it to stand for some time so that the
water comes to rest.
ii . Note the temperature of water.
iii. Partially open the outlet valve of the glass tube and allow the flow to take
place at a very low rate.
iv. Allow the flow to stabilize then open the valves at the inlet of the dye injector
and allow the dye to move through the tube. Observe the nature of the filament.
v. Measure the discharge by collecting water in the graduated cylinder for a
certain interval of time.
vi. Repeat the steps 3 and 5 for different discharges
vii. Again note the temperature of water

24. Thank You