Supersonic Flow (Ackeret Theory)

If M is greater than unity, Equation 5.56 changes to a hyperbolic partial differential equation, specifically, to the following wave equation.

where / and g represent arbitrary functions of their arguments. As an exercise, verify that Equation 5.69 satisfies Equation 5.68. ф is seen to be constant along families of straight lines defined by

x — By = constant x + By = constant

The slope of the lines represented by Equation 5.70a is

dy _ 1 dx~ В

= (M2-l)~m

But this is the tangent of the Mach wave angle as defined by Equation 5.45. Thus ф is constant along a Mach wave. In the case of Equation 5.706,

dy _ _І_ dx~ В

On the upper surface of a body, this would correspond to a disturbance being propagated forward in the flow, which is physically impossible in a supersonic flow. Thus Equation 5.706 is ruled out for the upper surface. However, on the lower surface of a body, g(x + By) is a physically valid flow and represents a disturbance being propagated rearward along a Mach wave. Similarly, f(x – By) is not allowed as a solution on the lower surface of a body. The net result is pictured in Figure 5.23, where it is seen that f(x – By) and g(x + By) are solutions to ф on the upper and lower surfaces, respec-

Flgure 5.23 Mach waves emanating from upper and lower surfaces of a body.

tively. Since ф is constant along Mach waves emanating from disturbances from the upper and lower surfaces, it follows that the properties of the flow (velocity and state) are also constant along these waves.

Along the surface of the body the flow must be tangent to the body. Thus, if Y(jc) represents the body surface, it follows, to a first order in the perturbation velocities, that

Consider the upper surface and let x – By = z. Then

/дф =d£d£

dy)u dz dy

= – Bf (5.72)

where /’ denotes dfldz. In addition, from Equation 5.64,

r = 2 дФ

p“ V„.dx

ні II	(5.73)
Thus, combining Equations 5.71 to 5.73, c =1^ p“ Bdx)u	(5.74)
Similarly, on the lower surface, 4,1 Bdx),	(5.75)

Thus, according to Ackeret’s linearized theory for supersonic flow around a slender body, the pressure locally on the body is determined by the slope of the surface at the particular location in question.

This simple result leads quickly to some interesting conclusions regarding the characteristics of thin, supersonic airfoils at low angles of attack. Since, for a unit chord,

C, = [ (CPI – CPu) dx

JO

you can quickly verify that

4a

C, = – g (5.76)

Thus, within the limitations of the linearized theory, the section lift coefficient of a supersonic airfoil depends only on its angle of attack. Camber is predicted to have no effect on Ct.

The wave drag coefficient is obtained by integrating the component of C„ in the drag direction around the airfoil.

(5.77)

If dyldx is expressed in the form

— a + e

where a is the angle of attack of the chord line and є is the slope of the surface relative to the chord line, Cdw becomes

Cd*=Ч~+1 і (£“2+e,2) dx (5 -78)

The wave drag coefficient can thus be viewed as the sum of two terms; the first results from lift and the second results from thickness and camber.

Cloi+sI (eu + €,) dx

The first term, CLa, is simply the streamwise component of the normal pressures integrated over the airfoil. In the case of a subsonic airfoil, this term is canceled by the leading edge suction force.

The pitching moment coefficient about the leading edge of a thin, super­sonic airfoil can be written

= j CPlx dx – j Cpx dx 2 2 f’

= ~ga + ffj0 + e^x dx

Table 5.1 compares the results of the linearized theory with the more exact predictions made earlier for the symmetrical wedge airfoil pictured in Figure 5.22. In this particular case, the linearized theory is seen to be somewhat optimistic with regard to lift and drag and predicts the center of pressure to be further aft than the position obtained from the more exact calculations. Nevertheless, the Ackeret theory is valuable for predicting trends. For example, for symmetrical airfoils, the expressions of Cdw and Cm reduce to

4

	Using Oblique Shock Wave and Prandtl-Meyer Relationships	Linearized Ackeret Theory
c,	0.0727	0.0806
Cd„	0.0358	0.0259
cm	-0.0287	-0.0403
Center of pressure	0.3950	0.5000

Table 5.1 Predicted Characteristics of a Symmetrical, 10% thick, Double-Wedge Airfoil at a 2° Angle of Attack and a Mach Number of 2.0

The center of pressure for a symmetrical airfoil in supersonic flow is thus predicted to be at the midchord point.

Figure 5.24 (taken from Ref. 5.6) provides a comparison between the linear theory and experiment for a 10% thick biconvex airfoil. This figure shows fairly good agreement of the theory with experiment with the differences being of the same signs as those in Table 5.1.