2 Cylinders, Spheres, and Other Bluff Shapes
Tải bản đầy đủ - 0trang
67706_07_ch07_p420-483.qxd
5/14/10
12:30 PM
Page 423
7.2 Cylinders, Spheres, and Other Bluff Shapes
FIGURE 7.3 Photographs of air flowing over a sphere. In the lower
picture a “tripping” wire induced early transition and delayed
separation.
Source: Courtesy of L. Prandtl and the Journal of the Royal Aeronautical Society.
U∞
θ
FIGURE 7.4 Streamlines for potential
flow over a circular cylinder.
Copyright 2011 Cengage Learning, Inc. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part.
423
67706_07_ch07_p420-483.qxd
12:30 PM
Page 424
Chapter 7 Forced Convection Over Exterior Surfaces
Pressure distribution
Theoretical
Supercritical
Subcritical
Cylinder diameter d = 25.0 cm
Resupercritical = 6.7 × 105
Resubcritical = 1.86 × 105
1.0
p
0
ρU2∞ /2gc
424
5/14/10
–1.0
–2.0
–3.0
0
30
60
90
120
150
180
210
240
270
300
330 360
θ
FIGURE 7.5 Pressure distribution around a circular cylinder in crossflow at various Reynolds numbers; p is the local pressure, U2q /2gc is
the free-stream impact pressure; is the angle measured from the
stagnation point.
Source: By permission from L. Flachsbart, Handbuch der Experimental Physik, Vol. 4, part 2.
Since the pressure distribution is symmetric about the vertical center plane of the
cylinder, it is clear that there will be no pressure drag in irrotational flow. However,
unless the Reynolds number is very low, a real fluid will not adhere to the entire surface of the cylinder, but as mentioned previously, the boundary layer in which the
flow is not irrotational will separate from the sides of the cylinder as a result of the
adverse pressure gradient. The separation of the boundary layer and the resultant
wake in the rear of the cylinder give rise to pressure distributions that are shown for
different Reynolds numbers by the dashed lines in Fig. 7.5. It can be seen that there
is fair agreement between the ideal and the actual pressure distribution in the neighborhood of the forward stagnation point. In the rear of the cylinder, however, the
actual and ideal distributions differ considerably. The characteristics of the flow pattern and of the boundary layer depend on the Reynolds number, U q D> , which
for flow over a cylinder or a sphere is based on the velocity of the oncoming free
stream U q and the outside diameter of the body D. Properties are evaluated at freestream conditions. The flow pattern around the cylinder undergoes a series of
changes as the Reynolds number is increased, and since the heat transfer depends
largely on the flow, we shall consider first the effect of the Reynolds number on the
flow and then interpret the heat transfer data in the light of this information.
The sketches in Fig. 7.6 illustrate flow patterns typical of the characteristic
ranges of Reynolds numbers. The letters in Fig. 7.6 correspond to the flow regimes
indicated in Fig. 7.7, where the total dimensionless drag coefficients of a cylinder
and a sphere, CD, are plotted as a function of the Reynolds number. The force term
Copyright 2011 Cengage Learning, Inc. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part.
67706_07_ch07_p420-483.qxd
5/14/10
12:30 PM
Page 425
7.2 Cylinders, Spheres, and Other Bluff Shapes
ReD < 1.0
ReD = 10
(a)
(b)
Turbulent
eddies wake
Vortex street
ReD = 100
Laminar
boundary layer
103 < ReD < 105
(c)
(d)
Small
turbulent wake
Laminar
boundary layer
Turbulent
boundry layer
ReD > 105
(e)
FIGURE 7.6 Flow patterns for cross-flow over a cylinder at
various Reynolds numbers.
100
80
60
40
a
b
c
d
e
CD
20
10
8
6
4
2
Cylinders
1
0.8
0.6
0.4
Spheres
0.2
0.1
0.1 0.2 0.5 1 2
5 10 20 50 102
103
104
105
106
ReD
FIGURE 7.7 Drag coefficient versus Reynolds number for
long circular cylinders and spheres in cross-flow.
Copyright 2011 Cengage Learning, Inc. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part.
425
67706_07_ch07_p420-483.qxd
426
5/14/10
12:30 PM
Page 426
Chapter 7 Forced Convection Over Exterior Surfaces
in the total drag coefficient is the sum of the pressure and frictional forces; it is
defined by the equation
CD =
where
drag force
Af (rU2q >2gc)
ϭ free-stream density
U q ϭ free-stream velocity
Af ϭ frontal projected area ϭ DL (cylinder) or D2> 4 (sphere)
D ϭ outside cylinder diameter, or diameter of sphere
L ϭ cylinder length
The following discussion strictly applies only to long cylinders, but it also gives a qualitative picture of the flow past a sphere. The letters (a) to (e) refer to Figs. 7.6 and 7.7.
(a) At Reynolds numbers of the order of unity or less, the flow adheres to the surface and the streamlines follow those predicted from potential-flow theory. The
inertia forces are negligibly small, and the drag is caused only by viscous forces,
since there is no flow separation. Heat is transferred by conduction alone.
(b) At Reynolds numbers of the order of 10, the inertia forces become appreciable
and two weak eddies stand in the rear of the cylinder. The pressure drag accounts
now for about half of the total drag.
(c) At a Reynolds number of the order of 100, vortices separate alternately from the
two sides of the cylinder and stretch a considerable distance downstream. These
vortices are referred to as von Karman vortex streets in honor of the scientist
Theodore von Karman, who studied the shedding of vortices from bluff objects.
The pressure drag now predominates.
(d) In the Reynolds number range between 103 and 105, the skin friction drag
becomes negligible compared to the pressure drag caused by turbulent eddies in
the wake. The drag coefficient remains approximately constant because the
boundary layer remains laminar from the leading edge to the point of separation, which lies throughout this Reynolds number range at an angular position
between 80° and 85° measured from the direction of the flow.
(e) At Reynolds numbers larger than about 105 (the exact value depends on the turbulence level of the free stream) the kinetic energy of the fluid in the laminar boundary layer over the forward part of the cylinder is sufficient to overcome the
unfavorable pressure gradient without separating. The flow in the boundary layer
becomes turbulent while it is still attached, and the separation point moves toward
the rear. The closing of the streamlines reduces the size of the wake, and the pressure drag is therefore also substantially reduced. Experiments by Fage and Falkner
[1, 2] indicate that once the boundary layer has become turbulent, it will not separate before it reaches an angular position corresponding to a of about 130°.
Analyses of the boundary layer growth and the variation of the local heat transfer coefficient with angular position around circular cylinders and spheres have been
only partially successful. Squire [3] has solved the equations of motion and energy
for a cylinder at constant temperature in cross-flow over that portion of the surface
to which a laminar boundary layer adheres. He showed that at the stagnation point
Copyright 2011 Cengage Learning, Inc. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part.
67706_07_ch07_p420-483.qxd
5/14/10
12:30 PM
Page 427
7.2 Cylinders, Spheres, and Other Bluff Shapes
427
and in its immediate neighborhood, the convection heat transfer coefficient can be
calculated from the equation
NuD =
hc D
rUq D
= C
k
C m
(7.1)
where C is a constant whose numerical value at various Prandtl numbers is tabulated
below:
Pr
C
0.7
1.0
0.8
1.05
1.0
1.14
5.0
2.1
10.0
1.7
Over the forward portion of the cylinder (0 Ͻ Ͻ 80°), the empirical equation for
hc(), the local value of the heat transfer coefficient at
Nu(u) =
hc1u2D
k
= 1.14a
rUq D 0.5 0.4
u 3
b Pr c1 - a
b d
m
90
(7.2)
has been found to agree satisfactorily [4] with experimental data.
Giedt [5] has measured the local pressures and the local heat transfer coefficients over the entire circumference of a long, 10.2-cm-OD cylinder in an airstream
over a Reynolds number range from 70,000 to 220,000. Giedt’s results are shown in
Fig. 7.8, and similar data for lower Reynolds numbers are shown in Fig. 7.9 (tooth
figures are shown on the next page). If the data shown in Figs. 7.8 and 7.9 are compared at corresponding Reynolds numbers with the flow patterns and the boundary
layer characteristics described earlier, some important observations can be made.
At Reynolds numbers below 100,000, separation of the laminar boundary layer
occurs at an angular position of about 80°. The heat transfer and the flow characteristics over the forward portion of the cylinder resemble those for laminar flow over a flat
plate, which were discussed earlier. The local heat transfer is largest at the stagnation
point and decreases with distance along the surface as the boundary layer thickness
increases. The heat transfer reaches a minimum on the sides of the cylinder near the
separation point. Beyond the separation point, the local heat transfer increases because
considerable turbulence exists over the rear portion of the cylinder, where the eddies
of the wake sweep the surface. However, the heat transfer coefficient over the rear is
no larger than that over the front because the eddies recirculate part of the fluid and,
despite their high turbulence, are not as effective as a turbulent boundary layer in mixing the fluid in the vicinity of the surface with the fluid in the main stream.
At Reynolds numbers large enough to permit transition from laminar to turbulent
flow in the boundary layer without separation of the laminar boundary layer, the heat
transfer coefficient has two minima around the cylinder. The first minimum occurs at the
point of transition. As the transition from laminar to turbulent flow progresses, the heat
transfer coefficient increases and reaches a maximum approximately at the point where
the boundary layer becomes fully turbulent. Then the heat transfer coefficient begins to
decrease again and reaches a second minimum at about 130°, the point at which the turbulent boundary layer separates from the cylinder. Over the rear of the cylinder, the heat
transfer coefficient increases to another maximum at the rear stagnation point.
Copyright 2011 Cengage Learning, Inc. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part.
67706_07_ch07_p420-483.qxd
5/14/10
12:30 PM
Page 428
800
700
600
ReD219,000
500
186,000
Nu(θ)
170,000
400
140,000
101,300
300
70,800
200
100
40
80
120
160
θ — Degrees from stagnation point
0
FIGURE 7.8 Circumferential variation of the dimensionless
heat transfer coefficient (Nu) at high Reynolds numbers
for a circular cylinder in cross-flow.
Source: Courtesy of W. H. Giedt, “Investigation of Variation of Point
Unit-Heat-Transfer Coeffient around a Cylinder Normal to an Air Stream”,
Trans. ASME, Vol. 71, 1949, pp. 375–381. Reprinted by permission of
The American Society of Mechanical Engineers International.
Direction of flow
θ
20,
0
50 100
0
0
0,0
=5
500
Re
00
4,0
Nu (scale)
Nuθ
FIGURE 7.9 Circumferential variation of the local
Nusselt number Nu() ϭ hc()Do/kf at low Reynolds
numbers for a circular cylinder in cross-flow.
Source: According to W. Lorisch, from M. ten Bosch,
Die Wärmeübertragung, 3d ed., Springer Verlag, Berlin, 1936.
428
Copyright 2011 Cengage Learning, Inc. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part.
67706_07_ch07_p420-483.qxd
5/14/10
12:31 PM
Page 429
7.2 Cylinders, Spheres, and Other Bluff Shapes
429
EXAMPLE 7.1
To design a heating system for the purpose of preventing ice formation on an aircraft
wing, it is necessary to know the heat transfer coefficient over the outer surface of
the leading edge. The leading-edge contour can be approximated by a half-cylinder
of 30-cm diameter, as shown in Fig. 7.10. The ambient air is at Ϫ34°C, and the surface temperature is to be no less than 0°C. The plane is designed to fly at 7500-m
altitude at a speed of 150 m/s. Calculate the distribution of the convection heat transfer coefficient over the forward portion of the wing.
SOLUTION
At an altitude of 7500 m the standard atmospheric air pressure is 38.9 kPa and the
density of the air is 0.566 kg/m3 (see Table 38 in Appendix 2).
The heat transfer coefficient at the stagnation point ( ϭ 0) is, according
to Eq. (7.2),
rUq D 0.5 0.4 k
b Pr
m
D
3
(0.566 kg/m ) * (150 m/s) * (0.30 m) 0.5
0.024 W/m K
= (1.14) a
b (0.72)0.4 a
b
-5
0.30 m
1.74 * 10 kg/m s
hc(u = 0) = 1.14a
ϭ 96.7 W/m2 °C
The variation of hc with is obtained by multipling the value of the heat transfer
coefficient at the stagnation point by 1 Ϫ (> 90)3. The results are tabulated below.
(deg)
hc()(W/m2 °C)
0
96.7
15
96.3
30
93.1
45
84.6
NAL
60
68.0
75
40.7
AIR
O
ATI
ERN
INT
Air
–34°C
150 m/s
30 cm
Leading edge
FIGURE 7.10 Approximation of the leading edge of an aircraft
wing for Example 7.1.
Copyright 2011 Cengage Learning, Inc. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part.
67706_07_ch07_p420-483.qxd
12:31 PM
Page 430
Chapter 7 Forced Convection Over Exterior Surfaces
3
2
Log NuD
430
5/14/10
Wire No. 1
Wire No. 2
Wire No. 3
Wire No. 4
Wire No. 6
Wire No. 7
Diameter
0.0189 mm
0.0245 mm
0.050 mm
0.099 mm
0.500 mm
1.000 mm
Tube No. 8
Tube No. 9
Tube No. 10
Tube No. 11
Tube No. 12
Diameter
2.99 mm
25.0 mm
44.0 mm
99.0 mm
150.0 mm
1
0
1
2
3
Log ReD
4
5
6
FIGURE 7.11 Average Nusselt number versus Reynolds number
for a circular cylinder in cross-flow with air.
Source: After R. Hilpert [6, p. 220].
It is apparent from the foregoing discussion that the variation of the heat transfer coefficient around a cylinder or a sphere is a very complex problem. For many
practical applications, it is fortunately not necessary to know the local value hc but
is sufficient to evaluate the average value of the heat transfer coefficient around the
body. A number of observers have measured mean heat transfer coefficients for flow
over single cylinders and spheres. Hilpert [6] accurately measured the average heat
transfer coefficients for air flowing over cylinders of diameters ranging from 19 m
to 15 cm. His results are shown in Fig. 7.11, where the average Nusselt qhc D> k is
plotted as a function of the Reynolds number Uq D> .
A correlation for a cylinder at uniform temperature Ts in cross-flow of liquids
and gases has been proposed by ZIukauskas [7]:
NuD =
qhc D
Uq D m n Pr 0.25
= Ca
b Pr a
b
n
k
Prs
(7.3)
where all fluid properties are evaluated at the free-stream fluid temperature except
for Prs, which is evaluated at the surface temperature. The constants in Eq. (7.3) are
given in Table 7.1. For Pr Ͻ 10, n ϭ 0.37, and for Pr Ͼ 10, n ϭ 0.36.
TABLE 7.1
Coefficients for Eq. (7.3)
ReD
C
m
1Ϫ40
40Ϫ1 ϫ 103
1 ϫ 103Ϫ2 ϫ 105
2 ϫ 105Ϫ1 ϫ 106
0.75
0.51
0.26
0.076
0.4
0.5
0.6
0.7
Copyright 2011 Cengage Learning, Inc. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part.
67706_07_ch07_p420-483.qxd
5/14/10
12:31 PM
Page 431
7.2 Cylinders, Spheres, and Other Bluff Shapes
431
For cylinders that are not normal to the flow, Groehn [8] developed the following correlation
0.36
NuD = 0.206 Re0.63
N Pr
(7.4)
In Eq. (7.4), the Reynolds number ReN is based on the component of the flow velocity normal to the cylinder axis:
ReN ϭ ReD sin
and the yaw angle, , is the angle between the direction of flow and the cylinder axis,
for example, ϭ 90° for cross-flow.
Equation (7.4) is valid from ReN ϭ 2500 up to the critical Reynolds number,
which depends on the yaw angle as follows:
Recrit
15°
30°
45°
Ͼ45°
2 ϫ 104
8 ϫ 104
2.5 ϫ 105
Ͼ2.5 ϫ 105
Groehn also found that, in the range 2 ϫ 105 Ͻ ReD Ͻ 106, the Nusselt number is
independent of yaw angle
0.36
NuD = 0.012 Re0.85
D Pr
(7.5)
For cylinders with noncircular cross sections in gases, Jakob [9] compiled data
from two sources and presented the coefficients of the correlation equation
NuD = B RenD
(7.6)
in Table 7.2 on the next page. In Eq. (7.6), all properties are to be evaluated at the
film temperature, which was defined in Chapter 4 as the mean of the surface and
free-stream fluid temperatures.
For heat transfer from a cylinder in cross-flow of liquid metals, Ishiguro et al.
[10] recommended the correlation equation
NuD = 1.125(ReDPr)0.413
(7.7)
in the range 1 Յ ReDPr Յ 100. Equation (7.7) predicts a somewhat lower NuD than that
of analytic studies for either constant temperature [NuD = 1.015(ReDPr)0.5] or constant
flux [NuD = 1.145(ReDPr)0.5]. As pointed out in [10], neither boundary condition was
achieved in the experimental effort. The difference between Eq. (7.7) and the correlation equations for the two analytic studies is apparently due to the assumption of inviscid flow in the analytic studies. Such an assumption cannot allow for a separated region
at large values of ReDPr, which is where Eq. (7.7) deviates from the analytic results.
Copyright 2011 Cengage Learning, Inc. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part.
67706_07_ch07_p420-483.qxd
432
5/14/10
12:31 PM
Page 432
Chapter 7 Forced Convection Over Exterior Surfaces
TABLE 7.2 Constants in Eq. (7.6) for forced convection
perpendicular to noncircular tubes
Flow Direction
and Profile
D
D
D
D
D
ReD
From
To
n
B
5,000
100,000
0.588
0.222
2,500
15,000
0.612
0.224
2,500
7,500
0.624
0.261
5,000
100,000
0.638
0.138
5,000
19,500
0.638
0.144
D
5,000
100,000
0.675
0.092
D
2,500
8,000
0.699
0.160
D
4,000
15,000
0.731
0.205
19,500
100,000
0.782
0.035
3,000
15,000
0.804
0.085
D
D
Quarmby and Al-Fakhri [11] found experimentally that the effect of the tube
aspect ratio (length-to-diameter ratio) is negligible for aspect ratio values greater
than 4. The forced air flow over the cylinder was essentially that of an infinite cylinder in cross-flow. They examined the effect of heated-length variations, and thus
aspect ratio, by independently heating five longitudinal sections of the cylinder.
Their data for large aspect ratios compared favorably with the data of ZIukauskas [7]
for cylinders in cross-flow. For aspect ratios less than 4, they recommend
D 0.85
NuD = 0.123 Re0.651
+ 0.00416a b Re0.792
D
D
L
(7.8)
in the range
7 ϫ 104 Ͻ ReD Ͻ 2.2 ϫ 105
Properties in Eq. (7.8) are to be evaluated at the film temperature. Equation (7.8)
agrees well with data of ZIukauskas [7] in the limit L> D : q for this relatively
small Reynolds number range.
Several studies have attempted to determine the heat transfer coefficient near the
base of a cylinder attached to a wall and exposed to cross-flow or near the tip of a
cylinder exposed to cross-flow. The objective of these studies was to more accurately
predict the heat transfer coefficient for fins and tube banks and the cooling of
Copyright 2011 Cengage Learning, Inc. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part.
67706_07_ch07_p420-483.qxd
5/14/10
12:31 PM
Page 433
7.2 Cylinders, Spheres, and Other Bluff Shapes
433
electronic components. Sparrow and Samie [12] measured the heat transfer coefficient at the tip of a cylinder and also for a length of the cylindrical portion (equal to
1/4 of the diameter) near the tip. They found that heat transfer coefficients are 50%
to 100% greater, depending on the Reynolds number, than those that would be predicted from Eq. (7.3). Sparrow et al. [13] examined the heat transfer near the attached
end of a cylinder in cross-flow. They found that in a region approximately one diameter from the attached end, the heat transfer coefficients were about 9% less than
those that would be predicted from Eq. (7.3).
Turbulence in the free stream approaching the cylinder can have a relatively
strong influence on the average heat transfer. Yardi and Sukhatme [14] experimentally determined an increase of 16% in the average heat transfer coefficient as the
free-stream turbulence intensity was increased from 1% to 8% in the Reynolds number range 6000 to 60,000. On the other hand, the length scale of the free-stream turbulence did not affect the average heat transfer coefficient. Their local heat transfer
measurements showed that the effect of free-stream turbulence was largest at the
front stagnation point and diminished to an insignificant effect at the rear stagnation
point. Correlations given in this chapter generally assume that the free-stream turbulence is very low.
7.2.1 Hot-Wire Anemometer
The relationship between the velocity and the rate of heat transfer from a single
cylinder in cross-flow is used to measure velocity and velocity fluctuations in turbulent flow and in combustion processes through the use of a hot-wire anemometer.
This instrument consists basically of a thin (3- to 30-m diameter) electrically
heated wire stretched across the ends of two prongs. When the wire is exposed to a
cooler fluid stream, it loses heat by convection. The temperature of the wire, and
consequently its electrical resistance, depends on the temperature and the velocity of
the fluid and the heating current. To determine the fluid velocity, either the wire is
maintained at a constant temperature by adjusting the current and determining the
fluid speed from the measured value of the current, or the wire is heated by a constant current and the speed is deduced from a measurement of the electrical resistance or the voltage drop in the wire. In the first method, the constant-temperature
method, the hot wire forms one arm in the circuit of a Wheatstone bridge, as shown
in Fig. 7.12(a) on the next page. The resistance of the rheostat arm, Re, is adjusted
to balance the bridge when the temperature, and consequently the resistance, of the
wire has reached some desired value. When the fluid velocity increases, the current
required to maintain the temperature and resistance of the wire constant must also
increase. This change in the current is accomplished by adjusting the rheotstat in
series with the voltage supply. When the galvanometer indicates that the bridge is in
balance again, the change in current, read on the ammeter, indicates the change in
speed. In the other method, the wire current is held constant, and the fluctuations in
voltage drop caused by variations in the fluid velocity are impressed across the input
of an amplifier, the output of which is connected to an oscilloscope. Figure 7.12(b)
schematically shows an arrangement for the voltage measurement. Additional
information on the hot-wire method is given in Dryden and Keuthe [15] and Pearson
[16]. Although the circuitry required to maintain constant wire temperature is more
Copyright 2011 Cengage Learning, Inc. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part.