Displacement Thickness Assignment Help | Displacement Thickness Homework Help

Displacement Thickness

It is the distance perpendicular to the boundary, by which the boundary should be displaced outward to compensate for reduction in the discharge in the boundary layer formation.

δ  =  ∫  ( 1 - u/U) dy


Consider the fluid flow having free-stream velocity ‘U’ over a thin smooth plate as shown in.

Consider elementary strip at a distance ‘y’ from the plate and ‘dy’ is thickness.

Let u be the velocity at elemental strip and b is width of plate.
Mass per second through strip

= ρ . V. A
= ρ . u. (dy x b)
= ρu . b dy

Mass per second through strip if no plate is placed i.e. u = U.

= ρ U b. dy

Total reduction of mass / second of strip,

= ρ U b. dy - ρ  ub dy

= ρ b dy (U - U)
Total reduction of mass / second for whole boundary laye = ∫  ρ b dy (U -η)

Let the plate displaced by a distance ( ) and velocity of flow for the distance ( ) is free stream velocity ‘U’.

Loss of mass / second flowing through distance δ*

= ρ x Velocity x Area
= ρ x U x b δ

Equating Equations

ρ U b δ*  = ∫  ρ b dy (U - u)
δ* = ∫ U - u / U dy

δ* = ∫ ( 1 - u / U) dy

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