Broadband Perturbations Research Articles

Freeze out of multi-mode Richtmyer–Meshkov instability using particles

Richtmyer–Meshkov instability (RMI) occurs when a shock wave traverses an interface separated by two fluids with different densities. Achieving “freeze out” (i.e., “killing” of RMI), a critical objective in RMI research for engineering applications, remains an open problem in the context of multi-mode RMI. Here, we introduce particles into the flow field to achieve freeze out, which is attributed to the momentum non-equilibrium effect inherent in the gas–particle phases. This effect facilitates the transfer of momentum and energy from the fluid to the particles, thereby mitigating the amplification of initial perturbations within the mixing zone. We developed a one-dimensional model to predict the velocities of the mixing zone boundaries in multiphase RMI. The growth of RMI was suppressed by controlling the velocities of the mixing zone boundaries through particle effects. A non-dimensional freeze out criterion was derived, incorporating the gas–particle coupling along with the particle volume fraction effect. The condition for freezing a multi-mode RMI was specially designed to estimate the required particle volume fraction to achieve the freeze out. A series of simulations were conducted using a well-verified compressible multiphase particle-in-cell method to validate the realization of freeze out. Further analysis reveals that the designed condition exhibits applicability across a spectrum of multi-mode perturbations, including both broadband and narrowband perturbations, as well as various initial Mach numbers.

Simulations and models for the Richtmyer–Meshkov instability with broadband perturbations

The Richtmyer–Meshkov instability (RMI) is shock driven and affects many phenomena from inertial fusion to supernova explosions. The behavior of single-modes in the RMI has been studied extensively but less is known with the broadband perturbations that occur in applications. Here, we describe extensive numerical simulations and modeling of the RMI with broadband perturbations with an initial power spectrum of the form P ∝ km, where k is the wavenumber. The hydrodynamic condition is the same as in the θ-Group Collaboration [Thornber et al., “Late-time growth rate, mixing, and anisotropy in the multimode narrowband Richtmyer–Meshkov instability: The θ-group collaboration,” Phys. Fluids 29, 105107 (2017)] with a Mach ∼1.86 shock and gamma-law = 5/3 fluids with Atwood number A = −0.5. The bubble amplitude hB is found to grow in two stages. Initially, hB undergoes a phase-inversion and grows linearly in time (t) at a rate consistent with a linear theory. Asymptotically, hB grows as a power law ∼tθ when k|hB| > O(1), where k is the average wavenumber for the initial spectrum. The RMI behavior in simulations and modeling agree over a wide range of exponent m, spectral width, initial amplitude, and time. The agreement is quantified objectively using statistical analysis.

Mechanism and Solution of Inaccurate Perturbation Output of DC-AC Type IMU

Impedance measurement unit (IMU) with injection of broadband perturbations is widely used to obtain grid and converter impedance. The accuracy of measured impedance and the power quality of the measured system rely on the amplitude control of the injected perturbations. However, the accuracy of output perturbation of IMU is lack of attention especially around the fundamental frequency of the measured system. In this paper, a three-phase DC-AC type IMU is introduced and the mechanism and solution of inaccurate perturbation output of the IMU is study. First, the DC-AC type IMU with PR current controllers, DC voltage controller and phase locked loop (PLL) is introduced. Next, the mechanism of inaccurate perturbation output is analysis, which is caused by the introduction of DC voltage ripple. Then, improved DC voltage control solutions with notch filter or delay filter are proposed to achieve the accurate control of single-frequency and multi-frequency perturbation current respectively. Finally, control hardware-in-the-loop experiment results are given to demonstrate and verify the effectiveness and feasibility of the proposed method.

Numerical simulation of an idealised Richtmyer–Meshkov instability shock tube experiment

The effects of initial conditions on the evolution of the Richtmyer–Meshkov instability (RMI) at early to intermediate times are analysed, using numerical simulations of an idealised version of recent shock tube experiments performed at the University of Arizona (Sewell et al., J. Fluid Mech., vol. 917, 2021, A41). The experimental results are bracketed by performing both implicit large-eddy simulations of the high-Reynolds-number limit as well as direct numerical simulations (DNS) at Reynolds numbers lower than those observed in the experiments. Various measures of the mixing layer width $h$ , known to scale as ${\sim }t^\theta$ at late time, based on both the plane-averaged turbulent kinetic energy and volume fraction profiles are used to explore the effects of initial conditions on $\theta$ and are compared with the experimental results. The decay rate $n$ of the total fluctuating kinetic energy is also used to estimate $\theta$ based on a relationship that assumes self-similar growth of the mixing layer. The estimates for $\theta$ range between 0.44 and 0.52 for each of the broadband perturbations considered and are in good agreement with the experimental results. Decomposing the mixing layer width into separate bubble and spike heights $h_b$ and $h_s$ shows that, while the bubbles and spikes initially grow at different rates, their growth rates $\theta _b$ and $\theta _s$ have equalised by the end of the simulations. Overall, the results demonstrate important differences between broadband and narrowband surface perturbations, as well as persistent effects of finite bandwidth on the growth rate of mixing layers evolving from broadband perturbations. Good agreement is obtained with the experiments for the different quantities considered; however, the results also show that care must be taken when using measurements based on the velocity field to infer properties of the concentration field.

Input Spectrum Design for Identification of a Thermostat System

This paper considers the shaping of the amplitude spectra of perturbation signals for the identification of a thermostat system. The current approach in control engineering practice utilizes flat spectrum signals, which may not result in the highest possible accuracy. This research aims to investigate the effectiveness of optimal signals with amplitude spectra designed using two state-of-the-art software approaches, namely the model-based optimal signal excitation 2 (MOOSE2) design and the optimal excitation (optexcit) design, in improving estimation accuracy. Such a comparison on a real system is currently lacking. In particular, there exists a research gap on how the combined choice of signal and model structure affects performance measures. In this research, two model structures are used, which are the autoregressive with exogenous input (ARX) and the output error (OE) model structures. Four performance measures are compared, namely the determinant of the covariance matrix of the parameter estimates and the minimum error, mean error and maximum error in the frequency response. Results show that the optimal signals are effective in reducing the determinant of the covariance matrix and the maximum error in the frequency response for the thermostat system, when applied in combination with the ARX model structure. The flat spectrum signal remains very useful as a general broadband perturbation signal as it provides a good overall fit of the frequency response. The findings from this work highlight the benefits of applying optimal signals especially if the identification results are to be used for control, since these signals improve key performance measures which have direct implications on controller design.

Coronal Quasi-periodic Fast-mode Propagating Wave Trains

QFP wave trains in the corona have been studied intensively in the past decade, thanks to the full-disk, high spatiotemporal resolution, and wide-temperature coverage observations taken by the SDO/AIA. In AIA observations, QFP wave trains are seen to consist of multiple coherent and concentric wavefronts emanating successively near the epicenter of the accompanying flares; they propagate outwardly either along or across coronal loops at fast-mode magnetosonic speeds from several hundred to more than 2000 km/s, and their periods are in the range of tens of seconds to several minutes. Based on the distinct different properties of QFP wave trains, they might be divided into two distinct categories including narrow and broad ones. For most QFP wave trains, some of their periods are similar to those of quasi-periodic pulsations (QPPs) in the accompanying flares, indicating that they are probably different manifestations of the same physical process. Currently, candidate generation mechanisms for QFP wave trains include two main categories: pulsed energy excitation mechanism in association with magnetic reconnection and dispersion evolution mechanism related to the dispersive evolution of impulsively generated broadband perturbations. In addition, the generation of some QFP wave trains might be driven by the leakage of three and five minute oscillations from the lower atmosphere. As one of the new discoveries of SDO, QFP wave trains provide a new tool for coronal seismology to probe the corona parameters, and they are also useful for diagnosing the generation of QPPs, flare processes including energy release and particle accelerations. This review aims to summarize the main observational and theoretical results of the spatially-resolved QFP wave trains in extreme ultraviolet observations, and states briefly a number of questions that deserve further investigations.

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.

Dependence of Enstrophy Transport and Mixed Mass on Dimensionality and Initial Conditions in the Richtmyer–Meshkov Instability Induced Flows1

AbstractThis paper presents a comparative study of the enstrophy budget and mixed mass between two- and three-dimensional flows induced by Richtmyer–Meshkov instability (RMI). Specifically, the individual contributions to the enstrophy budget due to the production from baroclinicity and from vortex stretching (which vanishes in two-dimensional (2D) flow) are delineated. This is enabled by a set of two- and three-dimensional computations at Atwood 0.5 having both narrow- and broad-band perturbations. A further three-dimensional (3D) computation is conducted at Atwood 0.9 using an identical narrowband perturbation to the Atwood 0.5 case to examine the sensitivity to density ratio. The mixed mass is also considered with the goal to obtain insight on how faithfully a simplified calculation performed in two dimensions can capture the mixed mass for an inertial confinement fusion (ICF) or other practical application. It is shown that the late time power law decay of variable density enstrophy is substantially different in two and three dimensions for the narrowband initial perturbation. The baroclinic production term is negligible in three dimensions (aside from the initial shock interaction), as vortex stretching is larger by two orders of magnitude. The lack of vortex stretching considerably reduces the decay rate in both narrowband and broadband perturbations in two dimensions. In terms of mixed mass, the lack of vortex stretching reduces the mixed mass in two dimensions compared to three in all cases. In the broadband cases, the spectral bandwidth in the 2D case is wider; hence, there is a longer time period of sustained linear growth which reduces the normalized mixed mass further.

The influence of initial perturbation power spectra on the growth of a turbulent mixing layer induced by Richtmyer–Meshkov instability

This paper investigates the influence of different broadband perturbations on the evolution of a Richtmyer–Meshkov turbulent mixing layer initiated by a Mach 1.84 shock traversing a perturbed interface separating gases with a density ratio of 3:1. Both the bandwidth of modes in the interface perturbation, as well as their relative amplitudes, are varied in a series of carefully designed numerical simulations at grid resolutions up to 3.2×109 cells. Three different perturbations are considered, characterised by a power spectrum of the form P(k)∝km where m=−1, −2 and −3. The growth of the mixing layer is shown to strongly depend on the initial conditions, with the growth rate exponent θ found to be 0.5, 0.63 and 0.75 for each value of m at the highest grid resolution. The asymptotic values of the molecular mixing fraction Θ are also shown to vary significantly with m; at the latest time considered Θ is 0.56, 0.39 and 0.20 respectively. Turbulent kinetic energy (TKE) is also analysed in both the temporal and spectral domains. The temporal decay rate of TKE is found not to match the predicted value of n=2−3θ, which is shown to be due to a time-varying normalised dissipation rate Cϵ. In spectral space, the data follow the theoretical scaling of k(m+2)∕2 at low wavenumbers and tend towards k−3∕2 and k−5∕3 scalings at high wavenumbers for the spectra of transverse and normal velocity components respectively. The results represent a significant extension of previous work on the Richtmyer–Meshkov instability evolving from broadband initial perturbations and provide useful benchmarks for future research.

Amplitude Design of Perturbation Signal in Frequency-Domain Analysis of Grid-Connected Systems

The rise of renewable electricity production has driven the power grid to a remarkable transformation, where a large share of the electricity production is integrated to the grid through power-electronic inverters. The inverters have fast internal dynamics and no inherent inertia, which makes the power grid prone to stability issues. The stability analysis to ensure system robustness can be performed based on the impedance ratio of the inverter and power grid. The grid impedance is often an unknown parameter, and methods for grid impedance measurements are required. Past studies have presented a number of measurement methods based on a broadband perturbation, such as pseudo-random binary sequence (PRBS), and Fourier techniques for obtaining the grid impedance. However, only a little attention has been paid to the injection-amplitude design, and most often, the amplitude has been selected based on trial and error. This work presents an algorithm based on the total harmonic distortion (THD) levels of grid currents and voltages for choosing a suitable perturbation amplitude. The proposed method makes it possible to fully automate the stability analysis of a grid-connected system. Experimental results based on a three-phase grid-connected inverter are presented and used to demonstrate the effectiveness of the proposed method.

Three-dimensional simulations of turbulent mixing in spherical implosions

High-resolution large-eddy simulations of turbulent mixing at the inner surface of a dense shell which undergoes forced compression by a spherically imploding shock wave are presented. Perturbations on the inner surface grow as a result of Richtmyer-Meshkov and Rayleigh-Taylor instabilities and effects related to geometric convergence and compressibility. Three different cases with different initial surface perturbations, one broadband and two narrowband, are considered. The perturbation power spectrum is related to the mode number via Pℓ ∝ ℓn, where the case with broadband perturbations has n = −2, and modes in the range ℓ = 6–200. The narrowband perturbations have n = 0 and modes in the range ℓ = 50–100 and ℓ = 100–200. The simulations are carried out in spherical coordinates using the PLUTO hydrodynamics code. Results on the mix layer width, molecular mix, and turbulent kinetic energy distribution are presented, demonstrating clearly the impact of the amplitude and spectral form of the initial perturbation on the evolution of integral properties. A recently developed model predicting the growth of single mode perturbations in spherical implosions including shock waves is extended to predict mix layer amplitudes for broadband and narrowband cases, along with a model proposed by Mikaelian [“Rayleigh-Taylor and Richtmyer-Meshkov instabilities and mixing in stratified spherical shells,” Phys. Rev. A 42, 3400–3420 (1990)]. The resultant layer amplitude predictions from the new model are in good agreement with the numerical results while the longest wavelengths are not yet saturated, while Mikaelian’s model agrees well where the initial modes are saturated.

Weakly-nonlinear interactions of broadband Tollmien–Schlichting waves in a non-self-similar boundary layer on an airfoil

Abstract The present investigation is devoted to initial nonlinear stages of laminar–turbulent transition initiated by broadband disturbances in an essentially non-self-similar boundary layer, developing on a two-dimensional airfoil. The measurements are performed at fully controlled disturbance conditions with respect to the distribution of the initial spectra of excited 3D (in general) Tollmien–Schlichting (TS) waves. The main attention is devoted to the investigation of resonant (and non-resonant) interactions and amplification of broadband low-amplitude 3D instability waves in the presence (and in absence) of one or two primary harmonic 2D TS-waves. The broadband, “noise-like” TS-waves are composed of controlled perturbations, pseudo-random in time and space, with a continuous frequency–wavenumber spectrum. These perturbations simulate typical background disturbances usually observed at so-called “natural” disturbance conditions. The harmonic primary TS-wave simulates a primary instability mode, which has been formed by the linear instability mechanism. In fact, this paper represents the third step in a set of detailed investigations (Wurz et al., 2012 [1] ) and Wurz et al., 2012 [2] ) of weakly-nonlinear stages of transition performed in the same base flow (on an airfoil) but at increased complexity of initial disturbance conditions. In contrast to the studies of periodic resonant regimes described in refs. Wurz et al. (2012) [1] and Wurz et al. (2012) [2] , no specific (preinstalled) disturbance frequencies or/and spanwise wavenumbers were excited in the boundary layer (except for the primary wave). So, the flow was free to amplify almost any mode of the frequency–wavenumber spectrum by means of any available mechanisms of TS-wave interaction. Such disturbance conditions are the closest to the so-called “natural” ones, i.e. to those observed typically for uncontrolled external perturbations. Four main groups of regimes of disturbance excitation were studied: (i) two harmonic fundamental waves and a pair of frequency detuned subharmonics, (ii) the low-amplitude broadband disturbances only and (iii) the low-amplitude broadband disturbances together with harmonic 2D primary TS-waves, and (iv) the high-amplitude broadband disturbances only. It is found that in case (i) the superposition of the resonances takes place in a nonlinear way and joint efforts of the two fundamental waves are similar to a fundamental-mode amplitude enhancement, which always leads to the increase of the subharmonic amplification rates. In case (ii) the perturbations develop downstream in agreement with the linear stability mechanism. In case (iii) strong resonant interactions between the broadband perturbations and the harmonic primary TS-wave is observed in a wide frequency range. In case (iv) very similar strong resonant interactions are observed between 2D and 3D spectral modes starting already at significantly lower amplitudes of the 2D modes in the frequency range of the primary fundamental modes. In cases (i), (iii) and (iv) the amplification is double-exponential (exponent in exponent), similar to that found in isolated tuned and detuned resonances of deterministic modes in papers Wurz et al. (2012) [1] and Wurz et al. (2012) [2] . For all resonantly interacting disturbances the phase synchronism conditions are found to be satisfied. Based on the cumulative results obtained in the framework of this intensive investigation, the most important observation is the predominant role of resonant TS-wave interactions in the non-self-similar boundary layer developing on a natural laminar flow airfoil section.

Hydrodynamic instability growth of three-dimensional modulations in radiation-driven implosions with “low-foot” and “high-foot” drives at the National Ignition Facility

Hydrodynamic instability growth has been studied using three-dimensional (3-D) broadband modulations by comparing “high-foot” and “low-foot” spherical plastic (CH) capsule implosions at the National Ignition Facility (NIF) [E. M. Campbell et al., AIP Conf. Proc. 429, 3 (1998)]. The initial perturbations included capsule outer-surface roughness and capsule-mounting membranes (“tents”) that were similar to those used in a majority of implosions on NIF. The tents with thicknesses of 31-nm, 46-nm, and 109-nm were used in the experiments. The outer-surface roughness in the “low-foot” experiment was similar to the standard specification, while it was increased by ∼4 times in the “high-foot” experiment to compensate for the reduced growth. The ablation-front instability growth was measured using a Hydrodynamic Growth Radiography platform at a convergence ratio of ∼3. The dominant capsule perturbations, generated by the tent mountings, had measured perturbation amplitudes comparable to the capsule thickness with the “low-foot” drive. These tent perturbations were reduced by ∼3 to 10 times in implosions with the “high-foot” drive. Unexpectedly, the measured perturbations with initially thinner tents were either larger or similar to the measured perturbations with thicker tents for both “high-foot” and “low-foot” drives. While the measured instability growth of 3-D broadband perturbations was also significantly reduced by ∼5 to 10 times with the “high-foot” drive, compared to the “low-foot” drive, the growth mitigation was stronger than expected based on previous “growth-factor” results measured with two-dimensional modulations [D. T. Casey et al., Phys. Rev. E 90, 011102 (2014)]. One of the hypotheses to explain the results is based on the 3-D modulations of the oxygen content in the bulk of the capsule having a stronger effect on the overall growth of capsule perturbations than the outer-surface capsule roughness.

Quantum ratchet driven by broadband perturbation

Motion of an atomic ensemble trapped into a deep optical lattice is considered. We propose a novel approach to construct an atomic ratchet by superimposing two additional lattices whose amplitudes are small and subjected to broadband modulation. The broadband modulation is modeled by means of harmonic noise. Directed atomic transport occurs with the properly chosen phase shift between the signals modulating the amplitudes of the additional lattices. It is shown that efficiency of the ratchet depends non-monotonously on the parameter determining the spectral width of modulating signals.

Observation of Self-Similarity in the Magnetic Fields Generated by the Ablative Nonlinear Rayleigh-Taylor Instability

Magnetic fields generated by the nonlinear Rayleigh-Taylor growth of laser-seeded three-dimensional broadband perturbations were measured in laser-accelerated planar targets using ultrafast proton radiography. The experimental data show self-similar behavior in the growing cellular magnetic field structures. These observations are consistent with a bubble competition and merger model that predicts the time evolution of the number and size of the bubbles, linking the cellular magnetic field structures with the Rayleigh-Taylor bubble and spike growth.

Characterization of the dynamic mechanical stability of liquid-filled lenses

In this paper, the dynamic mechanical stability of the liquid-filled lenses was studied, in which acoustic excitation was used as broad band perturbation sources and the resultant response of the lens was characterized using non-contact laser Doppler vibrometer. To the best of our knowledge, it's the first time that the mechanical stability of liquid-filled lenses was experimentally reported. Both experimental results and theoretical analysis demonstrate that the resonance of the lens will shift to higher frequency while the vibration velocity as well as its magnitude will be reduced accordingly when the pressure in the lens cavity is increased to shorten the focal length. All of these results will provide useful references to help researchers design their own liquid-filled lenses for various applications.

Frequency- and Amplitude-Dependent Transmission of Stress Waves in Curved One-Dimensional Granular Crystals Composed of Diatomic Particles

We study the stress wave propagation in curved chains of particles (granular crystals) confined by bent elastic guides. We report the frequency- and amplitude-dependent filtering of transmitted waves in relation to various impact conditions and geometrical configurations. The granular crystals studied consist of alternating cylindrical and spherical particles pre-compressed with variable static loads. First, we excite the granular crystals with small-amplitude, broadband perturbations using a piezoelectric actuator to generate oscillatory elastic waves. We find that the linear frequency spectrum of the transmitted waves creates pass- and stop-bands in agreement with the theoretical dispersion relation, demonstrating the frequency-dependent filtering of input excitations through the diatomic granular crystals. Next, we excite high-amplitude nonlinear pulses in the crystals using striker impacts. Experimental tests verify the formation and propagation of highly nonlinear solitary waves that exhibit amplitude-dependent attenuation. We show that the wave propagation can be easily tuned by manipulating the pre-compression imposed to the chain or by varying the initial curvature of the granular chains. We use a combined discrete element (DE) and finite element (FE) numerical model to simulate the propagation of both dispersive linear waves and compactly-supported highly nonlinear waves. We find that the tunable, frequency- and amplitude-dependent filtering of the incoming signals results from the close interplay between the granular particles and the soft elastic media. The findings in this study suggest that hybrid structures composed of granular particles and linear elastic media can be employed as new passive acoustic filtering materials that selectively transmit or mitigate excitations in a desired range of frequencies and amplitudes.

Growth of a Richtmyer-Meshkov turbulent layer after reshock

This paper presents a numerical study of a reshocked turbulent mixing layer using high-order accurate Implicit Large-Eddy-Simulations (ILES). Existing theoretical approaches are discussed, and the theory of Youngs (detailed in Ref. 1) is extended to predict the behaviour of a reshocked mixing layer formed initially from a shock interacting with a broadband instability. The theory of Mikaelian2 is also extended to account for molecular mixing in the single-shocked layer prior to reshock. Simulations are conducted for broadband and narrowband initial perturbations and results for the growth rate of the reshocked layer and the decay rate of turbulent kinetic energy show excellent agreement with the extended theoretical approach. Reshock causes a marginal decrease in mixing parameters for the narrowband layer, but a significant increase for the broadband initial perturbation. The layer properties are observed to be very similar post-reshock, however, the growth rate exponent for the mixing layer width is higher in the broadband case, indicating that the reshocked layer still has a dependence (although weakened) on the initial conditions. These results have important implications for Unsteady Reynolds Averaged Navier Stokes modelling of such instabilities.

Experimental detection of deterministic turbulence

This paper is devoted to an experimental investigation of transitional and turbulent, post-transitional boundary layer initiated by fully controlled 3D broadband perturbations. A self-similar laminar boundary layer with an adverse pressure gradient was used as a primary base flow. The excited controlled disturbances initially represented Tollmien–Schlichting (TS) waves, which had a spectral component that was random at small and moderate time scales but periodical at very large time scales, during which the flow passed the experimental model several times. Detailed measurements were carried out by means of a hot-wire anemometer in a broad spatial region starting with stages of small-amplitude instability waves and ending with stage of a developed post-transitional turbulent flow. It was found that average characteristics of the produced post-transitional flow corresponded to the developed wall turbulence; meanwhile, the instantaneous flow structure there still remained deterministic, i.e. basically reproducible from one long signal realization to another. This instantaneous structure was measured in detail and compared with the one known from previous investigations of turbulent and transitional flows. A conclusion was drawn that the deterministic (reproducible) wall turbulence did exist in nature and could be generated experimentally in boundary layers. Besides important philosophical meaning, this result provided investigators with a very powerful methodological tool for subsequent turbulence research.

The influence of initial conditions on turbulent mixing due to Richtmyer–Meshkov instability

This paper investigates the influence of different three-dimensional multi-mode initial conditions on the rate of growth of a mixing layer initiated via a Richtmyer–Meshkov instability through a series of well-controlled numerical experiments. Results are presented for large-eddy simulation of narrowband and broadband perturbations at grid resolutions up to 3 × 109 points using two completely different numerical methods, and comparisons are made with theory and experiment. It is shown that the mixing-layer growth is strongly dependent on initial conditions, the narrowband case giving a power-law exponent θ ≈ 0.26 at low Atwood and θ ≈ 0.3 at high Atwood numbers. The broadband case uses a perturbation power spectrum of the form P(k) ∝ k−2 with a proposed theoretical growth rate of θ = 2/3. The numerical results confirm this; however, they highlight the necessity of a very fine grid to capture an appropriately broad range of initial scales. In addition, an analysis of the kinetic energy decay rates, fluctuating kinetic energy spectra, plane-averaged volume fraction profiles and mixing parameters is presented for each case.