Inclined Flat Plate

The simplest case of a lifting-airfoil profile is the inclined flat plate. The angle between the direction of the incident flow and the direction of the plate is called angle of attack a of the plate.

The flow about the inclined flat plate is obtained as shown in Fig. 2-8, by superposition of the plate in parallel flow (a) and the plate in normal flow (b). The resulting flow

(c) = (a) + (b)

does not yet produce lift on the plate because identical flow conditions exist at the leading and trailing edges. The front stagnation point is located on the lower surface and the rear stagnation point on the upper surface of the plate.

Figure 2-8 Flow about an inclined flat plate, (a) Flat plate in parallel flow, (b) Flat plate in normal (stagnation) flow, (c) Inclined flat plate without lift, (c) = (a) + (b), (d) Pure circulation flow. (e) Inclined flat plate with lift (Kutta condition), (<?) = (c) + (d).

To establish a plate flow with lift, a circulation Г according to Fig. 2-М must be superimposed on (c). The resulting flow

0) = (c) + (d) = (a) + (b) + (d)

is the plate flow with lift. The magnitude of the circulation is determined by the condition of smooth flow-off at the plate trailing edge; for example, the rear stagnation point lies on the plate trailing edge (Kutta condition). By superposition of the three flow fields, a flow is obtained around the circle of radius a with its center at z = 0. It is approached by the flow under the angle a with the jc axis, a being arctan (v^/Uoo). The complex stream function of this flow, taking Eqs. (2-18) and (2-19) into account, becomes

F(z) — (Uoo — і Voo) z + (««, – j – г Vcc) 4- + т£іпг (2-24)

For the mapping, the Joukowsky transformation function from Eq. (2-21) was chosen. This function transforms the circle of radius a in the z plane into the plate of length c = 4a in the f plane. The velocity distribution about the plate is obtained with the help of Eq. (2-23) after some auxiliary calculations as

00 ъ o-r

= (2-25)

The magnitude of the circulation Г is now to be determined from the Kutta condition. Smooth flow-off at the trailing edge requires that there—that is, at $ = +2л—the velocity remains finite. Therefore, the nominator of the fraction in Eq. (2-25) must vanish for f = 2a. Hence, because of 4a — c,

Г = 47raUoo = rtcvaо

The + sign applies to the upper surface, the — sign to the lower surface. With w«, the resultant of the incident flow, and a, the angle of attack between plate and incident flow resultant, the flow components are given by = w«, cos a and vx = sin a.

At the plate leading edge, f = —c/2, the velocity is infinitely high. The flow around the plate comes from below, as seen from Fig. 2-8c. On the plate trailing edge, f = 4-е/2, the tangential velocity has the value u = vm cos a. At an arbitrary station of the plate, the tangential velocities on the lower and upper surfaces have a difference in magnitude Ли = uu —щ. At the trailing edge, Л и = 0 (smooth flow-off). The nondimensional pressure difference between the lower and upper surfaces, related to the dynamic pressure of the incident flow qx = (o/2)wi, is [see Eq. (2-8)]

where uu and щ stand for the velocities on the upper and lower surfaces of the plate, respectively. This load distribution on the plate chord is demonstrated in Fig. 2-9c. The loading at the plate leading edge is infinitely large, whereas it is zero at the trailing edge. By integration, the force resulting from the pressure distribution on the surface can he computed in principle [see Eq. (2-9)]. In the present case, the result is obtained more simply by introducing Eq. (2-266) into Eq. (2-15). With

L = QTtbcwl, sin a (2-29)

the lift coefficient becomes

This equation establishes the basic relationship between the lift coefficient and the angle of attack of a flat plate in two-dimensional flow. The so-called lift slope for small a is

dcL

da

Figure 2-10 gives a comparison, based on Eq. (2-30), between theory and experimental measurements for a flat plate and a very thin symmetric profile. Up to about a-6°, the agreement is quite good, although it is somewhat better for the plate than for the profile. At angles of attack in excess of 8°, the experimental curves lie considerably below the theoretical curve, a deviation due to the effect of viscosity. When the angle of attack exceeds 12°, flow separation sets in. Flows around profiles with and without separation are shown in Fig. 2-11. Naumann [42] reports measurements on a profile over the total possible range of angle of attack, that is, for 0° < a < 3’60°.

Without derivation, the pitching moment coefficient about the plate leading edge (tail-heavy taken to be positive) is given by

cm = —f— = — x sin2i* (2-32)

bc*q0о *

From Eqs. (2-30) and (2-32), the distance of the lift center of application from the leading edge at small angles of attack is obtained (see Fig. 2-9) as

_ _£M = 1

C CL 4

Since lift and moment depend exclusively on the angle of attack, the center of lift (= center of application of the load distribution in Fig. 2-9c) is identical to the neutral point (see Sec. 1-3-3).

An astounding result of the just computed inviscid flow about an infinitely thin

Figure 2-10 Lift coefficient vs. angle of attack a for a flat plate and a thin symmetric profile. Comparison of theory, Eq. (2-30), and experimental measure­ments, after Prandtl and Wieselsberger [47].

Figure 2-11 Photographs of the flow about airfoil, after Prandtl and Wieselsberger [47]. (a) Attached flow. (b) Separated flow.

inclined flat plate is the fact that the resultant L of the forces is not perpendicular to the plate, but perpendicular to the direction of the incident flow (Fig. 2-9a). Since only normal forces (pressures) are present on the plate surface in a frictionless flow, it could appear to be likely that the resultant of the forces acts normal to the plate, too. Besides the normal component Py=Lcosa, however, there is a tangential component Px = —L sin a that impinges on the plate leading edge. Together with the normal component Py, the resultant force L acts normal to the direction of the incident flow. For the explanation of the existence of a tangential component Px in an inviscid flow—we shall call it suction force—a closer look at the flow process is required. The suction force has to do with the flow at the plate nose, which has an infinitely high velocity. Consequently, an infinitely high underpressure is produced. This condition is easier to see in the case of a plate of finite but small thickness and rounded nose (Fig. 2-12a). Here the underpressure at the nose of the plate is finite and adds up to a suction force acting parallel to the plate in the forward direction. The detailed computation shows that the magnitude of this suction force is independent of plate thickness and nose rounding. It remains, therefore, the value of S = L sin a in the limiting case of an infinitely thin plate.

In real flow (with friction) around very sharp-nosed plates, an infinitely high underpressure does not exist. Instead, a slight separation of the flow (separation bubble) forms near the nose (Fig. 2-12b). For small angles of attack, the flow reattaches itself farther downstream and, therefore, on the whole is equal to the frictionless flow. The suction force is missing, however, and the real flow around an inclined sharp-edged plate produces drag acting in the direction of the incident flow. Also, this analysis shows that it is very important for keeping the drag small that the leading edge of wing profiles is well rounded. Figure 2-13 shows (a) the polar curves (cL vs. Cq) and (b) the glide angles e= cDjcL of a thin sharp-edged flat plate and of a thin symmetric profile. In the range of small to moderate angles of attack, the thin profile with rounded nose has a markedly smaller drag than the sharp-edged flat plate. Within a certain range of angles of attack, є is smaller than а (є < a) for

Figure 2-12 Development of the suction force S on the leading edge of a profile, (a) Thin, symmetric profile with rounded nose, suction force present. (b) Flat plate with sharp nose, suction force missing.

0 0,02 0№ 0.06 008 010 0.12 № 0.16 0° 2° <t° 6° 8° !0° 12° a CQ —► b a ^ Figure 2-13 Aerodynamic coefficients of a sharp-edged flat plate and a thin symmetric profile for Re = 4′ 10s, A = », from Prandtl and Wieselsberger [47]. {a) Polar curves, vs. cjj. (Ъ) Glide angle, є = cd/cd-

thin profiles; the resultant of the aerodynamic forces is inclined upstream relative to the direction normal to the profile chord. This must be attributed to the effect of the suction force.