(a) If the boundary layer velocity profile for y ≤ δ is given by where U is the velocity at a distance y from the surface, δ is the boundary layer thickness and Ue is the freestream velocity. (i) Find the ratio of the displacement thickness to the boundary layer thickness (this is a number). (ii) Find the ratio of the momentum thickness to the boundary layer thickness (this is a number). (b) Air enters a two-dimensional duct with a uniform velocity profile. As the boundary layers on the top and bottom walls grow with downstream distance, the velocity in the freestream tends to increase. However, if the walls diverged with downstream distance so that the freestream velocity remained constant, express the angle of divergence of the walls in terms of the boundary layer displacement thickness δ∗ and the distance along the duct x. (c) A laminar boundary layer is observed to grow on a flat plate of width w and length x = L such that the pressure is constant everywhere. If the skin friction coefficient Cf is given by: Here F is the total frictional force acting on one side of the plate, ρ is the fluid density, and Ue is the freestream velocity.
(a) If the boundary layer velocity profile for y ≤ δ is given by
where U is the velocity at a distance y from the surface, δ is the boundary layer thickness and Ue is the freestream velocity.
(i) Find the ratio of the displacement thickness to the boundary layer thickness (this is a number).
(ii) Find the ratio of the momentum thickness to the boundary layer thickness (this is a number).
(b) Air enters a two-dimensional duct with a uniform velocity profile. As the boundary layers on the top and bottom walls grow with downstream distance, the velocity in the freestream tends to increase. However, if the walls diverged with downstream distance so that the freestream velocity remained constant, express the angle of divergence of the walls in terms of the boundary layer displacement thickness δ∗ and the distance along the duct x.
(c) A laminar boundary layer is observed to grow on a flat plate of width w and length x = L such that the pressure is constant everywhere. If the skin friction coefficient Cf is given by:
Here F is the total frictional force acting on one side of the plate, ρ is the fluid density, and Ue is the freestream velocity.
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