Page 210 - Computational Fluid Dynamics for Engineers

198 6. Inviscid Flow Equations for Incompressible Flows

/-
- C — a=0°
P6- r - - a = 4 °
i -•-a=10°
5- "i
4-
\
3- " \
2- \ ' v.
1- - ^ ~ ^. __ — -- . -. ._
0- { „ -
l -
- 1 - '.^: —^ \— — I 1 1
0.0 0.2 0.4 0.6 0.8 1.0
X / C
Fig. 6.8. Distribution of pressure coefficients on the NACA 0012 airfoil at three angles of
attack.

hand, the region of accelerated flow increases with incidence angle which leads
to regions of more laminar flow than turbulent flow.
These results indicate that the use of inviscid flow theory becomes increas-
ingly less accurate at higher angles of attack since, due to flow separation, the
viscous effects neglected in the panel method become increasingly more im-
portant. This is indicated in Fig. 6.10, which shows the calculated inviscid lift
coefficients for this airfoil together with the experimental data reported in [3]
6
for chord Reynolds numbers, R c (= VOQC/U), of 3 X 10 6 and 9 x 10 . As can
be seen, the calculated results agree reasonably well with the measured values
at low and modest angles of attack. With increasing angle of attack, the lift
coefficient reaches a maximum, called the maximum lift coefficient, (c/) max , at
an angle of attack, a, called the stall angle. After this angle of attack, while the
experimental lift coefficients begin to decrease with increasing angle of attack,
the calculated lift coefficient, independent of Reynolds number, continuously
increases with increasing a. The inviscid lift slope is not influenced by i? c , but
Qmax is dependent upon R c.
Figure 6.11 shows the moment coefficient c m about the aerodynamic center.
In general, moments on an airfoil are a function of angle of attack. However,
there is one point on the airfoil about which the moment is independent of a;
this point is referred to as the aerodynamic center. As illustrated by Fig. 6.11,
the moment coefficient is insensitive to R c except at higher angles of attack.

6.6.2 Flow Over a Circular Cylinder

The computer program of Section 6.5 can also be used for two-dimensional
geometries other than airfoils. To demonstrate this, consider flow over a cir-
cular cylinder of radius ro and compute its external velocity distribution with
the panel program. The Kutta-Joukowski theorem states that the lift per unit