Slender Body Theory

The potential theory is also a tool for studying the flow about slender bodies. Here the aim is to calculate the aerodynamic forces and moments exerted on the body by the flowfield. Since the body is slender and the angle of attack considered is small, the small perturbation method is applicable for the case of compressible flows. For this reason, the equation we are going to use the linearized compressible small perturbation equation given with 2.24. Let us express Eq. 2.24 in terms of Mach number explicitly for the perturbation potential /.

—Фи + Фхі + (M – 1)/xx = Фуу + /zz (5-94)

a2 a

For simple harmonic motion in terms of the amplitude and the frequency the perturbation potential becomes

Ф(x; y, z, t) = Ф(х, у, z)elmt-

Equation 5.94 in terms of the amplitude and the frequency reads as

x2 2M 2

—- 2 Ф + Фх + (M – 1)Фхх = Фуу + фzz – (5-95)

a2 a

The nondimensional form of Eq. 5.94 is more convenient to interpret each term in terms of its order of magnitude. We can perform nondimensionalization using Fig. 5.18.

Let l be the length of the body and s be the maximum lateral dimension of the body. The nondimensional coordinates than defined as П = x/l, g = у/s, f = z/s. If we write Eq. 5.95 for the nondimensional potential defined as F(£, g, f) = Ф(x, у, z)/(Ul), the equation becomes

x2 2MU, ,

Ul F + ix Fn + M2

a2 a

The right hand side of the equation above contains the Laplacian in transformed coordinates. Rearranging it gives

If we redefine the reduced frequency as k = ml/U, Eq. 5.96 becomes

Fnn + Fgg = Q2 [-k2M2F + 2ikM2Fn + (M2 – 1)FK] (5.97)

For the slender bodies by definition ratio of the lateral dimension to its length is small, therefore (s/l)2 << 1. In addition if the following parameters k2 M2, k M2 and M2 is not too large, Eq. 5.97 becomes

Fgg + Fre = 0. (5.98)

Equation 5.98 is the Laplace’s equation written involving lateral coordinates only. That is according to 5.98, the cross flow created by the slender body is incompressible. Equation 5.98 does not contain any term related to n, meaning that it seems the potential is independent of flow direction. However, we have to specify the boundary condition along the body surface which makes our potential depend on the low direction. The cross flow behaving incompressible enables us to use Munk’s airship theory. This theory is used for finding the aerodynamic forces created by the momentum of the air parcel displaced by the body itself. Let us see the application of Munk’s airship theory for steady and unsteady flows.