Front Speed Research Articles

Flame propagation modes for iron particle clusters in air — Part I: Transition from continuous to discrete propagation mode under weak convection effects

Iron powder can serve as an energy carrier and help generate heat and power in a clean and sustainable manner. In this paper, we analyze one of the challenges related to iron combustion: the mode of flame propagation. Depending on the conditions, an iron dust flame can propagate in either the continuous mode, with neighboring particles burning simultaneously, or in a discrete mode, with each particle burning separately, i.e. as a percolating wave. We use particle-resolved simulations including the heterogeneous reaction of iron with oxygen and detailed transport models. Simulations are carried out with multiple particles in a single row with hot gas at the inlet for initiating combustion under weak convection effects. Continuous and discrete flame propagation modes are identified and the particle distance, at which the transition from the continuous to the discrete propagation mode occurs, is calculated for different particle diameters. It is found that the transition from one mode to another occurs close to stoichiometry and the maximum flame reaction front speed is within the range of experiments with similarly sized particles. The flame propagation mode transitions to the discrete mode for larger particles at slightly lower particle distances than smaller particles due to higher thermal inertia. The discreteness parameter is modified to take into account the effect of heating from burnt particles on the unburnt ones. This results in an overall adequate prediction of the numerically found transitional distances under weak convection effects.Novelty and significance: As a carbon-free energy carrier, there is a great requirement for exploring the fundamental and application aspects of iron combustion. One critical issue for the heterogeneous combustion of iron is the mode of flame propagation. While previous studies have relied on simplified models and simplifying assumptions, in this work, we carry out detailed boundary layer resolved simulations of multiple particles with varying distances between the particles. Beyond a critical particle spacing, the mode of flame propagation transitions from the continuous mode where multiple particles burn simultaneously, to the discrete mode where the particles burn standalone. The scaling of such a transition is improved with the help of appropriate temperature based properties and the length scale. Our work is significant in understanding the fundamentals of flame propagation in iron clusters and can help in the development of Euler–Lagrangian models where tens of thousands of particles need to be simulated.

Scaling and mechanism of the propagation speed of the upstream turbulent front in pipe flow

The scaling and mechanism of the propagation speed of turbulent fronts in pipe flow with the Reynolds number has been a long-standing problem in the past decades. Here, we derive an explicit scaling law for the upstream front speed, which approaches a power-law scaling at high Reynolds numbers, and we explain the underlying mechanism. Our data show that the average wall distance of low-speed streaks at the tip of the upstream front, where transition occurs, appears to be constant in local wall units in the wide bulk-Reynolds-number range investigated, between 5000 and 60 000. By further assuming that the axial propagation of velocity fluctuations at the front tip, resulting from streak instabilities, is dominated by the advection of the local mean flow, the front speed can be derived as an explicit function of the Reynolds number. The derived formula agrees well with the speed measured by front tracking. Our finding reveals a relationship between the structure and speed of a front, which enables a close approximation to be obtained of the front speed based on a single velocity field without having to track the front over time.

Structure of rotating detonation in mixtures of natural gas and oxygen

The rotating detonation combustor is investigated as a potential implementation of pressure-gain combustion. Achieving net pressure-gain has proven to be extremely challenging in most combustors reported in the literature, regardless of their operating parameters. The performance losses observed in rotating detonation, compared to ideal predictions, can be attributed to mechanisms such as vitiation by residual burnt gases, partial detonation, and pre-combustion of reactants. The literature frequently discusses the impact of these mechanisms on measurable properties such as thrust and wave speed, but their effects on the front structure have not been fully understood. This study proposes using the rotating detonation structure for the first time to evaluate the vitiation quantitatively. The front structure of a compressed natural gas — oxygen-fueled rotating detonation is characterized using broadband chemiluminescence, and properties such as oblique shock angle, expansion angle, and front height are reported. The experimental values are then compared to the ideal predictions derived based on mass flow rates. The experimental front heights are found to be much higher than their ideal predicted counterparts. Analytical calculations that include pre-burning and vitiation show an increase in front height and a reduction in expansion angle, consistent with the chemiluminescence levels observed. The vitiated mass fraction is then deduced from the experimental front height, and the remaining velocity deficit is then used to evaluate the partial detonation mass fraction. Vitiated front angles compare favorably to the predicted values.Novelty and SignificanceThis study reports high-quality chemiluminescence imaging of natural gas/oxygen rotating detonation front enabling the measurement of critical geometrical front features such as the front height, oblique shock angle, and expansion angle. The conventional front mass flow rate balance is extended to account for the re-circulation of burnt gases effects and is used to estimate the experimental vitiation. Most notably, vitiation of the fresh mixture, which was already known to reduce front speed and pressure rise, is also shown to strongly decrease the expansion angle resulting in further losses of axial thrust.

Experimental study on the effects of ignition position and inhomogeneous concentration on vented hydrogen deflagrations in a 7 m3 chamber

Vented inhomogeneous hydrogen deflagrations are conducted in a 7 m3 chamber. The effects of ignition position and inhomogeneous concentration are studied. The results show that a coupling flame structure of jet fire and flame bubble is first observed. In the case of top ignition, a M-shaped flame structure forms for average hydrogen concentration C¯H2≤ 10.14 vol%. In the cases of center ignition and bottom ignition, the flame bubble rapidly expands towards the walls and the horizontal propagation speed of upper part of flame bubble is obviously higher than that of lower part of flame bubble for C¯H2≥ 11.27 vol%. Moreover, the speed of horizontal flame front quickly decreases as the flame front moves away from the centerline of hydrogen jet. Three overpressure peaks appear in all scenarios, namely Popen, Phel and Pvib, which results from the vent failure, Helmholtz-type oscillations and thermo-acoustic oscillations respectively. The maximum overpressure dominated by Popen is nearly insensitive to C¯H2for top ignition. However, the maximum overpressure induced by center ignition or bottom ignition significantly increases when C¯H2≥ 10.14 vol%. For a given C¯H2, Pvib induced by bottom ignition is always the largest.

ИССЛЕДОВАНИЕ КУМУЛЯТИВНЫХ И УДАРНО-ВОЛНОВЫХ ПРОЦЕССОВ В ВОЗДУШНОЙ СРЕДЕ В СВАРОЧНОМ ЗАЗОРЕ ПРИ СВАРКЕ ВЗРЫВОМ

A photoelectric method is proposed for determining the velocity of the shock wave front in air in the gap between colliding plates during explosion welding. Using traps placed in the welding gap, dispersed metal particles of cumulative origin were discovered, filling the area of shock-compressed air in the gap and significantly affecting the speed of the shock wave front in the air.

Nonlocal effect on shear wave propagation in a fiber-reinforced poroelastic layered structure subjected to interfacial impulsive disturbance

The present study illuminates the influence of impulsive line source on Horizontally polarized shear (SH) waves propagating in a stratified structure consisting of two distinct nonlocal fiber-reinforced layers with voids over a foundation with nonlocal functionally graded fiber-reinforced substrate with voids. The governing equations for a nonlocal fiber-reinforced poroelastic medium are established. The proposed mathematical model incorporates the Green’s function technique and the Fourier transformation to yield the complex dispersion relation of the propagating wave. The study discloses the existence of two wave fronts of SH-waves propagating with different speeds through the layered structure. The second wave front arises as a result of existence of void pores, while the first wave front defines SH-wave propagation in a nonlocal fiber-reinforced layered structure. Each wave front possesses individual critical frequency beyond which they fail to propagate. Speeds of both wave fronts are influenced by the appearance of nonlocality parameter. Using MATHEMATICA, several graphs have been plotted to demonstrate the impact of key parameters on the dispersion and attenuation of both wave fronts. Some notable cases have been derived which validate the present mathematical model.

Fitness Costs of Antibiotic Resistance Impede the Evolution of Resistance to Other Antibiotics.

Antibiotic resistance is a major threat to global health, claiming the lives of millions every year. With a nearly dry antibiotic development pipeline, novel strategies are urgently needed to combat resistant pathogens. One emerging strategy is the use of sequential antibiotic therapy, postulated to reduce the rate at which antibiotic resistance evolves. Here, we use the soft agar gradient evolution (SAGE) system to carry out high-throughput in vitro bacterial evolution against antibiotic pressure. We find that evolution of resistance to the antibiotic chloramphenicol (CHL) severely affects bacterial fitness, slowing the rate at which resistance to the antibiotics nitrofurantoin and streptomycin emerges. In vitro acquisition of compensatory mutations in the CHL-resistant cells markedly improves fitness and nitrofurantoin adaptation rates but fails to restore rates to wild-type levels against streptomycin. Genome sequencing reveals distinct evolutionary paths to resistance in fitness-impaired populations, suggesting resistance trade-offs in favor of mitigation of fitness costs. We show that the speed of bacterial fronts in SAGE plates is a reliable indicator of adaptation rates and evolutionary trajectories to resistance. Identification of antibiotics whose mutational resistance mechanisms confer stable impairments may help clinicians prescribe sequential antibiotic therapies that are less prone to resistance evolution.

Characterization of refill region and mixing state immediately ahead of a hydrogen-air rotating detonation using LES

The extraordinary complexity of real Rotating Detonation Combustors (RDC) demands a deep knowledge of each phenomenon involved in the wave development and propagation. Since the reactants are typically injected separately, a key element for the combustor design optimization is understanding the refilling process and the reactants mixing. However, due to the harsh environment and high working frequencies of RDCs, the experimental diagnostics is usually limited, so high-fidelity simulations represent an essential tool to complement the measurements with detailed insights into the flow. In the present work, the non-premixed RDC installed at TU Berlin is simulated with the AVBP code by solving the fully-compressible, spatially-filtered reactive Navier-Stokes equations. The complete hydrogen and air injection system is included in the numerical model to accurately describe both the reactants mixing and the complex turbulent flow field in the resulting refill region. This study shows how both the injection system configuration and its transient interaction with the wave are fundamental for the reactants mixing, as they directly influence the refilled gas properties. Limiting the imbalances between the blockage dynamics of the fuel and oxidizer ducts and optimizing the fuel injectors can improve considerably the homogeneity of the fresh mixture, and consequently the leading shock strength and reasonably the pressure gain. In fact, the detailed analysis of the detonation front speed shown a higher instability near the chamber base for the periodic presence of unmixed reactants. Nevertheless, the unstable root of the front does not affect the whole wave speed, and remaining part propagates steadily thanks to the tangential mixture uniformity. Moreover, the local speed distribution does not appear directly related to the small-scale mixture properties, indicating a higher sensibility to the annulus curvature rather than to the local gas state.

Experimental and numerical investigation on the premixed methane/air flame propagation in duct with obstacle gradients

This work experimentally and numerically investigates the effect of the obstacle gradient on the characteristics of the methane/air explosion in obstructed ducts. The obstacle gradient is expressed through various blockage ratios, and three different obstacle gradients are investigated, i.e., C357, C555, and C753. Noted that C357 means that the blockage ratios of three obstacles arranged sequentially in the duct are 0.3, 0.5 and 0.7 respectively, and so do C555 and C753. A two-dimensional (2D) model is adopted and the Scale-Adaptive Simulation method with the thickened flame model is considered. Experimental results show that the obstacle gradient significantly affects the flame evolution structure, flame propagation speed and overpressure. The obstacle gradient is accountable for the “tulip flames” appearing downstream of the obstacle. For the case with a fixed methane volume fraction, the average flame front speed is fixed. However, with the increasing obstacle gradient, the maximum flame front speed increases until it achieves the maximum at C357, and so does the maximum overpressure. The numerical simulation can predict the flame evolution behaviour precisely. It is evident that the generation of different flame shapes appearing is derived from the flow field evolution of unburned gas downstream of the flame front.

Experimental and numerical study on explosion behavior of hydrogen-air mixture in an obstructed closed chamber

The flame evolution, flame front speed and pressure dynamics are investigated through experimental and simulation analysis. The effects of the obstacle on velocity vector field and vorticity field are analyzed. The results indicate that a rise in blockage ratio causes a rapid rise in flame front speed. The turbulent combustion, flame backflow, and overpressure are significantly enhanced as obstacle distance and blockage ratio increase. The jet flame results in a surge in flame front speed, and a momentary increase in pressure rise. The flame-vortex interaction causes strong turbulent combustion and a rapid overpressure increase. The overpressure displays regular periodic fluctuations and high-frequency oscillations with high amplitudes, which can be attributed to the dynamic and impulsive responses. The vortex results in the formation of a curled flame with high localized vorticity. The backflow flame, exhibiting an extremely high vorticity magnitude, is induced by the overall pressure gradient and flow field recirculation.

Validation of RANS-Based Turbulence Models Against High-Resolution Experiments and DNS for Buoyancy-Driven Flow with Stratified Fronts

To provide computational fluid dynamics (CFD)–grade experimental data for studying stratification, measurements on the High-Resolution Jet (HiRJet) facility at the University of Michigan have been conducted with density differences of 1.5 % and − 1.5 % , respectively. Fluid with a density different from the fluid initially present in the HiRJet tank was injected, and the propagation of the time-dependent density stratification was captured on a two-dimensional plane with the aid of the wire-mesh sensor technique for Reynolds numbers near 5000 and Richardson numbers near 0.29. Direct numerical simulations (DNSs) of the two cases have also been conducted to expand the multifidelity database. The novel experimental and DNS data were then used to assess the predictive capabilities of the Standard k − ε (SKE) model and the Reynolds Stress Transport (RST) model. In particular, the propagation speed and thickness of the stratification fronts were assessed by comparing the CFD results against the experimental and DNS data. It was found that the general trends of the stratified density fronts were well predicted by the CFD simulations; however, slight overprediction of the thickness of the stratification layer was found with the SKE model while the RST model gave a larger overprediction of the mixing. Examination of the turbulent statistics showed that the turbulent viscosity was largely overpredicted by the RST model compared to the SKE model.

A comparison of mechanistic models for the combustion of iron microparticles and their application to polydisperse iron-air suspensions

Metals can serve as carbon-free energy carriers, e.g., in innovative metal-metal oxide cycles as proposed by Bergthorson (Prog. Energy Combust. Sci., 2018). For this purpose, iron powder is a suitable candidate since it can be oxidized with air, exhibits a high energy density, is non-toxic and abundant. Nevertheless, the combustion of iron powder in air is challenging especially with respect to flame stabilization which depends on the particle size distribution and the morphology of the iron microparticles among other factors. Models for the prediction of reaction front speed in iron-air suspensions can contribute to overcoming this challenge. To this end, three different models for iron particle oxidation are integrated into a laminar flame solver for simulating such reaction fronts. The scientific objective of this work is to elucidate the influence of polydispersity of the iron particles on the reaction front speed, which is still not satisfactorily understood. In a systematic approach, cases with successively increasing complexity are considered: The investigation starts with single particle combustion and then proceeds with iron-air suspensions prescribing binary particle size distributions (PSDs), parametrized generic PSDs, and an exemplary PSD measured for a real iron powder sample. The simulations show that, dependent on the kinetic particle model, the particles’ thermal inertia, and the PSD, particles undergo thermochemical conversion in a sequential manner according to their size and every particle fraction exhibits an individual combustion environment (surrounding gas phase temperature and oxygen concentration). The local environment can be leaner or richer than the overall iron-to-air ratio would suggest and can be very different from single particle experiments. The contribution of individual particle fractions to the overall reaction front speed depends on its ranking within the PSD. The study further demonstrates, that although the three particle models show good agreement for single particle combustion, they lead to very different reaction front speeds. This is due to the different ignition behavior predicted by the particle models, which is shown to strongly influence the reaction front characteristics. Overall, the present work illustrates the complex relationship between characteristics of single particle ignition and combustion, polydispersity, and properties of reaction fronts in iron-air suspensions.

Effect of moisture on the granulation and sintering characteristics of sinter materials with carbide slag as a flux

Carbide slag is a solid waste generated during acetylene production mostly not exceeding 200 μm, mainly composed of Ca(OH)2. Using carbide slag as a flux for iron ore sintering is a promising utilization method. The effect of moisture on the granulation and sintering process of sinter mixtures with carbide slag as a flux was studied in this study. The results indicate that carbide slag suppresses the deterioration of bed compressive strength with increasing moisture compared with our previous work. As the granulation moisture increases from 5.73% to 7.66%, the Sauter mean diameter of the granules increases from 1.48 mm to 3.35 mm, and the flame front speed increases from 20.55 mm/min to 34.09 mm/min. However, excessive moisture worsens the thermal pattern of the sintering bed, causing a decrease in the peak temperature from 1377 °C to 1340 °C. There is an optimal moisture content that maximizes the comprehensive index.

Electromagnetic drift wave instability in tokamak plasmas with strong pedestal gradient

The linear eigenmode characterizations and the nonlinear turbulence energy spreading of the drift waves in a tokamak plasma with strong pedestal gradient are numerically investigated based on an electromagnetic Landau fluid model. By the linear eigenmode analysis, it is found that the dominant instability in the low β regime is the ion-temperature-gradient (ITGc) mode and the electron drift wave instability (eDWI p ) in the core and edge region with strong density gradient, respectively. Multiple eigenstates of the eDWI p with different peak locations in the poloidal direction can be obtained by the eigenvalue problem solver. The dominant one is the high order eDWI p corresponding to the unconventional ballooning mode structure with multiple peaks in the poloidal position, in contrast to the conventional modes that peak at the outboard mid-plane, and has been verified through initial value simulation. In the high β regime, the dominant eigenmodes in the core and edge region are the conventional and unconventional kinetic ballooning modes respectively. In the nonlinear simulation, an inward turbulence spreading phenomenon during the quasi-saturation phase of the edge turbulence is clearly observed. The inward speed of the turbulence energy front in the high β regime is much faster than that in the low β regime. It is interestingly found that the speed of the turbulence energy front increases with the increase of the plasma β in the low β regime, while it is almost unchanged in the high β regime. It is identified that the turbulence spreading in the low and high β regimes are determined by the nonlinear dynamics and the linear toroidal coupling respectively.

Experimental and simulation study of NH3–H2-Air flame dynamics at elevated temperature in a closed duct

Ammonia allows for long-distance and large-scale storage transport of renewable energy sources. The reactivity of NH3-Air flame can be enhanced by doping with hydrogen and by increasing the initial temperature. The flame combustion characteristics of NH3–H2-Air are studied in a rectangular closed pipe with a mole fraction of 45% H2, an equivalence ratio (φ) of 0.8–1.2 and 0.1 MPa. The initial temperatures (Tu) of the mixture are 300 K, 375 K and 450 K. Experimental studies of flame front speed, flame tip structure evolution, and overpressure dynamics are performed. Laminar burning velocity (SL), peak mole fraction of key radicals, and flame instability of NH3–H2-Air flames are simulated in Chemkin Pro-software using the Mathieu and Petersen mechanism. The results show that the relatively low equivalence ratio and initial temperature promote the structural changes of the flame tip, forming a “tulip” flame and a distorted “tulip”. As the equivalence ratio increases, the flame front speed and overpressure increase and the flame oscillates slightly. The distortion in the lip of the “tulip” is more pronounced. The increase in temperature promotes flame propagation in the early stages and the pressure decreases. Reactive activity increases with increasing temperature, the peak mole fraction of NH2 and H radicals increases, and SL increases. At normal temperature, the flame is affected by thermo-diffusive instability only when lean mixture. The Darrieus-Landau instability increases as the equivalence ratio increases. Conversely, the Darrieus-Landau instability decreases with increasing temperature. Therefore, the current experimental and simulation results help to understand the laminar combustion characteristics of NH3–H2-Air at high temperatures.

Upper and lower bounds for the speed of fronts of the reaction diffusion equation with Stefan boundary conditions

We establish two integral variational principles for the spreading speed of the one dimensional reaction diffusion equation with Stefan boundary conditions. The first principle is valid for monostable reaction terms and the second principle is valid for arbitrary reaction terms. These principles allow to obtain several upper and lower bounds for the speed. In particular, we construct a generalized Zeldovich–Frank–Kamenetskii type lower bound for the speed and upper bounds in terms of the speed of the standard reaction diffusion problem. We construct asymptotically exact lower bounds previously obtained by perturbation theory.

Polymer co-crystallization from LLE: Crystallization kinetics of POCB hydrate from two-phase mixtures of POCB and water

Polyoxacyclobutane (POCB, –[CH2CH2CH2O−]n) has the rare ability to co-crystallize with water to form a hydrate. Above the melting point of the hydrate (∼37 °C), mixtures exist in liquid-liquid equilibrium (LLE) between a POCB-rich and a water-rich liquid phase over a wide range of POCB:water ratios. Such phase behavior, which combines liquid-liquid equilibrium (LLE) and co-crystallization of a polymer with a small molecule, appears to be unique to POCB-water mixtures. Here, we examine the kinetics of isothermal hydrate co-crystallization when mixtures that are initially in liquid-liquid equilibrium (LLE) are cooled to below the hydrate melting point. Hydrate co-crystallization requires both POCB and water, and hence, it is coupled with transport of species between the liquid phases. Optical microscopy shows that hydrate crystallization occurs within the POCB-rich domains, and hydrate spherulites grow at constant speed. The temperature-dependence of the speed of hydrate growth front is consistent with the Hoffman Lauritzen model for homopolymer crystallization. Dilatometric measurements of bulk co-crystallization kinetics are conducted with constant stirring to avoid gravitational separation of the liquid phases. The temperature-dependence of bulk crystallization kinetics of these stirred mixtures in LLE is found to be far weaker than that of spherulitic growth kinetics, or of the quiescent bulk crystallization kinetics of mixtures starting from homogeneous conditions.

Gravity currents of viscous fluids in a vertically widening and converging fracture

We are investigating flows in the viscous-buoyancy balance regime in a converging channel with an upward increase in the width, with the gap of the channel varying according to a xkzr power function, being x and z the horizontal and vertical coordinate, respectively, and with 0<k<1 and 0<r<1 in order to be consistent with the model. The fluid rheology is described according to the Ostwald–de Waele model, with a power-law relationship between shear stress and shear rate and with application for shear-thinning, shear-thickening and, as a special case, Newtonian fluids. While for the case of flow in the direction of widening of the horizontal channel, a self-similar solution of the first kind can be expected, for flow toward the origin, with channel narrowing horizontally, the solution is self-similar of the second kind, with the space and a reduced time coupled in a self-similar independent variable but with an unknown parameter of the transformation group that makes the differential problem invariant. The solution is found in the phase plane by numerical integration of the paths connecting pairs of singular points, separately for the pre-closure phase, in which the current front advances invading the channel, and for the post-closure or leveling phase, in which the fluid has reached the origin and the front no longer propagates, while the level progressively increases canceling the pressure gradient. Integration is performed with a trial and error procedure by modifying the unknown parameter, generally named eigenvalue and specifically critical eigenvalue when the path has been successfully integrated. The overall effect of an increasing permeability upward is that of an increase in the front speed, with the current profile also becoming locally steeper. The effect of an increase in the fluid-behavior index is mixed, as it reduces the speed of the front but still increases the steepness of the local current profile. In any case, the model implies that the eigenvalue tends to infinity for k→1 even in the presence of an increase in the vertical permeability (r > 0).

Sucrose transport inside the phloem: Bridging hydrodynamics and geometric characteristics

In plants, the delivery of the products of photosynthesis is achieved through a hydraulic system labeled as phloem. This semi-permeable plant tissue consists of living cells that contract and expand in response to fluid pressure and flow velocity fluctuations. The Münch pressure flow theory, which is based on osmosis providing the necessary pressure gradient to drive the mass flow of carbohydrates, is currently the most accepted model for such sucrose transport. When this hypothesis is combined with the conservation of fluid mass and momentum as well as sucrose mass, many simplifications must be invoked to mathematically close the problem and to resolve the flow. This study revisits such osmotically driven flows by developing a new two-dimensional numerical model in cylindrical coordinates for an elastic membrane and a concentration-dependent viscosity. It is demonstrated that the interaction between the hydrodynamic and externally supplied geometrical characteristic of the phloem has a significant effect on the front speed of sucrose transport. These results offer a novel perspective about the evolutionary adaptation of plant hydraulic traits to optimize phloem soluble compounds transport efficiency.

Interaction of a moving shock wave with a turbulent boundary layer

In the present study, the influence of a uniformly moving impinging shock on the resulting shock wave–turbulent boundary layer interaction is numerically investigated. The relative Mach number of the shock front travelling above a flat-plate model varied between 0 and 2.3, while the quasi-steady inflow conditions remained constant with a Mach number of 3. To quantitatively evaluate the effect of shock travelling speed, a well-known scaling method for interaction length in quasi-steady flows was applied as a reference after significant improvements in modelling the effects of Reynolds number and wall temperature using new and existing data. Moreover, previously obtained experimental results for a limited range of travelling speeds were employed to validate the obtained numerical results. Three ranges of shock travelling speeds with distinctly different properties were extracted and quantitatively described using a developed correlation-based approach built on the extended quasi-stationary scaling law. In the first range, the scaled interaction length reaches its maximum for the given interaction strength and can be directly described by the scaling law obtained for quasi-stationary interactions. In the second travelling-speed range, the dependence of the interaction length on the interaction strength is explicitly influenced by the shock movement. With increasing shock travelling speed, the scaled interaction length here decreases significantly faster than in the quasi-stationary reference case. The end of this speed range is reached when the absolute shock front speed has caught up with the maximum speed of sound in the interaction zone, and thus the interaction length has fallen to zero. This travelling-speed limit signifies the transition to the third range, where upstream influence is no longer possible.