This document discusses turbulent fluid flow. It defines turbulence as an irregular flow with random variations in time and space that can be expressed statistically. Turbulence occurs above a critical Reynolds number when the kinetic energy of the flow is enough to sustain random fluctuations against viscous damping. Characteristics of turbulent flow include fluctuating velocities and pressures, and more uniform velocity distributions compared to laminar flow. Turbulence can be generated by solid walls or shear between layers, and can be categorized as homogeneous, isotropic, or anisotropic. Transition from laminar to turbulent flow is also discussed.
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
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
It includes details about boundary layer and boundary layer separations like history,causes,results,applications,types,equations, etc.It also includes some real life example of boundary layer.
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
Fluid Mechanics Chapter 4. Differential relations for a fluid flowAddisu Dagne Zegeye
Introduction, Acceleration field, Conservation of mass equation, Linear momentum equation, Energy equation, Boundary condition, Stream function, Vorticity and Irrotationality
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.
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.
It includes details about boundary layer and boundary layer separations like history,causes,results,applications,types,equations, etc.It also includes some real life example of boundary layer.
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
Fluid Mechanics Chapter 4. Differential relations for a fluid flowAddisu Dagne Zegeye
Introduction, Acceleration field, Conservation of mass equation, Linear momentum equation, Energy equation, Boundary condition, Stream function, Vorticity and Irrotationality
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.
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.
A fluid is a state of matter in which its molecules move freely and do not bear a constant relationship in space to other molecules.
In physics, fluid flow has all kinds of aspects: steady or unsteady, compressible or incompressible, viscous or non-viscous, and rotational or irrotational to name a few. Some of these characteristics reflect properties of the liquid itself, and others focus on how the fluid is moving.
Fluids are :-
Liquid : blood, i.v. infusions)
Gas : O2 , N2O)
Vapour (transition from liquid to gas) : N2O (under compression in cylinder), volatile inhalational agents (halothane, isoflurane, etc)
Sublimate (transition from solid to gas bypassing liquid state) : Dry ice (solid CO2), iodine
Fluid Mechanics Chapter 3. Integral relations for a control volumeAddisu Dagne Zegeye
Introduction, physical laws of fluid mechanics, the Reynolds transport theorem, Conservation of mass equation, Linear momentum equation, Angular momentum equation, Energy equation, Bernoulli equation
Fluid Flow, Heat and Mass Transfer at Bodies of Different Shapes: Numerical Solutions presents the current theoretical developments of boundary layer theory, a branch of transport phenomena. Also, the book addresses the theoretical developments in the area and presents a number of physical problems that have been solved by analytical or numerical method. It is focused particularly on fluid flow problems governed by nonlinear differential equations. The book is intended for researchers in applied mathematics, physics, mechanics and engineering.
fluid Motion in the presence of solid particlesUsman Shah
This slide will explain you the chemical engineering terms .Al about the basics of this slide are explain in it. The basics of fluid mechanics, heat transfer, chemical engineering thermodynamics, fluid motions, newtonian fluids, are explain in this process.
Quality defects in TMT Bars, Possible causes and Potential Solutions.PrashantGoswami42
Maintaining high-quality standards in the production of TMT bars is crucial for ensuring structural integrity in construction. Addressing common defects through careful monitoring, standardized processes, and advanced technology can significantly improve the quality of TMT bars. Continuous training and adherence to quality control measures will also play a pivotal role in minimizing these defects.
Event Management System Vb Net Project Report.pdfKamal Acharya
In present era, the scopes of information technology growing with a very fast .We do not see any are untouched from this industry. The scope of information technology has become wider includes: Business and industry. Household Business, Communication, Education, Entertainment, Science, Medicine, Engineering, Distance Learning, Weather Forecasting. Carrier Searching and so on.
My project named “Event Management System” is software that store and maintained all events coordinated in college. It also helpful to print related reports. My project will help to record the events coordinated by faculties with their Name, Event subject, date & details in an efficient & effective ways.
In my system we have to make a system by which a user can record all events coordinated by a particular faculty. In our proposed system some more featured are added which differs it from the existing system such as security.
Industrial Training at Shahjalal Fertilizer Company Limited (SFCL)MdTanvirMahtab2
This presentation is about the working procedure of Shahjalal Fertilizer Company Limited (SFCL). A Govt. owned Company of Bangladesh Chemical Industries Corporation under Ministry of Industries.
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.
Forklift Classes Overview by Intella PartsIntella Parts
Discover the different forklift classes and their specific applications. Learn how to choose the right forklift for your needs to ensure safety, efficiency, and compliance in your operations.
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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.
Water scarcity is the lack of fresh water resources to meet the standard water demand. There are two type of water scarcity. One is physical. The other is economic water scarcity.
Explore the innovative world of trenchless pipe repair with our comprehensive guide, "The Benefits and Techniques of Trenchless Pipe Repair." This document delves into the modern methods of repairing underground pipes without the need for extensive excavation, highlighting the numerous advantages and the latest techniques used in the industry.
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TECHNICAL TRAINING MANUAL GENERAL FAMILIARIZATION COURSEDuvanRamosGarzon1
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The Single Aisle is the most advanced family aircraft in service today, with fly-by-wire flight controls.
The A318, A319, A320 and A321 are twin-engine subsonic medium range aircraft.
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2. Introduction
1-2
Turbulent motion is an irregular motion.
Turbulent fluid motion can be considered as an irregular condition of flow in which
various quantities (such as velocity components and pressure) show a random
variation with time and space in such a way that the statistical average of those
quantities can be quantitatively expressed.
It is postulated that the fluctuations inherently come from disturbances (such as
roughness of a solid surface) and they may be either dampened out due to viscous
damping or may grow by drawing energy from the free stream.
At a Reynolds number less than the critical, the kinetic energy of flow is not
enough to sustain the random fluctuations against the viscous damping and in such
cases laminar flow continues to exist.
At somewhat higher Reynolds number than the critical Reynolds number, the
kinetic energy of flow supports the growth of fluctuations and transition to
turbulence takes place.
3. Characteristics of Turbulent Flow
1-3
The most important characteristic of turbulent motion is the fact that velocity and
pressure at a point fluctuate with time in a random manner.
Fig. Variation of horizontal components of velocity for laminar and turbulent flows at a point P
The mixing in turbulent flow is more due to these fluctuations. As a result we can
see more uniform velocity distributions in turbulent pipe flows as compared to the
laminar flows.
Fig. Comparison of velocity profiles in a pipe for (a) laminar and (b) turbulent flows
4. Characteristics of Turbulent Flow
1-4
Turbulence can be generated by -
1. frictional forces at the confining solid walls
2. the flow of layers of fluids with different velocities over one another
The turbulence generated in these two ways are considered to be different
Turbulence generated and continuously affected by fixed walls is designated as
wall turbulence , and turbulence generated by two adjacent layers of fluid in absence
of walls is termed as free turbulence .
Turbulence can be categorized as below -
1. Homogeneous Turbulence: Turbulence has the same structure quantitatively in
all parts of the flow field.
2. Isotropic Turbulence: The statistical features have no directional preference and
perfect disorder persists.
3. Anisotropic Turbulence: The statistical features have directional preference and
the mean velocity has a gradient.
5. Characteristics of Turbulent Flow
1-5
Homogeneous Turbulence : The term homogeneous turbulence implies that the
velocity fluctuations in the system are random but the average turbulent characteristics
are independent of the position in the fluid, i.e., invariant to axis translation.
Consider the root mean square velocity fluctuations
In homogeneous turbulence, the rms values of u', v' and w' can all be
different, but each value must be constant over the entire turbulent field. Note that
even if the rms fluctuation of any component, say u' s are constant over the entire field
the instantaneous values of u necessarily differ from point to point at any instant.
6. Characteristics of Turbulent Flow
1-6
Isotropic Turbulence : The velocity fluctuations are independent of the axis of
reference, i.e. invariant to axis rotation and reflection. Isotropic turbulence is by its
definition always homogeneous . In such a situation, the gradient of the mean velocity
does not exist, the mean velocity is either zero or constant throughout.
In isotropic turbulence fluctuations are independent of the direction of
reference and
It is reemphasized that even if the rms fluctuations at any point are same,
their instantaneous values necessarily differ from each other at any instant.
7. Laminar-Turbulent Transition
1-7
For a turbulent flow over a flat plate,
The turbulent boundary layer continues to grow in thickness, with a small region
below it called a viscous sub layer. In this sub layer, the flow is well behaved, just as
the laminar boundary layer
Fig. Laminar - turbulent transition
8. Laminar-Turbulent Transition
1-8
Observe that at a certain axial location, the laminar boundary layer tends to
become unstable. Physically this means that the disturbances in the flow grow in
amplitude at this location.
Free stream turbulence, wall roughness and acoustic signals may be among the
sources of such disturbances. Transition to turbulent flow is thus initiated with the
instability in laminar flow
The possibility of instability in boundary layer was felt by Prandtl as early as
1912. The theoretical analysis of Tollmien and Schlichting showed that unstable
waves could exist if the Reynolds number was 575. The Reynolds number was
defined as
9. Laminar-Turbulent Transition
1-9
Taylor developed an alternate theory, which assumed that the transition is caused
by a momentary separation at the boundary layer associated with the free stream
turbulence. In a pipe flow the initiation of turbulence is usually observed at Reynolds
numbers in the range of 2000 to 2700.
The development starts with a laminar profile, undergoes a transition, changes over
to turbulent profile and then stays turbulent thereafter (Fig.). The length of
development is of the order of 25 to 40 diameters of the pipe.
Fig. Development of turbulent flow in a circular duct
10. Correlation Functions
1-10
A statistical correlation can be applied to fluctuating velocity terms in turbulence.
Turbulent motion is by definition eddying motion. A high degree of correlation exists
between the velocities at two points in space, if the distance between the points is
smaller than the diameter of the eddy. Conversely, if the points are so far apart that the
space, in between, corresponds to many eddy diameters (Figure), little correlation can
be expected
Fig. Velocity Correlation
11. Correlation Functions
1-11
Consider a statistical property of a random variable (velocity) at two points
separated by a distance r. An Eulerian correlation tensor (nine terms) at the two points
can be defined by
In other words, the dependence between the two velocities at two points is
measured by the correlations, i.e. the time averages of the products of the quantities
measured at two points. The correlation of the components of the turbulent velocity of
these two points is defined as
It is conventional to work with the non-dimensional form of the correlation, such as
12. Correlation Functions
1-12
A value of R(r) of unity signifies a perfect correlation of the two quantities
involved and their motion is in phase. Negative value of the correlation function
implies that the time averages of the velocities in the two correlated points have
different signs. Figure shows typical variations of the correlation R with increasing
separation r .
The positive correlation indicates that the fluid can be modeled as travelling in
lumps. Since swirling motion is an essential feature of turbulent motion, these lumps
are viewed as eddies of various sizes. The correlation R(r) is a measure of the strength
of the eddies of size larger than r. Essentially the velocities at two points are correlated
if they are located on the same eddy.
13. Correlation Functions
1-13
To describe the evolution of a fluctuating function u'(t), we need to know the
manner in which the value of u' at different times are related. For this purpose the
correlation function
Between the values of u' at different times is chosen and is called
autocorrelation function.
The correlation studies reveal that the turbulent motion is composed of eddies
which are convected by the mean motion . The eddies have a wide range variation in
their size. The size of the large eddies is comparable with the dimensions of the flow
passage.
The size of the smallest eddies can be of the order of 1 mm or less. However, the
smallest eddies are much larger than the molecular mean free paths.
14. Reynolds decomposition of turbulent flow
1-14
The Experiment: In 1883, O. Reynolds conducted experiments with pipe flow by
feeding into the stream a thin thread of liquid dye. For low Reynolds numbers, the dye
traced a straight line and did not disperse. With increasing velocity, the dye thread got
mixed in all directions and the flowing fluid appeared to be uniformly colored in the
downstream flow.
The Inference: It was conjectured that on the main motion in the direction of the
pipe axis, there existed a superimposed motion all along the main motion at right
angles to it. The superimposed motion causes exchange of momentum in transverse
direction and the velocity distribution over the cross-section is more uniform than in
laminar flow. This description of turbulent flow which consists of superimposed
streaming and fluctuating (eddying) motion is well known as Reynolds decomposition
of turbulent flow.
Here, we shall discuss different descriptions of mean motion. Generally, for
velocity u , the following two methods of averaging could be obtained.
15. Reynolds decomposition of turbulent flow
1-15
Time average for a stationary turbulence:.
Space average for a homogeneous turbulence:
For a stationary and homogeneous turbulence, it is assumed that the two averages
lead to the same result: and the assumption is known as the ergodic hypothesis.
In our analysis, average of any quantity will be evaluated as a time average . Take a
finite time interval t1. This interval must be larger than the time scale of turbulence.
Needless to say that it must be small compared with the period t2 of any slow variation
(such as periodicity of the mean flow) in the flow field that we do not consider to be
chaotic or turbulent .
16. Reynolds decomposition of turbulent flow
1-16
Thus, for a parallel flow, it can be written that the axial velocity component is
As such, the time mean component determines whether the turbulent
motion is steady or not. The symbol signify any of the space variables
While the motion described by Fig.(a) is for a turbulent flow with steady mean
velocity the Fig.(b) shows an example of turbulent flow with unsteady mean velocity.
The time period of the high frequency fluctuating component is t1 whereas the time
period for the unsteady mean motion is t2 and for obvious reason t2>>t1. Even if the
bulk motion is parallel, the fluctuation u ' being random varies in all directions.
Fig. Steady and unsteady mean motions in a turbulent flow
17. Reynolds decomposition of turbulent flow
1-17
The continuity equation, gives us
Invoking Eq.(1) in the above expression, we get
Since, Eq.(2) depicts that y and z components of velocity exist even for the
parallel flow if the flow is turbulent. We have-
18. Reynolds decomposition of turbulent flow
1-18
However, the fluctuating components do not bring about the bulk displacement of a
fluid element. The instantaneous displacement is u’dt, and that is not responsible for
the bulk motion. We can conclude from the above
Due to the interaction of fluctuating components, macroscopic momentum transport
takes place. Therefore, interaction effect between two fluctuating components over a
long period is non-zero and this can be expressed as
Taking time average of these two integrals and write
19. Reynolds decomposition of turbulent flow
1-19
Now, we can make a general statement with any two fluctuating parameters, say,
with f ' and g' as
The time averages of the spatial gradients of the fluctuating components also follow
the same laws, and they can be written as
The intensity of turbulence or degree of turbulence in a flow is described by the
relative magnitude of the root mean square value of the fluctuating components with
respect to the time averaged main velocity. The mathematical expression is given by
20. Reynolds decomposition of turbulent flow
1-20
For isotropic turbulence,
In this case, it is sufficient to consider the oscillation u' in the direction of flow and to
put
This simpler definition of turbulence intensity is often used in practice even in
cases when turbulence is not isotropic.
Following Reynolds decomposition, it is suggested to separate the motion into a
mean motion and a fluctuating or eddying motion. Denoting the time average of the u
component of velocity by and fluctuating component as , we can write down
the following,
21. Reynolds decomposition of turbulent flow
1-21
By definition, the time averages of all quantities describing fluctuations are equal to
zero.
The fluctuations u', v' , and w' influence the mean motion, in such a way
that the mean motion exhibits an apparent increase in the resistance to deformation. In
other words, the effect of fluctuations is an apparent increase in viscosity or
macroscopic momentum diffusivity .
Rules of mean time - averages :
If f and g are two dependent variables and if s denotes anyone of the independent
variables x, y
22. Governing Equations for Turbulent Flow
1-22
For incompressible flows, the Navier-Stokes equations can be rearranged in the
form
Express the velocity components and pressure in terms of time-mean values and
corresponding fluctuations. In continuity equation, this substitution and subsequent
time averaging will lead to
24. Governing Equations for Turbulent Flow
1-24
It is evident that the time-averaged velocity components and the fluctuating
velocity components, each satisfy the continuity equation for incompressible flow.
Imagine a two-dimensional flow in which the turbulent components are
independent of the z -direction. Eventually, Eq.(3b) tends to
On the basis of condition (4), it is postulated that if at an instant there is an increase
in u' in the x -direction, it will be followed by an increase in v' in the negative y -
direction. In other words, is non-zero and negative. (Figure 1)
25. Governing Equations for Turbulent Flow
1-25
Fig 1 Each dot represents uν pair at an instant
Invoking the concepts of into the equations of motion (eqn
1a,b,c), we obtain expressions in terms of mean and fluctuating components. Now,
forming time averages and considering the rules of averaging we discern the following
26. Governing Equations for Turbulent Flow
1-26
The terms which are linear, such as and vanish when they are averaged
(Refer Equation 6)
The same is true for the mixed like
But the quadratic terms in the fluctuating components remain in the equations.
After averaging, they form
Perform the aforesaid exercise on the x-momentum equation, we obtain
28. Governing Equations for Turbulent Flow
1-28
Introducing simplifications arising out of continuity Eq. (3a), we shall obtain.
Performing a similar treatment on y and z momentum equations, finally we obtain
the momentum equations in the form
29. Governing Equations for Turbulent Flow
1-29
Comments on the governing equation
I. The left hand side of Eqs (5a)-(5c) are essentially similar to the steady-state
Navier-Stokes equations if the velocity components u, v and w are replaced by
II. The same argument holds good for the first two terms on the right hand side of
Eqs (5a)-(5c).
III. However, the equations contain some additional terms which depend on turbulent
fluctuations of the stream. These additional terms can be interpreted as
components of a stress tensor.
30. Governing Equations for Turbulent Flow
1-30
Now, the resultant surface force per unit area due to these terms may be considered
as
Comparing Eqs (5) and (6), we can write
31. Governing Equations for Turbulent Flow
1-31
It can be said that the mean velocity components of turbulent flow satisfy the same
Navier-Stokes equations of laminar flow. However, for the turbulent flow, the
laminar stresses must be increased by additional stresses which are given by the
stress tensor (Eq. 7)
These additional stresses are known as apparent stresses of turbulent flow or
Reynolds stresses
Since turbulence is considered as eddying motion and the aforesaid additional
stresses are added to the viscous stresses due to mean motion in order to explain the
complete stress field, it is often said that the apparent stresses are caused by eddy
viscosity . The total stresses are now
and so on
The apparent stresses are much larger than the viscous components, and the viscous
stresses can even be dropped in many actual calculations
32. Turbulent Boundary Layer Equations
1-32
For a two-dimensional flow (w = 0)over a flat plate, the thickness of turbulent
boundary layer is assumed to be much smaller than the axial length and the order of
magnitude analysis may be applied. As a consequence, the following inferences are
drawn:
33. Turbulent Boundary Layer Equations
1-33
The turbulent boundary layer equation together with the equation of continuity
becomes
A comparison of Eq. (10) with laminar boundary layer Eq. depicts that: u, v and p
are replaced by the time average values
Laminar viscous force per unit volume is replace by
34. Prandtl’s Mixing Length Theory
1-34
Consider a fully developed turbulent boundary layer, quite away from leading edge
so that it may be considered s fully developed.
Such flow are classified as one dimension parallel flow. The velocity varies only
from one stream line to next. The main flow direction is assumed to be parallel to X
axis.
35. Prandtl’s Mixing Length Theory
1-35
The time average component of the velocities are given by
The fluctuating component of transverse velocity v’ transports mass and
momentum across a plane at a distance y1 from the wall. The shear stress due to
fluctuation is equal to the rate of x-momentum transport in y direction per unit area,
and so.
The time mean value of the turbulent shear stress is
In the above equation so the resulting equation is
Considering the direction of force associated with the shear stress, we can write,
36. Prandtl’s Mixing Length Theory
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A lump of fluid, which comes to the layer y1 from a layer (y1 -l) has a positive value
of v’.
If a lump of fluid retain its original value of momentum then its velocity at its
current location y1 is smaller then the velocity prevailing there. The difference in
velocity is then,
The above expression is obtained by expanding the function
in a Taylor series and neglecting all higher order terms and higher ordered derivatives.
The term l is small length scale and called Prandt’s mixing length. Prandtl’s
proposed that transverse displacement of any fluid particle is on an average equal to l.
Let’s consider another lump of fluid with a negative value of v’. This arrives at y1
from a layer (y1 +l).
If this lump of fluid retain its original value of momentum then its velocity at its
current location y1 will be somewhat more than the original velocity. This difference is
given by,
37. Prandtl’s Mixing Length Theory
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The velocity differences caused by the transverse motion can be regarded as the
turbulent velocity component at y1.
The time average of the absolute value of this fluctuation is given by,
Based on the experiments and simplification Prandlt’s observe that
Along with the condition that the moment at which u’ is positive, v’ is more likely
to be negative, and conversely when u’ is negative. Based on that we can write,
38. Prandtl’s Mixing Length Theory
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Where C1 and C2 are different proportionality constant. If C2 is included in the
unknown mixing length then the above equation can simplified as
From the expression of turbulent shear stress τt, we may write
Where µt is the turbulent viscosity or eddy viscosity and it is given by
The velocity gradient was express in modulus to adjust its sign according to sign of
shear stress.
39. Prandtl’s Mixing Length Theory
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The turbulent viscosity and consequently the mixing length are not properties of
fluid.
They are purely local functions and dependent on the nature of the turbulent flow
field.
The problem of determining the mixing length in terms of the velocity field has not
been resolved and it is a topic of continuous research. Several correlation using
experimental results for τt have been proposed to determine l.
Variation of mixing length with the boundary layer is shown in figure. It is zero in
the sublayer, equal to χy over about 20 percent of δ and proportional to δ elsewhere.
Parameter χ, the von Karman constant has a value of 0.4.
40. Prandtl’s Mixing Length Theory
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Where y is the distance from the wall, λ is a mixing length constant (=0.09) and δ is
the layer width.