Terms Associated with Boundary Layer MCQ Quiz - Objective Question with Answer for Terms Associated with Boundary Layer - Download Free PDF

Explanation:

Boundary Layer Thickness in Incompressible Flow

Definition: In fluid dynamics, the boundary layer is the thin region adjacent to a solid surface where the effects of viscosity are significant. For incompressible flow over a flat plate with zero pressure gradient, the boundary layer thickness depends on the Reynolds number and the velocity of the fluid. The Reynolds number (Re) is a dimensionless quantity that represents the ratio of inertial forces to viscous forces in the flow.

For laminar flow over a flat plate with zero pressure gradient, boundary layer thickness is inversely proportional to the square root of the Reynolds number, i.e.,

$\delta \propto \frac{1}{\sqrt{Re}}$

Calculation:

Given:

Initial Reynolds number, $R{e}_1=1000$

Initial boundary layer thickness, ${\delta }_1=1\text{ mm}$

Velocity increased by a factor of 4 ⇒ $U_2=4U_1$

Reynolds number is directly proportional to velocity, so $R{e}_2=4\times R{e}_1=4000$

Now,

$\frac{{\delta }_2}{{\delta }_1}=\sqrt{\frac{R{e}_1}{R{e}_2}}=\sqrt{\frac{1000}{4000}}=\frac{1}{2}$

${\delta }_2=1\times \frac{1}{2}=0.5\text{mm}$

Concept: 

For laminar boundary on a flat plate,

The boundary layer thickness at a distance x from leading-edge is given as

$\frac{\delta }{x}=\frac{5}{\sqrt{R{e}_x}}\Rightarrow \delta =\frac{5x}{\sqrt{\frac{\rho Vx}{\mu }}}$

⇒ $\delta \propto \frac{1}{\sqrt{R{e}_x}}$

$\Rightarrow \frac{{\delta }_1}{{\delta }_2}=\sqrt{\frac{R{e}_2}{R{e}_1}}$

$R{e}_x=\frac{\rho Vx}{\mu }=\frac{Vx}{\nu }$

Re ∝ V

$\therefore \frac{{\delta }_1}{{\delta }_2}=\sqrt{\frac{R{e}_2}{R{e}_1}}=\sqrt{\frac{V_2}{V_1}}$

Calculation:

Given:

δ₁ = 1 mm, δ₂ = ?, V₁ = V, V₂ = 4V

$\therefore \frac{{\delta }_1}{{\delta }_2}=\sqrt{\frac{V_2}{V_1}}$

$\therefore \frac{1}{{\delta }_2}=\sqrt{\frac{4V}{V}}$

$\therefore \frac{1}{{\delta }_2}=2$

δ2 = 0.5 mm

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.

Given as

${\delta }^{\ast }=\underset{0}{\overset{\delta }{\int }}\left(1-\frac{u}{U}\right)dy$

(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

$\theta =\underset{0}{\overset{\delta }{\int }}\frac{u}{U}\left(1-\frac{u}{U}\right)dy$

(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

${\delta }_E=\underset{0}{\overset{\delta }{\int }}\frac{u}{U}\left(1-\frac{u^2}{U^2}\right)dy$

Explanation:

It is known as the Momentum thickness distance at which the whole loss of momentum per second would be equivalent to passing through a stationary plate.

Concept:

For a particular velocity profile (u/U):

1) Displacement thickness (δ*)

${\delta }^{\ast }=\underset{\mathrm{o}}{\overset{\delta }{\int }}\left(1-\frac{\mathrm{u}}{\mathrm{U}}\right)\mathrm{d}\mathrm{y}$

2) Momentum thickness (θ)

$\theta =\underset{\mathrm{o}}{\overset{\delta }{\int }}\frac{\mathrm{u}}{\mathrm{U}}\left(1-\frac{\mathrm{u}}{\mathrm{U}}\right)\mathrm{d}\mathrm{y}$

3) Energy thickness (δ**)

${\delta }^{\ast \ast }=\underset{o}{\overset{\delta }{\int }}\frac{u}{U}\left[1-{\left(\frac{u}{U}\right)}^2\right]dy$

Calculation:

For the given velocity profile:

$\frac{\mathrm{u}}{\mathrm{U}}={\left(\frac{\mathrm{y}}{\delta }\right)}^{1/7}$

Displacement thickness (δ*)

${\delta }^{\ast }=\underset{\mathrm{o}}{\overset{\delta }{\int }}\left[1-\frac{\mathrm{u}}{\mathrm{U}}\right]\mathrm{d}\mathrm{y}$

${\delta }^{\ast }=\underset{\mathrm{o}}{\overset{\delta }{\int }}\left[1-{\left(\frac{y}{\delta }\right)}^{1/7}\right]\mathrm{d}\mathrm{y}$

${\delta }^{\ast }=\underset{\mathrm{o}}{\overset{\delta }{\int }}(y-\frac{7}{8}\times \frac{y^{\frac{8}{7}}}{{\delta }^{\frac{1}{7}}})\mathrm{d}\mathrm{y}$

Putting the value of upper and lower limits, we get

δ^* = δ - 7δ / 8

δ^* = δ / 8

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 = $\underset{0}{\overset{\delta }{\int }}\rho u_{\mathrm{\infty }}dy$

The mass flow rate of real fluid flow = $\underset{0}{\overset{\delta }{\int }}\rho udy$

The loss is compensated by displacement layer thickness,

$\Rightarrow \rho {\delta }^{\ast }{u}_{\mathrm{\infty }}=\underset{0}{\overset{\delta }{\int }}\rho u_{\mathrm{\infty }}dy-\underset{0}{\overset{\delta }{\int }}\rho udy$

$\Rightarrow {\delta }^{\ast }=\underset{0}{\overset{\delta }{\int }}\left(1-\frac{u}{u_{\mathrm{\infty }}}\right)dy$

(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

$\theta =\underset{0}{\overset{\delta }{\int }}\frac{u}{u_{\mathrm{\infty }}}\left(1-\frac{u}{u_{\mathrm{\infty }}}\right)dy$

(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

${\delta }_E=\underset{0}{\overset{\delta }{\int }}\frac{u}{u_{\mathrm{\infty }}}\left(1-\frac{u^2}{u_{\mathrm{\infty }}^2}\right)dy$

The sequence of the above parameter is given by 

δ > δ* > δE > θ

Explanation:

• 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.

• For laminar boundary layers, the boundary layer thickness is proportional to the square root of the distance from the surface. Therefore, the maximum value of the boundary layer thickness occurs at the surface, where the distance is zero.

• The maximum thickness of the boundary layer in a pipe of radius R is R.

• For turbulent boundary layers, the boundary layer thickness grows more quickly, but it still has a maximum value of about R/2. This is because, at this point, the turbulence intensity is such that the momentum diffusing effect of the turbulent fluctuations balances the momentum loss due to viscous effects, resulting in a maximum velocity gradient at the edge of the boundary layer.

The correct answer is 5.5 cm.

• The total kinetic energy possessed by the block goes into the potential energy of the spring and the work done against friction.
• K.E. supplied = Work done against friction + P.E. of spring  
  • $\frac{1}{2}m{v}^2=Fx+\frac{1}{2}k{x}^2$
• Suppose x be the compression of the spring.
• Here:
  • mass=2 kg,υ = 4 m/s
  • Force of kinetic friction, F = 15 N
  • spring constant, K = 10000 N/m
• $\frac{1}{2}\times 2\times 4^2=15x+\frac{10000}{2}{x}^2$
• $5000x^2+15x-16=0$
• $x=0.055m=5.5cm$

Additional Information

• Kinetic energy, the form of energy that an object or a particle has by reason of its motion.
  • If work, which transfers energy, is done on an object by applying a net force, the object speeds up and thereby gains kinetic energy.
•  Kinetic friction is defined as a force that acts between moving surfaces.
  • A body moving on the surface experiences a force in the opposite direction of its movement.
  • The magnitude of the force will depend on the coefficient of kinetic friction between the two materials.

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.

Given as

${\delta }^{\ast }=\underset{0}{\overset{\delta }{\int }}\left(1-\frac{u}{U}\right)dy$

(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

$\theta =\underset{0}{\overset{\delta }{\int }}\frac{u}{U}\left(1-\frac{u}{U}\right)dy$

(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

${\delta }_E=\underset{0}{\overset{\delta }{\int }}\frac{u}{U}\left(1-\frac{u^2}{U^2}\right)dy$

Explanation:

It is known as the Momentum thickness distance at which the whole loss of momentum per second would be equivalent to passing through a stationary plate.

Explanation:

Boundary-Layer

When a fluid of ambient velocity flows over a flat stationary plate, the bottom layer of fluid directly contacts with the solid surface and its velocity reaches zero. Due to the cohesive forces between two layers, the bottom layer offers resistance to the adjacent layer and due to this reason, the velocity gradient develops in a fluid. A thin region over a surface velocity gradient is significant, known as the boundary layer.

The Laminar boundary layer thickness for the flat plate given by Blasius equation is :

$\delta =\frac{5x}{\sqrt{R{e}_x}}$  

$\Rightarrow \delta \propto (R{e}_x{)}^{-1/2}$

Note:

$Re=\frac{\rho Ux}{\mu }\Rightarrow Re\propto x$

$\delta =\frac{5x}{\sqrt{Re}}$

$\delta =\frac{5x}{\sqrt{\frac{\rho Vx}{\mu }}}$

$\delta \propto \frac{x}{\sqrt{x}}$

∴ $\delta \propto x^{1/2}$

Concept:

Momentum thickness (θ): It is the distance perpendicular to the boundary of the body over which flow occurs by which the boundary should be displaced to compensate for the reduction in the momentum of the flowing fluid due to boundary layer formation. It is given by

$\theta ={\int }_0^{\delta }\frac{u}{U_{\mathrm{\infty }}}\left(1-\frac{u}{U_{\mathrm{\infty }}}\right)dy$

Where u is the velocity at height y above the surface and $U_{\mathrm{\infty }}$ is free stream velocity of flow

Calculation:

Given, $\frac{u}{U_{\mathrm{\infty }}}={\left(\frac{y}{\delta }\right)}^{1/2}$

$\theta ={\int }_0^{\delta }(\frac{y}{\delta }{)}^{1/2}\left(1-(\frac{y}{\delta }{)}^{1/2}\right)={\int }_0^{\delta }((\frac{y}{\delta }{)}^{1/2}-\frac{y}{\delta })dy$

$\theta =\frac{2\delta }{3}-\frac{\delta }{2}=\frac{\delta }{6}$

Therefore,

$\frac{\theta }{\delta }=\frac{1}{6}=0.166$

Displacement Thickness (δ):

• It is the distance perpendicular to the boundary of the body over which flow occurs by which the boundary should be displaced to compensate for the reduction in the flow rate of the flowing fluid due to boundary layer formation. It is given by
• $\delta ={\int }_o^{\delta }\left(1-\frac{u}{U_{\mathrm{\infty }}}\right)dy$
• Energy Thickness (δ**):
• It is the distance perpendicular to the boundary of the body over which flow occurs by which the boundary should be displaced to compensate for the reduction in the kinetic energy of the flowing fluid due to boundary layer formation. It is given by
• ${\delta }^{\ast \ast }={\int }_0^{\delta }\frac{u}{U_{\mathrm{\infty }}}\left(1-\frac{u^2}{U_{\mathrm{\infty }}^2}\right)dy$

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

${\left(P_r\right)}^{\frac{1}{3}}=\frac{{\delta }_{hy}}{{\delta }_{th}}\Rightarrow \frac{{\delta }_{hy}}{{\delta }_{th}}=1$

∴ δ_hy = δ_th

$Re=\frac{\rho VD}{\mu }=\frac{VD}{v}=\frac{10\times 0.5}{30\times {10}^{-6}}=1.66\times {10}^5$ 

∵ Re ≤ 5 × 10⁵

∴ Flow is laminar flow

∴ Blasius equation for laminar flow

${\delta }_{hy}=\frac{5x}{\sqrt{R_{ex}}}=\frac{5\times 0.5}{\sqrt{1.66\times {10}^5}}\times 1000mm$ 

δ_hy = 6.12 mm

δ_hy = δ_th = 6.12 mm

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
• $\frac{\mathbf{k}}{\delta }<0.25$
• Boundary transition condition, when
• $0.25<\frac{\mathbf{k}}{\delta }<6$
• Hydrodynamically rough when
• $\frac{\mathbf{k}}{\delta }>6$

Explanation:

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).

Concept:

Displacement thickness is given by

${\delta }^{\ast }=\underset{0}{\overset{\delta }{\int }}\left(1-\frac{\mathrm{u}}{{\mathrm{U}}_{\mathrm{\infty }}}\right)\mathrm{d}\mathrm{y}$

Where,

u – velocity of the fluid

U∞ - Free stream velocity

Calculations:

Given:

$\frac{u}{u_{\mathrm{\infty }}}=\sin \left(\frac{xy}{2\delta }\right)$

$DisplacementThickness{\delta }^{\ast }=\underset{0}{\overset{\delta }{\int }}\left(1-\frac{u}{u_{\mathrm{\infty }}}\right)dy$

${\delta }^{\ast }=\underset{0}{\overset{\delta }{\int }}\left(1-\sin \frac{\pi y}{2\delta }\right)dy$

${\delta }^{\ast }={\left[y+\frac{\cos \frac{\pi y}{2\delta }}{\frac{\pi }{2\delta }}\right]}_0^{\delta }$

${\delta }^{\ast }=\delta +\frac{2\delta }{\pi }\left(0\right)-0-\frac{2\delta }{\pi }$

${\delta }^{\ast }=\delta -\frac{2\delta }{\pi }\Rightarrow \frac{{\delta }^{\ast }}{\delta }=1-\frac{2}{\pi }$

Thickness of Laminar Boundary layer:

$\delta =\frac{5x}{\sqrt{R{e}_x}}$

Where,

x = distance from the leading edge

Re_x = local Reynold's Number = $R{e}_x=\frac{\rho Vx}{\mu }=\frac{Vx}{\nu }$

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.

$\delta =\frac{5x}{\sqrt{\frac{\rho Vx}{\mu }}}\propto \frac{x}{\sqrt{x}}$ thus, δ ∝ x^1/2