External Flows
The subject of external ﬂows involves both low Reynolds-number ﬂows and high Reynolds-number ﬂows. Low Reynolds-number ﬂows are not of interest in most engineering applications and will not be considered; ﬂow around spray droplets, river sediment, ﬁlaments, and red blood cells would be examples that are left to the specialists. High Reynolds-number ﬂows, however, are of interest to many engineers and include ﬂow around airfoils, vehicles, buildings, bridge cables, stadiums, turbine blades, and signs, to name a few.
It is quite difﬁcult to solve for the ﬂow ﬁeld external to a body, even the simplest of bodies like a long cylinder or a sphere. We can, however, develop equations that allow us to estimate the growth of the thin viscous layer, the boundary layer, which grows on a ﬂat plate or the rounded nose of a vehicle. Also, coefﬁcients have been determined experimentally that allow the drag and the lift to be approximated with sufﬁcient accuracy.
8.1 Basics
The ﬂow around a blunt body involves a separated region, a region in which the ﬂow separates from the body and forms a recirculating region downstream, as
Figure 8.1 The details of a ﬂow around a blunt body.
shown in Fig. 8.1. A wake, a region inﬂuenced by viscosity, is also formed; it is a diffusive region that continues to grow (some distance downstream the velocity is less than the free-stream velocity V ). A laminar boundary layer exists near the front of the body followed by a turbulent boundary layer as shown in Fig. 8.1. An inviscid ﬂow, often referred to as the free stream, exists on the front of the body and outside the boundary layer, separated region, and wake. The ﬂow around a streamlined body has all the same components as that of Fig. 8.1 except it does not have a signiﬁcant separated region, and the wake is much smaller.
The free-stream inviscid ﬂow is usually irrotational although it could be a rotational ﬂow with vorticity, e.g., the ﬂow of air near the ground around a tree trunk or water near the ground around a post in a river; the water digs a depression in the sand in front of the post and the air digs a similar depression in snow in front of the tree, a rather interesting observation. The vorticity in the approaching air or water accounts for the observed phenomenon.
It should be noted that the boundary of the separated region is shown at an aver- age location. It is, however, highly unsteady and is able to slowly exchange mass with the free stream, even though the time-average streamlines remain outside the separated region. Also, the separated region is always located inside the wake.
Interest in the ﬂow around a blunt object is focused on the drag, the force the ﬂow exerts on the body in the direction of the ﬂow.1 Lift is the force exerted normal
to the ﬂow direction and is of interest on airfoils and streamlined bodies. The and lift F are speciﬁed in terms of the drag coefﬁcient C and lift coefﬁcient C , respectively, by
where, for a blunt body, the area A is the area projected on a plane normal to the ﬂow direction, and for an airfoil the area A is the chord (the distance from the nose to the trailing edge) times the length.
The force due to the lower pressure in the separated region dominates the drag force on a blunt body, the subject of Sec. 8.2. The viscous stress that acts on and parallel to each boundary element is negligible and thus little, if any, attention is paid to the boundary layer on the surface of a blunt body. The opposite is true for an airfoil, the subject of Sec. 8.3; the drag force is primarily due to the viscous stresses that act on the boundary elements. Consequently, there is considerable interest in the boundary layer that develops on a streamlined body.
The basics of boundary-layer theory will be presented in Sec. 8.5. But ﬁrst, the inviscid ﬂow outside the boundary layer (see Fig. 8.1) must be known. So, inviscid ﬂow theory will be presented in Sec. 8.4. The boundary layer is so thin that it can be ignored when solving for the inviscid ﬂow. The inviscid ﬂow solution provides the lift, which is not signiﬁcantly inﬂuenced by the viscous boundary layer, and it also provides the pressure distribution on the body’s surface as well as the velocity on that surface (since the inviscid solution ignores the effects of viscosity, the ﬂuid does not stick to the boundary but slips by the boundary). Both the pressure and the velocity at the surface are needed in the boundary-layer solution.
8.2 Flow Around Blunt Bodies
DRAG COEFFICIENTS
The primary ﬂow parameter that inﬂuences the drag around a blunt body is the Reynolds number. If there is no free surface, the drag coefﬁcients for both smooth and rough spheres and long cylinders are presented in Fig. 8.2; the values for streamlined cylinders and spheres are also included.
Separation always occurs in the ﬂow of a ﬂuid around a blunt body if the Reynolds number is sufﬁciently high. However, at low Reynolds numbers (it is
Figure 8.2 Drag coefﬁcients for ﬂow around spheres and long cylinders (E. Achenbach, J. “Fluid Mech.,” v.54, 1972).
called a Stokes ﬂow if Re < 5), there is no separation and the drag coefﬁcient, for a sphere, is given by
Separation occurs for Re ≥ 10 beginning over a small area on the rear of the sphere until the separated region reaches a maximum at Re ≅ 1000. The drag coefficient is then relatively constant until a sudden drop occurs in the vicinity of Re = 2 × 105. This sudden drop is due to the transition of the boundary layer just before separation undergoing transition from a laminar ﬂow to a turbulent ﬂow. A turbulent boundary layer contains substantially more momentum and is able to move the separation region further to the rear (see the comparison in Fig. 8.3). The sudden decrease in drag could be as much as 80 percent. The surface of an object can be roughened to cause the boundary layer to undergo transition prematurely; the dimples on a golf ball accomplish this and increase the ﬂight by up to 100 per- cent when compared to the ﬂight of a smooth ball. After the sudden drop occurs, the drag coefﬁcient again increases with increased Reynolds number. Experimental data does not provide the drag coefﬁcients for either the sphere or the cylinder for high Reynolds numbers. The values of 0.4 for
Figure 8.3 Laminar and turbulent velocity proﬁles for the same boundary layer thickness.
long smooth cylinders and 0.2 for smooth spheres for Reynolds numbers exceeding 106 are often used (contrary to the data of Fig. 8.2).
Streamlining can substantially reduce the drag coefﬁcients of blunt bodies. The drag coefﬁcients for streamlined cylinders and spheres are also shown in Fig. 8.2. The included angle at the trailing edge should not exceed about 20° if the separated region is to be minimized. The drag due to the shear stress acting on the enlarged surface will certainly increase for a streamlined body, but the drag due to the low pressure will be reduced much more so that the total drag will be less. Also, stream- lining eliminates the vortices that cause vibrations when shed from a blunt body.
For cylinders of ﬁnite length with free ends, the drag coefﬁcient must be reduced using the data of Table 8.1. If a ﬁnite-length cylinder has one end ﬁxed to a solid
Table 8.1 Drag Coefﬁcients for Finite-Length Circular Cylinders* with Free Ends†
L/D | C_{D }/C _{D∞} |
∞ | 1 |
40 | 0.82 |
20 | 0.76 |
10 | 0.68 |
5 | 0.62 |
3 | 0.62 |
2 | 0.57 |
1 | 0.53 |
surface, the length of the cylinder is doubled. Note that the L/D of a cylinder with free ends has to be quite large before the end effects are not signiﬁcant.
Drag coefﬁcients for a number of common shapes that are insensitive to high Reynolds numbers are presented in Table 8.2.
EXAMPLE 8.1
A 5-cm-diameter, 6-m-high pole ﬁxed in concrete supports a ﬂat, circular 4-m- diameter sign. For a wind speed of 30 m/s, estimate the maximum moment that must be resisted by the concrete.
Solution
To obtain the maximum moment, the wind is assumed normal to the sign. From Table 8.2 the drag coefﬁcient for a disc is 1.1. The moment due to the drag force, which acts at the center of the sign, is
assuming a Reynolds number of Re = 30 × 0.05 / 1.5 × 10^{−5} = 10^{−5} and high- intensity ﬂuctuations in the air ﬂow, i.e., a rough cylinder. The factor from
Table 8.1 was not used since neither end was free. The moment that must be resisted by the concrete base is
M = M_{1} + M_{2 }= 60 700 + 346 = 61 000 N ⋅ m
VORTEX SHEDDING
Long cylindrical bodies exposed to a ﬂuid ﬂow can exhibit the phenomenon of vortex shedding at relatively low Reynolds numbers. Vortices are shed from electrical wires, bridges, towers, and underwater communication wires, and can actually experience signiﬁcant damage. We will consider the vortices shed from a long circular cylinder. The shedding occurs alternately from each side of the cylinder, as shown in Fig. 8.4. The shedding frequency f, in hertz, is given by the Strouhal number,
Figure 8.4 Vortices shed from a cylinder.
If this shedding frequency is the same, or a multiple of a structure’s frequency, then there is the possibility that damage may occur due to resonance.
The shedding frequency cannot be calculated from equations; it is determined experimentally and shown in Fig. 8.5. Note that vortex shedding initiates at Re ≈ 40 and for Re ≥ 300 the Strouhal number is essentially independent of Reynolds number and is equal to about 0.21. The vortex-shedding phenomenon disappears for Re > 10^{4}.
Figure 8.5 Strouhal number for vortex shedding from a cylinder.
EXAMPLE 8.2
A 6-cm-diameter cylinder is used to measure the velocity of a slow-moving air stream. Two pressure taps are used to determine that the vortices are shed with a frequency of 4 Hz. Determine the velocity of the air stream.
Solution
Assume the Strouhal number to be in the range 300 < Re < 10 000. Then
It is quite difﬁcult to measure the velocity of an air stream this low. The measurement of the shed vortices is one method of doing so.
CAVITATION
When a liquid ﬂows from a region of relatively high pressure into a region of low pressure, cavitation may occur, that is, the pressure may be sufﬁciently low so that the liquid vaporizes. This can occur in pipe ﬂows in which a contraction and expansion exists: in the vanes of a centrifugal pump, near the tips of propellers, on hydrofoils, and torpedoes. It can actually damage the propellers and the steel shafts (due to vibrations) on ships and cause a pump to cease to function properly. It can, however, also be useful in the destruction of kidney stones, in ultrasonic cleaning devices, and in improving the performance of torpedoes.
Cavitation occurs whenever the cavitation number s, deﬁned by
is less than the critical cavitation number scrit, which depends on the geometry and the Reynolds number. In Eq.(8.4), p is the absolute pressure in the free stream and is the vapor pressure of the liquid. The drag coefﬁcient of a body that experiences cavitation is given by
where C (0) is given in Table 8.3 for several bodies for Re ≅ 10^{5}.
The hydrofoil, an airfoil-type shape that is used to lift a vessel above the water surface, invariably cannot operate without cavitation. The area and Reynolds number are based on the chord length. The drag and lift coefﬁcients along with the critical cavitation numbers are presented in Table 8.4.
A 2-m-long hydrofoil with chord length of 40 cm operates at 30 cm below the water’s surface with an angle of attack of 6°. For a speed of 16 m/s determine the drag and lift and decide if cavitation exists on the hydrofoil.
Solution
The pressure p_{∞} must be absolute. It is
Assuming the water temperature is about 15°C, the vapor pressure is 1600 Pa (see Table C.1) so that the cavitation number is
This is less than the critical cavitation number of 1.2 given in Table 8.4 so cavitation is present. Note: we could have used p = 0, as is often done, with sufﬁcient accuracy. The drag and lift are
8.3 Flow Around Airfoils
Airfoils are streamlined so that separation does not occur. Airfoils designed to oper- ate at subsonic speeds are rounded at the leading edge, whereas those designed for supersonic speeds may have sharp leading edges. The drag on an airfoil is primarily due to the shear stress that acts on the surface. The boundary layer, in which all the shear stresses are conﬁned, that develops on an airfoil is very thin (see in Fig. 8.6) and can be ignored when solving for the inviscid ﬂow surrounding the airfoil. The pressure distribution that is determined from the inviscid ﬂow solution is inﬂuenced very little by the presence of the boundary layer. Consequently, the lift is estimated on an airfoil by ignoring the boundary layer and integrating the pressure distribution of the inviscid ﬂow. The inviscid ﬂow solution also provides the velocity at the outer edge of the thin boundary layer, a boundary condition needed when solving
Figure 8.6 Flow around an airfoil at an angle of attack a.
the boundary-layer equations; the solution of the boundary-layer equations on a ﬂat plate will be presented in Sec. 8.5.
The lift and drag on airfoils will not be calculated from the ﬂow conditions but from graphical values of the lift and drag coefﬁcients. These are displayed
in Fig. 8.7 for a conventional airfoil with Re ≅ 9 × 106. The lift and drag coefﬁcients are deﬁned as
Conventional airfoils are not symmetric and are designed to have positive lift at zero angle of attack, as shown in Fig. 8.7. The lift is directly proportional to the angle of attack until just before stall is encountered. The drag coefﬁcient is also
directly proportional to the angle of attack up to about 5°. The cruise condition is at an angle of attack of about 2°, where the drag is a minimum at CL = 0.3 as noted.
Mainly, the wings supply the lift on an aircraft. But an effective length is the tip-to- tip distance, the wingspan, since the fuselage also supplies lift.
The drag coefﬁcient is essentially constant up to a Mach number of about 0.75. It then increases by over a factor of 10 until a Mach number of one is reached at which point it begins to slowly decrease. So, cruise Mach numbers between 0.75 and 1.5 are avoided to stay away from the high drag coefﬁcients. Swept-back airfoils are used since it is the normal component of velocity that is used when calculating the Mach number; that allows a higher plane velocity before the larger drag coefﬁcients are encountered.
Figure 8.7 Lift and drag coefﬁcients for a conventional airfoil at Re ≅ 9 × 106.
Slotted ﬂaps are also used to provide larger lift coefﬁcients during takeoff and landing. Air ﬂows from the high-pressure region on the bottom of the airfoil through a slot to energize the slow-moving air in the boundary layer on the top side of the airfoil thereby reducing the tendency to separate and stall. The lift coefﬁcient can reach 2.5 with a single-slotted ﬂap and 3.2 with two slots.
EXAMPLE 8.4
Determine the takeoff speed for an aircraft that weighs 15 000 N including its cargo if its wingspan is 15 m with a 2-m chord. Assume an angle of attack of 8° at takeoff.
Solution
Assume a conventional airfoil and use the lift coefﬁcient of Fig. 8.7 of about 0.95. The velocity is found from the equation for the lift coefﬁcient:
The answer is rounded off to two signiﬁcant digits, since the lift coefﬁcient of is approximated from the ﬁgure.