External Flows - Faculty of Engineering and Applied Science · 2 External Flows • Considers...
Transcript of External Flows - Faculty of Engineering and Applied Science · 2 External Flows • Considers...
![Page 1: External Flows - Faculty of Engineering and Applied Science · 2 External Flows • Considers bodies fully immersed in a fluid stream. • In external flows we are concerned with](https://reader031.fdocuments.net/reader031/viewer/2022020303/5b4713347f8b9a824f8bb178/html5/thumbnails/1.jpg)
1
External Flows
![Page 2: External Flows - Faculty of Engineering and Applied Science · 2 External Flows • Considers bodies fully immersed in a fluid stream. • In external flows we are concerned with](https://reader031.fdocuments.net/reader031/viewer/2022020303/5b4713347f8b9a824f8bb178/html5/thumbnails/2.jpg)
2
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
![Page 3: External Flows - Faculty of Engineering and Applied Science · 2 External Flows • Considers bodies fully immersed in a fluid stream. • In external flows we are concerned with](https://reader031.fdocuments.net/reader031/viewer/2022020303/5b4713347f8b9a824f8bb178/html5/thumbnails/3.jpg)
External Flows • Consider the following system:
3
ENGR 5961 Fluid Mechanics I: Dr. Y.S. Muzychka
![Page 4: External Flows - Faculty of Engineering and Applied Science · 2 External Flows • Considers bodies fully immersed in a fluid stream. • In external flows we are concerned with](https://reader031.fdocuments.net/reader031/viewer/2022020303/5b4713347f8b9a824f8bb178/html5/thumbnails/4.jpg)
4
External Flows
ENGR 5961 Fluid Mechanics I: Dr. Y.S. Muzychka
![Page 5: External Flows - Faculty of Engineering and Applied Science · 2 External Flows • Considers bodies fully immersed in a fluid stream. • In external flows we are concerned with](https://reader031.fdocuments.net/reader031/viewer/2022020303/5b4713347f8b9a824f8bb178/html5/thumbnails/5.jpg)
External Flows 5
• Pressure distribution around a car. Drag is the integrated force opposing motion.
ENGR 5961 Fluid Mechanics I: Dr. Y.S. Muzychka
![Page 6: External Flows - Faculty of Engineering and Applied Science · 2 External Flows • Considers bodies fully immersed in a fluid stream. • In external flows we are concerned with](https://reader031.fdocuments.net/reader031/viewer/2022020303/5b4713347f8b9a824f8bb178/html5/thumbnails/6.jpg)
6
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
![Page 7: External Flows - Faculty of Engineering and Applied Science · 2 External Flows • Considers bodies fully immersed in a fluid stream. • In external flows we are concerned with](https://reader031.fdocuments.net/reader031/viewer/2022020303/5b4713347f8b9a824f8bb178/html5/thumbnails/7.jpg)
7
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
![Page 8: External Flows - Faculty of Engineering and Applied Science · 2 External Flows • Considers bodies fully immersed in a fluid stream. • In external flows we are concerned with](https://reader031.fdocuments.net/reader031/viewer/2022020303/5b4713347f8b9a824f8bb178/html5/thumbnails/8.jpg)
8
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
![Page 9: External Flows - Faculty of Engineering and Applied Science · 2 External Flows • Considers bodies fully immersed in a fluid stream. • In external flows we are concerned with](https://reader031.fdocuments.net/reader031/viewer/2022020303/5b4713347f8b9a824f8bb178/html5/thumbnails/9.jpg)
9
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
![Page 10: External Flows - Faculty of Engineering and Applied Science · 2 External Flows • Considers bodies fully immersed in a fluid stream. • In external flows we are concerned with](https://reader031.fdocuments.net/reader031/viewer/2022020303/5b4713347f8b9a824f8bb178/html5/thumbnails/10.jpg)
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
10
€
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
![Page 11: External Flows - Faculty of Engineering and Applied Science · 2 External Flows • Considers bodies fully immersed in a fluid stream. • In external flows we are concerned with](https://reader031.fdocuments.net/reader031/viewer/2022020303/5b4713347f8b9a824f8bb178/html5/thumbnails/11.jpg)
11
External Flows
ENGR 5961 Fluid Mechanics I: Dr. Y.S. Muzychka
![Page 12: External Flows - Faculty of Engineering and Applied Science · 2 External Flows • Considers bodies fully immersed in a fluid stream. • In external flows we are concerned with](https://reader031.fdocuments.net/reader031/viewer/2022020303/5b4713347f8b9a824f8bb178/html5/thumbnails/12.jpg)
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:
12
€
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
![Page 13: External Flows - Faculty of Engineering and Applied Science · 2 External Flows • Considers bodies fully immersed in a fluid stream. • In external flows we are concerned with](https://reader031.fdocuments.net/reader031/viewer/2022020303/5b4713347f8b9a824f8bb178/html5/thumbnails/13.jpg)
13
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
![Page 14: External Flows - Faculty of Engineering and Applied Science · 2 External Flows • Considers bodies fully immersed in a fluid stream. • In external flows we are concerned with](https://reader031.fdocuments.net/reader031/viewer/2022020303/5b4713347f8b9a824f8bb178/html5/thumbnails/14.jpg)
External Flows • Effect of Turbulence
14
ENGR 5961 Fluid Mechanics I: Dr. Y.S. Muzychka
![Page 15: External Flows - Faculty of Engineering and Applied Science · 2 External Flows • Considers bodies fully immersed in a fluid stream. • In external flows we are concerned with](https://reader031.fdocuments.net/reader031/viewer/2022020303/5b4713347f8b9a824f8bb178/html5/thumbnails/15.jpg)
External Flows • Turbulent Skin Friction:
– Fully turbulent boundary layer (x > 0)
– Initially laminar and proceeding to turbulent (x >0), see slide #6.
15
€
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
![Page 16: External Flows - Faculty of Engineering and Applied Science · 2 External Flows • Considers bodies fully immersed in a fluid stream. • In external flows we are concerned with](https://reader031.fdocuments.net/reader031/viewer/2022020303/5b4713347f8b9a824f8bb178/html5/thumbnails/16.jpg)
External Flows • Flat Plate Skin Friction
16
ENGR 5961 Fluid Mechanics I: Dr. Y.S. Muzychka
![Page 17: External Flows - Faculty of Engineering and Applied Science · 2 External Flows • Considers bodies fully immersed in a fluid stream. • In external flows we are concerned with](https://reader031.fdocuments.net/reader031/viewer/2022020303/5b4713347f8b9a824f8bb178/html5/thumbnails/17.jpg)
17
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
![Page 18: External Flows - Faculty of Engineering and Applied Science · 2 External Flows • Considers bodies fully immersed in a fluid stream. • In external flows we are concerned with](https://reader031.fdocuments.net/reader031/viewer/2022020303/5b4713347f8b9a824f8bb178/html5/thumbnails/18.jpg)
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.
18
€
CD =FD /A12 ρU
2
ENGR 5961 Fluid Mechanics I: Dr. Y.S. Muzychka
![Page 19: External Flows - Faculty of Engineering and Applied Science · 2 External Flows • Considers bodies fully immersed in a fluid stream. • In external flows we are concerned with](https://reader031.fdocuments.net/reader031/viewer/2022020303/5b4713347f8b9a824f8bb178/html5/thumbnails/19.jpg)
19
External Flows • Sphere
€
CD =24ReD
+6
1+ ReD+ 0.4 0.01 < ReD < 250,000
ENGR 5961 Fluid Mechanics I: Dr. Y.S. Muzychka
![Page 20: External Flows - Faculty of Engineering and Applied Science · 2 External Flows • Considers bodies fully immersed in a fluid stream. • In external flows we are concerned with](https://reader031.fdocuments.net/reader031/viewer/2022020303/5b4713347f8b9a824f8bb178/html5/thumbnails/20.jpg)
20
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
![Page 21: External Flows - Faculty of Engineering and Applied Science · 2 External Flows • Considers bodies fully immersed in a fluid stream. • In external flows we are concerned with](https://reader031.fdocuments.net/reader031/viewer/2022020303/5b4713347f8b9a824f8bb178/html5/thumbnails/21.jpg)
21
External Flows • Cylinder
€
CD =10ReD
2 / 3 +1 0.1 < ReD < 250,000
ENGR 5961 Fluid Mechanics I: Dr. Y.S. Muzychka
![Page 22: External Flows - Faculty of Engineering and Applied Science · 2 External Flows • Considers bodies fully immersed in a fluid stream. • In external flows we are concerned with](https://reader031.fdocuments.net/reader031/viewer/2022020303/5b4713347f8b9a824f8bb178/html5/thumbnails/22.jpg)
22
External Flows • Plate Normal to Flow
ENGR 5961 Fluid Mechanics I: Dr. Y.S. Muzychka
![Page 23: External Flows - Faculty of Engineering and Applied Science · 2 External Flows • Considers bodies fully immersed in a fluid stream. • In external flows we are concerned with](https://reader031.fdocuments.net/reader031/viewer/2022020303/5b4713347f8b9a824f8bb178/html5/thumbnails/23.jpg)
23
External Flows
ENGR 5961 Fluid Mechanics I: Dr. Y.S. Muzychka
![Page 24: External Flows - Faculty of Engineering and Applied Science · 2 External Flows • Considers bodies fully immersed in a fluid stream. • In external flows we are concerned with](https://reader031.fdocuments.net/reader031/viewer/2022020303/5b4713347f8b9a824f8bb178/html5/thumbnails/24.jpg)
24
External Flows • Streamlining: the trade off between
pressure (form) drag and skin friction:
ENGR 5961 Fluid Mechanics I: Dr. Y.S. Muzychka
![Page 25: External Flows - Faculty of Engineering and Applied Science · 2 External Flows • Considers bodies fully immersed in a fluid stream. • In external flows we are concerned with](https://reader031.fdocuments.net/reader031/viewer/2022020303/5b4713347f8b9a824f8bb178/html5/thumbnails/25.jpg)
25
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