Maarten Hornikx

The extended Fourier pseudospectral time-domain method for atmospheric sound propagation

An extended Fourier pseudospectral time-domain (PSTD) method is presented to model atmospheric sound propagation by solving the linearized Euler equations. In this method, evaluation of spatial derivatives is based on an eigenfunction expansion. Evaluation on a spatial grid requires only two spatial points per wavelength. Time iteration is done using a low-storage optimized six-stage Runge-Kutta method. This method is applied to two-dimensional non-moving media models, one with screens and one for an urban canyon, with generally high accuracy in both amplitude and phase. For a moving atmosphere, accurate results have been obtained in models with both a uniform and a logarithmic wind velocity profile over a rigid ground surface and in the presence of a screen. The method has also been validated for three-dimensional sound propagation over a screen. For that application, the developed method is in the order of 100 times faster than the second-order-accurate FDTD solution to the linearized Euler equations. The method is found to be well suited for atmospheric sound propagation simulations where effects of complex meteorology and straight rigid boundary surfaces are to be investigated.

A multi-domain Fourier pseudospectral time-domain method for the linearized Euler equations

The Fourier pseudospectral time-domain (F-PSTD) method is computationally one of the most cost-efficient methods for solving the linearized Euler equations for wave propagation through a medium with smoothly varying spatial inhomogeneities in the presence of rigid boundaries. As the method utilizes an equidistant discretization, local fine scale effects of geometry or medium inhomogeneities require a refinement of the whole grid which significantly reduces the computational efficiency. For this reason, a multi-domain F-PSTD methodology is presented with a coarse grid covering the complete domain and fine grids acting as a subgrid resolution of the coarse grid near local fine scale effects. Data transfer between coarse and fine grids takes place utilizing spectral interpolation with super-Gaussian window functions to impose spatial periodicity. Local time stepping is employed without intermediate interpolation. The errors introduced by the window functions and the multi-domain implementation are quantified and compared to errors related to the initial conditions and from the time iteration scheme. It is concluded that the multi-domain methodology does not introduce significant errors compared to the single-domain method. Examples of scattering from small scale density scatters, sound reflecting from a slitted rigid object and sound propagation through a jet are accurately modelled by the proposed methodology. For problems that can be solved by F-PSTD, the presented methodology can lead to a significant gain in computational efficiency.

Urban background noise mapping: the general model

Surveys show that inhabitants of dwellings exposed to high noise levels benefit from having access to a quiet side. However, current practice in noise prediction often underestimates the noise levels at a shielded facade. Multiple reflections between facades in street canyons and inner yards are commonly neglected and facades are approximated as perfectly flat surfaces yielding only specular reflection. In addition, sources at distances much larger than normally taken into account in noise maps might still contribute significantly. Since one of the main reasons for this is computational burden, an efficient engineering model for the diffraction of the sound over the roof tops is proposed, which considers multiple reflections, variation in building height, canyon width, facade roughness and different roof shapes. The model is fitted on an extensive set of full-wave numerical calculations of canyon-to-canyon sound propagation with configurations matching the distribution of streets and building geometries in a typical historically grown European city. This model allows calculating the background noise in the shielded areas of a city, which could then efficiently be used to improve existing noise mapping calculations. The model was validated by comparison to long-term measurements at 9 building facades whereof 3 were at inner yards in the city of Ghent, Belgium. At shielded facades, a strong improvement in prediction accuracy is obtained.

Application of the Fourier pseudospectral time-domain method in orthogonal curvilinear coordinates for near-rigid moderately curved surfaces

The Fourier pseudospectral time-domain method is an efficient wave-based method to model sound propagation in inhomogeneous media. One of the limitations of the method for atmospheric sound propagation purposes is its restriction to a Cartesian grid, confining it to staircase-like geometries. A transform from the physical coordinate system to the curvilinear coordinate system has been applied to solve more arbitrary geometries. For applicability of this method near the boundaries, the acoustic velocity variables are solved for their curvilinear components. The performance of the curvilinear Fourier pseudospectral method is investigated in free field and for outdoor sound propagation over an impedance strip for various types of shapes. Accuracy is shown to be related to the maximum grid stretching ratio and deformation of the boundary shape and computational efficiency is reduced relative to the smallest grid cell in the physical domain. The applicability of the curvilinear Fourier pseudospectral time-domain method is demonstrated by investigating the effect of sound propagation over a hill in a nocturnal boundary layer. With the proposed method, accurate and efficient results for sound propagation over smoothly varying ground surfaces with high impedances can be obtained.

A model of sound scattering by atmospheric turbulence for use in noise mapping calculations

Sound scattering due to atmospheric turbulence limits the noise reduction in shielded areas. An engineering model is presented, aimed to predict the scattered level for general noise mapping purposes including sound propagation between urban canyons. Energy based single scattering for homogeneous and isotropic turbulence following the Kolmogorov model is assumed as a starting point and a saturation based on the von Karman model is used as a first-order multiple scattering approximation. For a single shielding obstacle the scattering model is used to calculate a large dataset as function of the effective height of the shielding obstacle and its distances to source and receiver. A parameterisation of the dataset is used when calculating the influence of single or double canyons, including standardised air attenuation rates as well as facade absorption and Fresnel weighting of the multiple facade reflections. Assuming a single point source, an aver aging over three receiver positions and that each ground reflection causes energy doubling, the final engineering model is formulated as a scattered level for a shielding building without canyon plus a correction term for the effect of a single or a double canyon, assuming a flat rooftop of the shielding building. Input parameters are, in addition to geometry and sound frequency, the strengths of velocity and temperature turbulence.

Flow and geometrical effects on radiated noise from exhausts computed by a hybrid extended Fourier PSTD method

In urban areas, a significant source of noise from road traffic i s the contribution by the sound radiated from exhaust pipes. The current work has initialized a systematic numerical study on the combined effects of the geometry surrounding the exhaust pipe, i.e. the ground surface and the automotive body, as well as the influence of the non-uniform velocity and temperature from the exhaust jet on the directivity and power radiated from the pipe. The problem is treated by first solving the RANS equations to obtain the mean velocity and temperature solution of the exhaust jet. Then, the acoustic radiation is computed by solving the linearized Euler (LEE) equations. For an efficient solution of the LEE, a recently developed multi-domain extended Fourier pseudospectral time-domain (PSTD) methodology is used. This method combines the favorable spectral accuracy of the PSTD method with a local grid refinement in the region with high gradients of the mean flow and temperature fields. The numerical methodology is successfully verified for a case with a hot flow by a comparison with a reference numerical methodology. The presence of a rigid ground surface and simplified automotive body is shown to increase the radiated sound power by 6 dB for the lower frequency region, and the effect of the body on the directivity is largest for the higher frequencies. Flow effects slightly increase the shielding effect of the body for all frequencies, but have a main impact behind the exhaust pipe, where low frequencies experience higher levels and a cone of low sound levels characterizes the high frequencies.

The extended Fourier pseudospectral time-domain (PSTD) method for fluid media with discontinuous properties

The Fourier pseudospectral time-domain (PSTD) method is an attractive method to efficiently model wave propagation through a weakly inhomogeneous fluid medium. The method fails, however, for fluid media with discontinuous properties. This failure is due to the well known Gibbs phenomenon which arises when Fourier transforming a discontinuous function. The extended Fourier PSTD method developed here can accurately and efficiently model wave propagation through weakly inhomogeneous fluid media with discontinuities in the media properties. Rather than using Fourier transforms to calculate the spatial derivatives in the wave equation, the method developed here uses a generalized eigenfunction expansion for which no Gibbs phenomenon arises. Two approaches to solving the resulting time-domain problem are explored and the accuracy of the method is demonstrated. Finally, results from an example calculation of transmission through the water-air interface are shown.

An eigenfunction expansion method to efficiently evaluate spatial derivatives for media with discontinuous properties

Pseudo‐Spectral methods are often used as an alternative to the Finite Difference Time Domain (FDTD) method to model wave propagation in heterogeneous moving media. The FDTD method is robust and accurate but is numerically expensive. Pseudo‐Spectral methods make use of the wavelike nature of the solution to obtain more efficient time‐domain algorithms. The most straightforward of the Pseudo‐Spectral methods is the Fourier method in which a spatial Fourier transform is used to evaluate the spatial derivatives in the wave equation. Whereas this method is accurate for a weakly heterogeneous moving medium, it degenerates for media with discontinuous properties. The eigenfunction expansion method presented here is a way to accurately and efficiently evaluate spatial derivatives in media with interfaces. As in the Fourier method, transforms may be calculated using FFTs and spatial sampling is limited only by the Nyquist condition. The performance of the method is shown in a time‐domain implementation for media...

Urban background noise mapping : the multiple-reflection correction term

Mapping of road traffic noise in urban areas according to standardized engineering calculation methods systematically results in an underestimation of noise levels at areas shielded from direct exposure to noise, such as inner yards. In most engineering methods, road traffic lanes are represented by point sources and noise levels are computed utilizing point-to-point propagation paths. For a better prediction of noise levels in shielded urban areas, an extension of engineering methods by an attenuation term Acan has been proposed, including multiple reflections of the urban environment both in the source and in the receiver area. The present work has two main contributions for the ease of computing Acan. Firstly, it is shown by numerical calculations that Acan may be divided into independent source and receiver environment terms, As and Ar. Based on an equivalent free field analogy, the distance dependence of these terms may moreover be expressed analytically. Secondly, an analytical expression is proposed to compute As and Ar for 3D configurations from using 2D configurations only. The expression includes dependence of the street width-to-height ratio, the difference in building heights and the percentage of facade openings in the horizontal plane. For the expression to be valid, the source should be separated from the receiver environment by at least four times the street width.

A scale model study of parallel urban street canyons

The access to quiet areas in cities is of increasing importance. Recently, the equivalent sources method for a two dimensional situation of parallel urban street canyons has been developed. One canyon represents a busy road, whereas the other is one without traffic; the quiet side. With the model, the transfer function between the two canyons can be calculated, as well as the influence of diffusion, absorption, and atmospheric turbulence on the transfer function. A scale model study of two parallel canyons has now been executed. A scale of 1:40 has been chosen and the maximum length sequence technique has been applied using the MLSSA system. Results of the scale model study have been compared to calculations with the equivalent sources method. The difference between a two-dimensional and a three-dimensional quiet side, between a coherent and an incoherent line source and the influence of absorption and diffusion has been investigated. The scale model study also gives insight in the evolution of the sound field in the time domain. [Work supported by the Swedish Foundation for Strategic Environmental Research (MISTRA).]