The Taylor–Maccoll (T–M) equations are the governing equations for steady inviscid irrotational axisymmetric conical flow, and have been widely applied to the design of waveriders and intakes. However, only four classic solutions have been reported: external conical flow (ECF), Busemann flow and internal conical flow of types A and B (ICFA and ICFB). In this work, the analysis of the T–M equations clarifies all possible solutions and reveals their relations. The domain where elementary solutions exist is divided into four domains. The classic Busemann and ICFB solutions share the same elementary solution as the template in a domain called the pre-shock domain, while the classic ECF and ICFA solutions belong to a domain named the ECF domain. Two new solutions, the inner flow of ECF (IECF) and degenerate conical flow (DCF), are found in the domains named after the corresponding solutions, namely the IECF and DCF domains. The IECF behaves as the mass injection supporting the classic ECF on an imaginary cone surface, while the DCF behaves as the conical expansion of a uniform flow. Furthermore, possible combinations of pre-shock solutions and supersonic post-shock solutions are clarified. The classic solutions are special cases where the pre-/post-shock solutions are combined with uniform flows. In general, the Busemann and ICFB solutions can be combined with any post-shock solutions in accord with the shock relations, including the ECF, ICFA, IECF and DCF solutions. In addition, numerical analyses are conducted to verify the validity of the two new solutions, DCF, IECF and one combined solution Busemann–ECF.

Isolated-roughness-induced transitions controlled by local wall heating strips are studied via direct numerical simulation and BiGlobal linear stability analysis. The transition mechanisms are studied first with different wall temperatures. The separated shear layer–counter-rotating vortex system is found to be the main source for transitions. Symmetric and antisymmetric modes are observed in the wake, and the former is dominant. The local wall heating strip can delay the transition, and this effect is enhanced with higher heating temperature, wider strip and a combination of upstream and downstream control strips. The upstream strip lifts up the inlet flow and weakens the wake system in an indirect manner. The antisymmetric mode gradually vanishes, while the symmetric mode always exists but becomes weaker. The downstream strip exhibits a more effective transition delay by directly weakening the separated shear layer and vortex system in the wake. Vorticity transport analysis suggests that the downstream strip increases dissipation for streamwise vorticity and transfers it into wall-normal and spanwise vorticity. BiGlobal analyses indicate that the downstream strip shows less influence on the peak growth rate of the symmetric mode but significantly shrinks its unstable region. Analyses of the disturbance energy production indicate that the upstream strip wakens the wall-normal and spanwise shear at the same time, but the downstream strip mainly wakens the wall-normal one. More simulations are performed with different roughness heights, point-source disturbance and different roughness shapes. The results show that the current method remains effective enough in delaying transitions at a wide range of conditions.

Supersonic internal flows often exhibit multiple reflected shocks within a limited distance. These shocks can interact with each other in a complex manner due to the characteristics of the shock wave–turbulent boundary layer interaction (STBLI), including flow distortion and the relaxing boundary layer. This study aims to characterise this type of interaction and to clarify its fluid physics. A separated STBLI zone was established either upstream or downstream, and another weaker STBLI was established in the opposing position to serve as a perturbation. Time-resolved measurements were employed to characterise the mean separation and unsteadiness as the two regions approached each other, as well as their relationship. The experimental results indicated that the STBLI could affect the separation and reattachment of the other STBLI through either the decelerated or relaxing boundary layer. Despite a small deflection angle, the incident shock can amplify the low-frequency oscillations in the downstream STBLI region. Additionally, the interaction in the downstream region can be influenced by both low- and high-frequency oscillations associated with the upstream STBLI through a relaxing boundary layer. Despite the limited correlation observed between the low-frequency fluctuations in the downstream region and the boundary layer flow not far upstream, there still exists some degree of correlation between the low-frequency shock motions even when they are widely separated. Both the ‘upstream mechanism’ and ‘downstream mechanism’ have been observed, and the significance of low-frequency dynamics in the separated flow, relative to that of the upstream flow, is closely associated with interaction intensity.

Regular reflection (RR) to Mach reflection (MR) transitions (${\rm RR}\leftrightarrow {\rm MR}$) on long wedges in steady supersonic flows have been well studied and documented. However, in a short wedge where the wedge length is small, the transition prediction becomes really challenging owing to the interaction of the expansion fan emanating from the trailing edge of the wedge with the incident shock and the triple/reflection point. The extent of this interaction depends on the distance between the wedge trailing edge and the symmetry line (Ht). This distance is a geometric combination of the distance of the wedge leading edge from the symmetry line $(H)$, the wedge angle ($\theta$) and the wedge length $(w)$. In the present study, we used the method of characteristics to model the complete wave interactions which accurately predicted shock curvatures and the reflection configurations for all ranges of the incoming flow Mach number. In the case of short wedges, the transition criterion strongly depends on the wedge length, which can be so adjusted even to eliminate the ${\rm RR}\rightarrow {\rm MR}$ transitions till the wedge angle reaches the no-reflection domain. Transition lines for both the detachment criterion and von Neumann criterion are also drawn to investigate the dual solution domain, and the reflection configurations were verified experimentally for the first time on short wedges. By using proper input configuration parameter ($w/H$), various types of shifts in the dual solution domain for short wedges are studied and categorised into three types, namely Type I, Type II and Type III.

The complex behaviours of supersonic turbulent boundary-layer flows interacting with combustion is explored through the use of direct numerical simulations. The chosen flow model is a non-premixed hydrogen–air flame ignited within a three-dimensional supersonic turbulent flat-plate boundary layer operating at a Mach number of 2.33 and a friction Reynolds number of approximately 1000. The simulation involves a finite-rate model for the hydrogen–air reaction including 9 species and 19 steps with adiabatic non-catalytic wall conditions. The inlet flow consists of preheated air in the main stream and a hydrogen film injected in proximity to the cold wall, inducing mixing and ignition in the outer layer. The specific inlet configuration leads to two successive transition stages of distinct mechanisms, with the first stage related to the mixing-layer instability and the second one to the boundary-layer instability. The excess Reynolds stresses resulting from the transition exhibit a downstream decrease, with values being lower than canonical adiabatic profiles, particularly in the outer layer. This difference is attributed to the reduced local friction Reynolds numbers, associated with the non-classical wall-normal distributions of density and viscosity. The effect of combustion on the recovery process and skin friction is analysed in comparison with a supplemented chemistry-frozen setting. Velocity–temperature and velocity–species correlations are further examined. The wall-normal profiles of turbulent Prandtl number tend to classical non-reacting values and the turbulent Schmidt number is only slightly affected by the reaction. The strong Reynolds analogy linking velocity and temperature fluctuations is found to be invalid in the outer layer due to the presence of large-scale temperature fluctuations and the inhomogeneity of hydrogen gas distribution, as revealed by a spectrum analysis. A statistical analysis of elementary reactions at varying wall-normal distances is provided, highlighting the dominance of hydrogen atom depletion in the inner region and the prevalence of water vapour production in the outer region, together contributing to the chemical heat release across the boundary layer. Turbulence–chemistry interaction is assessed through a comparison of mean turbulent and laminar chemical heat release rates, showing significant differences of up to an order of magnitude. Using a simple diagnostic tool, the modelling of fluctuations associated with hydrogen gas and two intermediate products is found to be crucial for turbulent chemistry closure.

As shown by Wenzel et al. (J. Fluid Mech., vol. 930, 2022, A1), the Eckert number $Ec$ defined using the difference between recovery temperature $\bar{T}_r$ and wall temperature $\bar{T}_w$ can be understood as a meaningful quantity to compare heat-transfer effects inside compressible turbulent boundary layers (for a calorically perfect gas), no matter whether these are caused by different Mach-number or wall-temperature conditions. While the named study deduced this comparative behaviour of $Ec$ from an integral perspective in a strict sense, Cogo et al. (J. Fluid Mech., vol. 974, 2023, A10) performed a systematic parameter study based on the previous findings to look at wall-normal profiles. They have shown that the diabatic parameter $\varTheta$, being equivalent to $Ec$, is capable of categorizing heat-transfer effects for cases at different Mach numbers, even to some extent for some of the wall-normal profiles. Building on this progress, the present paper provides a comprehensive classification of both existing and newly computed super- and hypersonic direct numerical simulation data at various wall temperature conditions into heated cases, adiabatic cases or weakly/moderately/strongly/quasi-incompressibly cooled cases. Hereby, the classification is largely based on the wall-normal position of the temperature peak occurring in cooled boundary-layer cases, which is one of the determining factors for the topological characteristics of diabatic boundary-layer profiles. Integrating high-enthalpy data into the analysis allowed us to confirm the reliability of the proposed classification also in more complex scenarios, where the calorically perfect gas assumption no longer applies and additional heat-transfer mechanisms come into play. While the Eckert number is shown to well characterize heat-transfer effects on most important temperature-related quantities for a wide range of Mach numbers and $\bar {T}_w/\bar {T}_r$ conditions, also the local Reynolds number $Re_{\tau }$ is shown to notably affect the strength of heat-transfer effects. Since both $Ec$ and $Re_{\tau }$ can be determined in advance – or estimated to a reasonable extent – a key advantage of the classification scheme is to allow for an effective a priori estimation of the extent to which heat-transfer effects are to be expected for a given compressible turbulent boundary-layer configuration.

The unsteady mechanism of unstart flow for an inlet with rectangular-to-elliptical shape transition (REST) under the off-design condition at a Mach of 4 is investigated using the delay detached eddy simulation method. With the help of numerical simulations, the unsteady dynamics, especially the low-frequency characteristics of the REST inlet unstart flow, as well as the self-sustaining mechanism, is investigated. The instantaneous flow illustrates the unsteady phenomena of the REST unstart flow, including the interaction between the cowl-closure leading edge (CLE) shock and the shear layer, breathing of the separation bubble, flapping of the separation shock, instability of the shear layer and vortex shedding along the shear layer. The spectral analysis reveals that the lower frequency dynamics is associated with the breathing of the separation bubble and the flapping motion of the separation shock wave, while the higher frequency is related to the instability of the shear layer affected by cowl-closure leading edge shock and the formation of shedding vortices. Further, coherence analysis shows that the contribution of these flow structures dominating the low-frequency dynamics couple with each other. Based on the dynamic mode decomposition results, the characteristics that contribute to the unsteady behaviour of unstart flow are summarized. The streamwise vortices downstream of the separation and the shedding vortices are believed to be the main driving force of the global low-frequency unsteadiness of the REST inlet unstart flow under the off-design condition. Moreover, the CLE shock plays an important role in the process during the dominant flow structure conversion from the backflow within the separation bubble into elongated streamwise structures.

We investigate Reynolds number effects in strong shock-wave/turbulent boundary-layer interactions (STBLI) by leveraging a new database of wall-resolved and long-integrated large-eddy simulations. The database encompasses STBLI with massive boundary-layer separation at Mach $2.0$, impinging-shock angle $40^{\circ }$ and friction Reynolds numbers ${\textit {Re}}_\tau$ $355$, $1226$ and $5118$. Our analysis shows that the shape of the reverse-flow bubble is notably different at low and high Reynolds number, while the mean-flow separation length, separation-shock angle and incipient plateau pressure are rather insensitive to Reynolds number variations. Velocity statistics reveal a shift in the peak location of the streamwise Reynolds stress from the separation-shock foot to the core of the detached shear layer at high Reynolds number, which we attribute to increased pressure transport in the separation-shock excursion domain. Additionally, in the high Reynolds case, the separation shock originates deep within the turbulent boundary, resulting in intensified wall-pressure fluctuations and spanwise variations associated with the passage of coherent velocity structures. Temporal spectra of various signals show energetic low-frequency content in all cases, along with a distinct peak in the bubble-volume spectra at a separation-length-based Strouhal number $St_{L_{sep}}\approx 0.1$. The separation shock is also found to lag behind bubble-volume variations, consistent with the acoustic propagation time from reattachment to separation and a downstream mechanism driving the shock motion. Finally, dynamic mode decomposition of three-dimensional fields suggests a Reynolds-independent statistical link among separation-shock excursions, velocity streaks and large-scale vortices at low frequencies.

Three-dimensional effects of sidewalls on the low-frequency unsteadiness of the shock-wave/boundary-layer interaction (SBLI) are of academic and practical importance but not yet well understood. Considerable attention has been paid to the viscous effect of sidewalls, whereas the potential inviscid confinement effect of sidewalls has received little attention. The present work provides experimental evidence of multiscale spanwise travelling waves crossing the separation front under the confinement of sidewalls. Global pressure measurements were made for a sidewall-confined 24$^\circ$ compression ramp interaction in Mach-2.83 flow using fast-responding pressure-sensitive paint. The unsteady pressure in a statistically two-dimensional intermittent region suggests that in addition to the canonical streamwise oscillation, the separation front exhibits significant low-frequency, multiscale spanwise distortion. Modal analysis further reveals that multiscale spanwise unsteadiness has higher intensity and frequency than the streamwise oscillation. Such strong spanwise unsteadiness calls attention to the low-frequency unsteadiness in previous sidewall-confined SBLI experiments and encourages further study on the mechanism of the confinement effect.

This paper investigates the linear and nonlinear evolution of radiating modes in supersonic boundary layers under the influence of impinging sound waves. It is found that the ensuing boundary-layer response is extraordinarily large for a subset of the sound frequency and incident angle, and the resonant over-reflection, corresponding to the reflection coefficient becoming infinite, occurs at a particular pairing of frequency and incident angle. At this point, the reflected wave coincides with a locally neutral radiating mode, which emits spontaneously sound in the form of Mach waves. A fundamental resonance takes place between the incident wave and the radiating mode. Viewed in a developing boundary layer, the response is rendered finite by introducing non-parallelism and nonlinear effects near the neutral location of the radiating mode, where the sound wave directly excites the radiating mode and/or acts on the pre-existing radiating mode. Inhomogeneous amplitude equations are derived to describe the excitation as well as the nonlinear development of the radiating mode in the two regimes where non-equilibrium and non-parallelism play a leading-order role, respectively. A composite amplitude equation is then constructed to take into account both non-parallelism and non-equilibrium effects. This amplitude equation is, with an appropriate initial condition, solved to quantify the impact of the impinging sound wave on the linear and nonlinear instability characteristics of the radiating mode. The far-field analysis shows that the Mach wave field of the radiating mode is changed significantly due to the incident sound.

This paper extends the work of Tamano & Morinishi (J. Fluid Mech., vol. 548, 2006, pp. 361–373) by simulating supersonic turbulent channel flow with asymmetric thermal walls using a larger computational domain and a finer mesh. Direct numerical simulation is carried out for four cases with different thermal wall boundaries at the top wall at fixed $Ma=1.5$, $Re=6000$ and $Pr=0.72$, while the bottom wall is maintained at a constant temperature of $T_L$ equal to the reference temperature. These cases are referred to as the adiabatic case TAd, where the top wall is adiabatic; the pseudo-adiabatic case T32, where the top wall is isothermal with temperature $T_{w,t}=T_A$; the sub-adiabatic case T25, with $T_{w,t}=0.77T_A$; and the super-adiabatic case T40, with $T_{w,t}=1.24T_A$. Here, $T_A=3.234$ is the mean temperature at the adiabatic wall in the TAd case. The objective of this study is to compare and contrast the TAd case with its corresponding T32 case, and to investigate the effect of the wall temperature difference between the two isothermal walls. Comparisons of the basic turbulent statistics, the heat transfer between the Favre-averaged mean-flow kinetic energy, the Favre-averaged turbulent kinetic energy and the Favre-averaged mean internal energy, as well as the wall heat transfer properties, indicate that the TAd case and its corresponding T32 case are generally equivalent. The only discernible difference is in the region very close to the top wall for the temperature-fluctuation-related quantities. The analysis reveals that the asymmetry of the thermal walls causes asymmetry in the flow and thermal fields. In addition, the transfer of the heat generated by the pressure dilatation and the viscous stress is facilitated by the turbulent heat flux term and the mean molecular heat flux term.

Supersonic backward-facing step (BFS) flow is numerically studied using direct numerical simulation (DNS) and global stability analysis (GSA) with a free stream Mach number of 2.16 and a Reynolds number of 7.938 × 105 based on the flat-plate length L and free stream conditions. Two-dimensional BFS flow becomes unstable to three-dimensional perturbations as the step height h exceeds a certain value, while no two-dimensionally unstable mode is found. Global instability occurs with the fragmentation of the primary separation vortex downstream of the step. Two stationary modes and one oscillatory unstable mode are obtained at a supercritical ratio of L/h = 32.14, among which the two stationary modes originate from the coalescence of a pair of conjugate modes. The most unstable mode manifests itself as streamwise streaks in the reattached boundary layer, which is similar to that in shock-induced separated flow, although the flow separation mechanisms are different. Without introducing any external disturbances, the DNS captures the preferred perturbations and produces a growth rate in agreement with the GSA prediction in the linear growth stage. In the quasi-steady stage, the secondary separation vortex breaks up into several small bubbles, and the number of streamwise streaks is doubled. A low-frequency unsteadiness that may be associated with the oscillatory mode is also present.

Two-dimensional compressible flows in radial equilibrium are investigated in the ideal dilute-gas regime and the non-ideal single-phase regime close to the liquid–vapour saturation curve and the critical point. Radial equilibrium flows along constant-curvature streamlines are considered. All properties are therefore independent of the tangential streamwise coordinate. A differential relation for the Mach number dependency on the radius is derived for both ideal and non-ideal conditions. For ideal flows, the differential relation is integrated analytically. Assuming a constant specific heat ratio $\gamma$, the Mach number is a monotonically decreasing function of the radius of curvature for ideal flows, with $\gamma$ being the only fluid-dependent parameter. In non-ideal conditions, the Mach number profile also depends on the total thermodynamic conditions of the fluid. For high molecular complexity fluids, such as toluene or hexamethyldisiloxane, a non-monotone Mach number profile is admissible in single-phase supersonic conditions. For Bethe–Zel'dovich–Thompson fluids, non-monotone behaviour is observed in subsonic conditions. Numerical simulations of subsonic and supersonic turning flows are carried out using the streamline curvature method and the computational fluid dynamics software SU2, respectively, both confirming the flow evolution from uniform flow conditions to the radial equilibrium profile predicted by the theory.

Supersonic flow over a hollow cylinder/flare with a free-stream Mach number of 2.25 is numerically investigated in this study. Axisymmetric computational fluid dynamics simulations and global stability analysis (GSA) are performed for a wide range of cylinder radii and flare deflection angles. The onset of incipient and secondary separation is delayed as the cylinder radius is decreased due to the axisymmetric effects. The GSA reveals that a decrease in cylinder radius also postpones the emergence of global instability. The GSA results agree well with the results of direct numerical simulations for a supercritical case in the linear stage. The saturated flow exhibits pairs of unsteady streamwise streaks downstream of reattachment. The criterion of the global stability boundary established for supersonic flow over a compression corner (Hao et al., J. Fluid Mech, vol. 919, 2021, A4) is extended to its axisymmetric counterpart.

This paper presents a vortex breakdown study due to the interaction between a Batchelor vortex and the crossing of oblique shock waves with Mach numbers of 3.5 and 5.0, favourable for supersonic mixing and combustion, respectively. Numerical simulations were conducted to investigate the effects of the circulation intensity and shock angle on vortex breakdown. The results indicate that a breakdown occurs at the shock angle $\beta \geqslant 45^\circ$ or the vortex circulation $q = 0.32$, and the configuration is a bubble structure with a recirculation region; most of the breakdowns possess a stagnation point. Furthermore, the structure differs from that of a normal shock wave and vortex interaction because the bubble region is subsonic and does not comprise a normal shock wave on the inside. Additionally, this vortex breakdown shows that the momentum flux on the centreline decreases once at the tip of the bubble owing to a sudden drop in velocity in the subsonic region. In addition, the enstrophy production resulting from vortex stretching and tilting is found to have a significant advantage in the interaction region. Based on these results, the threshold required for a bubble vortex breakdown was theoretically derived as an inequality. The numerical simulation results support the theoretical criterion obtained from the proposed inequality. Therefore, a streamwise vortex breakdown resulting from the interaction between the vortex and intersecting oblique-shocks should be reasonably predicted.

The choking condition of a one-dimensional steady-state diabatic flow, with wall friction, of a perfect gas through a constant cross-section pipe has been analysed by applying a recent analytical solution. Such a choking condition can be achieved for an initially subsonic flow and a supersonic one, even when the heat flux goes from the fluid to the wall. Entropy–enthalpy diagrams have been analysed, and a universal choking condition for compressible diabatic flows with wall friction has been determined. The analytical solution of a supersonic flow, in which a normal shock occurs, has been obtained for a diabatic flow with wall friction and compared with the results of a numerical model.

The shock wave–turbulent boundary layer interaction over a compression corner is studied using global stability analysis (GSA) and resolvent analysis based on a separation of scales between the low-frequency, large-scale motions and the turbulent fluctuations. The GSA identifies a leading stationary mode, which becomes globally unstable as the ramp angle is beyond a critical value. For globally stable flows, the resolvent analysis captures two-dimensional and three-dimensional local maxima in optimal gain, both of which are due to modal resonance between the forcing and the leading global mode. Notably, the frequency-premultiplied optimal gain associated with two-dimensional disturbances peaks at a low frequency. For different interaction strengths, the peak frequencies collapse onto a universal value of 0.015 when non-dimensionalized using the length of the separation region and the free-stream velocity. A numerical simulation perturbed with the corresponding optimal forcing reveals that the response is in the form of a back-and-forth shock motion.

In this paper, we present a systematic study of the nonlinear evolution of the travelling Mack modes in a Mach 3 supersonic boundary layer over a rotating cone with a $7^{\circ }$ half-apex angle using the nonlinear parabolic stability equation (NPSE). To quantify the effect of cone rotation, six cases with different rotation rates are considered, and from the same streamwise position, a pair of oblique Mack modes with the same frequency but opposite circumferential wavenumbers are introduced as the initial perturbations for NPSE calculations. As the angular rotation rate $\varOmega$ increases such that $\bar \varOmega$ (defined as the ratio of the rotation speed of the cone to the streamwise velocity at the boundary-layer edge) varies from 0 to $O(1)$, three distinguished nonlinear regimes appear, namely the oblique-mode breakdown, the generalised fundamental resonance and the centrifugal-instability-induced transition. For each regime, the mechanisms for the amplifications of the streak mode and the harmonic travelling waves are explained in detail, and the dominant role of the streak mode in triggering the breakdown of the laminar flow is particularly highlighted. Additionally, from the linear stability theory, the dominant travelling mode undergoes the greatest amplification for a moderate $\varOmega$, which, according to the $e^N$ transition-prediction method, indicates premature transition to turbulence. However, this is in contrast to the NPSE results, in which a delay of the transition onset is observed for a moderate $\varOmega$. Such a disagreement is attributed to the different nonlinear regimes appearing for different rotation rates. Therefore, the traditional transition-prediction method based on the linear instability should be carefully employed if multiple nonlinear regimes may appear.

Steady shock reflection where the incident shock is free of interaction with other waves has been well studied. In this paper, we consider the less studied shock reflection problem where the incident shock interacts with the wedge trailing-edge expansion fan, which occurs when the wedge trailing-edge height surpasses a threshold. The influence of this interaction on the advance of transition from Mach reflection to regular reflection is quantified in terms of the wedge trailing-edge height ratio. The wave pattern, including primary and reflected Mach waves, for Mach reflection with interaction is clarified using computational fluid dynamics (CFD) and the method of characteristics. Those reflected Mach waves having an important effect on Mach reflection are identified. A simplified Mach stem model that accounts for the direct role of the interaction on the incident shock and its indirect role on the reflected shock and slipline is built up on a past model without interaction. Both theory and CFD show that the Mach stem height decreases nonlinearly with increasing trailing-edge height.

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.