A physical theory of .supersonic aerofoils in unsteady flow By W. J. Strang, King's College, University of London {Communicated by G. Temple, F.R.S.—Received 21 May 1948) Solutions for point and line sources are given, for the * transient ’ case only, in both stationary and moving co-ordinate systems. These are then integrated to give the solutions for a supersonic aerofoil (a) entering a sharp-edged gust, (b) following a sudden change of incidence.
246 W. J. Strang a = velocity of sound.
A physical theory of supersonic aerofoils in unsteady flow 247 The radial velocity is u = — d<f>/dr. This cannot really be drawn in a graph, but the flux through a fixed sphere of radius r up to time t is F = 47 rr2 judr = V{H(at — r) + r8(at — (see figure 1).
248 W. J. Strang (ii) Although we have used the linearized theory, which requires small disturb ances, the solutions do not satisfy this condition. Nevertheless, if we use these solutions to calculate others in which the disturbances are small, these latter solutions will be valid approximations.
A physical theory of supersonic aerofoils in unsteady flow 249 asymptotic to , Figure 3. Flux for line emission for time tx.
250 W. J. Strang It will be convenient in what follows to regard r2) as co-ordinates of P. This is ambiguous in that one pair of values of and refers to two points P, but since the solutions are symmetrical, this is harmless. Note that (i) rx and real and distinct, P lies within the Mach angle; (ii) rL and r2 complex, P lies outside the Maoh angle; (iii) rx and r2 equal, P lies on the Mach hues.
A physical theory of supersonic aerofoils in unsteady flow 251 and the Mach wedge joining it to the source. Within the cylinder, ^ is a function °f rx, r2 and t, and the equipotential surfaces are elliptic cylinders touching the circular cylinder along its lines of contact with the Mach wedge. As the cylinder travels downstream, it leaves behind it a region in which (j> = const, and the flow consists of a velocity surge across the wave fronts. This is the final flow pattern spreading downstream—which is obviously necessary since changes cannot be propagated upstream if U > a.
252 W. J. Strang It may be remarked that in both methods the aerofoil in the second diagram is supposed to lie everywhere very near the axis so that the solution is only valid for small values of v. In practice, interest centres on gusts in which the velocity v is not small, but varies gradually through a ‘gradient distance’. Solutions for gusts of this type, obtained from the sharp-edged gust solution by use of a superposition integral, are valid provided that the variation in v over the chord of the aerofoil is small.
A physical theory of supersonic aerofoils in unsteady flow 253 4*3. Solution by means of stationary sources Physically, we can replace the aerofoil by stationary sources which spring into operation as the leading edge of the aerofoil passes over them—a notion exploited by Prandtl. The appropriate diagram is figure 9.
254 W. J. Strang 4*4. Lift and pressure-growth functions for sharp-edged gust Figure 8 gives a picture of what is going on—simple Aokeret flow in the shaded region (rx and r2 less than t) and a rather complicated transition zone with a cylindrical boundary. As the aerofoil penetrates the gust the picture grows in size, but does not change otherwise until the trailing edge enters the disturbance. A new pattern then springs from the rear of the aerofoil, but has no effect on the aerofoil itself. If tandem aerofoils become the fashion, it will be necessary to extend the calculations to this field, but for the moment this can be omitted.