The critical nozzle is defined as a device to measure the mass flow with only the nozzle supply conditions making use of the flow choking phenomenon at the nozzle throat. The discharge coefficient and critical pressure ratio of the gas flow through the critical nozzle are strongly dependent on the Reynolds number, based on the diameter of the nozzle throat and nozzle supply conditions. Recently a critical nozzle with a small diameter has been extensively used to measure mass flow in a variety of industrial fields. For low Reynolds numbers, prediction of the discharge coefficient and critical pressure is very important since the viscous effects near walls significantly affect the mass flow through the critical nozzle, which is associated with working gas consumption and operation conditions of the critical nozzle. In the present study, computational work using the axisymmetric, compressible, Navier-Stokes equations is carried out to predict the discharge coefficient and critical pressure ratio of gas flow through the critical nozzle. In order to investigate the effect of the working gas and turbulence model on the discharge coefficient, several kinds of gases and several turbulence models are employed. The Reynolds number effects are investigated with several nozzles with different throat diameters. The diffuser angle is varied in order to investigate the effects on the discharge coefficient and critical pressure ratio. The computational results are compared with the previous experimental ones. It is known that the standard k-ε turbulence model with the standard wall function gives the best prediction of the discharge coefficient. The discharge coefficient and critical pressure ratio are given by functions of the Reynolds number and boundary layer integral properties. It is also found that the diffuser angle affects the critical pressure ratio.
The present study addresses a computational result of unsteady gas flow through a critical nozzle. The axisymmetric unsteady compressible Navier-Stokes equations are solved using a finite volume method that makes use of the second-order upwind scheme for spatial derivatives and the multi-stage Runge-Kutta integral scheme for time derivatives. The steady solutions of the governing equation system are validated with the previous experimental data to ensure that the present computational method is valid to predict the critical nozzle flows. In order to simulate the effects of back-pressure fluctuations on the critical nozzle flows, an excited pressure oscillation with an amplitude and frequency is assumed downstream of the exit of the critical nozzle. The results obtained show that, for low Reynolds numbers, the unsteady effects of the pressure fluctuations can propagate upstream of the throat of the critical nozzle, thus giving rise to the applicable fluctuations in mass flow rate through the critical nozzle, while, for high Reynolds numbers, the pressure signals occurring at the exit of the critical nozzle do not propagate upstream beyond the nozzle throat. For a low Reynolds number, it is found that the sonic line near the throat of the critical nozzle markedly fluctuates with time, providing an important mechanism for pressure signals to propagate upstream of the nozzle throat, even in choked flow conditions. The present study is the first investigation to clarify the unsteady effects on the critical nozzle flows.
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