Nonlinear Saturation Amplitude Research Articles

The Effect of Hydrogen on Nonlinear Flame Saturation

Abstract We investigate the effect of increasing levels of hydrogen enrichment on the nonlinear response and saturation of premixed bluff-body stabilized methane/hydrogen flames submitted to acoustic forcing. The thermal power is kept approximately constant to preserve the nozzle velocity while increasing the flame speed through hydrogen enrichment. The flame describing function (FDF) is measured for a fixed frequency and three hydrogen–methane blends ranging from 10% to 50% by power, corresponding to 25% to 75% by volume. We show that when the flame is forced at the same frequency at similar power and bulk velocities, increasing levels of hydrogen enrichment increase the saturation amplitude of the flame. To provide insight into the flame dynamics responsible for the change in the global nonlinear response and saturation amplitude, the flames were investigated using high-speed imaging in combination with OH planar laser-induced fluorescence (OH-PLIF) at a range of forcing amplitudes. At lower hydrogen concentrations, the flame is stabilized along the inner shear layer and saturation in the heat release rate (HRR) occurs at lower forcing amplitudes due to large-scale flame–vortex interactions causing flame annihilation as observed in several previous studies. At increased levels of hydrogen enrichment, distinctly different flame dynamics are observed. In these cases, the flame accelerates and propagates across to the outer shear layer, which acts to suppress large-scale flame annihilation during roll-up of both the inner and outer shear layers. This results in a coherent increase in flame surface area with forcing amplitudes significantly increasing the saturation amplitude of the flame. These results show that high levels of hydrogen increase the amplitude response to acoustic forcing leading to higher saturation amplitudes. This suggests that substituting natural gas with hydrogen in gas turbines increases the risk of much higher limit-cycle amplitudes if self-excited instabilities occur.

Effects of radial electric field on kinetic ballooning mode in toroidal plasma

Global gyrokinetic particle simulations show that the radial electric field (Er) shear can suppress the kinetic ballooning mode (KBM) in a toroidal plasma. The linear KBM growth rate reaches a maximum when the toroidal rotation induced by the ion diamagnetic shear is canceled by the E × B flow shear. High toroidal-mode-number (high-n) KBMs are more sensitive to the Er shear than low-n KBMs. Nonlinear simulations find that both the Er shear and a self-generated zonal flow can reduce the nonlinear KBM saturation level with smaller particle and ion heat transport. Meanwhile, the zonal flow can weaken the suppressing effects of the Er shear on KBM nonlinear saturation amplitude. The radial correlation length of the turbulence is reduced by the Er shear and the zonal flow.

Effects of compressibility and guiding field on the Kelvin–Helmholtz instability in weakly reversed magnetic shear configurations

The effects of compressibility and guiding field on the Kelvin–Helmholtz (KH) instability in a double resonant surface system are numerically investigated by using a compressible magnetohydrodynamics model. The linear growth rate and the nonlinear saturation amplitude are analyzed for different parameters. A transition from double tearing modes to the KH instability occurs depending on the compressibility and guiding field with a weak magnetic shear. These results may deepen our understanding of the properties of the KH instability with double resonant surfaces in both space and laboratory plasmas.

Weakly nonlinear incompressible Kelvin–Helmholtz instability in plane geometry

A weakly nonlinear (WN) theoretical model for the two-dimensional incompressible Kelvin–Helmholtz instability (KHI) is proposed. Its solution form is the complete expansion in real space. The transition from linear to nonlinear growth is analytically studied via third-order solutions of plane KHI initiated by a single-mode surface perturbation. The difference between the WN growth of the Rayleigh–Taylor instability (RTI) in plane geometry and the WN growth of the KHI in plane geometry is discussed. It is found that there are resonance solutions in the higher harmonics of KHI but not in RTI. The vertex of spikes and bubbles is deflected because of the shearing effect in KHI compared with that in RTI. The spike velocity increases with the Atwood number before a particular time and then inversely after that time. There is no such reversal in RTI. However, the bubble velocity with the Atwood number of KHI has the same evolution rule as that of RTI. In addition to the influence of initial perturbation on the nonlinear saturation amplitude, the influence of Atwood number on the nonlinear saturation amplitude of the fundamental mode is obtained. Finally, compared with the numerical simulations, at the same initial conditions, the analytical solutions up to the third harmonics are well consistent with the simulated datum in the linear and weakly nonlinear stages for a widely varied Atwood number.

Global gyrokinetic simulation of neoclassical ambipolar electric field and its effects on microturbulence in W7-X stellarator

Global neoclassical simulations of a model equilibrium of the W7-X stellarator find an ambipolar electric field with either an ion root or an electron root solution depending on the temperature ratio between electrons and ions. The ambipolar electric field is then used as an equilibrium field in the turbulence simulations of ion temperature gradient (ITG) instability. The shear of the ambipolar electric field has modest effects on the ITG linear instability, nonlinear saturation amplitude, and turbulent transport in the ion root case. However, in the electron root case, the ambipolar electric field significantly reduces the linear ITG growth rate, turbulence intensity, and radial correlation length, resulting in an ion heat conductivity comparable to the neoclassical transport level in the strong shear region.

A modal wave-packet model for the multi-mode Richtmyer–Meshkov instability

A model for multimode perturbations subject to the Richtmyer–Meshkov (RM) instability is presented and compared with simulations and experiments for conditions relevant to inertial confinement fusion. The model utilizes the single mode response to the RM impulse whereby its amplitude h(k, t) first grows with an initial velocity V0 ∝ kh(k, 0) that eventually decays in time as 1/kV0t. Both the growth and saturation stages are subject to nonlinearities since they depend explicitly on the initial amplitude. However, rather than using the individual mode amplitude h(k, t), nonlinearity is taken to occur when the root-mean-square amplitude hrms(k, t) of a wave-packet within wavenumbers k ± δk becomes comparable to 1/k. This is done because nearby sidebands can act in unison for an auto-correlation distance 1/δk beyond nonlinearity as observed in the beam-plasma instability. Thus, the nonlinear saturation amplitude for each mode is reduced from the usual 1/k by a phase space factor that depends on the physical dimensionality, as in the Haan model for the Rayleigh–Taylor instability. In addition, for RM, the average value of khrms for the initial spectrum is used to calculate a nonlinear factor FNL that reduces V0, as observed for single modes. For broadband perturbations, the model describes self-similar growth ∝tθ as successively longer wavelength modes reach saturation. The growing and saturated modes must be discerned because only the former promote θ and are enhanced by reshock and spherical convergence. All of these flows are described here by the model in good agreement with simulations and experiments.

Interface Width Effect on the Weakly Nonlinear Rayleigh–Taylor Instability in Spherical GeometrySupported by the National Natural Science Foundation of China (Grant Nos. 11575033, 11675026, and 11975053).

Interface width effect on the spherical Rayleigh–Taylor instability in the weakly nonlinear regime is studied by numerical simulations. For Legendre perturbation mode Pn with wave number kn and interface half-width L, the commonly adopted empirical linear growth rate formula is found to be sufficient in spherical geometry. At the weakly nonlinear stage, the interface width affects the mode coupling processes. The development of the mode P2n is substantially influenced by the interface width. Moreover, the nonlinear saturation amplitude increases with the interface width.

Simulation of the Weakly Nonlinear Rayleigh-Taylor Instability in Spherical Geometry**Supported by the National Natural Science Foundation of China under Grant Nos. 11575033, 11675026, and 11975053.

The Rayleigh–Taylor instability at the weakly nonlinear (WN) stage in spherical geometry is studied by numerical simulation. The mode coupling processes are revealed. The results are consistent with the WN model based on parameter expansion, while higher order effects are found to be non-negligible. For Legendre mode perturbation Pn(cosθ), the nonlinear saturation amplitude (NSA) of the fundamental mode decreases with the mode number n. When n is large, the spherical NSA is lower than the corresponding planar one. However, for large n, the planar NSA can be recovered by applying Fourier transformation to the bubble/spike near the equator and calculating the NSA of the converted trigonometric harmonic.

Coupling between velocity and interface perturbations in cylindrical Rayleigh–Taylor instability**Project supported by the National Natural Science Foundation of China (Grant Nos. 11275031, 11475034, 11575033, and 11274026) and the National Basic Research Program of China (Grant No. 2013CB834100).

Rayleigh–Taylor instability (RTI) in cylindrical geometry initiated by velocity and interface perturbations is investigated analytically through a third-order weakly nonlinear (WN) model. When the initial velocity perturbation is comparable to the interface perturbation, the coupling between them plays a significant role. The difference between the RTI growth initiated only by a velocity perturbation and that only by an interface perturbation in the WN stage is negligibly small. The effects of the mode number on the first three harmonics are discussed respectively. The low-mode number perturbation leads to large amplitudes of RTI growth. The Atwood number and initial perturbation dependencies of the nonlinear saturation amplitude of the fundamental mode are analyzed clearly. When the mode number of the perturbation is large enough, the WN results in planar geometry are recovered.

Limits on magnetic field amplification from the r -mode instability

At second order in perturbation theory, the unstable r-mode of a rotating star includes growing differential rotation whose form and growth rate are determined by gravitational-radiation reaction. With no magnetic field, the angular velocity of a fluid element grows exponentially until the mode reaches its nonlinear saturation amplitude and remains nonzero after saturation. With a background magnetic field, the differential rotation winds up and amplifies the field, and previous work where large mode amplitudes were considered suggests that the amplification may damp out the instability. A background magnetic field, however, turns the saturated time-independent perturbations corresponding to adding differential rotation into perturbations whose characteristic frequencies are of order the Alfv\'en frequency. As found in previous studies, we argue that magnetic- field growth is sharply limited by the saturation amplitude of an unstable mode. In contrast to previous work, however, we show that if the amplitude is small, i.e., of order 10^(-4), then the limit on the magnetic-field growth is stringent enough to prevent the loss of energy to the magnetic field from damping or significantly altering an unstable r-mode in nascent neutron stars with normal interiors and in cold stars whose interiors are type II superconductors. We show this result first for a toy model, and we then obtain an analogous upper limit on magnetic field growth using a more realistic model of a rotating neutron star. Our analysis depends on the assumption that there are no marginally unstable perturbations, and this may not hold when differential rotation leads to a magnetorotational instability.

Nonlinear Saturation Amplitude in Classical Planar Richtmyer–Meshkov Instability**Supported by the National Natural Science Foundation of China under Grant Nos. 11472278 and 11372330, the Scientific Research Foundation of Education Department of Sichuan Province under Grant No. 15ZA0296, the Scientific Research Foundation of Mianyang Normal University under Grant Nos. QD2014A009 and 2014A02, and the National High-Tech ICF Committee

The classical planar Richtmyer–Meshkov instability (RMI) at a fluid interface supported by a constant pressure is investigated by a formal perturbation expansion up to the third order, and then according to definition of nonlinear saturation amplitude (NSA) in Rayleigh–Taylor instability (RTI), the NSA in planar RMI is obtained explicitly. It is found that the NSA in planar RMI is affected by the initial perturbation wavelength and the initial amplitude of the interface, while the effect of the initial amplitude of the interface on the NSA is less than that of the initial perturbation wavelength. Without marginal influence of the initial amplitude, the NSA increases linearly with wavelength. The NSA normalized by the wavelength in planar RMI is about 0.11, larger than that corresponding to RTI.

Effects of variations in electron thermal velocity on the whistler anisotropy instability: Particle-in-cell simulations

This paper investigates how the physics of the whistler anisotropy instability (WAI) is affected by variations in the electron thermal velocity vte, referred to here in terms of the ratio v̂te=vte/c, where c is the speed of light. The WAI is driven by the electron condition RT>1, where RT=Te⊥/Te∥ is the temperature anisotropy ratio and ⊥/∥ signify directions perpendicular/parallel to the background magnetic field B0. While a typical value of v̂te in the solar wind is ∼0.005, electromagnetic (EM) particle-in-cell (PIC) simulations often use a value near 0.1 in order to maximize the computational time step. In this study, a two-dimensional (2D) Darwin particle-in-cell (DPIC) code, MDPIC2, is used. The time step in the DPIC model is not affected by the choice of v̂te, making DPIC suited for this study. A series of simulations are carried out under the condition that the electron βe is held fixed, while v̂te is varied over the range 0.1≥v̂te≥0.025. The results show that, with βe held fixed, the linear dispersion properties and the nonlinear saturation amplitude and pitch angle scattering rates associated with the WAI are insensitive to the value of v̂te. A supplementary investigation is conducted which characterizes how the WAI model is affected at various values of v̂te by noise associated with the limited number of particles in a typical PIC simulation. It is found that the evolution of the WAI is more strongly influenced by electrostatic noise as v̂te is decreased. The electrostatic noise level is inversely proportional to the number of particles per computational cell (Nc); this implies that the number of particles required to remove nonphysical effects from the PIC simulation increases as v̂te decreases. It is concluded that PIC simulations of this instability which use an artificially large value of v̂te accurately reproduce the response of a cooler plasma as long as a realistic value of βe is used. Moreover, for a fixed number of particles, the quantitative accuracy of the model improves with increasing v̂te.

Structure of wave-particle resonances and Alfvén mode saturation

The dynamics of beta-induced Alfvén eigenmodes driven by anisotropic co-passing or counter-passing fast ions, in a low-shear magnetic equilibrium, is investigated by self-consistent hybrid MHD-particle simulations with the XHMGC code. Though the modes exhibit similar structure and frequency in both cases and the linear growth rate is 10% larger for counter-passing ions than for co-passing ions, the nonlinear saturation amplitude is much larger in co-passing case. Moreover, different scalings for the saturation amplitude with increasing growth rates are observed in the two cases. It is shown that these differences are caused by the different radial dependence of resonance frequencies of co-passing and counter-passing fast ions: flat in the former case, steep in the latter case, so that the resonance width is, respectively, larger (in the former case) or smaller (in the latter case) than the mode width.

Nonlinear analysis of q ≳ 1 non-resonant fishbone modes in tokamak plasmas with weakly reversed magnetic shear

The nonlinear properties of q ≳ 1 non-resonant fishbone (NRF) instabilities excited by energetic ions are analyzed theoretically for weakly reversed shear in this work. Nonlinear radial displacement of the NRF mode ξ0 is derived by the method of matched asymptotic expansions. It is found that ξ0 depends on the energetic ion beta in a power law of ∼βhα with α=1/3, α=1/7, and α≪1, corresponding to a finite qs″, qs″=0, and an extremely flattened q-profile, respectively. The scaling dependence of ξ0 on the linear growth rate ∼γ1/4 is also different from that of ∼γ in a conventional positive shear configuration. The scaling suggests that ξ0 weakly depends on βh and γ when the q-profile is flattened. Nonlinear saturation amplitude of the mode for an ITER-like plasma is also estimated by numerical analysis.

Effects of initial radius of the interface and Atwood number on nonlinear saturation amplitudes in cylindrical Rayleigh-Taylor instability

Nonlinear saturation amplitudes (NSAs) of the first two harmonics in classical Rayleigh-Taylor instability (RTI) in cylindrical geometry for arbitrary Atwood numbers have been analytically investigated considering nonlinear corrections up to the fourth-order. The NSA of the fundamental mode is defined as the linear (purely exponential) growth amplitude of the fundamental mode at the saturation time when the growth of the fundamental mode (first harmonic) is reduced by 10% in comparison to its corresponding linear growth, and the NSA of the second harmonic can be obtained in the same way. The analytic results indicate that the effects of the initial radius of the interface (r0) and the Atwood number (A) play an important role in the NSAs of the first two harmonics in cylindrical RTI. On the one hand, the NSA of the fundamental mode first increases slightly and then decreases quickly with increasing A. For given A, the smaller the r0/λ (with λ perturbation wavelength) is, the larger the NSA of the fundamental mode is. When r0/λ is large enough (r0≫λ), the NSA of the fundamental mode is reduced to the prediction of previous literatures within the framework of third-order perturbation theory [J. W. Jacobs and I. Catton, J. Fluid Mech. 187, 329 (1988); S. W. Haan, Phys. Fluids B 3, 2349 (1991)]. On the other hand, the NSA of the second harmonic first decreases quickly with increasing A, reaching a minimum, and then increases slowly. Furthermore, the r0 can reduce the NSA of the second harmonic for arbitrary A at r0≲2λ while increase it for A ≲ 0.6 at r0≳2λ. Thus, it should be included in applications where the NSA has a role, such as inertial confinement fusion ignition target design.

Nonlinear saturation amplitude of cylindrical Rayleigh—Taylor instability

The nonlinear saturation amplitude (NSA) of the fundamental mode in the classical Rayleigh—Taylor instability with a cylindrical geometry for an arbitrary Atwood number is analytically investigated by considering the nonlinear corrections up to the third order. The analytic results indicate that the effects of the initial radius of the interface (r0) and the Atwood number (A) play an important role in the NSA of the fundamental mode. The NSA of the fundamental mode first increases gently and then decreases quickly with increasing A. For a given A, the smaller the r0/λ (λ is the perturbation wavelength), the larger the NSA of the fundamental mode. When r0/λ is large enough (r0 ≫ λ), the NSA of the fundamental mode is reduced to the prediction in the previous literatures within the framework of the third-order perturbation theory.

On the laminar–turbulent transition of the rotating-disk flow: the role of absolute instability

AbstractThis paper describes a detailed experimental study using hot-wire anemometry of the laminar–turbulent transition region of a rotating-disk boundary-layer flow without any imposed excitation of the boundary layer. The measured data are separated into stationary and unsteady disturbance fields in order to elaborate on the roles that the stationary and the travelling modes have in the transition process. We show the onset of nonlinearity consistently at Reynolds numbers, $R$, of $\sim $510, i.e. at the onset of Lingwood’s (J. Fluid Mech., vol. 299, 1995, pp. 17–33) local absolute instability, and the growth of stationary vortices saturates at a Reynolds number of $\sim $550. The nonlinear saturation and subsequent turbulent breakdown of individual stationary vortices independently of their amplitudes, which vary azimuthally, seem to be determined by well-defined Reynolds numbers. We identify unstable travelling disturbances in our power spectra, which continue to grow, saturating at around $R=585$, whereupon turbulent breakdown of the boundary layer ensues. The nonlinear saturation amplitude of the total disturbance field is approximately constant for all considered cases, i.e. different rotation rates and edge Reynolds numbers. We also identify a travelling secondary instability. Our results suggest that it is the travelling disturbances that are fundamentally important to the transition to turbulence for a clean disk, rather than the stationary vortices. Here, the results appear to show a primary nonlinear steep-fronted (travelling) global mode at the boundary between the local convectively and absolutely unstable regions, which develops nonlinearly interacting with the stationary vortices and which saturates and is unstable to a secondary instability. This leads to a rapid transition to turbulence outward of the primary front from approximately $R=565$ to 590 and to a fully turbulent boundary layer above 650.

Experimental investigation on the influence of the dopant ratio on ablative Rayleigh-Taylor instability growth

Ablative Rayleigh-Taylor growth was measured with single-mode modulated planar CH foil with different ratio of Br dopant at Shenguang Ⅱ laser facility. Results show that CH (6% Br) sample first enters the nonlinear regime and has the largest perturbation amplitude of second harmonic. The reason is that the density gradient effects can suppress the generation of the second harmonic, the more the Br is doped, the smaller the density gradient scale length can be achieved. The density gradient effects also suppress the feedback of third-order harmonic to the fundamental mode, which induces the nonlinear saturation amplitude to exceed 0.1λ, as the classical prediction shows.

Weakly nonlinear incompressible Rayleigh-Taylor instability growth at cylindrically convergent interfaces

A weakly nonlinear (WN) model has been developed for the incompressible Rayleigh-Taylor instability (RTI) in cylindrical geometry. The transition from linear to nonlinear growth is analytically investigated via a third-order solutions for the cylindrical RTI initiated by a single-mode velocity perturbation. The third-order solutions can depict the early stage of the interface asymmetry due to the bubble-spike formation, as well as the saturation of the linear (exponential) growth of the fundamental mode. The WN results in planar RTI [Wang et al., Phys. Plasmas 19, 112706 (2012)] are recovered in the limit of high-mode number perturbations. The difference between the WN growth of the RTI in cylindrical geometry and in planar geometry is discussed. It is found that the interface of the inward (outward) development spike/bubble is extruded (stretched) by the additional inertial force in cylindrical geometry compared with that in planar geometry. For interfaces with small density ratios, the inward growth bubble can grow fast than the outward growth spike in cylindrical RTI. Moreover, a reduced formula is proposed to describe the WN growth of the RTI in cylindrical geometry with an acceptable precision, especially for small-amplitude perturbations. Using the reduced formula, the nonlinear saturation amplitude of the fundamental mode and the phases of the Fourier harmonics are studied. Thus, it should be included in applications where converging geometry effects play an important role, such as the supernova explosions and inertial confinement fusion implosions.

Nonlinear Rayleigh-Taylor instability of rotating inviscid fluids

It is demonstrated theoretically that the nonlinear stage of the Rayleigh-Taylor instability can be retarded at arbitrary Atwood numbers in a rotating system with the axis of rotation normal to the acceleration of the interface between two uniform inviscid fluids. The Coriolis force provides an effective restoring force on the perturbed interface, and the uniform rotation will always decrease the nonlinear saturation amplitude of the interface at any disturbance wavelength.