Boundary Layer Thickness MCQ Quiz in தமிழ் - Objective Question with Answer for Boundary Layer Thickness - இலவச PDF ஐப் பதிவிறக்கவும்

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
• \(\frac{{\bf{k}}}{{\bf{\delta }}} < 0.25\)
• Boundary transition condition, when
• \(0.25 < \frac{{\bf{k}}}{{\bf{\delta }}} < 6\)
• Hydrodynamically rough when
• \(\frac{{\bf{k}}}{{\bf{\delta }}} > 6\)

Concept:

According to the Blasius equation, for a laminar flow

Boundary layer thickness (δ) can be calculated by:

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

where Re_x = Reynold's number = \(\frac{{\rho Vx}}{\mu }\) {V = Velocity}

\(\therefore \delta \propto \frac{1}{V}\)

Calculation:

Given:

δ₁ = 1 mm, V₂ = 4V₁

\(As,\,\delta \propto \frac{1}{V}\)

\(\frac{{{\delta _1}}}{{{\delta _2}}} = \sqrt {\frac{{{V_2}}}{{{V_1}}}} \)

\(\frac{1}{{{\delta _2}}} = \sqrt {\frac{4}{1}} = 2\)

 δ₂ = 0.5 mm

Concept:

The thickness of the boundary layer is given by \(\delta = \frac{{5x}}{{\sqrt {R{e_x}} }}\)

For turbulent flow \(\delta = \frac{{0.379\;x}}{{R{e^{\frac{1}{5}}}}}\)

Where δ = Boundary layer thickness, x = Distance of boundary layer from the leading edge, Rex = Reynold’s number at the distance x from the leading edge

Calculation:

Since in this given case, Reynold’s number is 3200000 which is in the range of 5 × 10⁵ to 10⁷, so the flow is turbulent.

Given, x = 10 cm = 0.1 m

∴ \(\delta = \frac{{0.379\;x}}{{R{e^{\frac{1}{5}}}}} = \frac{{0.379\;\times 0.1}}{{3200000^{\frac{1}{5}}}}=1.895 \; mm \)

For laminar flow:

\(\delta \propto \frac{1}{{Re_x^{\frac{1}{2}}}}\)

Re_x = U_x/ν ⇒ Re_x ∝ U 

\(\delta \propto \frac{1}{{{U^{\frac{1}{2}}}}}\)

\({\delta _1}U_1^{\frac{1}{2}} = {\delta _2}U_2^{\frac{1}{2}}\)

U₂ = 5U₁

_{\(\left( 2 \right)U_1^{\frac{1}{2}} = {\delta _2}{\left( {5{U_1}} \right)^{\frac{1}{2}}}\)}

\({\delta _2} = \frac{2}{{{5^{\frac{1}{2}}}}} = \frac{2}{{\sqrt 5 }} = 0.89\:mm\)

Concept:

Displacement thickness is given by

\({{\rm{\delta }}^{\rm{*}}} =\int\limits_0^{\rm{\delta }} \left( {1 - \frac{{\rm{u}}}{{{{\rm{U}}_\infty }{\rm{\;}}}}} \right){\rm{dy}}\)

Where,

u – velocity of the fluid

U_∞ - Free stream velocity

Calculation:

Given:

\(\dfrac{u}{{U_\infty }} = \sin \left( {\frac{\pi }{2}\frac{y}{\delta }} \right)\)

Solution:

\({\delta ^*} = \mathop \smallint \nolimits_0^\delta \left[ {1 - \sin \left( {\dfrac{\pi }{2}\dfrac{y}{\delta }} \right)} \right]\;dy\)

\({\delta ^*} = \left[ y \right]_0^\delta - \left[ {\frac{{\begin{array}{*{20}{c}} { - \cos \left( {\dfrac{\pi }{2}\dfrac{y}{\delta }} \right)}\\ \; \end{array}}}{{\dfrac{\pi }{{2\delta }}}}} \right]_0^\delta \;\)

\({\delta ^*} = \left[ {\delta - 0} \right] - \left[ {\dfrac{{\begin{array}{*{20}{c}} { - \cos \left( {\dfrac{\pi }{2}\dfrac{\delta }{\delta }} \right)}\\ \; \end{array}}}{{\dfrac{\pi }{{2\delta }}}} +\dfrac{1}{{\dfrac{\pi }{{2\delta }}}}} \right]\)

\({\delta ^*} = \delta - \left[ {\dfrac{{\begin{array}{*{20}{c}} { - \cos \left( {\dfrac{\pi }{2}\dfrac{\delta }{\delta }} \right)}\\ \; \end{array}}}{{\dfrac{\pi }{{2\delta }}}} + \dfrac{1}{{\dfrac{\pi }{{2\delta }}}}} \right]\)

\({\delta ^*} = \delta - \dfrac{{2\delta }}{\pi }\)

Concept:

For a turbulent boundary layer 

\(\delta ~=\frac{0.37x}{R{{e}^{\frac{1}{5}}}}=\frac{0.37x}{{{(\frac{\rho vx}{\mu })}^{\frac{1}{5}}}}\Rightarrow \delta \propto {{x}^{\frac{4}{5}}}\)

Concept:

Laminar boundary layer thickness is given by

\( {\rm{δ \;}} = {\rm{\;}}5√ {\frac{{{\rm{x\nu }}}}{{\rm{V}}}}\)

From here we can see that

δ ∝ √x

Calculation:

Given:

δ_A = 2 cm, δ_B = 3 cm

Now,

\(\begin{array}{l} {\rm{δ \;}} = {\rm{\;}}5√ {\frac{{{\rm{x\nu }}}}{{\rm{V}}}} \\ \frac{{{δ _A}}}{{{δ _B}}}\; = \;√ {\frac{{{x_A}}}{{{x_B}}}} \\ \Rightarrow \frac{2}{3}\; = \;√ {\frac{{{x_A}}}{{{x_A}\; + \;1}}\;} \\ \Rightarrow \frac{4}{9}\; = \;\frac{{{x_A}}}{{{x_A}\; + \;1}} \end{array}\)

∴ 5x_A = 4

∴ x_A = 0.8 m

δ is the distance from the centre - line to the edge of boundary layer where the change in speed is 99% of maximum change in speed from centreline of outer flow. δ is not constant but varies along x. δ(x) increases with x.

But there are certain flows such as rapidly accelerating enter flow along a wall, in which δ(x) decreases with x.

Pressure drop is the driving force to push fluid to flow through a pipe, that means as far as there is no pressure difference between two points along the pipe length there is no flow. Generally pressure drop takes place even radially, but the value is too smaller than that in length, therefore we just consider the pressure drop in pipe length. When we have a fully developed flow it doesn't mean that there is no pressure drop, it just means there is no velocity difference along the axis where we have fully developed flow.

 Shear stress also decreases along plate direction.

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_x)

Where:

• x = Length of the plate (m)
• Re_x = Reynolds number at distance x, given by:

Re_x = (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^-5 m²/s

Step 1: Calculate the Reynolds number (Re_x):

Using the formula:

Re_x = (U * x) / ν

Substitute the values:

Re_x = (4.2 * 3.7) / (1.54 × 10^-5)

Re_x = 15.54 / (1.54 × 10^-5)

Re_x = 1.009 × 10⁶

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

Using the formula:

δ = 5.0 * (x / √Re_x)

Substitute the values:

δ = 5.0 * (3.7 / √(1.009 × 10⁶))

Calculate the square root of Re_x:

√(1.009 × 10⁶) ≈ 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 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^1/2)

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^1/2, 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.