The thermal effect on discharge coefficient of sonic nozzle based on reconstructed body temperature distribution by improved SOR method

For the sonic nozzles, the discharge coefficient ( $C_{d}$ ) is an essential parameter of flow characteristics, which is obviously influenced by the thermal effect determined by the nozzle body temperature distribution. In this study, two types of thermocouple arrays for temperature acquisition of the nozzle body were designed to investigate the thermal effect. The temperature isothermal maps of the nozzle body were reconstructed by improved successive over-relaxation (SOR) method combining the simple Kriging interpolation method and particle swarm optimization (SKIM-PSO), which can effectively eliminate the phenomenon of bull’s eye and nonsmoothness of the SKIM. And then, the accuracy of the new algorithm was checked by cross validation with test sets of $d = 5.25$ -mm nozzles with different structures. It is found that the maximum error $\Delta e_{\mathrm {max}}$ of the new algorithm is less than 0.2 °C. The thermal effect on discharge coefficient was analyzed with the above-reconstructed temperature distribution, which is classified into two aspects, the thermal boundary layer correction factor $C_{\mathrm{ T}}$ and the constrained thermal deformation correction factor $C_{\mathrm{ a}}$ related to the throat area. The $C_{\mathrm{ T}}$ is $1- 3.845\text {Re}_{\mathrm{ t}}^{-0.5}\,\,\Delta T/T$ from a similar solution to the boundary layer. Unlike the free expansion, the slopes of the $C_{\mathrm{ a}}$ of constrained expansion for three AISI304 nozzles with $d = 2.00$ , 5.25, and 7.45 mm are $- 1.24 \times 10^{-5}$ , $+ 1.98 \times 10^{-5}$ , and $+ 2.49 \times 10^{-5}$ , respectively. With the correction of $C_{\mathrm{ T}}$ and $C_{\mathrm{ a}}$ , the prediction error of discharge coefficient is reduced from 0.14%–0.15% to 0.08%–0.1% for $d = 2.00$ - and 7.45-mm two nozzles.