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The Kirchhoff Formulas for Moving Surfaces in Aeroacoustics - The Subsonic and Supersonic Cases
One of the active areas of computational aeroacoustics is the application of the Kirchhoff formulas to the problems of the rotating machinery noise predictions. The original Kirchhoff formula was derived for a stationary surface. In 1988, Farassat and Myers derived a Kirchhoff Formula obtained originally by Morgans using modem mathematics. These authors gave a formula particularly useful for applications in aeroacoustics. This formula is for a surface moving at subsonic speed. Later in 1995 these authors derived the Kirchhoff formula for a super-sonically moving surface. This technical memorandum presents the viewgraphs of a day long workshop by the author on the derivation of the Kirchhoff formulas. All necessary background mathematics such as differential geometry and multidimensional generalized function theory are discussed in these viewgraphs. Abstraction is kept at minimum level here. These viewgraphs are also suitable for understanding the derivation and obtaining the solutions of the Ffowcs Williams-Hawkings equation. In the first part of this memorandum, some introductory remarks are made on generalized functions, the derivation of the Kirchhoff formulas and the development and validation of Kirchhoff codes. Separate lists of references by Lyrintzis, Long, Strawn and their co-workers are given in this memorandum. This publication is aimed at graduate students, physicists and engineers who are in need of the understanding and applications of the Kirchhoff formulas in acoustics and electromagnetics.
An Analytical Comparison of the Acoustic Analogy and Kirchhoff Formulation for Moving Surfaces
The Lighthill acoustic analogy, as embodied in the Ffowcs Williams-Hawkings (FW-H) equation, is compared with the Kirchhoff formulation for moving surfaces. A comparison of the two governing equations reveals that the main Kirchhoff advantage (namely nonlinear flow effects are included in the surface integration) is also available to the FW-H method if the integration surface used in the FW-H equation is not assumed impenetrable. The FW-H equation is analytically superior for aeroacoustics because it is based upon the conservation laws of fluid mechanics rather than the wave equation. This means that the FW-H equation is valid even if the integration surface is in the nonlinear region. This is demonstrated numerically in the paper. The Kirchhoff approach can lead to substantial errors if the integration surface is not positioned in the linear region. These errors may be hard to identify. Finally, new metrics based on the Sobolev norm are introduced which may be used to compare input data for both quadrupole noise calculations and Kirchhoff noise predictions.
The Kirchhoff Formula for a Supersonically Moving Surface
The Kirchhoff formula for radiation from stationary surfaces first appeared in 1882, and it has since found many applications in wave propagation theory. In 1930, Morgans extended the formula to apply to surfaces moving at speeds below the wave propagation speed; we refer to Morgans formula as the subsonic formulation. A modern derivation of Morgans result was published by Farassat and Myers in 1988, and it has now been used extensively in acoustics, particularly for high speed helicopter rotor noise prediction. Under some common conditions in this application, however, the appropriate Kirchhoff surface must be chosen such that portions of it travel at supersonic speed. The available Kirchhoff formula for moving surfaces is not suitable for this situation. In the current paper we derive the Kirchhoff formula applicable to a supersonically moving surface using some results from generalized function theory. The new formula requires knowledge of the same surface data as in the subsonic case. Complications that arise from apparent singularities in the new formulation are discussed briefly in the paper.
A Fast Method of Deriving the Kirchhoff Formula for Moving Surfaces
The Kirchhoff formula for a moving surface is very useful in many wave propagation problems, particularly in the prediction of noise from rotating machinery. Several publications in the last two decades have presented derivations of the Kirchhoff formula for moving surfaces in both time and frequency domains. Here we present a method originally developed by Farassat and Myers in time domain that is both simple and direct. It is based on generalized function theory and the useful concept of imbedding the problem in the unbounded three-dimensional space. We derive an inhomogeneous wave equation with the source terms that involve Dirac delta functions with their supports on the moving data surface. This wave equation is then solved using the simple free space Green's function of the wave equation resulting in the Kirchhoff formula. The algebraic manipulations are minimal and simple. We do not need the Green's theorem in four dimensions and there is no ambiguity in the interpretation of any terms in the final formulas. Furthermore, this method also gives the simplest derivation of the classical Kirchhoff formula which has a fairly lengthy derivation in physics and applied mathematics books. The Farassat-Myers method can be used easily in frequency domain.
The Rankine-Kirchhoff approximations for moist thermodynamics
The Rankine-Kirchhoff equation for the saturation vapor pressure has been derived independently many times, sometimes with the presumption of novelty. As a result, the equation is referred to in the literature by a bewildering number of different names and, often, by no name at all. In this work, the history of the Rankine-Kirchhoff equation is reviewed. The three approximations used to derive it (ideal gas, constant heat capacities, and zero-volume condensates) can be used to derive a whole system of accurate and analytic expressions of relevance to the atmosphere. Consequently, it is proposed that this set of approximations be elevated to a named entity, e.g., the Rankine-Kirchhoff approximations.
Extension of Kirchhoff's formula to radiation from moving surfaces
Kirchhoff's formula for radiation from a closed surface has been used recently for prediction of the noise of high speed rotors and propellers. Because the closed surface on which the boundary data are prescribed in these cases is in motion, an extension of Kirchhoff's formula to this condition is required. In this paper such a formula, obtained originally by Morgans for the interior problem, is derived for regions exterior to surfaces moving at speeds below the wave propagation speed by making use of some results of generalized function theory. It is shown that the usual Kirchhoff formula is a special case of the main result of the paper. The general result applies to a deformable surface. However, the special form it assumes for a rigid surface in motion is also noted. In addition, Morgans' result is further extended by showing that edge line integrals appear in the formula when applied to a surface that is piecewise smooth. Some possible areas of application of the formula to problems of current interest in aeroacoustics are discussed.
Extension of Kirchhoff's formula to radiation from moving surfaces
Kirchhoff's formula for radiation from a closed surface has been used recently for prediction of the noise of high speed rotors and propellers. Because the closed surface on which the boundary data are prescribed in these cases is in motion, an extension of Kirchhoff's formula to this condition is required. In this paper such a formula, obtained originally by Morgans for the interior problem, is derived for regions exterior to surfaces moving at speeds below the wave propagation speed by making use of some results of generalized function theory. It is shown that the usual Kirchhoff formula is a special case of the main result of the paper. The general result applies to a deformable surface. However, the special form it assumes for a rigid surface in motion is also noted. In addition, Morgans' result is further extended by showing that edge line integrals appear in the formula when applied to a surface that is piecewise smooth. Some possible areas of application of the formula to problems of current interest in aeroacoustics are discussed.
Scattering of light by crystals - A modified Kirchhoff approximation
A modified Kirchhoff approximation (MKA) is developed for the scattering of light by randomly oriented crystals. The reflected and transmitted near fields are calculated from ray tracing; the corresponding far fields are then obtained via the vector Kirchhoff integral. On the shadow side of the particle, an additional near field exactly cancels the incident field and causes the forward diffraction. MKA contains a particle size dependence, which is not included in ray optics treatments, and satisfactory results can be obtained for size parameters larger than ten. The scattering phase functions and degrees of linear polarization are calculated for some hexagonal and cubic water ice crystals using MKA. The Kirchhoff approximation for particles other than crystals is discussed, and attention is paid to the backscattering enhancement due to the cyclic passage of internally or multiply externally reflected electromagnetic waves.
Electromagnetic fluctuations for anisotropic media and the generalized Kirchhoff's law
In this paper the polarimetric emission parameters for anisotropic media are derived using the generalized Kirchhoff's law for media with a uniform temperature and the fluctuation-dissipation theory for media with a temperature profile. Both finite-size objects and half-space media are considered. When the object has a uniform temperature across its body, the Kirchhoff's law, based on the condition of energy conservation in thermal equilibrium is generalized to obtain the emission parameters of an anisotropic medium, which can be interpreted as the absorptivity or the absorption cross section of the complementary object with a permittivity that is the transpose of the original object. When the medium has a nonuniform temperature distribution, the fluctuation-dissipation theory is applied for deriving the covariances between vector components of the thermal currents and, consequently, the covariances of the polarizations of electric fields radiated by the thermal currents. To verify the formulas derived from the fluctuation-dissipation theory, we let the temperature of the object be a constant and show that the results reduce to those obtained from the generalized Kirchhoff's law.
The Use of Kirchhoff's Method in Jet Aeroacoustics
Supersonic jet aeroacoustics will be studied using computational techniques. In the study, a Kirchhoff method is used to predict flow generated noise in the mid- and far-fields. This type of method shows promise because it is based on surface integrals and not the volume integrals found in traditional acoustic prediction methods. The Kirchhoff method is dependent on accurate prediction of flow variables in the near-field. Here, computational fluid dynamics (CFD) programs are used for these predictions. Specifically, an existing large eddy simulation (LES) code will be modified for aeroacoustic applications. Issues involved in the implementation of the Kirchhoff method as well as the coupling with the CFD code will be discussed. Important physical noise parameters will be identified and investigated in the study.
Simulation of Supersonic Jet Noise with the Adaptation of Overflow CFD Code and Kirchhoff Surface Integral
An acoustic prediction capability for supersonic axisymmetric jets was developed on the basis of OVERFLOW Navier-Stokes CFD (Computational Fluid Dynamics) code of NASA Langley Research Center. 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 methods are validated by a comparison of the predictions of sound pressure levels with the available data for an axisymmetric turbulent supersonic (Mach 2) perfectly expanded jet.
Simulation of Jet Noise with OVERFLOW CFD Code and Kirchhoff Surface Integral
An acoustic prediction capability for supersonic axisymmetric jets was developed on the basis of OVERFLOW Navier-Stokes CFD (Computational Fluid Dynamics) code of NASA Langley Research Center. 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 methods are validated by a comparison of the predictions of sound pressure levels with the available data for an axisymmetric turbulent supersonic (Mach 2) perfectly expanded jet.
Analysis of all-frequency variational behavior of the Kirchhoff approximation for a classic surface-scattering model
In testing a stochastic variational principle at high frequencies by using a Kirchhoffean trial function in an idealized model for surface scattering - a randomly embossed plane - we have found not only the predicted high-frequency improvement but also an unexpected low-frequency improvement in the calculated scattering amplitudes. To investigate systematically the all-frequency variational behavior, we consider here the deterministic one-boss case - Rayleigh's classic model whose exact solution is available for comparison - over all wavelengths, polarizations, and configurations of incidence and scattering. We examine analytically in particular the long-wave limit of the variational-Kirchhoff amplitudes; the results demonstrate improvements in both wavelength and angle depedence for horizontal (TM) polarization and some variational improvements for vertical (TE) polarization. This low-frequency behavior in tandem with the foreseen high-frequency improvement leads to good variational-Kirchhoff results through the intermediate resonance-frequency regime for this model.
Prediction of high-speed rotor noise with a Kirchhoff formula
A new methodology has been developed to predict the impulsive noise generated by a transonic rotor blade. The formulation uses a full-potential finite-difference method to obtain the pressure field close to the blade. A Kirchhoff integral formulation is then used to extend these finite-difference results into the far-field. This Kirchhoff formula is written in a blade-fixed coordinate system. It requires initial data across a plane at the sonic radius. This data is provided by the finite-difference solution. Acoustic pressure predictions show excellent agreement with hover experimental data for two hover cases of 0.88 and 0.90 tip Mach number, the latter of which has delocalized transonic flow. These results represent the first successful prediction technique for peak pressure amplitudes using a computational code.
Thermal-infrared remote sensing and Kirchhoff's Law. 1: Laboratory measurements
By the end of this century the Earth Observing System (EOS) will provide worldwide, thermal infrared, multispectral images of the Earth, presenting geologists with a new kind of remote sensing data for interpretation. Thus it has become essential to understand the spectral emittance behavior of terrestrial surface materials. Perhaps the most fundamental question to be answereed is the extent to which such materials follow Kirchhoff's law (epsilon = 1 -R) under laboratory and field conditions, especially when a sample displays a thermal gradient. We present the first rigorous quantitative comparison of directional and hemispherical reflectance and directional emittance of rock and soil samples in the laboratory, with thermal gradients induced by heating them from below and allowing them to radiate to a colder background. The results show that only an extemeley low density sample composed of fine particles sifted into a 'fairy castle' structure displays a thermal gradient steep enough within the infrared skin depth to cause significant (6%) departure from Kirchhoff's law. There is no detectable effect on the more normal terrestrial samples, such as soils and rocks measured in the laboratory, even when semitransparent coatings are involved. Thus both emittance and reflectance measurements can be used to calculate sample emissivity for most terrestrial surface materieals. However, the effect on Kirchhoffian behavior of different field environments, which may induce a steeper thermal gradient in particulate samples, has yet to be determined, and some low-density surface materials like newly fallen snow, frost, and efflorescent salts on playas have yet to be measured in emittance.
Isogeometric Bézier dual mortaring: The Kirchhoff–Love shell problem
Here in this paper we develop an isogeometric Bézier dual mortar method for coupling multi-patch Kirchhoff–Love shell structures. The proposed approach weakly enforces the continuity of the solution at patch interfaces through a dual mortar method and can be applied to both conforming and non-conforming discretizations. As the employed dual basis functions have local supports and satisfy the biorthogonality property, the resulting stiffness matrix is sparse. In addition, the coupling accuracy is optimal because the dual basis possesses the polynomial reproduction property. We also formulate the continuity constraints through the Rodrigues’ rotation operator which gives a unified framework for coupling patches that are intersected with G 1 continuity as well as patches that meet at a kink. Several linear and nonlinear examples demonstrated the performance and robustness of the proposed coupling techniques.
Violating Kirchhoff’s Law of Thermal Radiation in Semitransparent Structures
Kirchhoff’s law of thermal radiation imposes a constraint on photon-based energy harvesting processes since part of the incident energy flux is inevitably emitted back to the source. By breaking the reciprocity of the system, it is possible to overcome this restriction and improve the efficiency of energy harvesting. Here, we design and analyze a semitransparent emitter that fully absorbs normally incident energy from a given direction with zero backward and unity forward emissivity. The nearly ideal performance with wavelength-scale thickness is achieved due to the magneto-optical effect and the guided-mode resonance engineered in the emitter structure. We derive the general requirements for the nonreciprocal emitter using the temporal coupled mode theory and the symmetry considerations. As a result, we provide a realistic emitter design based on a photonic crystal slab consisting of a magnetic Weyl semimetal and silicon.