Hydraulic System
November 21, 2018

## ENERGY THICKNESS IN BOUNDARY LAYER

We were discussing the basics of Boundary
layer theory, laminar
boundary layer and turbulent
boundary layer, in the subject of fluid mechanics, in our recent
posts.

###

After understanding the fundamentals of boundary layer thickness, displacement thickness and momentum thickness, we will go
ahead now to find out the basics of energy thickness with the help of this
post.

###
**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**.

Let us consider the fluid flow over the plate as
displayed here in following figure. Let us assume one section 1-1 at a distance
x from the leading edge.

Mass of the fluid flowing per second through the
elemental strip of thickness dy at a distance y from the plate will be given by
following equation.

Mass of the fluid flowing per second = ρubdy

Where,

u = Velocity of the fluid at the elemental strip

b = width of the plate

Area of the elemental strip = b dy

Kinetic energy of this fluid with boundary layer = (1/2)
x (ρubdy) x u

^{2 }
Kinetic energy of this fluid without boundary layer
= (1/2) x (ρubdy) x U

^{2 }
Loss of kinetic energy through elemental strip = (1/2)
x ρub x [U

^{2}- u^{2}] x dy
Now we will integrate the above equation from 0 to δ
to secure the equation for loss of kinetic energy and we will have following
equation as mentioned here.

As we have discussed that energy thickness is
basically 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.

So let us assume that we are displacing boundary by δ**
to compensate for the reduction in kinetic energy of the flowing fluid on
account of boundary layer formation.

Loss of kinetic energy due to the displacement of
boundary by δ** = (1/2) x (ρUb
δ**) x U

^{2 }
Now we will equate the both equation of loss of
kinetic energy and we will have expression for the energy thickness in boundary
layer.

Further we will go ahead to start a new topic i.e.,
in the subject of fluid mechanics, with the help of our next post.

Do you have any suggestions? Please write in comment
box.

**Reference:**

Fluid mechanics, By R. K. Bansal

Image courtesy: Google

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