. Supersonic linearized theory (Ackeret’s rule)

Before proceeding to considerations of solution to the supersonic form of the simplified (small perturbation) equation of motion, Eqn (6.118), i. e. where the Mach number is everywhere greater than unity, it is pertinent to review the early work of Ackeret[31] in this field. Notwithstanding the intrinsic historical value of the work a fresh reading many decades later still has interest in the general development of first-order theory.

Making obvious simplifications, such as assuming thin sharp-edged wings of small camber at low incidence in two-dimensional frictionless shock-free supersonic flow, briefly Ackeret argued that the flow in the vicinity of the aerofoil may be likened directly to that of the Prandtl-Меуег expansion round a corner. With the restrictions imposed above any leading-edge effect will produce two Mach waves issuing from the sharp leading edge (Fig. 6.39) ahead of which the flow is undisturbed. Over the upper surface of the aerofoil the flow may expand according to the two-dimensional solution of the flow equations originated by Prandtl and Meyer (see p. 314). If the same restrictions apply to the leading edge and lower surface, then providing the inclinations are gentle and no shock waves exist the Prandtl-Meyer solution may still be used by employing the following device. Since the undisturbed flow is supersonic it may be assumed to have reached that condition by expanding through the appropriate angle vp from sonic conditions, then any isentropic compressive deflection 6 will lead to flow conditions equivalent to an expansion of (vp – 6) from sonic flow conditions.

Thus, providing that nowhere on the surface will any compressive deflections be large, the Prandtl-Meyer values of pressure may be found by reading off the values1^ appropriate to the flow deflection caused by the aerofoil surface, and the aerody­namic forces etc. obtained from pressure integration.

Referring back to Eqn (6.118) with > 1:

Подпись: (6.141)

Подпись: мт

(M2

(M°° ‘W dy2 0


. Supersonic linearized theory (Ackeret's rule)

Fig. 6.39

 

. Supersonic linearized theory (Ackeret's rule)

Fig. 6.40 Supersonic flow over as wedge: The schlieren method was used to obtain this flow visualization. A parallel light beam is refracted by the density differences in the flow field. It is then focused on to a knife edge and gives a flow visualization in the image on the photographic film, which takes the form of bright or dark patterns, depending on the direction the beam is bent by refraction. The main features of the flow field are the oblique bow shock wave which is slightly rounded at the nose (see Fig. 7.54, page 479), the expansion fans at the trailing edge, followed by recompression shock waves which form downstream in the wake. These last are slightly curved owing to the interaction with the expansion waves from trailing edge. (The photograph was taken by D. J. Buckingham at the School of Engineering, University of Exeter, UK.)

 

. Supersonic linearized theory (Ackeret's rule)

This wave equation has a general solution

ф = Fi{x – – 1 y)+F2{x + yjM2, – 1 y)

where F and F2 are two independent functions with forms that depend on the boundary conditions of the flow. In the present case physical considerations show that each function exists separately in well-defined regions of the flow (Figs 6.40, 6.41 and 6.42).

By inspection, the solution ф = F(x- y/M*, – 1 y) allows constant values of ф along the lines x – у/Ml, – 1 у = C, i. e. along the straight lines with an inclination of arc tan – 1 to the x axis (Fig. 6.42). This means that the disturbance originat­

ing on the aerofoil shape (as shown) is propagated into the flow at large, along the straight lines x = y/M – 1 у + C. Similarly, the solution ф = F2{x + T y)

. Supersonic linearized theory (Ackeret's rule)

allows constant values of ф along the straight lines x = —yjM^ — 1 y + C with inclinations of

to the axis.

It will be remembered that Mach lines are inclined at an angle

. Supersonic linearized theory (Ackeret's rule)fj, = arc tan

to the free-stream direction and thus the lines along which the disturbances are propagated coincide with Mach lines.

. Supersonic linearized theory (Ackeret's rule) Подпись: (6.142)

Since disturbances cannot be propagated forwards into supersonic flow, the appropriate solution is such that the lines incline downstream. In addition, the effect of the disturbance is felt only in the region between the first and last Mach lines and any flow conditions away from the disturbance are a replica of those adjacent to the body. Therefore within the region in which the disturbance potential exists, taking the positive solution, for example:

and

Подпись: ду д(х — y/Ml, - 1 у) ду^ дф dFy Э(х-Ум^Ту)

Подпись: (6.143)v = – yjMi – if;

. Supersonic linearized theory (Ackeret's rule)

Fig. 6.41 Transonic flow through a turbine cascade: The holographic interferogram shows fringes corresponding to lines of constant density. The flow enters from the right and exits at a Mach number of about 1.3 from the left. The convex and concave surfaces of the turbine blades act as suction and pressure surfaces respectively. Various features of the flow field may be discerned from the interferogram: e. g. the gradual drop in density from inlet to outlet until the formation of a sharp density gradient marking a shock wave where the constant-density lines fold together. The shock formation at the trailing edge may be compared with Fig. 7.51 on page 476. (The phototgraph was taken by P. J. Bryanston-Cross in the Engineering Department, University of Warwick, UK.)

. Supersonic linearized theory (Ackeret's rule)

Fig. 6.42

. Supersonic linearized theory (Ackeret's rule) Подпись: (6.144)

Now the physical boundary conditions to the problem are such that the velocity on the surface of the body is tangential to the surface. This gives an alternative value for V, i. e.

where df(x)/dx is the local surface slope, f(x) the shape of the disturbing surface and Ux the undisturbed velocity. Equating Eqns (6.143) and (6.144) on the surface where у = 0:

Подпись: Ґ(х)[^l]v = 0 —

or

On integrating:

ф = № =)-^=f(x – JmI – 1 y) (6.145)

VM^-1 V

With the value of ф (the disturbance potential) found, the horizontal perturbation velocity becomes on the surface, from Eqn (6.142):

– Uoo dfx

At x from the leading edge the boundary conditions require the flow velocity to be tangential to the surface

Подпись: =k l Подпись: -U=o

Equating Eqns (6.143) and (6.144) on the surface where у = 0, Eqn (6.145) gives:

)

where є may be taken as +ve or —ve according to whether the flow is compressed or expanded respectively. Some care is necessary in designating the sign in a particular case, and in the use of this result the angle є is always measured from the undisturbed stream direction where the Mach number is M, and not from the previous flow direction if different from this.

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