Kirchhoff Surface Research Articles

A GPU-Based Lattice Boltzmann Method for Predicting Near- and Far-Field Jet Noise

Jet flow and noise are simulated using the lattice Boltzmann method (LBM). The D3Q19 lattice model calculates the flow field with a high-order distribution function to overcome LBM limitations in solving high Mach number flows. Multiple relaxation times (MRT) and large eddy simulation (LES) are used in this study. Compressible LBM was applied to the simulation jet flow to demonstrate the ability of the fifth-order equilibrium distribution function to simulate compressible flows efficiently. The results from the LBM simulation were then employed in Kirchhoff's surface integral approach to predict far-field jet noise. In this research, we used the compute unified device architecture (CUDA) to implement the LBM in a graphics processing unit (GPU), thus creating a hybrid code, LBM-MRT-LES. The obtained results agreed with the experimental and numerical results in the literature.

Resolvent-based modeling of turbulent jet noise.

Resolvent analysis has demonstrated encouraging results for modeling coherent structures in jets when compared against their data-educed counterparts from high-fidelity large-eddy simulations (LES). We formulate resolvent analysis as an acoustic analogy that relates the near-field resolvent forcing to the near- and far-field pressure. We use an LES database of round, isothermal, Mach 0.9 and 1.5 jets to produce an ensemble of realizations for the acoustic field that we project onto a limited set of resolvent modes. In the near-field, we perform projections on a restricted acoustic output domain, r/D=[5,6], while the far-field projections are performed on a Kirchhoff surface comprising a 100-diameter arc centered at the nozzle. This allows the LES realizations to be expressed in the resolvent basis via a data-deduced, low-rank, cross-spectral density matrix. We find that a single resolvent mode reconstructs the most energetic regions of the acoustic field across Strouhal numbers, St=[0-1], and azimuthal wavenumbers, m=[0,2]. Finally, we present a simple function that results in a rank-1 resolvent model agreeing within 2 dB of the peak noise for both jets.

Acoustic time-dependent energy from vibrating surfaces via a generalized radiation impulse response approach.

A vibrating surface in contact with a fluid experiences a reaction force that is related to the velocity of the surface. In the case of a harmonic vibration, the acoustic radiation impedance provides a useful measure of the harmonic force resulting from a harmonic velocity of the surface. More generally, the radiation impulse response, which is the temporal Fourier transform of the radiation impedance, provides the basis for a convolution approach to evaluate both the time-dependent force and the energy transfer into the fluid resulting from a time-dependent velocity of a fluid-loaded vibrating surface for interior and external problems. Important properties of the radiation impulse responses are presented using the Kirchhoff Surface Integral Equation and Fourier transforms. A universal form of the radiation impulse response, which consists of a weighted Dirac delta function and a baseband function, is presented. The time-dependent force and energy transfer into a fluid resulting from the space-time normal velocity of a spherical surface is presented to simply illustrate the generalized radiation impulse response approach for exterior three-dimensional radiation problems.

Lattice Thermal Conductivity in Nanowires: Coupling the Bechmann–Kirchhoff Boundary Scattering Model With a Monte Carlo Framework

In this article, the effect of rough surface on phonon transport has been studied for silicon nanowire structures. The diffusive boundary scattering has been treated using the Beckmann–Kirchhoff (B-K) surface roughness scattering (SRS) model, considering the effect of roughness height, correlation length, phonon wavelength, and incident/reflected angles. The model is more comprehensive and accurate than the conventional approaches, where surface roughness is usually modeled based on experimental fitting parameters or only phonon wavelength. The B-K SRS model has been integrated within a particle-based Monte Carlo phonon transport (MCPT) simulator to study the thermal conductivity of different nanowire structures. The simulator is benchmarked against the experimental data for both bulk and nanowire devices. The reduction of thermal conductivity in nanowires as a function of the degree of roughness has been discussed. It is found that for a 70-nm-width, 6- $\mu \text{m}$ -long silicon structure, using 2.3-nm roughness height and 8.9-nm correlation length, 89% reduction in thermal conductivity occurs at 300 K. The sensitivity of the developed simulator with varying degree of roughness and the effect of SRS on different phonon spectral branches have been studied. These observations can be useful in designing materials with low thermal conductivity for thermoelectric cooling applications.

A Numerical Study on Fluid Flow and Acoustic Characteristics of a Supersonic Impinging Jet Using Vorticity Confinement

The objective of this work is to numerically study the fluid flow and acoustic field of a supersonic impinging jet by applying the vorticity confinement (VC) method. For this aim, the three-dimensional compressible Navier-Stokes equations with the incorporation of the VC method are considered and the resulting system of equations is solved by using the sixth-order compact finite-difference scheme. To eliminate the numerical instability, a low-pass high-order filter is used. The nonreflective boundary conditions are applied for all the free boundaries and the radiated sound field is obtained by the Kirchhoff surface integration. Comparisons of the present results with the experimental data and other numerical simulations show that the solution methodology adopted based on the application of the VC method with the high-order compact finite-difference scheme provides a good prediction of the fluid flow and the acoustic field of the impingement region on coarser grids than that usually required in the LESs, and thus, the calculations of coarse grid LESs are improved.

An explicit marching-on-in-time scheme for solving the time domain Kirchhoff integral equation.

A fully explicit marching-on-in-time (MOT) scheme for solving the time domain Kirchhoff (surface) integral equation to analyze transient acoustic scattering from rigid objects is presented. A higher-order Nyström method and a PE(CE)m-type ordinary differential equation integrator are used for spatial discretization and time marching, respectively. The resulting MOT scheme uses the same time step size as its implicit counterpart (which also uses Nyström method in space) without sacrificing from the accuracy and stability of the solution. Numerical results demonstrate the accuracy, efficiency, and applicability of the proposed explicit MOT solver.

Pre‐stack Kirchhoff depth migration local space‐shift imaging condition: synthetic and data examples

ABSTRACTExtracting accurate common image angle gathers from pre‐stack depth migrations is important in the generation of any incremental uplift to the amplitude versus angle attributes and seismic inversions that can lead to significant impacts in exploration and development success. The commonly used Kirchhoff migration outputs surface common offset image gathers that require a transformation to angle gathers for amplitude versus angle analysis. The accuracy of this transformation is one of the factors that determine the robustness of the amplitude versus angle measurements. Here, we investigate the possibility of implementing an extended imaging condition, focusing on the space‐lag condition, for generating subsurface reflection angle gathers within a Kirchhoff migration. The objective is to determine if exploiting the spatial local shift imaging condition can provide any increase in angle gather fidelity relative to the common offset image gathers. The same restrictions with a ray‐based approach will apply using the extended imaging condition as both the offset and extended imaging condition method use travel times derived from solutions to an Eikonal equation. The aims are to offer an alternative ray‐based method to generate subsurface angle gathers and to understand the impact on the amplitude versus angle response. To this end, the implementation of the space‐shift imaging condition is discussed and results of three different data sets are presented. A layered three‐dimensional model and a complex two‐dimensional model are used to assess the space shift image gathers output from such a migration scheme and to evaluate the seismic attributes relative to the traditional surface offset common image gathers. The synthetic results show that the extended imaging condition clearly provides an uplift in the measured amplitude versus angle over the surface offset migration. The noise profile post‐migration is also improved for the space‐lag migration due to the double summation inside the migration. Finally, we show an example of a space‐lag gather from deep marine data and compare the resultant angle gathers with those generated from an offset migration and a time‐shift imaging condition Kirchhoff migration. The comparison of the real data with a well log shows that the space‐lag result is a better match to the well compared to the time‐lag extended imaging condition and the common offset Kirchhoff migration. Overall, the results from the synthetics and real data show that a Kirchhoff migration with an extended imaging condition is capable of generating subsurface angle gathers with an incremental improvement in amplitude versus angle fidelity and lower noise but comes at a higher computational cost.

Prediction of fluid flow and acoustic field of a supersonic jet using vorticity confinement.

In this study, the numerical simulation of the fluid flow and acoustic field of a supersonic jet is performed by using high-order discretization and the vorticity confinement (VC) method on coarse grids. The three-dimensional Navier-Stokes equations are considered in the generalized curvilinear coordinate system and the high-order compact finite-difference scheme is applied for the space discretization, and the time integration is performed by the fourth-order Runge-Kutta scheme. A low-pass high-order filter is applied to stabilize the numerical solution. The non-reflecting boundary conditions are adopted for all the free boundaries, and the Kirchhoff surface integration is utilized to obtain the far-field sound pressure levels in a number of observer locations. Comparisons of the jet mean flow and jet aeroacoustics results with the other numerical and experimental data at similar flow conditions are made and show a reasonable agreement. The study shows that the proposed solution methodology based on the high-order compact finite-difference scheme in conjunction with the VC method can reasonably predict the near-field flow and the far-field noise of high Reynolds number jets with a fairly coarser grid than that used in the large eddy simulations and, thus, the computational cost can be significantly decreased.

A method of evaluating the three-dimensional damage effect of strong electromagnetic pulse on the box based on the triangular-element FDTD method

Based on the FDTD/TDPO hybrid theory, a fast FDTD mesh modeling method by triangulation is established. Near scattering field calculated by the FDTD method is calculated to the surface triangular facets of the target by using the Kirchhoff surface method. FDTD way is used to simulate and compute the power density and energy in the key positions of the computer box; some laws are also concluded through the research. The power density coupled in the box can form the oscillations. The time coupled from the start to the maximum for energy is approximately equal to the pulse width. And then the far field scattering field is approximated by the TDPO method. The node, power supply, CD-ROM and mainboard of the computer can be disturbed and even damaged by the strong electromagnetic pulse, especially the electronic components and chips of the mainboard and power. The coupled energy is concentrated nearby the motherboard, power supply, CD-ROM and hard disk, etc. Especially for the cutting-edge places, greater interference and damage can be generated. A simple method of research on the damage efficiency is introduced in this paper. Results provides theoretical basis for the reasonable arrangement of the components, so as to improve the shielding effectiveness.

Analysis of Residual Stresses on the Vibration of a Circular Sensor Diaphragm with Surface Effects

AbstractResonant micro-biochemical sensors play important roles in a wide range of emerging applications to detect biochemical molecules. As the resonators of micro-biochemical sensors, the vibration characteristics of circular sensor diaphragms are important for the design of diaphragm-based resonant micro-biochemical sensors. In this paper, the influence of residual stresses on the vibration of a circular sensor diaphragm with surface effects is analyzed. Based on the Kirchhoff's plate theory and surface elasticity theory, the governing equation is presented. The material characteristic lengths for different surface effects are obtained. The influences of residual stresses on the effective flexural rigidity and natural frequency of the diaphragm with surface effects are discussed. Results show that the influence of residual stresses on the effective flexural rigidity becomes obvious with the increasing of residual stresses. The first order natural frequency increases rapidly when the tension parameter is larger than 30 for the stiffened surfaces, while for the softened surfaces the value is 10. Moreover, surface effects can influence the transition range of diaphragm from the plate behavior to membrane behavior in terms of the tension parameter. The transition range can be enlarged by the stiffened surface and be shortened by the softened surface. The analysis and results are helpful for the design of sensor diaphragm-based resonant micro-biochemical sensors and some related researches.

Tunneling effects in electromagnetic wave scattering by nonspherical particles: A comparison of the Debye series and physical-geometric optics approximations

The accuracy of the physical-geometric optics (PG-O) approximation is examined for the simulation of electromagnetic scattering by nonspherical dielectric particles. This study seeks a better understanding of the tunneling effect on the phase matrix by employing the invariant imbedding method to rigorously compute the zeroth-order Debye series, from which the tunneling efficiency and the phase matrix corresponding to the diffraction and external reflection are obtained. The tunneling efficiency is shown to be a factor quantifying the relative importance of the tunneling effect over the Fraunhofer diffraction near the forward scattering direction. Due to the tunneling effect, different geometries with the same projected cross section might have different diffraction patterns, which are traditionally assumed to be identical according to the Babinet principle. For particles with a fixed orientation, the PG-O approximation yields the external reflection pattern with reasonable accuracy, but ordinarily fails to predict the locations of peaks and minima in the diffraction pattern. The larger the tunneling efficiency, the worse the PG-O accuracy is at scattering angles less than 90°. If the particles are assumed to be randomly oriented, the PG-O approximation yields the phase matrix close to the rigorous counterpart, primarily due to error cancellations in the orientation-average process. Furthermore, the PG-O approximation based on an electric field volume-integral equation is shown to usually be much more accurate than the Kirchhoff surface integral equation at side-scattering angles, particularly when the modulus of the complex refractive index is close to unity. Finally, tunneling efficiencies are tabulated for representative faceted particles.

Instability Waves in High-Speed Jets: Near-and Far-Field DNS/LES Data Analysis

Solutions from inviscid linear stability theory (LST) and linear parabolized stability equations (PSE) were used to quantify the extent to which instability waves represent actual near-field dynamics in high-speed forced turbulent jets. The mean and unsteady flow data were taken from a well-validated large-eddy simulation (LES) database (Bodony and Lele, 2005). The jets and their operating conditions were heated high subsonic ( M j = 0.97, T j/ T∞ = 2.3, Re D = 84,000) and unheated supersonic ( M j = 1.95, T j/T∞ = 0.56, Re D = 336,000) jets with the same jet exit velocities. The coherent components of the turbulent jet flow fields were extracted using a filter based on proper orthogonal decomposition (POD). To examine the effects of nonlinear interactions and to test the decomposition method, forced laminar jets were also considered; the flow data were obtained from a direct numerical simulation (DNS) database for an unheated M j = 0.90 and Re D = 4,100 jet (Ryu, 2010). Two methods were used to extract instability waves from the nonlinear flow fields: first, bi-orthogonal relations between regular and adjoint instability waves were formulated from linearized Euler equations to determine the amplitude coefficients of the instability waves in nonlinear flows; then, a least-squares fit was applied to solutions from the PSE, POD, and LES data for the same purpose. The noise radiated by the deduced instability wave modes, POD, and LES were computed using the Kirchhoff surface integral method. Frequencies St = fD j/ U j = 0.1 ∼ 0.5 at azimuthal modes m = 0, 1, and 2 were compared. The adjoint decomposition method showed promising results only for the laminar jet. The eigenfunction shape in the radial direction and the growth and decay of energy in the streamwise direction were well captured. However, this method produced errors in analyses of turbulent jets due to sensitivity in regions where the mode had saturated and was decaying. The results of PSE and POD showed reasonably good agreement in the near field for the first helical mode ( m = 1). However, there was only limited agreement in the far field, and among other modes, in contrast to the good agreements described in recent studies for supersonic unforced jets. This is due to the strong nonlinear effects in forced turbulent jets considered in the present study. This finding confirms a limitation of linear stability theories when predicting the wavepacket dynamics of forced turbulent jets. The PSE solution for the subsonic jet showed very weak acoustic radiation, and the far-field noise displayed strong sensitivity to the location of the Kirchhoff surface. This suggests a limitation of direct propagation approaches for the PSE solutions of moderately heated subsonic jets.

Determination of roughness and correlation length of dielectric surfaces in nano/micro scales using Kirchhoff Approximation method

In this paper determination of surface roughness and correlation length for dielectrics were extensively investigated by applying Kirchhoff Approximation method for second order. The technique is based on two dimensional analysis of scattered light intensity from a dielectric rough surface. The result show that the scattered intensities from a dielectric rough surface for the second order approximation are almost the same as for the first order. The results also show that by using Kirchhoff Approximation method, both surface roughness and correlation length simultaneously can be determined, if the scattered light intensity for two different incident angles is known. The investigations also show that the accuracy for evaluating the ratio of scattered light intensity to incident light intensity can influence the results.

Some modifications in evaluation of the size effects related to surface stresses in nanostructures

In this work, addressing some contradictions, it is tried to interpret some gaps between surface stress theories and the size effects observed in experiments for nanowires through different examples. Due to mandatory self-equilibrium state of nanostructures at different states, in a generalized model, a balancing factor is defined for the surface residual stress and duly the clamped nanowires are classified into suspended and etched types. The claims are confirmed by observing similar results for bending and tensile tests of Ag nanowires that addresses alternative sources for size effects beside the surface stresses. In addition, the size effects and surface material properties are identified lower at larger deformation ranges and regarding tremendous gap between atomistic simulation and continuum core–shell models, it is verified that the surface elasticity may not be the entire source for size effects. In extension, due to anisotropicity of single crystals, two orientation dependent parameters are defined for nanoplates that are modeled based on Kirchhoff plate, von-Karman strains and surface stress models. It is shown that orientation of (100)-nanoplates changes the size effects for more than 70%. Meanwhile, some test setups are recommended for characterization of the size effects of nanowires and nanoplates.

Wavepacket models for supersonic jet noise

AbstractGudmundsson and Colonius (J. Fluid Mech., vol. 689, 2011, pp. 97–128) have recently shown that the average evolution of low-frequency, low-azimuthal modal large-scale structures in the near field of subsonic jets are remarkably well predicted as linear instability waves of the turbulent mean flow using parabolized stability equations. In this work, we extend this modelling technique to an isothermal and a moderately heated Mach 1.5 jet for which the mean flow fields are obtained from a high-fidelity large-eddy simulation database. The latter affords a rigourous and extensive validation of the model, which had only been pursued earlier with more limited experimental data. A filter based on proper orthogonal decomposition is applied to the data to extract the most energetic coherent components. These components display a distinct wavepacket character, and agree fairly well with the parabolized stability equations model predictions in terms of near-field pressure and flow velocity. We next apply a Kirchhoff surface acoustic propagation technique to the near-field pressure model and obtain an encouraging match for far-field noise levels in the peak aft direction. The results suggest that linear wavepackets in the turbulence are responsible for the loudest portion of the supersonic jet acoustic field.

Analysis of the possibilities of the methods for calculating turbulent jet noise

Two methods for calculating the noise of turbulent exhaust jets of civil aircraft nozzles are considered. The first method is chiefly intended for engineering mass-volume calculations and is based on the solution of the averaged Navier-Stokes equations closed by a two-equation turbulence model. The second method uses direct numerical simulation of large eddies in a turbulent jet and the Kirchhoff surface for calculating noise spectra and radiation patterns in the far field. The possibilities and certain important restrictions of these methods are analyzed. The results obtained using these methods are compared with experimental data.

On the far field in the Lorenz–Mie theory and T-matrix formulation

The far field within the context of the Lorenz–Mie theory and the T-matrix formulation is usually expressed on the basis of the asymptotic properties of vector spherical waves. The radiation condition is taken into account by employing proper vector spherical functions as the expansion basis of the scattered field. The asymptotic behavior of the Hankel function is obtained from differential equations. The asymptotic far field can also be obtained from the Kirchhoff surface integral equation, in which the radiation condition has been implemented when it is derived from the Maxwell equations. This note is to present an explicit establishment of the relationship between the asymptotic far field and the near field in the Lorenz–Mie theory and the T-matrix formulation through the Kirchhoff surface integral.

A Study of an AIEM Model for Bistatic Scattering From Randomly Rough Surfaces

In this paper, we study the bistatic scattering using an advanced integral equation model (AIEM). By keeping all the surface current terms in the Kirchhoff surface fields, the bistatic scattering coefficients are obtained. For simplification, the complete Kirchhoff field did not cast into the derivation of the complementary field. We compare varied updated versions of IEM-based models with the small perturbation model, geometrical optics model, and Kirchhoff approximation standard models at respective regions of validity. The results indicate that the new AIEM provides much more accurate predictions for bistatic scattering.

Simulation of Supersonic Jet Noise with Nonlinear CFD and Kirchhoff Surface Integral

An acoustic prediction capability for supersonic axisymmetric jets was developed on the basis of the OVERFLOW Navier-Stokes CFD (Computational Fluid Dynamics) code. Reynolds-averaged turbulent stresses in the flow field are modeled with the aid of Spalart-Allmaras one-equation turbulence model. Appropriate acoustic and outflow boundary conditions were implemented to compute time-dependent acoustic pressure in the nonlinear source-field. Based on the specification of acoustic pressure, its temporal and normal derivatives on the Kirchhoff surface, the near-field and the far-field sound pressure levels are computed via Kirchhoff surface integral, with the Kirchhoff surface chosen to enclose the nonlinear sound source region described by the CFD code. The method is validated by a comparison of the predictions of sound pressure levels with the available data for axisymmetric turbulent supersonic perfectly expanded jets.

Prediction of Flowfield and Acoustic Signature of a Coaxial Jet Using RANS-Based Methods and Large-Eddy Simulation

The feasibility of using large-eddy simulation (LES) for jet noise predictions has been discussed in several publications. The vast majority of these studies are, however, restricted to fairly simple geometries and moderate Reynolds numbers flows. Recent studies have shown that LES-based methods can be used for realistic flows and complex geometries with promising results. With continuously increased computer capacity and with the possibility to perform computations in parallel on PC-clusters, the possibility of using LES for industrial applications is increasing. However, although, the complexity of configurations for which LES may be used is approaching that of a real jet engine, due to high demands for computational power and long turn-over times LES is still not feasible for industrial use. For low-noise nozzle design, engineering tools with short turn-over time, accurate enough for prediction of noise trends are therefore needed. In the development of such methods, LES can provide data suitable for method validation. In the present work, the high-subsonic flow in an axisymmetric coaxial nozzle/jet configuration is studied. Flowfield predictions obtained using an axisymmetric RANS solver for compressible flows and predicted acoustic signature obtained using a method based on RANS and Lighthill's acoustic analogy are compared with flowfield and radiated sound predictions obtained using LES combined with Kirchhoff surface integration. In the predicted flow statistics, the most noticeable difference between RANS and LES results is that the RANS calculation gives somewhat lower levels of turbulence kinetic energy in the shear layers in the near-nozzle regions and hence the LES jet mixes somewhat faster and thus the potential core region is shorter in this simulation.