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

Explanation:

Hydrodynamic Boundary Layer:

• The hydrodynamic boundary layer is the thin region adjacent to a solid surface where the effects of viscous forces are significant, and the fluid velocity changes from zero (due to the no-slip condition at the solid surface) to approximately the free stream velocity of the fluid. The boundary layer thickness is a key parameter in fluid dynamics and is often used to understand flow behavior near surfaces.

Boundary Layer Thickness:

• The boundary layer thickness is defined as the distance from the surface where the velocity of the fluid reaches a specific percentage (typically 99%) of the free stream velocity. This definition helps to quantify the region affected by viscous forces and is crucial for analyzing drag, heat transfer, and other flow-related phenomena.
• The hydrodynamic boundary layer thickness is the distance from the solid surface where the velocity of the fluid reaches 99% of the local external velocity (also referred to as the free stream velocity). The local external velocity is the velocity of the fluid outside the boundary layer, where viscous effects are negligible, and the flow can be approximated as inviscid.

Mathematical Representation:

Let u represent the velocity of the fluid at a distance y from the surface, and U represent the free stream velocity (local external velocity). The boundary layer thickness, denoted by δ, is defined such that:

u = 0.99 × U at y = δ

This equation indicates that the velocity at the edge of the boundary layer is 99% of the free stream velocity. Beyond this point, the effects of viscosity are negligible, and the velocity remains nearly constant at the free stream value.

Applications:

• Analyzing drag forces on vehicles, aircraft, and ships to improve aerodynamic and hydrodynamic performance.
• Designing heat exchangers and optimizing heat transfer in thermal systems.
• Predicting flow separation and its impact on lift and drag in aerodynamic surfaces.
• Studying pollutant dispersion in environmental engineering and modeling fluid flow in natural systems.

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, \( Re_1 = 1000 \)

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 \( Re_2 = 4 \times Re_1 = 4000 \)

Now,

\( \frac{\delta_2}{\delta_1} = \sqrt{\frac{Re_1}{Re_2}} = \sqrt{\frac{1000}{4000}} = \frac{1}{2} \)

\( \delta_2 = 1 \times \frac{1}{2} = 0.5~\text{mm} \)

Explanation:

The boundary layer over a flat plate develops from the leading edge. As fluid flows, three main regions form:

• Laminar Boundary Layer: Region close to the leading edge, where flow is orderly and smooth.
• Transition Region: Flow becomes unstable and gradually transitions to turbulence.
• Turbulent Boundary Layer: Downstream region where flow becomes fully turbulent and chaotic.

In the second figure, "Region A" is the zone immediately after the leading edge and before the transition point — hence, it represents the Laminar Boundary Layer.

Explanation:

Thickness of Laminar Boundary layer:

\(δ = \frac{{5x}}{{\sqrt {R{e_x}} }}\)

Where,

x = distance from the leading edge

Rex = local Reynold's Number = \(R{e_x} = \frac{{ρ Vx}}{μ } = \frac{{Vx}}{ν }\)

where, ρ = density of fluid in kgIm3, V = average velocity in mIs

μ = dynamic viscosity in NsIm2 and ν = kinematic viscosity in m2Is.

\(δ = \frac{{5x}}{{\sqrt {\frac{{\rho Vx}}{\mu }} }} ∝ \frac{x}{{\sqrt x }}\) thus, δ ∝ x1I2

∴ \(\delta_t \propto \sqrt{x}\)

Explanation:

The image depicts a Rankine half-body subjected to a two-dimensional uniform flow. The body shape is formed by a combination of a uniform flow and a source located at point 
O. Streamlines are shown as dotted lines, indicating the flow pattern around the half-body. Point A lies on the surface of the half-body.

Flow Characteristics at Point A:
Velocity Components:

• At point A, which is on the surface of the body, the velocity vector has components due to both the uniform flow and the source.
• The velocity is tangential to the surface of the body at this point, meaning that there is no normal component of velocity that would penetrate the body.

Stagnation Point:

• Point A could also potentially be a stagnation point (where the velocity is zero) if it's positioned correctly in the flow field.
• This is typical in cases where the source strength and uniform flow balance in such a way that flow velocity at point A becomes zero.

Explanation:

Streamlined body

It is defined as that body whose surface does coincide with the streamline when the body is placed in a flow.

In this case, separation of flow will take place only at the trailing edge or farthest downstream part of the body.

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 to 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 thickness of the boundary layer is given by

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

\(\delta =\frac{5x}{\sqrt{Re}}\)  

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

\(\delta =\frac{5x}{\sqrt{Re}} \Rightarrow \delta \propto \sqrt x\)

Where, x = distance where the boundary layer is to be found, Re = Reynolds no, ρ = density of the fluid, V = velocity of the fluid

µ = dynamic viscosity fluid

for laminar flow, Reynolds number should be less than 5 × 10⁵.
The Turbulent boundary layer for flat plate is given as

For turbulent flow  Re > 5 x 10⁵

\(\begin{array}{l} \delta = \frac{{0.379x}}{{{{\left( {Re} \right)}^{\frac{1}{5}}}}}\\ \delta \propto {x^{1 - \frac{1}{5}}}\\ \delta \propto {x^{\frac{4}{5}}} \end{array}\)

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 = \(\mathop \smallint \limits_0^δ \rho {u_\infty }dy\)

The mass flow rate of real fluid flow = \(\mathop \smallint \limits_0^δ \rho udy\)

The loss is compensated by displacement layer thickness,

\( \Rightarrow \rho {δ ^*}{u_\infty } = \mathop \smallint \limits_0^δ \rho {u_\infty }dy - \;\mathop \smallint \limits_0^δ \rho udy\)

\(\Rightarrow {δ ^*} = \mathop \smallint \limits_0^δ \left( {1 - \frac{u}{{{u_\infty }}}} \right)dy\)

(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

\(\theta = \mathop \smallint \limits_0^δ \frac{u}{{{u_\infty }}}\left( {1 - \frac{u}{{{u_\infty }}}} \right)dy\)

(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

\({δ _E} = \mathop \smallint \limits_0^δ \frac{u}{{{u_\infty }}}\left( {1 - \frac{{{u^2}}}{{u_\infty ^2}}} \right)dy\)

Shape factor (H)= \(\frac{\delta^*}{\theta}\)

Calculation:

Given:

Velocity profile: \(\frac{u}{U_{\infty}}~=~\frac{y}{δ}\)

Displacement thickness (δ*):

\(\Rightarrow {δ ^*} = \mathop \smallint \limits_0^δ \left( {1 - \frac{u}{{{u_\infty }}}} \right)dy\)

\(\eqalign{ & {\delta ^ \star } = \mathop \smallint \limits_0^\delta \left( {1 - {u \over {{u_\infty }}}} \right)dy \cr & = \mathop \smallint \limits_0^\delta \left( {1 - {y \over \delta }} \right)dy \cr & = \left[ {y - {{{y^2}} \over {2\delta }}} \right]_0^\delta = \delta - {\delta \over 2} = {\delta \over 2} \cr}\)

Concept:

Boundary layer theory:

• 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 to 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 thickness of the boundary layer is given by

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

\(\delta =\frac{5x}{\sqrt{Re}}\)  

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

\(\delta =\frac{5x}{\sqrt{Re}} \Rightarrow \delta \propto \sqrt x\)

Where, x = distance where the boundary layer is to be found, Re = Reynolds no, ρ = density of the fluid, V = velocity of the fluid

µ = dynamic viscosity fluid

for laminar flow, Reynolds number should be less than  2 × 105.
The Turbulent boundary layer for flat plate given by

For turbulent flow  Re > 107

\(\begin{array}{l} \delta = \frac{{0.379x}}{{{{\left( {Re} \right)}^{\frac{1}{5}}}}}\\ \delta \propto {x^{1 - \frac{1}{5}}}\\ \delta \propto {x^{\frac{4}{5}}} \end{array}\)

Concept:

Drag Force (F_d) is applied to the body when it moves through and liquid media.

Drag is defined as forceful pull experienced by the flat plate while the fluid flows over it.

Pressure drag comes from the eddying motions that are set up in the fluid by the passage of the body; This drag is associated with the formation of the wake in the flow.

Frictional drag comes from friction between the fluid and the surfaces over which it is flowing.

\({{\rm{F}}_{\rm{d}}} = \frac{{{{\rm{C}}_{\rm{d}}} \times {\rm{\rho }} \times {\rm{A}} \times {{\rm{V}}^2}}}{2}\)

\({\rm{Drag\;Coefficient}},{\rm{\;}}{{\rm{C}}_{\rm{d}}} = \frac{{24}}{{{{\rm{R}}_{\rm{e}}}}}\) when R_e < 0.2

Drag Coefficient, C_d = \({24\over R_e}({1+{3\over 16R_e}})\) when 0.2 < R_e < 5

Where C_d is the drag coefficient and R_e is Reynold’s Number.

Calculation:

\({{\rm{R}}_{\rm{e}}} = \frac{{{\rm{V}} \times {\rm{d}}}}{{\rm{\nu }}} = \frac{{0.02 \times 0.02}}{{10 \times {{10}^{ - 4}}}} = 0.4\)

\({{\rm{C}}_{\rm{d}}} = \frac{{24}}{{0.4}} = 60\)

Note:

the The given answer is as per official  exam of SSC JE.

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  
  • \({1\over2} mv^2= Fx+{1\over2} kx^2\)
• 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
• \({1\over 2}×2×4^2=15x+{10000\over 2}x^2\)
• \( 5000x^2+15x−16=0\)
• \(x=0.055m=5.5cm\)

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.

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 0.99 times the free stream velocity (U). It s denoted by the symbol (δ).

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 = \(\mathop \smallint \limits_0^\delta \rho {u_\infty }dy\)

The mass flow rate of real fluid flow = \(\mathop \smallint \limits_0^\delta \rho udy\)

The loss is compensated by displacement layer thickness,

\( \Rightarrow \rho {\delta ^*}{u_\infty } = \mathop \smallint \limits_0^\delta \rho {u_\infty }dy - \;\mathop \smallint \limits_0^\delta \rho udy\)

\(\Rightarrow {\delta ^*} = \mathop \smallint \limits_0^\delta \left( {1 - \frac{u}{{{u_\infty }}}} \right)dy\)

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

\(\theta = \mathop \smallint \limits_0^\delta \frac{u}{{{u_\infty }}}\left( {1 - \frac{u}{{{u_\infty }}}} \right)dy\)

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

\({\delta _E} = \mathop \smallint \limits_0^\delta \frac{u}{{{u_\infty }}}\left( {1 - \frac{{{u^2}}}{{u_\infty ^2}}} \right)dy\)

A bluff body is defined as that body whose surface is not aligned with the stream-lines, when placed in the flow.

Therefore, the body offers lesser resistance in terms of Viscous or Frictional drag.

There is very a large pressure drag, due to eddy formation after the body leading to a large wake region.

While, a streamlined body is defined as that body whose surface is aligned with the streamlines, when the body is placed in the flow.

Concept:

Pressure gradient in the direction of flow is constant and negative.

Flow separation occurs when the pressure gradient is positive and the velocity gradient is negative.

A favourable pressure gradient is one in which the pressure decreases in the flow direction (i.e., dp/dx < 0)

On the other hand, an adverse pressure gradient is one in which pressure increases in the flow direction (i.e., dp/dx > 0)