Thin-Airfoil Theory

The basic approximate expression [cf. Equation (15.19)]

(15.46) |

specifies the relative change in pressure across either a weak oblique shock (see Section 15.4) or a weak expansion fan (see Exercise vi) that deflects flow of Mach number through an angle . Because, in the weak wave approximation, the pressure, , never greatly differs from the upstream pressure, , and the Mach number, , never differs appreciably from the upstream Mach number, , we can write

(15.47) |

which is correct to first order in . Here, is the deflection angle relative to the upstream flow.

It is convenient to define a dimensionless quantity known as the *pressure coefficient*:

where , and and are the upstream density and flow speed, respectively. Given that the upstream sound speed is , and , we obtain

(15.49) |

which yields

This is the fundamental formula of thin-airfoil theory. It states that the pressure coefficient is proportional to the local deflection of the flow from the upstream direction.

Consider the flat-plate airfoil shown in Figure 15.10. The deflection angle is on the upper surface of the airfoil, and on the lower surface. (A positive deflection angle corresponds to compression, and a negative deflection angle to expansion.) Thus, the pressure coefficients on the upper and lower surfaces are

and

respectively. It is convenient to define the dimensionless

and the dimensionless

(15.54) |

It follows that

Making use of Equations (15.51) and (15.52), as well as the conventional small angle approximations and , we obtain

(15.57) | ||

(15.58) |

The focus (see Section 9.3) of the airfoil is at the midchord. Moreover, the ratio is independent of .

As a second example, consider the diamond-section airfoil pictured in Figure 15.11. This airfoil has a nose angle , and zero angle of attack. The pressure coefficient on the front face of the airfoil is

(15.59) |

whereas that on the rear face is

(15.60) |

It follows that the pressure difference is

(15.61) |

giving a drag

(15.62) |

where and are the thickness and chord-length of the airfoil, respectively. (See Figure 15.11.) Thus,

(15.63) |

Figure 15.12 shows the cross-section of an arbitrary airfoil. The cross-section is assumed to be uniform in the
-direction, with the upstream flow parallel to the
-axis. The upper surface of the
airfoil corresponds to the curve
, the lower surface to the curve
, and the *camber line* (i.e., the centerline) to the curve
. Furthermore, the leading and
trailing edges of the airfoil lie at
and
, respectively. Hence,
and
.
By definition,

for . It is helpful to define the

for . Note that . We can also define the

Thus, we can write

where . Here,

is the local angle of attack of the camber line. Thus, the

Hence, the airfoil shape is completely specified by the thickness function, , the camber function, , and the mean angle of attack, .

The pressure coefficients on the upper and lower surfaces of the airfoil are [see Equations (15.50)]

(15.71) | ||

(15.72) |

respectively. It follows from Equations (15.68)-(15.70) that

Now, the lift and drag per unit transverse length acting on the airfoil are given by [cf., Equations (15.53)-(15.56)]

(15.75) | ||

(15.76) |

Thus, it follows from Equations (15.71)-(15.74) that

(15.77) | ||

(15.78) |

It is helpful to define the chord-average operator:

(15.79) |

where is the chord-length. Taking the average of Equation (15.73), making use of Equation (15.66), as well as the fact that , we obtain

(15.80) |

However, the average of Equation (15.68) yields

(15.81) |

which implies that . We can also write

(15.82) |

Hence, the coefficients of lift and drag, and , are written

(15.83) | ||

(15.84) |

respectively. Thus, in thin-airfoil theory, the lift only depends on the mean angle of attack, whereas the drag splits into three components. Namely, a drag due to thickness, a drag due to lift, and a drag due to camber.