Noise Source Term Research Articles

Entropy generation mechanisms from exothermic chemical reactions in laminar, premixed flames

This paper analyzes the mechanisms of entropy production that stem from exothermic chemical reactions in laminar, premixed flames. This paper is particularly motivated by the problem of indirect combustion noise, where acceleration of entropy fluctuations through a nozzle leads to acoustic fluctuations. In general, entropy generation in reacting systems occurs through molecular transport and chemical reaction, with the reaction terms being dominant in combustion systems. However, there are multiple mechanisms for entropy production from exothermic reactions, including heat release, change in number of moles of species, and changes in species. Most past analyses of indirect combustion noise have equated the heat release term with the flames entropy production, implicitly neglecting the other reaction-induced entropy sources. This paper shows the decomposition of the chemical source term from the entropy equation into three sub-terms, and then use detailed kinetic calculations from a one-dimensional reacting flow solver to calculate their magnitudes. Illustrative results are presented in the paper for several different fuels with varying thermodynamic properties methane, propane, octane, n-dodecane and hydrogen. Calculations are also performed for both air and oxygen as an oxidizer as well as at different parameter sweeps, in order to vary the flames fractional molar production and levels of product dissociation. These results show that heat release is by far the dominant production term in air-fueled flames. However, in oxygen fired systems, heat release has comparable magnitude as the other reaction terms. An important practical implication of this result is that indirect combustion noise source terms can be equated with the heat release terms for air-breathing systems but cannot be for rockets.

Aeroacoustic prediction based on large-eddy simulation and the Ffowcs Williams\u2013Hawkings equation

A hybrid noise computation method is presented in this paper. Large-eddy simulation with wall-model equation is proposed to compute the flow field. With a stress-balanced wall-model equation, the near-wall computation cost of large eddy simulation was effectively reduced. The instantaneous flow variables obtained by the large-eddy simulation were used to compute the noise source terms of the Ffowcs Williams–Hawkings equation. The present method was investigated with two test cases: a single cylinder at Re = 10,000 and a rod-airfoil at Re = 480,000. The flow quantities and aeroacoustic characteristics were compared with the reference data. The mean velocity profiles and spectra of the flow fluctuations were consistent with data from the literature. When compared with the reference data, the noise computation error was less than 3 dB. The computation results demonstrate the present wall-modeled large eddy simulation is efficient for the noise computation of complex vortex shedding flows.

Trailing-Edge Noise Comparability in Open, Closed, and Hybrid Wind Tunnel Test Sections

Open-jet, hard-wall, and hybrid test section configurations are typically used in wind tunnel tests, aiming to represent the ideal free-flight condition. Each test section type requires special adaptations of the experimental hardware and specific correction methodologies that account for systematic measurement errors. Differently from previous research, this paper investigates the comparability of measurements performed in three test section configurations in the same wind tunnel. With this approach, systematic errors related to the facility or boundary-layer tripping methodology are minimized. Basic 2D aerodynamic boundary corrections are evaluated using a DU97W300 airfoil. The corrected lift curve collapsed well while differences in stall behavior and distribution at larger angles of attack are nonnegligible. The trailing-edge noise produced by a NACA-0012, NACA-0018, and NACA-63018 served as reference aeroacoustic sources in this comparability analysis. Moreover, the noise reduction from trailing-edge serrations served as an additional reference for the comparability of relative noise levels. The chord-based Reynolds number of the experiments ranged from 260,000 to 660,000. Unsteady wall pressure measurements performed near the trailing-edge provided a reference for the aeroacoustic noise source terms, demonstrating a negligible change between the different test section types. The applied experimental hardware and acoustic corrections yielded aeroacoustic sources of comparable absolute far-field noise level in the three test section types within . The relative noise levels are comparable within . These results show that aeroacoustic measurements of trailing-edge noise performed in different test section configurations are equivalent, provided that adequate experimental and postprocessing methodologies are systematically implemented.

A general framework for SPDE-based stationary random fields

This paper presents theoretical advances in the application of the Stochastic Partial Differential Equation (SPDE) approach in geostatistics. We show a general approach to construct stationary models related to a wide class of linear SPDEs, with applications to spatio-temporal models having non-trivial properties. Within the framework of Generalized Random Fields, a criterion for existence and uniqueness of stationary solutions for this class of SPDEs is proposed and proven. Their covariance are then obtained through their spectral measure. We present a result relating the covariance of the solution in the case of a White Noise source term with the covariance in a generic case through convolution. Then, we obtain a variety of SPDE-based stationary random fields. In particular, well-known results regarding the Matérn Model and Markovian models are recovered. A new relationship between the Stein model and a particular SPDE is obtained. New spatio-temporal models obtained from evolution SPDEs of arbitrary temporal derivative order are then obtained, for which properties of separability and symmetry can be controlled. We also obtain results concerning stationary solutions for physically inspired models, such as solutions to the heat equation, the advection-diffusion equation, some Langevin’s equations and the wave equation.

Prediction of Combustion Noise in a Model Combustor Using a Network Model and a LNSE Approach

The reduction of pollution and noise emissions of modern aero engines represents a key concept to meet the requirements of the future air traffic. This requires an improvement in the understanding of combustion noise and its sources, as well as the development of accurate predictive tools. This is the major goal of the current study where the low-order thermo-acoustic network (LOTAN) solver and a hybrid computational fluid dynamics/computational aeroacoustics approach are applied on a generic premixed and pressurized combustor to evaluate their capabilities for combustion noise predictions. LOTAN solves the linearized Euler equations (LEE) whereas the hybrid approach consists of Reynolds-averaged Navier–Stokes (RANS) mean flow and frequency-domain simulations based on linearized Navier–Stokes equations (LNSE). Both solvers are fed in turn by three different combustion noise source terms which are obtained from the application of a statistical noise model on the RANS simulations and a post-processing of incompressible and compressible large eddy simulations (LES). In this way, the influence of the source model and acoustic solver is identified. The numerical results are compared with experimental data. In general, good agreement with the experiment is found for both the LOTAN and LNSE solvers. The LES source models deliver better results than the statistical noise model with respect to the amplitude and shape of the heat release spectrum. Beyond this, it is demonstrated that the phase relation of the source term does not affect the noise spectrum. Finally, a second simulation based on the inhomogeneous Helmholtz equation indicates the minor importance of the aerodynamic mean flow on the broadband noise spectrum.

Combustion Noise Prediction Using Linearized Navier–Stokes Equations and Large-Eddy Simulation Sources

In modern aeroengines, combustion noise has become a significant source to the overall noise, particularly at approach conditions. This requires further advances in understanding and predicting combustion noise of turbulent flames. This is the aim of the present study, where a hybrid computational fluid dynamics/computational aeroacoustics approach is applied on a generic premixed and pressurized combustor to assess its accuracy for combustion noise predictions. The hybrid approach consists of Reynolds-averaged Navier–Stokes (RANS) or large-eddy simulations (LES) mean flow and frequency-domain simulations based on linearized Navier–Stokes equations that are fed by combustion noise source terms. The latter are obtained from both the application of a statistical noise model on the RANS simulations and a postprocessing of incompressible LES, in a first step. The acoustic simulation results are compared with experimental pressure measurements conducted by the Centre National de la Recherche Scientifique. Very good agreement is found over the entire frequency range if the LES source model is applied. Sensitivity studies with respect to feeding lines, mean flowfield, and mesh were performed. The resulting comparisons of the linearized Navier–Stokes equation simulations based on the RANS and LES flowfields revealed that the combustion noise spectrum is mainly governed by the heat release spectrum but not by the aerodynamic combustor flowfield. However, this issue needs further investigation.

The contribution of intrinsic thermoacoustic feedback to combustion noise and resonances of a confined turbulent premixed flame

The influence of Intrinsic Thermoacoustic (ITA) feedback on the combustion noise spectrum produced by a confined, turbulent, premixed, swirl flame is investigated. The analysis is based on the understanding that sound is generated by unsteady heat release resulting from turbulent fluctuations on the one hand, and from the response of the flame to incoming acoustic perturbations on the other. The former effect is described by a source term for combustion noise, i.e. a spectral distribution of unsteady heat release rate, the latter by the flame transfer function. Both quantities are identified from time series data for fluctuating velocity and heat release rate, generated with large eddy simulation of premix swirl burner. The combustion noise source term and the flame transfer function are then introduced in an acoustic network model of the test rig in order to compute the spectral distribution of the sound pressure level at a certain location in the combustion chamber. Results for the noise spectrum are in good agreement with experiment, showing a broadband component and well-defined peaks. The frequencies of the peaks correspond to either acoustic cavity or ITA resonances. The acoustic network model is used for parametric studies, where the acoustic reflection coefficient at the combustor exit is varied. Remarkably, it is found that the magnitude of the ITA peak increases with decreasing values of the acoustic reflection coefficient, and vice versa. Furthermore, the influence of combustion chamber length on resonance frequencies is explored. It is observed that the frequency of the ITA resonance is insensitive to combustor length. This behaviour is observed qualitatively also in experiments.

Numerical solutions of some stochastic hyperbolic wave equations including sine-Gordon equation

Many wave equations, including Klein–Gordon, Liouville’s and the sine-Gordon equation, with added space–time white noise can be transformed to second order partial differential equations with mixed derivatives of the form Yxt=F(Y)+σWxt. Such equations are related to what Zimmerman (1972) called a diffusion equation. For such equations an explicit numerical scheme is employed in both deterministic and stochastic examples which is checked for accuracy against known exact analytical deterministic solutions. The accuracy is further tested in the stochastic case with F=0 by comparing statistics of solutions with those for the Brownian sheet. Generally the boundary conditions are chosen to be values of Y(x,0) and Y(0,t) on boundaries of the first quadrant or subsets thereof, existence and uniqueness of solutions having been established for such systems. For the linear case, solutions are compared at various grid sizes and wave-like solutions were found, with and without noise, for non-zero initial and boundary conditions. Surprisingly, putative wave-like structures seemed to emerge with zero initial and boundary conditions and purely noise source terms. Equations considered with nonlinear F included quadratic and cubic together with the sine-Gordon equation. For the latter, wave-like structures were apparent with σ≤0.25 but they tended to be shattered at larger values of σ. Previous work on stochastic sine-Gordon equations is briefly reviewed.

Homogenization of the evolution Stokes equation in a perforated domain with a stochastic Fourier boundary condition

The evolution Stokes equation in a domain containing periodically distributed obstacles subject to Fourier boundary condition on the boundaries is considered. We assume that the dynamic is driven by a stochastic perturbation on the interior of the domain and another stochastic perturbation on the boundaries of the obstacles. We represent the solid obstacles by holes in the fluid domain. The macroscopic (homogenized) equation is derived as another stochastic partial differential equation, defined in the whole non perforated domain. Here, the initial stochastic perturbation on the boundary becomes part of the homogenized equation as another stochastic force. We use the two-scale convergence method after extending the solution with 0 in the holes to pass to the limit. By Itô stochastic calculus, we get uniform estimates on the solution in appropriate spaces. In order to pass to the limit on the boundary integrals, we rewrite them in terms of integrals in the whole domain. In particular, for the stochastic integral on the boundary, we combine the previous idea of rewriting it on the whole domain with the assumption that the Brownian motion is of trace class. Due to the particular boundary condition dealt with, we get that the solution of the stochastic homogenized equation is not divergence free. However, it is coupled with the cell problem that has adivergence free solution. This paper represents an extension of the results of Duan and Wang (Comm. Math. Phys. 275:1508--1527, 2007), where a reaction diffusion equation with a dynamical boundary condition with a noise source term on both the interior of the domain and on the boundary was studied, and through a tightness argument and a pointwise two scale convergence method the homogenized equation was derived.

Progress towards primary frequency standard

The cesium fountain clock as primary frequency standard is widely used in the areas, such as time-keeping system, satellite navigation, fundamental physics research, etc. The principle of operation of cesium fountain clock is introduced. The noise source and frequency shift term are ananlyzed. The major noise source influencing frequency stability are cold atom loading time, microwave phase noise related to Dick effect, and detection laser frequency noise. The major frequency bias influencing frequency uncertainty is blackbody radiation frequency shift,cold atom collision frequency shift,distributed cavity phase frequency shift and microwave leakage frequency shift.The key technique to achieve highperformance cesium fountain clock is sumerized. The application of cesium fountain clock is presented. The status of space cesium clock and future primary frequency standard of optical clock are shown.

Investigation of combustion noise using a LES/CAA hybrid approach

A hybrid approach combining large eddy simulation with computational aeroacoustics (LES/CAA) to describe combustion noise in the near field of a turbulent flame is presented. The approach is based on a LES of the reactive flow field using a low Mach number approximation and the solution of the wave equation with the appropriate combustion noise source term and inhomogeneous fluid properties. Two well established solvers are combined into a single tool for the description of the acoustics in the vicinity of the flame, exploiting the disparity of scales between the fluid flow and the acoustics. The complete approach is tested for a non-premixed turbulent jet flame and provides an insight into the combustion noise phenomenon. The LES is shown to yield good agreement with measurements for the velocity, the mixture fraction, as well as the temperature. The comparison of the acoustic results to an experimentally determined noise intensity spectrum allows to judge on the prediction quality of the presented LES/CAA hybrid approach.

Acoustic source terms study for non-isothermal flows using an aeroacoustic hybrid approach

This paper presents a numerical study of noise source term in non-isothermal flows in the context of an aeroacoustic hybrid technique at low Mach numbers. Asymptotic analysis applied to the fully compressible Navier–Stokes equations provides separated sets of equations for the dynamic of the flow and the production and propagation of acoustic waves. Comparisons with analytical dipole and quadrupole distributions are performed, confirming the dipole type of non-isothermal source distribution. This paper is a preliminary work for some more extensive studies on the topic. To cite this article: F. Golanski, C. Prax, C. R. Mecanique 333 (2005).

Influence of magnetic field on 1/ f noise and thermal noise in multi-terminal homogeneous semiconductor resistors and discrimination between the number fluctuation model and the mobility fluctuation model for 1/ f noise in bulk semiconductors

We have derived an accurate noise current density equation for homogeneous semiconductors under a constant magnetic field with three noise source terms which are the thermal noise source term and the two excess noise source terms due to either carrier number fluctuation or carrier mobility fluctuation. Based on Hooge’s empirical relations for 1/ f noise, the two excess noise source terms are shown to become equal to each other under zero magnetic field. Using the characteristic potential method and employing Hooge’s empirical relations for 1/ f noise, we have derived the formulas for the short-circuit terminal 1/ f noise and thermal noise currents and the open-circuit terminal 1/ f noise and thermal noise voltages of multi-terminal homogeneous semiconductor resistors with arbitrarily-shaped 2-D geometries under a constant magnetic field. We have shown that the derived formulas can explain the measured 1/ f noise and thermal noise of n-GaAs rectangularly-shaped Hall devices and n-GaAs Corbino disks under magnetic field from 0 to 8 T at room temperature. It is also shown that the magnetic field dependence of 1/ f noise in bulk semiconductors should be explained by the number fluctuation model rather than by the mobility fluctuation model.

Fan interaction noise reduction using a wake generator: experiments and computational aeroacoustics

A control grid (wake generator) aimed at reducing rotor–stator interaction modes in fan engines when mounted upstream of the rotor has been studied here. This device complements other active noise control systems currently proposed. The compressor model of the instrumented ONERA CERF-rig is used to simulate suitable conditions. The design of the grid is drafted out using semi-empirical models for wake and potential flow, and experimentally achieved. Cylindrical rods are able to generate a spinning mode of the same order and similar level as the interaction mode. Mounting the rods on a rotating ring allows for adjusting the phase of the control mode so that an 8 dB sound pressure level (SPL) reduction at the blade passing frequency is achieved when the two modes are out of phase. Experimental results are assessed by a numerical approach using computational fluid dynamics (CFD). A Reynolds averaged Navier–Stokes 2-D solver, developed at ONERA, is used to provide the unsteady force components on blades and vanes required for acoustics. The loading noise source term of the Ffowcs Williams and Hawkings equation is used to model the interaction noise between the sources, and an original coupling to a boundary element method (BEM) code is realized to take account of the inlet geometry effects on acoustic in-duct propagation. Calculations using the classical analytical the Green function of an infinite annular duct are also addressed. Simple formulations written in the frequency domain and expanded into modes are addressed and used to compute an in-duct interaction mode and to compare with the noise reduction obtained during the tests. A fairly good agreement between predicted and measured SPL is found when the inlet geometry effects are part of the solution (by coupling with the BEM). Furthermore, computed aerodynamic penalties due to the rods are found to be negligible. These results partly validate the computation chain and highlight the potential of the wake generator system proposed.

COMPUTATIONAL AEROACOUSTICS EXAMPLES SHOWING THE FAILURE OF THE ACOUSTIC ANALOGY THEORY TO IDENTIFY THE CORRECT NOISE SOURCES

Lighthill's Acoustic Analogy has been the dominant theory of aeroacoustics, especially jet aeroacoustics for almost fifty years. As yet, except for the u8 scaling law, which was derived by dimensional analysis, jet noise prediction based on the Acoustic Analogy approach has not been particularly successful. This paper examines some of the weaknesses and ambiguities in the formulation of the Acoustic Analogy theories. It is concluded that if the analogy is carried out completely, in the sense that the full wave propagation terms are retained in the propagation part of the equations of the analogy, then the theory offers no sensible noise source terms. To demonstrate that the Acoustic Analogy can fail to identify the correct noise sources, four examples are considered. They include an initial value problem, a boundary problem, the problem of weak solution and the problem of sound generation by instability waves in jets and mixing layers. These examples show clearly how, in each case, the Acoustic Analogy theory identifies the wrong noise source. Indeed, the Acoustic Analogy could provide, if not careful, misleading interpretation of the physics of sound generation. This paper is dedicated to Professor David G. Crighton, outstanding applied mathematician, world famous acoustician and a much respected friend.

Theory of 1/f noise currents in semiconductor devices with one-dimensional geometry and its application to Si Schottky barrier diodes

A theory of the short-circuit 1/f noise currents in semiconductor devices with one-dimensional geometry is derived using the noise current density equation with the 1/f noise source term based on the Hooge's empirical 1/f noise relation. It is shown that both the number fluctuation model and the mobility fluctuation model for 1/f noise give the same result. The newly derived 1/f formula is shown to explain the measured 1/f noise currents of Si Schottky barrier rectifiers under both forward and reverse bias conditions with a single set of Hooge parameters of electrons and holes for a given device.

Quantum noise of a mode-locked laser

The master equation of mode locking written in operator notation is supplemented with noise-source terms that conserve commutator brackets. The noise sources are associated with the reservoirs responsible for loss and gain. The output of a mode-locked laser with the least possible quantum noise is determined.

Calcul du rayonnement acoustique d'une couche de mélange à l'aide des équations d'Euler linéarisées

The noise radiated by a mixing layer is computed by an hybrid approach using linearized Euler's equations (LEE) as a wave operator. The acoustic source terms are obtained by substituting the aerodynamic fluctuating velocity field computed by the Large Eddy Simulation (LES) method into the right-hand side of the LEE. The radiated sound field as calculated in this way is in good agreement with the direct calculation obtained by LES. This study provides support to the validity of the acoustic analogy based on the LEE, both for the noise source terms and the mean flow effects on sound propagation.

Nonsingular Collapse of a Perfect Fluid Sphere Within a Dilaton-Gravity Reformulation of the Oppenheimer Model

A generic four-dimensional dilaton gravity is considered as a basis for reformulating the paradigmatic Oppenheimer–Synder model of a gravitationally collapsing star modelled as a perfect fluid or dust sphere. Initially, the vacuum Einstein scalar-tensor equations are modified to Einstein–Langevin equations which incorporate a noise or micro-turbulence source term arising from Planck scale conformal, dilaton fluctuations which induce metric fluctuations. Coupling the energy-momentum tensor for pressureless dust or fluid to the Einstein–Langevin equations, a modification of the Oppenheimer–Snyder dust collapse model is derived. The Einstein–Langevin field equations for the collapse are of the form of a Langevin equation for a non-linear Brownian motion of a particle in a homogeneous noise bath. The smooth worldlines of collapsing matter become increasingly randomised Brownian motions as the star collapses, since the backreaction coupling to the fluctuations is non-linear; the input assumptions of the Hawking–Penrose singularity theorems are then violated. The solution of the Einstein–Langevin collapse equation can be found and is non-singular with the singularity being smeared out on the correlation length scale of the fluctuations, which is of the order of the Planck length. The standard singular Oppenheimer–Synder model is recovered in the limit of zero dilaton fluctuations.

Kinky beam paths inside photorefractive crystals

Beam paths inside photorefractive crystals under conditions of beam fanning or self-pumped conjugation that uses crystal corners are often kinky—i.e., they consist of short, straight line segments as opposed to smoothly curved paths. We show that kinky paths occur naturally when the beam-coupling equations including noise source terms are solved to all Bragg orders without phase-matching approximations. Higher-order gratings arise in high-gain two-wave mixing without noise but are apparently unobservable because fanning always dominates in real crystals. In the presence of noise and fanning the residual signature of the higher-order gratings is kinky beam paths.