Engineering Papers⌕ Search

SEARCH · Engineering Papers

Results for “Plasma Instabilities”

Search indexed NASA NTRS and DOE OSTI research on propulsion, heat transfer, battery materials and energy systems. Follow report and document links to the original sources.

Quote a phrase for an exact phrase match. Source license links do not imply unrestricted reuse.

At least 721 records · Page 40
All sourcesNASA NTRSDOE OSTI
DOE OSTI2021

Suppressing the enhancement of stimulated Raman scattering in inhomogeneous plasmas by tuning the modulation frequency of a broadband laser

The stimulated Raman scattering (SRS) instability can inhibit the performance of laser driven inertial confinement fusion (ICF) implosions by scattering light into unwanted directions or by generating hot electrons that preheat the target fuel. In principle, ICF target designs can avoid parameter regimes conducive to large, linear SRS gains. In practice, kinetic inflation—the nonlinear enhancement of SRS due to electron trapping in the excited plasma wave—makes this difficult. Here we show that laser bandwidth in the form of frequency modulation can either decrease or increase the inflationary SRS (iSRS) threshold in inhomogeneous plasmas depending on the maximum chirp of the laser pulse. The threshold, mapped out by a series of particle-in-cell simulations, exhibits a minimum when the frequency change within the pulse cancels the spatial detuning due to density inhomogeneities along the trajectory of the scattered light. In conclusion, by tuning the pump laser parameters away from this minimum, the iSRS threshold can be larger than at zero bandwidth, providing a path to mitigating kinetic inflation in ignition relevant plasmas.

70 PLASMA PHYSICS AND FUSION TECHNOLOGY↗
DOE OSTI2025

Tokamak Disruption Simulation (Final Technical Report)

The Tokamak Disruption Simulation collaboration was a SciDAC (Scientific Discovery through Advanced Computing) program led by Xianzhu Tang of the Los Alamos National Laboratory but with separate grants to each institution. I was the PI for Columbia University and the only person at Columbia directly funded by this SciDAC. Disruptions are a sudden loss of confinement in tokamaks, and a reliable method of preventing disruptions must be developed before tokamak fusion power plants become feasible. A related plasma confinement concept, the stellarator, is the only way known way of preventing disruptions with the required reliability. The largest devices, both existing and under construction, for confining plasmas for fusion using magnetic fields are tokamaks, so it is important to solve their disruption issues. The sudden loss of confinement in tokamaks can be from a bit of material falling into the plasma, which causes the plasma energy to be quickly lost by radiation, or by an instability causing a breakup of the confining surfaces formed by the magnetic field. This breakup can occur more than a million times faster than one would naively think possible. The sudden breakup of surfaces is called a fast magnetic reconnection and also results in a sudden cooling of the plasma.

70 PLASMA PHYSICS AND FUSION TECHNOLOGY↗
DOE OSTI2024

The collisional particle-in-cell method for the Vlasov–Maxwell–Landau equations

We introduce an extension of the particle-in-cell method that captures the Landau collisional effects in the Vlasov–Maxwell–Landau equations. The method arises from a regularisation of the variational formulation of the Landau equation, leading to a discretisation of the collision operator that conserves mass, charge, momentum and energy, while increasing the (regularised) entropy. The collisional effects appear as a fully deterministic effective force, thus the method does not require any transport–collision splitting. The scheme can be used in arbitrary dimension, and for a general interaction, including the Coulomb case. We validate the scheme on scenarios such as the Landau damping, the two-stream instability and the Weibel instability, demonstrating its effectiveness in the numerical simulation of plasma.

Bailo, Rafael (ORCID:0000000180183799)↗
DOE OSTI2020

A conservative approach to scaling magneto-inertial fusion concepts to larger pulsed-power drivers

The Magnetized Liner Inertial Fusion (MagLIF) experimental platform [M. R. Gomez et al., Phys. Rev. Lett. 113, 155003 (2014)] represents the most successful demonstration of magneto-inertial fusion (MIF) techniques to date in pursuit of ignition and significant fusion yields. The pressing question remains regarding how to scale MIF concepts like MagLIF to more powerful pulsed-power drivers while avoiding significant changes in physical regimes that could adversely impact performance. In this work, we propose a conservative approach for scaling general MIF implosions, including MagLIF. Underpinning our scaling approach is a theoretical framework describing the evolution of the trajectory and thickness of a thin-walled, cylindrical, current-driven shell imploding on preheated, adiabatic fuel. By imposing that scaled implosions remain self-similar, we obtain a set of scaling rules expressing key target design parameters and performance metrics as functions of the maximum driver current I max . We identify several scaling paths offering unique, complementary benefits and trade-offs in terms of physics risks and driver requirements. Remarkably, when scaling present-day experiments to higher coupled energies, these paths are predicted to preserve or reduce the majority of known performance-degrading effects, including hydrodynamic instabilities, impurity mix, fuel energy losses, and laser-plasma interactions, with notable exceptions clearly delineated. In the absence of α heating, our scaling paths exhibit neutron yield per-unit-length scaling as $\tilde{Y}$∝[I$3\atop{max}$,I${4.14}\atop{max}$] and ignition parameter scaling as χ∝[I max ,I${2.14}\atop{max}$]. By considering the specific physics risks unique to each scaling path, we provide a roadmap for future investigations to evaluate different scaling options through detailed numerical studies and scaling-focused experiments on present-day facilities. Overall, these results highlight the potential of MIF as a key component of the national ignition effort.

70 PLASMA PHYSICS AND FUSION TECHNOLOGY↗
DOE OSTI2023

Estimates of the wavenumber wavelet power spectrum of magnetic fluctuations during magnetic reconnection

Fluctuation analyses of experimental observations generally lack high temporal resolution and are in frequency-space f, contrary to theoretical efforts in wavenumber-space k. This is due to the inherent limits of the Fourier transform, though it is prominent due to the ease of diagnostic implementation. Advances in wavelet-based analysis have provided relief due to its temporal resolution, but in its common use, is still hard to compare to theoretical models. By using the two-point correlation technique in conjunction with large data sets, a wavelet power spectrum in wavenumber-space can be created. Dubbed the wavenumber wavelet power spectrum, this spectrum relates wavenumber to power in time. Further, this analysis technique more closely connects characterizations of experimentally observed fluctuations with other system parameters and theoretical predictions. In this article, we develop the wavenumber wavelet power spectrum using magnetic fluctuations caused by tearing instability driven magnetic reconnection in reproducible, high temperature laboratory plasmas. These dynamic magnetic fluctuations generated in reversed field pinch plasmas are broadband, ranging from the low frequency, 10's of kHz, up to the ion gyroradii frequencies, 100's of kHz. The dominant fluctuations have poloidal and toroidal mode numbers (m,n)=(1,6−10) and can grow to 2%–3% of the mean magnetic field. During these reconnection events, ions, and electrons are energized, magnetic fluctuation amplitudes increase, plasma flow is halted, and the toroidal magnetic flux increases, all on a semi-periodic basis. The newly developed spectrum provides better temporal resolution of spectrum characteristics to correlate with these particle energization phenomena.

70 PLASMA PHYSICS AND FUSION TECHNOLOGY↗
DOE OSTI2024

Eigenmode twist by energy flux during fast-ion driven instabilities

Twisting of the spatial structure of eigenmodes by energy fluxes generated due to the spatial channeling (SC)—the fast-ion energy and momentum transfer across the magnetic field by destabilized modes—is considered. It is revealed that there is a correlation between the direction of the radial wave energy flux, orientation of the mode twist (MT), and the direction of mode rotation. The mode twist parameter is introduced and relations connecting it with the energy flux are established. It is shown the energy flux transforms zeros of the radial profile of the mode amplitude into minima (i.e., zeros disappear). It is found that the radial group velocity and phase velocity of reversed shear Alfvén eigenmodes (RSAEs) have opposite directions. These findings can be used for diagnostics of the energy fluxes during fast-ion driven instabilities and understanding whether the SC degrades or improves plasma performance. Specific calculations are carried out for fast magnetoacoustic modes, FMMs (known also as compressional Alfvén eigenmodes, CAEs), and Alfvén eigenmodes, AEs. A DIII-D experiment where RSAEs were observed is considered. It is concluded that an outward SC took place in this experiment. Peculiarities of various modes are discussed, which may explain why twisting of AEs, but not of FMMs, was observed experimentally in the DIII-D and NSTX tokamaks.

Kolesnichenko, Ya. I. (ORCID:0000000323219681)↗
DOE OSTI2023

Perturbative analysis of low-frequency instabilities in high-field ST40 experiments

Abstract Linear and nonlinear perturbative analyses of low-frequency instabilities are presented for ST40 spherical torus high field plasmas, with simultaneous neutral beam heating due to two tangential sources at 25 keV and 55 keV. Initial plasma profiles on which the TRANSP and NUBEAM codes were based were obtained using an integrated analysis of several diagnostics including line-of-sight integrated and volume average measurements, as well as limited profile information from a charge-exchange-recombination spectrometer. Magnetohydrodynamics (MHD) analysis using the NOVA/NOVA-K codes suggests that the modes in the experiments are core-localized n = 1 eigenmodes with mixed acoustic and electromagnetic Alfvénic polarization. The global transport combined with the perturbative mode analysis indicates that there is greater tendency for bursty chirping response as the level of background micro-turbulence decreases in time and those predictions are shown to be consistent with the experimental data. This finding is interpreted in terms of the suppression of coherent phase-space structures that support chirping when the resonant fast ion dynamics is dominated by frequent stochastic, orbit-decorrelating events.

70 PLASMA PHYSICS AND FUSION TECHNOLOGY↗
DOE OSTI2022

Parametric dependencies of resonant layer responses across linear, two-fluid, drift-MHD regimes

Non-axisymmetric magnetic fields arising in a tokamak either by external or internal perturbations can induce complex non-ideal MHD responses in their resonant surfaces while remaining ideally evolved elsewhere. This layer response can be characterized in a linear regime by a single parameter called the inner-layer Delta, which enables outer-layer matching and the prediction of torque balance to non-linear island regimes. Here, we follow strictly one of the most comprehensive analytic treatments including two-fluid and drift MHD effects and keep the fidelity of the formulation by incorporating the numerical method based on the Riccati transformation when quantifying the inner-layer Delta. The proposed scheme reproduces not only the predicted responses in essentially all asymptotic regimes but also with continuous transitions as well as improved accuracies. In particular, the Delta variations across the inertial regimes with viscous or semi-collisional effects have been further resolved, in comparison with additional analytic solutions. The results imply greater shielding of the electromagnetic torque at the layer than what would be expected by earlier work when the viscous or semi-collisional effects can compete against the inertial effects, and also due to the intermediate regulation by kinetic Alfven wave resonances as rotation slows down. These are important features that can alter the nonaxisymmetric plasma responses including the field penetration by external fields or island seeding process in rotating tokamak plasmas.

MHD instability↗
DOE OSTI2023

NIF 3-pager FY24: Mshock [Slides]

Complete initial condition scan examining surface roughness alteration of larger mode growth (begin FY23 Q2). Previous reshock experiments have shown that we can measure the change in growth rate of a larger carrier mode as a diagnostic for the small-scale non-linear instability growth.

70 PLASMA PHYSICS AND FUSION TECHNOLOGY↗
DOE OSTI2022

Disruption Prediction and Avoidance in High Beta Long Pulse KSTAR Plasmas

This research is a concentrated and coordinated experimental and physics analysis program by a small U.S. team, focused on the topical area of “Long Pulse Control” as stated in the proposal solicitation. The overall goal of the research is to produce world-class analysis regarding forecasting and avoidance of disruptions in a long-pulse, superconducting tokamak at high stability parameters (e.g. normalized beta) with direct, quantifiable figures of merit (i.e. plasma disruption rate - “disruptivity”) and its reduction based on improved stability physics and control. The research targets the backbone of the problem – the basic, main issues – to produce this quantitative analysis by: (1) Creating and applying a new physics-based paradigm to understand how to eliminate disruptions: disruption event characterization, and event chain analysis – a direct manifestation of the approach suggested in the DOE Transients Report, (2) Ensuring that the components required to produce high-quality measurements and analysis needed for disruption event characterization for the tokamak in question (here, KSTAR) exist (e.g. pressure profile peaking, high-accuracy pitch angle data, compensated low frequency toroidal mode number n = 1 mode measurement) to create a primarily physics-based disruption forecasting capability, and (3) Addressing disruption avoidance for the first time in KSTAR by attacking primary disruption ‘event chains’, utilizing the most important associated plasma measurements (dynamic n = 1 field). The focus of this work will be avoidance of macroscopic instabilities, especially global modes.

70 PLASMA PHYSICS AND FUSION TECHNOLOGY↗
DOE OSTI2023

ETG turbulence in a tokamak pedestal

This paper explores the fundamental characteristics of electron-temperature-gradient (ETG)-driven turbulence in the tokamak pedestal. The extreme gradients in the pedestal produce linear instabilities and nonlinear turbulence that are distinct from the corresponding ETG phenomenology in the core plasma. The linear system exhibits multiple (greater than ten) unstable eigenmodes at each perpendicular wave vector, representing different toroidal and slab branches of the ETG instability. Proper orthogonal decomposition of the nonlinear fluctuations reveals no clear one-to-one correspondence between the linear and nonlinear modes for most wave vectors. Moreover, nonlinear frequencies deviate strongly from those of the linear instabilities, with spectra peaking at positive frequencies, which is opposite the sign of the ETG instability. The picture that emerges is one in which the linear properties are preserved only in a narrow range of k-space. Outside this range, nonlinear processes produce strong deviations from both the linear frequencies and eigenmode structures. This is interpreted in the context of critical balance, which enforces alignment between the parallel scales and fluctuation frequencies. We also investigate the nonlinear saturation processes. We observe a direct energy cascade from the injection scale to smaller scales in both perpendicular directions. However, in the bi-normal direction, there is also nonlocal inverse energy transfer to larger scales. Neither streamers nor zonal flows dominate the saturation.

70 PLASMA PHYSICS AND FUSION TECHNOLOGY↗
DOE OSTI2022

Near-cancellation of up- and down-gradient momentum transport in forced magnetized shear-flow turbulence

Visco-resistive magnetohydrodynamic turbulence, driven by a two-dimensional unstable shear layer that is maintained by an imposed body force, is examined by decomposing it into dissipationless linear eigenmodes of the initial profiles. The down-gradient momentum flux, as expected, originates from the large-scale instability. However, continual up-gradient momentum transport by large-scale linearly stable but nonlinearly excited eigenmodes is identified and found to nearly cancel the down-gradient transport by unstable modes. The stable modes effectuate this by depleting the large-scale turbulent fluctuations via energy transfer to the mean flow. This establishes a physical mechanism underlying the long-known observation that coherent vortices formed from nonlinear saturation of the instability reduce turbulent transport and fluctuations, as such vortices are composed of both the stable and unstable modes, which are nearly equal in their amplitudes. The impact of magnetic fields on the nonlinearly excited stable modes is then quantified. Furthermore, even when imposing a strong magnetic field that almost completely suppresses the instability, the up-gradient transport by the stable modes is at least two-thirds of the down-gradient transport by the unstable modes, whereas for weaker fields, this fraction reaches up to 98%. These effects are persistent with variations in magnetic Prandtl number and forcing strength. Finally, continuum modes are shown to be energetically less important, but essential for capturing the magnetic fluctuations and Maxwell stress. A simple analytical scaling law is derived for their saturated turbulent amplitudes. It predicts the falloff rate as the inverse of the Fourier wavenumber, a property which is confirmed in numerical simulations.

70 PLASMA PHYSICS AND FUSION TECHNOLOGY↗
DOE OSTI2023

Observation of impurity accumulation and its compatibility with high plasma performance in W7-X

At the W7-X stellarator, the bolometer system has measured an intensive radiation zone in the inner plasma region (at a normalized radius $ρ$ ~ 0.3–0.4) in the hydrogen plasma generated by electron cyclotron resonance heating; it differs from the normal plasma radiation distribution with an edge-localized emission zone. Spectroscopic diagnostics have recorded high-Z elements such as iron. This phenomenon happens in the plasma phases after gas supply turn-off, which results in all impurity relevant diagnostic signals increasing for several seconds. Despite the enhancement of the core radiation, the plasma energy confinement is improved. A transport analysis shows that this impurity radiation behavior is associated with a low diffusion coefficient ($D$~ 0.02 m 2 s -1 ) and a reversal of the convection around the radial position of the emission peak, which, under normal conditions, separates the zones of outward convection in the central (|$V$| ~ 0.1 m s -1 ) and inward convection in the outer region (|$V$| ~ 0.3 m s -1 ). An impurity accumulation around this radial position has been identified. The transport coefficients obtained are comparable with the theoretical predictions of collisional impurity transport. In the plasma phases studied, both impurity and energy confinement are enhanced. The mechanism responsible for the improvement is believed to be a reduction of micro-instabilities associated with the observed steepening of the density profile, initiated by a low edge plasma density (<1.0 × 10 19 m -3 ) after switching off the gas fueling. The normalized temperature and density gradients fulfil the condition for the suppression of ITG turbulence.

suppression of ITG turbulence↗
DOE OSTI2023

Simulated impact of fill tube geometry on recent high-yield implosions at the National Ignition Facility

Inertial confinement fusion capsules fielded at the National Ignition Facility are filled with deuterium and tritium fuel by means of a fill tube. The fill tube introduces a low-density pathway into the fuel region of the capsule that allows high Z contaminant to invade the hot spot during the course of the implosion. A recent series of nominally identical high-yield implosions on the NIF has exhibited significant variability in performance. We evaluate the impact of the fill tube in these implosions computationally to determine whether variations in fill tube geometry could have contributed to this variability. The main contrast between the fill tube geometry in the six shots was the outer diameter of the capsule bore hole, a conical hole into which the fill tube is inserted. In our simulations, the geometry of the bore hole can play a significant role in the development of nonlinear flows seeded by the fill tube. Additionally, we find that the amount of space between the bore hole and the fill tube is the primary factor that determines the amount of contaminant jetted into the hot spot by the fill tube and, in turn, the level of yield reduction due to the fill tube in our simulations. As a consequence, some capsules with 5 μm fill tubes are predicted to outperform capsules with 2 μm fill tubes. We also find that micrometer-scale changes to bore hole size can impact fusion yields by up to four times near the ignition threshold. Nevertheless, simulation trends do not reproduce experimental yield trends, suggesting that the fill tube geometry was not the primary factor contributing to the observed variability in performance and that the fill tube could be masking sensitivity to other asymmetries such as other micrometer-scale capsule defects like voids that were not included in our simulations.

70 PLASMA PHYSICS AND FUSION TECHNOLOGY↗
DOE OSTI2025

Observation and characterisation of trapped electron modes in Wendelstein 7-X

In the past, quasi coherent (QC) modes were reported for nearly all tokamaks. The general definition describes modes as QC when the magnitude squared coherence is in the range of 0.3 to 0.6. QC modes are observed in the plasma core as well as in the plasma edge and can have quite different physical origins. The one in the core are observed in plasmas with low collisionality, where the electron temperature exceeds the ion temperature in the plasma core. This is the case for electron cyclotron heating in general. The origin of these modes are electrons trapped within a magnetic mirror, as reported in the past from various fusion devices. The so-called trapped-electron modes (TEMs) belong to drift wave instabilities and can be destabilized by electron-temperature gradients in the plasma core. From the diagnostic point of view, QC modes appear as fluctuations in electron density and temperature. Therefore, the microwave reflectometer is very well suited to monitor these modes. This paper describes experiments, conducted at the Wendelstein 7-X stellarator (W7-X), which aim at detecting QC modes at low wave numbers. A poloidal correlation reflectometer installed at W7-X, is able to measure low wave numbers ($k_\perp\unicode{x2A7D}{3.5}\,\textrm{cm}^{-1}$). For medium line-averaged densities ($\int n_\textrm{e}\unicode{x2A7D}{6 \times10^{19}}\,\textrm{m}^{-2}$) the plasma core is accessible for this diagnostic. For different magnetic configurations and plasma parameters, broad QC structures are observed in the coherence spectra. From the analysis of the rotation and the poloidal structure, these QC modes show the properties of electron-temperature-gradient driven TEMs. A linear relation between the mode velocity and the rotation frequency is found. The relation is uniform and confirms the nature of QC-mode observation as TEM in tokamaks, too.

70 PLASMA PHYSICS AND FUSION TECHNOLOGY↗
DOE OSTI2024

A Gaussian process based surrogate approach for the optimization of cylindrical targets

Simulating direct-drive inertial confinement experiments presents significant computational challenges, both due to the complexity of the codes required for such simulations and the substantial computational expense associated with target design studies. Machine learning models, and in particular, surrogate models, offer a solution by replacing simulation results with a simplified approximation. In this study, we apply surrogate modeling and optimization techniques that are well established in the existing literature to one-dimensional simulation data of a new cylindrical target design containing deuterium–tritium fuel. These models predict yields without the need for expensive simulations. We find that Bayesian optimization with Gaussian process surrogates enhances sampling efficiency in low-dimensional design spaces but becomes less efficient as dimensionality increases. Nonetheless, optimization routines within two-dimensional and five-dimensional design spaces can identify designs that maximize yield, while also aligning with established physical intuition. Optimization routines, which ignore constraints on hydrodynamic instability growth, are shown to lead to unstable designs in 2D, resulting in yield loss. However, routines that utilize 1D simulations and impose constraints on the in-flight aspect ratio converge on novel cylindrical target designs that are stable against hydrodynamic instability growth in 2D and achieve high yield.

70 PLASMA PHYSICS AND FUSION TECHNOLOGY↗
DOE OSTI2025

Nonideal stability analysis of differentially rotating plasmas with global curvature effects

The linear stability of global nonaxisymmetric modes in differentially rotating, magnetized, nonideal plasma is critical to classifying turbulence and transport phenomena. We investigate the competition between the local magneto-rotational instability (MRI) and the magneto-curvature instability (MCI)—a distinct nonaxisymmetric low-frequency curvature-driven global branch that appears alongside MRI. Here, to accomplish this, we developed a nonideal global spectral method, which is validated against NIMROD code simulations. This spectral approach allows for the direct derivation of an extended effective potential formalism and a resistive Alfvénic resonance condition, providing a framework for direct analysis of energy contributions and confinement mechanisms. Our study reveals that the global, low-frequency MCI persists at low magnetic Reynolds numbers (Rm), whereas the localized, high-frequency MRI is stabilized by diffusive broadening of its structure around its Alfvénic resonances. Consequently, we identify the global MCI branch as the primary onset mechanism for nonaxisymmetric magnetohydrodynamic instability in systems with finite curvature, e.g., astrophysical rotators. We establish distinct parameter regimes for mode dominance: MCI prevails in geometrically moderate-thickness disks with intermediate curvature and radial gaps, while MRI dominates in thin, low-curvature disks with large radial gaps. Mode competition is also highly sensitive to the flow profile, particularly vorticity and its gradient, with nonuniform shear profiles exhibiting more robust instability due to flow curvature (i.e., the second derivative of the flow profile) and shear contributions. A key outcome is the development of spectral diagrams derived from the global spectral method. These diagrams comprehensively map dominant instabilities and their characteristics, offering a predictive tool for critical onset parameters (i.e., flow curvature, magnetic field, and Rm) and facilitating the interpretation of experimental and simulation results. Notably, these diagrams demonstrate that the global MCI is generally the sole unstable mode at the initial onset of nonaxisymmetric instability.

Haywood, Alexander [Princeton Univ., NJ (United St↗
DOE OSTI2025

Kelvin–Helmholtz instability under stabilizing parallel magnetic field in nonhomogeneous compressible MHD flows

We study the Kelvin–Helmholtz instability (KHI) for the general case of a compressible, nonhomogeneous, magnetized plasma flow. The study is limited to a vortex sheet interface with an imposed parallel magnetic field. We introduce a new formalism based on a convective Mach number M c , a convective Alfvénic Mach number M Ac , and a total convective Mach number that combines the two. We derive an analytic expression of the KHI growth rate for a homogeneous flow (i.e., zero Atwood number, A=0) that converges toward both the expression for unmagnetized compressible flow and Chandrasekhar's expression for magnetized incompressible flow. Otherwise, the dispersion relation is solved numerically and allows deriving general stability diagrams of magnetized KHI for the triplet (A, M c , β −plasma) parameters. We show these parameters uniquely define all configurations for a parallel magnetic field. We also construct diagrams with respect to the convective Alfvénic Mach number, the β − plasma parameter, or the magnetic field showing which magnetic field strength is required for stabilizing a given shear flow. The theoretical growth rates are compared with 18 simulations made with the GAMERA code, currently used for 3D magnetospheric simulations. Finally, we apply our results to the analysis of a past KHI experiment performed at the OMEGA laser facility, showing linear theory succeeds to provide accurate estimates of the growth rate at early times. We further discuss how our results can inform future experiments in the high-Mach magnetized regime at the National Ignition Facility. Possible limitations of the study due to resistive, mixing, or turbulence effects are discussed.

compressible flows↗