Growth Rate Of Disturbances Research Articles

A new translating quasigeostrophic V-state

A new nonlinear translating V-state of the barotropic quasigeostrophic equations with topography in the form of an infinitely long escarpment (or step) is computed numerically. Before the numerical computation, with the assumption of small excursions by fluid columns across the step, the linear equations of motion are solved analytically, demonstrating the possibility of constructing a translating V-state. The linear V-state has zero total circulation and self-propagates parallel to the step. It consists of two line vortices with positive circulation located on the deep side of the step and a finite-area patch of constant negative vorticity on the shallow side comprising of fluid which has crossed the step from the deep side. The linear solution motivates the numerical search for a nonlinear translating V-state comprising of two line vortices and a finite-area patch of constant vorticity. An algorithm based on contour dynamics and Newton's method is used to find such a V-state. Time-dependent contour dynamics is used to compute the evolution of the V-state and it is found to be unstable. However, the growth rate of disturbances is sufficiently small for the V-state to survive several turn-over times.

Backward wave oscillator regime of the whistler cyclotron instability in an inhomogeneous magnetic field

A linear theory for the backward wave oscillator generation regime of whistler waves in the Earth’s magnetosphere is presented. Using a parabolic profile of the magnetic field and a linear expression for the resonant current in the case of a zeroth order distribution function with a step discontinuity in the velocity component parallel to the magnetic field, the modes of the system are investigated by means of a search procedure. The existence of at least one mode exponentially growing in time is indicative of absolute instability, and such modes have been found. Therefore, an earlier prediction of such a regime, based on the homogeneous magnetic field model, is confirmed. The dependence of growth rates on the frequency mismatch and energetic electron density has been studied. These results yield the characteristic spatial profile and temporal growth rate of small-amplitude whistler-wave disturbances, which are likely to be the seeds for chorus emissions.

Hydrodynamic instability of a thin viscous film between two drops

Linear stability analysis is employed to derive analytical expressions for the growth rate of disturbances applied to a thin plane-parallel film trapped between two drops. From these expressions, the band of unstable wavenumbers and the “most dangerous” wavenumber are identified for systems in the absence and presence of insoluble surfactant. Marangoni effects are shown to exert a stabilizing influence and reasonably good agreement with experimental observations is found. Subsequent nonlinear analysis indicated amplification of the disturbance growth rate beyond that suggested by linear theory as the film proceeded toward rupture.

Influence of attractive van der Waals interactions on the optimal excitations in thermocapillary-driven spreading

Recent investigations of microfluidic flows have focused on manipulating the motion of very thin liquid films by modulating the surface tension through an applied streamwise temperature gradient. The extent to which the choice of contact line model affects the flow and stability of such thermocapillary-driven films is not completely understood. Regardless of the contact line model used, the linearized disturbance operator corresponding to the evolution of the film height is non-normal, and a generalized non-modal stability analysis is required. Surprisingly, early predictions of frontal instability that stemmed from conventional modal analysis of thermocapillary flow on a flat, infinite precursor film showed excellent agreement with experiment. Within the more rigorous framework provided by a generalized stability analysis, this work investigates the transient dynamics and amplification of optimal disturbances subject to a finite precursor film generated by attractive van der Waals forces. Convergence of the disturbance growth rates and perturbed shapes to the asymptotic solutions obtained by conventional linear stability analysis occurs early in the spreading process. In addition, the level of transient disturbance amplification is minimal. The equations governing thermocapillary-driven spreading exhibit a small degree of non-normality, which explains the source of agreement between modal theory and experiment. The more rigorous generalized stability analysis presented here, however, affords critical insight into the types of disturbances leading to maximum unstable growth and the exact influence of the contact line model used.

Origin of Turbulence in Near-Wall Flows

In experiments by Klingmann, Boiko, Westin, Kozlov and Alfredsson (1993) it became possible to remove the suction peak near the leading edge and to reduce the pressure gradient region. The experimental theoretical neutral disturbance data are given in Fig. 1. As seen, for the low frequencies, the difference between the theoretical curves appeared to be practically negligible. The experiments showed also that the linear parallel stability theory predicts well the distributions of amplitudes, wave numbers, growth rates and neutral stability of the two-dimensional Tollmien–Schlichting waves. The assumption of the boundary layer parallelity allows to acquire quite precise results for the disturbance growth rates, except for the most upper part of the neutral stability curve, where it is necessary to take into account the effects of flow nonparallelity which are quite small. The classical analysis of the linear stability treats the disturbances as separate modes of the Orr–Sommerfeld and Squire equations with exponential growth rates. However, the approach does not take into account the fact that these equations are not self-adjoint, i.e. the modes are not orthogonal. The phenomenon of the ‘lift-up’ effect (a redistribution of the streamwise momentum by small velocity perturbations in the direction normal to

Three-Dimensional Nonlinear Rupture Theory of Thin Liquid Films on a Cylinder

The three-dimensional nonlinear rupture theory of thin liquid films on a cylinder is presented and studied in this note. The thin liquid film with the effect of intermolecular forces was modeled by a continuum theory, and the three-dimensional evolution equation of liquid films on a cylindrical surface was derived based on a long wavelength approximation. Both linear stability theory and nonlinear numerical method were adopted to solve this evolution equation. The linear stability analysis fails to distinguish the three-dimensional mode from the two-dimensional one in terms of maximum disturbance growth rate and always yields a rupture time larger than the nonlinear solution. In contrast, the nonlinear numerical results clearly show that among three disturbance modes, the two-dimensional annular disturbance one yields the longest rupture time, the two-dimensional axisymmetric disturbance one yields the second longest, and the three-dimensional disturbance one does the shortest. Accordingly, it can be concluded that the rupture status of thin films on cylinder is most likely evolved in the three-dimensional disturbance mode as predicted.

An instability mechanism in the atmosphere over a water body

¶The previously unexplored mechanism of the hydrodynamic instability of the atmospheric boundary layer over a water body is theoretically investigated. In air, stratified with respect to moisture, vertical motions produce variations in specific humidity (mixing ratio) near the interface surface. This, in turn, causes variation in evaporation from the water surface and horizontal thermal inhomogeneities result on the surface which, under certain conditions, can strengthen the initial vertical motions. In this paper the linear problem of the stability of the system under consideration is solved. Boundaries of the unstable region in the space of physical parameters are defined, and specific values of growth rates of disturbances are investigated. The results show the possibility of the development of disturbances with horizontal scales of several hundred metres for a period of about one hour even for the stable stratified atmospheric layer over a water surface and in the absence of destabilizing shears of velocity. The horizontal sizes of the most rapidly growing modes, as a rule, are an order of magnitude larger than the vertical ones.

On the stability of a forced-free boundary layer flow with viscous heating

The inviscid instability of an accelerating forced-free convection boundary layer with viscous dissipation is investigated. The boundary layer equations for the flow with free-stream velocity U∞xn admit self-similar solutions for n = 1 only. The “overshooting” of the free-stream value, characteristic of accelerating buoyant boundary layers, is found to increase with increasing viscous heating. The scaled temperature profile is also found to exceed its plate value close to the plate where the viscous dissipation effects are strongest. For Eckert number Ec = 0 only one single inviscid unstable mode exists. For the flow with viscous dissipation, three unstable modes have been identified. The secondary modes may be associated with the combined effect of thermal buoyancy and viscous heating. There is evidence of these modes crossing at relatively higher wavenumbers. The disturbance growth rate is found to increase with increasing buoyancy and viscous heating.

Linear stability analysis of the solidification of a supercooled liquid in a half-space

The instability of the solidifying front of a supercooled liquid in a half-space is investigated by introducing a small disturbance at the solid–liquid interface. A relationship between the thermal properties of the material and the disturbance growth rate is obtained using the heat balance equation at the interface, including the effects of surface curvature on the equilibrium temperature. The heat balance equation is solved numerically and compared to the analytical solution obtained by neglecting the effects of surface curvature. The results show that the thermal gradients increase the growth rates of disturbances at the solid–liquid interface and that the effect of surface curvature results in a decrease in the disturbance growth rates. Further analysis shows that marginal stability occurs in both the longer wavelength and capillary regions.

Advanced Fluid Information. A Study on Three-Dimensional Disturbances in a Compressible Shear Layer.

Early stages of transition to turbulence of compressible shear layers were investigated in the viewpoint of a linear stability analysis. The primary stage of the transition is by which instability of a laminar flow (primary instability), in which disturbances develop into spanwise vortices and form a secondary flow. The secondary stage of the transition is by which instability of the secondary flow (secondary instability), in which disturbances develop into streamwise vortices. In the present study, the primary and secondary instability problems were formulated as eigenvalue problems and solved numerically using spectral methods. At first, the primary instability problems were solved. From the eigenvectors of the primary instability, we obtain density, velocity, temperature and pressure fields of the secondary flow. Using these quantities, the instability of the secondary flow was calculated. For the secondary instability, it is shown that (1) growth rates of disturbances are larger than those in the primary instability, (2) growth rates of disturbances have a peak when the ratio of wavelength of the primary and secondary instability is approximately 0.8 and (3) growth rates of disturbances are less affected by flow compressibility than those in the primary instability.

固体壁の影響を受ける超音速せん断層の不安定性に関する研究

Effects of solid wall on instability of a supersonic shear layer were investigated with the viewpoint of linear stability analysis. It is known that Mach waves become to be radiated from shear layers, as flow Mach number increases. In this case, because closed lower pressure regions of pressure perturbation, which grows to be core of vortex, cannot be formed in the shear layers, growth rates of disturbances are decreased extremely. In the present study, solid wall was set parallel to a supersonic shear layer. It is shown that, with the solid wall, outgoing Mach waves are reflected and turn back to the supersonic shear layer. The outgoing and incoming Mach waves form standing waves, and closed lower pressure regions are formed in the supersonic shear layer. It is shown that, with the solid wall, growth rates of disturbances are increased compared with that for without the solid wall.

Wrinkling of spherically expanding flames

The onset of instability and subsequent development of cells on spherically expanding flames is examined theoretically. The model used accounts for both hydronamic and diffusive-thermal effects and, in contrast to earlier theories, is valid for variable transport properties over a wide range of equivalence ratios. The analysis yields predictions for a number of flame properties, including growth rate of small disturbances, critical flame size for the instability onset, cell size beyond the threshold, and an estimate of the speed of the developing turbulent flame. It is shown that results using the more realistic temperature-dependent transport coefficients are more commensurate with experimental data concerning the critical conditions, that is, flame size or Peclet number, at the transition from one burning regime to another.

On the short-wave instability of natural convection boundary layers

This paper considers the stability of the one‐dimensional boundary layer generated by sudden heating of an infinite vertical wall. A quasi‐steady approximation is used to obtain the asymptotic form of the growth rate and phase speed of disturbances whose wavelength is comparable with the boundary‐layer width. Results for the inviscid modes governed by Rayleigh9s equation are obtained for several values of the Prandtl number and are compared with solutions of the full stability equations. As the wavelength increases, the phase speed of disturbances approaches the maximum flow speed of the boundary layer and a five‐tier structure extends across and outside the boundary layer. This intermediate regime, where viscous effects are important within a critical layer centred on the position of maximum flow speed, provides a link with an earlier long‐wave analysis of the problem.

Sound propagation in a planar channel in the presence of a two-layer flow

Expressions describing the field of a point source in a planar channel with admittance walls enclosing a two-layer nonuniform flow are obtained. The dispersion equation that determines the eigenvalues in a wide range of flow velocities in the layers (including supersonic velocities) is studied. The effect of the admittance of the channel walls on the growth rate of unstable disturbances is considered for different frequencies. It is established that the effect of the admittance of the channel walls on the growth rate of the instability waves decreases with increasing frequency and essentially depends on the type of admittance. It is shown that, in the presence of the admittance, new unstable disturbances are formed with a growth rate that can exceed that of the Kelvin—Helmholtz instability wave.

Damping of mesh-induced errors in Free-Lagrange simulations of Richtmyer-Meshkov instability

Mesh-induced errors at material interfaces are identified as a source of unphysical behaviour in Lagrangian numerical simulations of Richtmyer-Meshkov instability. The mesh geometry introduces interface perturbations with wavelengths of the same order as the mesh resolution. When a shock propagates through the interface, these perturbations can grow, severely contaminating the predicted interface development. Here an algorithm is presented which damps small-scale interface perturbations. A body force is applied at the interface which depends upon the disturbance amplitude and growth rate, and which resembles surface tension. Using this technique, qualitative improvements are obtained in Free-Lagrange simulations of single-mode Richtmyer-Meshkov instability. Growth rate behaviour and the evolution of the instability are seen to agree well with previously published results.

Linear analysis of the temporal instability of axisymmetrical non-Newtonian liquid jets

The temporal instability behavior of non-Newtonian liquid jets moving in an inviscid gaseous environment is investigated theoretically for axisymmetrical disturbances. The corresponding dispersion relation between the wave growth rate and the wave number is derived. The linearized stability analysis shows that a jet of a viscoelastic fluid exhibits a larger growth rate of axisymmetric disturbances than a jet of a Newtonian fluid with the same Ohnesorge number, indicating that non-Newtonian liquid jets are more unstable than their Newtonian counterparts. This is a well-known effect for small perturbations of the jet surface. For non-Newtonian liquid jets the instability behavior is influenced by the interaction of the liquid viscosity and elasticity effects, in which the liquid viscosity tends to dampen the instability, whereas the elasticity results in an enhancement of instability for small perturbations. The validity of the theoretical results for the growth rate spectra and breakup lengths of viscoelastic liquid jets is tested against experimental results from the literature. The comparisons confirm that the linearized theory fails to describe the nonlinear phenomena involved in viscoelastic jet breakup correctly, but it yields good results for the growth rate of disturbances in a regime of low jet Weber numbers and small deformations. The limits of validity of linear theories for viscoelastic jet instability are quantified, taking also into account the onset of non-axisymmetric deformations due to bending.

Nonparallel and nonlinear stability of supersonic jet flow

Linear and nonlinear evolution of disturbances in an axisymmetric, supersonic, low Reynolds number jet is studied using parabolized stability equations approach. Both axisymmetric and helical modes are considered and nonparallel effect is found to increase the disturbance growth rate, although there is very little effect on the wavenumber. Nonlinear interaction of the helical modes, which are the dominant instability modes of the jet, results in disturbance saturation, spectrum filling and large mean flow distortions. Similar to that for the supersonic boundary layer flow (Chang, C.-L., Malik, M.R., Oblique-mode breakdown and secondary instability in supersonic boundary layers. J. Fluid Mech. 273 (1994) 323–359), interaction of the helical modes induces streamwise vortices which cause significant mean flow distortion and growth of other harmonics. The computed evolution of disturbances is in reasonably good agreement with the experimental data of Morrison and McLaughlin (Morrison, G.L., McLaughlin, D.K., Instability process in low Reynolds number supersonic jets. AIAA J. 18(7) (1980) 793–800). The present marching scheme is able to compute evolution of supersonic disturbances without any numerical reflections and the induced far-field pressure disturbance is found to be almost in phase with the acoustic wave at ambient conditions.

Effects of confinement on the temporal instability of gas jets injected in viscous liquids

A temporal linear stability analysis of an inviscid incompressible gas jet injected into a coflowing viscous liquid confined within a tube was conducted to study the growth of small disturbances that lead to the breakup of the gas jet. For low-viscosity liquids, the confinement of the liquid jet did not affect the growth rates of the gas-jet disturbances. However, for high viscosity liquids, confining the liquid jet results in a significant increase in the disturbance growth rate and the range of unstable wave numbers. Also, for small ratios of the liquid jet diameter to gas jet diameter, the change in disturbance growth rate with Weber number is small. The effect of Weber number becomes significant only at large values of the diameter ratio.

Flame instability effects on the smallest wrinkling scale and burning velocity of high-pressure turbulent premixed flames

To explore the mechanism determining the smallest scale of flame wrinkles and turbulent burning velocity in a high-pressure environment, OH planar laser-induced fluorescence (PLIF) images of turbulent and non-turbulent premixed flames stabilized in a high-pressure chamber were analyzed for CH4/air and C3H8/air mixtures. Fractal analysis was performed to investigate the characteristics of the scale and complexity of the flame wrinkles. It was found that fractal dimension increased with increasing u′/SL for the whole pressure range in the experiments. The increase in the dimension was rapid at higher pressure. The fractal inner cutoff decreased with u′/SL and pressure. However, at high pressure, the variation of the fractal inner cutoff with u′/SL was very small, showing that the inner cutoff is almost constant over the wide range of u′SL. Comparison between the inner cutoff and various characteristic scales of turbulent flames was made. It was proved that, at high pressure, a significant correlation exists between the inner cutoff and the characteristic scale of flame instability, that is, Darrieus-Landau instability combined with diffusive thermal effects, where the characteristic instability scale is defined based on the wavenumber at the maximum growth rate of flame disturbances. Flame instability of the high-pressure flame without flow turbulence also was observed. The nominal buring velocity enlarged by the flame-area increase due to the flame instability and its variation with pressure was measured using a mean angle method for OH-PLIF images at pressure up to 3.0 MPa, and the pressure exponent was found to be 0.4. Based on these results, a concept to explain the pressure effects that appeared in the general correlation of the turbulent buring velocity obtained by Kobayashi et al. was proposed. That is, flame instability which produces small-scale wrinkles is significant in a high-pressure environment and overlaps with flame-area increase due to the turbulence, causing larger ST/SL.

Stability of Non-Homentropic, Inviscid, Compressible Shear Flows

For non-homentropic, inviscid, compressible shear flows, the equivalent of Squire's theorem is proved. It is shown that a shear free basic flow does not support subsonic modes. Further, it is shown that the instability region for subsonic disturbances is a semi-ellipse type region, which depends on the Mach number, wave number, and depth of the fluid layer. Under an approximation, two estimates for the growth rate of an unstable subsonic mode are obtained. For unbounded flows, a sufficient condition for stability to supersonic disturbances and an estimate for the growth rate of an unstable supersonic disturbance are given.