Small Perturbation Solution Research Articles

On inhibition of the Rayleigh–Taylor instability by a horizontal magnetic field in ideal MHD fluids with velocity damping

It is still open whether the inhibition phenomenon of the Rayleigh–Taylor (RT) instability by a horizontal magnetic field can be mathematically verified for a non-resistive magnetohydrodynamic (MHD) fluid in a two-dimensional (2D) horizontal slab domain, since it was roughly verified in the linearized case by Wang in [43]. In this paper, we show that this inhibition phenomenon can be rigorously verified in the (nonlinear) inhomogeneous, incompressible, inviscid case with velocity damping. More precisely, we show that there is a critical number mC, such that if the strength |m| of a horizontal magnetic field is bigger than mC, then the small perturbation solution around the magnetic RT equilibrium state is exponentially stable in time. Moreover, we also provide a nonlinear instability result for the case |m|∈(0,mC). Our instability result reveals that a horizontal magnetic field can not inhibit the RT instability, if it's strength is too small.

Antenna geometry effect on the reconstruction of the unsteady rotating forces of axial fan blade

The aim of this paper is to investigate the influence of the geometrical arrangement of acoustic pressure measurement sensors on the reconstruction of the unsteady rotating forces acting on the fluid by the fan’s blade. A development based on the tonal noise generated by an axial fan and validated with a directivity experience is used to derive a discretized form of the direct problem and to simulate acoustic pressures at known spatial positions in the radiated field. The inverse problem is usually ill-posed and requires optimization technique to stabilize the solution for small perturbations in the measured acoustic input data. The reconstruction shows that the inverse problem conditioning depends on the aeroacoustic source and the far-field sensor number and geometrical distribution as well as on the studied frequency. Tikhonov regularization can provide an appropriate regularization parameter leading to a satisfactory reconstruction of imposed unsteady rotating forces even in the presence of measurement noise. Simulations are conducted for three geometrical arrangements of microphones: line, arc of a circle and square.

Small perturbation solutions for nonlocal elliptic equations

ABSTRACTWe present a small perturbation result for nonlocal elliptic equations, which says that for a class of nonlocal operators, the solutions are in Cσ+α for any α∈(0,1) as long as the solutions are small. This is a nonlocal generalization of a celebrated result of Savin in the case of second order equations.

Optimal edge preserving restoration with efficient regularisation

Deblurring is an ill-posed inverse problem naturally and becomes more complex if the noise is also tainting the image. Classical approaches of inverse filtering and linear algebraic restorations are now obsolete due to the additive noise, as the problem is very sensitive to the small perturbation in the data. To cater the sensitivity of solution for small perturbations, smoothness constraints are generally added in the classical approaches. Previously neural networks and gradient based approaches have widely been used for optimisation; however, due to improper regularisation and computational cost, the problem is still an active research problem. In this paper, a new simple, efficient and robust method of regularisation based on the difference of grey scale average of image and each element of the estimated image is proposed. Constrained least square error is considered for optimisation and gradient based steepest descent algorithm is designed to estimate the optimal solution for restoration iteratively. The visual results and statistical measures of the experiments are presented in the paper which shows the effectiveness of the approach as compared to the state of art and recently proposed techniques.

Reconstruction of the unsteady rotating forces of fan’s blade from far-field sound pressure

The aim of this paper is to investigate an inverse method used to evaluate the unsteady rotating forces acting on the fluid by the fan’s blade. A simple model based on the tonal noise produced by an axial fan and validated with a directivity experience is used to derive a discretized form of the direct problem and to simulate acoustic pressures at known spatial positions in the radiated field. The inversion of this direct problem is ill-conditioned and requires a regularization technique to stabilize the solution for small perturbations in the measured acoustic pressures. The reconstruction reveals that the conditioning of the inverse problem depends on the aeroacoustic source and the far-field microphones number as well as on the studied frequency. Tikhonov regularization can provide an appropriate regularization parameter leading to an accurate reconstruction of imposed unsteady rotating forces even with the presence of measurement noise.

Small perturbation solutions for parabolic equations

Let $\phi$ be a smooth solution of the parabolic equation $F(D^2u, Du, u, x, t)- u_{t} = 0$: Assume $F$ is uniform elliptic only in a neighborhood of $(D^2\phi, D\phi, \phi, x, t)$, we prove that any solution obtained from small L1-perturbation of $\phi$ remains smooth.

Transformation of Rice’s small perturbation solution for rough surface scattering into the high frequency reciprocal physical and geometric optics solutions

A step by step transformation of the low frequency small height and slope perturbation solution into the high frequency reciprocal and dual, physical and geometrical optic solutions is presented. The familiar Kirchhoff approximations for the fields impressed by the incident plane wave upon the rough surfaces results in nonreciprocal solutions. It is shown that the surface element scattering coefficients based on the Kirchhoff approximations agree with the corresponding reciprocal physical optics solutions only at the stationary phase, specular points on the rough surfaces. While the Kirchhoff approximations and physical optics approximations are based on the characterization of the surface fields by Fresnel reflection coefficients, the corresponding surface element scatter coefficient derived for the small perturbation solution and the full wave solutions are based on the imposition of boundary conditions for the tangential components of the electric and magnetic fields. A flow graph schematically depicting the relationships between these solutions for the scattered fields is also presented.

Estimating basal properties of ice streams from surface measurements: a non-linear Bayesian inverse approach applied to synthetic data

Abstract. We propose a new approach to indirectly estimate basal properties of ice streams, i.e. bedrock topography and basal slipperiness, from observations of surface topography and surface velocities. We demonstrate how a maximum a posteriori estimate of basal conditions can be determined using a Bayesian inference approach in a combination with an analytical linearisation of the forward model. Using synthetic data we show that for non-linear media and non-linear sliding law only a few forward-step model evaluations are needed for convergence. The forward step is solved with a numerical finite-element model using the full Stokes equations. The Fréchet derivative of the forward function is approximated through analytical small-perturbation solutions. This approximation is a key feature of the method and the effects of this approximation on model performance are analyzed. The number of iterations needed for convergence increases with the amplitude of the basal perturbations, but generally less than ten iterations are needed.

Multidimensional Hele‐Shaw flows modelling Stokesian fluids

AbstractWe consider here an n‐dimensional periodic flow describing the motion of an incompressible Stokesian fluid in a Hele‐Shaw cell. The free surface separating the fluid from air, at pressure normalized to be zero, is moving under the influence of gravity and surface tension.We prove the existence of a unique classical Hölder solution for small perturbations of cylinders. Moreover, we evidence the existence of a single steady state and prove its exponential stability. Copyright © 2008 John Wiley & Sons, Ltd.

Small Perturbation Solutions for Elliptic Equations

We consider viscosity solutions of fully nonlinear elliptic equations for which u ≡ 0 is a solution. We assume F is smooth and uniformly elliptic only in a neighborhood of the points (0, 0, 0, x), and show that u is smooth in the interior if ‖u‖ L ∞ is sufficiently small.

Control of tonal noise from subsonic axial fan. Part 1: reconstruction of aeroacoustic sources from far-field sound pressure

An inverse method is investigated to evaluate the unsteady rotating forces (dipole strength distribution) acting by the fan on the fluid from far-field acoustic pressure measurements. A development based on the tonal noise generated by a propeller is used to derive a discretized form of the direct problem. The inversion of this direct problem is ill-posed and requires optimization technique to stabilize the solution for small perturbations in the measured acoustic input data. The reconstruction reveals that the conditioning of the inverse model depends on the aeroacoustic source and far-field sensor locations as well as on the frequency under investigation. Simulations show that an adequate choice of a regularization parameter leads to a satisfactory reconstruction of imposed unsteady rotating forces in the presence of measurement noise, and a correct localization of acoustic “hot spots” on the radiation surface. Preliminary experimental results also show the ability to extrapolate the radiated sound field at blade passage frequency (BPF), and harmonics, from the reconstructed forces. These data are exploited in the second part of this paper to evaluate various active control strategies for tonal fan noise.

Fourth- and higher-order small-perturbation solution for scattering from dielectric rough surfaces

A recursive solution of the small-perturbation method for rough surface scattering is presented. These results permit fourth- and higher-order corrections to rough surface scattering coefficients to be determined in a form that explicitly separates surface and electromagnetic properties. Sample results are presented for the fourth-order correction to the specular reflection coefficient of a rough surface and the sixth-order correction to incoherent scattering cross sections.

Reductive perturbation method of super KdV equations

By using reductive perturbation method, super KdV equations are changed into ordinary KdV equations, small amplitude perturbation solutions are obtained.

Radar backscatter from stationary breaking waves

Radar backscatter measurements from stationary breaking waves were used to examine how the surface roughness generated by wave breaking affects radar backscatter at moderate incidence angles. Stationary breaking waves were generated by submerging a stationary hydrofoil in a uniform flow. X band radar backscatter measurements were made at numerous streamwise positions along the stationary breaking waves from an incidence angle of 45° for horizontal transmit and receive polarization (HH) and vertical transmit and receive polarization (VV) looking both upwave and downwave. The radar returns increased substantially, and the HH‐to‐VV polarization ratio approached unity near the breaking crests. This radar signature is consistent with those observed in the field. Detailed optical measurements of the breaking surfaces revealed that the observed radar returns near the breaking crests were the result of increased incoherent backscatter from the small‐scale surface roughness generated by the breaking waves, although surface tilt effects also modified the radar return. Scattering models based upon the small perturbation solution performed well in the wake of the stationary breaking crest, but they significantly underestimated the HH‐to‐VV polarization ratio near the breaking crest. More advanced scattering solutions such as the integral equation method produced more accurate results in regions containing the largest surface roughness. These findings suggest that incoherent backscatter from surface disturbances produced by deep water breaking waves may be the source of the high radar returns and small polarization ratios observed from the ocean at moderate incidence angles.

Probleme de cauchy pour des equations de kirchhoff generalisees

We consider the global Cauchy problem for generalized Kirchhoff equations with small non-linear terms or small data. We solve this problem in the space of functions which are twice differentiable with respect to time coordinate and uniformly analytic with respect to other coordinates. We determine, in two different situations, estimates of lifespan of solutions for some problems with perturbations and we give stability result of the solution for small perturbations.

Influence of Fluid Film Nonlinearity on the Experimental Determination of Dynamic Stiffness and Damping Coefficients for Three-Lobe Journal Bearings©

Recently, several papers have been published reporting measured journal bearing characteristics including the linear dynamic force coefficients. In some instances, analyses were performed to give confidence levels for the data based on uncertainties in the measured data. However, the effect of fluid film nonlinearity on the measured data used to calculate the dynamic coefficients has not previously been reported. This paper presents results of a nonlinear simulation to assess these effects for one particular bearing test rig previously documented. The nonlinearity arises from the finite size orbital motion imposed on the journal or housing during the measurement process to obtain response data used to extract the dynamic coefficients from a linear model of the journal bearing dynamic system. The orbit size may be large enough to introduce nonlinearity in the fluid film forces which may in turn affect the resulting calculation of the linearized force coefficients. The nonlinear simulation includes a time-transient solution of the Reynolds equation for a three-lobe bearing and replicates the testing procedure used in an experimental journal bearing test rig previously reported upon. Comparisons of orbital motion and dynamic force coefficients to data obtained from that rig are reported herein. The fluid film nonlinearity was found to produce an uncertainty in the coefficients of up to 20 percent compared to the linearized coefficients obtained from a small perturbation solution of the Reynolds equation which are commonly used in rotating machinery design and analysis. A further objective of the simulation is to evaluate the effect of different external forcing functions to assist guiding the test procedure in the actual test rig to reduce the influence of fluid film nonlinearity on the test results. Presented as a Society of Tribologists and Lubrication Engineers paper at the ASME/STLE Tribology Conference in San Francisco, California, October 13–17, 1996

Non-Fickian Transport in Heterogeneous Saturated Porous Media

Existing theories of flow and contaminant transport in aquifers are either based on Monte Carlo simulations or small perturbation solutions of the governing stochastic partial differential equations, which limit the applications to cases of small variances in the physical parameters. In most cases the “smallness” is a subjective statement from the modeler or is forced by considering the logarithm of the random quantities. This article constitutes a preliminary attempt to reanalyze the problem of flow and contaminant transport in a hypothetical heterogeneous aquifer without the usual assumptions of small perturbation, logarithmic transformation, a specific probability law, and disregard for the underlying hydrologic problem. Statistical properties of the pore velocity are derived from the inherent ground water flow problem; a non-Fickian dispersion equation is derived by assuming two scales, a small and a large one; and a solution of the dispersion equation is obtained. A general analytic procedure, the decomposition method is used in the solution of the flow and dispersion equations. Finally, some comparisons with existing results are presented.

Effective impedance of an irregular interface of media: Case of deterministic perturbations

A rigorous equation is formulated and a small perturbation solution is found for the effective impedance operator of an irregular interface in an arbitrary linear stationary medium. Effective boundary conditions for an electromagnetic field are developed in the case of an arbitrarily shaped interface with slight deterministic roughness in a generally anisotropic inhomogeneous medium. Possible extensions of the results obtained are discussed.

Radar scatter cross sections for two-dimensional random rough surfaces – full wave solutions and comparisons with experiments

Abstract In this paper, the full wave expressions for the radar scattering cross sections for two-dimensional random rough surfaces are obtained. The rough-surface height/slope correlations are accounted for in this analysis. Analytical and numerical comparisons of the full wave solution with the small perturbation and physical optics solutions are made for isotropic, homogeneous random rough surfaces with Gaussian probability density function. The full wave results are also compared with experimental results.

Transmission scatter cross sections across two-dimensional random rough surfaces – full wave solutions and comparison with numerical results

Abstract In this paper, the full wave expressions for the bistatic transmission scattering cross sections across two-dimensional random rough surfaces are obtained. The full wave analysis accounts for the surface height/slope correlations. Analytical and numerical comparisons of the full wave solutions with the small perturbation and physical optics solutions are made for isotropic random rough surfaces. The full wave results are also compared with the numerical results based on Monte Carlo simulations of one-dimensional random rough surfaces. Detailed consideration is given to illustrating the relationship between these full wave solutions and the original full wave solutions including the impact of accounting for the height/slope correlations in this analysis.