SKIN FRICTION DRAG

Figure 4.1 depicts a thin, flat plate aligned with the free-stream velocity. Frequently the drag of a very streamlined shape such as this is expressed in terms of a skin friction drag coefficient, Q, defined by,

(4.1)

where Sw is the wetted surface area that is exposed to the flow. This coefficient is presented in Figure 4.1 as a function of Reynolds number for the two cases where the flow in the boundary layer is entirely laminar or entirely turbulent over the plate. Here the Reynolds number is based on the total length of the plate in the direction of the velocity. In a usual application, the boundary layer is normally laminar near the leading edge of the plate undergoing transition to a turbulent layer at some distance back along the surface, as described in Chapter Two. The situation is pictured in Figure 4.1, where the velocity profile through the layer is shown. In order to illustrate it, the thickness of the layer is shown much greater than it actually is.

As shown in this figure, a laminar boundary layer begins to develop at the leading edge and grows in thickness downstream. At some distance from the leading edge, the laminar boundary becomes unstable and is unable to suppress disturbances imposed on it by surface roughness or fluctuations in the free stream. In a short distance the boundary layer undergoes transition to a turbulent boundary layer. Here the layer suddenly increases in thickness and is characterized by a mean velocity profile on which a random fluctuating velocity component is superimposed. The distance, x, from the leading edge

•Turbulent boundary layer

Laminar boundary layer

of the plate to the transition point can be calculated from the transition Reynolds number, Rx. Rx is typically, for a flat plate, of the order of 3 x 105, Rx being defined by

For very smooth plates in a flow having a low level of ambient turbulence, Rx can exceed 1 x Ю6.

Since the velocity profile through the boundary layer approaches the velocity outside the layer asymptotically, the thickness of the layer is vague. To be more definitive, a displacement thickness, 8*, is frequently used to measure the thickness of the layer. 5* is illustrated in Figure 4.2 and is defined mathematically by

where у is the normal distance from the plate at any location such that, without any boundary layer, the total flow past that location would equal the flow for the original plate with a boundary layer. To clarify this further, let 8 be the boundary layer thickness where* for all intents and purposes, и = V.

Figure 4.2 Displacement thickness.

-8*)= I J (

Then

Observe that relatively speaking, the turbulent boundary layer is more uniform, with 5* being only one-eighth of 8 as compared to one-third for the laminar layer.

In order to clarify the use of Figure 4.1 and Equations 4.5 to^4.8, let us consider the horizontal tail of the Cherokee pictured in Figure 3.62 at a velocity of 60.4 m/s (135 mph) at a 1524 m (5000 ft) standard altitude. We will assume that the tail can be approximately treated as a flat plate at zero angle of attack.

From Figure 3.62, the length of the plate is 30 in or 0.762 m. The total wetted area, taking both sides and neglecting the fuselage, is 4.65 m2 (50 ft2). From Figure 2.3, at an altitude of 1.52 km, p= 1.054 kg/m3 and v = 1.639 x 10~5 m2/s. We will assume that the transition Reynolds number is equal to 3 x 105.

The distance from the leading edge to the transition point is found from Equation 4.2.

x =

= 1.639 xl(T5x3x

= 0.0814 m (3.2 in.)

The Reynolds number based on the total length will be equal to

V

60.4 (0.762)

1.639 xlO’5

= 2.81 x 106

If the flow over the tail were entirely turbulent then, from Figure 4.1,

C, =0.455 (logioK,)’2 58 (4.9)

= 0.00371

The dynamic pressure q for this case is

q = pV2ll

1.054 (60.4)2
2

= 1923 N/m2

Hence the total skin friction drag would be

D = qSwCf

=4923^4.65X0.00371)

= 33.17 N

However, the leading portion of the plate is laminar. The wetted area of this portion is equal to 0.497 m2. For laminar flow over this portion,

Cf = 1.328R112 (4.10)

= 1.328 (3 x 105Г,/2 = 0.00242

Hence the drag of this portion of the plate is equal to

D = qCfSw

= 1923(0.0024^)(0.497)

-2.31 N

К the flow were turbulent over the leading portion of the plate, its Q would be

Cf = 0.455 (logw-R)"258 = 0.455 (togio 3 x 105r2 58 = 0.00566

Thus its drag for a turbulent boundary layer would be

D = qCfSw

= (1923)(0.00566)(0.497)

= 5.35 N

The above is 5.35-2.31, or 3.04 N higher than the actual drag for laminar flow. Hence this difference must be subtracted from the total drag of 33.17 N previously calculated assuming the boundary layer to be turbulent over the entire plate. Hence the final drag of the total horizontal tail is estimated to be

D = 33.17-3.04 = 30.13 N = 6.77 lb.

The thickness, 5, of the laminar boundary layer at the beginning of transition can be calculated from Equation 4.5.

8 = 5.2 (0.0814X3 x 105)1/2 = 7.728 xl0~4m = 0.0304 in.

The thickness of the turbulent layer right after transition is found from Equation 4.7 assuming the layer to have started at the leading edge.

8 = 0.37(0.0814)(3 x 105)1/5 = 2.418 x 10 3 m = 0.0952 in.

At the trailing edge, the thickness of the turbulent layer will be

8 = 0.37(0.762X2.81 x 106Г,/5 = 0.0145 m = 0.5696 in.

The displacement thickness at the trailing edge is thus only 0.0018 m (0.071 in.).

Before leaving the topic of skin friction drag, the importance of surface roughness should be discussed. Surface roughness can have either a beneficial or adverse effect on drag. If it causes premature transition, it can result in a

reduced form drag by delaying separation. This is explained more fully in the next section. Adversely, surface roughness increases the skin friction coefficient. First, by causing premature transition, the resulting turbulent Q is higher than Cf for laminar flow, in accordance with Figure 4.1. Second, for a given type of flow laminar or turbulent, Q increases as the surface is roughened.

It is difficult to quantify the increment in Q as a function of roughness, since roughness comes in many forms. For some information on this, refer to the outstanding collection of drag data noted previously (e. g., Ref. 4.4). Generally, if a roughness lies well within the boundary layer thickness, say of the order of the displacement thickness, then its effect on Q will be minimal. Thus, for the preceding example of the horizontal tail for the Cherokee, the use of flush riveting near the trailing edge is probably not justified.

An approximate estimate of the effect of roughness, at least on stream­lined bodies, can be obtained by examining the airfoil data of Reference 3.1. Results are presented for airfoils having both smooth and rough surfaces. The NACA “standard” roughness for 0.61-m (2-ft) chords consisted of 0.028-cm (0.011-in.) carborundum grains applied to the model surface starting at the leading edge and extending 8% of the chord back on both the upper and lower surfaces. The grains were spread thinly to cover 5 to 10% of the area.

An examination of the drag data with and without the standard roughness discloses a 50 to 60% increase in airfoil drag resulting from the roughness. It is difficult to say how applicable these results are to production aircraft. Probably the NACA standard roughness is too severe for high-speed aircraft employing extensive flush riveting with particular attention to the surface finish. In the case of a production light aircraft for general aviation usage, the standard roughness could be quite appropriate.