The document discusses fluid mechanics focusing on boundary layer theory, detailing the concepts of laminar and turbulent boundary layers, their characteristics, and the effects of separation. It explains how fluid velocity changes near a solid boundary and introduces important parameters such as boundary layer thickness, displacement thickness, and momentum thickness. Additionally, it addresses drag and lift forces experienced by bodies in fluid flow and introduces the Magnus effect, which influences the trajectories of spinning objects in fluids.

1. Geethanjali College of Engineering and Technology
Cheeryala(V) Keesara(M), Medchal Dist.
Telangana, INDIA Pin Code-501301.
FLUID MECHANICS

2. BOUNDARY LAYER THEORY
CONTENTS:-
 Boundary layer concept
 Characteristics of boundary layer along a thin flat
plate,
 Von Karman momentum integral equation,
 Laminar and Turbulent Boundary layers
 Separation of Boundary Layer,
 Control of Boundary Layer,
 flow around submerged objects-
 Drag and Lift- Expression
 Magnus effect.

3. BOUNDARY LAYER
 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.
 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 v
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.

4.  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.

5. 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 (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.

6. 1. Laminar boundary layer
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.
BASIC DEFINITIONS
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

7. 2.Turbulent boundary layer
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.

9.  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. For laminar and turbulent zone it is denoted as
BOUNDARY LAYER THICKNESS

10. 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

11. 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.
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**.

12. Characteristics of boundary layer
along a thin flat plate

13. boundary layer Seperation
Effect of Pressure Gradient on Boundary Layer Separation
The flow separation depends upon factors such as
(i) The curvature of the surface
(ii) The Reynolds number of flow
(iii) The roughness of the surface

14. The following are some of the methods generally adopted to retard or arrest the flow
separation:
1. Streamlining the body shape
2. Tripping the boundary layer from laminar to turbulent by provision of surface
roughness
3. Sucking the retarded flow
4. Injecting high velocity fluid in the boundary layer
5. Providing slots near the leading edge
6. Guidance of flow in a confined passage
7. Providing a rotating cylinder near the leading edge
8. Energizing the flow by introducing optimum amount of swirl in the incoming
flow
Methods of preventing the Separation of Boundary Layer

15. When a fluid is flowing over a stationary body, a force is exerted by the fluid on the
body. Similarly, when a body is moving in a stationary fluid, a force is exerted by the
fluid on the body. Also, when both the body and fluid are moving at different velocities, a
force is exerted by the fluid on the body. Some of the examples of the fluids flowing over
stationary bodies or bodies moving in a stationary fluid are:
(a) Flow of air over buildings,
(b) Flow of water over bridges
(c) Submarines, ships, airplanes and automobiles moving through water and air

16. Force Exerted by a Flowing fluid on Stationary Bodies
Consider a body held stationary in a real fluid which is flowing at a uniform velocity U as
shown in the figure below
Force on a stationary body
The fluid will exert a force on the stationary body. The total force (FR) exerted by the
fluid on the body is perpendicular to the surface of the body. Thus the total force is
inclined to the direction of motion.
The total force can be resolved into two components, or in the direction of motion and the
other perpendicular to the direction of motion.

17. DRAG
When a body is immersed in a fluid and is in relative motion with respect to it, the
drag is defined as that component of the resultant or total force (FR) acting on the
body which is in the direction of the relative motion. This is denoted by FD
LIFT
The component of the total or resultant force (FR) acting in the direction normal or
perpendicular to the relative motion is called lift i.e. the force component
perpendicular to drag. This component is denoted by FL. Lift force occurs only
when the axis of the body is inclined to the direction of fluid flow. If the axis of
the body is parallel to the direction of fluid flow, lift force is zero. In that case only
drag force acts. If the fluid is assumed ideal and

18. EXPRESSION FOR DRAG AND LIFT

19. 1. A flat plate 1.5m x 1.5m moves as 50km/hr in stationary air of density 1.15kg/m3. If
the
coefficients of drag and lift are 0.15 and 0.75 respectively. Determine:
(i) The lift force
(ii) The drag force
(iii) The resultant force and
(iv) The power required to keep the plate in motion
2. Find the difference in drag force exerted on a flat plate of size 2m x 2m when the
plate is moving at a speed of 4m/s normal to its plane in (i) water (ii) air of density
1.24kg/m3. Coefficient of drag is given as 1.15.

20. Magnus effect.
 The force exerted on a fast spinning cylinder or sphere travelling through air or
another fluid in a direction perpendicular to the axis of spin is known as the
Magnus force.
 Magnus effect, generation of a sidewise force on a spinning cylindrical or
spherical solid immersed in a fluid (liquid or gas) when there is relative motion
between the spinning body and the fluid.
 Named after the German physicist and chemist H.G. Magnus, who first (1853)
experimentally investigated the effect, it is responsible for the “curve” of a
served tennis ball or a driven golf ball and affects the trajectory of a spinning
artillery shell.
 The Magnus effect explains how a football player may bend the ball into a goal
around a five-person wall.
 Magnus effect is an application
of Newton’s third law of motion.
As a result of the object pushing
the air in one direction, the air
pushes the object in the
opposite direction.

21. Applications of Boundary Layer

22. U
Stationary body
FL
FD
FR