Subharmonic Perturbations Research Articles

Finger competition dynamics in rotating Hele-Shaw cells

We report analytical results for the development of interfacial instabilities in rotating Hele-Shaw cells. We execute a mode-coupling approach to the problem and examine the morphological features of the fluid-fluid interface at the onset of nonlinear effects. The impact of normal stresses is accounted for through a modified pressure jump boundary condition. A differential equation describing the early nonlinear evolution of the interface is derived, being conveniently written in terms of three relevant dimensionless parameters: viscosity contrast A , surface tension B , and gap spacing b . We focus our study on the influence of these parameters on finger competition dynamics. It is deduced that the link between finger competition and A , B , and b can be revealed by a mechanism based on the enhanced growth of subharmonic perturbations. Our results show good agreement with existing experimental and numerical investigations of the problem both in low and high A<0 limits. In particular, it is found that the condition of vanishing A suppresses the dynamic competition between fingers, regardless of the value of B and b . Moreover, our study enables one to extract analytical information about the problem by exploring the whole range of allowed values for A , B , and b . Specifically, it is verified that pattern morphology is significantly modified when the viscosity contrast -1< or =A< or =1 varies: increasingly larger values of A>0 (A<0) lead to enhanced competition of outward (inward) fingers. Within this context the role of B and b in determining different finger competition behaviors is also discussed.

Symmetry and chaos in the complex Ginzburg–Landau equation. II. Translational symmetries

The complex Ginzburg–Landau (CGL) equation on a one-dimensional domain with periodic boundary conditions has a number of different symmetries, and solutions of the equation may or may not be fixed by the action of these symmetries. We investigate the stability of chaotic solutions that are spatially periodic with period L with respect to subharmonic perturbations that have a spatial period kL for some integer k>1. This is done by considering the isotypic decomposition of the space of solutions and finding the dominant Lyapunov exponent associated with each isotypic component. We find a region of parameter space in which there exist chaotic solutions with spatial period L and homogeneous Neumann boundary conditions that are stable with respect to perturbations of period 2 L. On varying the parameters it is possible to arrange for this solution to become unstable to perturbations of period 2 L while remaining chaotic, leading to a supercritical subharmonic blowout bifurcation. For a large number of parameter values checked, chaotic solutions with spatial period L were found to be unstable with respect to perturbations of period 3 L. We conclude that while periodic boundary conditions are often convenient mathematically, we would not expect to see chaotic, spatially periodic solutions forming starting with an arbitrary, non-periodic initial condition.

Numerical simulation of interactions between rigid rod-like polymers and coherent structures in a mixing layer

The Galerkin method is used to solve the diffusion equation of the distribution function for rigid rod-like polymers and the pseudo-spectral method is used to solve the momentum equation that includes the stress tensor determined by distribution function of polymers. The properties of the coherent structure in a two-dimensional time-developing viscoelastic mixing layer are given and the most probable orientations of the multibead rods as well as the stresses of polymers are obtained. The results indicate that adding polymers to the Newtonian fluid will result in a stronger vorticity diffusion, accompanied by weaker fundamental and subharmonic perturbations, less intense vortex roll-up of the mixing layer, lower oscillation frequency of fundamental energy, slower rotational motion of neighboring vortices and shorter life times of eddies. The results also show that the stretching of large eddies will have a great effect on the behavior of rod-like molecules, since the most probable orientations of rods seem to coincide with the local fluid stretching directions.

Radial fingering in a Hele-Shaw cell: a weakly nonlinear analysis

The Saffman-Taylor viscous fingering instability occurs when a less viscous fluid displaces a more viscous one between narrowly spaced parallel plates in a Hele-Shaw cell. Experiments in radial flow geometry form fan-like patterns, in which fingers of different lengths compete, spread and split. Our weakly nonlinear analysis of the instability predicts these phenomena, which are beyond the scope of linear stability theory. Finger competition arises through enhanced growth of sub-harmonic perturbations, while spreading and splitting occur through the growth of harmonic modes. Nonlinear mode-coupling enhances the growth of these specific perturbations with appropriate relative phases, as we demonstrate through a symmetry analysis of the mode coupling equations. We contrast mode coupling in radial flow with rectangular flow geometry.

Weakly Nonlinear Investigation of the Saffman–Taylor Problem in a Rectangular Hele–Shaw Cell

We analyze the Saffman–Taylor viscous fingering problem in rectangular geometry. We investigate the onset of nonlinear effects and the basic symmetries of the mode coupling equations, highlighting the link between interface asymmetry and viscosity contrast. Symmetry breaking occurs through enhanced growth of sub-harmonic perturbations. Our results explain the absence of finger tip-splitting in the early flow stages, and saturation of growth rates compared with the predictions of linear stability.

Oscillating Tidal Barriers and Random Waves

The interaction between sea waves and the oscillating gates designed to close the three inlets of Venice lagoon and to protect the city from the phenomenon of high waters is studied. Previous studies of this topic, which considered monochromatic waves, are extended to consider random waves characterized by a narrow-band wave spectrum, such that the incoming wave strongly resembles a monochromatic wave, slowly modulated by a random function of time and space. A linear stability analysis of basic synchronous response of the barrier to the incident wave shows that the excitation of subharmonic oscillations of the gates is possible. When the width 𝒮* of the incoming wave spectrum tends to zero, the results of previous studies which considered an oscillatory forcing of constant amplitude tends to be recovered. Unexpectedly, as the width of the spectrum increases, the unstable regions in the parameter space widen. However, a weakly nonlinear stability analysis shows that the regime configuration, i.e., the configuration attained by gate oscillations for large time, is characterized by values of the amplitude of the subharmonic oscillations that decrease as the width of the spectrum increases. Present results suggest that for large band spectra the subharmonic oscillations cannot be detected. These findings result from linear and nonlinear amplitude equations characterized by time-dependent coefficients that introduce new interesting features in the time development of the amplitude of subharmonic perturbations of the gate oscillations.

Global investigation of the influence of the phase of subharmonic excitation of a driven system

Adding a small subharmonic perturbation to the driving signal of a modulated laser allows one to control its dynamics. The influence of the phase of such a perturbation has been investigated both experimentally and by numerical simulations on a model of a fiber laser subjected to pump modulation. Depending on the operating conditions, the laser destabilizes through a period-doubling cascade or via quasiperiodicity. Suitably phased subharmonic perturbations can stabilize periodic orbits, shift the whole bifurcation diagram, and induce new crises. Experimental observations are in excellent agreement with the predictions from a simple model of the fiber laser.

Periodic solutions for a complex Hamiltonian system: New standing water-waves

The interaction of several (up to 39) two-dimensional, deep water, gravity, standing wave modes is studied to high order (up to 18) using a Hamiltonian formulation. Four new time-periodic solutions are found. The stability of periodic standing waves to superharmonic perturbations is studied. A new instability of the standing Stokes wave is found at moderate steepness (0.266). Two of the new waves are weakly unstable. Stability of the Stokes wave to subharmonic perturbation is also studied. The long time evolution of the perturbed waves is examined. The Hamiltonian method gives most accurate results. Recurrence of the unstable Stoke waves is revealed. The periodic solutions are useful for understanding the dynamical system of the standing wave modes.

Effect of elastic scattering on stochastic ion heating by electrostatic waves

The dynamics of an ion interacting with an electrostatic wave propagating perpendicularly to a static magnetic field in the presence of elastic ion-neutral scattering is investigated. Superharmonic (ω/ω0∈I) and subharmonic (ω/ω0=r/s≤1) perturbations are studied. Here, ω and ω0 are the wave and the ion cyclotron frequency, respectively, and r and s are relatively prime integers. Without elastic collisions, the ion is found to exhibit either quasi-regular or stochastic motion in phase space depending on the wave frequency, wave amplitude, and the initial position and velocity of the particle. Quasi-regular and stochastic behavior are found to be exclusive with respect to each other. In the presence of elastic collisions, the entire phase space becomes accessible to the ion irrespective of the initial conditions. The heating rate is generally found to depend on the ratio γm/ω (where γm is the momentum loss rate) and is enhanced compared to the collisionless case.

The effect of vortex pairing on particle dispersion and kinetic energy transfer in a two-phase turbulent shear layer

The transport of heavy, polydispersed particles and the inter-phase transfer of kinetic energy due to the viscous drag forces is measured experimentally in a turbulent shear layer. To study the effect of the large-scale vortex pairing event, the shear layer is forced simultaneously with a fundarmental and subharmonic perturbation. It is shown that vortex pairing plays a homogenizing role on the particulate field, but hte amount of homogenization is strongly dependent upon the particle's viscous relaxtion time, the eddy turnover time, as well as the time the particles interact with each scale prior to a pairing event. Thus, even though the smaller size particles become well-mixed across the large eddies, the larger sizes are still dispersed in an inhormogeneous fashion. It is also found that the kinetic energy transfer between the phases occurs inhomogeneously with energy being exchanged predominantly in a sublayer just outside the region of maximum turbulence intensity. The kinetic energy transfer is shown to exhibit notable positive and negative peaks located beneath the cores and stagnation points of the large-scale eddy field, and these peaks are shown to result from the irrotational velocity perturbations created by the vortices. This energy exchange mechanism remains a prominent process as long as the Stokes number of the particles relative to the vortices is of order unity.

Transient statistics in stabilizing periodic orbits

The statistics of chaotic and periodic transient time intervals preceding the stabilization of a given periodic orbit have been experimentally studied in a ${\mathrm{CO}}_{2}$ laser with modulated losses, subjected to a small subharmonic perturbation. As predicted by the theory, an exponential tail has been found in the probability distribution of chaotic transients. Furthermore, a fine periodic structure in the distributions of the periodic transients, resulting from the interaction of the control signal and the local structure of the chaotic attractor, has been revealed.

Unsteady models for the nonlinear evolution of the mixing layer.

We investigate several unsteady two-dimensional models of increasing complexity for the nonlinear evolution of the temporally growing mixing layer. In the simplest model, we use a subharmonically perturbed point vortex row which serves initially to identify certain key dynamical and kinematic aspects of the transition process, in which small scales in the flow are rapidly generated as a result of the vortex pairing process. A slightly more realistic vortex blob model is used at the next level. Finally, Navier-Stokes calculations are performed in order to identify which features are artifacts of the earlier models, and which ones are likely to be inherent to the physics. An important kinematic aspect of the transition, present in all of the models for small subharmonic perturbations, is a change in the topology of the streamline pattern when viewed in a rotating reference frame. This topological change contributes to the formation of spiral arm structures seen in other work and consequently to the intense production of small scales. For larger subharmonic perturbation amplitudes, this effect does not occur. We attempt to establish a connection between the aspects of our two-dimensional models and the findings of other authors in three-dimensional computations and experiments.

The accumulation and dispersion of heavy particles in forced two-dimensional mixing layers. I. The fundamental and subharmonic cases

This paper presents detailed computational results for the dispersion of heavy particles in transitional mixing layers forced at both the fundamental and subharmonic frequencies. The results confirm earlier observations of particle streaks forming in the braid region between successive vortices. A scaling argument based on the idealization of the spatially periodic mixing layer as a row of point vortices shows that the formation of these concentrated particle streaks proceeds with optimum efficiency for St≂1. It thereby provides a quantitative basis for experimental and numerical observations of preferential particle dispersion at Stokes numbers of order unity. Both the model and full simulation furthermore exhibit oscillatory particle motion, as well as the formation of two bands of high particle concentrations, for larger Stokes numbers. The particle dispersion as a function of time and the Stokes number is quantified by means of two different integral scales. These show that the number of dispersed particles does not reach a maximum for intermediate Stokes number. However, when the distance is weighted, optimum dispersion is observed for Stokes numbers around unity. By tracing the dispersed particles backwards in time, they are found to originate in inclined, narrow bands that initially stretch from the braid region into the seeded free stream. This suggests that particle dispersion can be optimized by phase coupling the injection device with the forcing signal for the continuous phase. In the presence of a subharmonic perturbation, enhanced particle dispersion is observed as a result of the motion of the vortices, whereby a larger part of the flow field is swept out.

The flow of suspensions in channels: Single files of drops

The flow of a periodic suspension of two-dimensional viscous drops between two parallel plane walls is considered, and a framework for performing dynamic simulations with suspensions of increasingly complex structure is developed. The computations are based on an improved implementation of the boundary integral method which is coined the method of interfacial dynamics. An elementary configuration is considered in detail in which the suspension is composed of a periodic array of suspended drops arranged on a single file, and the flow is driven by the relative motion of the two walls. The motion is studied as a function of the capillary number, the viscosities of the fluids, and the drop size. The transition from the small-drop behavior, where the effects of the walls are insignificant, to the large-drop behavior, where the drops occupy almost the whole of the clearance between the walls, is illustrated. It is found that, in all cases, the suspension exhibits a shear-thinning behavior and some type of elasticity. The single-file arrangement is unstable to subharmonic perturbations in which the drops are displaced in an alternating fashion across the centerline of the channel. The evolution of the perturbed array may lead to oscillatory motions including orbiting and bypassing, or to formation of a double file of nearly stationary drops. The effects of drop interactions on the effective properties of the suspension are discussed.

Circle maps with symmetry-breaking perturbations

Renormalization group methods are used to describe the transition to chaos that occurs in dissipative quasiperiodic systems in the presence of a small subharmonic perturbation. This is accomplished by studying the circle map with the symmetry θ → θ + 2 π/ n, and looking at crossover exponents for perturbations obeying the usual 2π symmetry. In order to interpret these exponents, the standard renormalization group analysis must be extended to include extra functions. This has been worked out in detail for the case n = 2.

Standing waves in deep water: Their stability and extreme form

A stable and accurate numerical method to calculate the motion of an interface between two fluids is used to calculate two-dimensional standing water waves. The general method calculates arbitrary time-dependent motion of an interface, possibly including interfacial tension and different density ratios between the fluids. Extremely steep standing waves are determined, significantly steeper than has been previously reported. The peak crest acceleration is used as the determining parameter rather than the wave steepness as the wave steepness is found to have a maximum short of the most extreme wave. Profiles with crest accelerations up to 98% of gravity are calculated (a sequence of raster images of this profile as it evolves in time over one period may be obtained upon application to the authors: e-mail gmercer@spam.ua.oz.au or aroberts@spam.ua.oz.au), and the shape of these extreme standing wave profiles are discussed. The stability of the standing waves is examined and growth rates of the unstable modes are calculated. It is found that all but very steep standing waves are generally stable to harmonic perturbations. However, standing waves are typically unstable to subharmonic perturbations via a sideband-type instability.

On the pairing process in an excited plane turbulent mixing layer

The flow field in a plane, turbulent mixing layer disturbed by a small oscillating flap was investigated. Three experiments were carried out: one in which the flap oscillated sinusoidally at a single frequency: a second in which the flap oscillated at two frequencies, a fundamental and a subharmonic, but the ensuing motion was dominated by the fundamental perturbation; and a third in which the amplitude of the subharmonic perturbation was increased until a distortion in the mean flow was noticeable. Two velocity components were measured at all phase angles relative to the subharmonic oscillation of the flap at densely spaced intervals. The data were used to map the phase-locked spanwise vorticity component and the phase-locked streakline patterns for the purpose of assessing the relevance of the latter to the understanding of the dynamical process involved.

On the role of subharmonic perturbations in the far wake

The possibility of an excitation of individual subharmonic perturbations in each of the shear layers forming the far wake is investigated numerically. Principal considerations allow for the existence of two equivalent subharmonic modes which by opposite routes can lead to a doubling of the wavelength in the wake. Since vortical disturbances in the far wake are amplified only convectively, the simultaneous existence of both modes in the flow field is possible, which could provide an explanation for the group structure observed experimentally in the far wake. These considerations also provide a logical explanation of the finding of a very regular vortex pairing process in forced wakes.Two-dimensional numerical simulations assuming incompressible flow and almost inviscid dynamics illustrate the opposite developments of regions dominated by the two different modes and also confirm the possibility of a resulting group structure. As an important result it is demonstrated that, if vortex pairing plays an important role in the growth of the far-wake structure, this does not have to be related to the excitation of the subharmonic peak in the frequency spectrum. Quite the contrary, it is to be expected that the subharmonic itself is of minor importance and that instead a small frequency and its multiples related to the group structure of the flow dominate the spectrum. In the light of these considerations measurements by Cimbala (1984) are discussed and frequency spectra recorded by him are analysed more closely. Various properties of these spectra seem to indicate that vortex pairing might be significant with respect to the evolution of the far-wake structure.

Numerical study of the dynamics of a forced shear layer

Vortex simulation is used to study a confined two-dimensional, two-stream, spatially developing shear layer, focusing on the effect of upstream forcing on the development of the vorticity field and the entrainment of irrotational fluid into the mixing zone. In the unforced case, the layer rolls up at the most amplified mode as determined from the linear theory, forming eddies that equilibrate to the neutrally stable mode by increasing the local thickness. Subharmonic perturbation returns the state to the most amplified frequency and pairing starts, repeating the sequence of events and leading to the observed self-similarity. In the forced case, eddy interactions follow several stages. First, the layer rolls up at the harmonic of the forcing frequency closest to the most amplified mode. A process of accelerated pairing follows yielding eddies in tune with the forcing frequency. Pairing among resonant eddies, which represent the neutrally stable mode, is disabled and the growth of the vorticity layer is impaired. Finally, the effect of forcing diminishes and pseudorandom pairing is resumed. Velocity fluctuations are strongly affected by forcing, and the sign of Reynolds stresses is reversed following pairing. Entrainment of passive particles is commensurate with the development of the vorticity layer.

Nonlinear oscillations of non-spherical cavitation bubbles in acoustic fields

The nonlinear stability of gas bubbles in acoustic fields is studied using a multiple-scale type of expansion. In particular the development of a subharmonic or a synchronous perturbation to the flow is investigated. It is shown when an equilibrium non-spherical shape oscillation of a bubble is stable. If the amplitude of the sound field is ε then it is shown that subharmonic perturbations of order ε½ can exist and be stable. Furthermore synchronous perturbations of order ε can exist and be stable. It is shown that synchronous perturbations, unlike the subharmonic case where the bifurcation is symmetric, bifurcate transcritically when the driving frequency is varied and also undergo secondary bifurcations. It is further shown that, in certain cases, the latter properties of the synchronous modes cause the flow to exhibit a hysteresis phenomenon when the driving frequency is varied.