This study aimed to examine the acoustic field radiated by propellers and underwater vehicles. For the verification of sound radiation in underwater vehicles, numerical methods are widely used in addition to experiments and propeller blade frequencies for calculation and validation. Numerical convergence and accuracy are more important for near-field and far-field problems. This paper uses the boundary element method (BEM) to assess the convergence of the finite volume method (FVM). In this study, the FVM, including the Reynolds-averaged Navier–Stokes method and the Ffowcs Williams–Hawkings (FW-H) acoustic model, is used to investigate the influence of various geometric inflows on the hydrodynamic and noise performance of the propeller. Then, the sound radiation of the FVM is compared with the BEM at the far field to determine the number of meshed elements. Furthermore, spectral analysis is being conducted to examine the noise generated by the underwater vehicle and propeller. The objective is to investigate the influence of the free surface on propeller efficiency. After verifying the numerical simulation, the results indicate that a relationship can be established between water pressure and propeller thrust under specific conditions. This relationship can be used to estimate the magnitude of propeller thrust at different water depths. The simulated results of propeller thrust, torque coefficient, propulsion efficiency, and sound radiation in this study are consistent with experimental values. This demonstrates the accuracy and practicality of the findings of numerical procedures in engineering applications.

This paper proposes a solution to the sideways heat conduction problem (SHCP) based on the time and space integration direction. Conventional inverse problems depend highly on the available data, particularly when the observed data are contaminated with measurement noise. These perturbations may lead to significant oscillations in the solution. The uniqueness of the solution in this SHCP requires revaluation when boundary conditions (BCs) or initial conditions (ICs) are missing. First, the spatial gradient between two points resolves the missing BCs in the computational domain by a one-step Lie group scheme. Further, the SHCP can be transformed into a backward-in-time heat conduction problem (BHCP). The second-order backward explicit integration can be applied to determine the ICs using the two-point solution at each time step. The performance of the suggested strategy is demonstrated with three numerical examples. The exact solution and the numerical results correspond well, despite the absence of some boundary and initial conditions. The only method of preventing numerical instability in this study is to alter the direction of numerical integration instead of relying on regularization techniques. Therefore, a numerical formula with two integration directions proves to be more accurate and stable compared to existing methods for the SHCP.