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Wednesday, 21 November 2018

ENERGY THICKNESS IN BOUNDARY LAYER

We were discussing the basics of Boundary layer theorylaminar 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
Kinetic energy of this fluid without boundary layer = (1/2) x (ρubdy) x U
Loss of kinetic energy through elemental strip = (1/2) x ρub x [U2- u2] 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

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.drag and lift force in fluid mechanics, 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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