The Calculation of the Discharge Coefficient of Profiled Choked Nozzles and the Optimum Profile for Absolute Air Flow Measurement

TLDR

A choked nozzle with an appropriately contoured wall can achieve a discharge coefficient close to unity, enabling its use as an absolute air‑flow meter. The study proposes a circular‑arc throat profile with curvature about twice the throat diameter to optimize the discharge coefficient and calls for experimental investigation of boundary‑layer transition under unsteady flow and vibration. By showing that at sonic conditions the momentum equation reduces to ∂(ρv)/∂p=0, the authors derive simplified expressions for boundary‑layer displacement and pressure‑induced flow reduction.

Abstract

Summary A choked nozzle with an appropriate wall contour has adischarge coefficient, C D , so close to unity that a theoretical calculation of (I—C D ) would allow the nozzle to be used as an absolute meter for air flow. The high discharge coefficient results basically from the fact that ∂(ρv)∂p=0 at M=1. Simplified calculations yield formulae for the boundary layer displacement thickness and for the flow reduction resulting from the variation in static pressure across the throat. The optimum profile for the wall at the throat of an absolute meter is suggested to be a circular arc of radius of curvature equal to about twice the throat diameter. For such a meter the theoretical discharge coefficient is found to be within ¼ per cent of 0·995 over a wide range of Reynolds numbers. The uncertainty in the discharge coefficient for a steady flow at Reynolds numbers of 10 6 and over appears to be less than ±0·15 per cent, both when the boundary layer is known to be entirely turbulent and when it is known to be entirely laminar. When the state of the boundary layer is not known the corresponding figure appears to be ±0·25 per cent. Experimental information might therefore be helpful on transition—under the appropriate conditions of flow unsteadiness and rig vibration. Available experimental results with known boundary layers tend to confirm the theoretical discharge coefficients down to a Reynolds number of 0·4x10 6 . A pressure ratio of about 1·1/1 or less would probably be sufficient to establish fully supersonic flow if the nozzle were followed by a suitable diffuser.