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.
Principles of Heat Transfer (Activate Learning with these NEW titles from Engineering!)
8th Edition
ISBN:9781305387102
Author:Kreith, Frank; Manglik, Raj M.
Publisher:Kreith, Frank; Manglik, Raj M.
Chapter5: Analysis Of Convection Heat Transfer
Section: Chapter Questions
Problem 5.6P: 5.6 A fluid flows at 5 over a wide, flat plate 15 cm long. For each from the following list,...
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