Answered: Problem 6. A boundary layer is…

Elements Of Electromagnetics

7th Edition

ISBN:9780190698614

Author:Sadiku, Matthew N. O.

Publisher:Sadiku, Matthew N. O.

ChapterMA: Math Assessment

Section: Chapter Questions

Problem 1.1MA

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$**Problem 6**: A boundary layer is developing for uniform flow of air at STP over a flat plate as shown. The free-stream flow outside the boundary layer has an undisturbed velocity of $ U = 50 \, \text{m/s} $. The plate has a width of $ b = 3.0 \, \text{m} $ perpendicular to the schematic. A control volume (CV) with control surfaces (ab), (bc), (cd), and (ad) is defined as shown. Assume that the flow in the boundary layer is turbulent at surface (cd) with a $ 1/7^{\text{th}} $ power-law velocity profile and a boundary layer disturbance thickness of $ \delta = 2.0 \, \text{cm} $. For surfaces (ab), (bc), and (cd), calculate (a) mass flow rate and (b) momentum transported (in Newtons). (c) Determine the drag for the flat plate at surface (ad). **Diagram Explanation:** - **Control Volume:** The dashed blue lines outline the control volume around the boundary layer region over the flat plate. The control surfaces are labeled as (ab), (bc), (cd), and (ad). - **Boundary-Layer Region:** This is the region above the flat plate where the flow is affected by the presence of the plate, represented by the dashed curve showing how velocity varies from the edge of the boundary layer to the surface of the plate. - **Velocity Profile:** $ u(y) $ represents the velocity distribution across the boundary layer, indicated by the curved dashed line within the boundary-layer region. - **Velocity Indicators:** Solid blue arrows indicate the uniform velocity $ U $ outside the boundary layer, while they gradually become dotted within the boundary-layer region, indicating the variation in velocity. - **Coordinate System:** The $ x $-axis is aligned with the flat plate, and the $ y $-axis is perpendicular to it. This problem requires calculations of mass flow rate, momentum transportation, and drag force, by considering the effects of the turbulent boundary layer and power-law velocity profile.$

Transcribed Image Text:**Problem 6**: A boundary layer is developing for uniform flow of air at STP over a flat plate as shown. The free-stream flow outside the boundary layer has an undisturbed velocity of $ U = 50 \, \text{m/s} $. The plate has a width of $ b = 3.0 \, \text{m} $ perpendicular to the schematic. A control volume (CV) with control surfaces (ab), (bc), (cd), and (ad) is defined as shown. Assume that the flow in the boundary layer is turbulent at surface (cd) with a $ 1/7^{\text{th}} $ power-law velocity profile and a boundary layer disturbance thickness of $ \delta = 2.0 \, \text{cm} $. For surfaces (ab), (bc), and (cd), calculate (a) mass flow rate and (b) momentum transported (in Newtons). (c) Determine the drag for the flat plate at surface (ad). **Diagram Explanation:** - **Control Volume:** The dashed blue lines outline the control volume around the boundary layer region over the flat plate. The control surfaces are labeled as (ab), (bc), (cd), and (ad). - **Boundary-Layer Region:** This is the region above the flat plate where the flow is affected by the presence of the plate, represented by the dashed curve showing how velocity varies from the edge of the boundary layer to the surface of the plate. - **Velocity Profile:** $ u(y) $ represents the velocity distribution across the boundary layer, indicated by the curved dashed line within the boundary-layer region. - **Velocity Indicators:** Solid blue arrows indicate the uniform velocity $ U $ outside the boundary layer, while they gradually become dotted within the boundary-layer region, indicating the variation in velocity. - **Coordinate System:** The $ x $-axis is aligned with the flat plate, and the $ y $-axis is perpendicular to it. This problem requires calculations of mass flow rate, momentum transportation, and drag force, by considering the effects of the turbulent boundary layer and power-law velocity profile.

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