Boundary layer

Boundary layer theory, Boundary layer separation and different methods to control boundary layer separation.

1. BOUNDARY LAYER THEORY PREPARED BY MR. CHAUDHARI. V. S DEPARTMENT OF CIVIL ENGINEERING SRES’S SANJIVANI COE , KOPARGAON

2. Boundary layer…..

3. Concept of boundary layer Boundary layer over flat plate Laminar and Turbulent boundary layer Concept of displacement thickness Boundary layer separation Method to control boundary layer separation Contents

4. The concept of boundary layer was first introduced by L. Prandtl in 1904 and since then it has been applied to several fluid flow problems. No Slip condition occurs when real fluid past over solid body/wall. In the immediate vicinity of the boundary surface, the velocity of the fluid increases gradually from zero at boundary surface to the velocity of the mainstream . This region is known as BOUNDARY LAYER. Introdution

5. Flow is divided in to two regions Boundary layer (Inside) – Here velocity gradient (du/ dy ) and shear stress exist. Means Boundary layer (Outside) – Velocity is constant ( Free stream velocity) Means du/ dy = 0 i.e Ʈ = 0.

6. Development of Boundary layer over flat plat Laminar B.L - For smooth upstream boundaries the boundary layer starts out as a laminar boundary layer in which the fluid particles move in smooth layers. Turbulent B.L - As the laminar boundary layer increases in thickness, it becomes unstable and finally transforms into a turbulent boundary layer in which the fluid particles move in haphazard paths. Laminar Sub layer - When the boundary layer has become turbulent, there is still a very than layer next to the boundary layer that has laminar motion . It is called the laminar sub layer .

7. Boundary layer growth over a flat plate As the fluid passes over the plate action of viscous shear stress retats fluid more and more in the lateral direction The thickness of boundary layer goes on increasing Due to shear stress the flow in the boundary layer is rotational and the flow outside boundary layer is irrotational Reynolds's number for calculating distance from leading edge is

8. Boundary layer thickness, δ The boundary layer thickness is defined as the vertical distance from a flat plate to a point where the flow velocity reaches 99 per cent of the velocity of the free stream.

9. Displacement Thickness ( ) Let, At Section 1-1 consider elemental strip. ρ = Density of fluid y = Distance of elemental strip from plate d y = Thickness of elemental srtip u = Velo . of fluid at elemental strip b = Width of plate . The displacement thickness can be defined as follows: “It is the distance, measured perpendicular to the boundary, by which the main/free stream is displayed on account of formation of boundary layer

10. Area of strip = b * dy Mass of fluid flowing thr ’ elemental strip per sec. = ρ * Velocity* Area = ρ u b dy (1) If there has been no plate then fluid would be flowing with free stream velocity say ‘U’ Mass flow per second through the elementary strip if the plate were not there = ρ U b dy (2) As U is more than u hence due to presence of plate and consequently due to formation of boundary layer there will be reduction in mass/sec. thr ’ strip. Reduction of mass flow rate through the elementary strip = ρ b (U – u) dy [The difference (U – u) is called velocity of defect]. Total reduction of mass flow rate due to introduction of plate thr ’ BC (3)

11. (if the fluid is incompressible) Let the plate is displaced by a distance δ* and velocity of flow for the distance δ* is equal to main/free stream velocity (i.e. U). Then, loss of the mass of the fluid/sec. flowing through the distance δ * (4) Equate eq. 3 and 4

12. Momentum Thickness ( ) Let, At Section 1-1 consider elemental strip. ρ = Density of fluid y = Distance of elemental strip from plate d y = Thickness of elemental srtip u = Velo . of fluid at elemental strip b = Width of plate . The Momentum thickness may be defined as the distance measured perpendicular to the boundary of the solid body , by which the boundary should be displaced to compensate for reduction in momentum of the flowing fluid on account ofboundary layer formation.

13. Area of strip = b * dy Mass of fluid flowing thr ’ elemental strip per sec. = ρ * Velocity* Area = ρ u b dy Momentum/sec. of this fluid inside the boundary layer = ρ ubdy × u = (1) If there has been no plate then fluid would be flowing with free stream velocity say ‘U’ momentum of this mass flow per second through the elementary strip if the plate were not there = ρ ubdy × U As U is more than u hence due to presence of plate and consequently due to formation of boundary layer there will be reduction in momentum/sec. thr ’ strip. Reduction of momentum flow rate through the elementary strip (3) = [The difference (U – u) is called velocity of defect]. (2)

14. Total reduction of momentum flow rate due to introduction of plate thr ’ BC ( if the fluid is incompressible ) Let the plate is displaced by a distance ϴ and velocity of flow for the distance ϴ is equal to main/free stream velocity (i.e. U). Then , loss of the momentum of the fluid/sec. flowing through the distance ϴ with velo . U (4) (5)

15. Equate eq. 4 and 5

16. Energy Thickness Let, At Section 1-1 consider elemental strip. ρ = Density of fluid y = Distance of elemental strip from plate d y = Thickness of elemental srtip u = Velo . of fluid at elemental strip b = Width of plate . The Energy thickness may be defined as the distance measured perpendicular to the boundary of the solid body , by which the boundary should be displaced to compensate for reduction in kinetic energy of the flowing fluid on account of boundary layer formation.

17. Area of strip = b * dy Mass of fluid flowing thr ’ elemental strip per sec. = ρ * Velocity* Area = ρ u b dy Kinetic Energy/sec . of this fluid inside the boundary layer = ½* ρ ubdy × u 2 If there has been no plate then fluid would be flowing with free stream velocity say ‘U’ Kinetic energy of this mass flow per second through the elementary strip if the plate were not there = ½* ρ ubdy × U 2 Loss of K.E Through strip

18. Total loss of K.E flow rate due to introduction of plate thr ’ BC Let the plate is displaced by a distance and velocity of flow for the distance is equal to main/free stream velocity (i.e. U). Then , loss of the K.E of the fluid/sec. flowing through the distance with velo . U (1) (2)

19. Equate eq. 1 and 2

20. Boundary Layer Separation

21. In a flowing fluid when a solid body is immersed, a thin layer of fluid called the boundary layer is formed adjacent to the solid body. The forces acting on the fluid in the boundary layer are : Inertia force Viscous force Pressure force When the pressure gradient in the direction of flow is negative i.e. when the pressure decreases in the direction of flow, the flow is accelerated. In this case, the pressure force and inertia force add together and jointly tend to reduce the effect of viscous forces in the boundary layer. This results in a decrease in the thickness of boundary layer in the direction of flow, as a consequence of which there are low losses and high efficiencies in accelerating When the pressure increases in the direction of flow the pressure forces act opposite to the direction of flow and further increase the retarding effect of the viscous forces. Subsequently the thickness of the boundary layer increases rapidly in the direction of flow. If these forces act over a long stretch, the boundary layer gets separated from the surface and moves into the main stream. This phenomenon is called separation. The point of the body at which the boundary layer is on the verge of separation from the surface is called “point of separation’

22. Method of controlling separation of boundary layer 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 . .

23. 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. . 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. 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

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