Boundary Layer Theory 6b4650
This document was ed by and they confirmed that they have the permission to share it. If you are author or own the copyright of this book, please report to us by using this report form. Report 3i3n4
Overview 26281t
& View Boundary Layer Theory as PDF for free.
More details 6y5l6z
- Words: 1,921
- Pages: 28
1
BOUNDARY LAYER THEORY
Introduction
2
The concept of boundary layer was first introduced by a German engineer, Prandtl in 1904. According to Prandtl theory, when a real fluid flows past a stationary solid boundary, the flow will be divided into two regions. i) A thin layer ading the solid boundary where the viscous force and rotation cannot be neglected. ii) An outer region where the viscous force is very small and can be neglected. The flow behaviour is similar to the upstream flow.
When a real fluid flow past a solid boundary, a layer of fluid which comes in s with the boundary surface adheres to it on of viscosity. Since this layer of fluid can not slip away from the boundary surface it attains the same velocity as that of the boundary.
This is also called as no slip condition.
Contd..,
3
When a real fluid flow past a solid body or a solid wall, the fluid particles adhere to the boundary and condition of no slip occurs. This means that the velocity of fluid close to the boundary will be same as that of the boundary.
If the boundary is stationary, the velocity of fluid at the boundary will be zero. Farther away from the boundary, the velocity will be higher and as a result of this variation of velocity, the velocity gradient du/dy will exit. The velocity of fluid increases from zero velocity on the stationary boundary to free stream velocity (U) of the fluid in the
direction normal to the boundary.
Because of this velocity gradient the fluid exerts a shear stress on the wall in the direction of motion. The value of shear stress is given by
Laminar & turbulent zones in boundary layer
Laminar zone: Near the leading edge of the surface of plate, where the thickness is small, the flow in the boundary layer is laminar. This is shown by AB.
Transition zone:
The short length over which the boundary layer flow changes from laminar to turbulent is called transition zone. This is shown by BC.
Turbulent zone: For down stream to the transition zone , the boundary layer is turbulent and continuous to grow in thickness. This layer is called turbulent. This is shown by CD.
4
Boundary layer thickness
5
It is defined as the distance from the boundary of the solid body measure in the ydirection to the point, where the velocity of the fluid is approximately equal to 0.99 times
the free stream velocity of the fluid.
This variation of velocity from zero to 99%free stream velocity in the direction normal to the boundary takes place in a narrow region in the vicinity of solid boundary. This narrow
region of the fluid is called boundary layer. The theory dealing with boundary layer flows is called boundary layer theory.
6
Displacement Thickness , Momentum Thickness & Energy Thickness
Boundary Layers
A boundary layer is the layer of fluid in the immediate vicinity of a bounding surface where the effects of viscosity are significant.
7
There are Four main definitions of boundary layers : 1. Boundary layer thickness 2. Displacement thickness
3. Momentum thickness 4. Energy thickness
8
9
Assumptions The boundary layer equations require several assumptions about the flow in the boundary layer.
1. All of the viscous effects of the flow field are confined to the boundary layer, adjacent to the wall . Outside of the boundary layer, viscous effects are not important, so that flow
can be determined by in viscid solutions such as potential flow or Euler equations. 2. The viscous layer is thin compared to the wall.
3. The boundary conditions of the boundary layer region are the no-slip condition at the wall.
Boundary Layer
10
Thickness
It is defined as the distance by which the boundary should be displaced to compensate for the reduction in flow rate of mass of the flowing fluid on of boundary layer formation. δ(x) is the boundary layer thickness when u(y) =0.99V V is the free-stream velocity
The purpose of the boundary layer is to allow the fluid to change its velocity from the upstream value of V to zero on the surface
Displacement Thickness
There is a reduction in the flow rate due to the presence of the boundary layer
This is equivalent to having a theoretical boundary layer with zero flow
11
12 Because of the velocity deficit, within the boundary layer, the flow rate across section b–b is less than that across section a–a. However, if we displace the plate at section a–a by an appropriate amount the boundary layer displacement thickness, the flow rates across each section will be identical.
Mathematically :
Where b is plate width
Momentum Thickness
13
Momentum thickness is a measure of the boundary layer thickness.
It is defined as the distance by which the boundary should be displaced to compensate for the reduction in momentum of the flowing fluid on of boundary layer formation
The momentum thickness, symbolized by Ө is the distance that, when multiplied by the square of the free-stream velocity, equals the integral of the momentum defect, across the boundary layer.
14 It is often used when determining the drag on an object. Again because of the velocity deficit U-u, in the boundary layer, the momentum flux across section b–b in Fig. 9.8 is less than that across section a–a. This deficit in momentum flux for the actual boundary layer flow on a plate of width b is given by
Mathematically :
which by definition is the momentum flux in a layer of uniform speed U and thickness Ө.That is,
15
16 It is often used when determining the drag on an object. Again because of the velocity deficit U-u, in the boundary layer, the kinetic energy across section b–b in Fig. 9.8 is less than that across section a–a. This deficit in kinetic energy for the actual boundary layer flow on a plate of width b is given by
Energy Thickness
17
Momentum thickness is a measure of the boundary layer thickness.
It is defined as the distance by which the boundary should be displaced to compensate for the reduction in kinetic energy of the flowing fluid on of boundary layer formation
The energy thickness, symbolized by Ө** is the distance that, when multiplied by the cube of the free-stream velocity, equals the integral of the energy defect, across the boundary layer.
18
Separation of boundary layer
As the flow proceed over a soil surface , the boundary layer thickness increases .
The velocity profile change from parabolic to logarithmic .
The fluid layer adjacent to the solid surface has to do work against surface friction by consuming some kinetic energy. This loss of kinetic energy recovered from adjacent fluid layer through momentum exchange process.
Thus the velocity of the layer goes on decreasing.
Along the length of solid body, at a certain point a stage may come when the boundary layer may not be able to keep sticking to the solid body .
In other words , the boundary layer will be separated from the surface . This phenomenon is called the boundary layer separation.
The point on the body at which the boundary layer is on the verge of separation from the surface is called point of separation.
Disadvantage of boundary layer separation
Separation of boundary layer greatly affect the flow as a whole.
In particular the formation of a weak zone of disturbed fluid on the downstream, in which the pressure is approximately constant and much less than that on the upstream, gives rise to boundary forces.
Thus, the separation of boundary layer gives to additional resistance to flow.
Separation of boundary layer from the surface of a body is a accompained by reversed flow in the vicinity of the body.
Reversal of flow and consequent eddy formation are generally undesirable because considerable amount of energy is lost in the process of eddying.
It is therefore necessary to control the separation of boundary layer as far as possible.
Method of controlling separation of boundary layer 1. Acceleration of the fluid in the boundary layer:
This method consist of supplying additional energy to the particle of fluid which are being retarded in the boundary layer.
This may be achieved by injecting fluid into the region of boundary layer from the interior of the body with the help of some suitable device shown in fig.
2. Suction of the fluid from the boundary layer :
In this method the slow moving fluid in the boundary layer is removed by suctions through slots, so that on the downstream of the point of suction a new boundary layer starts developing which is able to withstand an adverse pressure gradient and hence separation is prevented.
3.Motion of solid boundary :
The formation of the boundary layer is due to the difference between the velocity of the flowing fluid and that of the solid boundary.
As such it is possible to eliminate the formation of boundary layer by causing the solid
boundary to move with the flowing fluid. 4. When the flow take place round a bend, a pressure gradient is generated and there is a tendency of separation at the inner radius of the bend.
Control of separation.
Streamlined body
25
It is defined as the type of body whose surface coincides with the stream – lines, when the body is placed in a flow. In this case the flow separation does not occur up to rear most of the body.
Thus behind a stream lined body, wake formation zone will be very small. The total drag on the streamlined body drag will be very small.
A streamlined body is a shape that lowers the friction drag between a fluid, like air and water, and an object moving through that fluid.
Drag is a force that slows down motion; friction drag is a special kind of drag.
It occurs when the fluid closest to the object sticks to its surface, exerting a force that opposes the object’s motion.
Many animals, such as birds and dolphins, and many machines such as airplanes and submarines, have streamlined bodies to reduce friction drag as they move through either air or water.
Bluff body
26
It is defined ass that type of body whose surface does not coincide with the streamlines, when placed in a flow.
Bodies subjected to fluid flow are classified as being streamlined or blunt/bluff, depending on their overall shape.
A bluff body can be defined as a body that, as a result of its shape, has separated flow over a substantial part of its surface. Any body which when kept in fluid flow, the fluid does not touch the whole boundary of the object. An important feature of a bluff body flow is that there is a very strong interaction between the viscous and inviscid regions.
Cylinders and spheres are considered bluff bodies because at large Reynolds numbers the drag is dominated by the pressure losses in the wake.
Therefore, when the drag is dominated by a frictional component, the body is called a streamlined body; whereas in the case of dominant pressure drag, the body is called a bluff body.
27
28
Thank You
BOUNDARY LAYER THEORY
Introduction
2
The concept of boundary layer was first introduced by a German engineer, Prandtl in 1904. According to Prandtl theory, when a real fluid flows past a stationary solid boundary, the flow will be divided into two regions. i) A thin layer ading the solid boundary where the viscous force and rotation cannot be neglected. ii) An outer region where the viscous force is very small and can be neglected. The flow behaviour is similar to the upstream flow.
When a real fluid flow past a solid boundary, a layer of fluid which comes in s with the boundary surface adheres to it on of viscosity. Since this layer of fluid can not slip away from the boundary surface it attains the same velocity as that of the boundary.
This is also called as no slip condition.
Contd..,
3
When a real fluid flow past a solid body or a solid wall, the fluid particles adhere to the boundary and condition of no slip occurs. This means that the velocity of fluid close to the boundary will be same as that of the boundary.
If the boundary is stationary, the velocity of fluid at the boundary will be zero. Farther away from the boundary, the velocity will be higher and as a result of this variation of velocity, the velocity gradient du/dy will exit. The velocity of fluid increases from zero velocity on the stationary boundary to free stream velocity (U) of the fluid in the
direction normal to the boundary.
Because of this velocity gradient the fluid exerts a shear stress on the wall in the direction of motion. The value of shear stress is given by
Laminar & turbulent zones in boundary layer
Laminar zone: Near the leading edge of the surface of plate, where the thickness is small, the flow in the boundary layer is laminar. This is shown by AB.
Transition zone:
The short length over which the boundary layer flow changes from laminar to turbulent is called transition zone. This is shown by BC.
Turbulent zone: For down stream to the transition zone , the boundary layer is turbulent and continuous to grow in thickness. This layer is called turbulent. This is shown by CD.
4
Boundary layer thickness
5
It is defined as the distance from the boundary of the solid body measure in the ydirection to the point, where the velocity of the fluid is approximately equal to 0.99 times
the free stream velocity of the fluid.
This variation of velocity from zero to 99%free stream velocity in the direction normal to the boundary takes place in a narrow region in the vicinity of solid boundary. This narrow
region of the fluid is called boundary layer. The theory dealing with boundary layer flows is called boundary layer theory.
6
Displacement Thickness , Momentum Thickness & Energy Thickness
Boundary Layers
A boundary layer is the layer of fluid in the immediate vicinity of a bounding surface where the effects of viscosity are significant.
7
There are Four main definitions of boundary layers : 1. Boundary layer thickness 2. Displacement thickness
3. Momentum thickness 4. Energy thickness
8
9
Assumptions The boundary layer equations require several assumptions about the flow in the boundary layer.
1. All of the viscous effects of the flow field are confined to the boundary layer, adjacent to the wall . Outside of the boundary layer, viscous effects are not important, so that flow
can be determined by in viscid solutions such as potential flow or Euler equations. 2. The viscous layer is thin compared to the wall.
3. The boundary conditions of the boundary layer region are the no-slip condition at the wall.
Boundary Layer
10
Thickness
It is defined as the distance by which the boundary should be displaced to compensate for the reduction in flow rate of mass of the flowing fluid on of boundary layer formation. δ(x) is the boundary layer thickness when u(y) =0.99V V is the free-stream velocity
The purpose of the boundary layer is to allow the fluid to change its velocity from the upstream value of V to zero on the surface
Displacement Thickness
There is a reduction in the flow rate due to the presence of the boundary layer
This is equivalent to having a theoretical boundary layer with zero flow
11
12 Because of the velocity deficit, within the boundary layer, the flow rate across section b–b is less than that across section a–a. However, if we displace the plate at section a–a by an appropriate amount the boundary layer displacement thickness, the flow rates across each section will be identical.
Mathematically :
Where b is plate width
Momentum Thickness
13
Momentum thickness is a measure of the boundary layer thickness.
It is defined as the distance by which the boundary should be displaced to compensate for the reduction in momentum of the flowing fluid on of boundary layer formation
The momentum thickness, symbolized by Ө is the distance that, when multiplied by the square of the free-stream velocity, equals the integral of the momentum defect, across the boundary layer.
14 It is often used when determining the drag on an object. Again because of the velocity deficit U-u, in the boundary layer, the momentum flux across section b–b in Fig. 9.8 is less than that across section a–a. This deficit in momentum flux for the actual boundary layer flow on a plate of width b is given by
Mathematically :
which by definition is the momentum flux in a layer of uniform speed U and thickness Ө.That is,
15
16 It is often used when determining the drag on an object. Again because of the velocity deficit U-u, in the boundary layer, the kinetic energy across section b–b in Fig. 9.8 is less than that across section a–a. This deficit in kinetic energy for the actual boundary layer flow on a plate of width b is given by
Energy Thickness
17
Momentum thickness is a measure of the boundary layer thickness.
It is defined as the distance by which the boundary should be displaced to compensate for the reduction in kinetic energy of the flowing fluid on of boundary layer formation
The energy thickness, symbolized by Ө** is the distance that, when multiplied by the cube of the free-stream velocity, equals the integral of the energy defect, across the boundary layer.
18
Separation of boundary layer
As the flow proceed over a soil surface , the boundary layer thickness increases .
The velocity profile change from parabolic to logarithmic .
The fluid layer adjacent to the solid surface has to do work against surface friction by consuming some kinetic energy. This loss of kinetic energy recovered from adjacent fluid layer through momentum exchange process.
Thus the velocity of the layer goes on decreasing.
Along the length of solid body, at a certain point a stage may come when the boundary layer may not be able to keep sticking to the solid body .
In other words , the boundary layer will be separated from the surface . This phenomenon is called the boundary layer separation.
The point on the body at which the boundary layer is on the verge of separation from the surface is called point of separation.
Disadvantage of boundary layer separation
Separation of boundary layer greatly affect the flow as a whole.
In particular the formation of a weak zone of disturbed fluid on the downstream, in which the pressure is approximately constant and much less than that on the upstream, gives rise to boundary forces.
Thus, the separation of boundary layer gives to additional resistance to flow.
Separation of boundary layer from the surface of a body is a accompained by reversed flow in the vicinity of the body.
Reversal of flow and consequent eddy formation are generally undesirable because considerable amount of energy is lost in the process of eddying.
It is therefore necessary to control the separation of boundary layer as far as possible.
Method of controlling separation of boundary layer 1. Acceleration of the fluid in the boundary layer:
This method consist of supplying additional energy to the particle of fluid which are being retarded in the boundary layer.
This may be achieved by injecting fluid into the region of boundary layer from the interior of the body with the help of some suitable device shown in fig.
2. Suction of the fluid from the boundary layer :
In this method the slow moving fluid in the boundary layer is removed by suctions through slots, so that on the downstream of the point of suction a new boundary layer starts developing which is able to withstand an adverse pressure gradient and hence separation is prevented.
3.Motion of solid boundary :
The formation of the boundary layer is due to the difference between the velocity of the flowing fluid and that of the solid boundary.
As such it is possible to eliminate the formation of boundary layer by causing the solid
boundary to move with the flowing fluid. 4. When the flow take place round a bend, a pressure gradient is generated and there is a tendency of separation at the inner radius of the bend.
Control of separation.
Streamlined body
25
It is defined as the type of body whose surface coincides with the stream – lines, when the body is placed in a flow. In this case the flow separation does not occur up to rear most of the body.
Thus behind a stream lined body, wake formation zone will be very small. The total drag on the streamlined body drag will be very small.
A streamlined body is a shape that lowers the friction drag between a fluid, like air and water, and an object moving through that fluid.
Drag is a force that slows down motion; friction drag is a special kind of drag.
It occurs when the fluid closest to the object sticks to its surface, exerting a force that opposes the object’s motion.
Many animals, such as birds and dolphins, and many machines such as airplanes and submarines, have streamlined bodies to reduce friction drag as they move through either air or water.
Bluff body
26
It is defined ass that type of body whose surface does not coincide with the streamlines, when placed in a flow.
Bodies subjected to fluid flow are classified as being streamlined or blunt/bluff, depending on their overall shape.
A bluff body can be defined as a body that, as a result of its shape, has separated flow over a substantial part of its surface. Any body which when kept in fluid flow, the fluid does not touch the whole boundary of the object. An important feature of a bluff body flow is that there is a very strong interaction between the viscous and inviscid regions.
Cylinders and spheres are considered bluff bodies because at large Reynolds numbers the drag is dominated by the pressure losses in the wake.
Therefore, when the drag is dominated by a frictional component, the body is called a streamlined body; whereas in the case of dominant pressure drag, the body is called a bluff body.
27
28
Thank You
Related Documents 3h463d
Boundary Layer Theory 6b4650
April 2022 0
Boundary Layer Theory Schlichting 3vi5v
December 2019 133
Fluid-mechanics Boundary Layer 2b4q3o
November 2021 0
Boundary Layer Control j402m
November 2021 0
Flat Plate Boundary Layer 4q2s1j
November 2019 59
Schlichting H. - Boundary Layer Theory Springer 4f3j2m
November 2019 75More Documents from "VITAL" 4n3z5r
Boundary Layer Theory 6b4650
April 2022 0
4p345f
November 2019 47
Examen 5 Salud Ocupacional Respuestas 1y4xc
December 2019 70
4p345f
November 2019 47
4p345f
April 2022 0