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In turbulent flow over a smooth bed, the flow is typically separated into two regions. Near the wall, viscosity damps turbulent motions such that the flow is laminar (this is known as the "viscous sublayer", and its size is greatly exaggerated in the figure below). Further from the wall (in the "turbulent layer"), turbulence dominates the vertical transport of momentum. The shear stress is roughly constant across the two layers, taking a value equal to the stress at the wall (tw). The shear stress at the wall is often expressed as a shear velocity (u,), defined as u. = √Tw/P. z=H Z turbulent layer z=0 6, viscous sublayer (a) For horizontal flow over a long, wide wall with no applied pressure gradient, solve the relevant governing equation to show that the mean velocity profile in the viscous sublayer is given by u(z) = u²z/v. (b) The thickness of the viscous sublayer (8) is observed to be roughly 5v/u,. Determine the mean velocity at the top of the viscous sublayer. (c) In the turbulent layer, the eddy velocity scales on u, and the size of the largest eddy scales on the distance from the wall (z). Consequently, the eddy viscosity has the form v+(z) : = Kuz, where x is an O(1) constant.

In turbulent flow over a smooth bed, the flow is typically separated into two regions. Near the wall, viscosity damps turbulent motions such that the flow is laminar (this is known as the "viscous sublayer", and its size is greatly exaggerated in the figure below). Further from the wall (in the "turbulent layer"), turbulence dominates the vertical transport of momentum. The shear stress is roughly constant across the two layers, taking a value equal to the stress at the wall (tw). The shear stress at the wall is often expressed as a shear velocity (u,), defined as u. = √Tw/P. z=H Z turbulent layer z=0 6, viscous sublayer (a) For horizontal flow over a long, wide wall with no applied pressure gradient, solve the relevant governing equation to show that the mean velocity profile in the viscous sublayer is given by u(z) = u²z/v. (b) The thickness of the viscous sublayer (8) is observed to be roughly 5v/u,. Determine the mean velocity at the top of the viscous sublayer. (c) In the turbulent layer, the eddy velocity scales on u, and the size of the largest eddy scales on the distance from the wall (z). Consequently, the eddy viscosity has the form v+(z) : = Kuz, where x is an O(1) constant.

Elements Of Electromagnetics
Elements Of Electromagnetics
7th Edition
ISBN:9780190698614
Author:Sadiku, Matthew N. O.
Publisher:Sadiku, Matthew N. O.
ChapterMA: Math Assessment
Section: Chapter Questions
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Please do not rely too much on chatgpt, because its answer may be wrong. Please consider it carefully and give your own answer. You can borrow ideas from gpt, but please do not believe its answer.Very very grateful!Please do not rely too much on chatgpt, because its answer may be wrong. Please consider it carefully and give your own answer. You can borrow ideas from gpt, but please do not believe its answer.

and and Very very grateful!

In turbulent flow over a smooth bed, the flow is typically separated into two regions. Near the wall, viscosity damps turbulent motions such that the flow is laminar (this is known as the "viscous sublayer", and its size is greatly exaggerated in the figure below). Further from the wall (in the "turbulent layer"), turbulence dominates the vertical transport of momentum. The shear stress is roughly constant across the two layers, taking a value equal to the stress at the wall (tw). The shear stress at the wall is often expressed as a shear velocity (u.), defined as u. = √Tw/P. z=H Z z=0 turbulent layer 8, viscous sublayer (a) For horizontal flow over a long, wide wall with no applied pressure gradient, solve the relevant governing equation to show that the mean velocity profile in the viscous sublayer is given by u(z) = u²z/v. (b) The thickness of the viscous sublayer (8.) is observed to be roughly 5v/u. Determine the mean velocity at the top of the viscous sublayer. (c) In the turbulent layer, the eddy velocity scales on u, and the size of the largest eddy scales on the distance from the wall (z). Consequently, the eddy viscosity has the form v,(z): = Ku,z, where к is an O(1) constant.
Transcribed Image Text:In turbulent flow over a smooth bed, the flow is typically separated into two regions. Near the wall, viscosity damps turbulent motions such that the flow is laminar (this is known as the "viscous sublayer", and its size is greatly exaggerated in the figure below). Further from the wall (in the "turbulent layer"), turbulence dominates the vertical transport of momentum. The shear stress is roughly constant across the two layers, taking a value equal to the stress at the wall (tw). The shear stress at the wall is often expressed as a shear velocity (u.), defined as u. = √Tw/P. z=H Z z=0 turbulent layer 8, viscous sublayer (a) For horizontal flow over a long, wide wall with no applied pressure gradient, solve the relevant governing equation to show that the mean velocity profile in the viscous sublayer is given by u(z) = u²z/v. (b) The thickness of the viscous sublayer (8.) is observed to be roughly 5v/u. Determine the mean velocity at the top of the viscous sublayer. (c) In the turbulent layer, the eddy velocity scales on u, and the size of the largest eddy scales on the distance from the wall (z). Consequently, the eddy viscosity has the form v,(z): = Ku,z, where к is an O(1) constant.
Using your answer to (b) as a boundary condition for the velocity in the turbulent layer (and the fact that the shear stress is constant), determine the mean velocity profile in the turbulent layer. Explain why this layer is typically referred to as the "logarithmic layer".
Transcribed Image Text:Using your answer to (b) as a boundary condition for the velocity in the turbulent layer (and the fact that the shear stress is constant), determine the mean velocity profile in the turbulent layer. Explain why this layer is typically referred to as the "logarithmic layer".
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