The Study Of High Accuracy Numerical Method For Computational Aeroacoustics Based On The Anlaysis Of Dispersion Relation

Aeroacoustic problems are very different from standard aerodynamic and fluid mechanic problems naturally. It is important to recognize that the numerical algorithms for computational aeroacoustics (CAA) are more critical than for computational fluid dynamical (CFD). In order to accurately capture the dispersion, dissipation, phase velocity and group velocity characteristics of aeroacoustic wave, the numerical scheme must exhibit a minimum of dispersion and dissipation error. Furthermore, non-reflecting boundary condition is also a critical component for open and half open computational boundary problems. Therefore, high resolution numerical schemes and high powered non-reflecting boundary conditions are two key problems of CAA.Based on the technical requirements of CAA, in view of the dispersion relation between numerical schemes and controlled equations, the error behavior of high order accuracy finite difference schemes, high order accuracy finite difference schemes and non-reflecting boundary conditions are studied in this paper. Mainly includes:the error analysis of high order accuracy finite difference scheme, the optimization of low-dispersion explicit finite difference scheme, the construction of high order accuracy compact finite difference scheme on non-uniform meshes, the construction of linear and nonlinear perfectly matched layer (PML) absorbing boundary condition for oblique flow calculation. The accuracy of proposed schemes and non-refection PML absorbing boundary conditions is demonstrated by CAA benchmark problems. The detailed research works and results of this dissertation are listed as follows:(1) An optimization method for constructing high order accuracy explicit finite difference scheme is proposed. In order to obtain high order accuracy numerical schemes and improve the accuracy for computing the propagation of aeroacoustic wave in long time or long distance. First, the finite difference scheme is transformed into frequency and wavenumber space by applying Fourier and Laplace transformation method. Then the mathematical formula of error construction, error propagation and error accumulation of high order finite difference schemes are derived in detail using Neumann error analysis method and correction error propagation analysis method. This can provide a theoretical basis for developing low dissipation and low dispersion higher order accuracy numerical schemes. Based on the idea of dispersion relation preserving optimization, a new strategy to optimize finite difference schemes in spectral domain is proposed considering the effects of group velocity on dispersion characteristics. The objective of optimization is minimizing the integrated absolute error with a strict tolerance for dispersion, dissipation and group velocity errors. The scheme coefficients and optimized wavenumber domain are determined using iterative calculations and sequential quadratic programming method. The7-point,9-point and11-point finite difference schemes are optimized using the new strategy, and the dispersion, dissipation and group characteristics of these optimized schemes are analyzed. At last, the results of error analysis and optimization method are validated by CAA benchmark problems.(2) A new method for constructing high order accuracy compact finite difference scheme on non-uniform meshes is proposed. When the classic compact finite difference scheme was applied to practical problems using non-uniform meshes, the spurious numerical oscillations would be excited. Based on the smooth techniques, the general mathematical expressions of inner and boundary point scheme’s coefficients are derived through Taylor expansion method. Then a sixth order tridiagonal compact scheme is presented based on the theory expressions. The numerical dispersion and dissipation on different mesh types are analyzed by applying Fourier and Laplace transformation to the sixth order tridiagonal compact scheme. This paper mainly studies the influence of uniform mesh, perturbed mesh, stretched mesh and sudden mesh on the numerical dispersion and dissipation, especially for the perturbed factor. Furthermore, the asymptotic stability of semi discrete scheme on different mesh types is researched using eigenvalue method. At last, the resolution performance of the scheme is demonstrated using CAA benchmark problems. Numerical results show that the numerical solutions for CAA benchmark problems agree well with the theoretical solutions. The compact difference scheme proposed in this paper is stability, accurate and simple for CAA problems on non-uniform meshes, which shows advantages in the simulation of CAA problems on non-uniform meshes.(3) A seven steps general method for constructing PML absorbing boundary conditions in case of convection flow is proposed. First, the dispersion relation of linear controlled equations is established by employing Fourier and Laplace transform method. Then the propagation directions of the phase and group velocities of the physical waves are analyzed according to the dispersion relation. Two kinds of proper space-time transformations are determined under the hypothesis of unchanged frequency (UCF) and changed frequency (CF), respectively. UCFPML and CFPML absorbing boundary conditions for linear Euler equations in two dimensions are developed when mean velocities strike the boundary at an arbitrary angle, and the UCFPML and CFPML equations for the side layers and corner layers of a rectangular domain will be derived independently. Furthermore, the stability of linear wave in Hu’s split PML, UCFPML and CFPML layers are analyzed using eigenvalve method. The influence of mean flow velocities, wavenumbers, absorption parameters and added absorption term on stability is studied in detail. Finally, these PML absorbing boundary conditions for linear Euler equations in the case of oblique mean flow are validated by computing the CAA benchmark problem. The phase velocity should be consistent with the direction of group velocities of the physical waves, which is a necessary but not sufficient condition for stable PML absorbing boundary conditions. For the three PML absorbing boundary conditions, Hu’s split PML equations are instability, UCFPML equations are stable with some conditions, and CFPML equations are very stable. Moreover, the acoustic, vorticity and entropy wave will be exponentially decreasing with minimizing boundary reflections in the PML domain. Therefore, the CFPML absorbing boundary conditions are effective for computing CAA problems in case of oblique flow.(4) Three kinds of nonlinear PML absorbing boundary conditions in two dimensional are proposed in case of oblique flow. First, the pseudo mean flow is introduced for formulating the time-dependent fluctuation nonlinear equations in conservation and primitive form. Then these time-dependent fluctuation nonlinear equations are transformed into new space-time coordinates using CF space-time transformations proposed in section4. For nonlinear Euler equations in oblique flow in two dimensions, the primitive PML (PPML), unsplit conservation PML (USCPML) and split conservation PML (SCPML) absorbing boundary conditions are established by applying the PML complex change of time-dependent fluctuation nonlinear equations in new space-time coordinates. These PML equations for the side layers and corner layers of a rectangular domain are also formulated independently. The nonlinear stability analysis of nonlinear PML equations is translated into linear stability analysis using linearity hypothesis, the influence of mean flow velocities, wavenumbers, absorption parameters and added absorption term on stability is studied in detail, and the importance of added absorption term is emphasized. Finally, these PML absorbing boundary conditions for nonlinear Euler equations in the case of oblique mean flow are validated by computing the CAA benchmark problems. Numerical results show that the imaginary part of eigenvalue is nonpositive for every eigenvalue when the absorption terms are added to corner PML equations. Therefore, these PML equations will be dynamically stable. Otherwise, these PML equations will be not stable if absorption terms are not added to corner PML equations. Among the three nonlinear PML absorbing boundary conditions, PPML is the best, SCPML is better, USCPML is the worst one. The three nonlinear PML absorbing boundary conditions are effective for computing CAA problems in case of oblique flow due to no boundary reflections in the PML domain, so the method for developing nonlinear PML absorbing boundary conditions in case of oblique flow is effective and feasible.(5) The linear and nonlinear PML absorbing boundary conditions in three dimensions for the more general case of an oblique mean flow are proposed. The goal of this work is to further extend the two dimensional linear and nonlinear PML methodology in three dimensions. The three dimensional PML computational domain is divided into twenty six subdomains and seven categories. According to the path of dispersion relationship of physical waves, a series of appropriate space-time transformations for correcting the inconsistencies in phase and group velocity is presented with the hypothesis of unchanged frequency. Based on the space-time transformations, the derivations of linear PML, primitive nonlinear PML (PPML), unsplit conservation nonlinear PML (USCPML) and split conservation nonlinear PML (SCPML) equations are illustrated in details using PML complex transformation method, and the adding principle of absorption term for corner PML equations is emphasized. In addition, the influence of mean flow velocities, wavenumbers, absorption parameters and added absorption term on stability of corner PML equations is studied. Finally, the validity and efficiency of proposed equations as linear and nonlinear PML absorbing boundary conditions are demonstrated by numerical examples. The numerical results show that the acoustic wave and vortex ring will be exponentially decreasing with minimizing boundary reflections in the PML domain. Therefore, the proposed PML absorbing boundary conditions are effective and stable for computing CAA problems when mean velocities strike the boundary at an arbitrary angle.