Normal Mode Method Research Articles

Theoretical model of linear instability for condensation flow in circular channels

The paper presents a theoretical model for liquid-vapor interface instabilities for R134a condensation flow in tubes. The model is based on linear Kelvin-Helmholtz instability and capillary instability theories that neglect the effect of gravity and take into account the surface tension and shear stress effects. The normal mode method was used to analyze the system instability reactions to various perturbation wave lengths and physical conditions. Three modes were obtained with one giving the greatest instability wave length for each condition. The cooling temperature had little effect on the greatest instability. The greatest instability wave length increased with decreasing R134a mass flux and quality, with jumps between two instability states. This instability state transition can be used to predict condensation flow regime transitions for intermittent flow. The dimensionless intermittent flow criterion is in good agreement with available experimental data in the literature.

Non-iterative Rauscher method for 1-DOF system: a new approach to studying non-autonomous system via equivalent autonomous one

Rauscher method becomes the matter of interest because in combination with the method of nonlinear normal vibration modes it allows to calculate steady forced vibrations in the system with multiple degrees of freedom (DOF) via reduction in the number of DOFs. However, modern realizations of that approach have drawbacks such as iterative nature and the need to have initial approximation for the solution. The primary principle of Rauscher method is in obtaining periodic solutions of a non-autonomous system via studying some equivalent autonomous one. In the paper, a new non-iterative variant of Rauscher method is considered. In its current statement, the method can be used in analysis of forced harmonic oscillations in a nonlinear system with one degree of freedom. The primary goals of the study were to find out what kind of equivalent autonomous systems could be built for a given non-autonomous one and how they can be used for the construction of periodic solutions and/or periodic phase plane orbits of the initial system. It is shown that three different types of equivalent autonomous dynamical systems can be built for a given 1-DOF non-autonomous one. The system of 1st type is a fourth-order dynamical system. Technically it can be considered as a 2-DOF system where additional “DOF” is explicitly “responsible” for forced oscillations. The system of 2nd type is a third-order dynamical system. Its periodic orbits are exactly the same as in the initial system. Using the invariant manifold of the system of 1st type, the system of 2nd type can be reduced to the form $$W(x,x')=0$$ (which is called here the equivalent system of the 3rd type). It is important that the function $$W(x,x')$$ can be built a priori. Once $$W(x,x')$$ is found: (i) one can obtain different periodical orbits corresponding to forced oscillations in the initial system; (ii) one can estimate amplitudes of vibrations for these regimes; (iii) one can track bifurcations of periodical regimes of the initial system with respect to change in amplitude of external excitation f. As shown in the paper, periodical orbits of the initial non-autonomous system can be obtained via two different approaches: (i) as set of points on phase plane satisfying the condition $$W(x,x')=0$$ ; (ii) via the application of harmonic balance method to the equivalent system of 1st type using system’s energy level as a continuation parameter. This approach has advantage over application of harmonic balance method to initial system because the latter requires good initial guess for expansion coefficients, while the new approach does not and always starts from zero initial guess.

Forced Response Analysis of Pipes Conveying Fluid by Nonlinear Normal Modes Method and Iterative Approach

The forced vibration of gyroscopic continua is investigated by taking the pipes conveying fluid as an example. The nonlinear normal modes and a numerical iterative approach are used to perform numerical response analysis. The nonlinear nonautonomous governing equations are transformed into a set of pseudo-autonomous ones by using the harmonic balance method. Based on the pseudo-autonomous system, the nonlinear normal modes are constructed by the invariant manifold method on the state space and substituted back into the original discrete equations. By repeating the above mentioned steps, the dynamic responses can be numerically obtained asymptotically using such iterative approach. Quadrature phase difference between the general coordinates is verified for the gyroscopic system and traveling waves instead of standing waves are found in the time-domain complex modal analysis.

Bayesian-Maximum Entropy method applied to seabed geoacoustic models using broadband and narrowband acoustic measurements

Conditional probability distributions P(H|D,M) are computed using a Bayesian-Maximum Entropy method applied to acoustic measurements collected during the Seabed Characterization Experiment in the spring of 2017. Marginal distributions are computed for both seabed geoacoustic and source parameter values. A prior range-dependent seabed model M is derived from CHIRP survey measurements made in 2015. The prior bounds of parameter values forming the hypothesis vector H are obtained from an extensive set of sediment core measurements made in the region. The acoustic data, D, were produced by small explosive and combustive sources and towed tonal sources. Two acoustic models are employed to sample the N-dimensional H space: Range-dependent parabolic equation RAM and a normal mode method that includes shear wave effects. The dimensionality of M is optimized via a Gaussian Mixture Model (GMM) and compared to the Bayesian Information Criterion (BIC) and the Akaike Information Criterion (AIC). [Work supported by Office of Naval Research.]

A new hydrodynamic boundary condition simulating the effect of rough boundaries on the onset of Rayleigh-Bénard convection

This paper introduces a new hydrodynamic boundary condition which enables the simulation of the effects caused by rough boundaries. The classical Rayleigh-Bénard stability analysis is performed here to investigate the onset of thermal convection in a parallel-plate channel with rough boundaries. The hydrodynamic boundary conditions are modified, from the classical treatment, in order to consider channel boundaries characterised by non-negligible roughness. This roughness is simulated as a shallow fluid saturated porous medium and the Saffman interface condition is thus employed to model the hydrodynamic boundary conditions. The normal mode method is employed and the obtained eigenvalue problem is solved numerically. The Principle of Exchange of Stabilities is proved and the critical values of the Rayleigh number and of the wave number are obtained.

Effects of a semi-infinite stratification on the Rayleigh-Taylor instability in an interface with surface tension

The Rayleigh-Taylor instability (RTI) in an infinite slab where a constant density lower fluid is initially separated from an upper stratified fluid is discussed in linear regime. The upper fluid is of increasing exponential density and surface tension is considered between both of them. It was found useful to study stability by using the initial value problem approach (IVP), so that we ensure the inclusion of certain continuum modes, otherwise neglected. This methodology includes the branch cut in the complex plane, consequently, in addition to discrete modes (surface RTI modes), a set of continuum modes (internal RTI modes) also appears. As a result, the usual information given by the normal mode method is now complete. Furthermore, a new role is found for surface tension: to transform surface RTI modes (discrete spectrum) into internal RTI modes belonging to a continuous spectrum at a critical wavenumber. As a consequence, the cut-off wavenumber disappears: i.e. the growth rate of the RTI surface mode does not decay to zero at the cut-off wavenumber, as previous researchers used to believe. Finally, we found that, due to the continuum, the asymptotic behavior of the perturbation with respect to time is slower than the exponential when only the continuous spectrum exists.

Nonlinear Normal Modes of Strongly Nonlinear Periodically Excited Piecewise Linear Systems

A method for the numerical analysis of nonlinear normal modes of forced vibrations in strongly nonlinear systems with piecewise linear elastic characteristics is proposed. The approach is based on the combination of the Shaw–Pierre method of nonlinear normal modes with the Rauscher technique. As a result of application of this approach, the nonautonomous piecewise linear system is transformed into an autonomous system. For this system, we determine the Shaw–Pierre nonlinear normal modes. We also study the nonlinear torsional vibrations of the power transmission in a three-cylinder transport engine.

Normal Mode Approach to the Stability Analysis of Rossby-Haurwitz Wave

The stability of the steady Rossby-Haurwitz wave (R-H wave) in the nondivergent barotropic model (NBM) on the sphere was investigated with the normal mode method. The linearized NBM equation with respect to the R-H wave was formulated into the eigenvalue-eigenvector problem consisting of the huge sparse matrix by expanding the variables with the spherical harmonic functions. It was shown that the definite threshold R-H wave amplitude for instability could be obtained by the normal mode method. It was revealed that some unstable modes were stationary, which tend to amplify without the time change of the spatial structure. The maximum growth rate of the most unstable mode turned out to be in almost linear proportion to the R-H wave amplitude. As a whole, the growth rate of the unstable mode was found to increase with the zonal- and total-wavenumber. The most unstable mode turned out to consist of more-than-one zonal wavenumber, and in some cases, the mode exhibited a discontinuity over the local domain of weak or vanishing flow. The normal mode method developed here could be readily extended to the basic state comprised of multiple zonalwavenumber components as far as the same total wavenumber is given.

A step function density profile model for the convective stability of CO $$_2$$ 2 geological sequestration

The convective stability associated with carbon sequestration is usually investigated by adopting an unsteady diffusive basic profile to account for the space and time development of the carbon-saturated boundary layer instability. The method of normal modes is not applicable due to the time dependence of the nonlinear base profile. Therefore, the instability is quantified either in terms of critical times at which the boundary layer instability sets in or in terms of long-time evolution of initial disturbances. This paper adopts an unstably stratified basic profile having a step function density with top heavy carbon-saturated layer (boundary layer) overlying a lighter carbon-free layer (ambient brine). The resulting configuration resembles that of the Rayleigh–Taylor problem with buoyancy diffusion at the interface separating the two layers. The discontinuous reference state satisfies the governing system of equations and boundary conditions and pertains to an unstably stratified motionless state. Our model accounts for anisotropy in both diffusion and permeability and chemical reaction between the carbon dioxide-rich brine and host mineralogy. We consider two cases for the boundary conditions, namely an impervious lower boundary with either a permeable (one-sided model) or poorly permeable upper boundary. These two cases possess neither steady nor unsteady unstably stratified equilibrium states. We proceed by supposing that the carbon dioxide that has accumulated below the top cap rock forms a layer of carbon-saturated brine of some thickness that overlies a carbon-free brine layer. The resulting stratification remains stable until the thickness, and by the same token, the density, of the carbon-saturated layer is sufficient to induce the fluid to overturn. The existence of a finite threshold value for the thickness is due to the stabilizing influence of buoyancy diffusion at the interface between the two layers. With this formulation for the reference state, the stability calculations will be in terms of critical boundary layer thickness instead of critical times, although the two formulations are homologous. This approach is tractable by the classical normal-mode analysis. Even though it yields only conservative threshold instability conditions, it offers the advantage for an analytically tractable study that puts forth expressions for the carbon concentration convective flux at the interface and explores the flow patterns through both linear and weakly nonlinear analyses.

NOLB: Nonlinear Rigid Block Normal-Mode Analysis Method

We present a new conceptually simple and computationally efficient method for nonlinear normal-mode analysis called NOLB. It relies on the rotations-translations of blocks (RTB) theoretical basis developed by Y.-H. Sanejouand and colleagues [ Durand et al. Biopolymers 1994 , 34 , 759 - 771 . Tama et al. Proteins: Struct., Funct., Bioinf . 2000 , 41 , 1 - 7 ]. We demonstrate how to physically interpret the eigenvalues computed in the RTB basis in terms of angular and linear velocities applied to the rigid blocks and how to construct a nonlinear extrapolation of motion out of these velocities. The key observation of our method is that the angular velocity of a rigid block can be interpreted as the result of an implicit force, such that the motion of the rigid block can be considered as a pure rotation about a certain center. We demonstrate the motions produced with the NOLB method on three different molecular systems and show that some of the lowest frequency normal modes correspond to the biologically relevant motions. For example, NOLB detects the spiral sliding motion of the TALE protein, which is capable of rapid diffusion along its target DNA. Overall, our method produces better structures compared to the standard approach, especially at large deformation amplitudes, as we demonstrate by visual inspection, energy, and topology analyses and also by the MolProbity service validation. Finally, our method is scalable and can be applied to very large molecular systems, such as ribosomes. Standalone executables of the NOLB normal-mode analysis method are available at https://team.inria.fr/nano-d/software/nolb-normal-modes/ . A graphical user interface created for the SAMSON software platform will be made available at https://www.samson-connect.net .

Interaction entropy for protein-protein binding.

Protein-protein interactions are at the heart of signal transduction and are central to the function of protein machine in biology. The highly specific protein-protein binding is quantitatively characterized by the binding free energy whose accurate calculation from the first principle is a grand challenge in computational biology. In this paper, we show how the interactionentropy approach, which was recently proposed for protein-ligand binding free energy calculation, can be applied to computing the entropic contribution to the protein-protein binding free energy. Explicit theoretical derivation of the interactionentropy approach for protein-protein interaction system is given in detail from the basic definition. Extensive computational studies for a dozen realistic protein-protein interaction systems are carried out using the present approach and comparisons of the results for these protein-protein systems with those from the standard normal mode method are presented. Analysis of the present method for application in protein-protein binding as well as the limitation of the method in numerical computation is discussed. Our study and analysis of the results provided useful information for extracting correct entropic contribution in protein-protein binding from molecular dynamics simulations.

Local thermal non-equilibrium effects in the Horton-Rogers-Lapwood problem with a free surface

The onset of thermoconvective instability in a modified Horton-Rogers-Lapwood problem is here investigated. Since the local thermal non equilibrium model is employed, two temperatures, one for the solid phase and one for the fluid phase, are considered. The porous layer is saturated by a Newtonian fluid and the lower plate is impermeable. A horizontal free surface is assumed as top boundary. The free surface is subject to a uniform pressure condition and a third kind boundary condition rules the heat transfer with the external environment. The lower boundary is subject to a uniform heat flux modelled by means of Model A proposed by Amiri et al. [1]. The linear stability of the basic state is investigated by means of normal modes method. An eigenvalue problem characterised by ordinary differential equations is obtained. This eigenvalue problem is governed by a number of parameters. This feature gives the chance of investigating different limiting cases. Some of these cases are solved analytically. These analytical solutions are employed as benchmark and as guess values for the numerical solver employed to solve the general case: a fourth order Runge-Kutta method coupled with the shooting method. The critical values of the stability parameter for the onset of convective instability are obtained for a number of cases.

Instability of aerosol boundary accelerated in a direction perpendicular to its plane

Theory of Rayleigh–Taylor instability (RTI) is extended for the case of a dilute aerosol system. Equations of interpenetrating continua are investigated on stability to small perturbations by the method of normal modes. In contrast with homogeneous, equations of two-phase, two-velocity medium possess three types of disturbances: an entropy-vorticity one along with two acoustic ones. The difference in the amplitude constant signs of these types is able to affect the disturbance shape and phase convection inside mixing layer. The acoustic branch of solution is responsible for mechanism of instability, while the entropy-vorticity one determines damping mechanism. Instability of a plane accelerating surface, which separates aerosol and homogeneous incompressible fluid, is then investigated. Dispersion relation of the boundary-value problem for perturbations has two aperiodic unstable roots; one of them coincides with the classic R–T root in the particle absence limit, another is able to compete with only at huge accelerations. Instability is governed by modified Shields and Atwood numbers. As applications, systems with different aerosol densities are considered: atmospheric dust, mist around atomizing drop and explosively dispersed powder. Conclusion is done that a suspension somewhat destabilizes the system at small particle volume concentrations, while at a denser content it has essential damping influence on RTI.

Effect of thermal loading due to laser pulse on thermoelastic porous medium under G-N theory

Abstract The aim of this paper is to study the wave propagation of generalized thermoelastic medium with voids under the effect of thermal loading due to laser pulse with energy dissipation. The material is a homogeneous isotropic elastic half-space and heated by a non-Gaussian laser beam with the pulse duration of 0.2 ps. A normal mode method is proposed to analyse the problem and obtain numerical solutions for the displacement components, stresses, temperature distribution and the change in the volume fraction field. The results of the physical quantities have been illustrated graphically by comparison between both types II and III of Green-Naghdi theory for two values of time, as well as with and without void parameters.

Effect of magnetic field on piezo-thermoelastic medium with three theories

Abstract In the present paper, we use the classical dynamical coupled theory, Lord-Shulman theory with one relaxation time, and Green-Lindsay with two relaxation times of thermoelasticity to study the influence of magnetic field on piezo-thermoelastic medium. The normal mode method is used to transfer partial differential equation to an ordinary differential equation. Numerical computations are performed for a specific material and the results obtained are represented graphically. A comparison is made between the three theories in the absence and presence of a magnetic field. The presence of magnetic field plays an important role on all physical quantities.

Numerical simulation of underwater acoustical field with directional sources based on the normal modes model

In this paper, beamforming and normal modes method are combined to calculate the underwater acoustical field. Firstly, beamforming technology is used to form directional sources by controlling the phase and relative amplitude. Secondly, considering the ocean speed, surface and bottom boundary, the point source sound field is calculated based on the normal modes model. Then underwater acoustical field with directional sources is achieved by adding all point source field. Taken the 11 elements vertical linear array as an example, the special directional source is formed by chebyshev amplitude weighting, and then underwater acoustical field is calculated in the Munk profile of different grazing angles. The results indicate that beam grazing angle is very essential to its sound field, and changing the grazing angle can enhance the sound intensity of the shadow zone correspondingly.

MAGNETO-CONVECTIVE INSTABILITY IN A HORIZONTAL VISCOELASTIC NANOFLUID SATURATED POROUS LAYER

Nanofluids have been shown experimentally to have high thermal conductivity. In this study, the convective instabilities in a horizontal viscoelastic nanofluid saturated by porous layer under the influences of gravity and magnetic field are investigated. The linear stability theory is used for the transformation of the partial differential equations to system of ordinary differential equations through infinitesimal perturbations, scaling, linearization and method of normal modes with two-dimensional periodic waves. The system is solved analytically for the closed form solution of the thermal Darcy-Rayleigh number by using the Galerkin-type weighted residuals method to investigate the onset of both stationary and oscillatory convection. The effects of the scaled stress relaxation parameter, scaled strain retardation parameter and Chandrasekhar number on the stability of the system are investigated. The scaled strain retardation parameter stabilizes while the scaled stress relaxation parameter destabilizes the nanofluid system. The system in the presence of magnetic field is more stable than the system in the absence of magnetic field.

Effect of Hall current on an initially-stressed two-temperature thermoelastic material under G-N theory

The influence of the initial stress and the Hall current on a two-temperature thermoelastic material is investigated in the framework of the modified Ohm’s law and Green-Naghdi theory. Analytical expressions for the physical quantities are obtained by using the normal mode method and are presented graphically. Finally, the resulting physical quantities are discussed with and without the initial stress and the Hall current. The results of this study have applications in a wide range of physical problems, geophysical applications, and radiation physics.

Morphological instability of an evaporation front moving in a geothermal reservoir

The evaporation front moving with a constant velocity through a high-temperature geothermal reservoir is investigated. The solution in the form of a traveling wave is represented. The existence of a thermodynamically consistent solution and the solution with superheating of water ahead of the phase transition front is demonstrated. A boundary separating two types of the solutions is found in the domain of constitutive parameters. Stability of the solutions obtained is investigated by means of the normal mode method. It is shown that the criterion of superheating of water ahead of the boiling front and the criterion of linear stability of the front do not coincide.

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