BOUNDARY LAYERS

                                                  BOUNDARY LAYER FLOWS

 

Introduction:

When 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, velocity of fluid at the boundary will be zero. Further away from the boundary, the velocity will be higher and as a result of this variation of velocity,

 the velocity gradient du/dy  will exist. 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 boundary condition take place in a narrow region in the vicinity of solid boundary. This narrow region of fluid is called boundary layer. The theory dealing with boundary layer flows is called boundary layer theory, the flows of fluid in neighborhood of solid boundary may be divided into two regions as shown in the figure.

                                                                

 

1.     A very thin layer of fluid, called the boundary layer, in the intermediate neighborhood of the solid boundary, where the variation of velocity from zero at the solid boundary to free-stream velocity in the direction normal to the boundary take place. In this region, the velocity gradient du/dy exists and hence the fluid exerts a shear on the wall in the direction of motion. The value of shear stress is given by    

2.     The reaming fluid which is outside the boundary layer. The velocity outside the boundary layer is constant and equal to free-stream velocity. As there is no variation of velocity in this region, the velocity gradient  becomes zero. As result of this the shear stress is also zero.

Now let’s see some definitions related to boundary layer:

1.     Laminar boundary layer:

For defining the boundary layer (i.e. , laminar boundary layer or turbulent layer) consider the flow of a fluid, having free-stream velocity (U), over a smooth thin plate which is flat and placed parallel to the direction for free stream of fluid as shown in below figure. Let we consider the flow with zero pressure gradient on one side of the plate, which is stationary.

 

 

 

The length of the plate from leading edge, up to which the laminar boundary layer exists, is called laminar zone. This is shown by distance AB. The distance of B from leading edge is obtained from Reynold number equal to 5×10^5 for plate. Because up to this Reynold number the boundary layer is laminar. The Reynold number is given by (Re)x= u*x/v

Where   x = distance from leading edge,

              U = free-stream velocity of fluid,

              V = kinematics viscosity of fluid,

Hence for laminar boundary layer, we have 5×10^5 = u*x/v

   Turbulent boundary layer:

if the length of the plate is more than the distance x, calculated from the above equation, the thickness of boundary layer will go on increasing in the downstream direction. Then the laminar boundary layer becomes unstable an motion of fluid within it, is disturbed and irregular which leads to a transition from the laminar to turbulent boundary layer. This short length over which the boundary layer flow changes from laminar to turbulent is called transition zone this shown by distance BC in fig further downstream the transition zone, the boundary layer is turbulent and continues to grow in thickness. This layer of boundary is called turbulent boundary layer, which is shown by the portion FG in figure.

    Laminar sub-layer:

This is the region in turbulent boundary layer zone, adjacent to the solid surface of the plate as shown in fig. in this zone, the velocity variation is influenced only by viscous effects. Though the velocity distribution would be parabolic curve in the laminar sub layer zone, but in view of the very small thickness we can reasonably assume that the velocity variation is linear and so the velocity gradient can be consider constant. Therefore the shear stress in the laminar sub layer would be constant and equal to the boundary shear stress in the laminar sub layer would be constant and equal to the boundary shear stress . Thus, the shear stress in the sub layer is

   Boundary layer thickness 

It is defined as the distance from the boundary of the solid body measured in the y-direction to the point, where the velocity of the fluid is approximately equals to 0.99 times the free stream velocity (U) of the fluid. It is donated by the symbol For laminar and turbulent zone it is donated as :

·       

    Displacement thickness

It is 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 the flow rate on account of boundary layer formation. It is defined by

Expression for

                            

 

 

 6.              Momentum thickness (theta)

It is 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 the momentum of the flowing fluid on account of boundary layer  formation. It is denoted by (theta)

       Energy thickness (delta**)

It is defined as the distance measured perpendicular to  the boundary of the solid body, by which the boundary should be placed to compensate for the reduction in kinetic energy of the flowing fluid on account of boundary layer formation. It is denoted by delta**

                                                       

   Drag force on a flat plate due to boundary layer:

 Consider the flow of fluid having free-stream velocity equal to U, over a thin plate as shown in the fig, the drag force on plate can be determined if the velocity profile near the plate is known, consider a small length of the plate at a distance of x from the leading edge as shown in fig, the enlarged view of small length of the plate is shown in fig

 

 

                    

         the below equation is also known as  Von Karman momentum integral equation for boundary layer flows      

This applied to:

1.     Laminar boundary layers

2.     Transition boundary layers, and

3.     Turbulent boundary layer flows, 

                    The total drag on the plate of length L on one side is 

Local coefficient of drag (Cd*)  

It is defined as the ratio of the shear stress   to the quantity ( tau0)   It is denoted by 1/2 density U square

hence 

                                                    

Average coefficient of drag [Cd]

It is defined as the ratio of total drag force to the quantity 1/2 density A*U square. It is also called coefficient of drag and is denoted by Cd

hence 

Where   

  A = Area of the surface (or plate

  U = free stream velocity

   Ρ = mass density fluid

 

 

Boundary conditions for velocity profile:

The following are the boundary  conditions which must be satisfied by any velocity profile, whether it is in laminar boundary layer zone, or in turbulent boundary layer zone:

·       At  y = 0, u = 0 and du/dy has some finite value

·       At y = delta  u = U

·       At y = δ, du/dy = 0 

Turbulent boundary layer on a flat plate:

the thickness of the boundary layer, drag force on one side of the plate and coefficient of the drag due to turbulent boundary layer on smooth a smooth plate zero pressure gradient are determined as in case of laminar boundary layer provided the velocity profile is known. Blasius on the basis of experiments give the following velocity profile for turbulent boundary layer.

 

hence 

                                                          

above equation is not applicable very near the boundary, where the thin laminar sub layer of thickness delta  exist. Here velocity distribution is influenced only viscous effects. 

Analysis of turbulent boundary layer:

(a a)   If Reynold number is more than 5×10 raise to 5  and less than   10 raise to 7 the thickness of boundary layer and drag coefficient are given below as:  

      b) If the Reynold number is more than 10  raise to 7,  but less than 10 raise to 9, Schlichting gave the empirical equation as 

                                                        

Total drag on a flat plate due to laminar and turbulent boundary layer:

  

 

Let                                L = total length of the plate, b= width of the plate,

                                      A = length of laminar boundary layer

If the length of transition region is assumed negligible, then

                                      L-A = length of turbulent boundary layer.

We obtain the drag on plate for the laminar as well as turbulent boundary layer on the assumption that turbulent boundary layer starts from the leading edge. This assumption is valid only when the length of laminar boundary layer is negligible. But if the length of laminar boundary layers is calculated not negligible, then the  total drag on the plate due to laminar and turbulent boundary layer is calculated as,

1.      By finding the length from the leading edge up to which laminar boundary layer is exist,

2.     By finding drag using Blasius solution for the laminar boundary layer for length A,

3.      By finding the drag due to turbulent boundary layer for the whole length of the plate, also

4.     By finding the drag due to turbulent boundary layer for length A only

                 Then total drag on the plate =

·       Drag given by (2) + drag given by (3) – drag given by (4)

·       = drag due to laminar boundary layer for length A

·       + drag due to turbulent boundary layer for length L

·       −drag dure to turbulent boundary layer for length A

Separation of boundary layer:

When a solid body is immersed in flowing fluid, a thin layer of fluid called the boundary layer is formed adjacent to the solid body. In this thin layer of fluid, velocity varies from zero to free stream velocity in the direction to normal to solid body. Along the length of solid body, the thickness of boundary layer is increased. The fluid layer adjacent to the solid surface has to do work against surface friction at the expense of its kinetic energy. This loss of kinetic energy recovered from the intermediate fluid layer adjacent to the solid surface has to do work against surface  friction at the expense of its kinetic energy. The loss of the kinetic energy is to solid surface through momentum exchange process. Thus, the velocity goes on decreasing. Along the length of the solid body. At a certain point stage may come when the boundary layer may not be able to keep sticking to solid body of it cannot be provided kinetic energy to overcome the resistance offered by 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 separation.

 

Effect of pressure gradient on boundary layer separation:

Sure gradient dp/dx on boundary layer separation can be show by considering the flow over a curved surface ABCSD as shown in below figure. In the region ABC of the curved surface the area of the curved surface, the area of the flow decreases and hence velocity increases. This means that flows get accelerated in this region due to increases of the velocity, the pressure decreases in the direction of the flow and hence pressure gradient dp/dx is negative in this region. As long as dp/dx is smaller than the entire boundary layer moves forward as shown in below fig

 

Region CSD of the curved surface:

The pressure is minimum at the point C. along the region CSD of curved surface the area of flow velocity of the flow along the direction of fluid decreases. Due to decrease of velocity, the pressure increases in the direction of flow and hence the pressure gradient dp/dx is positive or dp/dx is smaller thus in the region CSD, the pressure gradient is positive and velocity of fluid layer along the direction of flow region decreases. As explained in below figure the velocity of the layer adjacent to the solid surface along the length of the solid surface goes on decreasing as kinetic energy is used to overcome the frictional resistance reduce the momentum of the fluid is enable to the surface. A stage comes, when the momentum of the fluid is unable to overcome the resistance and the boundary layer start separating from the surface at the point S downstream the point S, the flow is taking place in reverse direction and velocity gradient becomes negative

 Thus, the positive pressure gradient helps in separations of the boundary layer.

 

Methods of preventing the separation of the boundary layer:

When the boundary layer separates from the surface is shown in figure at a point S a certain portion adjacent to the surface has back flow and eddies are continuously formed in this region and hence continuous loss of energy take place. Thus, separation of boundary layer is undesirable and attempts should be made to avoid separation by various methods, the following are methods for preventing the separation of boundary layer:

1.     Suction of the slow-moving fluid by a suction slot.

2.     Supplying additional energy from a blower.

3.     Providing a bypass in the slotted wing.

4.     Rotating boundary in the direction of flow.

5.     Providing small divergence in a diffuser.

6.     Providing guide blades in bend.

Providing a trip wire ring in the laminar region for the flow over a sphere 

 

Contributors

 

Karan sanap

Vikas sanap

Sagar sanas

Durgesh sandhan

Aditya saravade