1. Boundary Layer
Laminar Flow Re ‹ 2000
Turbulent Flow Re › 4000

2. Boundary Layer Section I Section II Boundary Layer
Laminar and Turbulent boundary layer growth over flat plate
Von-Karman momentum integral equation-Separation of boundary Layer
Regimes of external flow-wakes and drag-Drag on immersed body-sphere-cylinder-
bluff body-Lift and Magnus effect

3. Section I

4. Boundary Layer:
When a real fluid flows past a solid body or a solid wall, the fluid particles adhere to the boundary and condition of no slip occurs.

5. Laminar Boundary Layer:
If the value of k is less then boundary (δ’ ) is known as smooth Boundary

6. Turbulent Boundary Layer:
If the length of plate is more than the distance x, the thickness of boundary layer will go on increasing in the downstream direction.
Then the laminar boundary layer will becomes unstable and motion of fluid within it, is disturbed and irregular which leads to a transition from laminar to turbulent boundary layer.

7. Laminar Sub-layer:
This is the region in the turbulent boundary layer zone, adjacent to the solid surface of the plate.
In this zone, the velocity variation is influenced only by viscous effects.

8. Boundary layer thickness (δ):
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 equal to 0.99 times the free stream velocity (U) of the fluid.
It denoted by the symbol δ

9. Displacement Thickness (δ*):
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 flow rate on account of boundary layer formation.
It is denoted by δ*
“ The distance perpendicular to the boundary, by which the free-stream is displaced due to the formation of boundary layer’

10. Consider the flow of fluid having free-stream velocity equal to U
Area of elemtal strip, dA= b x dy
2) Mass of fluid per second=
As U is more than u,
Reduction in mass/sec =
Total Reduction in mass of fluid/s through BC

11. Loss of the mass of the fluid/sec flowing through the distance δ*

12. Momentum Thickness (Ө):
Momentum thickness 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 momentum of the flowing fluid on account of boundary layer formation.
It is denoted by Ө
Momentum of this fluid = Mass x Velocity
2) Momentum of this fluid in absence of boundary layer =
Loss of momentum through elemental strip

13. Total loss of momentum/Sec through
Loss of momentum/sec of fluid flowing through distance Ө with U

14. Energy thickness (δ**):
It is defined as the distance measured perpendicular to the boundary of the solid body, by which the boundary should be displaced to compenste for the reduction in kinetic energy of the flowing fluid on account of boundary layer formation.
It is denoted by δ**
Kinetic energy of fluid in the absence of boundary layer
Loss of K.E. through elemental strip

15. Total loss of K.E. of fluid passing through BC
Loss of K.E. through δ ** of fluid flowing with velocity U

16. Equating the two losses of K.E., we get

17. 1) Find the displacement thickness, the momentum thickness and energy thickness for the velocity distribution in the boundary layer given by u/U = y/δ where u is the velocity at a distance y from the plate and u = U at y = δ where δ = boundary layer thickness. Also calculate the value of δ*/θ.
Answer: 1) δ/2 2) δ/6 3) δ/4 4) 3.0

18. 1) Find the displacement thickness, the momentum thickness and energy thickness for the velocity distribution in the boundary layer given by
Answer: 1) δ/3 2) )

19. Drag Force on Flat Plate due to Boundary Layer

20. Then drag force or shear force on small distance Δ x
Then mass rate of flow entering through the side AD

21. Mass rate of flow leaving the side BC
From Continuity equation for a steady incompressible fluid
Mass rate of flow entering DC = BC- AD

22. Momentum flux entering through AD
Momentum flux entering through side DC = DC x Velocity

23. As U is constant and so it can be taken (Differential and Int.)
Rate of change of moment of control Volume = Mom. Flux BC- Mon. flux AD- Mom flux DC

24. Total external force in the direction of rate of change of momentum
According to momentum Principal,

25. Von Karman Momentum Integral Equation

26. Local co-efficient of Drag
Average co-efficient of Drag

27. For the velocity profile for laminar boundary layer flows given as
Find an expression for boundary layer thickness δ, shear stress τo and co-efficient of drag CD in terms of Reynold number.

28. 2) For the velocity profile given in previous problem, find the thickness of boundary layer at the end of the plate and the drag force on one side of a plate 1 m long and 0.8 m wide when placed in water flowing with a velocity of 150 mm per second. Calculate the velocity of co-efficient of drag also. Take μ for water =0.01 poise.

29. Turbulent boundary layer on a flat plate:

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

31. Separation of Boundary Layer:

32. Methods of Preventing the Separation of Boundary Layer:

33. Ex A smooth pipe line of 100 mm diameter carries 2
Ex A smooth pipe line of 100 mm diameter carries 2.27 m3 per minute of water at 20oC with kinematic viscosity of stokes. Calculate the friction factor, maximum velocity as well as shear stress at the boundary.

34. Prepared by, Dr Dhruvesh Patel www.drdhruveshpatel.com
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