mech 221 fluid mechanics (fall 06/07) tutorial 9
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MECH 221 FLUID MECHANICS (Fall 06/07) Tutorial 9. Example 1. - PowerPoint PPT PresentationTRANSCRIPT
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MECH 221 FLUID MECHANICS(Fall 06/07)Tutorial 9
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Example 1
A viscous fluid flows past a flat plate such that the boundary layer thickness at a distance 1.3m from the leading edge is 12mm. Determine the boundary layer thickness at distances of 0.2m, 2.0m and 20m from the leading edge. Assume laminar flow.
If the upstream velocity of the flow is 1.5m/s, determine the kinematic viscosity of the fluid.
3
Example 1
From similarity solution and normalization analysis,
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5247.10
5247.10
)3.1(12
12,3.1
,
~
x
C
C
xgiven
Cxor
x
X (m) δ (mm)
0.2 4.707
2.0 14.884
20 47.068
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Example 1
From Eq. 9.15,
smx
U
mmmmxm/s.UGiven
x
U
U
x
/10646.6)3.1(25
)012.0)(5.1(
25
012.012,3.1 ,51
25
5
2622
2
5
Example 2
Water flows past a flat plate with an upstream velocity of U=0.02m/s. Determine the water velocity a distance of 10mm from the plate a distances of x=1.5m and x=15m from the leading edge.
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Example 2
flowlaminar ReRe
10510975.1)10519.1(
)15)(02.0(Re ,15
Re ,
)( where),('
plate,flat aon flowlayer boundary for Solution Blasius From
crx
556x
x
21
mxfor
UxSince
x
Uyf
U
u
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Example 2
smuu
U
uf
f
f
f
/006178.03089.002.0
)('
3089.0)(' ,937.0
ion,interpolatBy
3938.0)(' ,2.1
2647.0)(' ,8.0
9.1 TableIn
937.0))5.1)(10519.1(
)02.0()(01.0()
x
Uy(
0.01m,10mmy 1.5m,At x
applicabe. is 9.1 Table
plate,flat aon flowlayer boundary laminar for Solution Blasius From
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smuu
U
uf
f
f
f
/001965.00983.002.0
)('
0983.0)(' ,296.0
ion,interpolatBy
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0)(' ,0
9.1 TableIn
296.0))15)(10519.1(
)02.0()(01.0()
x
Uy(
0.01m,10mmy 15m,At x
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Example 3
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 the figure, 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.
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Example 3
mU
x
UL
B
B
0382.0)1(
)4)(101.46(55
i.e. L,at x hicknessBoundary t
flowlaminar 1074.2)101.46(
)4)(1(Re
/sm101.46 4m;L 1m/s;given U
Re
flow, theof Re
5-
5-5-L
25-
L
x
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Example 3
my
UyU
m
U
xwhere
UyU
BA
BA
BA
0251.001315.00382.0
)())((
01315.0)1(
)4)(1046.1(721.1
k)in textboo 9.16 (Eq. 721.1
)())((
B,-A streamlineor plate he through tflow no is thereSince
.amount an by displaced plate with the velocity uniform aby carried
that toequal definitionby islayer boundary actual by the carried flowrate The
*
*
5*
*
*
BA
*
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Example 3
xy
x
U
x
3
5
A*
A
*A
*A
A
1058.60251.0
)1(
)1046.1(721.10251.0y
721.1yyy
-yy
)-U(y)(U)(y
direction,-any xFor
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Example 4
Fluid flows past a triangular flat plate oriented parallel to the free stream as shown in the figure. Integrate the wall shear stress over the plate to determine the friction drag on one side of the plate. Assume laminar boundary layer flow.
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Example 4
xU
dA
xy
w
w
23
332.0 where
2 D dragFriction
5.00 ;5.0x0 :region Area
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Example 4
N
smUNs
xxU
dxx
xU
dydxx
U
dydxx
U
dA
x
x
x
x
xy
y
x
x
xy
y
w
0296.0D
))5.0(3
2()5.0)(2(5.0)1012.1)(999()2.0(664.0 D
2.0;m
1012.1;m
kg999 1.6, Table From
)3
2())(2(5.0664.0 D
5.0664.0 D
1664.0 D
332.02 D
2 D dragFriction
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3
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3
5.0
0
5.0
0
5.0
0
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