Kirchhoff Integral Method Research Articles

Simulation of Isolated and Installed Jet Noise at Mach = 0.9: Influence of Numerical Mesh and Physical Insights

AbstractThe noise radiated by an isothermal, single-stream jet with a Mach number M = 0.9 and diameter-based Reynolds number ReD = 106 is investigated numerically without and with the presence of a flat plate. Noise sources are predicted with Zonal Detached Eddy Simulations yielding Wall Modelled LES in attached boundary layers (so called ZDES mode 3) together with turbulence tripping inside the nozzle to recover an initially turbulent flow, while radiated pressure is extrapolated with integral methods. Numerical methodology, namely grid and statistical convergence of the signals, is assessed for the isolated jet. Noise levels are accurately simulated at least up to St = 8 and integrated pressure levels collapse within 1 dB with the experiments. In the presence of the plate, a noise radiation methodology based on both Ffowcs Williams Hawkings and Kirchhoff integral methods is proposed to reconstruct the pressure signals at microphone locations with a reduced numerical cost. The simulation compares very favorably with the experimental data, azimuthal noise variations induced by the plate are correctly captured and noise levels collapse within 1 dB. It is concluded the numerical methodology is mature enough for application in an industrial context.

Wavelet Entropy-Based Method for Migration Imaging of Hidden Microcracks by Using the Optimal Wave Velocity

Exploring the shape and direction of hidden cracks in a tunnel lining structure is one of the main objectives of ground penetrating radar (GPR) map interpretation. The most important factor that restricts the migration imaging of hidden cracks is the propagation velocity of electromagnetic waves. Determining the optimal electromagnetic wave velocity is the key to truthfully restoring the actual shape of hidden cracks. To study the GPR characteristic response signals of hidden cracks, forward simulation and model experiments of different cracks were performed. Subsequently, a method to determine the optimal electromagnetic wave velocity based on the wavelet entropy theory was proposed, and the frequency wavenumber domain migration (F-K) and Kirchhoff integral migration imaging method were combined. Horizontal, S-type, and inclined hidden fractures were examined by migration imaging. The results show that the radar characteristic response images of different cracks can be simulated forward by using the finite difference time domain method to write the fracture model instruction. Based on the wavelet entropy theory, the error range between the estimated value and true value was controlled within 4%. Taking the optimal electromagnetic wave velocity as the velocity parameter of the conventional migration method can make the migration more effective and suppress the interference of echo signals so that the diffraction wave converges, and the energy is more concentrated; thus, the real fracture morphology can be restored to the greatest extent. The research results can provide technical support for the fine detection of hidden quality defects in tunnel lining structures by GPR mapping.

Validation of Fresnel–Kirchhoff Integral Method for the Study of Volume Dielectric Bodies

In this work, we test a nondestructive optical method based on the Fresnel–Kirchhoff integral, which could be applied to different fields of engineering, such as detection of small cracks in structures, determination of dimensions for small components, analysis of composition of materials, etc. The basic idea is to apply the Fresnel–Kirchhoff integral method to the study of the properties of small-volume dielectric objects. In this work, we study the validity of this method. To do this, the results obtained by using this technique were compared to those obtained by rigorously solving the Helmholtz equation for a dielectric cylinder of circular cross-section. As an example of the precision of the method, the Fresnel–Kirchhoff integral method was applied to obtain the refractive index of a hair by fitting the theoretical curve to the experimental results of the diffraction pattern of the hair measured with a CCD camera. In a same manner, the method also was applied to obtain the dimensions of a crack artificially created in a piece of plastic.

The logarithmic mean method for Nonlinear Conductivities at Cell Faces

In this work, logarithmic mean method, which is based on the assumption that the diffusion coefficients profile is linear between nodes, is developed to calculate diffusion coefficients at cell interfaces. The new method is evaluated by solving one dimension steady diffusion equation and convection diffusion equation with strongly non-linear diffusion coefficients. Results indicate that the logarithmic mean method can achieve high precision as the Kirchhoff integral mean method even if the number of nodes is small, while the harmonic mean method needs more nodes to achieve the same accuracy. Furthermore, the logarithmic mean method only needs the diffusion coefficients of nodes, the disadvantage of Kirchhoff method which needs concrete expression of diffusion coefficients and numerous integral operation thus can be overcome.

Large‐eddy simulation of subsonic turbulent jets using the compressible lattice Boltzmann method

SummaryThe lattice Boltzmann method (LBM) is a powerful technique for the computational modeling of a wide variety of single‐s and multiphase flows involving complex geometries. Although the LBM has been demonstrated to be effective for the solution of incompressible flow problems, there are limitations when this methodology is applied to the solution of compressible flows, especially for flows at high Mach numbers. In this article, we investigate strategies to overcome some of the limitations associated with the application of LBM to compressible flows. To this purpose, one of the key contributions of this study is the synthesis and integration of previous efforts concerning the formulation of LBM for the large‐eddy simulation (LES) of compressible turbulent flows in the subsonic flow regime. It is shown how certain limitations of applying the LBM to compressible flows can be addressed by using either a higher order Taylor series expansion of the Maxwell–Boltzmann equilibrium distribution function or using the Kataoka and Tsutahara (KT) LBM model formulation for compressible flows. The proposed LBM/LES methodology for compressible flows has been combined with the Kirchhoff integral formulation for computational aeroacoustics and used to simulate the flow and acoustic fields of compressible jet flows at high subsonic speeds with practical relevance for providing a better understanding of problems associated with jet noise. In this context, simulations of the physics associated with the jet flow and concomitant noise in the near‐ and far‐field regimes were conducted using the proposed framework of a compressible LBM/LES and Kirchhoff integral method. The results of the subsonic isothermal and nonisothermal jet flow simulations for the flow and acoustic fields have been compared with available numerical and experimental results with generally good to excellent agreement.

The unsteady continuous adjoint method for minimizing flow-induced sound radiation

This paper develops the unsteady continuous adjoint method for aeroacoustic optimization problems governed by time-dependent turbulent flows. To predict the flow-induced sound radiation from a body in free-stream, an incompressible Improved Delayed Detached Eddy Simulation is firstly performed. The generated noise is then propagated with the Kirchhoff Integral method, which uses the pressure distribution on the body surface to compute the sound pressure at selected receivers. The continuous adjoint method for the aforementioned process is developed and presented for the first time. The differentiated Kirchhoff Integral is used to compute the boundary condition of the adjoint velocity on the body surface and, then, the unsteady adjoint equations are solved backwards in time. The time window during which the simulation is performed and the one during which the objective function is evaluated do not coincide. This is reflected on the adjoint boundary conditions along the body and the time integration of the sensitivity derivatives. To ensure the consistency of the continuous adjoint-based gradients, grid sensitivities are taken into account. By incorporating the grid displacement equations during the mathematical development of the adjoint method, the surface integral of the residuals of the flow PDEs, which is commonly omitted, is replaced with a volume integral of the grid sensitivities and gives rise to the adjoint grid displacement equations along with an accurate sensitivity derivatives expression. The proposed method is verified against finite differences in a 3D turbulent flow around a cylinder and, then, applied to a real-world test case, concerned with the flow-induced sound radiation of the side mirror of the generic SAE vehicle.

Multiple scattering effects for a solid cylinder with axis oblique relative to a nearby horizontal surface

It has been shown that when studying the acoustic scattering from a cylinder near a flat interface, there exists multiple paths by which sound can travel to, and subsequently scatter from, the cylinder. Specifically, sound can couple into a number of surface elastic waves (SEW), some of which give rise to an observed enhancement to the backscattering. Previously, the coupling conditions for these mechanisms have been derived, which are dependent on the relative angles between the target, the source/receiver and the interface [Plotnick, et al., JASA 140(3), 2016]. Here, a non-singular Helmholtz Kirchhoff Integral method has been implemented in a hybrid finite element/propagation model in order to predict the first and second order scattering mechanisms that exist when the cylinder is near an air-water interface, with its axis oblique with respect to the interface. This model, combined with the coupling conditions for SEWs, is used to dissect tank measurements that previously were only partially understood. Various multiple scattering mechanisms are identified and discussed. [Work supported by the Office of Naval Research.]

Analysis of Wave Scattering from a Viscoelastic Layer with Complex Shape

The Discretized Kirchhoff Integral method has been recently tested against laboratory experiments using a model with surface curvatures and sharp edges generating wave diffraction effects. Comparisons between numerical and laboratory data have exhibited a good quantitative fit in terms of time arrivals and amplitudes, except in the vicinity of secondary shadow boundaries created by the interaction of the edges of some topographical structures. Following this work, the effect of multiple scattering and the surface curvatures on the wavefield is studied here, using the so-called diffraction attenuation coefficient, in order to define the cases where these effects may be neglected in the numerical modeling without loss of accuracy.

Numerical investigation on radiation characteristics of aero-intake noise with ground effects

A hybrid method for the prediction of far-field Aero-intake noise from gas turbine engine with ground effects has been developed. The near-field predictions were obtained by solving the linearized Euler equations with computational aeroacoustic techniques composed of high-order finite difference scheme, non-reflecting boundary condition, overset grids, and parallel computational methods. Kirchhoff integral method was used for the prediction of far-field directivity patterns. In order to predict the noise propagation, ray theory is applied with Geographical Information System (GIS) terrain profile data. Then, the noise level on the ground has been predicted and analyzed, including the effects of air absorption, ground reflection, and diffraction.

The Effects of Heating on Noise Generation in Subsonic Transitional Jets

Large eddy simulation was performed for two jets, i.e. a cold jet and its heated counterpart, with the same exit Mach number. The far-field noise was predicted by Kirchhoff's integral method based on the near-field pressure fields on a controlled surface. The directivity and intensity of the far-field noise are found to be in good agreement with existed experiments. At polar angles ▪ < 70°, heating enhances the noise intensity, while the noise intensity for the cold jet at angles ▪ > 70° is higher. For the heated jet, the potential core length is shorter, and radial spreading is much faster. Moreover, the peak locations of the fluctuation intensity of the velocity on the centreline and nozzle lip line move upstream in the hot jet, and more rapid decay occurs. Based on the mean flow of simulation, linear parabolized stability equations (LPSE) are solved to study spatial evolution of instability waves. In the low-frequency range, m = 0 mode that absent in direct forcing is dominant in acoustic field, but the peak of N-factor of m = 0 mode predicted by LPSE is not the maximum. This should be associated with nonlinear interaction for instability waves with opposite signs of m.

Analyze the Effect of Thermal Conductivity of Cold Insulation Materials in Cryogenic Engineering

The pipe heat transfer models were constructed, respectively, to simulate the cryogenic insulated structure, nonlinear partial differential equations was obtained considering the temperature-dependent variation in thermal conductivity of insulation materials. In order to get the temperature profile of insulation layer, Kirchhoff integral method was introduced to process, program, and iteratively calculate the simplified energy equations. The influence of heat-transfer process related to the thermal conductivities of four cold insulators was analyzed. Results indicate that phenolic foam, hydrophobic perlite, and RPUR have superior low-temperature insulation at cryogenic environment, while foam glass has better low-temperature insulation at normal temperature. The thickness of inner cryogenic-insulation layer can be reduced largely by using a multicomponent cold insulator in the discharge pipe of LNG stations, which can save both materials and costs. Meanwhile, it offers a new method to solve variable thermal conductivities.

Numerical modeling of 3D zero-offset laboratory data by a discretized Kirchhoff integral method

Accurate simulation of seismic wave propagation in complex geologic structures is of particular interest nowadays. However, difficulties arise for complex geologic structures with great and rapid structural changes, due, for instance, to the presence of shadow zones, head waves, diffractions and/or edge effects. Different methods have thus been developed and are typically tested on synthetic configurations against analytical solutions for simple canonical problems, reference methods, or via direct comparison with real data acquired in situ. Such approaches have limitations, especially if the propagation occurs in a complex environment with strong-contrast reflectors and surface irregularities because it can be difficult to determine the method that gives the best approximation of the “real” solution or to interpret the results obtained without an a priori knowledge of the geologic environment. An alternative approach for seismics consists in comparing the synthetic data with data obtained in laboratory experiments. In contrast to in situ experiments, high-quality data are collected under controlled conditions for a known configuration. In contrast with numerical experiments, laboratory data possess many of the characteristics of field data because real waves propagate through models with no numerical approximations. Our main purpose was to test the approach of using laboratory data as reference data for benchmarking 3D numerical methods and techniques using the setup that we have designed for this study. We performed laboratory-scaled measurements of zero-offset reflection of broadband pulses from a strong topographic environment immersed in a water tank. We compared these measurements with numerical data simulated by means of a discretized Kirchhoff integral method. The comparisons of synthetic and laboratory data indicated a good quantitative fit in terms of time arrivals and acceptable fit in amplitudes. Thus, the first step of the approach was successfully applied.

Numerical modeling of zero-offset laboratory data in a strong topographic environment: results for a spectral-element method and a discretized Kirchhoff integral method

Accurate simulation of seismic wave propagation in complex geological structures is of particular interest nowadays. However conventional methods may fail to simulate realistic wavefields in environments with great and rapid structural changes, due for instance to the presence of shadow zones, diffractions and/or edge effects. Different methods, developed to improve seismic modeling, are typically tested on synthetic configurations against analytical solutions for simple canonical problems or reference methods, or via direct comparison with real data acquired in situ. Such approaches have limitations, especially if the propagation occurs in a complex environment with strong-contrast reflectors and surface irregularities, as it can be difficult to determine the method which gives the best approximation of the “real” solution, or to interpret the results obtained without an a priori knowledge of the geologic environment. An alternative approach for seismics consists in comparing the synthetic data with high-quality data collected in laboratory experiments under controlled conditions for a known configuration. In contrast with numerical experiments, laboratory data possess many of the characteristics of field data, as real waves propagate through models with no numerical approximations. We thus present a comparison of laboratory-scaled measurements of 3D zero-offset wave reflection of broadband pulses from a strong topographic environment immersed in a water tank with numerical data simulated by means of a spectral-element method and a discretized Kirchhoff integral method. The results indicate a good quantitative fit in terms of time arrivals and acceptable fit in amplitudes for all datasets.

地震波叠前逆时偏移——模型分析与应用

根据地震波逆时偏移技术与常规克希霍夫和单程波深度成像技术的算法差异，开展了针对逆时偏移方法关键技术的若干问题分析。从理论模型的脉冲响应出发，理清了低频背景逆时噪声的成因机理，采用上下行波波场分离相关逆时成像条件实现有效压制；采用基于扩散滤波的拉普拉斯算子实现了低频背景逆时噪声的横向相对保幅压制处理，其效果要优于常规的高通滤波方法；对比分析了不同阶数声波方程1D逆时偏移处理的成像波形差异以及与克希霍夫结果的差异。计算结果表明，综合考虑逆时噪声和波形差异成因及其对应的解决方法，在实际陆地地震资料逆时偏移中取得了较好的应用效果。 According to the algorithmic difference between the seismic wave reverse-time migration technology and the conventional Kirchhoff integral method’s depth migration and one-way wave depth migration technology, we carry out several analyses of key problem on specific aspects of the reverse-time migration method. The paper starts from the theoretical model’s impulse response, and finds out the origin mechanisms of the low-frequency reverse-time noise, effectively weakening the low-frequency noise by using correlative reverse-time imaging condition with up/down going wave-field separation processing. The low-frequency reverse-time noise is eliminated afterward with relatively horizontal-amplitude preserved Laplacian operator based on diffusion filter, and this method’s processing effect is better than that of conventional high-pass filter method. What’s more, 1D reverse-time migration is carried out using different order acoustic wave equations together with 1D Kirchhoff depth migration. The computational results show that perfect application results are achieved in the reverse-time migration processing of practical land’s seismic data by considering the above factors.

Kirchhoff integral approach to the calculation of Green’s functions beyond the normal neighborhood

We propose a new method for investigating the global properties of the retarded Green's function $G_R(x',x)$ for fields propagating on an arbitrary globally hyperbolic spacetime. Our method combines the Hadamard form for $G_R$ (this form is only valid within a normal neighbourhood of $x$) together with Kirchhoff's integral representation for the field in order to calculate $G_R$ outside the maximal normal neighbourhood of $x$. As an example, we apply this method to the case of a scalar field on a black hole toy-model spacetime, the Pleba{\'n}ski-Hacyan spacetime, $\mathbb{M}_2\times\mathbb{S}^2$. The method allows us to determine in an exact manner that the singularity structure of the `direct' term in the Hadamard form for $G_R(x',x)$ changes from a form $\delta(\sigma)$ to `$1/\pi\sigma$' after the null geodesic joining $x$ and $x'$ has crossed a caustic point, where $\sigma$ is the world function. Furthermore, there is a change of form from a $\theta(-\sigma)$ to a `$-\ln|\sigma|/\pi$' in the `tail' term, which has not been explicitly noted before in the literature. We complement the results from the Kirchhoff integral method with an analysis for large-$\ell$ of the Green function modes. This analysis allows us to determine the singularity structure after null geodesics have crossed an arbitrary number of caustics, although it raises a causality issue which the Kirchhoff integral method resolves. Because of the similarity in the caustic structure of the spacetimes, we expect our main results for wave propagation to also be valid on Schwarzschild spacetime.

Data analysis of z-scan experiment using Fresnel–Kirchoff integral method in colloidal TiO2 nanoparticles

In this study, a precise method to evaluate nonlinear optical absorption and refraction of materials using z-scan method based on Fresnel–Kirchhoff integral method (FK method) has been offered. The real electric field of a Gaussian beam passing through a nonlinear sample having both nonlinear absorption and refraction has been investigated using FK method. Subsequently, the z-scan curves have been studied. This is the first time that FK method has been used for calculating the nonlinear absorption coefficient. Additionally, an appropriate numerical curve-fitting method for calculating the nonlinear optical coefficients based on z-scan method has been suggested. Z-scan curves and nonlinear optical coefficients have been obtained for TiO2 nanoparticles in CW irradiation regime with the particle size ranges from 70 to 90 nm. This is the first experimental study which uses this analytical numerical method. Finally, all calculated results extracted from this new method have been compared with those of the previous methods.

Reprint of: Extension of the near acoustic field of a jet to the far field

The problem of extending the near acoustic field of a high-speed jet to the far field is considered. This is akin to the problem of analytic continuation in complex variable. Analytic continuation is the extension of an analytic function in a limited domain to a larger domain. The general continuation problem involving acoustic sources enclosed by a surface is first analyzed. It is shown that a unique continuation to the far field is possible provided either the pressure or the pressure gradient normal to the surface is given on the surface. In this paper, the continuation is carried out by means of an adjoint Green's function. One significant advantage of using adjoint Green's function is that for a given direction of radiation, one is required to solve only a single acoustic scattering problem. Because the present method requires measuring only the fluctuation pressure on the bounding surface whereas other popular methods such as the Kirchhoff integral method or the Ffowcs-Williams and Hawkings equation requires measuring three or more variables, it appears that the present method has definite advantage when it is used to extend an experimentally measured near acoustic field to the far field.

Extension of the near acoustic field of a jet to the far field

The problem of extending the near acoustic field of a high-speed jet to the far field is considered. This is akin to the problem of analytic continuation in complex variable. Analytic continuation is the extension of an analytic function in a limited domain to a larger domain. The general continuation problem involving acoustic sources enclosed by a surface is first analyzed. It is shown that a unique continuation to the far field is possible provided either the pressure or the pressure gradient normal to the surface is given on the surface. In this paper, the continuation is carried out by means of an adjoint Green’s function. One significant advantage of using adjoint Green’s function is that for a given direction of radiation, one is required to solve only a single acoustic scattering problem. Because the present method requires measuring only the fluctuation pressure on the bounding surface whereas other popular methods such as the Kirchhoff integral method or the Ffowcs-Williams and Hawkings equation requires measuring three or more variables, it appears that the present method has definite advantage when it is used to extend an experimentally measured near acoustic field to the far field.

Numerical investigation on radiation characteristics of discrete-frequency noise from scarf and scoop aero-intakes

Numerical studies have been carried out to investigate the detailed geometrical effects of full three-dimensional aero-intakes on the radiation of the discrete-frequency rotor–stator interaction noise. The near-field acoustic radiation characteristics and the far-field directivity patterns from the scarf and scoop aero-intakes with three different scarf/scoop angles are quantitatively analyzed and compared. The near-field predictions were obtained by solving the linearized Euler equations with computational aeroacoustic techniques consisting of high-order finite difference scheme, non-reflecting boundary conditions, overset grids and parallel computational methods. For the prediction of far-field directivity pattern, the Kirchhoff integral method was applied. By comparing the directivities of discrete-frequency noise radiating from the scarf and the scoop aero-intakes with that from an axisymmetric aero-intake, it is shown that the 7 dB noise reduction at downward peak radiation angle can be achieved by using the scoop aero-intake with scoop angle of 15°, and the 5 dB noise reduction by the scarf aero-intake with the scarf angle of 15°. The scattering of the radiating acoustic wave by the background mean flow around the aero-intakes shifts the peak lobe radiation angle toward ground and increases the amplitude of the acoustic pressure compared with the cases without mean flow effect. Overall, the scoop aero-intake was found to be more effective than the scarf and the axisymmetric aero-intakes in view of the lower noise radiation toward ground.

Methods and applications of 3-D wave equation common-azimuth prestack migration

To tackle the difficulties of a 3-D full volume prestack migration based on the double-square-root (DSR) one-way wave equation in practical applications, the common-azimuth migration approach is first discussed using dual-domain wave propagators under the theoretical frame of crossline common-offset migration. Through coordinate transforming, a common-azimuth prestack tau migration technology that recursively continues the source and receiver wavefields and picks up the migrated results in the two-way vertical traveltime (tau) direction is developed. The migrations of synthetic data sets of SEG/EAGE salt model prove that our common-azimuth migration approaches are effective in both depth and tau domains. Two real data examples show the advantages of wave-theory based prestack migration methods in accuracy and imaging resolution over the conventional Kirchhoff integral methods.