15-11-2012, 05:36 PM
DISPERSION RELATION PRESERVING SCHEMES FOR COMPUTATIONAL AEROACOUSTICS
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INTRODUCTION
Aeroacoustics is a sub-field of acoustics in which sound generated from flows is studied. In Computational Aero Acoustic (CAA), such sound is computed numerically. A computational Aeroacoustics (CAA) deal with the simulation of sound generated by unsteady flows and it is a rapidly growing area due to advances in computational power and the significant projected growth in global transportation. With the era of widespread supersonic flight and the proliferation of general aviation aircraft on future horizons, the noise generated by aircraft is of great concern for communities near airports, for passengers in the aircraft’s cabin, and for the structural integrity of the airframe. In addition, there are a number of situations that desire lower noise including underwater vehicles, wind turbines, and helicopter rotors. Understanding the source of the noise itself, its manifestation in the nearfield and propagation to the farfield are all critical in the development of future noise reduction technologies. When compared to conventional flow computations, CAA requires special treatment in the areas of numerical errors, low numerical noise, numerical dispersion, dissipation, non-reflective boundary conditions, methodologies to test boundary condition performance, and consideration of multiple scales. It must be acknowledged that CFD has been very successful in solving fluid and aerodynamics problems. CFD methods are generally designed for computing time independent solutions. Because of the tremendous success of CFD, it is tempting to use these methods to solve aeroacoustics problems as well.
Difference between CFD and CAA
Aeroacoustic problems are by nature very different from standard aerodynamics and fluid mechanics problems. Before discussing about aero acoustics problems numerically, an approach generally referred to as Computational Aeroacoustics (CAA), it is important to recognize and to have a good understanding of these differences given by Tam1 (1995). These differences pose a number of major challenges to CAA. A few of the important computational challenges are listed below.
1. Aeroacoustics problems, by definition, are time dependent, whereas aerodynamics and fluid mechanics problems are generally time independent or involve only low frequency unsteadiness.
Motivation for DRP Schemes
In computational aeroacoustics (CAA) there is a need to propagate waves over long distances, therefore the schemes using for these purpose must ideally be nondispersive and nondissipative. To meet these requirements, the use of high-order schemes becomes a necessity. Recently Tam and Webb2 (1993) developed a high-order finite difference method known as the Dispersion-Relation-Preserving (DRP) Finite Difference scheme specifically designed for CAA and wave propagation applications. This scheme is a central difference scheme and thus has no intrinsic numerical damping. Numerical dispersion is controlled by the group velocity of the numerical method. The stencil coefficients of the DRP scheme are chosen by Tam and Webb2 (1993) in order to ensure that the computation has a wide resolved bandwidth in wave number space but also the group velocity is equal to wave propagating velocity of the physical system. One unique characteristic of the scheme is that it would automatically preserve the dispersion relations of the partial differential equations in the resolved range of wave number and frequency. This assures that the wave modes supported by the original equations are reproduced faithfully by the finite difference equations.
Amplification Factor
Some dissipation and dispersion occur naturally in most physical systems described by PDEs. Errors in magnitude are termed dissipation and errors in phase are called dispersion. The term amplification factor is used to represent the change in the magnitude of a solution over time. Dissipation and dispersion can also be introduced when PDEs are discretized in the process of seeking a numerical solution. This introduces numerical errors. The accuracy of a discretization scheme can be determined by comparing the numeric amplification factor Gnumeric; with the analytical or exact amplification factor Gexact, over one time step.
Numerical Dispersion
In a numerical scheme, a situation where waves of different frequencies move at different speeds without a change in amplitude is called numerical dispersion - see Fig.2. Alternatively, the Fourier components of a wave can be considered to disperse relative to each other. It therefore follows that the effect of a dispersive scheme on a wave composed of different harmonics will be to deform the wave as it propagates. However the energy contained within the wave is not lost and travels with the group velocity. Generally, this results in higher frequency components travelling at slower speeds than the lower frequency components. The effect of dispersion therefore is that often spurious oscillations or wiggles occur in solutions with sharp gradients, discontinuities or shock effects, usually with high frequency oscillations trailing the particular effect, see Fig.3.Graham4 (2009) explained that the degree of dispersion can be determined by comparing the phase of the numeric amplification factor |Gnumeric|, with the phase of the exact amplification factor |Gexact|, over one time step. Dispersion represents phase shift and results from the imaginary part of the amplification factor.
NUMERICAL SCHEMES THAT MINIMIZE DISPERSION
The growing demand by aerospace, automotive and other industries for accurate and reliable noise prediction models has prompted the development of new computational aeroacoustic (CAA) methods. These are used not only as noise prediction tools, but also to evaluate new approaches for noise reduction and control. Different aeroacoustic problems often exhibit different flow physics. As a result, there is no single algorithm that can be used to simulate all problems with adequate resolution and accuracy. The quality of CAA predictions is affected by numerical dispersion and dissipation, the performance of acoustically transparent boundary conditions, the ability to simulate nonlinearities and to resolve disparate length scales. Any investigator developing a new CAA algorithm or applying an existing method must ensure that the method adequately addresses the above.