Spatial Stability Analysis Research Articles

Breakup of Viscous Liquid Sheets Subjected to Symmetric and Asymmetric Gas Flow

A study of the breakup of planar viscous liquid sheets subjected to gas flow on both sides was conducted. A linear spatial stability analysis was used to determine the instability wave characteristics. The analysis included the effects of liquid properties such as viscosity, density, and surface tension; the gas was treated as inviscid. Dispersion relations were obtained relating the wave growth rates to the frequency and other flow variables. The wave characteristics were determined by numerical solution of the governing dispersion relations for a wide range of operating conditions. In all cases, the gas velocity was found to be destabilizing; increases in the liquid density, viscosity, and surface tension were all found to have stabilizing effects. When the liquid sheet was exposed to unequal gas velocities, the wave propagation characteristics were found to be altered from the case of equal gas velocities.

The Stability of Radiatively Cooling Jets. I. Linear Analysis

The results of a spatial stability analysis of a two-dimensional slab jet, in which optically thin radiative cooling is dynamically important, are presented. We study both magnetized and unmagnetized jets at external Mach numbers of 5 and 20. We model the cooling rate by using two different cooling curves: one appropriate to interstellar gas, and the other to photoionized gas of reduced metallicity. Thus, our results will be applicable to both protostellar (Herbig-Haro) jets and optical jets from active galactic nuclei. We present analytical solutions to the dispersion relations in useful limits and solve the dispersion relations numerically over a broad range of perturbation frequencies. We find that the growth rates and wavelengths of the unstable Kelvin-Helmholtz (K-H) modes are significantly different from the adiabatic limit, and that the form of the cooling function strongly affects the results. In particular, if the cooling curve is a steep function of temperature in the neighborhood of the equilibrium state, then the growth of K-H modes is reduced relative to the adiabatic jet. On the other hand, if the cooling curve is a shallow function of temperature, then the growth of K-H modes can be enhanced relative to the adiabatic jet by the increase in cooling relative to heating in overdense regions. Inclusion of a dynamically important magnetic field does not strongly modify the important differences between an adiabatic jet and a cooling jet, provided the jet is highly supermagnetosonic and not magnetic pressure-dominated. In the latter case, the unstable modes behave more like the transmagnetosonic magnetic pressure-dominated adiabatic limit. We also plot fluid displacement surfaces associated with the various waves in a cooling jet in order to predict the structures that might arise in the nonlinear regime. This analysis predicts that low-frequency surface waves and the lowest order body modes will be the most effective at producing observable features in the jet.

Theory and Simulation of Asymmetrically Perturbed Radiatively Cooled Jets

AbstractResults of a spatial stability analysis and of numerical simulations of a “slab” jet in which optically thin radiative cooling is dynamically important are presented. Two different cooling curves are used. Unstable Kelvin-Helmholtz modes are significantly different from the adiabatic limit, and the form of the cooling function strongly affects the results. The numerical simulations are in excellent agreement with the linear stability analysis. In the non-linear regime growth of the surface wave at low frequencies results in sinusoidal oscillation which can disrupt the jet, while non-linear body waves produce low amplitude wiggles within the jet that can result in shocks within the jet. In cooling jets, these shocks can produce dense knots and filaments of cooling gas within the jet, and weak shock “spurs” in the ambient gas. Acceleration of ambient gas can be produced by these “spurs”, or by rapid entrainment if the jet is disrupted. For parameters typical of protostellar jets perturbations with a period of < 100 yrs should excite body waves which produce internal shocks and small amplitude wiggles. The lack of large amplitude wiggles in most observed systems is consistent with the suggestion that jets arise from the inner regions (r < 1 AU) of accretion disks.

A stability study of the developing mixing layer formed by two supersonic laminar streams

An inviscid, parallel, spatial linear stability analysis is performed on both bounded and semibounded developing mixing layers formed between a laminar Mach 8 stream and another at Mach 3. Three unstable modes have been found in the initial mixing zone of the unbounded flow. In the downstream region of this flow the slowest of these modes becomes stable, and the remaining two correspond to the fast and slow modes of the self-similar mixing layer. For the semibounded flow, a wall in the slow stream introduces a series of acoustic modes which replace the fast mode of the unbounded flow. For both the unbounded and the semibounded flows the largest growth rates belong to the slowest mode which resides in the developing region and is insensitive to the presence of the wall. The existence of this mode has been detected in a wind-tunnel experiment, at frequency ranges suggested by, and with maximum growth rates in agreement with, the theory.

A full-scale numerical study of interfacial instabilities in thin-film flows

Surface wave instabilities in a two-dimensional thin draining film are studied by a direct numerical simulation of the full nonlinear system. A finite element method is used with an arbitrary Lagrangian–Eulerian formulation to handle the moving boundary problem. Both temporal and spatial stability analysis of the finite-amplitude nonlinear wave regimes are done. As the wavenumber is decreased below the linear cut-off wavenumber, supercritical sinusoidal waves occur as reported earlier from weakly nonlinear analysis and experiments. Further reduction in wavenumber makes the Fourier spectrum broad-banded resulting in solitary humps. This transition from nearly sinusoidal permanent waveforms to solitary humps is found to go through a quasi-periodic regime. The phase boundaries for this quasi-periodic regime have been determined through extensive numerical parametric search. Complex wave interaction processes such as wave merging and wave splitting are discussed. In the exhaustive numerical simulations performed in this paper, no wave-breaking tendency was observed, and it is speculated that the complex wave-interaction processes such as wave merging and wave splitting curb the tendency of the film to break.

SPATIAL STABILITY ANALYSIS OF THIN-WALLED SPACE FRAMES

A clearly consistent finite element formulation for spatial stability analysis of thin-walled space frames is presented by applying linearized virtual work principle and introducing Vlasov's assumption. The improved displacement field for unsymmetric thin-walled cross-sections is introduced based on inclusion of second-order terms of finite rotations, and the potential energy corresponding to the semitangential moments is consistently derived. In the present formulation, displacement parameters of axial and bending deformations are defined at the centroid axis and parameters of lateral and torsional deformations at the shear centre axis, and all bending-torsional coupling effects due to unsymmetric cross-sections are taken into account. For finite element analysis, cubic Hermitian polynomials for the flexural beam with four types of end conditions are utilized as shape functions of Hermitian space frame element. Also, load correction stiffness matrices for off-axis point loadings are derived based on the second-order rotation terms. Finite element solutions for the spatial buckling analysis of thin-walled space frames are compared with available solutions and other researcher's results.

The Formation of Oceanic Eddies in Symmetric and Asymmetric Jets. Part I: Early Time Evolution and Bulk Eddy Transports

A numerical study of the life cycles of eddies formed in perturbed baroclinic jets, emphasizing the influence of spanwise asymmetry in the initial structure of the jet on the downstream evolution of the flow, is presented. In particular, attention is focused on the early time evolution of two such currents that are fully determined by their lateral/vertical cross sections of streamwise velocity and balanced temperature fields. The first jet profile is constructed so as to be representative of a typical observational cross section of the Gulf Stream off Cape Hatteras; it therefore has laterally asymmetric baroclinicity and barotropy. The second cross section consists of a jet with laterally symmetric velocity shear. Primitive equation nonseparable linear stability analyses of these two mean states are performed to determine the wavelengths, phase speeds, and growth rates of the fastest growing normal modes of temporal instability. The life cycles of the eddies that develop on these jets are investigated using a three-dimensional numerical model that employs the nonhydrostatic Boussinesq equations of motion in channel geometry with inflow/outflow streamwise boundary conditions. The wavelengths, phase speeds, and growth rates of the disturbances that develop during the early stages of flow evolution are compared to those predicted by the temporal stability analysis presented herein, and also to previous temporal and spatial stability analyses. Bulk spanwise eddy transports of heat and momentum are examined to assess the influence of asymmetry in the initial jet profile. The formation frequency and strength of warm core and cold core rings in the two simulations are compared and contrasted with observations of eddy formation in the Gulf Stream system. In a companion paper we explore the detailed dynamical mechanisms through which individual coherent vortical structures form and more fully analyze the associated horizontal and vertical mixing of heat and potential vorticity.

Destabilization of Strongly Magnetized Jets

view Abstract Citations (15) References (28) Co-Reads Similar Papers Volume Content Graphics Metrics Export Citation NASA/ADS Destabilization of Strongly Magnetized Jets Hardee, Philip E. ; Clarke, David A. Abstract Simulations of strongly magnetized slab jets in uniform expansion in an external atmosphere have been performed and compared to an unmagnetized expanding slab jet. A strongly magnetized super-Alfvénic jet with lower magnetosonic Mach number but with sonic Mach number equal to a purely fluid jet destabilizes in a much shorter distance than the fluid jet. A strongly magnetized sub-Alfvénic jet remains stable until the jet becomes super-Alfvénic. At the Alfvén point the jet is submagnetosonic and destabilizes abruptly. Results from a spatial stability analysis of the magnetized slab jet are compared to the numerical simulations. In general, predictions based on the theory are borne out by the simulations. In particular, the stability properties of a super-Alfvénic jet are confirmed to be a function of the magnetosonic Mach number, and the theoretical prediction of rapid jet destabilization at the Alfvén point transition between sub-Alfvénic and super-Alfvénic flow is confirmed. However, nonlinear effects associated with decreasing the jet density relative to the external density and with increasing the magnetic field strength can increase the destabilization-decollimation length and reduce decollimation, respectively. Some of the implications for three-dimensional jets are considered. Publication: The Astrophysical Journal Pub Date: August 1995 DOI: 10.1086/176037 Bibcode: 1995ApJ...449..119H Keywords: GALAXIES: INDIVIDUAL MESSIER NUMBER: M87; GALAXIES: JETS; HYDRODYNAMICS; INSTABILITIES; MAGNETOHYDRODYNAMICS: MHD full text sources ADS | data products SIMBAD (3) NED (1)

Spatial stability and free vibration of shear flexible thin‐walled elastic beams. I: Analytical approach

AbstractAn improved formulation for spatial stability and free vibration analysis of thin‐walled elastic beams is presented by applying Hellinger–Reissner principle and introducing Vlasov's assumption. It includes shear deformation effects due to flexural shear and restrained warping stress, rotary inertia effects and bendirsg–torsional coupling effects due to unsymmetric cross sections. Closed‐form solutions for determining flexural–torsional buckling loads and natural frequencies of unsymmetric simply supported beam‐columns subjected to eccentric axial force are newiy derived and also, the tangent stiffness matrix and stability functions for symmetric thin‐walled beam elements subjected to axial force are presented. In a companion paper,26 these analytic solutions are compared with the finite element solutions according to the increase of shear deformation effects.

Spatial stability and free vibration of shear flexible thin‐walled elastic beams. II: Numerical approach

AbstractIn a companion paper,1 equations of motion and closed‐form solutions for spatial stability and free vibration analysis of shear flexible thin‐walled elastic beams were analytically derived from the linearized Hellinger–Reissner principle. In this paper, elastic and geometric stiffness matrices and consistent mass matrix for finite element analysis are evaluated by using isoparametric and Hermitian interpolation polynomials. Isoparametric interpolation functions with 2, 3 and 4 nodes per element are utilized in isoparametric beam elements, and in Hermitian beam elements, the third‐ and fifth‐order Hermitian polynomials including shear deformation effects are newly derived and applied for the calculation of element matrices. In order to verify the validity of the finite element formulation, both analytic and numerical solutions for spatial buckling and free vibration problems including shear effects are presented and compared.

Mode coupling in the torsional—flexural buckling of regular multistorey buildings

AbstractAn analogy is established between the mode coupling of columns under concentrated end forces and those subjected to uniformly distributed axial forces. It is demonstrated that the cubic equation derived for the mode coupling of columns under end forces can be used directly for the more practical case of cantilevers under uniformly distributed axial load. Simple closed‐form formulae and design tables are given for the torsional—flexural buckling of cantilevers. These formulae can be used directly for the spatial stability analysis of core supported regular multistorey buildings. A numerical example is provided to show the step‐by‐step procedure to be followed in practical structural design for stability. The results of a series of tests on small‐scale multistorey building models are presented. The loadbearing system of the models consists of columns, shear walls and support cores. The results indicate that the contribution of the columns to the lateral and warping stiffness of the support system may be of considerable magnitude. The more sensitive the structure is to torsion, the greater the effect of the columns.

Linear disturbances in hypersonic, chemically reacting shock layers

The effects of equilibrium- and nonequilihrium-air chemical reactions on the linear stability of a Mach 25, 10-deg half-angle sharp-cone shock layer are investigated. First, the basic state is computed using the parabolized Navier-Stokes equations with a shock-fitting scheme. This eliminates spurious numerical oscillations that could adversely affect the stability analysis. Spatial stability analyses are then described for three different approximations of the physics: perfect gas, air in local chemical equilibrium, and air in chemical nonequilibrium. It is shown in both the equilibrium- and nonequilibrium-air calculations that the second mode of Mack is shifted to lower frequencies

Wavenumber selection and irregularity of spatially developing nonlinear Dean and Görtler vortices

Spatially developing vortices found in curved channels (Dean vortices) and in a concave boundary layer (Görtler vortices) asre studied numerically using Legendre spectral-element methods. The linear instability of these vortices with respect to spanwise perturbations (Eckhaus instability) is examined using parabolized spatial stability analysis. The nonlinear evolution of this instability is studied by solving the parabolized Navier–Stokes equations. When the energy level of Dean and Görtler vortices in the flow is low, the spatial growth of the vortices is governed by primary instability (Dean or Görtler instability). At this stage, vortices with different wavelengths can develop at the same time and do not interact with each other significantly. When certain vortices reach the nonlinear stage first and become the dominant wavelength, spatial Eckhaus instability sets in. For all cases studied, spatially developing Dean and Görtler vortices are found to be most unstable to spanwise disturbances with wavelength twice or $\frac{3}{2}$ times that of the dominant one. The nonlinear growth of these perturbations generates a small vortex pair in between two pairs of vortices with long wavelength, but forces two pairs of vortices with short wavelength to develop into one pair. For Görtler vortices, this is manifested mostly by irregular and deformed vortex structures. For Dean vortices, this is manifested by vortex splitting and merging, and spatial Eckhaus instability plays an important role in the wavenumber selection process.

Spatial stability of the magnetized slab jet

A spatial stability analysis of the Kelvin-Helmholtz instability of a magnetized slab jet is performed and compared to results of numerical simulations. Provided the jet is super-Alfvenic the dispersion relation describing the propagation and growth of Fourier components of some initial perturbation admits the same type of solutions as for a purely fluid jet. A growing fundamental sinusoidal mode at long wavelengths can be identified with an Alfven wave-like disturbance of the jet. Growing internal reflection modes at shorter wavelengths can be identified with fast magnetosonic waves reflecting off the jet's boundaries. Supermagnetosonic resonances comparable to the supersonic resonances of a strictly fluid jet exist where resonant wavelengths and growth lengths scale with the magnetosonic Mach number. The resonances disappear when the jet becomes transmagnetosonic but still super-Alfvenic and only the fundamental sinusoidal mode remains. The fundamental mode is stabilized when the jet is sub-Alfvenic, but for jet velocities slightly less than the slow magnetosonic speed reflection modes can grow even on a sub-Alfvenic jet. This behavior is qualitatively similar to the stability properties of a magnetized cylindrical jet. While the simulations reveal significant non-linear effects associated with increasing magnetic tension, the jet is not stabilized by nonlinear effects, and the linear analysis provides a reasonable description of the spatial stability properties of the jet. The results suggest that a jet which is initially sub-Alfvenic and stable to disruption will be doomed to disruption at the Alfven point if it becomes super-Alfvenic as a result of jet expansion.

Spatial stability analysis of Eigen's quasispecies model and the less than five membered hypercycle under global population regulation

Spatial stability analysis of Eigen's quasispecies model and the less than five membered hypercycle under global population regulation

Shear‐flexible models for spatial buckling of thin‐walled curved beams

AbstractThe governing non‐linear finite element equations for the spatial stability analysis of curved beams, using a simple two‐noded model, are derived based on the incremental form of a mixed variational principle with independent discretization for its generalized strain field and the reference line displacements as well as cross‐sectional warping and bending/twisting rotations. The formulation is valid for both open‐and closed‐type thin‐walled sections, and this is accomplished by the use of a kinematic description based on a generalized beam theory in which shear deformation due to both flexural‐and warping‐torsional actions is accounted for. The effect of finite rotations in space is included, resulting in a second‐order accurate geometric stiffness matrix and ensuring that all significant instability modes can be predicted. Finally, the results obtained in a number of numerical simulations for lateral‐torsional bifurcation buckling of circular arches are presented to illustrate the model effectiveness and practical usefulness, and to provide explanations for the source of discrepancies noted in the results obtained in previous investigations.

The effect of walls on instability waves in supersonic shear layers

A spatial stability analysis is performed to determine the effect of wall placement on instability waves in confined supersonic shear layers. It is shown that the growth rates of Kelvin–Helmholtz instability waves are independent of wall height. However, supersonic instability waves are found to exhibit peaks and valleys in their growth rate curve as a result of the reinforcement and cancellation of the reflected Mach waves at the shear layer. Finally, it is shown how the instability growth rates in a ducted shear layer may be maximized by the proper choice of the duct width to height ratio.

Spatial stability analysis of Eigen's quasispecies model and the less than five membered hypercycle under global population regulation

A generalization of the “constant overall organization” constraint of Eigen's quasispecies and hypercycle models, called herein “global population regulation”, is shown to lead to mathematically tractable spatial generalizations of these two models. The spatially uniform steady state of Eigen's quasispecies model is shown to be stable and globally attracting for all possible values of the mutation and replication rates. In contrast, the spatially and temporally uniform solutions to the hypercycle with fewer than five members, the only ones insensitive to stochastic perturbations, are shown to be unstable, and a lower bound to the spatial inhomogeneities is obtained. The prospect that the spatially localized hypercycle might be immune to various instabilities cited in the literature is then briefly considered. Although spatial localization makes possible a much richer dynamical repertoire than previously considered, it is also more difficult to understand how Darwinian selection of hypercycles could result in a unique genetic code.

Interpretation of Noise Coherence Function Measurements in Liquid-Metal Fast Breeder Reactor Critical Assemblies

An improved technique for inferring the eigenvalue separation, which is important in spatial stability analysis, was developed using the noise coherence function. It was applied to fast reactor critical assemblies of various sizes and compositions that exhibited a wide range of spatial decoupling. In each experiment, four lithium-glass detectors were used to measure noise coherence functions. Various ratios of the coherence functions were used to obtain the first two modes of separation considering higher modes and variations in detector efficiencies. The eigenvalue separation obtained by noise analysis agreed well with calculation.

Numerical methods for hypersonic boundary layer stability

Various numerical methods for the solution of linear stability equations for compressible boundary layers are compared. Both the global and local eigenvalue methods for temporal stability analysis are discussed. Global methods are used to compute all the eigenvalues of the discretized system. When a guess for the desired eigenvalue is available, local methods may be used both to purify the eigenvalue and compute the associated eigenfunctions. The extension to spatial stability analysis is also considered. The discretizations studied include: a second-order finite-difference method, a fourth-order accurate two-point compact difference scheme, and a Chebyshev spectral collocation method. Eigenvalue results are presented for Mach numbers up to 10. As the Mach number increases, the performance of the spectral method deteriorates due to the outward movement of the critical layer. To alleviate this problem, a multi-domain spectral collocation method is developed which exhibits better convergence. The overall performance of the fourth-order compact scheme is excellent. Our results also indicate that, in the limit of vanishing Mach number, there exist stable discrete modes in addition to the discrete modes of the Orr-Sommerfeld equation.