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Question 1
ప్రశ్న 1
Which statement best describes a velocity boundary layer formed when a viscous fluid flows over a solid surface?
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Explanation:
A velocity boundary layer develops adjacent to a solid surface because viscosity transmits the influence of the wall into the fluid. At a stationary wall the fluid satisfies the no-slip condition, whereas sufficiently far from the wall its streamwise velocity approaches the external-flow velocity. The change between these velocities produces a significant velocity gradient across a relatively thin region. Outside this region, viscous effects on the mean velocity field are usually much less important in classical boundary-layer analysis. Therefore Option B is correct.
వివరణ:
Question 2
ప్రశ్న 2
For flow over a stationary impermeable solid wall, which boundary condition is normally applied at the wall?
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Explanation:
The no-slip condition requires the fluid velocity tangent to a stationary wall to equal the wall velocity, which is zero. Impermeability also prevents fluid from passing through the solid surface, so the normal velocity is zero. These conditions create a strong streamwise velocity gradient near the wall when the external flow has nonzero velocity. The resulting viscous shear is central to boundary-layer formation. Therefore Option D is correct.
వివరణ:
Question 3
ప్రశ్న 3
How is the conventional velocity boundary-layer thickness δ commonly defined for a boundary layer with external velocity U?
Select an option first. || ముందుగా ఒక సమాధానాన్ని ఎంచుకోండి.
Explanation:
The boundary-layer velocity approaches the external velocity asymptotically rather than becoming exactly equal to it at a sharply defined location. A practical convention is therefore required to define the ordinary boundary-layer thickness. It is commonly taken as the distance from the wall where u is approximately 99 percent of the external velocity U. The precise 99 percent choice is conventional rather than a fundamental discontinuity in the flow. Therefore Option A is correct.
వివరణ:
Question 4
ప్రశ్న 4
For a zero-pressure-gradient boundary layer developing over a smooth flat plate, what is the usual qualitative behaviour of boundary-layer thickness as distance from the leading edge increases?
Select an option first. || ముందుగా ఒక సమాధానాన్ని ఎంచుకోండి.
Explanation:
At the leading edge, the region influenced by wall viscosity begins very thin. As the fluid travels downstream, viscous momentum diffusion acts over a progressively larger distance normal to the surface. The velocity-defect region therefore becomes thicker with increasing downstream distance. The precise growth law depends on whether the boundary layer is laminar or turbulent and on pressure-gradient conditions. Therefore Option C is correct.
వివరణ:
Question 5
ప్రశ్న 5
For an incompressible boundary layer with external velocity U, which expression defines displacement thickness δ*?
Select an option first. || ముందుగా ఒక సమాధానాన్ని ఎంచుకోండి.
Explanation:
Displacement thickness measures the mass-flow deficit caused by the reduction of velocity inside the boundary layer. For incompressible flow it is defined by δ* = ∫(1 - u/U)dy, with the integral extending through the velocity-defect region. The integrand vanishes in the outer flow where u approaches U. This makes displacement thickness an integral measure rather than an arbitrary 99-percent thickness. Therefore Option B is correct.
వివరణ:
Question 6
ప్రశ్న 6
What is the principal physical interpretation of boundary-layer displacement thickness?
Select an option first. || ముందుగా ఒక సమాధానాన్ని ఎంచుకోండి.
Explanation:
Fluid within a boundary layer moves more slowly than the hypothetical inviscid external stream. This creates a mass-flow deficit relative to uniform flow at velocity U adjacent to the original wall. Displacement thickness represents the amount by which an equivalent inviscid boundary would have to be shifted outward to reproduce that deficit. It therefore describes the effective thickening of the body as perceived by the outer flow. Therefore Option D is correct.
వివరణ:
Question 7
ప్రశ్న 7
For an incompressible boundary layer with external velocity U, which expression defines momentum thickness θ?
Select an option first. || ముందుగా ఒక సమాధానాన్ని ఎంచుకోండి.
Explanation:
Momentum thickness quantifies the momentum-flux defect associated with the boundary-layer velocity distribution. For incompressible flow it is defined by θ = ∫(u/U)(1 - u/U)dy. Both factors in the integrand are needed because the local mass flux and its velocity defect both contribute to momentum loss. The integrand approaches zero at both the stationary wall and the outer edge where u approaches U. Therefore Option A is correct.
వివరణ:
Question 8
ప్రశ్న 8
Which statement best describes the physical meaning of momentum thickness θ?
Select an option first. || ముందుగా ఒక సమాధానాన్ని ఎంచుకోండి.
Explanation:
Momentum thickness is an integral representation of the loss of streamwise momentum associated with the velocity defect. It is not the same quantity as displacement thickness, which is based on mass-flow deficit. For incompressible flow, the momentum deficit per unit span can be expressed in terms of ρU²θ under the standard definition. This makes momentum thickness especially important in momentum-integral boundary-layer methods and drag analysis. Therefore Option C is correct.
వివరణ:
Question 9
ప్రశ్న 9
For an incompressible boundary layer, which expression defines the kinetic-energy thickness δe?
Select an option first. || ముందుగా ఒక సమాధానాన్ని ఎంచుకోండి.
Explanation:
Kinetic-energy thickness is an integral measure associated with the kinetic-energy-flux defect produced by the velocity profile. For incompressible flow its standard definition is δe = ∫(u/U)[1 - (u/U)²]dy. It is sometimes denoted by δ3 in integral boundary-layer formulations. Its weighting differs from both displacement and momentum thickness because kinetic-energy flux contains a cubic velocity dependence. Therefore Option B is correct.
వివరణ:
Question 10
ప్రశ్న 10
What is the physical significance of kinetic-energy thickness δe?
Select an option first. || ముందుగా ఒక సమాధానాన్ని ఎంచుకోండి.
Explanation:
The fluid within the boundary layer carries less kinetic energy than it would if the same local mass flow moved at the outer-stream velocity. Kinetic-energy thickness provides an equivalent integral measure of this kinetic-energy-flux defect. Because kinetic-energy transport depends strongly on velocity, its integrand contains the factor 1 - (u/U)² in addition to u/U. It is therefore a distinct quantity from displacement and momentum thickness. Hence Option D is correct.
వివరణ:
Question 11
ప్రశ్న 11
For a boundary-layer profile satisfying 0 ≤ u/U ≤ 1 throughout the velocity-defect region, which inequalities necessarily follow from the incompressible integral definitions?
Select an option first. || ముందుగా ఒక సమాధానాన్ని ఎంచుకోండి.
Explanation:
Let r = u/U with 0 ≤ r ≤ 1. The displacement-thickness integrand is 1 - r, while the momentum-thickness integrand is r(1 - r), which cannot exceed 1 - r. The energy-thickness integrand is r(1 - r²) = r(1 - r)(1 + r), which is at least r(1 - r). Therefore δ* ≥ θ and δe ≥ θ for such a profile. Hence Option A is correct.
వివరణ:
Question 12
ప్రశ్న 12
For the idealized boundary-layer profile u/U = y/δ for 0 ≤ y ≤ δ and u = U outside the layer, what is the displacement thickness?
Select an option first. || ముందుగా ఒక సమాధానాన్ని ఎంచుకోండి.
Explanation:
Let η = y/δ, so that u/U = η and dy = δdη. The displacement thickness is δ* = δ∫0^1(1 - η)dη. The integral equals 1/2. Hence δ* = δ/2. Therefore Option C is correct.
వివరణ:
Question 13
ప్రశ్న 13
For the linear profile u/U = y/δ over 0 ≤ y ≤ δ, what is the momentum thickness θ?
Select an option first. || ముందుగా ఒక సమాధానాన్ని ఎంచుకోండి.
Explanation:
Using η = y/δ, the momentum thickness becomes θ = δ∫0^1 η(1 - η)dη. The integral is ∫0^1(η - η²)dη. This evaluates to 1/2 - 1/3 = 1/6. Therefore θ = δ/6 and Option B is correct.
వివరణ:
Question 14
ప్రశ్న 14
For the linear profile u/U = y/δ over 0 ≤ y ≤ δ, what is the kinetic-energy thickness δe?
Select an option first. || ముందుగా ఒక సమాధానాన్ని ఎంచుకోండి.
Explanation:
For the linear profile, u/U = η. The energy thickness is δe = δ∫0^1 η(1 - η²)dη. This is δ∫0^1(η - η³)dη. The integral equals 1/2 - 1/4 = 1/4. Therefore δe = δ/4 and Option D is correct.
వివరణ:
Question 15
ప్రశ్న 15
A boundary layer of conventional thickness δ = 12 mm is idealized by the linear profile u/U = y/δ. What are δ*, θ and δe respectively?
Select an option first. || ముందుగా ఒక సమాధానాన్ని ఎంచుకోండి.
Explanation:
For the linear profile, displacement thickness is δ/2. Momentum thickness is δ/6, and kinetic-energy thickness is δ/4. With δ = 12 mm, these values become 6 mm, 2 mm and 3 mm respectively. Each is an integral thickness and is smaller than the chosen geometric thickness for this profile. Therefore Option A is correct.
వివరణ:
Question 16
ప్రశ్న 16
An idealized boundary-layer velocity profile is u/U = 2(y/δ) - (y/δ)² for 0 ≤ y ≤ δ. What is its displacement thickness?
Select an option first. || ముందుగా ఒక సమాధానాన్ని ఎంచుకోండి.
Explanation:
Let η = y/δ, giving u/U = 2η - η². The displacement-thickness integrand becomes 1 - 2η + η². Integrating from zero to one gives 1 - 1 + 1/3 = 1/3. Hence δ* = δ/3. Therefore Option C is correct.
వివరణ:
Question 17
ప్రశ్న 17
For the velocity profile u/U = 2(y/δ) - (y/δ)² over 0 ≤ y ≤ δ, what is the momentum thickness?
Select an option first. || ముందుగా ఒక సమాధానాన్ని ఎంచుకోండి.
Explanation:
With η = y/δ, the profile is f = 2η - η². Momentum thickness is θ = δ∫0^1 f(1 - f)dη. Expanding and integrating the polynomial gives a value of 2/15. Therefore θ = 2δ/15. Hence Option A is correct.
వివరణ:
Question 18
ప్రశ్న 18
For the velocity profile u/U = 2(y/δ) - (y/δ)² over 0 ≤ y ≤ δ, what is the kinetic-energy thickness?
Select an option first. || ముందుగా ఒక సమాధానాన్ని ఎంచుకోండి.
Explanation:
For this profile, let f = 2η - η². Kinetic-energy thickness is δe = δ∫0^1 f(1 - f²)dη. Expanding the polynomial and integrating from zero to one gives 22/105. Thus δe = 22δ/105, which is approximately 0.2095δ. Therefore Option B is correct.
వివరణ:
Question 19
ప్రశ్న 19
The boundary-layer shape factor is defined as H = δ*/θ. For the profile u/U = 2(y/δ) - (y/δ)², what is H?
Select an option first. || ముందుగా ఒక సమాధానాన్ని ఎంచుకోండి.
Explanation:
For the stated quadratic profile, δ* = δ/3. Its momentum thickness is θ = 2δ/15. Therefore H = (δ/3)/(2δ/15). Cancelling δ gives H = 15/6 = 2.5. Hence Option D is correct.
వివరణ:
Question 20
ప్రశ్న 20
For the linear profile u/U = y/δ, what is the ratio of kinetic-energy thickness to momentum thickness, δe/θ?
Select an option first. || ముందుగా ఒక సమాధానాన్ని ఎంచుకోండి.
Explanation:
For the linear profile, kinetic-energy thickness is δ/4. Momentum thickness is δ/6. Their ratio is (δ/4)/(δ/6) = 6/4. This equals 1.50. Therefore Option C is correct.
వివరణ:
Question 21
ప్రశ్న 21
An incompressible boundary layer has external velocity U = 8 m/s, density ρ = 1000 kg/m³ and displacement thickness δ* = 3 mm. What is the mass-flow deficit per unit span relative to uniform flow at U?
Select an option first. || ముందుగా ఒక సమాధానాన్ని ఎంచుకోండి.
Explanation:
For an incompressible boundary layer, the mass-flow deficit per unit span is ρUδ*. The displacement thickness of 3 mm is 0.003 m. Substitution gives 1000 × 8 × 0.003. The result is 24 kg/(s·m). Therefore Option B is correct.
వివరణ:
Question 22
ప్రశ్న 22
An incompressible boundary layer in air has ρ = 1.2 kg/m³, external velocity U = 20 m/s and momentum thickness θ = 2.0 mm. What is the corresponding momentum-flux deficit per unit span, ρU²θ?
Select an option first. || ముందుగా ఒక సమాధానాన్ని ఎంచుకోండి.
Explanation:
The momentum-flux deficit represented by momentum thickness is ρU²θ per unit span. The thickness 2.0 mm equals 0.002 m. Substitution gives 1.2 × 20² × 0.002. This evaluates to 0.96 N/m. Therefore Option A is correct.
వివరణ:
Question 23
ప్రశ్న 23
An incompressible boundary layer has ρ = 1.2 kg/m³, external velocity U = 20 m/s and kinetic-energy thickness δe = 3.0 mm. Using the standard energy-thickness definition, what kinetic-energy-flux deficit per unit span is represented by (1/2)ρU³δe?
Select an option first. || ముందుగా ఒక సమాధానాన్ని ఎంచుకోండి.
Explanation:
The kinetic-energy-flux deficit represented by energy thickness is (1/2)ρU³δe per unit span under the stated definition. The thickness 3.0 mm equals 0.003 m. Substitution gives 0.5 × 1.2 × 20³ × 0.003. This evaluates to 14.4 W/m. Therefore Option C is correct.
వివరణ:
Question 24
ప్రశ్న 24
For the profile u/U = 2(y/δ) - (y/δ)², which statement about the streamwise velocity gradient is correct?
Select an option first. || ముందుగా ఒక సమాధానాన్ని ఎంచుకోండి.
Explanation:
Differentiating u = U[2(y/δ) - (y/δ)²] gives du/dy = (2U/δ)(1 - y/δ). At the wall, y = 0, so the gradient is 2U/δ. At y = δ, the factor 1 - y/δ becomes zero. The profile therefore approaches the outer velocity with zero gradient at the chosen edge. Hence Option D is correct.
వివరణ:
Question 25
ప్రశ్న 25
Consider the following statements about an incompressible velocity boundary layer: I. The conventional boundary-layer thickness is commonly based on u/U ≈ 0.99. II. Displacement thickness represents a mass-flow deficit. III. Momentum thickness represents a momentum-flux deficit. IV. Kinetic-energy thickness uses the integrand (u/U)[1 - (u/U)²]. Which option is correct?
Select an option first. || ముందుగా ఒక సమాధానాన్ని ఎంచుకోండి.
Explanation:
Statement I correctly describes the conventional practical definition of velocity boundary-layer thickness. Statement II correctly identifies the physical meaning of displacement thickness. Statement III correctly identifies momentum thickness as an integral measure of momentum-flux deficit. Statement IV is the standard incompressible definition of kinetic-energy thickness. Therefore all four statements are correct and Option D is the valid answer.
వివరణ:
Answer Key సమాధానాల పట్టిక
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Question 1 ప్రశ్న 1Answer: B. It is a region adjacent to the surface in which viscous effects produce a significant velocity gradient from the wall toward the outer flow సమాధానం: B.
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Question 2 ప్రశ్న 2Answer: D. Both tangential and normal fluid velocities are zero at the wall సమాధానం: D.
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Question 3 ప్రశ్న 3Answer: A. The distance from the wall at which u/U is approximately 0.99 సమాధానం: A.
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Question 4 ప్రశ్న 4Answer: C. It generally increases downstream as viscous momentum diffusion affects fluid farther from the wall సమాధానం: C.
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Question 5 ప్రశ్న 5Answer: B. δ* = ∫(1 - u/U) dy సమాధానం: B.
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Question 6 ప్రశ్న 6Answer: D. It represents the outward displacement of the external stream required to account for the mass-flow deficit caused by the boundary layer సమాధానం: D.
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Question 7 ప్రశ్న 7Answer: A. θ = ∫(u/U)(1 - u/U) dy సమాధానం: A.
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Question 8 ప్రశ్న 8Answer: C. It is an equivalent thickness associated with the momentum-flux deficit produced by the boundary-layer velocity defect సమాధానం: C.
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Question 9 ప్రశ్న 9Answer: B. δe = ∫(u/U)[1 - (u/U)²] dy సమాధానం: B.
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Question 10 ప్రశ్న 10Answer: D. It is an equivalent thickness representing the kinetic-energy-flux deficit caused by the boundary-layer velocity distribution సమాధానం: D.
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Question 11 ప్రశ్న 11Answer: A. δ* ≥ θ and δe ≥ θ సమాధానం: A.
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Question 12 ప్రశ్న 12Answer: C. δ/2 సమాధానం: C.
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Question 13 ప్రశ్న 13Answer: B. δ/6 సమాధానం: B.
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Question 14 ప్రశ్న 14Answer: D. δ/4 సమాధానం: D.
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Question 15 ప్రశ్న 15Answer: A. 6 mm, 2 mm and 3 mm సమాధానం: A.
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Question 16 ప్రశ్న 16Answer: C. δ/3 సమాధానం: C.
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Question 17 ప్రశ్న 17Answer: A. 2δ/15 సమాధానం: A.
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Question 18 ప్రశ్న 18Answer: B. 22δ/105 సమాధానం: B.
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Question 19 ప్రశ్న 19Answer: D. 2.50 సమాధానం: D.
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Question 20 ప్రశ్న 20Answer: C. 1.50 సమాధానం: C.
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Question 21 ప్రశ్న 21Answer: B. 24 kg/(s·m) సమాధానం: B.
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Question 22 ప్రశ్న 22Answer: A. 0.96 N/m సమాధానం: A.
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Question 23 ప్రశ్న 23Answer: C. 14.4 W/m సమాధానం: C.
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Question 24 ప్రశ్న 24Answer: D. du/dy at the wall is 2U/δ and it becomes zero at y = δ సమాధానం: D.
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Question 25 ప్రశ్న 25Answer: D. I, II, III and IV సమాధానం: D.