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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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ENGR 5961 Fluid Mechanics I: Dr. Y.S. Muzychka
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External Flows
ENGR 5961 Fluid Mechanics I: Dr. Y.S. Muzychka
External Flows 5
• Pressure distribution around a car. Drag is the integrated force opposing motion.
ENGR 5961 Fluid Mechanics I: Dr. Y.S. Muzychka
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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.
ENGR 5961 Fluid Mechanics I: Dr. Y.S. Muzychka
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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
ENGR 5961 Fluid Mechanics I: Dr. Y.S. Muzychka
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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:
ENGR 5961 Fluid Mechanics I: Dr. Y.S. Muzychka
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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
∫
€
θ =uU1− u
U⎛
⎝ ⎜
⎞
⎠ ⎟ dy ≈
0
∞
∫ uU1− u
U⎛
⎝ ⎜
⎞
⎠ ⎟ dy
0
δ
∫
ENGR 5961 Fluid Mechanics I: Dr. Y.S. Muzychka
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)
€
τwρ
=ddx
U 2θ( ) =U 2 ddx
uU1− u
U⎛
⎝ ⎜
⎞
⎠ ⎟ dy
0
δ
∫ U = constant
ENGR 5961 Fluid Mechanics I: Dr. Y.S. Muzychka
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External Flows
ENGR 5961 Fluid Mechanics I: Dr. Y.S. Muzychka
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
€
δx
=5Rex
Rex =ρUx
µ
€
C f =τ w12ρU 2
=1.328ReL
ReL =ρUL
µ
ENGR 5961 Fluid Mechanics I: Dr. Y.S. Muzychka
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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.
ENGR 5961 Fluid Mechanics I: Dr. Y.S. Muzychka
External Flows • Effect of Turbulence
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ENGR 5961 Fluid Mechanics I: Dr. Y.S. Muzychka
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
€
C f =τ w12ρU 2
=0.0742ReL
1/ 5 ReL > 500,000
€
C f =τ w12ρU 2
=0.0742ReL
1/ 5 −1740ReL
ReL > 500,000
€
δx≈0.38Rex
1/ 5
ENGR 5961 Fluid Mechanics I: Dr. Y.S. Muzychka
External Flows • Flat Plate Skin Friction
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ENGR 5961 Fluid Mechanics I: Dr. Y.S. Muzychka
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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.
ENGR 5961 Fluid Mechanics I: Dr. Y.S. Muzychka
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
ENGR 5961 Fluid Mechanics I: Dr. Y.S. Muzychka
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External Flows • Sphere
€
CD =24ReD
+6
1+ ReD+ 0.4 0.01 < ReD < 250,000
ENGR 5961 Fluid Mechanics I: Dr. Y.S. Muzychka
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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.
ENGR 5961 Fluid Mechanics I: Dr. Y.S. Muzychka
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External Flows • Cylinder
€
CD =10ReD
2 / 3 +1 0.1 < ReD < 250,000
ENGR 5961 Fluid Mechanics I: Dr. Y.S. Muzychka
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External Flows • Plate Normal to Flow
ENGR 5961 Fluid Mechanics I: Dr. Y.S. Muzychka
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External Flows
ENGR 5961 Fluid Mechanics I: Dr. Y.S. Muzychka
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External Flows • Streamlining: the trade off between
pressure (form) drag and skin friction:
ENGR 5961 Fluid Mechanics I: Dr. Y.S. Muzychka
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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.
ENGR 5961 Fluid Mechanics I: Dr. Y.S. Muzychka