How big a mouth in a flue pipe?

I had the luck to stumble on the wisdom of Hartmuth Ising (1971) when my interest in pipes started some 20 years ago. He did a lot of research on the jet mechanism in the flue pipe, including flow visualizations. This particular work was published in German in an extremely arcane publication.  So
it is no wonder it is never cited and probably not even known by all that should. Among other things it tells how to select the proper dimensions in the mouth area and how to estimate the sound power.

First the dimensions and constants we are talking about:

H [m] is the height of the cut-up, the distance the air jet travels across the mouth.
D [m] is the thickness of the base of the jet, the small dimension of the flue.
W [m] is the width of the mouth, the large dimension of the flue, usually somewhat less than the pipe diameter.
F [Hz] is the fundamental frequency of the pipe.
P [Pa=N/m^2] is the blowing pressure.
V [m/s] is the initial velocity of the jet.
rho [kg/m^3] is the density of air (rho=1.2 kg/m^3 under normal conditions).

You can use other consistent unit systems than SI, for instance cgs, but beware of peculiar additional constants if you try to drag in any inches.

The first simple basic relation is Bernoulli's law which tells you that

    V = sqrt(2*P/rho)  [m/s]

From this you immediately find the air consumption rate

    Q = V*W*D  [m^3/s]

And now comes what ought to be known as the Holy Gospel of voicing: Thou shalt make the intonation number I of Ising to be more than 2 but less than 3, where

    I = sqrt(2*P*D/(rho*H*H*H))/F    [-] [unit-less]

With I=2 you get maximum efficiency. With higher values you get stronger harmonics and when you go past about 3 the pipe will overblow. (If you add a frein you can stretch it).

So if you want to change the blowing pressure P for an already working pipe the obvious thing would be to change D in inverse proportion (if you want to keep H and F and the voicing characteristics).

The width W does not come in here, but will of course affect sound power and air consumption.

The reason this formula is so little known is no doubt that pipemakers do not start from scratch, they have their more or less secret tables and recipes from experience that will lead them about right from the beginning. The formula puts hard numbers to those voicing recommendations you find loosely formulated in the classical literature. (For instance in Audsley: The art of organ building, or Monette: The art of organ voicing).

Reference:  Ising H: Erforschung und Planung des Orgelklanges. Walcker Hausmitteilung no 42, Juni 1971, pp 38-57.