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Wednesday, 21 November 2018

November 21, 2018

ENERGY THICKNESS IN BOUNDARY LAYER

We were discussing the basics of Boundary layer theorylaminar boundary layer and turbulent boundary layer, in the subject of fluid mechanics, in our recent posts. 

After understanding the fundamentals of boundary layer thickness, displacement thickness and momentum thickness, we will go ahead now to find out the basics of energy thickness with the help of this post. 

Energy thickness 

Energy thickness is basically defined as the distance, measured perpendicular to the boundary of the solid body, by which the boundary should be displaced to compensate for the reduction in kinetic energy of the flowing fluid on account of boundary layer formation. 

Energy thickness will be displayed by the symbol δ**. 

Let us consider the fluid flow over the plate as displayed here in following figure. Let us assume one section 1-1 at a distance x from the leading edge. 
Mass of the fluid flowing per second through the elemental strip of thickness dy at a distance y from the plate will be given by following equation. 

Mass of the fluid flowing per second = ρubdy 

Where, 
u = Velocity of the fluid at the elemental strip 
b = width of the plate 
Area of the elemental strip = b dy 
Kinetic energy of this fluid with boundary layer = (1/2) x (ρubdy) x u
Kinetic energy of this fluid without boundary layer = (1/2) x (ρubdy) x U
Loss of kinetic energy through elemental strip = (1/2) x ρub x [U2- u2] x dy 

Now we will integrate the above equation from 0 to δ to secure the equation for loss of kinetic energy and we will have following equation as mentioned here. 
As we have discussed that energy thickness is basically the distance, measured perpendicular to the boundary of the solid body, by which the boundary should be displaced to compensate for the reduction in kinetic energy of the flowing fluid on account of boundary layer formation. 

So let us assume that we are displacing boundary by δ** to compensate for the reduction in kinetic energy of the flowing fluid on account of boundary layer formation. 

Loss of kinetic energy due to the displacement of boundary by δ** = (1/2) x (ρUb δ**) x U

Now we will equate the both equation of loss of kinetic energy and we will have expression for the energy thickness in boundary layer. 

Further we will go ahead to start a new topic i.e., in the subject of fluid mechanics, with the help of our next post. 

Do you have any suggestions? Please write in comment box.  

Reference: 
Fluid mechanics, By R. K. Bansal 
Image courtesy: Google  

Also read  

Sunday, 18 November 2018

November 18, 2018

BOUNDARY LAYER THICKNESS, DISPLACEMENT THICKNESS AND MOMENTUM THICKNESS

We were discussing the basics of Boundary layer theory, laminar boundary layer and turbulent boundary layer, in the subject of fluid mechanics, in our recent posts. 

Today we will be interested here to understand the basics of boundary layer thickness, displacement thickness and momentum thickness with the help of this post. 

Boundary layer thickness 

Boundary layer thickness is basically defined as the distance from the surface of the solid body, measured in the y-direction, up to a point where the velocity of flow is 0.99 times of the free stream velocity of the fluid. 

Boundary layer thickness will be displayed by the symbol δ. 

We can also define the boundary layer thickness as the distance from the surface of the body up to a point where the local velocity reaches to 99% of the free stream velocity of fluid. 

Displacement thickness 

Displacement thickness is basically defined as the distance, measured perpendicular to the boundary of the solid body, by which the boundary should be displaced to compensate for the reduction in flow rate on account of boundary layer formation. 

Displacement thickness will be displayed by the symbol δ*. 

We can also define the displacement thickness as the distance, measured perpendicular to the boundary of the solid body, by which the free stream will be displaced due to the formation of boundary layer. 

Momentum thickness 

Momentum thickness is basically defined as the distance, measured perpendicular to the boundary of the solid body, by which the boundary should be displaced to compensate for the reduction in momentum of the flowing fluid on account of boundary layer formation. 

Momentum thickness will be displayed by the symbol θ. 


Further we will go ahead to start a new topic i.e. Energy thickness in boundary layer, in the subject of fluid mechanics, with the help of our next post.  

Do you have any suggestions? Please write in comment box.  

Reference: 

Fluid mechanics, By R. K. Bansal 
Image courtesy: Google 

Also read  

Saturday, 17 November 2018

November 17, 2018

WHAT IS LAMINAR AND TURBULENT BOUNDARY LAYER?

We were discussing the basic concept of streamline and equipotential linedimensional homogeneityBuckingham pi theoremdifference between model and prototypebasic principle of similitude i.e. types of similarityvarious forces acting on moving fluid and Boundary layer theory in the subject of fluid mechanics, in our recent posts. 

Laminar boundary layer 

Before going to understand the basic principle and meaning of turbulent boundary layer, we will see here a brief introduction of laminar boundary layer and further we will understand here the basic concept of turbulent boundary layer with the help of this post. 

Let us consider the flow of a fluid over a smooth thin and flat plate. Let us assume that this thin and flat plate is located parallel to the direction of fluid flow as displayed here in following figure. 

Let us consider that fluid is flowing with free stream velocity U and with zero pressure gradient on one side of the stationary plate. 

As we have already discussed that when a real fluid will flow over a solid body or a solid wall, the particles of fluid will adhere to the boundary and there will be condition of no-slip. We can also conclude that the velocity of the fluid particles, close to the boundary, will have equal velocity as of the velocity of boundary.  

As we have assumed that plate is stationary and therefore the velocity of fluid flow over the surface of plate will be zero. 

If we move away from the plate, the velocity of fluid particles will also be increasing. Velocity of fluid particles will be changing from zero at the surface of stationary boundary to the free stream velocity (U) of the fluid in a direction normal to the plate. 

Therefore, there will be presence of velocity gradient, due to variation of velocity of fluid particles, near the surface of the fluid. 

This velocity gradient will develop shear resistance and this shear resistance will retard the fluid. Therefore, fluid with free stream velocity (U) is retarded in the surrounding area of the solid surface of the plate and boundary layer region will be started at the sharp leading edge. 

Once we will go away from the sharp leading edge, retardation of the fluid will be increased and therefore boundary layer region increases with increase in the retardation of the fluid. The increase in the region of boundary layer with increase in the retardation of the fluid will also be termed as growth of boundary layer. 

Near the leading edge of the surface of the plate, where thickness will be small, the flow in the boundary layer will be laminar and this layer of the fluid will be termed as laminar boundary layer. 

Let us see the above figure. AE indicates the laminar boundary layer. 

Laminar zone 

Length of the plate from the leading edge up to which laminar boundary layer exists will be termed as laminar zone. AB indicates the laminar zone in above figure. 

Length of the plate from the leading edge up to which laminar boundary layer exists i.e. laminar zone will be determined with the help of following formula as mentioned here. 
Where,
x = Distance from leading edge up to which laminar boundary layer exists
U = Free stream velocity of the fluid
v = Kinematic viscosity of the fluid 

Turbulent boundary layer fundamentals

If the length of plate is greater than the value of x which is determined from above equation, thickness of boundary layer will keep increasing in the downstream direction. 

Laminar boundary layer will become unstable and movement of fluid particles within it will be disturbed and irregular. It will lead to a transition from laminar to turbulent boundary layer. 

This small length over which the boundary layer flow changes from laminar to turbulent will be termed as transition zone. BC, in above figure, indicates the transition zone. 

Further downstream the transition zone, boundary layer will be turbulent and the layer of boundary will be termed as turbulent boundary layer. 

FG, in above figure, indicates the turbulent boundary layer and CD represent the turbulent zone. 

Further we will go ahead to start a new topic i.e. boundary layer thickness, in the subject of fluid mechanics, with the help of our next post. 

Do you have any suggestions? Please write in comment box. 

Reference: 

Fluid mechanics, By R. K. Bansal 
Image courtesy: Google  

Also read  

Wednesday, 14 November 2018

November 14, 2018

WHAT IS LAMINAR BOUNDARY LAYER ?

We were discussing the basic concept of streamline and equipotential linedimensional homogeneityBuckingham pi theoremdifference between model and prototypebasic principle of similitude i.e. types of similarityvarious forces acting on moving fluid and Boundary layer theory in the subject of fluid mechanics, in our recent posts. 

Now we will go ahead to start a new topic i.e. Laminar boundary layer, in the subject of fluid mechanics with the help of this post. 

Laminar boundary layer 

We will try to understand here the basic principle and meaning of laminar boundary layer and turbulent boundary layer. First, we will see here laminar boundary layer. 

Let us consider the flow of a fluid over a smooth thin and flat plate. Let us assume that this thin and flat plate is located parallel to the direction of fluid flow as displayed here in following figure. 

Let us consider that fluid is flowing with free stream velocity U and with zero pressure gradient on one side of the stationary plate. 
As we have already discussed that when a real fluid will flow over a solid body or a solid wall, the particles of fluid will adhere to the boundary and there will be condition of no-slip. We can also conclude that the velocity of the fluid particles, close to the boundary, will have equal velocity as of the velocity of boundary. 

As we have assumed that plate is stationary and therefore the velocity of fluid flow over the surface of plate will be zero. 

If we move away from the plate, the velocity of fluid particles will also be increasing. Velocity of fluid particles will be changing from zero at the surface of stationary boundary to the free stream velocity (U) of the fluid in a direction normal to the plate. 

Therefore, there will be presence of velocity gradient, due to variation of velocity of fluid particles, near the surface of the fluid. 

This velocity gradient will develop shear resistance and this shear resistance will retard the fluid. Therefore, fluid with free stream velocity (U) is retarded in the surrounding area of the solid surface of the plate and boundary layer region will be started at the sharp leading edge. 

Once we will go away from the sharp leading edge, retardation of the fluid will be increased and therefore boundary layer region increases with increase in the retardation of the fluid. The increase in the region of boundary layer with increase in the retardation of the fluid will also be termed as growth of boundary layer. 

Near the leading edge of the surface of the plate, where thickness will be small, the flow in the boundary layer will be laminar and this layer of the fluid will be termed as laminar boundary layer. 

Let us see the above figure. AE indicates the laminar boundary layer. 

Laminar zone 

Length of the plate from the leading edge up to which laminar boundary layer exists will be termed as laminar zone. AB indicates the laminar zone in above figure. 

Length of the plate from the leading edge up to which laminar boundary layer exists i.e. laminar zone will be determined with the help of following formula as mentioned here. 
Where, 
x = Distance from leading edge up to which laminar boundary layer exists 
U = Free stream velocity of the fluid 
v = Kinematic viscosity of the fluid 

Further we will go ahead to start a new topic i.e. Turbulent boundary layer, in the subject of fluid mechanics with the help of our next post. 

Do you have any suggestions? Please write in comment box. 

Reference: 

Fluid mechanics, By R. K. Bansal 
Image courtesy: Google 

Also read  

Sunday, 28 October 2018

October 28, 2018

BOUNDARY LAYER THEORY IN FLUID MECHANICS

We were discussing the basic concept of streamline and equipotential linedimensional homogeneityBuckingham pi theoremdifference between model and prototypebasic principle of similitude i.e. types of similarity and various forces acting on moving fluid in the subject of fluid mechanics, in our recent posts. 

Now we will go ahead to start a new topic i.e. Boundary layer theory, in the subject of fluid mechanics with the help of this post. 

Boundary layer theory 

When a real fluid will flow over a solid body or a solid wall, the particles of fluid will adhere to the boundary and there will be condition of no-slip. 

We can also conclude that the velocity of the fluid particles, close to the boundary, will have equal velocity as of the velocity of boundary. 

If we assume that boundary is stationary or velocity of boundary is zero, then the velocity of fluid particles adhere or very close to the boundary will also have zero velocity. 

If we move away from the boundary, the velocity of fluid particles will also be increasing. Velocity of fluid particles will be changing from zero at the surface of stationary boundary to the free stream velocity (U) of the fluid in a direction normal to the boundary. 

Therefore, there will be presence of velocity gradient (du/dy) due to variation of velocity of fluid particles. 

The variation in the velocity of the fluid particles, from zero at the surface of stationary boundary to the free stream velocity (U) of the fluid, will take place in a narrow region in the vicinity of solid boundary and this narrow region of the fluid will be termed as boundary layer. 

Science and theory dealing with the problems of boundary layer flows will be termed as boundary layer theory. 

According to the boundary layer theory, fluid flow around the solid boundary might be divided in two regions as mentioned and displayed here in following figure. 

First region 
A very thin layer of fluid, called the boundary layer, in the immediate region of the solid boundary, where the variation in the velocity of the fluid particles, from zero at the surface of stationary boundary to the free stream velocity (U) of the fluid, will take place. 

There will be presence of velocity gradient (du/dy) due to variation of velocity of fluid particles in this region and therefore fluid will provide one shear stress over the wall in the direction of motion.
Shear stress applied by the fluid over the wall will be determined with the help of following equation.

𝜏 = µ x (du/dy) 

Second region
Second region will be the region outside of the boundary layer. Velocity of the fluid particles will be constant outside the boundary layer and will be similar with the free stream velocity of the fluid. 

In this region, there will be no velocity gradient as velocity of the fluid particles will be constant outside the boundary layer and therefore there will be no shear stress exerted by the fluid over the wall beyond the boundary layer. 

Further we will go ahead to find out the some basic concepts and definitions in the respect of boundary layer theory in the subject of fluid mechanics, with the help of our next post. 

Do you have any suggestions? Please write in comment box. 

Reference: 

Fluid mechanics, By R. K. Bansal 
Image courtesy: Google 

Also read  

Sunday, 14 October 2018

October 14, 2018

WATER HAMMER IN PIPES

We were discussing the basics of shear stress in turbulent flowminor head losses in pipe flowhydraulic gradient and total energy line, basic concept and working of syphonflow through pipes in seriesflow through pipes in parallel, flow through branched pipes, power transmission through pipes and flow through nozzle, in the subject of fluid mechanics, in our recent posts. 

Now we will go ahead to see the concept of water hammer in pipes, in the subject of fluid mechanics, with the help of this post. 

Water hammer in pipes 

A sudden change of fluid flow rate in a large pipeline, due to sudden closing of valve or pump, may involve a great mass of water moving inside the pipe. 

Pressure will be increased in the pipe, due to sudden change of flow rate, greater than the normal static pressure in the pipe. 

Excessive pressure may fracture the pipe walls or cause other damage to the pipe line system. This phenomenon will be termed as water hammer phenomenon. 

Let us understand the basic concept of water hammer in pipes 

Let us consider, as displayed here in following figure, one tank filled with water and a pipe AB which is connected with water tank at one end. Let us think that water is filled in tank up to a height of H from the centre of pipe AB. 

One valve is provided at the other end of pipe in order to regulate the flow of fluid. Let us assume that valve is fully open and water is flowing at a velocity of V through the pipe. 

If valve is closed suddenly, momentum of flowing water will be destroyed and consequently a wave of high pressure will be set up. This wave of high pressure will be transmitted along the pipe with a velocity equal to the velocity of the sound wave and may create noise called knocking. 

There will be also hammering action over the pipe walls due to this wave of high pressure and hence this phenomenon will be termed as water hammer. 

Pressure rise due to water hammer will be dependent over the following factors as mentioned here 

  • Velocity of water flow in pipe 
  • Pipe length 
  • Time taken for closing the valve 
  • Elastic properties of the material of the pipe 

The sudden rise of pressure due to water hammer may be viewed as the result of the force developed in the pipe required to stop the flowing water column. Let us think that water column has a total mass M and it is changing its velocity at the rate of dV/dT. 

Now according to the Newton’s law of motion, we will have following equation as mentioned here
F = M dV/dT

If the velocity of entire water column is reduced to zero instantly then we will have following case 

F = M dV/dT = M (V0 – 0)/0 = ∞ 
F = Infinite 

The resulting force and hence pressure will be infinite, but it is not possible because mechanical valve will take a certain amount of time for complete closure of valve. 

In addition, neither the pipe walls nor the water column involved are perfectly rigid under large pressure. The elasticities of the material of the pipe and water column will also play an important role in the phenomenon of water hammer.  

Further we will go ahead to find out the concept of boundary layer theoryin the subject of fluid mechanics, with the help of our next post. 

Do you have any suggestions? Please write in comment box. 

Reference: 

Fluid mechanics, By R. K. Bansal 
Image courtesy: Google 

Also read