Test: Boundary Layer Thickness - Civil Engineering (CE) MCQ

Concept:
In flow over a flat plate, various types of thicknesses are defined for the boundary layer,
(i) Boundary layer thickness (δ): It is defined as the distance from the body surface in which the velocity reaches 99 % of the velocity of the mainstream (U∞)
(ii) Displacement thickness (δ* or δ⁺): It is defined as the distance by which the body surface should be shifted in order to compensate for the reduction in mass flow rate on account of boundary layer formation.
The mass flow rate of ideal fluid flow 
The mass flow rate of real fluid flow 
The loss is compensated by displacement layer thickness,


(iii) Momentum thickness (θ): It is defined as the distance from the actual boundary surface such that the momentum flux (momentum transferred per sec) corresponding to mainstream velocity (u_∞) through this distance θ is equal to the deficiency or loss in momentum due to the boundary layer formation.
Given as

(iv) Energy thickness (δE): It is defined as the distance from the actual boundary surface such that the energy flux corresponding to the mainstream velocity u∞ through the distance δE is equal to the deficiency or loss of energy due to the boundary layer formation.  
Given as

The sequence of the above parameter is given by 
δ > δ* > δ_E > θ

Concept:
Hydro-dynamically smooth:

• If the average height of irregularities (k) is much lesser than the thickness of the laminar sub-layer (δ), then the boundary is called hydro-dynamically smooth.

Hydro-dynamically rough:

• If the average height of irregularities (k) is much greater than the thickness of the laminar sub-layer (δ), then the boundary is called hydro-dynamically rough.
• According to NIKURDE's Experiment, the boundary is classified as:
• Hydrodynamically smooth when
•  k/δ < 0.25
• Boundary transition condition, when
•  0.25 < k/δ < 6
• Hydrodynamically rough when
•  k/δ > 6

Concept:
Displacement thickness is given by

Where,
u – velocity of the fluid
U_∞ - Free stream velocity
Calculation:
Given:

Concept:
Hydro-dynamically smooth:

• If the average height of irregularities (k) is much lesser than the thickness of the laminar sub-layer (δ), then the boundary is called hydro-dynamically smooth.

Hydro-dynamically rough:

• If the average height of irregularities (k) is much greater than the thickness of the laminar sub-layer (δ), then the boundary is called hydro-dynamically rough.
• According to NIKURDE's Experiment, the boundary is classified as:
• Hydrodynamically smooth when
•  k/δ < 0.25
• Boundary transition condition, when
•  0.25 < k/δ < 6
• Hydrodynamically rough when
•  k/δ > 6

Concept:
Displacement thickness is given by

Where,
u – velocity of the fluid
U_∞ - Free stream velocity
Calculations:
Given:

Concept:
Nominal boundary thickness as δ 
Displacement thickness is given as

Calculation:

∴ The ratio of displacement thickness to boundary layer thickness will be ​=  
Hence, option 2 is correct.

Boundary layer:
When a real fluid flows past a solid body or a solid wall, the fluid particles adhere to the boundary and the condition of no-slip occurs i.e velocity of fluid will be the same as that of the boundary.
Farther away from the boundary, the velocity will be higher and as a result of this variation, the velocity gradient will exist.

Boundary-Layer Thickness:
It is defined as the distance from the boundary of the solid body measured in the perpendicular direction to the point where the velocity of the fluid is approximately equal to 99% or 0.99 times the free stream velocity (U). It s denoted by the symbol (δ).

• The thickness of the boundary layer represented by δ is arbitrarily defined as that distance from the boundary surface in which the velocity reaches 99% of the velocity of the mainstream.
• ln, a pipe the free stream velocity is at the center of the pipe.
• Therefore the maximum thickness of the boundary layer is R.
  

Concept:
Since in question not any velocity profile is given so we will use Blasius equation.

Calculation: 
Pr = 1, T = 500 K, L = 1.5 m, T_∞ = 300 K

∵ Re ≤ 5 × 10⁵
∴ Flow is laminar flow
∴ Blasius equation for laminar flow

δ_hy = 6.12 mm
δ_hy = δ_th = 6.12 mm

Thickness of Laminar Boundary layer:

Where,
x = distance from the leading edge
Re_x = local Reynold's Number 

where, ρ = density of fluid in kg/m³, V = average velocity in m/s
μ = dynamic viscosity in N-s/m² and ν = kinematic viscosity in m²/s.