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2 Cylinders, Spheres, and Other Bluff Shapes

# 2 Cylinders, Spheres, and Other Bluff Shapes

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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.

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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

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–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

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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.

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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

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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.

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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.

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7.2 Cylinders, Spheres, and Other Bluff Shapes

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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

FIGURE 7.10 Approximation of the leading edge of an aircraft

wing for Example 7.1.

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Chapter 7 Forced Convection Over Exterior Surfaces

3

2

Log NuD

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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

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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.

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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

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7.2 Cylinders, Spheres, and Other Bluff Shapes

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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