Dominant Instability Mode Research Articles

Effect of material parameters on shear band spacing in work-hardening gradient dependent thermoviscoplastic materials

We study thermomechanical deformations of a viscoplastic body deformed in simple shear. The effect of material elasticity is neglected but that of work hardening, strain-rate hardening, thermal softening, and strain-rate gradients is considered. The consideration of strain-rate gradients introduces a material characteristic length into the problem. A homogeneous solution of the governing equations is perturbed at different values t 0 of time t, and the growth rate at time t 0 of perturbations of different wavelengths is computed. Following Wright and Ockendon's postulate that the wavelength of the dominant instability mode with the maximum growth rate at time t 0 determines the minimum spacing between shear bands, the shear band spacing is computed. It is found that for the shear band spacing to be positive, either the thermal conductivity or the material characteristic length must be positive. Approximate analytical expressions for locally adiabatic deformations of dipolar (strain-rate gradient-dependent) materials indicate that the shear band spacing is proportional to the square-root of the material charateristic length, and the fourth root of the strain-rate hardening exponent. The shear band spacing increases with an increase in the strain hardening exponent and the thermal conductivity of the material.

On the spectral distribution of the modes in nonlinear Görtler instability

Abstract Longitudinal vortices generated by Gortler instability in a concave wall boundary layer are studied experimentally. The vortices were triggered by a wire grid upstream of the test-section leading edge. The vortices are visualized and their velocity field is measured in four axial positions. The measurements show weakly nonlinear growth and also nonlinear saturation of perturbations. Wall-normal and spanwise gradients of the velocity field calculated from the velocity measurements show that the spanwise gradients grow faster than the wall-normal gradients. It is thus inferred that the sinuous mode associated with the spanwise gradient will be the dominant instability mode. Spectral analysis of the spanwise profiles of the streamwise velocity show that seven modes suffice to represent the velocity field in the nonlinear regime. However, the amplitudes of the modes present in the velocity profiles vary with distance from the wall as with distance from the leading edge.

Collective behavior and spacing of adiabatic shear bands

The failure of metals subjected to high strain rates is frequently related to the collective development of adiabatic shear bands which results in a patterning depending on the material properties and the loading conditions. In this paper, the spacing between adiabatic shear bands is characterized by analytical means in a one-dimensional formulation. Using a perturbation analysis, a dominant instability mode can be characterized, whose wavelength is related to the shear-band spacing. Explicit solutions are found for materials with no strain hardening. Asymptotic developments are used to obtain results that account for strain hardening. Comparisons are made with experimental results and other existing models.

On the transverse instability of solitary waves in the Kadomtsev-Petviashvili equation

One-dimensional solitary wave solutions of the Kadomtsev-Petviashvili (KP) equation were shown to be unstable to long-wavelength transverse disturbances by Kadomtsev and Petviashvili, in the positive dispersion case. Here we show that there is a short-wavelength cutoff for the instability, which is associated with a bifurcation to transversely modulated solitary waves, and we identify the dominant mode of instability, by finding explicitly all the exponentially unstable modes of the linearized equation for perturbations of the solitary wave. No unstable modes are found in the negative dispersion case.

Boundary-layer stability measurements in a hypersonic quiet tunnel

Hypersonic boundary-layer measurements were conducted over a flared cone in a quiet wind tunnel. The flared cone was tested at a freestream unit Reynolds number of 2.82 x 10 6 /ft in a Mach 6 flow. This Reynolds number provided laminar-to-transitional flow over the model in a low-disturbance environment. Point measurements with a single hot wire using a novel constant voltage anemometry system were used to measure the boundary-layer disturbances. Surface temperature and schlieren measurements were also conducted to characterize the laminar-to-transitional state of the boundary layer and to identify instability modes. Results suggest that the second mode is the dominant mode of instability. The integrated growth rates of the second mode compared well with linear stability theory in the linear stability regime. Furthermore, the existence of higher harmonics of the fundamental suggests that nonlinear disturbances are not associated with high freestream disturbance levels.

Interaction of oblique instability waves with weak streamwise vortices

This paper is concerned with the effect of a weak spanwise-variable mean-flow distortion on the growth of oblique instability waves in a Blasius boundary layer. The streamwise component of the distortion velocity initially grows linearly with increasing streamwise distance, reaches a maximum, and eventually decays through the action of viscosity. This decay occurs slowly and allows the distortion to destabilize the Blasius flow over a relatively large streamwise region. It is shown that even relatively weak distortions can cause certain oblique Rayleigh instability waves to grow much faster than the usual two-dimensional Tollmien–Schlichting waves that would be the dominant instability modes in the absence of the distortion. The oblique instability waves can then become large enough to interact nonlinearly within a common critical layer. It is shown that the common amplitude of the interacting oblique waves is governed by the amplitude evolution equation derived in Goldstein & Choi (1989). The implications of these results for Klebanoff-type transition are discussed.

Linear stability theory of two-layer fluid flow in an inclined channel

The linear stability of the two-layer flow of immiscible, incompressible fluids in an inclined channel is considered. In the long-wave limit, mechanisms for linear instability, and the consequences of competition between mechanisms, are identified. For arbitrary wave numbers, air–water and olive oil–water systems are considered, in order to determine the influence of the channel thickness and the mean interfacial height on the stability of the flow. This paper characterizes those physical situations in which the primary instability is to long-wave interfacial disturbances. The odd Orr–Sommerfeld shear mode within the water layer, which is necessarily stable in plane Poiseuille flow, is found to grow and even be the dominant mode of instability for the olive oil–water system. The consequences beyond linear stability are discussed.

Hydrodynamic instability of a fluid layer flowing down a rotating cylinder

In this paper the linear stability of a fluid layer flowing down the inside and outside of a rotating vertical cylinder is investigated. To this end, two approximations are made: the small wave number approximation and the small Reynolds number approximation. In the former, only the radial destabilizing effect of surface tension is important and may be counteracted by the centrifugal force at a critical value. In the latter, the analysis integrates the azimuthal modes different from m=0. It is shown that for flow outside the cylinder, the magnitude of centrifugal force and wave number may change the dominant mode of instability. For flow inside the cylinder, only the mode m=0 may be unstable. These results generalize those of Boudourides and Davis [Z. Angew. Math. Phys. 37, 597 (1986)] for swirling viscous flows.

Observations on transition in plane bubble plumes

Flow visualization studies of plane laminar bubble plumes have been conducted to yield quantitative data on transition height, wavelength and wave velocity of the most unstable disturbance leading to transition. These are believed to be the first results of this kind. Most earlier studies are restricted to turbulent bubble plumes. In the present study, the bubble plumes were generated by electrolysis of water and hence very fine control over bubble size distribution and gas flow rate was possible to enable studies with laminar bubble plumes. Present observations show that (a) the dominant mode of instability in plane bubble plumes is the sinuous mode, (b) transition height and wavelength are related linearly with the proportionality constant being about 4, (c) wave velocity is about 40% of the mean plume velocity, and (d) normalized transition height data correlate very well with a source Grashof number. Some agreement and some differences in transition characteristics of bubble plumes have been observed compared to those for similar single-phase flows.

Active control of unsteady combustion-induced oscillations

A loudspeaker-based active control system has been used to study a longitudinal combustion instability mode and its active control in a small scale premixed combustor. A simple gain/phase-shift type controller is shown to successfully suppress oscillations in the combustor at all equivalence ratios at low flow rates. It is found, however, that as the flow rate through the combustor is increased, the performance of the controller deteriorates rapidly, both in terms of allowable gain and phase margin and reduced effect on dominant instability such that beyond certain flow rates the controller is unable to suppress the oscillations for any values of gain and phaseshift. Nyquist stability diagram obtained from the frequency response data are used to understand the performance of the controller and help design an advanced controller with somewhat improved performance. The physical model developed by Lang et al. is adapted to help understand important features of the combustion instability and its active control. Without the controller the model is found to predict the frequency of the dominant instability mode and its variation with equivalence ratio fairly well. For off-stoichiometric operation the model predicts an increase in the number of potential instability modes as is also observed in the measured power spectra. With the controller on, the model shows that the controller interacts significantly with the combustion and in one case examined here, it is shown that the controller stabilizes the second longitudinal mode for one value of gain but at just a slightly higher value renders the previously stable third mode unstable. This offers plausible physical explanation for the observed limited gain and phase margin and the limited effectiveness of the controller at the higher flow rates. The model also predicts the stability boundary to have a reduced phase margin as observed experimentally.

The effects of nozzle exit lip thickness on plume resonance

The acoustic emission and initial shear layer of a cold jet issuing from an underexpanded sonic nozzle were measured where the lip thickness of the nozzle exit varied from 0·015 to 0·625 nozzle diameters. Because the amplitude of plume resonance (or screech) has been shown to be very sensitive to the initial conditions of the plume, the effects of nozzle lip thickness on this sound component in particular was investigated (although it will be shown that other noise components are also affected). Nearfield acoustic measurements have revealed the amplitude and frequency of certain modes of screech to be dependent on nozzle lip thickness. Fluctuating pressure measurements made on the nozzle exit surface have acoustic amplitudes in excess of near field microphone measurements. The dominant mode of instability that exists in the shear layer is also related to this geometric parameter. Detailed shear layer measurements were performed in an attempt to quantify any associated changes in the momentum thickness caused by altering the nozzle exit.

Effects of ambient turbulence on the decay of a trailing vortex wake

Experiments were conducted to investigate the effects of ambient turbulence on the evolution of a trailing vortex wake. The wake was generated by towing an NACA 0012 wing, with a span of 10.2 cm and a chord of 5.1 cm, at an incidence of —10 deg and a speed of 40 cm/s in a tow tank. The chord Reynolds number was 20,400. The ambient turbulence was generated by towing three grids, with square meshes of 1.45, 10.2, and 20.3 cm, upstream of the wing. Turbulence parameters were measured with crossed hot-film probes. The trailing vortex wake, tagged with a fluorescent dye, was visualized from different perspectives and its decay was derived from 16-mm movie records. For weak turbulence with large integral scales compared with the vortex separation, vortex linking is the dominant mode of instability. The dominant wavelength of the linking decreases with increasing turbulence intensity or dissipation rate. As the turbulence intensity increases, vortex bursting appears and eventually replaces linking as the dominant mode of instability. For turbulence with a small integral scale as compared with the vortex separation, vortex instability is predominantly of the bursting type.

In Situ Observations of Kelvin-Helmholtz Waves along a Frontal Inversion

Abstract An unusual set of observations of amplifying Kelvin-Helmholtz waves along a New England coastal front inversion is presented and compared to linear theory. The waves were observed by instrumented aircraft traversing the coastal front at various levels and measuring temperature and horizontal and vertical wind components at one-second intervals. The most prominent group of waves had amplitudes in the neighborhood of 100 m, comparable to the total depth of the inversion, and wavelengths near 1.7 km. Spectral analysis of data from an aircraft pass located just above the inversion indicates that Kelvin-Helmholtz billows of 1−2-km wavelength were present over a distance of at least 25 km. Numerical solution of the Taylor-Goldstein equation for observed and specified basic-state vertical profiles confirms that instability of the Kelvin-Helmholtz type is the dominant mode of instability. The wavelength of the most unstable mode is found to be strongly sensitive to the relative thicknesses of the shear a...

Two-dimensional finite difference analysis of shape instabilities in plates

A two-dimensional finite difference analysis is applied to surface diffusion-controlled instabilities of plates. Plates can evolve into “cylinders,” or if the plates have longitudinal internal boundaries, they may split into two segments. The evolution process of plates containing internal boundaries into equilibrium shapes depends on both the initial plate aspect ratio (plate width to thickness) and the ratio of the internal boundary energy to the plate-matrix interface energy. When the internal boundary energy is relatively low or the initial plate aspect ratio is relatively small, the transverse equilibrium cross-sectional area shape is composed of two circular segments, with an appropriate dihedral angle dictated by the ratio of the interface energy terms. As either the internal boundary energy or the initial aspect ratio increases, plate splitting, rather than cylinderization, becomes the dominant instability mode. The results of this work are compared to a recent theory of Courtney and Malzahn Kampe (CMK) on shape instability diagrams.[1] The complicated interactive effects between cylinderization and boundary splitting were not considered in the analytical CMK approach; thus, when they are minimal, the results of this finite difference calculation are in reasonable accord with the CMK results, as far as predicting instability times are concerned. However, when the interaction is significant, cylinderization and/or splitting times are markedly changed. The present accurate calculations allow refinement of the CMK plate instability diagrams.

Shape instabilities of plate-like structures—II. Analysis

In this paper, a companion to the previous one, a susceptibility analysis which delineates the primary instability mode of a terminated plate as it depends on plate width to thickness (aspect) ratio and the energy of internal boundaries of plates is presented. The analysis is based on a simple averaging of Fick's first law, and thus is based on geometrical, yet physically plausible, approximations. The analysis is carried out for both volume diffusion and surface diffusion control of the instability process. Direct cylinderization of plates is favored when the plate aspect ratio is small and the energy of internal boundaries that may be present is low. Boundary induced splitting of plates is favored when the boundary energy is high, whereas edge spheroidization is the dominant instability mode for plates with large aspect ratios containing low energy internal boundaries. The results of the analysis can be succintly represented in terms of plate instability diagrams which define the dominant instability mode in terms of plate aspect ratio and internal boundary energy. Lines of constant instability times can be superimposed on the diagrams, thereby increasing their utility.