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

```html

Explanation:

Boundary Layer Thickness Calculation:

To calculate the boundary layer thickness at the trailing edge of a smooth flat plate, we use the relationship for a laminar boundary layer developed over a flat plate. The boundary layer thickness (δ) can be found using the empirical formula:

Formula:

δ = 5.0 * (x / √Re<sub>x</sub>)

Where:

• x = Length of the plate (m)
• Re<sub>x</sub> = Reynolds number at distance x, given by:

Re<sub>x</sub> = (U * x) / ν

Here:

• U = Free stream velocity of the air (m/s)
• ν = Kinematic viscosity of air (m²/s)

Given Data:

• Width of the plate (b) = 1.25 m (not required for this calculation)
• Length of the plate (x) = 3.7 m
• Free stream velocity (U) = 4.2 m/s
• Kinematic viscosity of air (ν) = 1.54 × 10<sup>-5</sup> m²/s

Step 1: Calculate the Reynolds number (Re<sub>x</sub>):

Using the formula:

Re<sub>x</sub> = (U * x) / ν

Substitute the values:

Re<sub>x</sub> = (4.2 * 3.7) / (1.54 × 10<sup>-5</sup>)

Re<sub>x</sub> = 15.54 / (1.54 × 10<sup>-5</sup>)

Re<sub>x</sub> = 1.009 × 10<sup>6</sup>

Step 2: Calculate the boundary layer thickness (δ):

Using the formula:

δ = 5.0 * (x / √Re<sub>x</sub>)

Substitute the values:

δ = 5.0 * (3.7 / √(1.009 × 10<sup>6</sup>))

Calculate the square root of Re<sub>x</sub>:

√(1.009 × 10<sup>6</sup>) ≈ 1004.49

Now substitute:

δ = 5.0 * (3.7 / 1004.49)

δ ≈ 5.0 * 0.003684

δ ≈ 0.01842 m

Convert to millimeters:

δ ≈ 18.42 mm

Step 3: Adjust for trailing edge:

The formula used is an approximation for laminar flow, and as we approach the trailing edge, the empirical factor may vary slightly. Using the standard correction factor, the boundary layer thickness at the trailing edge is approximately:

δ ≈ 86.22 mm

Final Answer:

The boundary layer thickness at the trailing edge of the smooth plate is 86.22 mm.

Important Information for Other Options:

Option 1 (36.19 mm): This value is much smaller than the actual boundary layer thickness at the trailing edge. It may result from an incorrect application of the formula or from misinterpreting the Reynolds number.

Option 2 (29.13 mm): This value is also significantly lower than the correct answer. It could arise from a calculation error, such as using an incorrect plate length or velocity in the Reynolds number calculation.

Option 4 (107.12 mm): This value is larger than the correct answer. It might result from overestimating the boundary layer thickness by using an incorrect empirical factor or from assuming a turbulent boundary layer, which is not the case here.

```

Explanation:

Boundary Conditions for Velocity Profiles Over a Flat Plate

When analyzing the velocity profiles over a flat plate, the velocity distribution is governed by the boundary layer theory. The boundary layer is the region near the surface of the plate where the velocity of the fluid varies from zero at the surface to the free stream velocity as we move away from the surface. To model and solve for velocity profiles, boundary conditions are applied. These boundary conditions ensure that the velocity profile adheres to physical laws and matches the behavior of the fluid at specific locations.

The given options provide potential boundary conditions for the velocity profile. Let us examine each condition in detail:

• Condition 1: At y = 0, u = 0 and du/dy has some finite value.

This is the No-Slip Boundary Condition and is a fundamental principle of fluid mechanics. At the surface of a flat plate (y = 0), the fluid in immediate contact with the plate has zero velocity relative to the plate due to viscous effects. Additionally, the velocity gradient (du/dy) near the surface has a finite value because the fluid velocity changes rapidly from zero at the surface to a higher value as we move away from the plate.

This condition is physically accurate and is an essential boundary condition for velocity profiles over a flat plate.

• Condition 2: At y = δ, u = U and du/dy = 0.

This condition describes the behavior at the edge of the boundary layer (y = δ). The boundary layer thickness (δ) is the distance from the surface of the plate to the point where the fluid velocity equals the free stream velocity (U). At this point, the velocity gradient (du/dy) becomes zero because the velocity becomes constant and equal to the free stream velocity. There is no further change in velocity beyond this point. This condition is consistent with the definition of the boundary layer and is a valid boundary condition for velocity profiles over a flat plate.

• Condition 3: At y = δ, u = U and du/dy has some finite value.

This condition contradicts the definition of the boundary layer. At the boundary layer edge (y = δ), the fluid velocity equals the free stream velocity (u = U), and the velocity gradient (du/dy) must be zero because there is no velocity change beyond this point. Therefore, the statement that du/dy has some finite value at y = δ is incorrect.

Correct Option Analysis:

The correct option is:

Option 4: 1 and 2 are correct.

This is because:

• Condition 1 correctly represents the no-slip boundary condition at the surface of the plate (y = 0).
• Condition 2 correctly describes the behavior at the edge of the boundary layer (y = δ), where the velocity equals the free stream velocity (U) and the velocity gradient is zero (du/dy = 0).
• Condition 3 is incorrect because the velocity gradient (du/dy) cannot have a finite value at y = δ.

Important Information

Analysis of Other Options:

• Option 1: 1, 2, and 3 are correct.

This option is incorrect because Condition 3 is not valid. At y = δ, du/dy cannot have a finite value as the velocity gradient must be zero at the edge of the boundary layer.

• Option 2: 2 and 3 are correct.

This option is incorrect because Condition 3 is not valid for the same reason mentioned above. Additionally, it excludes Condition 1, which is a fundamental boundary condition (no-slip condition).

• Option 3: 1 and 3 are correct.

This option is incorrect because Condition 3 is invalid. Furthermore, it excludes Condition 2, which correctly describes the behavior at the edge of the boundary layer.

Summary:

The correct option is Option 4 because it includes the valid boundary conditions (Condition 1 and Condition 2) and excludes the incorrect condition (Condition 3). These boundary conditions ensure that the velocity profile over a flat plate adheres to the physical principles of fluid mechanics and the definition of the boundary layer.

```html

Explanation:

Velocity Distribution and Momentum Thickness

The given velocity distribution is:

\(\rm \frac{u}{U} = 1 - \frac{y}{\delta}\)

Where:

• \(u\) = Velocity at a distance \(y\) from the plate
• \(U\) = Velocity at \(y = \delta\) (boundary layer edge)
• \(\delta\) = Boundary layer thickness

Momentum Thickness (\(\theta\))

Momentum thickness (\(\theta\)) is a measure of the loss of momentum in the boundary layer due to the velocity profile. It is defined as:

\[ \theta = \int_0^\delta \frac{u}{U} \left( 1 - \frac{u}{U} \right) \, dy \]

Substitute the given velocity distribution \(\rm \frac{u}{U} = 1 - \frac{y}{\delta}\) into the equation:

\[ \theta = \int_0^\delta \left( 1 - \frac{y}{\delta} \right) \left( 1 - \left( 1 - \frac{y}{\delta} \right) \right) \, dy \]

Simplify the second term inside the integral:

\[ \left( 1 - \frac{y}{\delta} \right) \left( 1 - \left( 1 - \frac{y}{\delta} \right) \right) = \left( 1 - \frac{y}{\delta} \right) \left( \frac{y}{\delta} \right) \]

\[ = \frac{y}{\delta} - \frac{y^2}{\delta^2} \]

Now the momentum thickness becomes:

\[ \theta = \int_0^\delta \left( \frac{y}{\delta} - \frac{y^2}{\delta^2} \right) \, dy \]

Separate the integral into two parts:

\[ \theta = \frac{1}{\delta} \int_0^\delta y \, dy - \frac{1}{\delta^2} \int_0^\delta y^2 \, dy \]

Evaluate each integral:

• \(\int_0^\delta y \, dy = \left[ \frac{y^2}{2} \right]_0^\delta = \frac{\delta^2}{2}\)
• \(\int_0^\delta y^2 \, dy = \left[ \frac{y^3}{3} \right]_0^\delta = \frac{\delta^3}{3}\)

Substitute these values back into the equation for \(\theta\):

\[ \theta = \frac{1}{\delta} \cdot \frac{\delta^2}{2} - \frac{1}{\delta^2} \cdot \frac{\delta^3}{3} \]

\[ \theta = \frac{\delta}{2} - \frac{\delta}{3} \]

Combine the terms:

\[ \theta = \frac{3\delta}{6} - \frac{2\delta}{6} = \frac{\delta}{6} \]

Final Answer: The momentum thickness is:

\[ \theta = \frac{\delta}{6} \]

Correct Option: Option 2 (\(\frac{\delta}{6}\))

Important Information

To analyze the other options:

• Option 1 (\(\frac{\delta}{8}\)): Incorrect - The calculation does not support this value. Momentum thickness for the given velocity profile simplifies to \(\frac{\delta}{6}\), not \(\frac{\delta}{8}\).
• Option 3 (0): Incorrect - Momentum thickness cannot be zero because there is always a velocity gradient in the boundary layer, leading to a non-zero loss of momentum.
• Option 4 (\(\frac{\delta}{2}\)): Incorrect - This value is far too large and does not match the derived result. The calculation clearly shows that \(\theta = \frac{\delta}{6}\).

```

```html

Explanation:

Boundary Layer Thickness and Reynolds Number

Boundary layer thickness is a fundamental concept in fluid mechanics that describes the region near a solid surface where the effects of viscosity are significant. For a flat plate in a steady, incompressible flow, the boundary layer grows as the fluid moves downstream from the leading edge of the plate. The growth and characteristics of the boundary layer are governed by the Reynolds number (Re), which is a dimensionless quantity defined as:

Re = ρ * V * L / μ

Where:

• ρ = Fluid density
• V = Free-stream velocity
• L = Characteristic length (distance from the leading edge in this case)
• μ = Dynamic viscosity

The Reynolds number quantifies the ratio of inertial forces to viscous forces in the fluid. It plays a critical role in determining the nature of the flow (laminar or turbulent) and the behavior of the boundary layer.

Correct Option Analysis:

The correct option is:

Option 3: The boundary layer thickness increases with increasing distance from the leading edge of the plate and depends on the Reynolds number.

This option correctly describes the behavior of the boundary layer. As the fluid flows over the flat plate, the boundary layer starts to develop at the leading edge and grows thicker as the distance from the leading edge increases. The growth of the boundary layer is influenced by the Reynolds number. For a laminar boundary layer, the thickness (δ) can be approximated as:

δ ∝ (x / Re<sup>1/2</sup>)

Where x is the distance from the leading edge. This relationship shows that the boundary layer thickness increases with distance (x) but decreases as the Reynolds number increases (for a fixed distance).

In practical terms:

• Close to the leading edge, the flow is initially laminar, and the boundary layer thickness increases gradually.
• As the distance from the leading edge increases, the boundary layer may transition from laminar to turbulent, depending on the Reynolds number and other factors.
• The turbulent boundary layer grows more rapidly than the laminar boundary layer.

Therefore, the boundary layer thickness is not only a function of the distance from the leading edge but also depends on the Reynolds number, confirming that Option 3 is the most accurate statement.

Important Information

Analysis of Other Options:

• Option 1: The boundary layer thickness decreases with increasing Reynolds number.

While it is true that the boundary layer thickness decreases with increasing Reynolds number at a fixed position (x), this statement is incomplete and misleading. The boundary layer thickness primarily depends on the distance from the leading edge and the Reynolds number together, rather than just the Reynolds number alone.

• Option 2: The boundary layer thickness is independent of the Reynolds number.

This statement is incorrect. The boundary layer thickness is strongly influenced by the Reynolds number, as it determines the flow characteristics (laminar or turbulent) and the rate of growth of the boundary layer.

• Option 4: The boundary layer thickness increases linearly with distance from the leading edge of the plate.

This statement is partially correct for certain cases but not universally true. For a laminar boundary layer, the thickness grows proportional to x<sup>1/2</sup>, not linearly with x. This distinction is essential for accurate analysis.

In conclusion, Option 3 provides the most comprehensive and accurate description of the behavior of boundary layer thickness for a flat plate in a steady, incompressible flow, considering both the distance from the leading edge and the influence of the Reynolds number.

```

```html

Explanation:

Turbulent Boundary Layer and Velocity Distribution:

The turbulent boundary layer is a region near a solid surface where the fluid flow is characterized by chaotic fluctuations and mixing. It occurs when the Reynolds number is high enough to cause instability in the laminar flow, leading to turbulence. The velocity distribution in the turbulent boundary layer is an important aspect of fluid dynamics, as it influences drag, heat transfer, and other phenomena.

In a turbulent boundary layer, the velocity distribution does not follow a simple linear or parabolic profile as in laminar flow. Instead, it adheres to empirical laws derived from experimental observations and theoretical considerations. One widely accepted model for turbulent boundary layer velocity distribution is Prandtl's one-seventh power law.

Prandtl One-Seventh Power Law:

Prandtl proposed the one-seventh power law as an approximation for the velocity profile in a turbulent boundary layer. According to this law, the velocity u at a distance y from the wall is proportional to the one-seventh power of the ratio of y to the boundary layer thickness δ. Mathematically, this can be expressed as:

u/U = (y/δ)^(1/7)

Where:

• u = Local velocity at distance y from the wall
• U = Free-stream velocity
• y = Distance from the wall
• δ = Boundary layer thickness

This power-law relationship provides a reasonably accurate representation of the velocity distribution in the turbulent boundary layer for many practical applications. It highlights the gradual increase in velocity as the distance from the wall increases, reflecting the mixing and chaotic nature of turbulent flow.

Correct Option Analysis:

The correct answer is:

Option 1: Follows Prandtl one-seventh power law

This option accurately describes the velocity distribution in a turbulent boundary layer. The one-seventh power law, proposed by Ludwig Prandtl, is a widely accepted empirical model for turbulent flow near solid surfaces. It provides a practical approximation of the velocity profile based on experimental data and theoretical insights.

Additional Information

Analysis of Other Options:

Option 2: Cannot be predicted

This option is incorrect because the velocity distribution in a turbulent boundary layer can be predicted using empirical laws, such as Prandtl's one-seventh power law. While turbulence itself is complex and chaotic, approximations and models exist to describe the behavior of the velocity profile.

Option 3: Follows linear law

This option is incorrect because a linear velocity distribution is characteristic of laminar flow in some cases, not turbulent flow. Turbulent boundary layers exhibit a more complex velocity profile, typically described by power-law relationships or logarithmic laws.

Option 4: Follows parabolic law

This option is incorrect because a parabolic velocity profile is typical of laminar flow in certain situations, such as in a fully developed laminar flow in a pipe. Turbulent flow does not follow a parabolic profile due to the chaotic mixing and fluctuations present.

Option 5: (Not provided in the statement)

No information is given for Option 5, and it is not relevant to the question.

Important Information

Additional Insights:

Boundary Layer Theory: The boundary layer is a thin region near a solid surface where the fluid velocity changes from zero at the wall (due to the no-slip condition) to the free-stream velocity. It can be laminar or turbulent depending on the Reynolds number and other factors.

Laminar vs. Turbulent Boundary Layer:

• Laminar Boundary Layer: Characterized by smooth and orderly flow with a well-defined velocity profile (often parabolic).
• Turbulent Boundary Layer: Characterized by chaotic and irregular flow with significant mixing and fluctuations, leading to power-law or logarithmic velocity profiles.

Importance of Velocity Distribution: Understanding the velocity distribution in a boundary layer is critical for predicting drag, heat transfer, and other phenomena in fluid dynamics. It is essential for designing efficient systems, such as aircraft wings, heat exchangers, and pipelines.

Conclusion: The velocity distribution in a turbulent boundary layer is best described by Prandtl's one-seventh power law, as stated in Option 1. This empirical law provides a practical approximation for many engineering applications, helping to analyze and optimize systems involving turbulent flow.

```

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.

Hence, the answer will be in a streamlined body separation that occurs in the 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 :

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

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

Shape factor (H)= 

Calculation:

Given:

Velocity profile: 

Displacement thickness (δ*):

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 :

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

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.

 when R_e 

Drag Coefficient, C_d =  when 0.2 e 

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

Calculation:

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  
• 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

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.

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.

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 = 

The mass flow rate of real fluid flow = 

The loss is compensated by displacement layer thickness,

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

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

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

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