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For a turbulent boundary layer (under zero pressure gradient), the velocity profile is described by the one-fifth power law. What is the ratio of displacement thickness to boundary layer thickness?
- a)1/7
- b)1/6
- c)1/5
- d)1/4
Correct answer is option 'B'. Can you explain this answer?
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For a turbulent boundary layer (under zero pressure gradient), the vel...
δ is boundary layer thickness
δ* is displacement thickness
θ is momentum thickness Shape factor,
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For a turbulent boundary layer (under zero pressure gradient), the vel...
Explanation:
The displacement thickness (δ*) is defined as the distance by which the boundary has to be displaced in order to accommodate the same mass of fluid as exists in the actual boundary layer. It is a measure of the reduction in effective cross-sectional area of the flow, as compared to the flow without boundary layer.
The boundary layer thickness (δ) is defined as the distance from the wall where the velocity of the fluid is 99% of the free-stream velocity.
For a turbulent boundary layer under zero pressure gradient, the velocity profile is given by the one-fifth power law which states that:
u/u∞ = (y/δ)^1/5
where,
u = velocity at any distance y from the wall
u∞ = free-stream velocity
y = distance from the wall
δ = boundary layer thickness
Using the definition of boundary layer thickness, we can write:
u/u∞ = 0.99
y/δ = (0.99)^5
y/δ = 0.951
Now, using the definition of displacement thickness, we can write:
δ* = ∫(1-u/u∞)dy
δ* = (5/6)δ
Therefore, the ratio of displacement thickness to boundary layer thickness is:
δ*/δ = (5/6) / 1 = 5/6
Hence, the correct answer is option B, 1/6.
The displacement thickness (δ*) is defined as the distance by which the boundary has to be displaced in order to accommodate the same mass of fluid as exists in the actual boundary layer. It is a measure of the reduction in effective cross-sectional area of the flow, as compared to the flow without boundary layer.
The boundary layer thickness (δ) is defined as the distance from the wall where the velocity of the fluid is 99% of the free-stream velocity.
For a turbulent boundary layer under zero pressure gradient, the velocity profile is given by the one-fifth power law which states that:
u/u∞ = (y/δ)^1/5
where,
u = velocity at any distance y from the wall
u∞ = free-stream velocity
y = distance from the wall
δ = boundary layer thickness
Using the definition of boundary layer thickness, we can write:
u/u∞ = 0.99
y/δ = (0.99)^5
y/δ = 0.951
Now, using the definition of displacement thickness, we can write:
δ* = ∫(1-u/u∞)dy
δ* = (5/6)δ
Therefore, the ratio of displacement thickness to boundary layer thickness is:
δ*/δ = (5/6) / 1 = 5/6
Hence, the correct answer is option B, 1/6.
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For a turbulent boundary layer (under zero pressure gradient), the velocity profile is described by the one-fifth power law. What is the ratio of displacement thickness to boundary layer thickness?a)1/7b)1/6c)1/5d)1/4Correct answer is option 'B'. Can you explain this answer?
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