Effect of Reynolds Number on Drag

When the lift coefficient is small, the profile drag is caused essentially by friction. Its value depends on the position of the transition point and hence the lengths of laminar and turbulent stretches. The local velocities increase with angle of attack, leading to a slight rise of the profile-drag coefficient cDp. A further contributing factor is the increasing length of the turbulent boundary layer with a simultaneous shrinking of the length of the laminar layer. In the cLmax range, the profile drag rises steeply because of the strong increase in pressure drag caused by local separation. The Reynolds number has a very strong influence on the magnitude of the profile drag because both the pressure drag and the friction drag decrease with increasing Reynolds number (see Fig. 2-39b).

The dependence of the minimum drag coefficient cDmin on the Reynolds number [29] is plotted in Fig. 249 for several four-digit NACA profiles. Laminar separation causes quite high values of the minimum profile drag cDmin for small Reynolds numbers (Re <5 • 105). Symmetric profiles produce minimum drag at cL — 0, cambered profiles at the angle of smooth leading-edge flow. The value of cDmin decreases strongly when the Reynolds number grows. As soon as fully attached flow is established, the trend of the cDm-m curve is similar to that of the friction drag of the flat plate (see Fig. 248). In this range of Reynolds numbers (Re >8 • 10s), the minimum drag coefficient is raised more and more above the value of friction drag when the profile thickness grows (Fig. 249<z). The same behavior is found for the camber (Fig. 249b).

Peculiarities of the drag appear at laminar profiles (see Wortmann [75]). As an example, three-component measurements on the NACA 662 415 profile are plotted in Fig. 2-50 for various Reynolds numbers (after [29]). Over a limited range of small lift coefficients, the profile drag is constant, independent of the angle of attack. It is lower than that of a normal profile if the Reynolds number is large enough to prevent laminar separation. When the Reynolds number grows, cDp decreases; at the same time the dip in the drag curve, that is, the lift range for

mo

Figure 2-48 The friction law of the smooth flat plate (one wetted surface only) at zero incidence. Comparison of theory and experiment. Drag coefficient: Cf^D/q^bc. Theoretical curves: 1, laminar (Blasius); 2. turbulent (Prandtl); 3, turbulent (Prandtl, Sehlichting); 2a, transition laminar-turbulent (Prandtl); 4, turbulent (Schultz-Grunow).

Figure 249 Minimum drag of four-digit NACA profiles vs, Reynolds number, (a) Effect of thickness ratio. 0) Effect of camber ratio.

minimum drag, becomes narrower. When the angle of attack is increased, the pressure minimum shifts toward the nose and, in general, the transition point jumps upstream abruptly, causing a very strong increase in profile drag. This process is observed at reduced a when Re increases and at last, at very large Reynolds numbers, the dip in the drag curve disappears completely. A normal polar curve with an elevated Сдтin takes over (see [50]).

Computational determination of profile drag The profile drag of lifting wings can be determined theoretically by means of boundary-layer theory as long as the flow

Figure 2-50 Three-component measurements on the laminar profile NACA 662-415 at various Reynolds numbers.

is fully attached. Pretsch [48] and Squire and Young [62] were the first investigators to publish such methods, which were later improved by Cebeci and Smith [9]. The profile drag (= pressure drag plus friction drag) is obtained from the velocity distribution in the wake at large distance from the body in the form

+ CO

DP = oh f u(Uсо – u) dу (2-117)

// = – со

Here, b is the span of the wing profile, у is the coordinate normal to the incident flow direction, and u(y) is the velocity distribution in the wake. By defining the profile drag coefficient cDp by Dp = cDpbc(o/2)Ul, and introducing the momentum thickness 52oo, the drag of both sides of the profile with a fully turbulent boundary layer is given as

g

cDp = 2-у1 (2-1 l&z)

Here Re = uxcfv is the Reynolds number and U(x) is the velocity distribution over the profile as obtained for potential flow. The second relationship [Eq. (2-118b)] is derived from the findings of boundary-layer theory (see Schlichting [55]). For a plate in parallel flow, there is U(x) = UX= const.

For some symmetric wing profiles in chord-parallel flow, the coefficients for the profile drag from [62] are summarized in Fig. 2-51. The profile thickness varies from t/c = 0 (flat plate) to t/c = 0.25 and the Reynolds number ranges from Re = 106 to 108. The profile drag is strongly dependent on the location of the laminar-turbulent transition point xtT, which varies from xtr/c = 0 to 0.4. The increase in profile drag with thickness must be attributed essentially to a rising pressure drag.

Truckenbrodt [48] extended the drag formula, Eq. (2-118b), to contain explicitly the profile shape instead of the velocity distribution of potential flow. Application of this method to a large number of NACA profiles produces the simple relationship between the profile-drag coefficient and the thickness ratio tjc,

cDp 2Cft

Here Cft is the drag coefficient of the flat plate with a fully turbulent boundary layer. The constant C lies between C—2 and 2.5 (see also Scholz [48]).

The above statements apply to the profile drag at zero lift. The cD values determined in this way agree, in general, satisfactorily with experiments.

A comprehensive presentation of experimental data on the drag problem is found in Hoerner [24]. Truckenbrodt [69] summarized the decisive findings on drag of wing profiles. Progress in the development of profiles of low drag has been reported by Wortmann [76].

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Figure 2-51 Profile-drag coefficients of wing profiles vs. Reynolds number for several thickness ratios tic, from computations by Squire and Young [62]. xt, Location of transition point;c, chord.