Perturbation Equations Research Articles

Effect of vortex and entropy sources in sound generation for compressible cavity flow

The present study reports the effect of different source terms on the near and far-field acoustic characteristics of compressible flow over a rectangular cavity using hybrid computational aeroacoustics methodology. We use a low dispersive and dissipative compressible fluid flow solver in conjunction with an acoustic perturbation equation solver based on the spectral/hp element method. The hybrid approach involves calculating the base fields and the acoustic sources from a fluid simulation in the first step. In the next step, the acoustic solver utilizes the variables to predict the acoustic propagation due to the given sources. The validation of the methodology against benchmark cases provides quite accurate results while compared against the existing literature. The study is then extended to assess the importance of the entropy source term for the flow over a rectangular cavity. The predictions of hybrid simulations with vortex and entropy source terms reproduce the perturbation pressure values very close to the existing direct numerical simulation results. Moreover, the results suggest that the use of just the vortex source terms over-predicts the perturbation pressure near the source region. Finally, we have carried out detailed simulations with all the source terms to investigate the noise sources for compressible flow over the cavity for different Mach number ranges (M=0.4,0.5,0.6,0.7,1.5). The obtained acoustic spectra and the sound directivity are in close agreement with the reference experiment.

Gravitational and Electromagnetic Perturbations of a Charged Black Hole in a General Gauge Condition

We derive a set of coupled equations for the gravitational and electromagnetic perturbation in the Reissner–Nordström geometry using the Newman–Penrose formalism. We show that the information of the physical gravitational signal is contained in the Weyl scalar function Ψ4, as is well known, but for the electromagnetic signal, the information is encoded in the function χ, which relates the perturbations of the radiative Maxwell scalars φ2 and the Weyl scalar Ψ3. In deriving the perturbation equations, we do not impose any gauge condition and as a limiting case, our analysis contains previously obtained results, for instance, those from Chandrashekhar’s book. In our analysis, we also include the sources for the perturbations and focus on a dust-like charged fluid distribution falling radially into the black hole. Finally, by writing the functions on the basis of spin-weighted spherical harmonics and the Reissner–Nordström spacetime in Kerr–Schild type coordinates, a hyperbolic system of coupled partial differential equations is presented and numerically solved. In this way, we completely solve a system that generates a gravitational signal as well as an electromagnetic/gravitational one, which sets the basis to find correlations between them and thus facilitates gravitational wave detection via electromagnetic signals.

Hyperbolicity and causality of Einstein-Gauss-Bonnet gravity in warped product spacetimes

In Einstein-Gauss-Bonnet gravity, for a group of warped product spacetimes, we get a generalized master equation for the perturbation of tensor type. We show that the "effective metric" or "acoustic metric" for the tensor perturbation equation can be defined even without a static condition. Since this master equation does not depend on the mode expansion, the hyperbolicity and causality of the tensor perturbation equation can be investigated for every mode of the perturbation. Based on the master equation, we study the hyperbolicity and causality for all relavent vacuum solutions of this theory. For each solution, we give the exact hyperbolic condition of the tensor perturbation equations. Our approach can also applied to dynamical spacetimes, and Vaidya spacetime have been investigated as an example.

Coductivities in holographic two-currents model

In this letter, we decouple the perturbative equations of the gauge fields in the holographic two-currents model. In this way, we analytically work out the incoherent conductivities. These conductivities can be expressed in terms of the thermodynamic quantities of this system. Therefore, they are intrinsic quantities of the holographic two-currents model. As a by-product, we also discuss the mathematical equivalence between the two theories with coupling and without coupling between two gauge fields. In addition, when the coupling θ increases to θ=1, a soft gap emergences at the low frequency in Re[σA]/Re[σB], which is independent the doping χ.

Reducing the charging time of a lead–acid cell in the sense of linear stability analysis

Long charging time is one of the main disadvantages of lead–acid batteries. Although the higher charging voltage can shorten the charging time, it is limited in order to avoid thermal instability. In the present research, the charging time has shortened using a higher charging voltage while the stability of the system has remained the same through thermal stability analysis. The method of linear stability analysis is applied to derive governing equations of perturbations. The set of derived equations are rearranged to obtain the desired form of the eigenvalue problem. The obtained eigenvalue problem is solved numerically, which has resulted in a neutral curve separating the stable zone from the unstable one, and the critical point. In addition, the effects of some geometrical parameters on the position of the neutral curve are investigated. All charging processes are simulated using the computational battery dynamic method. The results show that on the one hand, the stable zone increases by increasing the maximum dimensionless volume. On the other hand, the stable zone decreases by increasing the charging voltage. In conclusion, an eight percent reduction in the charging time is observed through increasing both of the charging voltage and the maximum dimensionless volume at the same time.

Numerical study of flow and noise predictions for tandem cylinders using incompressible improved delayed detached eddy simulation combined with acoustic perturbation equations

The method of combining incompressible hybrid Reynolds average Navier–Stokes/large eddy simulation (RANS/LES) with an acoustic perturbation equation was studied to accurately predict the flow and noise performances in NASA tandem cylinder experiments. In this approach, acoustic waves simultaneously propagate in the time domain with an unsteady turbulent flow. The experiment was performed under the condition of a Reynolds number of 1.66 × 105, based on the cylinder diameter, which provided detailed surface pressure and flow field measurements. The improved delayed detached eddy simulation (IDDES) model established in this study was consistent with the experimental results, except for the flow field behind the rear cylinder, which was caused by an early separation relative to the experiment. The acoustic results showed that the rear cylinder dominated the noise radiation, which indicated that the pressure fluctuation caused by wake interference was greater than that caused by vortex shedding. Acoustic predictions were also obtained using an acoustic analogy approach based on the FH-W method. Three kinds of integral surfaces (PS-open, PS-close, and IPS) were made for the acoustic predictions using the Ffowcs Williams–Hawkings method. This study also showed that wavenumber decomposition could be utilised to quantify the mechanisms related to turbulent convection and sound propagation. An IDDES hybrid model for predicting near-field and far-field noise was established, which could be applied to practical engineering applications.

Modified gravity or imperfect dark matter: a model-independent discrimination

We analyze how to parametrize general modifications of the dark matter perturbations equations in a model-independent way. We prove that a general model with an imperfect and non-conserved dark matter fluid with bulk and shear viscosities and heat flux in a modified gravity scenario can be described with five general functions of time and scale. We focus on the sub-Hubble regime within the quasi-static approximation and calculate the observable power spectra of the galaxy distribution, galaxy velocities and weak lensing and find that these observables are only sensitive to three combinations of the initial five functions. Deviations of these three observable functions with respect to ΛCDM give us different characteristic signals which allow us to determine in which cases it is possible to discriminate a modification of gravity from an imperfect or non-conserved dark matter. Finally, we perform a Fisher forecast analysis for these three parameters and show an example for a particular model with shear viscosity.

Spherically symmetric linear perturbations of electrically counterpoised dust

We consider spherically symmetric linear perturbations of static spherically symmetric spacetimes where the matter content is electrically counterpoised dust. We show that the evolution equation for the fluid perturbation implies that the fluid elements move with constant velocities. Therefore there are neither oscillations nor exponential departure from the background solution. We present an explicit example showing that the perturbation could lead to the formation of a black hole.

Nonlinear analysis of the effect of viscoelasticity on ferroconvection

AbstractThis paper concerns a nonlinear analysis of the effects of viscoelasticity on convection in ferroliquids. We consider the Oldroyd model for the constitutive equation of the liquid. The linear stability analysis yields the critical value of the Rayleigh number for the onset of oscillatory convection in Maxwell and Jeffrey ferroliquids. The use of a minimal mode double Fourier series in the nonlinear perturbation equations yields a Khayat–Lorenz model for the ferromagnetic liquid, and that is scaled further to get the classical Lorenz model as a limiting case. The scaled Khayat–Lorenz model thus obtained is solved numerically and the solution is used to compute the time‐dependent Nusselt number, which quantifies the heat transport. The results are analyzed for the dependence of the time‐averaged Nusselt number on different parameters.

Coulomb interactions and renormalization of semi-Dirac fermions near a topological Lifshitz transition

We aim to understand how the spectrum of semi-Dirac fermions is renormalized due to long-range Coulomb electron-electron interactions at a topological Lifshitz transition, where two Dirac cones merge. At the transition, the electronic spectrum is characterized by massive quadratic dispersion in one direction, while it remains linear in the other. We have found that, to lowest order, the unconventional log squared (double logarithmic) correction to the quasiparticle mass in bare perturbation theory leads to resummation into strong mass renormalization in the exact full solution of the perturbative renormalization group equations. This behavior effectively wipes out the curvature of the dispersion and leads to Dirac cone restoration at low energy: the system flows towards Dirac dispersion which is anisotropic but linear in momentum, with interaction-depended logarithmic modulation. The Berry phase associated with the restored critical Dirac spectrum is zero - a property guaranteed by time-reversal symmetry and unchanged by renormalization. Our results are in contrast with the behavior that has been found within the large-$N$ approach.

Effects of Texture Bottom Profile on Static and Dynamic Characteristics of Journal Bearings

The purpose of this paper is to numerically study the effect of texture bottom profile on static, dynamic, and stability performance parameters of hydrodynamic journal bearings. The different performance parameters of square textured journal bearings with different bottom profiles are numerically investigated and compared with those of smooth journal bearing. There are five bottom profiles of this square texture: flat, curved, isosceles triangle (T1), oblique triangle (T2), and oblique triangle (T3). The static and dynamic coefficients are calculated by solving the steady‐state Reynolds equation and the perturbation equations with FDM numerical technique. The performance characteristics under different texture distribution, depth, and bottom profiles are studied, and the current numerical results show that the selection of texture parameters is crucial to improve the static, dynamic, and stability performances of hydrodynamic journal bearing. Meanwhile, it is also found that the square texture with a flat bottom profile has a higher improvement in the values of static performance parameters in comparison to those other bottom profiles. Moreover, the simulation results indicate that the dynamic and stability performances improvement of textured journal bearing is also significant, especially when the eccentricity ratio is smaller.

Numerical Study of Rayleigh-Benard Problem under the Effect of Magnetic Field

In this paper, a linear stability analysis is studied for Rayliegh-Benard problem with the effect of magnetic field, a perturbation equations is solved numerically by using spectral Chebyshev tau method, the boundaries are considered both are free, both are rigid, the lower is free and the upper is rigid, the results were illustrated graphically and compared with previous studies.

On perturbation theory and critical exponents for self-similar systems

Gravitational critical collapse in the Einstein-axion-dilaton system is known to lead to continuous self-similar solutions characterized by the Choptuik critical exponent gamma . We complete the existing literature on the subject by computing the linear perturbation equations in the case where the axion-dilaton system assumes a parabolic form. Next, we solve the perturbation equations in a newly discovered self-similar solution in the hyperbolic case, which allows us to extract the Choptuik exponent. Our main result is that this exponent depends not only on the dimensions of spacetime but also the particular ansatz and the critical solutions that one started with.

Vortex Dynamo in an Obliquely Rotating Stratified Nanofluid by Small-Scale Non-Helical Forces

In this work, a large-scale instability of the hydrodynamic -effect in an obliquely rotating stratified nanofluid taking into account the effects of Brownian diffusion and particle flux under the influence of a temperature gradient (thermophoresis) is obtained. The instability is caused by the action of an external small-scale non-spiral force, which excites small-scale velocity oscillations with zero helicity and a low Reynolds number. Nonlinear equations for large-scale motions are obtained using the method of multiscale asymptotic expansions by a small parameter (Reynolds number). A linear large-scale instability of hydrodynamic -effect is investigated depending on the parameters of rotation , temperature stratification , and concentration of nanoparticles . A new effect of the generation of large-scale vortex structures in nanofluid at is associated with an increase in the concentration of nanoparticles is obtained. The maximum instability increment is reached at inclination angles for the Prandtl numbers , and for the Prandtl numbers at inclination angles . It has been found that the frequency changing of the parametric impact will make it possible to control and track the generation of large-scale vortex structures. It is shown that circularly polarized Beltrami vortices appear in nanofluid as the result of new large-scale instability development. In this paper, the saturation regime of large-scale instability in an obliquely rotating stratified nanofluid with an external small-scale non-spiral force is investigated. In the stationary regime was obtained a dynamic system of equations for large-scale perturbations of the velocity field. Numerical solutions of this system of equations are obtained, which show the existence of localized vortex structures in the form of nonlinear Beltrami waves and kinks. The velocity profile of kink tends to be constant at large Z values.

Numerical and asymptotic flow stability analysis of vortex structures

Stability problem of an axisymmetric swirling flow of a viscous incompressible fluid with respect to nonaxisymmetric perturbations is considered. The system of ordinary differential equations for the amplitude functions is solved numerically by the Runge-Kutta method and orthogonalization procedure. Solutions of equations for perturbations at the neighborhood of singular points are obtained by the Frobenius method. The maximum of amplification coefficients and phase velocities of five unstable modes are calculated.

Bidispersive thermal convection with relatively large macropores and generalized velocity and temperature boundary conditions

In a fluid-saturated bidisperse porous medium at a single temperature, the issue of thermal convection when the Darcy theory is used in the micropores, particularly the effects of slip boundary conditions on the model stability, was examined (whereas the Brinkman theory is used in the macropores). In addition, the effect of general temperature boundary conditions was also addressed. The governing equations of motion are provided, followed by the way in which the related equations of perturbation were derived. In addition, the linear instability and nonlinear stability analyses of the system were introduced, with the numerical approach used to approximate the eigenvalue system resulting from our analysis. The threshold for linear instability was proved to be the same as the one for nonlinear stability, showing that the linear theory accurately captures the mechanism of the onset of thermal convection. The numerical results for stability/instability thresholds were also introduced. The researchers assumed that this was the first time a mixed Darcy–Brinkman model had been used in bidisperse flow with slip boundary conditions.

An intermediate inflation of the early Universe from f(R) gravity

In this work, we study on the inflationary era of the Universe in the context of f(R) gravity. For this purpose, we consider the f(R) gravity and its scalar perturbation equations. We start with an equation symmetry approach, a notation, which characterizes both the field equations of the gravity and evolution of the scalar perturbations. Throughout the study, to realize inflation we assume that the first slow-roll parameter $$\epsilon _{1}$$ takes a small value with the number of e-folds which has nearly a constant value as $$N\sim 60$$ . An intermediate inflation, which has an initial condition (the Bunch–Davies vacuum state), is obtained. Here, the resulting solution eventually causes the universe enters to the radiation era. Viability of this inflation scenario is investigated by calculating the power spectrum of the primordial curvature perturbations which is observed nearly scale-invariant with some parameters of the intermediate scale factor. Then, the two observable parameters (i.e., the scalar spectral index parameter and the tensor-to-scalar ratio) are obtained. The findings are compared with latest observational data, and we have observed that this inflation scenario for the early universe is possible.

Perturbations in Regularized Lovelock Gravity

In this paper we study the perturbation theory of the recently proposed Regularized Lovelock Gravity, on the curved FLRW space-time. We provide the first order perturbation equations both in the scalar and tensor sector in the presence of an additional minimally coupled scalar field. A general expression for the velocity of gravitational waves at the generic order is inferred. Moreover, we apply the results on the study of slow-roll inflation on flat FLRW background, at second order in the Regularized Lovelock Gravity, i.e. in the presence of the Gauss-Bonnet correction only. We derive the power spectra of scalar and tensor perturbations, and provide the equations for the inflation observable quantities: the spectral indices and the tensor-to-scalar spectra ratio.

Adjoint-Based Calculation of Parametric Thermoacoustic Maps of an Industrial Combustion Chamber

Abstract The aim of this study is to efficiently calculate parametric thermoacoustic maps of typical combustion chambers. Two configurations are considered: an academic configuration based on a Rijke tube, and an industrial combustion chamber, which is the core of a recently developed microturbine for power generation. Such maps can be understood as the collection of loci of thermoacoustic eigenfrequencies obtained under systematic variations of some defined parameters, while considering the Helmholtz equation as the thermoacoustic model of interest. In this study we consider variations on two parameters: the gain n and time-delay τ associated with a generic flame response model. We also show the feasibility of the proposed approach when considering more realistic flame responses. A straight-forward way to calculate such a thermoacoustic map is by solving the Helmholtz equation, and, thus, the corresponding nonlinear eigenvalue problem (NLEVP), one time per parameter combination. With that approach, the nonlinear eigenvalue problem needs to be solved hundreds or thousands of times if an adequate resolution of the thermoacoustic map is sought. Such a strategy may be computationally unaffordable. In order to overcome this difficulty, this work utilizes an adjoint-based, high-order perturbation method. The actual eigenvalue problem is only solved once at a baseline point. After applying the perturbation equations at that point, a polynomial rational function—the Padé approximant—is obtained to estimate the eigenfrequency drift that results for a small or large perturbation in the flame response. It is demonstrated, for both academic and industrial test cases, that the obtained maps are accurate. Additionally, it is shown that these maps reveal a large variety of thermoacoustic features, such as stability boundaries, intrinsic thermoacoustic modes, and exceptional points. The numerical costs for such calculations are negligible even for the industrial combustion chamber investigated.

EQUATIONS OF PLANETARY SYSTEMS MOTION

The study of the dynamically evolution of planetary systems is very actually in relation with findings of exoplanet systems. free spherical bodies problem is considered in this paper, mutually gravitating according to Newton's law, with isotropically variable masses as a celestial-mechanical model of non-stationary exoplanetary systems. The dynamic evolution of planetary systems is learned, when evolution's leading factor is the masses' variability of gravitating bodies themselves. The laws of the bodies' masses varying are assumed to be known arbitrary functions of time. When doing so the rate of varying of bodies' masses is different. The planets' location is such that the orbits of planets don't intersect. Let us treat this position of planets is preserve in the evolution course. The motions are researched in the relative coordinates system with beginning in the center of the parent star, axes that are parallel to corresponding axes of the absolute coordinates system. The canonical perturbation theory is used on the base aperiodic motion over the quasi-canonical cross-section. The bodies evolution is studied in the osculating analogues of the second system of canonical Poincare elements. The canonical equations of perturbed motion in analogues of the second system of canonical Poincare elements are convenient for describing the planetary systems dynamic evolution, when analogues of eccentricities and analogues of inclinations of orbital plane are sufficiently small. It is noted that to obtain an analytical expression of the perturbing function main part through canonical osculating Poincare elements using computer algebra is preferably. If in expansions of the main part of perturbing function is constrained with precision to second orders including relatively small quantities, then the equations of secular perturbations will obtained as linear non-autonomous system. This circumstance meaningful makes much easier to study the non-autonomous canonical system of differential equations of secular perturbations of considering problem.