FORM DRAG

In addition to skin friction drag, a body generally experiences some form drag. Unlike the skin friction drag that results from viscous shearing forces tangential to a body’s surface, form drag results from the distribution of pressure normal to the body’s surface. The extreme case of a flat plate normal to the flow is pictured in Figure 4.3. Here the drag is totally the result of an unbalance in the normal pressure distribution. There is no skin friction drag present in this case.

Generally, form drag is difficult to predict. For that matter, so is skin friction drag except for the simplest cases. Thus, in general cases, such as that pictured in Figure 4.4, where the total drag results from both normal and tangential stresses (or pressures) one must usually resort to experimental data to estimate the drag.

v

As with skin friction drag, form drag is generally dependent on Reynolds number.‘To see why, consider the flow around the circular cylinder pictured in Figure 4.5. In Figure 4.5a flow is pictured at a low Reynolds number. Here, beginning at the stagnation point, a laminar boundary layer develops. On the surface of the cylinder, the static pressure (normal) is highest at the stag­nation point and decreases to a minimum at the top and bottom. Moving around toward the rear, beyond these points, the static pressure increases, tending toward the stagnation pressure at the very rear. In the absence of viscosity the normal pressure distribution would be symmetrical (Equation 2.78) and there would be no drag. This is a clear example of D’Alembert’s paradox, which states that a body in a inviscid fluid will experience no drag. As the slower moving fluid in the laminar boundary layer moves beyond the minimum pressure point on the cylinder, its momentum is insufficient to move against the positive pressure gradient, referred to as an adverse gradient, and thus the flow separates just past the top and bottom locations on the cylinder. In the separated region over most of the rear portion of the cylinder the static pressure is constant and equal to the low pressure at the top and bottom. Thus the high pressure acting over the front and the low pressure over the rear result in a high form drag.

Figure 4.5 Flow over a circular cylinder, (a) Low Reynolds number. Separation occurs before transition. Large wake. (b) High Reynolds number. Transition occurs before separation. Small wake.

The high-Reynolds number case is shown in Figure 4.5b. Here the laminar boundary layer undergoes transition to a turbulent boundary layer before separating. The subsequent turbulent mixing increases the momentum and energy of the boundary layer so that it remains attached toward the rear of the cylinder, well beyond the separation point of the laminar layer. Thus, in this case, the separation region is much smaller and the static pressure is higher on the rear of the cylinder than for the laminar case. Therefore, because of reduced form drag, the drag coefficient of a cylinder isjower at higher Reynolds numbers.

Cd as a function of Reynolds number is presented in Figure 4.6 for both spheres and two-dimensional circular cylinders. Here, Q is based on the projected frontal area. Note the rapid drop in Q above an R value of approximately 2x 105. This is the so-called critical Reynolds number^ whet? the transition point is nearly coincident with the separation point. “Sub – critical” refers to flow at Reynolds numbers that are less than critical; “supercritical” denotes R values that are higher than critical. A body shape having a well-defined separation point will not exhibit a critical Reynolds number; neither will streamlined shapes.

Although not concerned with drag per se, Figure 4.6a also includes the

I

(b)

Figure 4.6 Drag coefficients of cylinders and spheres versus Reynolds number,

(a) Two-dimensional circular cylinders, (b) Spheres.

quantity fD/V, known as the Strouhal number, S. S characterizes an interes­ting behavior of bluff bodies with rounded trailing edges. As such a body first begins to move through a fluid, the vorticity in the boundary layer is shed symmetrically from the upper and lower surfaces to form two vortices of opposite rotation. However, the symmetrical placement of the vortex pair is unstable, so that succeeding vortices are then shed alternately from the upper and lower surfaces. The resulting flow pattern of periodically spaced vortices downstream of the body is known as a Karman vortex street.

In the definition of Strouhal number, / is the frequency at which the

vortices are shed. As a vortex is shed from one surface of the cylinder, it produces a momentary circulation around the cylinder opposite in direction to the vortex. From the Kutta-Joukowski law, a force on the cylinder normal to V results. As the next vortex is shed, the force reverses its direction, resulting in an alternating force on the cylinder. This particular phenomenon is the cause for the “singing” of telephone wires in the wind.

As an example of the use of Figure 4.6a, consider a wire 2 cm in diameter in a wind blowing at a speed of 8 m/s. Assuming standard sea level conditions,

R = ™ v

8(0.02)

1.456 X 10 5

= 1.099 x 104

From Figure 4.6a, for this Reynolds number,

Figure 4.7a Examples of shapes having Ca values nearly independent of Rey­nolds number.

If the ratio of the span to the height (or diameter) of a flat plate (or cylinder) normal to the flow is approximately 5 or less, Cd is nearly constant and equal to the 3-D value. For aspect ratios greater than 5, Cd varies approximately in the manner given by the normalized curve of Figure 4.7 b. This curve is based on data from several sources, including Reference 4.4.

A qualitative evaluation of the drag coefficient for a given shape can be made using some “educated intuition.” Referring to Figure 4.8, the drag

Crf = 1.55

X cd – 0.3

jected frontal areas becomes

Q(3-D) Sw(3-D) 4 Cd(2-D) S„(2-D) irD

where D is the maximum three-dimensional body diameter or the maximum thickness of the two-dimensional shape. For an elliptical two-dimensional shape compared to an ellipsoid, this becomes

Cd(3-D) – гг CA2-D) 2

This is close to the ratio from Figure 4.11 for a finess ratio of 8 and only slightly lower than the corresponding ratio given earlier for the form drag.

Minimum profile drag coefficients for NACA four – and five-digit airfoils are presented in Figure 4.12 as a function of thickness ratio at a Reynolds number of 6 x 106. Here, as is usual for airfoils, Q is based on the chord length. The several data points at each thickness ratio result from airfoils of different camber ratios. Note that Cdmi„ does not vary significantly with

camber. CdmiI1 appears to vary almost linearly with t/c and extrapolates to a value of 0.004 for a tic of zero. This corresponds to a Q value of 0.002. According to Figure 4.1, this would require laminar flow over these sections more extensive than one would expect. Probably, transition is delayed until approximately the 25% chord point, the location of maximum thickness. One would then expect a Cdmin value of about 0.005.

Figure 4.13 presents three-dimensional drag data directly comparable to Figure 4.11, but with more detail. Data representing practical fuselage and nacelle construction are included in Figure 4.13 together with Cd results from torpedo-shaped bodies. Assuming a reasonable relationship between the frontal and wetted areas of such bodies, expected Cd values for various values of Cf are also included on the figure. For a given Q value, the experimental results should approach one of these lines as the fineness ratio gets large.

For fully turbulent flow at an R of 25 x 106, Q for a flat plate would be 0.0026, whereas the data appears to be approaching a Q of 0.0032 to 0.0034. The higher skin friction drag on the bodies is probably the result of surface roughness.

It is interesting to examine the data of Figure 4.13 in terms of minimum drag for a given body volume. This is particularly important for airship and underwater applications. It is also of interest to the design of tip tanks, where minimum drag for a given volume of fuel is desirable. Denoting the volume by Vm, we will define another drag coefficient.

(4.12)

Cdv is related to Cd in Figure 4.13 by

Obviously, the ratio of the frontal area, A, to the 2/3 power of the volume depends, on the particular body shape. We will assume the body to be composed approximately of a hemispherical nose, a cylindrical midbody extending to the middle of the body, and a tail cone. For this particular shape,

(4.13)

Using this relationship and Figure 4.13, the graphs presented in Figure 4.14 were obtained. From this figure it can be seen that to enclose a given volume with a minimum drag body, its fineness ratio should be higher than the optimum values from Figure 4.13. Indeed, for fuselages, the drag for a given volume is nearly constant for lid values from 4 to 10.

I* Figure 4.14 Drag coefficients based on volume for bodies as a function of fineness ratio.

For certain applications, it is desirable to keep the rear portion of a fuselage as wide and bluff as possible without paying too much of a drag penalty. If the afterbody is tapered too abruptly, flow separation will occur over the rear, resulting in an unduly high form drag. Some guidance in this regard is provided by Figure 4.15 (taken from Ref. 4.8). Here, the increment in Q (based on frontal area) resulting from afterbody contraction is presented as a function of afterbody geometry. From this figure it appears that the ratio of the afterbody length to the equivalent diameter should be no less than approximately 2.0.

The importance of streamlining is graphically illustrated in Figure 4.16, which is drawn to scale. Conservatively (supercritical flow), the ratio of Cd for a circular cylinder to a two-dimensional streamlined shape having a fineness ratio of 4 is approximately 7.5. Thus, as shown in Figure 4.16, the height of the streamlined shape can be 7.5 times greater than the circular cylinder for the same drag. For subcritical flow the comparison becomes even more impressive, with the ratio increasing to approximately 25.

When two shapes intersect or are placed in proximity, their pressure distributions and boundary layers can interact with each other, resulting in a net drag of the combination that is higher than the sum of the separate drags. This increment in the drag is known as interference drag. Except for specific cases where data are available, interference drag is difficult to estimate accurately. Some examples of interference drag are presented in Figures 4.17,

4.18, and 4.19.

Figure 4.17 Effect of nacelle location on interference drag. (C. Keys and R. Wiesner, “Guidelines for Reducing Helicopter Parasite Drag”, 1975. Reprinted from the Journal of the American Helicopter Society, Vol. 20, No. 1 by permission of the American Helicopter Society.)

Figure 4.17 illustrates the drag penalty that is paid for placing an engine nacelle in proximity to a rear pylon on a tandem helicopter (like a CH-47). In this particular instance, the interference drag is nearly equal to the drag of the nacelle alone, because the nacelle is mounted very close to the pylon. For spacings greater than approximately one-half of a nacelle diameter, the interference drag vanishes.

Figure 4.18 presents the interference drag between the rotor hub and pylon for a helicopter. The trends shown in this figure are similar to those in the previous figure. In both instances the added interference drag is not necessarily on the appended member; probably, it is on the pylon.

Figure 4.19 shows a wing abutting the side of a fuselage. At the fuselage­wing juncture a drag increment results as the boundary layers from the two airplane components interact and thicken locally at the junction. This type of drag penalty will become more severe if surfaces meet at an angle other than 90°. In particular, acute angles between intersecting surfaces should be avoided. Reference 4.4, for example, shows that the interference drag of a 45% thick strut abutting a plane wall doubles as the angle decreases from 90°

Figure 4.18 Effect of hub/pylon gap on interference drag. (C. Keys and Ft. Wiesner, “Guidelines for Reducing Helicopter Parasite Drag”, 1975. Reprinted from the Journal of the American Helicopter Society, Vol. 20, No. 1 by permission of the American Helicopter Society.)

to approximately 60°. If acute angles cannot be avoided, filleting should be used at the juncture.

In the case of a high-wing configuration, interference drag results prin­cipally from the interaction of the fuselage boundary layer with that from the wing’s lower surface. This latter layer is relatively thin at positive angles of attack. On the other hand, it is the boundary layer on the upper surface of a

*•

low wing that interferes with the fuselage boundary layer. This upper surface layer is appreciably thicker than the lower surface layer. Thus the wing – fuselage interference drag for a low-wing configuration is usually greater than for a high-wing configuration.

The available data on wing-fuselage interference drag are sparse. Reference 4.4 presents a limited amount but, even here, there is no correlation with wing position or lift coefficient. Based on this reference, an approximate drag increment caused by wing-fuselage interference is estimated to equal 4% of the wing’s profile drag for a typical aspect ratio and wing thickness.

Although data such as those in Reference 4.4 may be helpful in estimat­ing interference drag, an accurate estimate of this quantity is nearly im­possible. For example, a wing protruding from a fuselage just forward of the station where the fuselage begins to taper may trigger separation over the rear portion of the fuselage.

Sometimes interference drag can be favorable as, for example, when one body operates in the wake of another. Race car drivers frequently use this to their advantage in the practice of “drafting.” Some indication of this favor-

able interference is provided by Figure 4.20, based on data obtained in Pennsylvania State University’s subsonic wind tunnel. Here the drag on one rectangular cylinder in tandem with another is presented as a function of the distance between the cylinders. The cylinders have a 2:1 fineness ratio. Tests were performed with the long side oriented both with and normal to the free-stream velocity. The drag is referenced with respect to D„, the drag on the one cylinder alone. The spacing is made dimensionless with respect to the dimension of the cylinder normal to the flow. The spacing, x, is positive when the cylinder on which the drag is being measured is downstream of the other. Notice that the cylinder’s drag is reduced significantly for positive x values and even becomes negative for small positive values of x. For small negative Values of x, the drag is increased slightly. Similar data for circular cylinders presented in Reference 4.4 show somewhat similar results, except that inter­ference on the forward cylinder is slightly favorable for spacings less than three diameters. For the downstream cylinder, the drag is reduced by a factor ®f 0.3 for spacings between three and nine diameters. For less than three diameters, the downstream drag is even less, and becomes negative for spacings less than approximately two diameters.