Effect of Multilateral Jet Mixing on Stability and Structure of Turbulent Partially-Premixed Flames

An experimental study was conducted to investigate the effects of multilateral jet mixing, using both three and four side-jets, on the structure and stability of turbulent partially-premixed flames. Particle Image Velocimetry and OH*-chemiluminescence were used to study the effects of geometry and operating conditions on the resulting flow-field and reaction zone structures, respectively. These effects were compared under varying ratios of side-jet to primary flow momentum, whilst keeping the bulk flow constant. It was found that the mixing regimes upstream of the nozzle exit affect the flame characteristics, i.e. an impinging regime is likely to generate a lifted flame whilst a backflow regime is likely to generate an attached flame. Unlike the 4 side-jets cases, the OH* images and v _{r m s} profiles for the 3 side-jets cases show distinct asymmetry, with intense OH* and low velocity fluctuations on the opposite sides of the fuel injection. It was also found that the flow and scalar fields become independent of the upstream conditions, for both 3 and 4 side-jets, after one diameter downstream of the nozzle exit.     

Measurements of Turbulent Jet Mixing in a Turbulent Co-Flow Including the Influence of Periodic Forcing and Heating

In this work, the turbulent mixing of a confined coaxial jet in air is investigated by means of simultaneous particle image velocimetry and planar laser induced fluorescence of the acetone seeded flow injection. The jet is injected into a turbulent duct flow at atmospheric pressure through a 90 ^° pipe bend. Measurements are conducted in a small scale windtunnel at constant mass flow rates and three modes of operation: isothermal steady jet injection at a Dean number of 20000 (R e _d =32000), pulsed isothermal injection at a Womersley number of 65 and steady injection at elevated jet temperatures of ΔT=50 K and ΔT=100 K. The experiment is aimed at providing statistically converged quantities of velocity, mass fraction, turbulent fluctuations and turbulent mass flux at several downstream locations. Stochastic error convergence over the number of samples is assessed within the outer turbulent shear layer. From 3000 samples the statistical error of time-averaged velocity and mass fraction is below 1 % while the error of Reynolds shear stress and turbulent mass flux components is in the of range 5-6 %. Profiles of axial velocity and turbulence intensity immediately downstream of the bend exit are in good agreement with hot-wire measurements from literature. During pulsed jet injection strong asymmetric growing of shear layer vortices lead to a skewed mass fraction profile in comparison with steady injection. Phase averaging of single shot PLIF-PIV measurements allows to track the asymmetric shear layer vortex evolvement and flow breakdown during a pulsation cycle with a resolution of 10^°. Steady injection with increased jet temperature supports mixing downstream from 6 nozzle diameters onward.     

Pre-Chamber Ignition Mechanism: Simulations of Transient Autoignition in a Mixing Layer Between Reactants and Partially-Burnt Products

The structure of autoignition in a mixing layer between fully-burnt or partially-burnt combustion products from a methane-air flame at ? = 0.85 and a methane-air mixture of a leaner equivalence ratio has been studied with transient diffusion flamelet calculations. This configuration is relevant to scavenged pre-chamber natural-gas engines, where the turbulent jet ejected from the pre-chamber may be quenched or may be composed of fully-burnt products. The degree of reaction in the jet fluid is described by a progress variable c (c = taking values 0.5, 0.8, and 1.0) and the mixing by a mixture fraction ξ (ξ = 1 in the jet fluid and 0 in the CH₄-air mixture to be ignited). At high scalar dissipation rates, N₀, ignition does not occur and a chemically-frozen steady-state condition emerges at long times. At scalar dissipation rates below a critical value, ignition occurs at a time that increases with N₀. The flame reaches the ξ = 0 boundary at a finite time that decreases with N₀. The results help identify overall timescales of the jet-ignition problem and suggest a methodology by which estimates of ignition times in real engines may be made.     

Experimental and Theoretical Sensitivity Analysis of Turbulent Flow Past a Square Cylinder

The Siemens SGT-800 3^rd generation DLE burner fitted to an atmospheric combustion rig has been numerically investigated. Pure methane and methane enriched by 80 vol% hydrogen flames have been considered. A URANS (Unsteady Reynolds Averaged Navier-Stokes) approach was used in this study along with the k ? ω SST and the k ? ω SST-SAS models for the turbulence transport. The chemistry is coupled to the turbulent flow simulations by the use of a laminar flamelet library combined with a presumed PDF. The effect of the mesh density in the mixing and the flame region and the effect of the turbulence model and reaction rate model constant are first investigated for the methane/air flame case. The results from the k ? ω SST-SAS along with flamelet libraries are shown to be in excellent agreement with experimental data, whereas the k ? ω SST model is too dissipative and cannot capture the unsteady motion of the flame. The k ? ω SST-SAS model is used for simulation of the 80 vol% hydrogen enriched flame case without further adjusting the model constants. The global features of the hydrogen enrichment are very well captured in the simulations using the SST-SAS model. With the hydrogen enrichment the time averaged flame front location moves upstream towards the burner exit nozzle. The results are consistent with the experimental observations. The model captures the three dominant low frequency unsteady motion observed in the experiments, indicating that the URANS/LES hybrid model indeed is capable of capturing complex, time dependent, features such as an interaction between a PVC and the flame front.     

Measurement of local forcing on a turbulent boundary layer using PIV

An experimental study was carried out to investigate the effect of periodic blowing and suction on a turbulent boundary layer. Particle image velocimetry (PIV) was used to probe the characteristics of the flow. Local forcing was introduced to the boundary layer via a sinusoidally-oscillating jet issuing from a thin spanwise slot. Three forcing frequencies (f⁺=0.44, 0.66 and 0.88) with a fixed forcing amplitude (A⁺=0.6) were employed at Re _θ =690. The effect of three different forcing angles (α=60°, 90° and l20°) was investigated under a fixed forcing frequency (f⁺=0.088). The PIV results showed that the wall-region velocity decreases on imposition of the local forcing. Inspection of the phase-averaged velocity profiles revealed that spanwise large-scale vortices are generated downstream of the slot and persist farther downstream. The highest reduction in skin friction was achieved at the highest forcing frequency (f⁺=0.088) and a forcing angle of α=120°. The spatial fraction of the vortices was examined to analyze the skin friction reduction.     

Operator method for solving the plane elasticity problem for a strip with periodic cuts

In the present paper, we use the conformal mapping z/c = ζ?2a sin ζ (a, c?const, ζ = u + iv) of the strip {|v| ≤ v ₀, |u| < ∞} onto the domain D, which is a strip with symmetric periodic cuts. For the domain D, in the orthogonal system of isometric coordinates u, v, we solve the plane elasticity problem. We seek the biharmonic function in the form F = C ψ ₀ + S ψ*₀ + x(C ψ ₁ ? S ψ ₂) + y(C ψ ₂ + S ψ ₁), where C(v) and S(v) are the operator functions described in [1] and ψ ₀(u), …, ψ ₂(u) are the desired functions. The boundary conditions for the function F posed for v = ±v ₀ are equivalent to two operator equations for ψ ₁(u) and ψ ₂(u) and to two ordinary differential equations of first order for ψ ₀(u) and ψ*₀(u) [2]. By finding the functions ψ _j (u) in the form of trigonometric series with indeterminate coefficients and by solving the operator equations, we obtain infinite systems of linear equations for the unknown coefficients. We present an efficient method for solving these systems, which is based on studying stable recursive relations. In the present paper, we give an example of analysis of a specific strip (a = 1/4, v ₀ = 1) loaded on the boundary v = v ₀ by a normal load of intensity p. We find the particular solutions corresponding to the extension of the strip by the longitudinal force X ^∞ and to the transverse and pure bending of the strip due to the transverse force Y ^∞ and the constant moment M ^∞, respectively. We also present the graphs of normal and tangential stresses in the transverse cross-section x = 0 and study the stress concentration effect near the cut bottom.     

Numerical Study of Turbulent Jet Ignition in a Lean Premixed Configuration

Direct numerical simulations (DNS) of a hot combustion product jet interacting with a lean premixed hydrogen-air coflow are conducted to fundamentally investigate turbulent jet ignition (TJI) in a three-dimensional configuration. TJI is an efficient method for initiating and controlling combustion in ultra-lean combustion systems. Fully compressible gas dynamics and species equations are solved with high order finite difference methods. The hydrogen-air reaction is simulated with a reliable detailed chemical kinetics mechanism. The physical processes involved in the TJI-assisted combustion are investigated by considering the flame heat release, temperature, species concentrations, vorticity, and Baroclinc torque. The complex turbulent flame and flow structures are delineated in three main: i) hot product jet, ii) burned-mixed, and iii) flame zones. In the TJI-assisted combustion, the flow structures and the flame features such as flame speed, temperature, and species distribution are found to be quite different than those in “standard” turbulent premixed combustion due to the existence of a high energy turbulent hot product jet. The flow structures and statistics are also found to be different than those normally seen in non-isothermal non-reacting jets.     

Turbulent Scalar Mixing in a Skewed Jet in Crossflow: Experiments and Modeling

Turbulent mixing of an inclined, skewed jet injected into a crossflow is investigated using MRI-based experiments and a high-fidelity LES of the same configuration. The MRI technique provides three-dimensional fields of mean velocity and mean jet concentration. The 30° skew of the jet relative to the crossflow produces a single dominant vortex which introduces spanwise asymmetries to the velocity and concentration fields. The turbulent scalar transport of the skewed jet is investigated in further detail using the LES, which is validated against the experimental measurements. Mixing is found to be highly anisotropic throughout the jet region. Isotropic turbulent diffusivity and viscosity are used to calculate an optimal value of the turbulent Schmidt number, which varies widely over the jet region and lies mostly outside of the typically accepted range 0.7 ≤ S c _t ≤ 0.9. Finally, three common scalar flux models of increasing complexity are evaluated based on their ability to capture the anisotropy and predict the scalar concentration field of the present configuration. The higher order models are shown to better represent the turbulent scalar flux vector, leading to more accurate calculations of the concentration field. While more complex models are better able to capture the turbulent mixing, optimization of model constants is shown to significantly affect the results.     

Does Density Ratio Significantly Affect Turbulent Flame Speed?

In order to experimentally study whether or not the density ratio σ substantially affects flame displacement speed at low and moderate turbulent intensities, two stoichiometric methane/oxygen/nitrogen mixtures characterized by the same laminar flame speed S _L = 0.36 m/s, but substantially different σ were designed using (i) preheating from T _u = 298 to 423 K in order to increase S _L , but to decrease σ, and (ii) dilution with nitrogen in order to further decrease σ and to reduce S _L back to the initial value. As a result, the density ratio was reduced from 7.52 to 4.95. In both reference and preheated/diluted cases, direct images of statistically spherical laminar and turbulent flames that expanded after spark ignition in the center of a large 3D cruciform burner were recorded and processed in order to evaluate the mean flame radius \(\bar {R}_{f}\left (t \right )\) and flame displacement speed \(S_{t}=\sigma ^{-1}{d\bar {R}_{f}} \left / \right . {dt}\) with respect to unburned gas. The use of two counter-rotating fans and perforated plates for near-isotropic turbulence generation allowed us to vary the rms turbulent velocity \(u^{\prime }\) by changing the fan frequency. In this study, \(u^{\prime }\) was varied from 0.14 to 1.39 m/s. For each set of initial conditions (two different mixture compositions, two different temperatures T _u , and six different \(u^{\prime })\), five (respectively, three) statistically equivalent runs were performed in turbulent (respectively, laminar) environment. The obtained experimental data do not show any significant effect of the density ratio on S _t . Moreover, the flame displacement speeds measured at u′/S _L = 0.4 are close to the laminar flame speeds in all investigated cases. These results imply, in particular, a minor effect of the density ratio on flame displacement speed in spark ignition engines and support simulations of the engine combustion using models that (i) do not allow for effects of the density ratio on S _t and (ii) have been validated against experimental data obtained under the room conditions, i.e. at higher σ.     

Effects of Fuel Lewis Number on Localised Forced Ignition of Turbulent Mixing Layers

The influences of fuel Lewis number Le _F on localised forced ignition of inhomogeneous mixtures are analysed using three-dimensional compressible Direct Numerical Simulations (DNS) of turbulent mixing layers for Le _F  = 0.8, 1.0 and 1.2 and a range of different root-mean-square turbulent velocity fluctuation u′ values. For all Le _F cases a tribrachial flame has been observed in case of successful ignition. However, the lean premixed branch tends to merge with the diffusion flame on the stoichiometric mixture fraction isosurface at later stages of the flame evolution. It has been observed that the maximum values of temperature and reaction rate increase with decreasing Le _F during the period of external energy addition. Moreover, Le _F is found to have a significant effect on the behaviours of mean temperature and fuel reaction rate magnitude conditional on mixture fraction values. It is also found that reaction rate and mixture fraction gradient magnitude \(\vert \nabla \xi \vert \) are negatively correlated at the most reactive region for all values of Le _F explored. The probability of finding high values of \(\vert \nabla \xi \vert \) increases with increasing Le _F . For a given value of u′, the extent of burning decreases with increasing Le _F . A moderate increase in u′ gives rise to an increase in the extent of burning for Le _F  = 0.8 and 1.0, which starts to decrease with further increases in u′. For Le _F  = 1.2, the extent of burning decreases monotonically with increasing u′. The extent of edge flame propagation on the stoichiometric mixture fraction ξ = ξ _st isosurface is characterised by the probability of finding burned gas on this isosurface, which decreases with increasing u′ and Le _F . It has been found that it is easier to obtain self-sustained combustion following localised forced ignition in case of inhomogeneous mixtures than that in the case of homogeneous mixtures with the same energy input, energy deposition duration when the ignition centre is placed at the stoichiometric mixture. The difficultly to sustain combustion unaided by external energy addition in homogeneous mixture is particularly prevalent in the case of Le _F  = 1.2.     

Vortical structures behind a transverse jet in a supersonic flow at high jet to crossflow pressure ratios

A three-dimensional supersonic turbulent flow with symmetric normal injection of circular jets from the channel walls is numerically simulated. The initial Favre-averaged Navier–Stokes equations closed by the k–ω turbulence model are solved by an algorithm based on an ENO scheme. The mechanism of the formation of vortical structures due to the interaction of the jet with the free stream is studied for jet to crossflow total pressure ratios ranging from 3 to 50. It is known from experiments reported in the literature that, for n ? 10, mixing of the jet with the high-velocity flow leads to the formation of a pair of vortices and of an additional separation zone near the wall behind the jet. It is demonstrated that the present numerical results are consistent with such findings and that the pressure distribution on the wall ahead of the jet in the plane of symmetry is also in reasonable agreement with available experimental data.     

Numerical Simulation of Two Phase Turbulent Flow of Nanofluids in Confined Slot Impinging Jet

In this article, a numerical investigation is performed on flow and heat transfer of confined impinging slot jet, with a mixture of water and Al₂O₃ nanoparticles as the working fluid. Two-dimensional turbulent flow is considered and a constant temperature is applied on the impingement surface. The k ? ω turbulence model is used for the turbulence computations. Two-phase mixture model is implemented to study such a flow field. The governing equations are solved using the finite volume method. In order to consider the effect of obstacle angle on temperature fields in the channel, the numerical simulations were performed for different obstacle angles of 0° ? 60°. Also different geometrical parameters, volume fractions and Reynolds numbers have been considered to study the behavior of the system in terms of stagnation point, average and local Nusselt number and stream function contours. The results showed that the intensity and size of the vortex structures depend on jet- impingement surface distance ratio (H/W) and volume fraction. The maximum Nusselt number occurs at the stagnation point with the highest values at about H/W = 1. Increasing obstacle angle, from 15° to 60°, enhances the heat transfer rate. It was also revealed that the minimum value of average Nusselt number occurs in higher H/W ratios with decreasing the channel length.     

Evaluations of SGS Combustion,Scalar Flux and Stress Models in a Turbulent Jet Premixed Flame

A newly developed fractal dynamic SGS (FDSGS) combustion model and a scale self-recognition mixed (SSRM) SGS stress model are evaluated along with other SGS combustion, scalar flux and stress models in a priori and a posteriori manners using DNS data of a hydrogen-air turbulent plane jet premixed flame. A posteriori tests reveal that the LES using the FDSGS combustion model can predict the combustion field well in terms of mean temperature distributions and peak positions in the transverse distributions of filtered reaction progress variable fluctuations. A priori and a posteriori tests of the scalar flux models show that a model proposed by Clark et al. accurately predicts the counter-gradient transport as well as the gradient diffusion, and introduction of the model of Clark et al. into the LES yields slightly better predictions of the filtered progress variable fluctuations than that of a gradient diffusion model. Evaluations of the stress models reveal that the LES with the SSRM model predicts the velocity fluctuations well compared to that with the Smagorinsky model.     

Vortex-enhanced mixing through active and passive flow control methods

This study aims to understand the underlying physics of vortex-enhanced mixing through active and passive flow control methods. To find a best flow control method that enhances turbulent mixing through the generation of streamwise vortices, an experimental investigation was carried out to compare active and passive flow control methods of an incompressible axisymmetric jet. For active flow control, the lip of the circular jet was equipped with a single small flap deflected away from the jet stream at an angle of 30° to the jet axis. The flap incorporated a flow control slot through which steady and oscillatory suction were implemented. The active flow control methods require power input to the suction devices. For passive flow control, the lip of the circular jet was equipped with a single small delta tab deflected into the jet stream at an angle of 30° to the jet axis. The chord lengths of the flap and delta tab were one-sixth of the jet diameter. The momentum of jet increased in the case of active flow control by entraining the ambient fluid, whereas momentum decreased in the case of passive flow control. The effect of steady suction saturated for volumetric suction coefficient values greater than 0.82 %. The strength of streamwise vortices generated by the flap were greater than those generated by the delta tab. Steady suction produced positive pressures just downstream of the flow control slot in the central portion of the flap and negative pressures at the flap edges. Oscillatory suction was highly dependent on dimensionless frequency (F ⁺) based on the distance from the flow control slot to the flap trailing edge; the pressures on the central portion of the flap increased for F ⁺ ≤ 0.11 and then decreased for greater F ⁺; finally attained negative pressures at F ⁺ = 0.44. The increase in jet momentum and turbulence intensity, combined with the induced streamwise vorticity, makes steady suction a potential concept for increasing propulsion efficiency through vortex-enhanced mixing. The flow control methods modify the jet flow, which in turn would alter the jet noise spectra.     

Decay of Almost Periodic Solutions¶of Conservation Laws

We consider the asymptotic behavior of solutions of systems of inviscid or viscous conservation laws in one or several space variables, which are almost periodic in the space variables in a generalized sense introduced by Stepanoff and Wiener, which extends the original one of H. Bohr. We prove that if u(x,t) is such a solution whose inclusion intervals at time t, with respect to ?>0, satisfy l _epsiv;(t)/t→0 as t→∞, and such that the scaling sequence u ^T (x,t)=u(T x,T t) is pre-compact as t→∞ in L _loc ¹(? ^{d +1} ₊, then u(x,t) decays to its mean value \(\), which is independent of t, as t→∞. The decay considered here is in L ¹ _loc of the variable ξ≡x/t, which implies, as we show, that \(\) as t→∞, where M _x denotes taking the mean value with respect to x. In many cases we show that, if the initial data are almost periodic in the generalized sense, then so also are the solutions. We also show, in these cases, how to reduce the condition on the growth of the inclusion intervals l _?(t) with t, as t→∞, for fixed ? > 0, to a condition on the growth of l _?(0) with ?, as ?→ 0, which amounts to imposing restrictions only on the initial data. We show with a simple example the existence of almost periodic (non-periodic) functions whose inclusion intervals satisfy any prescribed growth condition as ?→ 0. The applications given here include inviscid and viscous scalar conservation laws in several space variables, some inviscid systems in chromatography and isentropic gas dynamics, as well as many viscous 2 × 2 systems such as those of nonlinear elasticity and Eulerian isentropic gas dynamics, with artificial viscosity, among others. In the case of the inviscid scalar equations and chromatography systems, the class of initial data for which decay results are proved includes, in particular, the L ^∞ generalized limit periodic functions. Our procedures can be easily adapted to provide similar results for semilinear and kinetic relaxations of systems of conservation laws.     

Helical Structures in the Near Field of a Turbulent Pipe Jet

We perform a finely resolved Large-eddy simulation to study coherent vortical structures populating the initial (near-nozzle) zone of a pipe jet at the Reynolds number of 5300. In contrast to ‘top-hat’ jets featured by Kelvin-Helmholtz rings with the non-dimensional frequency S t≈0.3?0.6, no high-frequency dominant mode is observed in the near field of a jet issuing from a fully-developed pipe flow. Instead, in shear layers we observe a relatively wide peak in the power spectrum within the low-frequency range (S t≈0.14) corresponding to the propagating helical waves entering with the pipe flow. This is confirmed by the Fourier transform with respect to the azimuthal angle and the Proper Orthogonal Decomposition complemented with the linear stability analysis revealing that this low-frequency motion is not connected to the Kelvin-Helmholtz instability. We demonstrate that the azimuthal wavenumbers m=1?5 contain the most of the turbulent kinetic energy and that a common form of an eigenmode is a helical vortex rotating around the axis of symmetry. Small and large timescales are identified corresponding to “fast” and “slow” rotating modes. While the “fast” modes correspond to background turbulence and stochastically switch from co- to counter-rotation, the “slow” modes are due to coherent helical structures which are long-lived and have low angular velocities, in agreement with the previously described spectral peak at low S t.     

Effects of Synthetic jets on a D-Shaped Cylinder wake at a Subcritical Reynolds Number

Effects of synthetic jets on the wake of a D-shaped cylinder is investigated experimentally at a Reynolds number Re_H= 47,000, based on incoming free-stream velocity and the cylinder height (H). The synthetic jets are introduced immediately from the upper and lower trailing edges of the cylinder. The upper and lower synthetic jets are operated in an in-phase or anti-phase mode, and at a momentum ratio C_μ= 1.0% and perturbation frequency St_A= 0.11 ?0.37. The cylinder wake with perturbation is examined in detail and compared with that without, based on smoke-wire flow visualization, pressure transducer and hotwire rake measurements, and data analyses of spectra, tempo-spatial cross-correlation and proper orthogonal decomposition (POD). Large-scale vortical structures in the cylinder wake are significantly modified by the synthetic jets perturbations, exhibiting symmetric or asymmetric patterns, depending on the perturbation frequency and phase relationship of the synthetic jets. These observations are internally correlated with the drag force variations.     

Spray-Flame Dynamics in a Rich Droplet Array

We numerically study spray-flame dynamics. The initial state of the spray is schematized by alkane droplets located at the nodes of a face-centered 2D-lattice. The droplets are surrounded by a gaseous mixture of alkane and air. The lattice spacing s reduced by the combustion length scale is large enough to consider that the chemical reaction occurs in a heterogeneous medium. The overall spray equivalence ratio is denoted by ?_T, with ?_T = ?_L + ?_G, where ?_G corresponds to the equivalence ratio of the gaseous surrounding mixture at the initial saturated partial pressure, while ?_L is the so-called liquid loading. To model such a heterogenous combustion, the retained chemical scheme is a global irreversible one-step reaction governed by an Arrhenius law, with a modified heat of reaction depending on the local equivalence ratio. ?_T is chosen in the range 0.9 ≤ ?_T ≤ 2. Three geometries (s = 3, s = 6, s = 12) and four liquid loadings, ?_L = 0.3, ?_L = 0.5, ?_L = 0.7, ?_L = 0.85 are studied. In the rich sprays, our model qualitatively retrieves the recent experimental measurements: the rich spray-flames can propagate faster than the single-phase flames with the same overall equivalence ratio. To analyse the conditions for this enhancement, we introduce the concept of “spray Peclet number”, which compares the droplet vaporization time with the combustion propagation time of the single-phase flame spreading in the fresh surrounding mixture.     

Mach waves produced in the supersonic jet mixing layer by shock/vortex interaction

The noise emission of free jets has been extensively investigated for many decades. At subsonic jet velocities, coherent structures of the mixing layer move at subsonic speed and emit sound waves. Free jets blowing at supersonic speeds, however, can emit weak shock waves, called Mach waves. At supersonic speeds, two cases must be distinguished: the structures move either subsonically or supersonically relative to the inside and/or outside speed of sound. In the case of supersonic movement, the Mach waves exist inside as well as outside the jet. At subsonic speeds, no Mach waves appear. Although numerous theories have been established to find the origin of the Mach waves, to the authors’ best knowledge, the mechanism of the Mach wave formation has not yet been clearly explained. Recently another theory of Mach waves in supersonic jets was developed, as described herein, which outlines the causes for the Mach wave production and stability as well as their dynamics. The theory’s principle is that the Mach waves are initiated by vortices which move downstream at three speeds w, \({w}'\) and \({w}''\) inside of the mixing layer. These three types of vortices and Mach waves are described in a comprehensive manner by the theory and are called the “w-, \({w}'\)- and \({w}''\)-vortices” and “w-, \({w}'\)- and \({w}''\)-Mach waves,” respectively.