Open Access
Cheng, Rui
Graduate Program:
Aerospace Engineering
Doctor of Philosophy
Document Type:
Date of Defense:
December 20, 2006
Committee Members:
  • Kenneth Steven Brentner, Committee Chair
  • Philip John Morris, Committee Member
  • Lyle Norman Long, Committee Member
  • Victor Ward Sparrow, Committee Member
  • Sound Propagation
  • Parabolic Equation
A project has been undertaken to develop a numerical toolkit to simulate long-range propagation of wind turbine noise problems with the CFD (Computational Fluid Dynamics) and CAA (Computational Aeroacoustics) methods. This thesis, which is focused on developing a parabolic equation (PE) method for long-range noise propagation prediction in a complicated environment, is part of that project. The basic idea is marching an initial acoustic solution from the near field to the far field. This research has been performed in two steps: first, as a learning process a traditional two dimensional parabolic equation code is developed; second, a novel three dimensional parabolic equation model is developed, coded, and validated for sound propagation in an inhomogeneous arbitrary moving atmosphere. The 2D PE method is based on an axisymmetry assumption, where the homogeneous Helmholtz equation is split into an incoming wave and an outgoing wave. The outgoing wave satisfies a parabolic equation. In the process of the splitting, a square root operator is generated. Based on the order of the approximation scheme to the square root operator, the PE method is categorized as a narrow angle PE, a wide angle PE, or a higher-order PE. Because it is unconditionally stable in this application, the Crank-Nicholson finite difference method is used to solve the wide angle PE with second order accuracy. The discretization of the parabolic equation results in a linear system of equations, which is solved by an LU algorithm with appropriate initial field and boundary conditions. The numerical results of several example cases are compared to the analytical results, which validate the 2D CNPE method for sound propagation in an inhomogeneous atmosphere without wind. In order to handle sound propagating in an arbitrary moving inhomogeneous medium, a new formulation of the Helmholtz equation is derived in three dimensional cylindrical coordinates. Based on this new formulation, a new parabolic equation is constructed, which extends the homogeneous Parabolic Equation (PE) method to arbitrary moving media. The new PE is discretized by the Crank-Nicholson finite difference method and solved by the Generalized Minimum RESidual (GMRES) method. Numerical results for zero-wind and uniform wind cases above a rigid flat ground surface are presented and compared with benchmark analytical solutions to validate the methodology. Results of sound propagation problems in more realistic wind environments are compared to the 2D PE simulation with an effective speed of sound approximation. It is concluded that if the vertical distance between the acoustic source and observer is small, the effective sound speed approximation to the wind velocity component along the sound propagation direction is acceptable. However, as the vertical distance between them increases, the effective sound speed model will introduce as much as 20 dB errors at some long-range locations. Furthermore, the 2D PE cannot simulate the lateral refraction, thus cannot obtain correct phase information of the acoustic pressure. Therefore, in order to obtain accurate acoustic field in a complicated environment with arbitrary wind, it is necessary to use this 3D PE method.