Numerical solution of wave scattering problems in the parabolic approximation

Please cite this article in press as: A. Towne, T. Colonius, One-way spatial integration of hyperbolic equations, J. Comput. Phys. (2015), http://dx. AbstractIn this paper, we develop and demonstrate a method for constructing well-posed one-way approximations of linear hyperbolic systems. We use a semi-discrete approach that allows the method to be applied to a wider class of problems than existing methods based on analytical factorization of idealized dispersion relations. After establishing the existence of an exact one-way equation for systems whose coefficients do not vary along the axis of integration, efficient approximations of the one-way operator are constructed by generalizing techniques previously used to create nonreflecting boundary conditions. When physically justified, the method can be applied to systems with slowly varying coefficients in the direction of integration. To demonstrate the accuracy and computational efficiency of the approach, the method is applied to model problems in acoustics and fluid dynamics via the linearized Euler equations; in particular we consider the scattering of sound waves from a vortex and the evolution of hydrodynamic wavepackets in a spatially evolving jet. The latter problem shows the potential of the method to offer a systematic, convergent alternative to ad hoc regularizations such as the parabolized stability equations.

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“…Next, we apply the termination condition given by equation (25c). To do so, we write the left-hand-side of equation (30) in terms ofφ N β :…”

Section: Equation (25b) Can Then Be Writtenmentioning

confidence: 99%

“…This problem was previously investigated using a one-way wave equation [30] (which the author called a parabolic approximation) and direct numerical simulation [31]. We will compare our one-way Euler results to the later.…”

Section: Scattering Of Acoustic Waves By a Vortexmentioning

confidence: 99%

One-way spatial integration of hyperbolic equations

Towne

Colonius

2015

Journal of Computational Physics

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“…For a known vorticity distribution, rays are then computed either by solving Eq. (9) directly (or the associated Helmholtz equation [10][11][12]) or by using a ray-tracing scheme based on Eqs. (11) and (13).…”

Section: Geometrical Acousticsmentioning

confidence: 99%

Sound and vorticity interactions: transmission and scattering

Pinton

Brillant

2004

Theoret. Comput. Fluid Dynamics

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Abstract.We review several aspects of the propagation of sound in vortical flows. We restrict ourselves to isothermal, humidity-free flows at low Mach number M. Since vorticity plays a major role in vortex-flow interactions we focus on vortical flows. We consider two main canonical situations. The first concerns the transmission of sound. We analyze the evolution of acoustic wavefronts as they propagate across a single vortex. The second situation addresses the scattering of sound waves by nonstationary vortices. We study the evolution of the acoustic pressure emitted in the far field, at an angle with the initial direction of propagation. In this geometry one performs direct spectroscopy of the flow vorticity field. In each case, we review theoretical results and compare with experimental measurements and numerical simulations when available. We also briefly report how the following new acoustic techniques have recently been used to study complex or turbulent flows: time-resolved acoustic spectroscopy, speckle interferometry and Lagrangian particle tracking.

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“…In the general case where the flowfield is three dimensional or when the sound source is situated off the symmetry axis of the flow, the radiated Presented as Paper 81-2042 at the AIAA 7th Aeroacoustics Meeting, Palo Alto, Calif., Oct. [3][4][5]1981; submitted Oct. 16,1981; revision received April 17, 1982 wavefield may be calculated effectively by making use of standard approximations. One suitable method for these complicated problems is based on the geometrical approximation.…”

Section: Introductionmentioning

confidence: 99%

“…This method is well suitefi to the treatment of weakly inhomogeneous cases such as those encountered in scattering problems. 3 Now, in many practical situations one is interested in acoustic fields of low or intermediate frequencies propagating in strongly nonhomogeneous flows. While the general three-dimensional problems of this type cannot be solved without resorting to finite difference or finite element computations, it is much easier to deal with shear flow situations.…”

Section: Introductionmentioning

confidence: 99%

Sound source radiation in two-dimensional shear flow

Candel

1983

AIAA Journal

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A fundamental problem encountered in the analysis of aerodynamic noise is that of acoustic source radiation in nonhomogeneous flow. Exact numerical solutions have been obtained recently for source radiation near a plane interface and a shear discontinuity by directly synthesizing the wavefield from its spatial Fourier transform. This method is applied here to stratified flow configurations and the structure of the radiated field is obtained for several shear layer velocity profiles of practical interest. NomenclatureAj (a),Bj(ot) = amplitudes of outgoing and incoming waves in layerj c = sound speed d = layer thickness / = complex number, V -1 k = wavenumber f = spatial extension of the calculation domain L = layer index corresponding to a selected value of attenuation M = Mach number N = number of points in the discrete Fourier transform = pressure = Fourier transform of pressure = total number of shear discontinuities used to represent the flow = cumulated transfer matrix = layer transfer matrix = mean flow velocity = Cartesian coordinates '= spatial wavenumber, argument of Fourier transform = branch points of jj = phase or damping factor for waves in layer j = delta function = spatial sampling period = spatial wavenumber sampling period = interfacial displacement = polar angle measured with respect to the x axis = wavelength = angular frequency P P Q s T U x,z a. Ax 0 A CO Subscripts * j = reduced (dimensionless) quantity = layer index

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Numerical solution of wave scattering problems in the parabolic approximation

Cited by 57 publications

References 29 publications

One-way spatial integration of hyperbolic equations

One-way spatial integration of hyperbolic equations

Sound and vorticity interactions: transmission and scattering

Sound source radiation in two-dimensional shear flow

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