## Computational Aeroacoustics

Computational Aeroacoustics (CAA) is a relatively young research area. It began in earnest fewer than twenty years ago. During this time, CAA algorithms have developed rapidly. These methods soon found applications in many areas of aero – acoustics.

The objective of CAA is not simply to develop computational methods, but also to use these methods to solve real practical aeroacoustics problems. It is also a goal of CAA to perform numerical simulation of aeroacoustic phenomena. By analyzing the simulation data, an investigator can determine noise generation mechanisms and sound propagation processes. Hence, CAA offers a way to obtain a better understanding of the physics of a problem.

Computational Aeroacoustics is not the same as Computational Fluid Dynamics (CFD). In fact, CAA faces a different set of computational challenges, because aeroacoustics problems are intrinsically different from standard aerodynamics and fluid mechanics problems. By definition, aeroacoustics problems are time dependent, whereas aerodynamics and fluid mechanics problems are, in general, time independent or involve only low-frequency unsteadiness. The following list outlines some of the major computational challenges facing CAA:

1. Aeroacoustics problems typically involve a broad range of frequencies. Numerical resolution of the high-frequency waves with extremely short wavelengths becomes a formidable obstacle to accurate numerical simulation.

2. The amplitudes of acoustic waves are usually small, in particular, when compared with the mean flow. Oftentimes, the sound intensity of the acoustic waves is five to six orders smaller than that of the mean flow. To compute sound waves accurately, a numerical scheme must have high resolution and extremely low numerical noise.

3. In most aeroacoustics problems, interest is in the sound waves radiated to the far field. This requires a solution that is uniformly valid from the source region all the way to the measurement point many acoustic wavelengths away. Because of the long propagation distance, computational aeroacoustics schemes must have minimal numerical dispersion and dissipation. They should also propagate the waves at the correct wave speeds independent of the orientation of the computation mesh.

4. A computation domain is inevitably finite in size. For aerodynamics or fluid mechanics problems, flow disturbances tend to decay very quickly away from a body or their source of generation; therefore, the disturbances are usually small at the boundary of the computation domain. Acoustic waves, on the other hand, decay very slowly and actually reach the boundaries of a finite computation domain. To avoid the reflection of outgoing sound waves back into the computation domain and thus contaminating the solution, specially developed radiation and outflow or absorbing boundary conditions must be imposed at the artificial exterior boundaries to assist the waves in exiting smoothly. For standard CFD problems, such boundary conditions are usually not required.

5. Aeroacoustics problems are archetypical examples of multiscales problems. The length scale of the acoustic source is usually very different from the acoustic wavelength. That is, the length scale of the source region and that of the acoustic region can be vastly different. CAA methods must be designed to handle problems involving tremendously different length scales in different parts of the computational domain.

Many of these major computational issues and challenges to CAA have now been resolved or at least partly resolved. This book offers an overview of the methods, analysis, and new ideas introduced to overcome such challenges.

It should be clear, as elaborated above, that the nature of aeroacoustics problems is substantially different from that of traditional fluid dynamics and aerodynamics problems. As a result, standard CFD schemes, designed for applications to fluid mechanics problems, are generally not adequate for computing or simulating aero – acoustics problems accurately and efficiently. For this reason, there is a need for the independent development of CAA.

At the outset of CAA development, it was clear to investigators that a fundamental need in CAA was to develop algorithms that could resolve high-frequency short waves with the minimum number of mesh points per wavelength. Standard CFD second-order schemes often require 18 to 25 mesh points per wavelength to ensure adequate accuracy. This large number of mesh points is clearly not acceptable if the method is to be adopted for practical computation. It was soon recognized that a solution to the problem was to use large-stencil high-resolution CAA schemes. At present, high-resolution algorithms implemented on stencils of a reasonable size can resolve waves using about six to seven mesh points per wavelength.

The small amplitude of acoustic waves in comparison with that of the mean flow has raised a good deal of initial apprehension as to whether the inherent noise level of a numerical scheme used to compute the mean flow would overwhelm the actual radiated sound. Such apprehension is well founded, for experience has indicated that some CFD schemes do have a high intrinsic noise level. The development of new high-resolution CAA methods has proven that it is, indeed, possible to capture sound waves of minute amplitude with good accuracy.

One of the most significant differences between traditional CFD and CAA methodology is the method of error analysis. In CFD, the standard way to assess the quality of a scheme is by the order of Taylor series truncation. In general, it is assumed that a fourth-order scheme is better than a second-order scheme, which, in turn, is better than a first-order scheme, but all such assessments are qualitative not quantitative. There is no way to find out by order-of-magnitude analysis how many mesh points per wavelength are needed to achieve, say, a half-percent accuracy in a computation. Furthermore, traditional numerical analysis does not provide a way to quantify wave propagation errors. Dispersion and dissipation errors are often erroneously linked to the phase velocity and amplification factor.

The development of wave number analysis, through the use of Fourier-Laplace transforms, has provided a firm mathematical foundation for error analysis in CAA. Wave number analysis shows that the order of a scheme is not the most important factor in achieving high-quality results. Instead, the resolved bandwidth of a scheme in wave number space is the more important issue. As far as numerical wave propagation error is concerned, wave number analysis shows that the phase velocity is totally irrelevant. Rather, it is the dependence of the group velocity of the scheme on wave number that is important. In wave propagation, space and time play an important partnership role. The relationship is all encoded in the dispersion function. Thus, a dispersion-relation-preserving scheme (a numerical scheme having formally the same dispersion relation as the original partial differential equations) would not only automatically guarantee a numerically accurate solution, but it would also replicate the number of wave modes (acoustic, vorticity, and entropy) and the characteristics supported by the original partial differential equations. Wave number analysis provides an understanding of the existence and characteristics of spurious short waves. Such knowledge allows the design of very effective artificial selective damping stencils and filters. Wave number analysis, together with the dispersion relation of the discretized equations, offers a simple quantitative method for analyzing the numerical stability of CAA algorithms. Such an analysis is crucial when selecting the size of time marching step.

The development of numerical boundary conditions is also an integral part of CAA, and the importance of high-quality numerical boundary conditions for CAA can never be overemphasized. There is no exaggeration in saying that the development of high-quality numerical boundary conditions is as important as the development of a high-quality time marching scheme. The construction and analysis of numerical boundary conditions form a core part of the materials covered in this book.

A large difference between the size of a noise source and the acoustic wavelength would invariably result in a multiscale problem. However, very often, large length scale disparity arises because of a change in the dominant physics in different parts of the computational domain. For example, for the problem of sound waves dissipation by a resonant acoustic liner, the dominant physics adjacent to the solid surface and hole openings of the liner is viscous effect. An oscillatory Stokes layer, driven by the sound field, would develop. The thickness of the Stokes layer (the scale to be resolved) is very small even when excited by moderately low-frequency sound. Away from the wall, compressibility effect dominates. The length scale is the acoustic wavelength, which can be hundreds of times the thickness of the viscous Stokes layer. Another instance that would lead to severe length scale disparity is when sound waves propagate against a flow. The wavelength of an upstream propagating sound wave is drastically reduced over the region where the mean flow is transonic. Numerical treatment of multiscale problems is an important part of this book.

Once the appropriate physical model and computational methods have been developed to study a real phenomenon computationally, it is necessary to design a simulation code. Designing a computer code is very different from developing a computational algorithm. A computer code is formed by synthesizing many elements of basic computational methods. A good code should be stable, accurate, and efficient. Because many students of CAA may not have experience in designing computer code for real problems, an effort is made in this book to provide some guidance on code design. As a part of this effort, examples at a modest level of complexity are provided.

This book is written both as a text for graduate students and as a reference for researchers using CAA. Every effort has been made to make this book self – contained. No prior knowledge of numerical methods for solving partial differential equations is needed; however, a general understanding of partial differential equations and basic numerical analysis is assumed. Exercises are included at the end of many chapters; they are designed to be an integral part of the chapter content. In addition, sample computer programs are included to illustrate the implementation of the numerical algorithms.

This book has benefited from the work and publications of many investigators. In particular, the contributions of Drs. Jay C. Webb, Konstantin K. Kurbatskii, Nikolai N. Pastouchenko, Hongbin Ju, Hao Shen, Fang Q. Hu, Tom Dong, Laurent Auriault, Andrew T. Thies, and Philip P. LePoudre should be mentioned. Their work and that of others have made this book possible. The assistance of Dr. Sarah A. Parrish is greatly appreciated.

Tallahassee, Florida, 2011