External Flows - Faculty of Engineering and Applied Science · 2 External Flows • Considers...

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External Flows

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External Flows •  Considers bodies fully immersed in a fluid

stream. •  In external flows we are concerned with fluid

drag forces and lift forces. •  Analysis of lift and drag is usually facilitated by

defining lift and drag coefficients. •  These are often experimentally obtained, but

some laminar flow results are theoretical. •  These are used in momentum (force) balances.

ENGR 5961 Fluid Mechanics I: Dr. Y.S. Muzychka

External Flows •  Consider the following system:

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External Flows


External Flows 5

•  Pressure distribution around a car. Drag is the integrated force opposing motion.


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External Flows •  Boundary Layers: are thin fluid layers

attached to the surface of a body where changes in the velocity are greatest.


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External Flows •  The boundary layer is a layer of fluid with

variable thickness:

•  The shear stress is approximately obtained from:

€

τ(x) = µ∂u∂y wall

~ µUδ


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External Flows •  The flat plate immersed in a fluid stream is the

most fundamental problem of fluid mechanics. •  It can be analyzed using a control volume mass

and momentum balance:


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External Flows •  Boundary Layer Characteristics:

–  Displacement thickness –  Boundary layer thickness, δ –  Momentum thickness

€

δ* = 1− uU

⎛

⎝ ⎜

⎞

⎠ ⎟ dy ≈

0

∞

∫ 1− uU

⎛

⎝ ⎜

⎞

⎠ ⎟ dy

0

δ

∫

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θ =uU1− u

U⎛

⎝ ⎜

⎞

⎠ ⎟ dy ≈

0

∞

∫ uU1− u

U⎛

⎝ ⎜

⎞

⎠ ⎟ dy

0

δ

∫


External Flows •  Momentum Integral Equation:

–  Derived for a boundary layer with free stream velocity U(x)

–  When U = constant (parallel flow)

–  We assume profiles and solve approximately

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€

τwρ

=ddx

U 2θ( ) +δ *U dUdx

U =U(x)

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τwρ

=ddx

U 2θ( ) =U 2 ddx

uU1− u

U⎛

⎝ ⎜

⎞

⎠ ⎟ dy

0

δ

∫ U = constant


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External Flows


External Flows •  The drag coefficient is defined as (Rex >1000):

•  These are the exact solutions from Table 9.2. •  The mean drag (to get total drag) is obtained

by integrating the shear stress over the plate:

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Cf ,x =τx

12ρU 2

=0.664Rex

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δx

=5Rex

Rex =ρUx

µ

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C f =τ w12ρU 2

=1.328ReL

ReL =ρUL

µ


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Example - 31 •  A plate is of dimensions L x 5L is oriented

in a parallel stream in both the long direction and short direction. Determine the ratio of the total drag force assuming laminar flow.


External Flows •  Effect of Turbulence

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External Flows •  Turbulent Skin Friction:

–  Fully turbulent boundary layer (x > 0)

–  Initially laminar and proceeding to turbulent (x >0), see slide #6.

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Cf ,x =τx

12ρU 2

=0.0594Rex

1/ 5 Rex > 500,000

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C f =τ w12ρU 2

=0.0742ReL

1/ 5 ReL > 500,000

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C f =τ w12ρU 2

=0.0742ReL

1/ 5 −1740ReL

ReL > 500,000

€

δx≈0.38Rex

1/ 5


External Flows •  Flat Plate Skin Friction

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Example - 32 •  Compute the drag on the keel of a sail

boat moving through water at 0.5 m/s. The keel has a height H = 200 cm, and variable width. The width at the free end is 75 cm, while the width at its base (attached to the boat) is 150 cm. Solve assuming a mean width. Discuss how the exact solution is obtained using integration.


External Flows •  Drag in general consists of two components:

–  Skin Friction (shear stress) –  Profile or Form Drag (pressure forces)

•  We analyze total drag using a drag coefficient CD defined as:

–  A is usually the profile area (projected shadow area) •  Drag coefficients depend on Reynolds number,

surface roughness, and Mach number.

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CD =FD /A12 ρU

2


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External Flows •  Sphere

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CD =24ReD

+6

1+ ReD+ 0.4 0.01 < ReD < 250,000


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Example - 33 •  Compute the terminal velocity of a falling

sphere of 10 mm diameter, if only gravity, buoyancy, and friction are the dominant forces. Perform you calculations for air, water, and oil.


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External Flows •  Cylinder

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CD =10ReD

2 / 3 +1 0.1 < ReD < 250,000


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External Flows •  Plate Normal to Flow


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External Flows


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External Flows •  Streamlining: the trade off between

pressure (form) drag and skin friction:


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Example - 34 •  Find the drag force and bending moment

which are exerted on a large flag pole in a 50 km/hr wind, if the pole, spherical cap, and flag have the following dimensions: Dpole = 15 cm, Lpole = 15 m, Dcap = 25 cm, Wflag = 11 m, Hflag = 4.5 m. Make a sketch and label the system accordingly with all forces.


External Flows - Faculty of Engineering and Applied Science · 2 External Flows • Considers...

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