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NATIONAL UNIVERSITY OF SINGAPORE Department of Mechanical Engineering

ME2135 Fluid Mechanics II

Part 2 External Incompressible Viscous Flow

Tutorial 1 1. The average pressure and shear stress acting on the surface of the 1-m-square flat plate are as indicated in Fig. 1. Determine the lift and drag generated. Determine the lift and drag if the shear stress is neglected. Compare these two sets of results.

Fig. 1 Flow past an inclined flat plate; from [1]

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2. An atmospheric boundary layer is formed when the wind blows over the earth’s surface. Typically, such velocity profiles can be written as a power law: u = ayn, where the constants a and n depend on the roughness of the terrain. As is indicated in Fig. 2, typical values are n = 0.40 for urban areas, n = 0.28 for woodland or suburban areas, and n = 0.16 for flat open country. (a) If the velocity is 6 m/s at the bottom of the sail on your boat (y = 1.2 m), what is the velocity at the top of the mast (y = 9 m)? (b) If the average velocity is 16 km/h on the tenth floor of an urban building, what is the average velocity on the sixtieth floor?

Fig. 2 Atmospheric boundary layer flow over different terrains; from [1]

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3. The velocity profile in a laminar boundary layer is approximated by a sinusoidal function (Fig. 3):

sin2

u yU

πδ

=

Does this expression satisfy the boundary conditions applicable to the laminar boundary-layer velocity profile? Evaluate the non-dimensional displacement thickness δ*/δ

Fig 3. Laminar boundary layer velocity profiles; from [2]

Applicable boundary conditions are:

= 0 0 (no slip)

(continuity with freestream)

0

(no shear stress at freestream)

U at y

u U at y

du at ydy

δ

δ

=

= =

= =

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4. Air enters a square duct through a 0.3 m opening as is shown in Fig. 4. Because the boundary layer displacement thickness increases in the direction of flow, it is necessary to increase the cross-sectional size of the duct if a constant U = 0.6 m/s velocity is to be maintained outside the boundary layer. Plot a graph of the duct size, d, as a function of x for 0 ≤ x ≤ 3 m if U is to remain constant. Assume laminar flow with boundary layer displacement thickness δ* = 1.721 (νx/U)1/2, where kinematic viscosity of air is ν = 1.5 x 10-3 m2/s

Fig. 4 Flow through a square duct; adapted from [1]

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5. Because of the velocity deficit, U - u, in the boundary layer, the streamlines for flow past a flat plate are not exactly parallel to the plate. This deviation can be determined by use of the displacement thickness, δ*. For air blowing past the flat plate shown in Fig. 5, plot the streamline A - B that passes through the edge of the boundary layer (y = δB at x = ℓ ) at point B. That is, plot y = y(x) for streamline A - B. Assume laminar boundary layer flow with boundary layer thickness δ = 5 (νx/U)1/2 and

displacement thickness δ* = 1.721 (νx/U)1/2, where kinematic viscosity of air is ν = 1.5 x 10-3 m2/s.

Fig. 5 Boundary layer flow past a flat plate; from [1]