On Prandtl's Formulas for Supersonic Jet Cell Length

Abstract

In 1904 Prandtl developed a pioneering small perturbation theory to obtain the well-known “Prandtl Formula” λ = 1.306 d√(( wm/ c)2 − 1) for the wavelength (cell length) of an almost perfectly expanded circular supersonic jet, uniform everywhere and notably so on the bounding streamlines, where d = jet diameter, wm/ c = mean (unperturbed) speed (Mach number) of the jet. To relate to experiment and possibly provide a means of estimating jet speed from cell length measurements, he very tersely made a comparison of the results of his formula with those of Emden's formula, λ = 0.88 d√( R - Rc). This was based on extensive experiments with choked jets, with R = pressure ratio (critical value Rc), He rewrote his formula as λ = 1.306 dw√(( w/c)2 − 1), called herein “Prandtl's Formula II, where now dw = minimum diameter of the nozzle duct, i.e. exit diameter for Emden's choked jets, but the throat diameter of the convergent-divergent nozzle of uniform supersonic jets. He rewrote the foregoing equation, “my relationship,”” clearly considering that the nozzle exit diameter dw for the fully expanded flow would vary (according to the one-dimensional gas flow equations) so as to be consistent with the pressure ratio in Emden's formula. Unfortunately these simple but important notational changes for the comparison were made without comment and appear to have been totally ignored. Then converting his Formula II into Emden's form gave the well-known λ = 1.2 d√( R - Rc), a good approximation over a wide range of R. This fully satisfied Prandtl's objective, the discrepancy (1.2 vs. 0.88) being a constant factor easily taken into account, its magnitude apparently being of no particular significance considering the radically different jet flow structures. The coefficient 1.306 was improved to 1.22 by Pack who was able to compute up to 40 terms of the series solution, for which Prandtl had taken just the first term. Pack also provided the full series solution for the two-dimensional case. The circumstances of the origin of these formulas and their subsequent history over the next half-century are discussed in some detail, but a more general discussion of cell length is beyond the scope of this account.