Improved flow prediction of non-ideal sonic nozzles by CFD-based surrogate modelling

The use of sonic nozzles as a high-precision standard for gas flow measurements in various engineering and metrological applications is well-established. Due to tolerances during manufacturing, the local curvature radius near the nozzle throat often deviates from the prescribed specifications. Consequently, the resulting non-ideal nozzle shapes can have a considerable influence on the flow rate. In this work, a computational fluid dynamics-based surrogate model is developed for flow prediction in non-ideally shaped sonic nozzles. A parameterisation to represent realistic nozzle shapes is proposed, wherein the curvature radius near the nozzle throat can be varied independently of that at the nozzle entry. A comprehensive numerical study of 121 parameterised non-ideal nozzle geometries is conducted in a wide Reynolds number range, providing the input data for the surrogate model. For the establishment of the surrogate model, five surrogate modelling methods are compared and assessed by means of a leave-one-out cross-validation. Flow results of the resulting surrogate model reveal a systematic impact on the flow behaviour depending on the parameter combination of throat curvature radius and axial transition length. Subsequently, a flow prediction approach using the developed surrogate model is demonstrated for twelve real test nozzles. For this purpose, an approximation procedure based on a weighted non-linear least squares method is conducted to obtain parameterised contours adequately representing the real nozzle shapes. The flow predictions of the surrogate model for the aforementioned parameterised contours are in much better agreement with experimental data for the respective test nozzle contour, exhibiting an average root mean square error of 0.17% over all test nozzles compared to the associated standard, with an average root mean square error of 0.35%. The presented flow prediction approach allows to assess the applicability and effectiveness of the developed surrogate model to each test nozzle individually, which is beneficial for further improvements.