Slender Bodies with General Cross Sections
Consider a slender body, with a contour A at x location and contour B at x + dx, as shown in Fig. 7.16.
In the neighborhood of the body, the governing equation in terms of a small disturbance potential is the Laplace equation in the (y, z) cross plane
























In this approximation, д In r is neglected compared to ln r and r — л/y2 + z2. From the boundary condition

where S(x) is the cross sectional area.
The general solution is then
U
ф ~ S (x) ln r + g(x)
The pressure coefficient is
This is the same as the solution for an axisymmetric body having the same cross sectional area distribution as the actual body with a general cross section.
Oswatitsch andKeune [2], Ward [1820] andAshley, andLandhal [1] were among the first who derived the equivalence rule for transonic, subsonic and supersonic flows: “Far away from a general slender body, the flow becomes axisymmetric and equal to the flow around the equivalent body of revolution.” “Near the slender body, the flow differs from that around the equivalent body of revolution by a twodimensional constantdensity cross flow part that makes the tangency condition at the body surface satisfied.” (See also Harder and Klunker [21], Guderley [22] and Cole and Cook [23]).
One can solve the axisymmetric small disturbance equation for transonic, subsonic or supersonic flows over the equivalent body of revolution to provide the far field boundary condition for a twodimensional cross flow calculation at each cross section.
It can be shown that the pressure drag of the general slender body is equal to the drag of the equivalent body of revolution if the body ends in a point (S = 0) or in a cylindrical portion parallel to the free stream (S’ = 0 and ^ = 0). It is also true if the body ends with an axisymmetric portion.
Whitcomb [24] had verified these results experimentally, and the agreement was good in the transonic regime. Therefore, it is possible to reduce the wave drag of a wingbody combination by indenting the body such that the equivalent body of revolution has a smooth area distribution. This is the basis of “transonic area rule.”
Whitcomb has also shown experimentally that at high angle of attack, the lift contributes to the cross sectional area distribution of the equivalent body.
Cheng and Hafez [25], and Cheng [26], developed a theoretical justification in the transonic regime due to the nonlinearity of the governing equation, therefore in a good design, the fuselage will have a reduced waist at the location of the wing. See Fig. 7.17 for several arrangements.














The above expression yields an axisymmetric flow as r ^ to, confirming the area rule.
However, if t is much smaller than a, the cross flow solution breaks down before the source distribution prevails over the doublet distribution. in this case, nonlinear effects near the wing are important. The fuller equation governing the perturbation potential near the wing is
The nonlinear terms on the right hand side act as a source term in the cross flow equation (hence the doublet distribution contributes to the source distribution).
Away from the body, the threedimensional equation must be solved to fully account for these nonlinear effects, and the solution is determined principally by the doublet and the source distributions from the solution of the cross flow equation, including the nonlinear corrections.
The doublet distribution is still proportional to b2(x), as in linear theory, however the effective source distribution consists of two parts
dSe dS d (dD2
– = + a (7.115)
dx dx dx dx
The second term in the above expression is due to the nonlinear correction in the cross flow solution, where a is a parameter representing the relative importance of the lift.
Now, one can solve the three dimensional small disturbance equation away from the body
with
and
г дф 1 _ dSe 1 sin в
= т — a D(x)
Ub д г 2п dx 2п г
the solution of the above three dimensional problem provides the far field boundary conditions for the cross flow problems at each cross section, including the nonlinear effects as a forcing function and with the tangency condition at the solid surface. Notice iterations are needed to account for the forcing function.
If lift effects are not dominant, the three dimensional problem can be split into two axisymmetric ones where
and ф0 and ф12 are governed by
Both ф0 ^ 0, and ф2 ^ 0 as г ^гх>, and as approaching the axis, we have
г дф0 1 _ dSe r2 дфі2
Ub дr 2n dx’ Ub дr
Special treatment is required to enforce the perturbed shock jump condition for ф12 problem.
For more details, the reader is referred to Cheng and Hafez [25], particularly for the estimate of the size of the region in the neighborhood of the body, which is obtained via asymptotic analysis.
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