Normal Mode Method Research Articles

Finite amplitude long-wave instability of a thin viscoelastic fluid during spin coating

This paper investigates the stability of a thin incompressible viscoelastic fluid designated as Walters’ liquid B″ during spin coating. The long-wave perturbation method is proposed to derive a generalized kinematic model of the film flow. The method of normal mode is applied to study the linear stability. The amplitude growth rates and the threshold conditions are characterized subsequently and summarized as the by-products of the linear solutions. Using the multiple scales method, the weakly nonlinear stability analysis is studied for the evolution equation of a film flow. The Ginzburg–Landau equation is determined to discuss the threshold conditions of the various critical flow states. The study reveals that the rotation number and the radius of the rotating circular disk generate the destabilizing effects. Moreover, the viscoelastic parameter k indeed plays a more significant role in destabilizing the film flow than a thin Newtonian fluid during spin coating [27].

Magneto-convection in an anisotropic porous layer with Soret effect

Thermal instability in an electrically conducting two component Boussinesq fluid-saturated-porous medium has been investigated, in the presence of Soret coefficient. The porous medium is confined between two horizontal surfaces, and subjected to a constant vertical magnetic field. Flow in the porous medium is characterized by generalized Darcy model, which includes the time derivative term. Performing linear and non-linear stability analysis, the effect of magnetic field on the stability of flow through porous medium has been investigated. The normal mode method is used in linear stability analysis, while a weak non-linear analysis based on a minimal representation of double Fourier series method is used in non-linear analysis. The critical Rayleigh number, wave number for stationary and oscillatory modes, and frequency of oscillations are obtained analytically using linear theory. Effects of various parameters on stationary, oscillatory and finite amplitude convection, rate of heat and mass transfer have been obtained analytically and presented graphically.

Terahertz spectrum and normal-mode relaxation in pentaerythritol tetranitrate: Effect of changes in bond-stretching force-field terms

Terahertz (THz) active normal-mode relaxation in crystalline pentaerythritol tetranitrate (PETN) was studied using classical molecular dynamics simulations for energy and density conditions corresponding to room temperature and atmospheric pressure. Two modifications to the fully flexible non-reactive force field due to Borodin et al. [J. Phys. Chem. B 112, 734 (2008)] used in a previous study of THz-active normal-mode relaxation in PETN [J. Chem. Phys. 134, 014513 (2011)] were considered to assess the sensitivity of the earlier predictions to details of the covalent bond-stretching terms in the force field. In the first modification the harmonic bond-stretching potential was replaced with the Morse potential to study the effect of bond anharmonicity on the THz-region mode relaxation. In the second modification the C-H and nitro-group N-O bond lengths were constrained to constant values to mimic lower quantum occupation numbers for those high-frequency modes. The results for relaxation times of the initially excited modes were found to be insensitive to either force-field modification. Overall time scales for energy transfer to other modes in the system were essentially unaffected by the force-field modifications, whereas the detailed pathways by which the energy transfer occurs are more complicated for the Morse potential than for the harmonic-bond and fixed-bond cases. Terahertz infrared absorption spectra constructed using calculated normal-mode frequencies, transition dipoles, and relaxation times for THz-active modes were compared to the spectra obtained from the Fourier transform of the dipole-dipole time autocorrelation function (DDACF). Results from the two approaches are in near agreement with each other and with experimental results in terms of main peak positions. Both theoretical methods yield narrower peaks than observed experimentally and in addition predict a weaker peak at ω ∼ 50 cm(-1) that is weak or absent experimentally. Peaks obtained using the DDACF approach are broader than those obtained from the normal-mode relaxation method.

Toward benchmarks of low frequency reverberation level in a Pekeris waveguide: Insight from analytical solutions.

The requirement by modern navies to predict sonar performance in shallow water, whether for use in research, planning, or operations, led to an initiative for the validation of reverberation models in the form of two reverberation modeling Workshops at the University of Texas at Austin [J. S. Perkins and E. I. Thorsos, J. Acoust. Soc. Am. 126, 2208 (2009)]. The scenario considered here (Problem XI, from the first workshop) requires the computation of reverberation versus time in a Pekeris waveguide with Lambert scattering from the seabed. Simple analytical methods are presented that provide insight into the dominant propagation paths giving rise to the reverberation and hence establish regimes of validity for various computer models making different approximations or assumptions. Results from eigenray, normal mode, and hybrid continuum methods are compared with each other and with the analytical solutions. Numerical predictions are shown to overlap to within a few tenths of a decibel in regions where the different assumptions made by the various models are valid. These overlapping solutions are proposed as “benchmarks” in the sense of a baseline against which future model improvements can be assessed and quantified.

Stability Analysis of A Thin Micropolar Fluid Flowing on A Rotating Circular Disk

ABSTRACTThe stability analysis of a thin micropolar fluid flowing on a rotating circular disk is investigated numerically. The target is restricted to some neighborhood of critical value in the linear stability analysis. First, a generalized nonlinear kinematic model is derived by the long wave perturbation method. The method of normal mode is applied to the linear stability. After the weakly nonlinear dynamics of a film flow is studied by using the method of multiple scales, the Ginzburg-Landau equation is determined to discuss the necessary condition in terms of the various states of subcritical stability, subcritical instability, supercritical stability, and supercritical explosion for the existence of such flow pattern. The modeling results indicate that the rotation number and the radius of circular disk play the significant roles in destabilizing the flow. Furthermore, the micropolar parameter K serves as the stabilizing factor in the thin film flow.

FDTD 방법과 분수 함수 근사법을 이용한 다층 구조에서의 Green 함수 근사화

In this paper, a method to approximate a multi-layer Green`s function is proposed based on a FDTD scheme and a rational function approximation. For a given horizontal propagation wavenumber, time domain response is calculated and then Fourier transformed to the spectral domain Green`s function. Using the rational function approximation, the pole and residue of the Green`s function can be estimated, which are crucial for a calculation of a path loss. The proposed method can provide a wideband Green`s function, while the conventional normal mode method can be applied to a single frequency problem. To validate the proposed method, We consider two problems, one of which has a analytical solution. The other is about multi-layer case, for which the proposed method is compared with the known normal mode solution, Kraken.

Transient Response Analysis of a High Speed Rotor Using Constraint Real Mode Synthesis

In order to accurately calculate high-speed nonlinear transient phenomena of a large order rotor system such as rotor collision and rubbing with stator repeatedly due to blade loss, a very small incremental time step is required in numerical integration. But a small time step is known to become a cause of divergence or flutter type numerical instability when a direct integration such as Newmark-β method is applied in terms of physical coordinates. Normal mode method, which uses eigen modes instead of physical freedoms, is effective on this numerical stability, but the modes in the calculation are usually complex-modes and also need recalculation in accordance with varying rotational speed. Therefore, this method is inconvenient for an analysis of a rotor, especially speed varying system. This paper presents a new method using constraint real mode synthesis in which rotor modes are derived by neglecting gyroscopic moment as at 0 rpm and by fixing bearing connection interface freedoms to zero. This method is proved to be highly stable and also to have good accuracy and efficiency in the calculation.

A Normal Mode Solution for Low-Frequency Bottom-Refracted Wave Propagation

A normal mode solution for low-frequency bottom-sediment-refracted wave propagation is formulated for use with a normal mode method of solving the Helmholtz wave equation to describe the underwater sound field for a fixed point source in a plane multilayered medium. The approach utilizes a factorization of the Wronskian of Green's function which enables sediment reflecting and refracting modes to be summed independently. The sensitivity of propagation loss to the nature of mode interaction with the bottom is investigated for multiple frequencies in a sample environment, and results are compared to measurements of 100-Hz bottom returns in the Caribbean Basin.

Large‐scale comparison of protein essential dynamics from molecular dynamics simulations and coarse‐grained normal mode analyses

A large-scale comparison of essential dynamics (ED) modes from molecular dynamic simulations and normal modes from coarse-grained normal mode methods (CGNM) was performed on a dataset of 335 proteins. As CGNM methods, the elastic network model (ENM) and the rigid cluster normal mode analysis (RCNMA) were used. Low-frequency normal modes from ENM correlate very well with ED modes in terms of directions of motions and relative amplitudes of motions. Notably, a similar performance was found if normal modes from RCNMA were used, despite a higher level of coarse graining. On average, the space spanned by the first quarter of ENM modes describes 84% of the space spanned by the five ED modes. Furthermore, no prominent differences for ED and CGNM modes among different protein structure classes (CATH classification) were found. This demonstrates the general potential of CGNM approaches for describing intrinsic motions of proteins with little computational cost. For selected cases, CGNM modes were found to be more robust among proteins that have the same topology or are of the same homologous superfamily than ED modes. In view of recent evidence regarding evolutionary conservation of vibrational dynamics, this suggests that ED modes, in some cases, might not be representative of the underlying dynamics that are characteristic of a whole family, probably due to insufficient sampling of some of the family members by MD.

Non-Linear Two Dimensional Double Diffusive Convection in a Rotating Porous Layer Saturated by a Viscoelastic Fluid

Double diffusive convection in a rotating anisotropic porous layer, saturated by a viscoelastic fluid, heated from below and cooled from above has been studied making linear and non-linear stability analyses. The fluid and solid phases are considered to be in equilibrium. In momentum equation, we have employed the Darcy equation which includes both time derivative and Coriolis terms. The linear theory based on normal mode method is considered to find the criteria for the onset of stationary and oscillatory convection. A weak non-linear analysis based on minimal representation of truncated Fourier series analysis containing only two terms has been used to find the Nusselt number and Sherwood number as functions of time. We have solved the finite amplitude equations using a numerical scheme. 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 and unsteady streamlines, isotherms, and isohalines for various parameters.

Vibrations of a complex-shaped panel

The free vibrations of flexible shallow shells with complex planform are studied. To analyze the natural frequencies and modes of linear vibrations, the R-function and Rayleigh–Ritz methods are used. A discrete model is obtained using the Bubnov–Galerkin method. The nonlinear vibrations are studied by combining the nonlinear normal mode method and the multiple-scales method. Skeleton curves of natural vibrations are drawn

Theoretical analysis of transient waves in a simply-supported Timoshenko beam by ray and normal mode methods

The dynamic transient responses of a simply-supported Timoshenko beam subjected to an impact force are investigated by two theoretical approaches – ray and normal mode methods. The mathematical methodology proposed in this study for the ray method enable us to construct the solution for the interior source problem and to extend to solve the complicated problem for the multi span of the Timoshenko beam. Numerical results based on these two approaches are compared. The comparison in this study indicates that the normal mode method is more computationally efficient than the ray method except for very short time after the impact. The long-time transient responses are easily calculated using the normal mode method. It is shown that the average long-time transient response converges to the corresponding static value. The Timoshenko beam theory is more accurate than the Bernoulli–Euler beam theory because it includes shear and rotary inertia. This study also provides the slender ratio for which the Bernoulli–Euler beam can be used for the transient-response analysis of the displacement. Moreover, the resonant frequencies obtained from finite element calculation based on the three-dimensional model are compared with the results calculated using the Timoshenko beam and Bernoulli–Euler beam theories. It is noted in this study that the resonant frequency can be accurately determined by the Timoshenko beam theory if the slender ratio is larger than 100, and by the Bernoulli–Euler beam theory if the slender ratio is larger than 400.

Propagation models in the ocean acoustic field with hard soil.

This paper addresses the human interest for lossless data transmission by water sound waves, knowing that electromagnetic waves dramatically attenuate in sea water. A normal mode method is employed in order to determine varying frequency propagating modes and waveguide height to observe individual mode contribution. It considers a shallow water waveguide between the hard soil (sand) and the sea surface. Transmitter and receiver will be several kilometers away from each other, and the future intended applications include object detection and submarine communication.

Acoustic propagation in a river channel environment.

Acoustic propagation modeling in river environments would take advantage of available environmental data and be relevant to recent work regarding passive threat detection. Normal mode and ray methods will be applied to 3‐D geometric channels in order to investigate propagation characteristics such as convergence zones, cutoff frequencies, and transmission losses. Calculations for these types of environments must also account for 3‐D propagation effects including vertical and horizontal reflections, shallow bottoms, and sloping boundaries. Preliminary ray path results can be used to compare propagation behavior in ideal and fluid bottom linear waveguides with potential for incorporation of monitoring data and possible application to meandering river channels.

Thermal instability in a rotating anisotropic porous layer saturated by a viscoelastic fluid

Linear and non-linear thermal instability in a rotating anisotropic porous medium, saturated with viscoelastic fluid, has been investigated for free–free surfaces. The linear theory is being related to the normal mode method and non-linear analysis is based on minimal representation of the truncated Fourier series analysis containing only two terms. The extended Darcy model, which includes the time derivative and Coriolis terms has been employed in the momentum equation. The criteria for both stationary and oscillatory convection is derived analytically. The rotation inhibits the onset of convection in both stationary and oscillatory modes. A weak non-linear theory based on the truncated representation of Fourier series method is used to find the thermal Nusselt number. The transient behaviour of the Nusselt number is also investigated by solving the finite amplitude equations using a numerical method. The results obtained during the analysis have been presented graphically.

Weakly Nonlinear Stability Analysis of a Thin Magnetic Fluid during Spin Coating

This paper investigates the stability of a thin electrically conductive fluid under an applied uniform magnetic filed during spin coating. A generalized nonlinear kinematic model is derived by the long-wave perturbation method to represent the physical system. After linearizing the nonlinear evolution equation, the method of normal mode is applied to study the linear stability. Weakly nonlinear dynamics of film flow is studied by the multiple scales method. The Ginzburg-Landau equation is determined to discuss the necessary conditions of the various critical flow states, namely, subcritical stability, subcritical instability, supercritical stability, and supercritical explosion. The study reveals that the rotation number and the radius of the rotating circular disk generate similar destabilizing effects but the Hartmann number gives a stabilizing effect. Moreover, the optimum conditions can be found to alter stability of the film flow by controlling the applied magnetic field.

Surface manifestations of internal waves investigated by a subsurface buoyant Jet: 1. The mechanism of internal-wave generation

In a large test reservoir with artificial temperature stratification at the Institute of Applied Physics, Russian Academy of Sciences, we have performed a major laboratory simulation of the nonstationary dynamics of buoyant turbulent jets generated by wastewater flows from underwater collector diffusers. The interaction of buoyant jets with the pycnocline leads to an active generation of internal waves. An analysis of the dependence of wave amplitude on the control parameter proportional to the rate of liquid flow from the collector diffuser has indicated that this dependence is adequately described by a function that is characteristic for the presence in the Hopf bifurcation system, which occurs for a soft actuation mode of self-oscillations of the globally instable mode. To check the conditions for the actuation of the globally instable mode, we have performed an auxiliary experiment in a small reservoir with a salt stratification formulated similar to the experiment in the big reservoir. Using the particle image velocimetry (PIV) method, we have measured the velocity field in the buoyant jet and constructed the profiles of transverse velocity in several sections. When the jet approaches the pycnocline, a counterflow is generated at the edges. A stability analysis for the resulting profiles of flow velocities performed by the method of normal modes has revealed that, for the jet portions with counterflow, the condition of absolute instability by the Briggs criterion for axisymmetric jet oscillations is satisfied, which testifies to the fact that the globally instable mode is actuated. The estimates for oscillation frequencies of the globally instable mode are well consistent quantitatively with the measured spectrum of jet oscillations.

Hydroelastic behavior of a complex structure floating on waves

A numerical method is developed to solve the plane problem of the hydroelastic behavior of a complex structure floating on the surface of an ideal incompressible fluid of finite depth. The motion of the structure described by a deflection function is considered steady-state under the action of incident waves. The hydrodynamic part of the problem is solved using the proposed approach based on the normal-mode method for homogeneous plates. The problem is reduced to a system of linear algebraic equations by means of a transition matrix between representations of the required deflection in the form of expansion in the vibration eigenfunctions of the structure and the plate. It is shown that the results of the calculation performed are in good agreement with available calculation results for a two-part hinged structure at wavelengths comparable to the length of the structure.

Ray versus mode differences in reverberation modeling solutions for environments with high boundary scattering loss.

Several of the problems for the first Reverberation Modeling Workshop yielded interesting differences between solutions obtained with ray and normal mode methods. These particular problems were defined with high boundary scattering loss. A bottom reverberation case at 3.5 kHz with a down-refracting sound speed profile (Problem VI) will be considered as a case in point. The ray solutions show a “direct path” contribution unaffected by the bottom scattering loss as long as a direct path can reach the bottom, while the mode solutions obtained to date show a lower reverberation level during this period due to modal attenuation. These differences occur in both incoherent and coherent reverberation solutions for both rays and modes. Arguments will be presented that indicate the correctness of the ray solutions for this case. Suggestions will also be made on how the mode approach can be used to obtain solutions in agreement with the ray method. [Work supported by ONR and the Navy Program Executive Office C4I (PMW 120).]

Enhancing passive automation performance using an acoustic propagation simulation.

During an at‐sea encounter, signatures of interest can exhibit characteristics that differ from those observed in previously recorded data. These differences can occur due to variations in a number of factors including encounter geometry, propagation channel, and receiving sensor configuration. This paper presents a simulation technique that imposes low‐frequency propagation effects on a time‐domain signal using a normal mode method. High‐quality, time‐varying recorded signatures are used as inputs into the algorithm, which outputs band‐limited time‐series data for a selected geometry and environment. The output time‐series are phased to simulate the time‐varying pressure amplitudes that would be received by a towed array or any multielement passive sensor configuration operating in a realistic multipath environment. These capabilities enable the simulation of signatures of interest as captured under a broad range of littoral conditions by various passive sensors. These simulated data are used to augment scarce signature collections for training and assessing the performance of passive sonar automation.