Behavior Of Small Disturbances Research Articles

To The Mathematical Modeling of Nonstationary Transonic Flows. Analytical Approach

The transonic mode of gas flow has significant differences in comparison with both subsonic and supersonic regimes, combining both their general properties and specific features inherent in each of them, the possibility of penetration of weak disturbances upstream in subsonic flows. The behavior of small disturbances in subsonic and supersonic flows is qualitatively different. The paper proposes new analytical model for the mathematical modeling of unsteady transonic flows, including the effect of weak viscosity.

Competition between kinematic and dynamic waves in floods on steep slopes

We present a theoretical stability analysis of the flow after the sudden release of a fixed mass of fluid on an inclined plane formally restricted to relatively long time scales, for which the kinematic regime is valid. Shallow-water equations for steep slopes with bed stress are employed to study the threshold for the onset of roll waves. An asymptotic solution for long-wave perturbations of small amplitude is found on background flows with a Froude number value of 2. Small disturbances are stable under this condition, with a linear decay rate independent of the wavelength and with a wavelength that increases linearly with time. For larger values of the Froude number it is shown that the basic flow moves at a different scale than the perturbations, and hence the wavelength of the unstable modes is characterized as a function of the plane-parallel Froude numberFrpand a measure of the local slope of the free-surface height φ by means of a multiple-scale analysis in space and time. The linear stability results obtained in the presence of small non-uniformities in the flow, φ > 0, introduce substantial differences with respect to the plane-parallel flow with φ = 0. In particular, we find that instabilities do not occur at Froude numbersFrcrmuch larger than the critical value 2 of the parallel case for some wavelength ranges. These results differ from that previously reported by Lighthill & Whitham (Proc. R. Soc.A, vol. 229, 1955, pp. 281–345), because of the fundamental role that the non-parallel, time-dependent characteristics of the kinematic-wave play in the behaviour of small disturbances, which was neglected in their stability analyses. The present work concludes with supporting numerical simulations of the evolution of small disturbances, within the framework of the frictional shallow-water equations, that are superimposed on a base state which is essentially a kinematic wave, complementing the asymptotic theory relevant near the onset. The numerical simulations corroborate the cutoff in wavelength for the spectrum that stabilizes the tail of the dam-break flood.

Comparison Study of Dynamics in One-Sided and Two-Sided Solid-Combustion Models

Comparing two-sided and one-sided solid-combustion models, this paper concerns nonlinear transition behavior of small disturbances of front propagation and temperature as they evolve in time. Features include linear instability of basic solutions and weakly nonlinear evolution of small perturbations, as well as the complex dynamics of period doubling, quadrupling, and eventual chaotic oscillations. Both asymptotic and numerical methods are used for different solution regimes. First, multiscale weakly nonlinear analysis takes into account the cumulative effect of small nonlinearities to obtain a correct description of the evolution over long times. For a range of parameters, the asymptotic method with some dominant modes captures the formation of coherent structures. In other cases, numerical solutions reveal period-folding behaviors. In general, the one- and two-sided models agree qualitatively for all solution regimes, which is consistent with prior numerical comparisons and extends our results from [L. K. Gross and J. Yu, SIAM J. Appl. Math., 65 (2005), pp. 1708–1725].

Complex dynamic behavior during transition in a solid combustion model

AbstractThrough examples in a free‐boundary model of solid combustion, this study concerns nonlinear transition behavior of small disturbances of front propagation and temperature as they evolve in time. This includes complex dynamics of period doubling, and quadrupling, and it eventually leads to chaotic oscillations. Within this complex dynamic domain we also observe a period six‐folding. Both asymptotic and numerical solutions are studied.We show that for special parameters our asymptotic method with some dominant modes captures the formation of coherent structures. Finally,we discuss possible methods to improve our prediction of the solutions in the chaotic case. © 2009 Wiley Periodicals, Inc. Complexity, 2009

Experimental investigation of the flow instability near the attachment-line boundary layer on a yawed cylinder

The behavior of small disturbances in a 3-D laminar boundary layer on a yawed cylinder was experimentally investigated. This setup simulates the flow around the leading edge of swept wings. Since multiple instability modes appear near the attachment-line region, a point-source disturbance was artificially introduced to separate these modes. Amplitude and phase distributions of the disturbances originating from the point source were measured using a hotwire probe near the attachment-line flow to test existing theoretical predictions. Hotwire measurements show that two instability modes definitely coexist and overlap in the middle portion of the wedge-shaped region developing downstream of the point source. Decomposition by 2-D fast Fourier transform (FFT) analysis enables us to separate one oblique wave from the other. One of the oblique waves belongs to the cross-flow instability mode, which travels to the attachment line and grows even at Reynolds numbers that are slightly lower than the critical Reynolds number for the attachment-line instability. The origin of the other mode is not identifiable, because it has peculiar characteristics different from both the streamline-curvature instability mode and the cross-flow instability mode. This mode decays in the downstream direction for all frequencies examined. By investigating the spatial characteristics of the small disturbance, the critical Reynolds number for cross-flow instability was successfully determined in the off-attachment-line region. The value, Rc = 543, was lower than the critical Reynolds number of Rc = 583 for the attachment-line flow. Furthermore, the critical frequency and wavenumber were in good agreement with existing predictions based on linear stability theory.

Precision shooting: Sampling long transition pathways

The kinetics of collective rearrangements in solution, such as protein folding and nanocrystal phase transitions, often involve free energy barriers that are both long and rough. Applying methods of transition path sampling to harvest simulated trajectories that exemplify such processes is typically made difficult by a very low acceptance rate for newly generated trajectories. We address this problem by introducing a new generation algorithm based on the linear short time behavior of small disturbances in phase space. Using this "precision shooting" technique, arbitrarily small disturbances can be propagated in time, and any desired acceptance ratio of shooting moves can be obtained. We demonstrate the method for a simple but computationally problematic isomerization process in a dense liquid of soft spheres. We also discuss its applicability to barrier-crossing events involving metastable intermediate states.

An averaging method for the weakly unstable shallow water equations in a flat inclined channel

The asymptotic behavior of small disturbances as they evolve in space from boundary conditions in a flat inclined channel under weakly unstable conditions is determined using a method of averaging. The shallow water equations are first transformed to a normal form using a variation of parameters technique, and then a new method of averaging is defined. The method of averaging developed differs from averaging methods for ordinary differential equations and other partial differential equations by averaging along the characteristics of the linearized equations rather than along the time variable. It is found that the solution is dominated by the evolution of the solution along one characteristic. Both asymptotic and numerical results for periodic disturbances are presented. In addition, connections to dynamical systems concepts are described.

Nonlinear Spatial Evolution of Small Disturbances from Boundary Conditions in Flat Inclined-Channel Flow

The asymptotic behavior of small disturbances as they evolve spatially from boundary conditions in a flat inclined channel is determined. These small disturbances develop into traveling waves called roll waves, first discussed by Dressler in 1949. Roll waves exist if the Froude number F exceeds 2, which consist of a periodic pattern of bores, or discontinuities. After confirming the instability condition for F > 2 for the linearized equations in the boundary value case, the nonlinear boundary value problem for the weakly unstable region of F slightly larger than 2 is studied. Multiple scales and the Fredholm alternative theorem are applied to determine the evolution of the solution in space. It is found that the solution is dominated by the evolution of the disturbance along one characteristic. The shock conditions governing the asymptotic solution are determined and these conditions are used to determine the approximate shape of the resulting traveling wave from the solution. Both asymptotic and numerical results for periodic disturbances are presented.

Thermoelasticity with thermomechanical constraints

Equations are derived governing the behaviour of small disturbances superimposed on an underlying equilibrium configuration of a thermoelastic body. The body may be materially inhomogeneous, non-homogeneously prestrained and be subjected to a non-uniform temperature resulting in non-constant-coefficient partial differential equations. These equations are generalized to the case where thermomechanical constraints are present, both deformation-temperature and deformation-entropy constraints. It is known that the first of these types of constraints leads to material instabilities and the second does not. By examining nearly constrained materials, and taking an appropriate limit, we find that the instabilities associated with deformation-temperature constraints arise because the heat capacity at constant deformation becomes negative whilst deformation-entropy constraints are stable because the same heat capacity remains positive, though tending to zero in the limit of the constraint holding exactly. The ten important moduli of thermoelasticity are examined in the limit of each constraint holding exactly. It is found, for example, that for each type of constraint the heat capacity at constant stress remains positive and bounded away from zero. Results on wave propagation are also presented.

Reduced-Order Modeling of Unsteady Viscous Flow in a Compressor Cascade

A simultaneously coupled viscous-inviscid interaction (VII) analysis is used to model the unsteady viscous separated flow through a subsonic compressor. The inner viscous flow around the airfoil and in the wake is modeled using a finite difference discretization of the boundary-layer equations and a one-equation turbulence transport model. The outer inviscid flow is modeled using a variational finite element discretization of the compressible full potential equation. The viscous and inviscid regions are simultaneously coupled using a injection type boundary condition along the airfoil and wake. The resulting nonlinear unsteady equations are linearized about the nonlinear steady flow to obtain a set of linear equations that describe the unsteady small-disturbance behavior of the viscous flow through the cascade. The discretized small-disturbance VII equations are used to form a generalized, quadratic, non-Hermitian eigenvalue problem that describes the eigenmodes (natural modes) and eigenvalues (natural frequencies) of fluid motion about the cascade. Using a Lanczos algorithm, the eigeninformation is computed efficiently for various steady flow inflow angles and unsteady interblade phase angles

A Linearized Euler Analysis of Unsteady Transonic Flows in Turbomachinery

A computational method for efficiently predicting unsteady transonic flows in two-and three-dimensional cascades is presented. The unsteady flow is modeled using a linearized Euler analysis whereby the unsteady flow field is decomposed into a nonlinear mean flow plus a linear harmonically varying unsteady flow. The equations that govern the perturbation flow, the linearized Euler equations, are linear variable coefficient equations. For transonic flows containing shocks, shock capturing is used to model the shock impulse (the unsteady load due to the harmonic motion of the shock). A conservative Lax–Wendroff scheme is used to obtain a set of linearized finite volume equations that describe the harmonic small disturbance behavior of the flow. Conditions under which such a discretization will correctly predict the shock impulse are investigated. Computational results are presented that demonstrate the accuracy and efficiency of the present method as well as the essential role of unsteady shock impulse loads on the flutter stability of fans.

Deforming grid variational principle for unsteady small disturbance flows in cascades

A variational method for computing unsteady subsonic flows in turbomachinery blade rows is presented. A variational principle that describes the harmonic small disturbance behavior of the full potential equations about a nonlinear mean flow is developed. Included in this variational principle is the effect of a deforming computational grid that conforms to the motion of vibrating airfoils. Bilinear isoparametric finite elements are used to discretize the variational principle, and the resulting discretized equations are solved efficiently using lower-upper decomposition. The use of a deforming computational grid dramatically improves the accuracy of the method since no error-producing extrapolation is required to apply the upwash boundary conditions or to evaluate the unsteady pressure on the airfoil surfaces

Linearized Euler predictions of unsteady aerodynamic loads in cascades

A linearized Euler solver for calculating unsteady flows in turbomachinery blade rows due to both incident gusts and vibratory blade motion is presented. Using the linearized Euler technique, one decomposes the flow into a mean (or steady) flow plus an unsteady, harmonically varying, small-disturbance flow. Linear variable coefficient equations describe the small-disturbance behavior of the flow and are solved using a pseudotime time-marching Lax-Wendroff scheme. For the blade-motion problem, a harmonically deforming computational grid that conforms to the motion of vibrating blades eliminates large error producing mean flow gradient terms that would otherwise appear in the unsteady flow tangency boundary condition

Unipolar injection induced instabilities in plane parallel flows

The behavior of small disturbances in parallel flows of insulating liquids subjected to charge injection, with imposed electric field orthogonal to the primary flow, is studied. The resulting convective patterns and the instability thresholds, for plane Couette and Poiseuille flows, have been determined. Particular attention has also been given to the energy transfer between the primary flow and the secondary flow in order to gain physical insight into the way the inertial and the electric physical mechanisms act on the liquid. >

Bulk elastic wave propagation in partially saturated porous solids

The linear equations of motion that describe the behavior of small disturbances in a porous solid containing both liquid and gas are solved for bulk wave propagation. The equations have been simplified by neglecting effects due to changes in capillary pressure. With this simplifying assumption, the equations reduce to two coupled (vector) equations of the form found in Biot’s equations (for full saturation) but with more complicated coefficients. As in fully saturated solids, two shear waves with the same speed but different polarizations exist as do two compressional waves with distinct speeds. Attenuation effects can be enhanced in the partially saturated solid, depending on the distribution of gas in the pore space. Two models of the liquid/gas spatial distribution are considered: a segregated-fluids model and a mixed-fluids model. The two models predict comparable attentuation when the gas saturation is low, but the segregated-fluids model predicts a more rapid roll-off of attenuation as the gas saturation increases.

Linear Dynamic Poroelasticity With Microstructure for Partially Saturated Porous Solids

Linear equations of motion are derived to describe the behavior of small disturbances in a porous solid containing both liquid and gas. Appropriate boundary conditions are derived to guarantee the uniqueness of the solutions of these equations.

On a class of atmospheres occurring in stellar hydrodynamic theory

A derivation is given of the basic equation governing the behaviour of small disturbances in a stratified compressible atmosphere (with a locally unstable region bounded by semi-infinite stable regions) in which the time of radiative relaxation is constant.

Initial-value problem for boundary layer flows

The behavior of small disturbances in a boundary layer flow is studied. The initial-value problem is solved formally with Fourier–Laplace transforms, and an expression for the development of the velocity component normal to the wall is obtained. It is found that a disturbance evolves not only as discrete waves of the Tollmien–Schlichting type, but also has a portion described by a continuous spectrum. This portion is associated with a branch cut of the solution in the complex plane of the Laplace transform variable.

Wave motions of a liquid in tubes of viscoelastic material. Waves of small amplitude

The pliability of the walls of blood vessels is one of the essential factors determining the movement of the blood. Although there is an extensive literature devoted to this problem (see, for example, the collection [1] and the bibliography it contains), little attention has been devoted to the physiologically important specific effects related to the presence of muscular layers in the vascular wall. These effects may be roughly characterized as a significant variation in the rheological properties of the wall resulting from the activity of the vascular musculature. The latter, in turn, is controlled not only by external, nonmechanical factors but also by the stressed state of the wall itself (see, for example, [2]). Experiments indicate that in the vascular walls there are time-dependent effects characteristic of viscoelastic bodies [1] and a complex static “pressure-radius” characteristic, which may have a decreasing segment [3, 4]. Both of these facts can be reasonably interpreted within the framework of the general continual model for muscle tissue presented in [5]. In what follows, we shall study in a quasi-one-dimensional approximation the combined effect of the above-mentioned properties of the wall material on the behavior of small disturbances imposed on the stationary flow in the tube. We consider the possible types of instabilities. It is assumed that the inertial effects are insignificant and that the rheological law applicable to the wall is a nonlinear generalization of the Poynting model.

On the behaviour of small disturbances in plane Couette flow. Part 3. The phenomenon of mode-pairing

In two earlier papers (Gallagher & Mercer 1962, 1964) the results for the first four eigenvalues of the problem of the stability of plane Couette flow were given. The first twelve eigenvalues have been calculated by the same method and the results show that the manner in which the eigenvalues join to form complex pairs depends on α, the wavenumber of the disturbance. They also would appear to indicate that mode-crossing always occurs.