Theory of the momentum source method for synthetic turbulence

The interaction between turbulence and blade leading edges is known to have a significant impact on the aerodynamic and aeroacoustic performance of propellers. In addition to directly simulating turbulence, synthetic turbulence, such as the momentum source method, has been developed as a popular method for studying this interaction process in computational fluid dynamics and computational aeroacoustics. However, it is found that for non-periodic disturbances, although the induced velocity field is divergence-free, spurious noise may be generated in the source region and contaminate simulation results. To address this issue, the present work proposes adding a correction term so that the divergence-free condition is satisfied globally and the unwanted acoustic waves are suppressed, as an extension to our previous work for time-periodic gusts [H. Jiang, Phys. Fluids 35, 096115 (2023)]. The strength of the proposed approach lies in its simplicity, flexibility, and generality. First, it derives explicit source terms, which are straightforward for numerical implementations, to generate unsteady flow fluctuations. Second, the sources can be added inside the computational domain, saving computational costs for turbulence convection and being compatible with most existing boundary conditions. Third, the proposed method can obtain analytical expressions for the needed momentum source of the Navier–Stokes equation subject to any desired isotropic or anisotropic divergence-free turbulence fields. The method has been verified by examples of synthesizing harmonic gusts, Gaussian eddies, and random turbulence. The synthetic velocity results characterized by different spectral components are directly compared to target velocity fields, verifying the proposed approach and showing its capability. Parameters that influence the distribution of added sources are systematically investigated to identify an optimal combination for different scenarios. Finally, the model is employed to evaluate the aerodynamic interaction between an incoming turbulence and a thin airfoil. The obtained results exhibit good correspondence with analytical solutions.