Reynolds number and geometry concept, Momentum integral equations, Boundary layer equations, Flow over a flat plate, Flow over cylinder, Pipe flow, fully developed laminar pipe flow, turbulent pipe flow, Losses in pipe flow
Fluid Mechanics-Shear stress ,Shear stress distribution,Velocity profile,Flow Of Viscous Fluid Through The circular pipe ,Velocity profile for turbulent flow Boundary layer buildup in pipe,Velocity Distributions
B.TECH. DEGREE COURSE
SCHEME AND SYLLABUS
(2002-03 admission onwards)
MAHATMA GANDHI UNIVERSITY,mg university, KTU
KOTTAYAM
KERALA
Module 1
Introduction - Proprties of fluids - pressure, force, density, specific weight, compressibility, capillarity, surface tension, dynamic and kinematic viscosity-Pascal’s law-Newtonian and non-Newtonian fluids-fluid statics-measurement of pressure-variation of pressure-manometry-hydrostatic pressure on plane and curved surfaces-centre of pressure-buoyancy-floation-stability of submerged and floating bodies-metacentric height-period of oscillation.
Module 2
Kinematics of fluid motion-Eulerian and Lagrangian approach-classification and representation of fluid flow- path line, stream line and streak line. Basic hydrodynamics-equation for acceleration-continuity equation-rotational and irrotational flow-velocity potential and stream function-circulation and vorticity-vortex flow-energy variation across stream lines-basic field flow such as uniform flow, spiral flow, source, sink, doublet, vortex pair, flow past a cylinder with a circulation, Magnus effect-Joukowski theorem-coefficient of lift.
Module 3
Euler’s momentum equation-Bernoulli’s equation and its limitations-momentum and energy correction factors-pressure variation across uniform conduit and uniform bend-pressure distribution in irrotational flow and in curved boundaries-flow through orifices and mouthpieces, notches and weirs-time of emptying a tank-application of Bernoulli’s theorem-orifice meter, ventury meter, pitot tube, rotameter.
Module 4
Navier-Stoke’s equation-body force-Hagen-Poiseullie equation-boundary layer flow theory-velocity variation- methods of controlling-applications-diffuser-boundary layer separation –wakes, drag force, coefficient of drag, skin friction, pressure, profile and total drag-stream lined body, bluff body-drag force on a rectangular plate-drag coefficient for flow around a cylinder-lift and drag force on an aerofoil-applications of aerofoil- characteristics-work done-aerofoil flow recorder-polar diagram-simple problems.
Module 5
Flow of a real fluid-effect of viscosity on fluid flow-laminar and turbulent flow-boundary layer thickness-displacement, momentum and energy thickness-flow through pipes-laminar and turbulent flow in pipes-critical Reynolds number-Darcy-Weisback equation-hydraulic radius-Moody;s chart-pipes in series and parallel-siphon losses in pipes-power transmission through pipes-water hammer-equivalent pipe-open channel flow-Chezy’s equation-most economical cross section-hydraulic jump.
1. Introduction to Kinematics
2. Methods of Describing Fluid Motion
a). Lagrangian Method
b). Eulerian Method
3. Flow Patterns
- Stream Line
- Path Line
- Streak Line
- Streak Tube
4. Classification of Fluid Flow
a). Steady and Unsteady Flow
b). Uniform and Non-Uniform Flow
c). Laminar and Turbulent Flow
d). Rotational and Irrotational Flow
e). Compressible and Incompressible Flow
f). Ideal and Real Flow
g). One, Two and Three Dimensional Flow
5. Rate of Flow (Discharge) and Continuity Equation
6. Continuity Equation in Three Dimensions
7. Velocity and Acceleration
8. Stream and Velocity Potential Functions
Reynolds number and geometry concept, Momentum integral equations, Boundary layer equations, Flow over a flat plate, Flow over cylinder, Pipe flow, fully developed laminar pipe flow, turbulent pipe flow, Losses in pipe flow
Fluid Mechanics-Shear stress ,Shear stress distribution,Velocity profile,Flow Of Viscous Fluid Through The circular pipe ,Velocity profile for turbulent flow Boundary layer buildup in pipe,Velocity Distributions
B.TECH. DEGREE COURSE
SCHEME AND SYLLABUS
(2002-03 admission onwards)
MAHATMA GANDHI UNIVERSITY,mg university, KTU
KOTTAYAM
KERALA
Module 1
Introduction - Proprties of fluids - pressure, force, density, specific weight, compressibility, capillarity, surface tension, dynamic and kinematic viscosity-Pascal’s law-Newtonian and non-Newtonian fluids-fluid statics-measurement of pressure-variation of pressure-manometry-hydrostatic pressure on plane and curved surfaces-centre of pressure-buoyancy-floation-stability of submerged and floating bodies-metacentric height-period of oscillation.
Module 2
Kinematics of fluid motion-Eulerian and Lagrangian approach-classification and representation of fluid flow- path line, stream line and streak line. Basic hydrodynamics-equation for acceleration-continuity equation-rotational and irrotational flow-velocity potential and stream function-circulation and vorticity-vortex flow-energy variation across stream lines-basic field flow such as uniform flow, spiral flow, source, sink, doublet, vortex pair, flow past a cylinder with a circulation, Magnus effect-Joukowski theorem-coefficient of lift.
Module 3
Euler’s momentum equation-Bernoulli’s equation and its limitations-momentum and energy correction factors-pressure variation across uniform conduit and uniform bend-pressure distribution in irrotational flow and in curved boundaries-flow through orifices and mouthpieces, notches and weirs-time of emptying a tank-application of Bernoulli’s theorem-orifice meter, ventury meter, pitot tube, rotameter.
Module 4
Navier-Stoke’s equation-body force-Hagen-Poiseullie equation-boundary layer flow theory-velocity variation- methods of controlling-applications-diffuser-boundary layer separation –wakes, drag force, coefficient of drag, skin friction, pressure, profile and total drag-stream lined body, bluff body-drag force on a rectangular plate-drag coefficient for flow around a cylinder-lift and drag force on an aerofoil-applications of aerofoil- characteristics-work done-aerofoil flow recorder-polar diagram-simple problems.
Module 5
Flow of a real fluid-effect of viscosity on fluid flow-laminar and turbulent flow-boundary layer thickness-displacement, momentum and energy thickness-flow through pipes-laminar and turbulent flow in pipes-critical Reynolds number-Darcy-Weisback equation-hydraulic radius-Moody;s chart-pipes in series and parallel-siphon losses in pipes-power transmission through pipes-water hammer-equivalent pipe-open channel flow-Chezy’s equation-most economical cross section-hydraulic jump.
1. Introduction to Kinematics
2. Methods of Describing Fluid Motion
a). Lagrangian Method
b). Eulerian Method
3. Flow Patterns
- Stream Line
- Path Line
- Streak Line
- Streak Tube
4. Classification of Fluid Flow
a). Steady and Unsteady Flow
b). Uniform and Non-Uniform Flow
c). Laminar and Turbulent Flow
d). Rotational and Irrotational Flow
e). Compressible and Incompressible Flow
f). Ideal and Real Flow
g). One, Two and Three Dimensional Flow
5. Rate of Flow (Discharge) and Continuity Equation
6. Continuity Equation in Three Dimensions
7. Velocity and Acceleration
8. Stream and Velocity Potential Functions
In fluid dynamics, laminar flow is characterized by fluid particles following smooth paths in layers, with each layer moving smoothly past the adjacent layers with little or no mixing. At low velocities, the fluid tends to flow without lateral mixing, and adjacent layers slide past one another like playing cards.
Dimension less numbers in applied fluid mechanicstirath prajapati
In dimensional analysis, a dimensionless quantity is a quantity to which no physical dimension is assigned. It is also known as a bare number or pure number or a quantity of dimension one[1] and the corresponding unit of measurement in the SI is one (or 1) unit[2][3] and it is not explicitly shown. Dimensionless quantities are widely used in many fields, such as mathematics, physics, chemistry, engineering, and economics. Examples of quantities, to which dimensions are regularly assigned, are length, time, and speed, which are measured in dimensional units, such as meter , second and meter per second. This is considered to aid intuitive understanding. However, especially in mathematical physics, it is often more convenient to drop the assignment of explicit dimensions and express the quantities without dimensions, e.g., addressing the speed of light simply by the dimensionless number 1.
In fluid dynamics, laminar flow is characterized by fluid particles following smooth paths in layers, with each layer moving smoothly past the adjacent layers with little or no mixing. At low velocities, the fluid tends to flow without lateral mixing, and adjacent layers slide past one another like playing cards.
Dimension less numbers in applied fluid mechanicstirath prajapati
In dimensional analysis, a dimensionless quantity is a quantity to which no physical dimension is assigned. It is also known as a bare number or pure number or a quantity of dimension one[1] and the corresponding unit of measurement in the SI is one (or 1) unit[2][3] and it is not explicitly shown. Dimensionless quantities are widely used in many fields, such as mathematics, physics, chemistry, engineering, and economics. Examples of quantities, to which dimensions are regularly assigned, are length, time, and speed, which are measured in dimensional units, such as meter , second and meter per second. This is considered to aid intuitive understanding. However, especially in mathematical physics, it is often more convenient to drop the assignment of explicit dimensions and express the quantities without dimensions, e.g., addressing the speed of light simply by the dimensionless number 1.
Hybrid optimization of pumped hydro system and solar- Engr. Abdul-Azeez.pdffxintegritypublishin
Advancements in technology unveil a myriad of electrical and electronic breakthroughs geared towards efficiently harnessing limited resources to meet human energy demands. The optimization of hybrid solar PV panels and pumped hydro energy supply systems plays a pivotal role in utilizing natural resources effectively. This initiative not only benefits humanity but also fosters environmental sustainability. The study investigated the design optimization of these hybrid systems, focusing on understanding solar radiation patterns, identifying geographical influences on solar radiation, formulating a mathematical model for system optimization, and determining the optimal configuration of PV panels and pumped hydro storage. Through a comparative analysis approach and eight weeks of data collection, the study addressed key research questions related to solar radiation patterns and optimal system design. The findings highlighted regions with heightened solar radiation levels, showcasing substantial potential for power generation and emphasizing the system's efficiency. Optimizing system design significantly boosted power generation, promoted renewable energy utilization, and enhanced energy storage capacity. The study underscored the benefits of optimizing hybrid solar PV panels and pumped hydro energy supply systems for sustainable energy usage. Optimizing the design of solar PV panels and pumped hydro energy supply systems as examined across diverse climatic conditions in a developing country, not only enhances power generation but also improves the integration of renewable energy sources and boosts energy storage capacities, particularly beneficial for less economically prosperous regions. Additionally, the study provides valuable insights for advancing energy research in economically viable areas. Recommendations included conducting site-specific assessments, utilizing advanced modeling tools, implementing regular maintenance protocols, and enhancing communication among system components.
Automobile Management System Project Report.pdfKamal Acharya
The proposed project is developed to manage the automobile in the automobile dealer company. The main module in this project is login, automobile management, customer management, sales, complaints and reports. The first module is the login. The automobile showroom owner should login to the project for usage. The username and password are verified and if it is correct, next form opens. If the username and password are not correct, it shows the error message.
When a customer search for a automobile, if the automobile is available, they will be taken to a page that shows the details of the automobile including automobile name, automobile ID, quantity, price etc. “Automobile Management System” is useful for maintaining automobiles, customers effectively and hence helps for establishing good relation between customer and automobile organization. It contains various customized modules for effectively maintaining automobiles and stock information accurately and safely.
When the automobile is sold to the customer, stock will be reduced automatically. When a new purchase is made, stock will be increased automatically. While selecting automobiles for sale, the proposed software will automatically check for total number of available stock of that particular item, if the total stock of that particular item is less than 5, software will notify the user to purchase the particular item.
Also when the user tries to sale items which are not in stock, the system will prompt the user that the stock is not enough. Customers of this system can search for a automobile; can purchase a automobile easily by selecting fast. On the other hand the stock of automobiles can be maintained perfectly by the automobile shop manager overcoming the drawbacks of existing system.
Sachpazis:Terzaghi Bearing Capacity Estimation in simple terms with Calculati...Dr.Costas Sachpazis
Terzaghi's soil bearing capacity theory, developed by Karl Terzaghi, is a fundamental principle in geotechnical engineering used to determine the bearing capacity of shallow foundations. This theory provides a method to calculate the ultimate bearing capacity of soil, which is the maximum load per unit area that the soil can support without undergoing shear failure. The Calculation HTML Code included.
Cosmetic shop management system project report.pdfKamal Acharya
Buying new cosmetic products is difficult. It can even be scary for those who have sensitive skin and are prone to skin trouble. The information needed to alleviate this problem is on the back of each product, but it's thought to interpret those ingredient lists unless you have a background in chemistry.
Instead of buying and hoping for the best, we can use data science to help us predict which products may be good fits for us. It includes various function programs to do the above mentioned tasks.
Data file handling has been effectively used in the program.
The automated cosmetic shop management system should deal with the automation of general workflow and administration process of the shop. The main processes of the system focus on customer's request where the system is able to search the most appropriate products and deliver it to the customers. It should help the employees to quickly identify the list of cosmetic product that have reached the minimum quantity and also keep a track of expired date for each cosmetic product. It should help the employees to find the rack number in which the product is placed.It is also Faster and more efficient way.
Welcome to WIPAC Monthly the magazine brought to you by the LinkedIn Group Water Industry Process Automation & Control.
In this month's edition, along with this month's industry news to celebrate the 13 years since the group was created we have articles including
A case study of the used of Advanced Process Control at the Wastewater Treatment works at Lleida in Spain
A look back on an article on smart wastewater networks in order to see how the industry has measured up in the interim around the adoption of Digital Transformation in the Water Industry.
Overview of the fundamental roles in Hydropower generation and the components involved in wider Electrical Engineering.
This paper presents the design and construction of hydroelectric dams from the hydrologist’s survey of the valley before construction, all aspects and involved disciplines, fluid dynamics, structural engineering, generation and mains frequency regulation to the very transmission of power through the network in the United Kingdom.
Author: Robbie Edward Sayers
Collaborators and co editors: Charlie Sims and Connor Healey.
(C) 2024 Robbie E. Sayers
COLLEGE BUS MANAGEMENT SYSTEM PROJECT REPORT.pdfKamal Acharya
The College Bus Management system is completely developed by Visual Basic .NET Version. The application is connect with most secured database language MS SQL Server. The application is develop by using best combination of front-end and back-end languages. The application is totally design like flat user interface. This flat user interface is more attractive user interface in 2017. The application is gives more important to the system functionality. The application is to manage the student’s details, driver’s details, bus details, bus route details, bus fees details and more. The application has only one unit for admin. The admin can manage the entire application. The admin can login into the application by using username and password of the admin. The application is develop for big and small colleges. It is more user friendly for non-computer person. Even they can easily learn how to manage the application within hours. The application is more secure by the admin. The system will give an effective output for the VB.Net and SQL Server given as input to the system. The compiled java program given as input to the system, after scanning the program will generate different reports. The application generates the report for users. The admin can view and download the report of the data. The application deliver the excel format reports. Because, excel formatted reports is very easy to understand the income and expense of the college bus. This application is mainly develop for windows operating system users. In 2017, 73% of people enterprises are using windows operating system. So the application will easily install for all the windows operating system users. The application-developed size is very low. The application consumes very low space in disk. Therefore, the user can allocate very minimum local disk space for this application.
CFD Simulation of By-pass Flow in a HRSG module by R&R Consult.pptxR&R Consult
CFD analysis is incredibly effective at solving mysteries and improving the performance of complex systems!
Here's a great example: At a large natural gas-fired power plant, where they use waste heat to generate steam and energy, they were puzzled that their boiler wasn't producing as much steam as expected.
R&R and Tetra Engineering Group Inc. were asked to solve the issue with reduced steam production.
An inspection had shown that a significant amount of hot flue gas was bypassing the boiler tubes, where the heat was supposed to be transferred.
R&R Consult conducted a CFD analysis, which revealed that 6.3% of the flue gas was bypassing the boiler tubes without transferring heat. The analysis also showed that the flue gas was instead being directed along the sides of the boiler and between the modules that were supposed to capture the heat. This was the cause of the reduced performance.
Based on our results, Tetra Engineering installed covering plates to reduce the bypass flow. This improved the boiler's performance and increased electricity production.
It is always satisfying when we can help solve complex challenges like this. Do your systems also need a check-up or optimization? Give us a call!
Work done in cooperation with James Malloy and David Moelling from Tetra Engineering.
More examples of our work https://www.r-r-consult.dk/en/cases-en/
Democratizing Fuzzing at Scale by Abhishek Aryaabh.arya
Presented at NUS: Fuzzing and Software Security Summer School 2024
This keynote talks about the democratization of fuzzing at scale, highlighting the collaboration between open source communities, academia, and industry to advance the field of fuzzing. It delves into the history of fuzzing, the development of scalable fuzzing platforms, and the empowerment of community-driven research. The talk will further discuss recent advancements leveraging AI/ML and offer insights into the future evolution of the fuzzing landscape.
1. Mechanical Engineering Department
4th semester
Subject :- Fluid Mechanics
TOPIC :- TURBULENT FLOW
Prepared by :- Guided by :-
Sanjay Noelia (151120119022) prof. Dharmesh Jariwala
Prince Kumar (151120119031)
Abhishek Mishra (151120119001)
Niraj Pandey (151120119023)
2. • Introduction
• In the laminar flow the fluid particles moves along
straight parallel path in layer or lamina , such that the
path of individual fluid particles do no cross those of
neighboring particles .
• Laminar flow is possible only at low velocities and
when the fluid is highly viscous. But when the velocity
is increase or the fluid is less viscous , the fluid
particles do not move in straight path.
• The fluid particle moves in random manner resulting in
general mixing of the particles . This type of flow is
called turbulent flow.
3. • Introduction
• In the laminar flow the fluid particles moves along
straight parallel path in layer or lamina , such that the
path of individual fluid particles do no cross those of
neighboring particles .
• Laminar flow is possible only at low velocities and
when the fluid is highly viscous. But when the velocity
is increase or the fluid is less viscous , the fluid
particles do not move in straight path.
• The fluid particle moves in random manner resulting in
general mixing of the particles . This type of flow is
called turbulent flow.
4. A laminar flow changes to turbulent flow when :
1.Velocity is increase .
2.Diameter of pipe is increase .
3.Velocity of fluid is decreased .
O .Reynolds was the first to demonstrate that the
transition from laminar to turbulent depends not only
on the main velocity but not on the quantity .
• This quantity is a dimensionless quantity and is
called Reynolds number .
vD
eR
VD
5. In case of circular pipe if the flow is said
to be laminar and if then the floe is said to
be turbulent .
• If lies between 2000 to 4000 , the flow changes
from laminar to turbulent.i.e,
Laminar when Re < 2000.
Turbulent when Re > 4000.
Transient when 2000 < Re < 4000.
2000eR
4000eR
eR
6. 2.Reynolds experiment
The type of flow is determined from the Reynolds number
i.e.,
• This was demonstrated by O. Reynolds in 1883. His apparatus
in fig.
The apparatus consist of :
1) A tank contain water at constant head .
2) A small tank containing some dye.
3) A glass tube having a bell-mouthed entrance at one end and a
rectangular value at other end.
V d
7. The following observation were made by Raynold :
(a). When the velocity of the flow was low , the dye filament
in glass tube was in form of straight line.
(b). With the increase of velocity of flow , the dye filament
was no longer a straight- line it become a wavy one .
(c).With further increase of velocity of flow , the wavy dye-
filament broke up and finally diffused in water.
8. In case of laminar flow , the loss of pressure head was
found to be proportional to the velocity but in case of
turbulent flow , Reynolds observed that loss of head is
approximately proportional to the square of velocity .
More exactly the loss of head , where
varies from 1.75 to 2.0 .
n
fh v n
9. 3 . Frictional loss in pipe flow
When a liquid is flowing through a pipe ,the velocity of the liquid
layer adjacent to the pipe wall is zero the velocity of liquid is goes
to increasing from the wall and thus velocity gradient and hence
shear stress are produce in the whole liquid due to viscosity. This
viscous action causes loss of energy which is usually known as
frictional loss.
On the basis of his experiments, William Froude gave the laws of
fluid fraction for turbulent flow
The frictional resistance for the turbulent flow is :
(1) Proportional to V n ,where n varies from 1.5 to 2.0,
(2) Proportional to the density of fluid,
(3) Proportional to the area of surface in contact,
(4) Independent of pressure
(5) Dependent on the nature of the surface in contact.
10. 3.1 Expression for loss of head due to friction in pipes.
Consider a uniform horizontal pipe, having steady flow as
shown in fig. 10.3. Let 1-1 and 2-2 are two section of pipe.
Le P1 = pressure intensity at section 1-1,
V1 = Velocity of flow at section 1-1,
L = Length of pipe between section 1-1 and 2-2,
d = diameter of pipe
= frictional resistance per unit wetted area per unit velocity
h f = loss of head due to friction,
P2, V2 = are value of pressure intensity and velocity at
section 2-2,
f
11. Applying Bernoulli’s equation between section 1-1 and 2-2
Total heat at 1-1 = Total heat 2-2 + loss of head due to friction
between 1-1 and 2-2
Z1 = Z2 as pipe is horizontal
V1 = V2 as dia. Of pipe is same at 1-1 and 2-2
but is the head lost due to friction and hence intensity of
pressure will be reduced in the direction of flow by frictional
resistance.
22
1 2 2
1 2
2 2
f
p p vv
z z h
g g g g
1 2
f
p p
h
g g
fh
12. Now frictional resistance = frictional per unit wetted area per
unit velocity wetted area velocity2
The force acting on the fluid between section 1-1 and 2-2 are:
1. Pressure force at section 1-1 = P A
2. Pressure of at section 2-2 = P2 A
3. Frictional force f1 as shown in fig.
Resolving all forces in horizontal direction ,we have
2
1 'F f dL V
1 2 1 0p A p A F
2
f P L V
13. But from equation (1)
Equating the value of (p1-p2) we get
In equation (3)
2
1 2 1
2
1 2
'
'
P P A F f P L V
f P L V
P P
A
1 2 fp p gh
2
2
'
'
f
f
f p l v
gh
A
f P
h L V
g A
2
4
4
P Wettedperimeter d
A Area dd
14. Putting where f is known is co-efficient of
friction.
Equation (4) becomes as
Equations (4) is known as Darcy- Weisbach equation.
This equation commonly used for finding loss of head
due to friction in pipes.
Sometimes equation written as
Then f is known as friction factor.
2
2' 4 ' 4
f
f f LV
h L V
g d g d
'
,
2
f f
2 2
4. 4 . .
.
2 2
f
f LV f LV
h
g d d g
2
. .
2
f
f L V
h
d g
15. 3.2 Expression for co-efficient in terms of shear stress.
The equation gives forces acting between in section 1-1 and
2-2 in horizontal direction as
Force due to shear stress
shear stress*surface area
Cancelling πd from both sides we have
Equation can be written as
11 2
0A Ap p F
21 1
( )App F
2
1 2
4
p p d t d L
1 2
4
d
p p L
1 2 4
L
p p
d
2
1 2 4 . .
2f
p p f LV
g
g dh
16.
2
1 2
4. . .
2
f LV
p p g
d g
Equating the value of (P1-P2) in equation
2
4. . .
4
2
L f LV
g
d d g
2 2
2 2
fV g fV
g
g g
2
2
V
f
2
2
f
v
17. 4.Shear stress in turbulent flow
The shear stress in viscous flow is given by newton's law of
viscosity as
Similar to the expression for viscous shear ,J. Bossiness
expressed the turbulent shear in mathematical form as
τt= shear stress due to turbulence
ῃ=eddy viscosity
ū= average velocity at a distance y from boundary
The ratio of eddy viscosity and mass density is known as
kinematic eddy viscosity and is denoted by epsilon
v
du
dy
t
du
dy
18. If the shear stress due to viscous flow is also considered
then the total shear stress becomes
The value of n=0 for laminar flow
v t
du du
dy dy
19. 4.1 Reynolds Expression For Turbulent Shear Stress
Reynolds in 1886 developed an expression for turbulent shear
stress between two layer of fluid at a small distance apart,
which is given as
Where u’, v’ fluctuating component of a velocity in the
direction of x and y due to turbulence.
As u’and v’ are varying and hence will be also very. Hence
to find the shear stress, the time average on both the sides of
the equations (10.10) is taken. Then equation (10.10) becomes
as
' 'u v
u v
20. 4.2. Prandtl Mixing Length Theory for Turbulent Shear
Stress.
In equation the turbulent shear stress can be only calculated if
the value of . To overcome this difficulty, L. Prandtl in
1925, presented a mixing length hypothesis which can be used
to express the turbulent shear stress in terms of measurable
quantities.
According to Prandtl, the mixing length l, is that distance
between two layers in the transmission direction such that the
lamps of fluid particle from one layer could reach the another
layer and the particle are mixed to other layer in a such way
that the momentum of the particle in the dimension of x is
same. He also assumed that the velocity fluctuation in the x-
direction is related to the momentum and length l as
u v
u
du
u l
dy
21. and v’ the fluctuation of velocity in y direction is one of
the same order magnitude and hence
Now becomes as
Substituting the value of in equations, we get
the expression for shear stress for turbulent flow due to
Prandtl as
du
v l
dy
u v u v
2
2du du du
l l l
dy dy dy
u v
2
2 du
l
dy
22. Thus the total shear stress at any point turbulent flow
is the same of the shear stress in turbulent flow due to
Prandtl as
But the viscous shear stress is negligible except near
the boundary. Equations is used in most of turbulent
fluid flow problem for determining shear stress in
turbulent flow.
2
2du du
l
dy dy