Nonlinear Evolution Of Disturbances Research Articles

Nonlinear evolution of disturbances in higher time-derivative theories

We investigate the evolution of localized initial value profiles when propagated in integrable versions of higher time-derivative theories. In contrast to the standard cases in nonlinear integrable systems, where these profiles evolve into a specific number of N-soliton solutions as dictated by the conservation laws, in the higher time-derivative theories the theoretical prediction is that the initial profiles can settle into either two-soliton solutions or into any number of N-soliton solutions. In the latter case this implies that the solutions exhibit oscillations that spread in time but remain finite. We confirm these analytical predictions by explicitly solving the associated Cauchy problem numerically with multiple initial profiles for various higher time-derivative versions of integrable modified Korteweg-de Vries equations. In the case with the theoretical possibility of a decay into two-soliton solutions, the emergence of underlying singularities may prevent the profiles from fully developing or may be accompanied by oscillatory, chargeless standing waves at the origin.

Linear and Nonlinear Disturbance Evolution on the Frustum of Hypersonic Ogive Cylinders

Aspects of transition mechanisms on a 14° sharp-nosed ogive cylinder at Mach 6 are elucidated by considering linear and nonlinear disturbance evolution of freestream stochastic and wave packet forcing at different locations downstream of the ogive-cylinder junction. The spectral response of the stochastic forcing displays favorable agreement with experimental observations from Air Force Research Laboratory on which the configuration is based. Intermittent wave packets are generated from small-amplitude continuous freestream pressure forcing. Linear wave packet evolution reveals that Mack modes are the dominant primary instabilities, followed by relatively weaker first-mode waves. The nonlinear wavepacket displays a three-legged wall pressure perturbation signature, which is traced to spanwise curvature effects, and fundamental resonance is the dominant secondary instability mechanism. At higher amplitudes of excitation, weak signatures of subharmonic and oblique resonance phenomena are identified. Downstream placement of the small-amplitude disturbance diminishes amplification of first-mode waves; however, with high-amplitude forcing, the location of actuation does not significantly vary the disturbance evolution signature or the secondary instability mechanisms.

Parabolized Stability Equations analysis of nonlinear interactions with forced eigenmodes to control subsonic jet instabilities

Nonlinear evolution of disturbances in an axisymmetric, high subsonic, high Reynolds number hot jet with forced eigenmodes is studied using the Parabolized Stability Equations (PSE) approach to understand how modes interact with one another. Both frequency and azimuthal harmonic interactions are analyzed by setting up one or two modes at higher initial amplitudes and various phases. While single mode excitation leads to harmonic growth and jet noise amplification, controlling the evolution of a specific mode has been made possible by forcing two modes (m1, n1), (m2, n2), such that the difference in azimuth and in frequency matches the desired “target” mode (m1 − m2, n1 − n2). A careful setup of the initial amplitudes and phases of the forced modes, defined as the “killer” modes, has allowed the minimizing of the initially dominant instability in the near pressure field, as well as its estimated radiated noise with a 15 dB loss. Although an increase of the overall sound pressure has been found in the range of azimuth and frequency analyzed, the present paper reveals the possibility to make the initially dominant instability ineffective acoustically using nonlinear interactions with forced eigenmodes.

Evolution of disturbance wavepackets in an oscillatory Stokes layer

AbstractNumerical simulation results are presented for the linear and nonlinear evolution of disturbances in a flat Stokes layer. The response to a spatially localised impulsive forcing is investigated and it is found that the spatial–temporal development of the flow displays an intriguing family-tree-like structure, which involves the birth of successive generations of distinct wavepacket components. It is shown that some features of this unexpected structure can be predicted using the results of a linear stability analysis based on Floquet theory.

Nonlinear disturbance evolution in a two-dimensional boundary-layer along an elastic plate and induced radiated sound

The interaction between a boundary-layer flow and an elastic plate is addressed by direct numerical simulation, taking into account the full coupling between the fluid flow and the flexible wall. The convectively unstable flow state is harmonically forced and two-dimensional nonlinearly saturated wavelike disturbances are computed along archetype-plates with respect to stiffness and natural frequencies. In the aim of determining the low-Mach-number radiated sound for the system, the simulation data are used to solve the Lighthill's equation in terms of a Green's function in the wavenumber-frequency space. Different degrees of fluid-structure coupling are implemented in the radiated sound model and the resulting acoustic pressure levels are compared. The sound radiation levels are shown to be increased in the presence of flexible walls with however significant differences in the radiated pressure levels for different coupling assumptions.

Nonlinear evolution of Mack modes in a hypersonic boundary layer

In hypersonic boundary layer flows the nonlinear disturbance evolution occurs relatively slowly over a very long length scale and has a profound effect on boundary layer transition. In the case of low-level freestream disturbances and negligible surface roughness, the transition is due to the modal growth of exponentially growing Mack modes that are destabilized by wall cooling. Cross-bicoherence measurements, derived from hot-wire data acquired in a quiet hypersonic tunnel, are used to identify and quantify phase-locked, quadratic sum and difference interactions involving the Mack modes. In the early stages of the nonlinear disturbance evolution, cross-bicoherence measurements indicate that the energy exchange between the Mack mode and the mean flow first occurs to broaden the sidebands; this is immediately followed by a sum interaction of the Mack mode to generate the first harmonic. In the next stages of the nonlinear disturbance evolution, there is a difference interaction of the first harmonic, which is also thought to contribute to the mean flow distortion. This difference interaction, in the latter stages, is also accompanied by a difference interaction between Mack mode and first harmonic, and a sum interaction, which forces the second harmonic. Analysis using the digital complex demodulation technique, shows that the low-frequency, phase-locked interaction that is identified in the cross bicoherence when the Mack mode and first harmonic have large amplitudes, arises due to the amplitude modulation of Mack mode and first harmonic.

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.

Nonlinear Evolution of Singular Disturbances to the Bickley Jet

We consider the nonlinear evolution of a disturbance to the Bickley jet, and the critical layer for the disturbance is located very close to the nose of the jet rather than at the inflection points. Using a nonlinear critical layer analysis, equations governing the evolution of the disturbance are derived and discussed. When the critical layer is located exactly at the nose of the jet, we find that the disturbance cannot exist on a linear basis, even with weak viscosity present, but that nonlinear effects inside the critical layer do permit the disturbance to exist if both modes are present. However, when the phase velocity of the disturbance is perturbed sufficiently away from unity, so that we have a pair of critical layers slightly above and below the nose rather than a single critical layer, we find that the waves can exist on a linear basis, and again we derive equations governing the nonlinear evolution of the disturbance.

Simulation of early transition in compressible rotating Couette flow: Subharmonic development

The nonlinear temporal evolution of disturbances in compressible flow between infinitely long, concentric cylinders is investigated through direct numerical simulations of the full, three-dimensional Navier–Stokes and energy equations. Counter-rotating cylinders separated by wide gaps are considered with supersonic velocities of the inner cylinder. Initially, the primary disturbance grows exponentially, in accordance with linear stability theory. As the disturbances evolve, higher harmonics and subharmonics are generated in a cascading order, eventually reaching a saturation state. Subsequent highly nonlinear stages of the evolution are governed by the interaction of the disturbance modes, particularly the axial subharmonics. Nonlinear evolution of the disturbance field is characterized by the formation of high-shear layers, extending from the inner cylinder toward the center of the gap in the form of jets similar to the ejection events in transitional and turbulent wall-bounded shear flows.

The Nonlinear Evolution of Disturbances to a Parabolic Jet

Abstract It has been shown that the linearized equations for disturbances to a parabolic jet on a β plane, with curvature Un0(y) such that the basic-state absolute vorticity gradient β − Un0(y) is zero, ultimately become inconsistent in the neighborhood of the jet axis and that nonlinear effects become important. Numerical solutions of the nonlinear long-time asymptotic form of the equations are presented. The numerical results show that the algebraic decay of the disturbances as t−1/2 predicted by the linear equations is inhibited by the nonlinear formation of coherent vortices new the jet axis. These lead to a disturbance amplitude that decays only through the action of weak numerical diffusion but is otherwise as t0. The linear theory is extended to the case when the basic-state absolute vorticity gradient is nonzero but weak. When the gradient is weak and negative the decay is modified and is ultimately as t−3/2. When the gradient is weak and positive, on the other hand, a discrete eigenmode is excite...

Nonlinear Meniscus Instability Analysis of Viscous Fluid Between Circular Parallel Plates By Boundary Element Method. 1st Report. A Comparison with Experimental Results.

In the present paper nonlinear evolution of disturbances at the meniscus between parallel circular plates is simulated by a boundary element method. To simulate accurately such a nonlinear phenomenon, the thickness of liquid remaining on the plates is obtained numerically against the low capillary number Ca<1.0, which is defined by the viscosity, velocity and surface tension. An integral equation for the boundary element analysis is newly proposed by taking into account the thickness of liquid remaining on the plates. The results in the simulation for the curvature, velocity and the tip position of disturbances growing at the fastest speed were compared with the experimental ones from various points of view, and it was shown that both agreed fairly well.

A Comparison of the Weakly Nonlinear Instability of Westerly and Easterly Jets in a Two-Layer Beta-Plane Model

Abstract The nonlinear evolution of disturbances that emanate from unstable westerly and easterly jet profiles is examined with a two-layer quasi-geostrophic β-plane channel model. Significant differences arise between the solutions for westerly and easterly jets. For unstable narrow (width less than the deformation radius) westerly jets, the disturbance undergoes a sequence of life cycles characterized by barotropic growth and barotropic decay. Unstable easterly jets also give rise to a series of life cycles in which the disturbance grows and decays in a combined baroclinic/barotropic manner. In each of these cases, the disturbance structure remains close to its unstable normal mode form throughout the life cycle. This contrasts with earlier results of Feldstein and Held who find that disturbances emanating from unstable wide (width greater than the deformation radius) westerly jets undergo a change in meridional structure, enhanced lateral radiation, and a single life cycle consisting of baroclinic grow...

Truncation effect in a Fourier series expansion on the nonlinear evolution of disturbances in free convection between vertical parallel plates

The effect of truncation in a Fourier series expansion on the nonlinear evolution of a disturbance has been investigated numerically in free convection between vertical parallel plates with different temperatures. Qualitatively different solutions were obtained, depending on the number of terms involved in the expansion. Transition from a stationary wave to an oscillatory traveling wave associated with the breakdown of the spatial symmetry was also obtained

Electrohydrodynamic Convective Instability in a Horizontal Fluid Layer with Temperature Gradient

Electrohydrodynamic convective instability in a horizontal fluid layer with temperature gradient is studied. For smaller electric forces, convection is set in as a static mode. As the electric force becomes larger, convection occurs as a time-periodic motion. Transition point from a static mode to an overstable mode is determined by the Prandtl number. Nonlinear evolution of disturbances near the onset of convection is also considered. Special attention is paid to ranges of parameters falling in the neighborhood of the multiple bifurcation point. For a certain range of the Prandtl numbers, the hysteresis between static and overstable modes can appear. Subcritical instabilities are also possible for small or high enough values of the Prandtl number.

Instability and nonlinear evolution of a density-driven coastal current with a surface front in a two-layer ocean

Abstract Coupled equations describing the nonlinear evolution of a disturbance in a two-layer coastal current with a surface front are derived by assuming that the alongshore scale of the disturbance is much longer than the current width, the lower layer is much thicker than the upper layer and the potential vorticity in the upper layer is zero. Linear stability and the characteristics of the cnoidal wave and soliton solutions governed by these equations are examined analytically, and the nonlinear evolution of disturbance is discussed using numerical methods. The main results are as follows. (1) The current is linearly stable when the current speed is greater than a certain critical value. (2) Even in such a situation, there is no stable nonlinear solution whose amplitude lies between certain critical values, a c and a′c, which are functions of the wavelength and the basic flow speed. (3) A solitary disturbance whose amplitude lies between a c and a′c grows with an eddy-pair-like structure in the lower layer. After its amplitude exceeds a′c, a soliton with a large amplitude separates from the eddy-pair-like structure and propagates by itself. (4) In the linearly unstable case, a wave-like perturbation with small amplitude grows. However, this unstable mode does not grow indefinitely but reaches a maximum amplitude, having exceeded the critical amplitude, a′c. Its amplitude then oscillates with time. (5) When several unstable waves have been given as the initial condition, the unstable waves interact nonlinearly with each other and solitary disturbances are produced. These solitary disturbances also repeat the growth and decay, and a turbulence-like phenomenon occurs.