Small-Perturbation Theory

(15.102) | ||

(15.103) |

Suppose that a solid body, such as an airfoil, is placed in the aforementioned flow pattern. The cross-section of the body is assumed to be independent of the Cartesian coordinate . The body disturbs the flow pattern, and changes its velocity field, which is now written

(15.104) | ||

(15.105) |

where and are known as induced velocity components. We are interested in situations in which and .

Equation (15.101) can be combined with the previous two equations to give

(15.106) |

It then follows from Equation (15.100) that

The previous equation is exact. However, if and are small then it becomes possible to neglect many of the terms on the right-hand side. For instance, neglecting terms that are third-order in small quantities, we obtain

Furthermore, if we neglect terms that are second-order in small quantities then we get the linear equation

Note, however, than in so-called

On the other hand, subsonic (i.e., ) and supersonic flow (i.e., ) are both governed by Equation (15.109). Another situation in which certain terms on the right-hand side of Equation (15.108) must be retained is

The pressure coefficient is defined

(15.111) |

(See Section 15.9.) Equation (15.101) implies that

(15.112) |

where . Moreover, Equation (14.61) yields

(15.113) |

where . The previous two equations can be combined to give

(15.114) |

Hence, we obtain

(15.115) |

which reduces to

Using the binomial expansion on the expression in square brackets, and neglecting terms that are third-order, or higher, in small quantities, we obtain

(15.117) |

For two-dimensional flows, in the limit in which Equation (15.109) is valid, it is consistent to retain only first-order terms in the previous equation, so that

Let

(15.119) |

be the equation of the surface of the solid body that perturbs the flow. At the surface, the velocity vector of the flow must be perpendicular to the local normal: that is, the flow must be tangential to the surface. In other words,

(15.120) |

which reduces to

(15.121) |

Neglecting with respect to , we obtain

where is the slope of the surface, and the approximate slope of a streamline.

Now, the body has to be thin in order to satisfy our assumption that the induced velocities are relatively small. This implies that the coordinate differs little from zero (say) on the surface of the body. Hence, we can write

in the immediate vicinity of the surface. Within the framework of small-perturbation theory, it is consistent to neglect all terms on the right-hand side of the previous equation after the first. Hence, the boundary condition (15.122) reduces to

respectively.

Because a homenergic, homentropic flow pattern is necessarily irrotational (see Section 15.10), we can write

(15.125) |

where is the perturbed velocity potential. (See Section 4.15.) It follows that

(15.126) |

Hence, Equations (15.109), (15.118), and (15.124) become

respectively.