BOUNDARY LAYER
BOUNDARY LAYER
Fig. 1: Laminar and Turbulent Flow |
Introduction:
When a real fluid is flowing through a solid object, the particles
of fluid tend to adhere to the boundary. This gives rise to no slip condition.
This means that the velocity of fluid near the boundary would be same as
boundary of object. If the object is stationary then the velocity of fluid flowing
near the boundary has zero velocity. As we go further from the boundary
condition the velocity of fluid increases which results in variation of
velocity which give rise to velocity gradient. Let’s assume the
velocity of free stream is U in direction of normal to boundary. This variation
of velocity occurs in narrow region near to solid boundary. This narrow region
is called the boundary layer. The theory dealing with the study of the boundary
layer is called boundary layer theory.
According to the theory, there are two
regions that can be distinguished regarding flow of fluid near the solid
boundary.
1.
A very thin layer called the boundary layer,
which is in the immediate neighborhood of the solid boundary where the
variation of velocity from zero to U takes place in a direction normal to solid
boundary. In this region, velocity gradient exists and fluid exerts shear stress
on the object in direction of motion of the fluid.
The value of shear stress can be expressed as
2. The other part is the remaining fluid, which
is outside the boundary layer. There the velocity is constant and equal to U.
As there is no variation in velocity, velocity gradient does not exist in this
region. As a result, the shear stress is zero.
Laminar Boundary Flow:
Consider the flow of
fluid, having free stream velocity as U, over a flat plate which is placed parallel
in direction of the flow of fluid. Let us consider the plate is stationary.
Fig. 2: Boundary Layer Concept |
Fig. 3 |
As plate is stationary,
velocity of fluid near the surface of object is zero. But as we go further from
surface, there is certain velocity of fluid. The fluid with uniform velocity U
is retarded in the vicinity of the solid surface of the plate and boundary
layer region begins at the sharp leading edge. Near the leading edge of the
surface of plate where the thickness is small, the flow of boundary layer is
laminar though the main flow is turbulent. This layer of the fluid is said to
be laminar boundary layer (AE). The length of plate from the leading edge, up
to which laminar boundary layer exists, is called laminar zone. AB shown
is laminar zone. The distance of B is obtained from Reynold number which is
equal to 5*105 for plate. Because up to Reynolds number flow is
laminar.
x: Distance from leading edge
U: Free stream velocity
v: Kinematic viscosity
If we know values of U and viscosity of fluid, we can find the range of laminar flow.
Turbulent Boundary
Layer:
As we go further from
laminar zone, the thickness of boundary layer will go on increasing in
downstream direction. Here the motion of fluid is disturbed and unstable which
leads transition from laminar to turbulent boundary layer. This is called
transition zone (BC). The further downstream the layer grows thicker which is
called turbulent boundary layer (FG).
Laminar Sub-Layer:
This region is in turbulent
boundary layer zone, adjacent to solid surface of the plate. In this region,
the velocity variation is caused due to viscous force. Velocity variation is linear
and velocity gradient is constant.
Thus, the shear stress in the sub layer is
Boundary Layer
Thickness (δ):
Defined as distance from the boundary of object in y- direction to the point, where the velocity of fluid is approximately 0.99 times U.
Displacement Thickness
(δ*):
Defines as distance measured perpendicular to the boundary of object, by which the boundary should be displaced to compensate for the reduction in flow rate on account of boundary layer formation
Momentum Thickness (θ):
Defined as distance
measured perpendicular to the boundary, by which the boundary should be
displaced to compensate for the reduction in momentum of the flowing fluid on
account of boundary layer formation.
Energy Thickness (δ**):
Defined as the distance
measured perpendicular to the boundary of the object, 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.
The drag force on the plate can be determined if the velocity profile near the plate is known.
Fig. 4: Drag Force on Plate |
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