Normal Mode Method Research Articles

Linear stability of one‐dimensional non‐Darcy flow in broken rocks

SUMMARYConsidering the effect of non‐Darcy flow, the perturbation theory and normal mode method are introduced to analyze the linear stability of one‐dimensional non‐Darcy flow of gases in broken rocks. A stability criterion for linear systems is obtained theoretically when the steady states of pressure and velocity fields are perturbed, and the effects of the physical parameters on the linear governing system are studied theoretically and numerically. It is pointed out that the deviation coefficient from Darcy's law plays an important role in the governing system; the increasing absolute value of deviation coefficient from Darcy's law stabilizes the system, and the numerical results are shown graphically. Copyright © 2015 John Wiley & Sons, Ltd.

Megneto-convection in a Layer of Maxwell Visco-Elastic Fluid in a Porous Medium with Soret Effect

Double Diffusive convection in a horizontal layer of Maxwell visco-elastic fluid in a porous medium in the presence of constant vertical magnetic field and Soret coefficient is investigated. Flow in porous medium is characterized by Darcy model. The normal mode method is used to find linear stability analysis for the fluid layer confined between two free-free boundary surfaces. The stability criterions for stationary and oscillatory convection have been derived. Effects of various parameters on the stationary have been obtained analytically and graphically.

Nonlinear Response of Carbon-Carbon Laminated Panels to Random Acoustic Excitation under a Gradient Temperature Field

Future missions for aircraft will expose structures to severe thermal and acoustic loads. Efficient analysis methods for predicting nonlinear random response and fatigue life are urgently important. This paper presents a finite element model for analyzing nonlinear random dynamic behaviors of Carbon-Carbon composite panels under temperature gradients and Guassian excitations. The temperature distribution over the plate follows double sine curve. A finite element formulation combined with the equivalent linearization approach and normal mode method is established. The global system of equations is reduced to a set of nonlinear, coupled modal equations. Examples are given for an orthotropic C/C composite laminated panel at various combinations of temperatures gradients and sound pressure levels. Numerical results include RMS values of maximum deflection, time histories of deflection response and stress response, power spectrum densities, probability distribution functions and higher statistical moments. Numerical results verified all three types of panel motions for a simply supported orthotropic laminated plate: small-deflection random vibration about the initial equilibrium positions, snap-through motions between the two buckled positions, and nonlinear random response about new equilibrium positions after post-buckling. Numerical results will provide the important reference basis to aero engine structural integrity design and improving the structure dynamic strength and working life.

Rayleigh Taylor instability of two superposed compressible fluids in un-magnetized plasma

The linear Rayleigh Taylor instability of two superposed compressible Newtonian fluids is discussed with the effect of surface tension which can play important roles in space plasma. As in both the superposed Newtonian fluids, the system is stable for potentially stable case and unstable for potentially unstable case in the present problem also. The equations of the problem are solved by normal mode method and a dispersion relation is obtained for such a system. The behaviour of growth rate is examined in the presence of surface tension and it is found that the surface tension has stabilizing influence on the Rayleigh Taylor instability of two superposed compressible fluids. Numerical analysis is performed to show the effect of sound velocity and surface tension on the growth rate of Rayleigh Taylor instability. It is found that both parameters have stabilizing influence on the growth rate of Rayleigh Taylor instability.

Effect of Coriolis force on thermomagnetic convection in a ferrofluid saturating porous medium: A weakly nonlinear stability analysis

We investigate the influence of Coriolis force on the onset of thermomagnetic convection in ferrofluid saturating a porous layer in the presence of a uniform vertical magnetic field using both linear and weakly non-linear analyses. The modified Brinkman–Forchheimer-extended Darcy equation with Coriolis term has been used to describe the fluid flow. The linear theory based on normal mode method is considered to find the criteria for the onset of stationary thermomagnetic Convection and weakly non-linear analysis based on minimal representation of truncated Fourier series analysis containing only two terms has been used to find the Nusselt number Nu as functions of time. The range of thermal Rayleigh number R beyond which the bifurcation becomes subcritical increases with increasing Λ, Da−1 and Ta. The global quantity of the heat transfer rate decreases by increasing the Taylor number Ta. The results obtained, during the above analyses, have been presented graphically and the effects of various parameters on heat and mass transfer have been discussed. Finally, we have drawn the steady streamlines for various parameters.

A low-order reduced model for the long range propagation of infrasounds in the atmosphere.

This paper considers a class of low-order, range-dependent propagation models obtained from the normal mode decomposition of infrasounds in complex atmospheres. The classical normal mode method requires calculating eigenvalues for large matrices making the computation expensive even though some modes have little influence on the numerically obtained results. By decomposing atmospheric perturbations into a wavelet basis, it is shown that the most sensitive eigenvalues provide the best reduced model for infrasound propagation. These eigenvalues lie on specific curves in the complex plane that can be directly deduced from atmospheric data through a WKB approach. The computation cost can be reduced by computing the invariant subspace associated with the most sensitive eigenvalues. The reduction method is illustrated in the case of the Fukushima explosion (12 March 2011). The implicitly restarted Arnoldi algorithm is used to compute the three most sensitive modes, and the correct tropospheric arrival is found with a cost of 2% of the total run time. The cost can be further reduced by using a stationary phase technique. Finally, it is shown that adding uncertainties triggers a stratospheric arrival even though the classical criteria, based on the ratio of stratospheric sound speed to that at ground level, is not satisfied.

Initial value problem approach of two regions of system solar: the continuous spectrum

The standard mathematical procedures in Helioseismology field are based on normal mode approach for various models of solar interior and atmosphere. We consider a two region model of a system solar interior and solar atmosphere. For simplicity, the two different regions are assumed quasi-isothermal, semi-infinite, without magnetic fields and separated by a boundary z=0. It was found useful to phrase of stability as initial value problem (IPV) in order to ensure the inclusion of certain continuum modes otherwise neglected. In addition to discrete mode (f-mode), sets of continuum modes due to a branch cuts in the complex plane, not treated explicitly in the literature, appears. It will be seen that an ambiguity of the usual normal mode method is avoided.

Co-seismic change of length of day based on the point dislocation theory for a SNREI Earth

We developed a new approach for computing the changes in the length of day (LOD) due to earthquakes from both shear- and tensile-type faults. This approach was based on the point dislocation theory for a SNREI Earth, which was validated by comparing our results with the ones obtained by the normal mode method. We defined two numerical functions that vary with hypocenter depth. These functions allow us to compute the co-seismic change of the trace of the Earth's inertia tensor, and hence the co-seismic LOD change. For future applications, by adding a trace term, we corrected Lambeck's formulas (Lambeck, 1980), which are being commonly used for computing co-seismic LOD change. Finally, the new approach was used to compute the co-seismic LOD change from 1977 to 2011. The results show that the co-seismic trace and the co-seismic J2 changes contribute to the LOD change with the same magnitude (Gross and Chao, 2006). This means that, unlike other deformation mechanisms, the trace term cannot be neglected when modeling the co-seismic LOD change. The earthquakes from 1977 to 2011 decreased the LOD by about 12μs, and the rate of decrease was enhanced after the 2004 Sumatra earthquake due to several large earthquakes.

Effect of rotation on micropolar generalized thermoelasticity with two temperatures using a dual-phase lag model

In the present paper, we introduce the dual-phase lag theory to study the effect of the rotation on a two-dimensional problem of micropolar thermoelastic isotropic medium with two temperatures. A normal mode method is proposed to analyze the problem and obtain numerical solutions for the displacement, the conductive temperature, the thermodynamic temperature, the microrotation, and the stresses. The results of the physical quantities have been obtained numerically and illustrated graphically. The results show the effect of phase lag of the heat flux τq, a phase lag of temperature gradient τθ and two-temperature parameter on all the physical quantities.

Waves propagating along a channel with ice cover

Linear progressive waves in a channel covered with ice sheet are studied. The channel is of rectangular cross section. The ice sheet is clamped to the walls of the channel. The thickness of the ice plate is constant. Deflections of the ice sheet are described by the linear elastic plate equation. The hydroelastic waves in the channel are combinations of waves propagating along the channel and sloshing waves. The problem is formulated with respect to the wave profiles across the channel. The problem is solved by the normal mode method for a channel of finite depth and by using the shallow water approximation for a channel of small depth. The dispersion relations of the hydroelastic waves and the characteristics of these waves are determined. It is shown that the shallow water approximation predicts well the dispersion relations for long waves. The dispersion relation for the wave, which does not oscillate across the channel, is well approximated by the corresponding dispersion relation of one-dimensional hydroelastic waves in an unbounded ice sheet. The wave profiles across the channel and the distributions of strains in the ice sheet are investigated. It is shown that the strains are maximum at the walls for long waves and at the centreline of the channel for short waves. The bending stresses across the channel are higher than the stresses along the channel for the conditions of the present study.

Buoyancy-Opposed Darcy’s Flow in a Vertical Circular Duct with Uniform Wall Heat Flux: A Stability Analysis

An analytical and numerical study is presented to show that buoyancy-opposed mixed convection in a vertical porous duct with circular cross-section is unstable. The duct wall is assumed to be impermeable and subject to a uniform heat flux. A stationary and parallel Darcy’s flow with a non-uniform radial velocity profile is taken as a basic state. Stability to small-amplitude perturbations is investigated by adopting the method of normal modes. It is proved that buoyancy-opposed mixed convection is linearly unstable, for every value of the Darcy–Rayleigh number, associated with the wall heat flux, and for every mass flow rate parametrised by the Peclet number. Axially invariant perturbation modes and general three-dimensional modes are investigated. The stability analysis of the former modes is carried out analytically, while general three-dimensional modes are studied numerically. An asymptotic analytical solution is found, suitable for three-dimensional modes with sufficiently small wave number and/or Peclet number. The general conclusion is that the onset of instability selects the axially invariant modes. Among them, the radially invariant and azimuthally invariant mode turns out to be the most unstable for all possible buoyancy-opposed flows.

On the Origin and Impact of a Polygonal Eyewall in the Rapid Intensification of Hurricane Wilma (2005)

Abstract An analysis of a high-resolution dataset from a realistic simulation of Hurricane Wilma (2005) was performed to understand the mechanism for the formation of a prominent polygonal eyewall and mesovortices during the rapid intensifying stage of the hurricane. The impact of these asymmetries on the intensity change of the hurricane vortex was assessed using the empirical normal mode (ENM) method and Eliassen–Palm (EP) flux calculations. The results indicated that the eyewall of Wilma exhibited an early azimuthal wavenumber-4 (m = 4) asymmetry followed by a transition to lower-wavenumber asymmetries. The simulated reflectivity and the spatial structure of potential vorticity (PV) anomalies strongly suggest that barotropic instability is the most likely driving mechanism for these asymmetries. From the ENM analysis, it was found that the dominant modes for m = 4 and m = 3 asymmetries are vortex Rossby waves (VRWs) that possess characteristics of unstable modes, supporting the importance of barotropic instability. The EP flux calculations associated with these modes indicate that the VRWs act to decelerate the flow at the initial radius of maximum wind while they act to accelerate the flow radially inside and outside of this location, suggesting that VRWs may provide a positive impact to the intensification of the vortex. The present results complement previous findings of theoretical and highly idealized numerical studies that a polygonal eyewall and mesovortices are the result of barotropic instability, thereby furnishing a bridge between idealized studies and observations. The work also provides new insight on the role of asymmetries and VRWs in the intensification of hurricanes.

DOMINANT CONTRIBUTION OF DIPOLES IN THE TURBULENCE-GENERATED NOISE

A problem of noise generation by a compact region of turbulence in an infinite straight immovable rigid-walled pipe of circular cross-section is solved by the Green’s function technique and the normal mode method. A turbulence region is modeled by quadrupoles and dipoles. The cases of homogeneous and non-homogeneous turbulence are considered. The generated noise power is shown to be a sum of powers of the pipe acoustic modes. The acoustic mode power consists of the three parts. The first part is the power generated by the quadrupoles, the second one results from the dipoles, and the third one is due to interaction of the quadrupoles and dipoles. Also, a particular case of the problem is considered. In this case the situation is studied when the generated noise field is dominated by the contribution made by surface dipoles. Those flows and shapes of the local pipe narrowing are of concern, that result in large or small eddies distributed uniformly in the turbulence region behind the narrowing. The corresponding simplified expressions for the generated noise power are obtained in the considered cases. Also, estimates of these expressions are carried out for the characteristic scales in the turbulence region. ых диполей. При этом интерес представляют такие потоки и формы локальных сужений труб, при которых регион турбулизированного сужением течения занимают равномерно распределенные большие или малые вихри. Для этих случаев получены соответствующие упрощенные выражения для мощности сгенерированного шума и проведены их оценки для характерных масштабов в области турбулентности.

Onset of triply diffusive convection in a Maxwell fluid saturated porous layer

The linear stability of triply diffusive convection in a binary Maxwell fluid saturated porous layer is investigated. Applying the normal mode method theory, the criterion for the onset of stationary and oscillatory convection is obtained. The modified Darcy–Maxwell model is used as the analysis model, this allows us to make a thorough investigation of the processes of viscoelasticity and diffusions that causes the convection to set in through oscillatory rather than stationary. The effects of Vadasz number, normalized porosity parameter, relaxation parameter, Lewis number and solute Rayleigh number on the system are presented numerically and graphically.

Thermal instability in a rotating porous layer saturated by a non-Newtonian nanofluid with thermal conductivity and viscosity variation

The stability of a non-Newtonian nanofluid saturated horizontal rotating porous layer subjected to thermal conductivity and viscosity variation is investigated using linear and nonlinear stability analyses. The model used for the non-Newtonian nanofluid includes the effects of Brownian motion and thermophoresis. The Darcy law for the non-Newtonian nanofluid of the Oldroyd type is used to model the momentum equation. The linear theory based on the normal mode method, and the criteria for both stationary and oscillatory modes are derived analytically. A weak nonlinear analysis based on the minimal representation of truncated Fourier series method containing only two terms is used to compute the concentration and thermal Nusselt numbers. The results obtained during the analysis are presented graphically.

Combined influence of throughflow and Soret effect on the onset of Marangoni convection

The onset of Marangoni convection with throughflow and the Soret effect in a top-free and bottom-rigid horizontal fluid layer is studied using the normal mode method for different types of thermal and solutal boundary combinations. The bottom surface is either conducting or insulating to temperature and solute concentration perturbations. The resulting eigenvalue problem is solved exactly by assuming that stationary convection is exhibited at the neutral state. It is found that the destabilizing behavior of a small amount of throughflow described by Nield (J Fluid Mech 185:353–360, 1987) becomes more significant in the presence of Soret effect for some boundary combinations. The results are consistent with the existing results in the literature.

From transport to disorder: Thermodynamic properties of finite dust clouds

The quantities entropy and diffusion are measured for two- and three-dimensional (3D) dust clusters in the fluid state. Entropy and diffusion are predicted to be closely linked via unstable modes. The method of instantaneous normal modes is applied for various laser-heated clusters to determine these unstable modes and the corresponding diffusive properties. The configurational entropy is measured for 2D and 3D clusters from structural rearrangements. The entropy shows a threshold behavior at a critical temperature for the 2D clusters, allowing us to estimate a configurational melting temperature. Further, the entropic disorder increases for larger clusters. Finally, the predicted relation between entropy and unstable modes has been confirmed from our experiments for 2D systems, whereas 3D systems do not show such a clear correlation.

Stability Analysis of a Thin Pseudoplastic Fluid with Condensation Effects Flowing on a Rotating Circular Disk

This paper investigates the linear stability of a thin axisymmetric pseudoplastic fluid with condensation effects flowing on a rotating circular disk. Long-wave perturbation analysis is proposed to derive a generalized kinematic model of the physical system with a small Reynolds number. The method of normal mode is applied to study the linear stability. The neutral stability curve and the linear growth rate are obtained subsequently as the by-products of linear solution. The study reveals that the rotation number generates a destabilizing effect in pseudoplastic fluid. The degree of the flow index n plays a vital role in stabilizing the film flow.

Bayesian localization of an unknown number of ocean acoustic sources

This paper considers localizing an unknown number of ocean acoustic sources when properties of the environment are poorly known. A Bayesian formulation is developed in which environmental parameters, noise statistics, and the number, locations, and complex spectra (amplitudes and phases) of multiple sources are considered unknown random variables constrained by acoustic data and prior information. The number of sources is determined during a burn-in stage by minimizing the Bayesian information criterion using hybrid optimization with an efficient source birth/death scheme. Optimal estimates and marginal posterior probability distributions for source locations are computed employing a variety of sampling approaches. Environmental properties and source locations and are treated as explicit parameters and marginalized using Markov-chain Monte Carlo sampling methods. In particular, environmental parameters are treated using Metropolis-Hastings sampling applied efficiently in a principal-component space, and source locations are treated using Gibbs sampling since the corresponding conditional probability distributions can be computed efficiently using normal-mode methods. Source and noise spectra are sampled implicitly by applying analytic maximum-likelihood solutions expressed in terms of the explicit parameters. This represents an empirical Bayesian approximation within a hierarchical formulation, and significantly reduces the dimensionality and improves sampling efficiency in the inversion.

ANALYTICAL TIME DOMAIN NORMAL MODE SOLUTION OF AN ACOUSTIC WAVEGUIDE WITH PERFECTLY REFLECTING WALLS

Solution of wide-band underwater acoustic problems with the classical Normal Mode method in the frequency domain needs to solve the problem repeating for every frequency component of the wide-band source signal. In this paper, a direct and causal analytical Time Domain Normal Mode Method is presented for arbitrary time-dependent acoustic sources for a single layered isovelocity waveguide. An incomplete separation of variables technique is used to solve the inhomogeneous wave equation, directly in the time domain. Therefore, it becomes possible to calculate the time domain acoustic pressure in a single run.