## Abstract

We perform an inviscid, spatial stability analysis of supersonic, heated round jets with the mean properties assumed uniform on either side of the jet shear layer, modelled here via a cylindrical vortex sheet. Apart from the hydrodynamic Kelvin–Helmholtz (K–H) wave, the spatial growth rates of the acoustically coupled supersonic and subsonic instability waves are computed for axisymmetric conditions (*m*=0) to analyse their role on the jet stability, under increased heating and compressibility. With the ambient stationary, supersonic instability waves may exist for any jet Mach number *M*_{j}≥2, whereas the subsonic instability waves, in addition, require the core-to-ambient flow temperature ratio *T*_{j}/*T*_{o}>1. We show, for moderately heated jets at *T*_{j}/*T*_{o}>2, the acoustically coupled instability modes, once cut on, to govern the overall jet stability with the K–H wave having disappeared into the cluster of acoustic modes. Sufficiently high heating makes the subsonic modes dominate the jet near-field dynamics, whereas the supersonic instability modes form the primary Mach radiation at far field.

## 1. Introduction

The aeroacoustics of supersonic, turbulent jets can be complex as it potentially includes additional sound source mechanisms not available at the subsonic speeds [1–3]. Jets exhausting from engines of supersonic aircrafts have significantly higher core temperatures than the ambient, especially if fitted with afterburners, which can significantly broaden the spectrum of unstable modes, possibly leading to even more sources of sound. Traditionally, jet noise reduction engineering has been focused towards the higher-bypass-ratio subsonic turbofan engines of civilian aircrafts, which are subjected to increasingly stricter global noise regulations. Supersonic engines of military aircrafts or those that potentially could be used in the next-generation supersonic civilian transports are essentially of lower bypass ratios and thus a higher velocity ratio persists between the core supersonic jet and the subsonic coflow/ambient jet. Here, the role of fan coflow in suppressing the acoustic radiation efficiencies of the core jet unsteady disturbances is minimal [4], thus retaining the importance of jet noise and the need for understanding its basic noise source mechanisms at these conditions. Apart from its role in aural injury among the communities in closer proximity to airstrips serving supersonic military aircrafts, increasingly common in densely populated regions, supersonic jet noise is also a concern on board aircraft carriers where it may potentially deafen personnel.

Once the pressure mismatch between the core supersonic jet and ambient is absent, as may be the case during the steady, level flight phases of aircraft operation, turbulent mixing noise assumes prominence over other sources of supersonic jet noise [5]. Although, mixing noise may include some contributions from the smaller turbulent scales, it is now well-established that jet shear layers subjected to Kelvin–Helmholtz (K–H)-like instabilities create large-scale structures that are the dominant contributor to noise from subsonic and low-supersonic jets [6–8]. Thus, in spite of their basic turbulent nature, methods based on the linear and quasi-linear stability theories [9–11], including those based on parabolized stability equations (PSEs) [12,13] have provided reasonable mechanistic models for the dominant sound sources from the largest wave-like eddies of jet instabilities. This appears to be especially true for fluctuations that propagate with supersonic phase speeds, when the K–H instability modes radiate efficiently as Mach waves coinciding with the measured frequencies and polar directions of the peak noise from such jets [5,9].

It has been known that supersonic jets may support at least two additional type of acoustically coupled instability waves [1,2], of which the first type are the supersonic phase speed instability modes, whose first radial mode is known to exist once (*M*_{j}−*M*_{o})>2 [3], where *M*_{j} and *M*_{o} are the core and ambient jet Mach numbers, respectively, with the speed of sound assumed to be uniform everywhere. Whether these modes, once they exist, or the usual K–H mode form the dominant radiation component for cold supersonic jets will depend upon the magnitude of (*M*_{j}−*M*_{o}), but it is fair to say that the net radiation depends directly on their relative unstable growth rates and phase speeds. The second type of instability modes have subsonic phase speeds: the subsonic instability modes, which exist as upstream-propagating neutral modes at lower Mach numbers [2], whereas a portion of their spectrum seems to be downstream-propagating and unstable at higher Mach numbers [14]. These additional instabilities are not purely hydrodynamic, rather a coupling between the acoustic waves, repeatedly reflected off the flow interface at certain favourable angles, with the shear layer hydrodynamics leads to their growth [15]. We are particularly interested in how the spatial evolution of these new instability modes measure up against the K–H instability, especially at higher compressibility and heating conditions. This is important because, for example, at higher *M*_{j}, at least some of the supersonic instability modes are predicted to grow faster, as the spatial growth rate of the corresponding K–H mode drops rapidly with increased *M*_{j} [16]. However, it has also been suggested that at appropriately lower supersonic Mach numbers, K–H mode still dominates radiation as the other instability modes are either neutral or decaying so that their role in sound generation may be safely ignored [17]. In this work, all the unstable modes including the K–H mode, are labelled as (*m*,*n*), where *m* is the corresponding azimuthal mode number (only *m*=0 results are included here), whereas the radial mode number *n* is ordered in terms of the decreasing phase speed *c* of the modes at a chosen parametric configuration (see §3a).

As the core supersonic jet mean temperature *T*_{j} exceeds the ambient *T*_{o}, such that *κ*_{T} has on the growth rates of acoustically coupled instability modes. At the higher levels of jet compressibility (*M*_{j}−*M*_{o})>2, of interest here, increased core jet temperature would generally be expected to augment the compressibility effects beyond a point by reducing the growth rates of these instability modes [3]. Presumably, the suppression of hydrodynamic pressure interactions via the finite speed-of-sound effects [18,19] play a role in their respective growth rates, similar to what has been previously observed in fully developed turbulence [20,21]. However, at higher *κ*_{T}, the instability spectrum of acoustically coupled modes widens which complicates the overall picture as some of these are now downstream-propagating with subsonic phase speeds which has the potential of being more unstable, as we shall see in §3d.

Here, we mention that absolute instability of heated round jets is a concern only at low-subsonic Mach numbers [22–24], but as *M*_{j} is raised the region of absolute instability disappears even at moderate subsonic Mach numbers *M*_{j}>0.5, for negligible-thickness shear layers subjected to moderate to low heating *κ*_{T}<1.7 [24]. For the higher supersonic, Mach numbers of interest here, we exclude any possibility of absolute instability, because the sufficiently high temperature ratios needed to achieve such instability state is deemed impractical, while also beyond the scope of this work.

Numerically computing global modes of even cold supersonic jets can be expensive [25], more so for heated cases, restricting such methods to lower Reynolds numbers, while quasi-parallel methods like PSE suffers from unmanageable errors once strong heating is introduced [26]. Our requirement of tracking the evolution of all the major instability mechanisms in moderate to high-Mach-number heated supersonic jets (1<*M*_{j}<5 and 1<*κ*_{T}<3) at high Reynolds numbers (theoretically inviscid) necessitated the choice of simplified physical models. We consider an axisymmetric, inviscid supersonic jet in a low-Mach-number ambient flow, with uniform velocity and temperature profiles for each of the streams (figure 1), a reasonable approximation for uniformly heated high-speed jets, especially near the nozzle exit. Assumption of a homentropic base flow and the absence of any heat conduction or heat source eliminates the entropy waves with only the role of acoustic and vortical modes on the spatial stability of such jets to be investigated. It is to be noted that even if present, the entropy-to-acoustic wave conversion may only be possible for flows with strong mean velocity gradients, without which the role of entropy waves in the far-field sound is deemed insignificant [27].

Thickness of the shear layer does affect the stability properties of instability modes with direct consequences to the radiated sound, especially at the lower Strouhal numbers (*St*=*fD*/*U*_{j}, where *f* is the frequency in Hz, *D* the nozzle diameter and *U*_{j} the jet speed) of interest (*St*<0.4), where the peak radiation normally occurs. However, it has been recently shown that unless *St* is scaled by the momentum thickness (instead of *D*), thus directly including the effect of shear layer thickness, the differences in sound levels from several jets with varying shear layer thicknesses turned out to be not that significant even at lower frequencies [28]. Elsewhere, using a procedure described in Tam & Burton [10], growth rates for the supersonic and subsonic instability modes were computed by Tam & Hu [2], for a series of finite-thickness shear layer jets. For supersonic instability modes, the biggest difference was observed for the higher-order modes, especially at frequencies *St*>0.5, whereas the axisymmetric, lowest-radial-order mode (0,1) considered here gave correct predictions until *b*/*R*_{j}=0.1, where *R*_{j} was the jet radius and *b* a measure of shear layer thickness [10]. Note that the subsonic instability modes computed by these authors were neutral and upstream propagating [2], in contrast to the unstable modes obtained here for heated jets. In fact, for the lowest-azimuthal-order *m*=0 modes up to the maximum frequency (*St*<0.4) considered in this work, a simple zero-thickness model provides a reasonably good approximation until moderately high shear layer thicknesses of *b*≈0.05*R*_{j} [2]. This is also advantageous from a cost-of-computations viewpoint, especially when required to solve for a large number of cases to perform parametric studies like we do here. However, one consequence of zero-thickness shear layer model which will be apparent in §3 is the presence of sharper modal cut-offs, for example, as a function of *M*_{j}, which for finite-thickness shear layers are expected to be of a more gradual nature.

Detailed computations of the pressure eigenfunctions and dispersion relations of supersonic and subsonic modes were already obtained by Tam & Hu [2] and are not repeated here. Instead, we focus on how their spatial growth rates and phase speeds are modified at higher Mach numbers for a heated core jet, which has not been explored in detail. Elsewhere, Parras & Le Dizes [14] have performed a temporal stability analysis of axisymmetric supersonic jets of *M*_{j}=10, without considering the effect of temperature ratio, with a primary focus on providing detailed asymptotic descriptions of these modes.

In what follows, we describe the basic flow configuration and the equations in §§2a and 2b, respectively and derive the dispersion relation in §2c. The spectra of unstable modes for supersonic heated jets appear in §3a, whereas in §3b we introduce a classification methodology based on their respective phase speeds, their evolution for cold and heated core jet conditions described in §§3c and 3d, respectively. A summary of the main results and conclusions are in §4.

## 2. Formulation

### (a) Flow configuration

We consider an inviscid, compressible, non-thermal-conductive, axisymmetric free jet (figure 1) bounded by a zero-thickness mean vortex sheet with the following mean flow parameters:
*T*_{j} is typically much larger than the ambient flow temperature *T*_{o}, which is included by the parameter *κ*_{ρ}=*ρ*_{j}/*ρ*_{o}. Although, *κ*_{T} and *κ*_{ρ} may be distinct, on assuming continuity of static pressures across the interface for identical gases with constant specific heat ratios *γ*, the parameters are related as *R*_{j}, *c*_{j} and *c*_{j} being the speed of sound in the core jet. This yields the following non-dimensional numbers: Mach numbers *M*_{j}=*U*_{j}/*c*_{j} and *M*_{o}=*U*_{o}/*c*_{j}, for the respective streams and the Helmholtz number *ω*=*ω***R*_{j}/*c*_{j}, where *ω** is the dimensional angular frequency. The Strouhal number *St* can then be related to the Helmholtz number via *ω*=2*πM*_{j}*St*, which will be used here to select physically relevant frequencies. In this work, the core jet is always supersonic (*M*_{j}>1) and usually hot (*κ*_{T}≥1), whereas the ambient is subsonic (*M*_{o}<1).

### (b) Governing equations and boundary conditions

Advected wave equations, written in terms of a velocity potential *ϕ*(*r*,*θ*,*z*) such that ** u**=∇

*ϕ*, for both streams yield

*k*≡

*j*,

*o*, per the linearized unsteady Bernoulli equation.

The mean flows in the core jet and the ambient are matched using the kinematic and dynamic boundary conditions at the mean vortex sheet. The linearized kinematic continuity yields
*ξ*(*θ*,*z*) is the radial displacement of the vortex sheet from *r*=1, whereas the dynamic pressure continuity gives

A radiation/causality condition is needed to correctly identify the spatial growth/decay of waves in the solution to ensure all of them are outgoing and decaying at infinity. This is discussed in §2c in the context of selecting appropriate branch cuts for the solution.

### (c) Dispersion relation and spatial stability

A Fourier transform is first applied on (2.2)–(2.7) given by
*q* is *ϕ* or *ξ* and for a spatial analysis the frequency *ω* and azimuthal wavenumber *m* are specified real quantities, whereas the axial wavenumber *ωζ* is unknown and complex.

For equations (2.2a) and (2.2b), this procedure yields two Bessel equations of the form
*r*<1 and *r*>1, respectively. Here, the radial wavenumbers are defined as *ωt*), where *ω* in general can be complex, causality requires its imaginary part *ω*_{i}≥0 or, 0≤*δ*≤*π*/2, where *ω*=|*ω*|exp(*iδ*). This translates to Im [*λ*_{k}]>0 as

The boundary conditions are similarly transformed using (2.8) to yield
*k*≡*j*,*o*, respectively, and
*A*_{j}, *A*_{o},

Poles and zeros of *D*(*ω*,*ζ*) for real *ω* yields the complex spatial modes *ωζ*=*ωζ*_{r}+i*ωζ*_{i} of the free jet via (2.8). The spatial wavenumber is given by *ωζ*_{r}, whereas the growth rate is *ωζ*_{i}, with *ζ*_{i}<0 indicating unstable modes. The corresponding phase speed is then *c*=1/*ζ*_{r}. The poles of (2.16) represent a continuous spectrum of neutral modes (figure 2*a*), except at the chosen branch cuts, which are either upstream or downstream propagating depending upon the sign of their respective phase speeds. These neutral modes are equivalent to the duct modes in a doubly-infinite duct of radius *R*_{j} [32] and as *M*_{j} or *κ*_{T} is raised some of these are downstream-propagating and unstable. The zeros of (2.16) are non-neutral, with a positive sign of the imaginary part indicating an evanescent mode, whereas the signs of their real part indicate their direction of propagation. The unstable vortex-sheet and acoustically coupled hydrodynamic modes appear as zeros in the *R*^{−} plane (figure 2), and all of these are downstream propagating as we shall see §3. It is to be noted that semi-infinite ducted jets have modes identical to free jets, except for a continuous tail of poles and zeros parallel to the imaginary axis which models the acoustic scattering owing to the finite duct termination [3,31,32].

## 3. Results and discussions

The instability modes governed by the zeros and poles of the dispersion relation (2.16) are computed via a Newton–Raphson-based iterative scheme with correct treatment of the numerical derivatives [33], the details of which appear elsewhere [34]. In this section, we report their sensitivities with respect to a few flow parameters: *St*, *M*_{j} and *κ*_{T}, at different operating conditions. Of course, a generalized measure of compressibility like the convective Mach number *M*_{v} [35] can include the effects of *M*_{j} and *κ*_{T} simultaneously for a given *M*_{o}, because *M*_{v}=(*M*_{j}−*M*_{o})/(1+1/*κ*_{T}), but here we separately focus on *M*_{j} and *κ*_{T} to understand their respective roles in the instability mechanism.

### (a) Eigenspectra of supersonic jets

The classification of linear modes of supersonic jets in the context of a shrouded configuration, which also includes the free jet modes, has been discussed in detail elsewhere [3]. In this section, we focus on a few parametric configurations to understand the changes in modal spectra as compressibility and temperature effects are varied.

For supersonic jets (*M*_{j}>1), both the branch points corresponding to the core jet *κ*_{T} and for *κ*_{T}>1/*M*_{o}, all the four branch points are positive, yielding a second branch cut along the positive real axis connecting *M*_{o}=0.2 and restricted *κ*_{T}<3, sufficient for most practical applications where our model is expected to hold, yielding a

Figure 2*a* shows the lowest-azimuthal-order modal spectrum of our reference cold supersonic case at *M*_{j}=2.5 and *κ*_{T}=1, for a given frequency as indicated in the figure caption. The vertical shaded strip satisfies *M*_{v}>1, a condition necessary for acoustically coupled instability modes to be cut on. In fact, (3.1) is a generalized form of the original equation by Tam & Hu [2] that can also include the propagation of subsonic instability modes for heated jets *κ*_{T}>1 (see (3.2)). However, for the cold supersonic jet of figure 2*a*, (3.1) just predicts the two supersonic instability modes located inside the shaded vertical strip of the figure. At these parameters, the K–H mode is significantly more unstable with a higher unstable growth rate *ωIm*(*ζ*)<0 and could be spotted near the bottom of figure 2*a*. In this work, the unstable radial modes, on which we focus, are continuously indexed in order of increasing Re(*ζ*) (decreasing phase speeds), but as any of *St*, *M*_{j} or *κ*_{T} are changed the corresponding phase speeds may show different variations, making the modal indices meaningless. This is avoided by fixing the modal radial index *n* with respect to the reference cold jet configuration of figure 2*a*, with new unstable modes indexed when they first appear, for example, in figure 2*b* as *κ*_{T} is increased. Following this scheme, the K–H mode in figure 2*a* is the (0,1) mode while the supersonic instability modes are the (0,2) and (0,3) modes, respectively. The neutral duct modes appear as poles of (2.16) and a majority of them are upstream propagating with phase speed *Re*(1/*ζ*)<0 (similar to observed elsewhere [3]), while two are downstream propagating and the rest are cut off. The remaining zeros of figure 2*a* are the decaying acoustic modes (*Im*(*ζ*)>0) which radiate to the far-field if with supersonic phase speeds. The free jet spectrum differs from a shrouded jet by the presence of additional (theoretically infinite in number) decaying acoustic modes for the latter that physically model the scattering at the shroud lip [3,31,32].

As the core jet temperature is raised to *a* fixed, the modal spectrum changes significantly as evident from figure 2*b*. The vertical strip, still satisfying (3.1), widens as *κ*_{T} (see (2.11b)) and some of the otherwise cut-off neutral modes (zeros) of figure 2*a* are now cut on and unstable, appearing inside the shaded strip in addition to the unstable modes of figure 2*a*: labelled as the (0,4), (0,5) and (0,6) mode, respectively. Here, it may be noted that the absence of an unstable mode outside the shaded strip indicates that none could be clearly identified as a K–H mode, as discussed later in §3d. Note that the (0,1) mode of figure 2*a* (filled square) is no longer the unstable mode with the highest phase speed in figure 2*b* but continuing our indexing scheme, we label this mode as per its phase speed in the former (reference) case. Moreover, because *ζ*_{n} such that *Im*(*ζ*_{n})<0 is unstable with a subsonic phase speed, two of which are the (0,5) and (0,6) modes, respectively, in figure 2*b*. In fact, it follows from (2.11b), that subsonic instability modes exist if
*κ*_{T} and *M*_{j}, respectively, for a given ambient flow *M*_{o}. If the ambient is static (*M*_{o}=0), subsonic instability modes are predicted in any heated jet (*κ*_{T}>1) of the requisite supersonic Mach number *M*_{j}>1+1/*κ*_{T}.

Figure 2*c*,*d* shows the changes in the spectrum of figure 2*b* as one of the parameters is raised from its reference value to *M*_{j}=4.5 and *ω*=15, respectively. Both yield an overall increase in the number of modes, including the unstable ones, while the vertical strip further widens in figure 2*c* as *M*_{j} is apparent in figure 2*c*, where in spite of a larger number of unstable modes their respective growth rates are largely diminished compared with figure 2*b*.

### (b) Characterization of instability waves

In this section, we introduce a set of phase speed-based criteria to characterize the three set of instability waves discussed in §3a. Such a condition is easily obtained upon realizing that for the model supersonic jet considered here, the branch point *c*, between the K–H and acoustically coupled instability modes. With changing parametric conditions, as either family of modes approach this boundary, the gradual disappearance of their distinctive features would make it impossible to isolate them. Thus, we classify an *unstable* mode into a K–H mode if it propagates with a phase speed of *c*>(*M*_{j}−1) (see (2.11)*a*), a supersonic instability mode if (*M*_{j}−1)>*c*>*M*_{o}+1/*κ*_{T}>1 and a subsonic instability mode if 1>*c*>*M*_{o}+1/*κ*_{T}, where the last limit corresponds to *b*). It may also be noted that because phase speed is mostly unaffected by the shear layer thickness [2], the same classification may also be used to identify these modes in supersonic jets with finite-thickness shear layer models.

### (c) Stability of cold jets

The acoustically coupled instability modes of supersonic cold jets have already been studied in significant detail [2], including their asymptotic representations [14]. The aim of this section is to establish certain basic trends of these modes, especially with regard to their spatial growth rates, which is expected to serve as a necessary foundation for analysing the heated jet configurations discussed next in §3d, the main objective of this work.

Figures 3 and 4 track the three unstable modes: the (0,1), (0,2) and (0,3), respectively, for the reference case corresponding to figure 2*a*. Figure 3 shows the phase speed *c* and the growth rate −*α*_{i} as a function of Strouhal numbers *St*, where the thicker grey lines in figure 3*a* are *c*=*M*_{j}−1 and *c*=*M*_{o}+1/*κ*_{T}, corresponding to *St* is raised, phase speeds of all the tracked modes asymptotically approach the upper bound such that *St*>2 the individual phase speeds are almost indistinguishable. The (0,1) mode with a phase speed *c*>(*M*_{j}−1) at all *St*, particularly for *St*<1, yields *κ*_{T}). However, this mode switches to an acoustically coupled instability mode with changing *St* and *M*_{j}, as we shall see in figure 4*g*. The (0,2) and (0,3) modes, each propagating only beyond a particular frequency, are acoustically coupled supersonic instability modes for all frequencies whose phase speeds lie within the marked bounds of figure 3*a*. Here, supersonic phase speeds of all the unstable modes make sure they are radiating, as is always the case in cold supersonic jets. Note that as *St* is raised there would be other instability modes in the mix once cut on (similar to figure 2*d*), not shown here.

Because, as remarked in §1, the acoustically coupled instability modes are sustained via the repeated reflections of acoustic waves inside the core jet, easier to attain at lower frequencies (i.e. longer wavelengths) via perturbing the vortex sheet over longer coherent distances, this is expected to yield higher growth rates at lower frequencies. This is precisely observed for the spatial growth rates of the (0,2) and (0,3) supersonic instability modes of figure 3*b*, which rapidly attain their respective peak growth rates after being cut on, eventually leading to gradual decays at higher *St*. Between these two instability modes, the slower-phase-speed mode cuts on at a higher frequency, reaches a higher peak growth rate and remains relevant over a longer *St* range. In case of the (0,1) K–H mode, we recover here the classical result for zero-thickness vortex sheet model where the growth rate increases unbounded (∝*ω*), as seen in figure 3*b*. Note that for a finite-thickness shear layer model, the sharper modal cut-offs for the supersonic instability modes of figure 3*b* are expected to be more gradual, perhaps asymptotically approach zero growth over diminishing *St*.

Figure 4 shows the effect of increased flow compressibility (via increasing *M*_{j}) over a range of Strouhal numbers *St*=0.1−0.4, when the acoustic radiation is expected to be the most efficient. In the phase speed plots of figure 4, the thicker grey lines are the same phase speed bounds of figure 3*a*, which in this case intersect at the cut-on Mach number *M*_{c}=*M*_{o}+1+1/*κ*_{T} (see §3b) for the acoustically coupled instability modes to propagate, independent of *St* and modal orders (*m*,*n*).

As *M*_{j}<*M*_{c}, only the (0,1) mode, a K–H instability wave propagates with *c*>*M*_{j}−1, seen from figure 4*a*,*c*,*e* and *g*. As *M*_{j} approaches sonic speed, for *St*≥0.2, this mode has a subsonic phase speed and hence does not radiate, similar to observed elsewhere via a temporal analysis [14]. Clearly, peak growth rates of the K–H mode, as evident from figure 4*b*,*d*,*f* and *h*, show a rising trend with increased *St* (compare also with figure 3*b*), whereas the corresponding phase speeds are asymptotic to the *c*=*M*_{j}−1 line progressively earlier in *M*_{j}, so that at *St*=0.3, for example, the (0,1) mode ceases to be distinctly of K–H type beyond *M*_{j}>2.7 (figure 4*e*). Interestingly, at *St*=0.4, this (0,1) mode switches over to an acoustically coupled supersonic instability mode near *M*_{j}≈2.6 (figure 4*g*), where instead the (0,2) mode is the one with the highest phase speed and asymptotic to the *c*=*M*_{j}−1 curve.

The growth rate of the (0,1) K–H mode drops the fastest among all the instability modes, which is exceeded by the (0,2) supersonic instability mode at *M*_{j}>2.8 for *St*≤0.3 (figure 4*b*,*d*,*f*) to transform the latter into the most unstable jet mode. At *St*=0.4, the (0,3) mode after cut-on dominates the other instability modes, including the (0,1) mode beyond *M*_{j}>3.0, which by this point has itself switched to a supersonic instability mode (figure 4*g*,*h*). Also note that, because the decay rate of the modes depend on their respective phase speeds, the (0,1) mode owing to a lower phase speed at *St*=0.4 decays slower when compared to at *St*≤0.3.

### (d) Stability of hot jets

Although heating of jets bound by thinner shear layers yields more instability as *κ*_{T} with other parameters held constant boosts the speed of sound leading to an effective drop in *M*_{j}, perhaps yielding a state amenable to more instabilities. In this section, we investigate this effect of core heating on highly compressible jets with reference to the case of figure 2*b* at *c*=1 line (figure 5*a*), which necessitates these instability modes to pass through a subsonic phase speed state post cut-on, when these downstream-directed, non-radiating instability modes are labelled as the subsonic instability modes, unique for such heated supersonic jets.

In figure 5*a*, over the entire range of *St*, the (0,1) mode is an acoustically coupled instability, unlike in the cold jet case of figure 3*a*, where it was of K–H type. Instead, the (0,2) mode is a K–H-type instability below *St*<0.26, above which there appears to be no mode distinctly of K–H type.

The growth rate curves of figure 5*b* establish the dominance of the acoustically coupled instability modes in heated jets over K–H instability, if at all present, which in this case exceeds the latter at a low *St*=0.06. All the acoustically coupled instability modes show steep rise in growth rates at subsonic phase speeds until reaching a peak near the sonic phase speed point, followed by a relatively flat growth region, which beyond a certain *St*, as their respective phase speeds approach

The effect of jet compressibility (*M*_{j}) appears in figure 6 where the (0,2) mode is the K–H instability at lower *St*<0.3 (figure 6*a*,*c*,*e*), whereas the (0,1) mode clearly takes this role at the higher frequencies (figure 6*g*). Once the acoustically coupled instabilities are cut on at subsonic phase speeds *c*=*M*_{o}+1/*κ*_{T}, quite expectedly, as in figure 5, their growth rates show rapid rise with increased *M*_{j} (also typical of the vortex sheet model) until the respective phase speeds reach supersonic, beyond which they decay. This yields a distinct possibility in heated supersonic jets dominated by the acoustically coupled modes, where the peak radiated sound is not from the most unstable mode present at a particular *M*_{j}, because frequently this mode is of subsonic phase speed, especially at the higher radial orders. In figure 6*f*, for example, all of the (0,3)−(0,6) instability modes have already started their decaying phases once supersonic *c*>1.

On revisiting the growth rates for the cold jet case of figure 4, we note the corresponding K–H mode ceases to be the most unstable beyond *M*_{j}>2.8, whereas the same limit for figure 6 is *M*_{j}>2.2, due primarily to the increased dominance of the acoustically coupled instability modes in heated jets even at lower supersonic Mach numbers. As we shall see in figure 7, one reason for this is the greater stabilizing effect that increased core jet temperature has on the K–H mode, compared with the other types of instability waves. Note that at the higher Mach numbers of figure 6 there are, of course, additional instability modes of higher radial orders *n*>6 present in the spectrum (see also figure 2*c*), but these higher-order modes with slower phase speeds have even lower growth rates and are not expected to have a role in the jet stability below *M*_{j}<5, considered here.

Figure 7 shows the effect of core jet heating on the instability modes. The phase speeds of the acoustically coupled modes, tracked inside the same limits defined in §3b and marked by the thicker grey lines in figure 7*a*,*c*,*e*,*g*, appear to be almost independent of *κ*_{T}, whereas that of the K–H mode drops gradually with increasing *κ*_{T} until it turns into a supersonic instability mode. In fact, the phase speed of K–H mode travelling at the layer interface depends upon the corresponding convective Mach number *M*_{v} at the vortex sheet, which changes slightly with the relative change of core jet temperature compared with the ambient even if *M*_{j} is held constant, yielding in the observed small change in phase speed. As this mode gets transformed to an acoustically coupled mode, differing mechanistic details contributes to the near-flattening of the phase speed curve, which in this case depends primarily on the Mach number of the core jet, held to be a constant in figure 7. A major consequence of this insensitivity to temperature change being once a mode is cut on at subsonic phase speeds will remain subsonic for a fixed *St*, independent of *κ*_{T}, and hence non-radiating. For example, at *St*=0.1, the (0,1) and (0,3) modes are cut on (figure 7*a*) as non-radiating modes while the (0,2) mode radiates, a K–H mode at these parameters, in spite of being mostly less unstable than either of the acoustically coupled modes (figure 7*b*).

The insensitivity of the phase speed is also reflected in the corresponding growth rate curves (figure 7*b*,*d*,*f*,*h*), where the acoustically coupled modes, especially the subsonic ones, are largely unaffected by the temperature ratios once attaining their respective peak growth rates. Interestingly, the various subsonic instability modes seem to follow growth rate curves which are largely similar after being cut on, including the peak growths they attain, which points to them being similarly affected by the heating. For a given *St*, this yields similar maximum instability states (growth rates) from all unstable modes in supersonic heated jets, when they are largely independent to the degree of heating. For example, at *St*=0.3, as *κ*_{T} is raised beyond *κ*_{T}>1.4, the growth rate of the most unstable mode present at any *κ*_{T} seems to be −*α*_{i}≈0.6. Note that at *St*=0.4, the (0,1) mode recovers its role as a K–H mode at the lowest *κ*_{T} values (figure 7*g*) similar to what has been observed at the lowest *M*_{j} (figure 6*g*).

We mention here that for the heated supersonic jets considered, the K–H mode seems to all but disappear from being distinctly identifiable at temperature ratios that are lower than observations made elsewhere, e.g. in Tam & Hu [2]. For the Mach numbers of figure 7, we note the following temperature ratios when the respective K–H mode transforms into an acoustically coupled instability: *κ*_{T}>1.16 for *St*=0.3 and *κ*_{T}>1.11 for *St*=0.4 compared with *κ*_{T}≈1.45 in Tam & Hu [2], whose results are apparently independent of *St*.

In view of practical applications where perturbations are frequently of multimodal nature, maximum growth rate curves are important in assessing the most unstable state of a system, which we do in the remainder of this work by focusing on the quantity −*α*_{i}|_{max}, the maximum −*α*_{i} over all perturbation frequencies *St*. In figures 8 and 9, we track its variation with respect to *M*_{j} and *κ*_{T}, respectively.

Although cylindrical vortex sheet models are known to support unbounded growth rates for the hydrodynamic mode, as a function of the perturbation frequency, introduction of compressibility imparts a distinct lengthscale to the system via a finite speed of sound, which might yield a modal growth that is bounded, whose possibility we investigate here. For the heated supersonic jet of figure 2*b*, on considering only the first four modes from (0,1) to (0,4), figure 8*a* shows each of these modes to support finite maximum growths except at specific *M*_{j}, where their growth rates seem unbounded. Except the (0,2) mode, which is unbounded at *M*_{j}=2.26, the remaining modes seem to show this behaviour near *M*_{j}≈2.37. Here, we stress that the presence of these unbounded growth rates must not be interpreted as any onset to absolute instability, because as discussed in §1, at these high Mach numbers such instability states are unlikely to be attained for the kind of moderate temperature ratios we consider. More importantly, although the individual growth rates do seem to be unbounded at these points, the corresponding changes in phase speeds over the same range of *M*_{j} are finite, without any sharp drop in their magnitudes (or, appearance of a cusp), as is required for the group velocity to reach a stationary state, per the Briggs–Bers absolute instability criterion [37]. Instead, these locations are likely where the respective acoustically coupled modes attain their limiting instability states, corresponding to the equivalent incompressible K–H mode of theoretically unbounded growth. Here, we may hypothesize such states to be reached via some sort of acoustic resonance inside the core jet, when the acoustic waves reflect at special resonant angles to maximize the hydrodynamic perturbations at the shear layer. As *M*_{j} is lowered from these limiting-growth points, the maximum growth rates drop sharply, with the corresponding *b*) signifying the approach of the cut-off Mach number (*M*_{c}≈1.78 in figure 8), when all the respective phase speeds converge to *c*=1/*κ*_{T}+*M*_{o}. As the jet Mach number is raised beyond the unbounded-growth zones, the maximum growth rates drop too, albeit more gradually, but the maximum Strouhal numbers in this branch (*St*_{max}<0.6) are probably of greater relevance from the point of acoustic radiation. Note that the (0,2) mode which mostly follows the *c*|_{max}=*M*_{j}−1 line has a near-constant *St*|_{max} at these higher Mach numbers.

Figure 9*a* shows the variation of maximum growth rate curves as a function of temperature ratio *κ*_{T}. As *b*), but as *κ*_{T} is raised this mode gets morphed into an acoustically coupled mode with finite instability beyond *κ*_{T}>1.48, from which point it is tracked in figure 9. Also, for the cold (*κ*_{T}=1) jet, we note from figure 9*c* that the maximum phase speed for all the unstable modes converges to *c*|_{max}=0.5(*M*_{j}+*M*_{o}), while the corresponding maximum growth rates drop to magnitudes typically lower than heated jets (*κ*_{T}<3) at Strouhal numbers that are relatively higher (*St*|_{max}>0.6), except for the (0,2) mode.

The unbounded growths for the (0,3) and (0,4) modes appear near *κ*_{T}≈1.52, except the (0,2) mode which is singular near *κ*_{T}∼1.31. As expected, the decay rates of −*α*_{i}|_{max}, as *κ*_{T} is raised beyond these points, are slower than when *M*_{j} is increased (see figure 8 but also compare figures 6 and 7) which points to their decreased sensitivity with *κ*_{T}. The maximum phase speed *c*|_{max}, which for a cold jet is always *c*|_{max}≥1, drops to subsonic speeds as *κ*_{T} is raised (figure 9*c*) in a manner roughly proportional to the lower phase-speed limit (marked by the lower thicker grey line in the figure), except for the (0,2) mode which briefly rises and then is virtually constant, following the upper phase speed limit. Thus, beyond a heating level of the core jet (specifically, *κ*_{T}>2.26 in figure 9*c*) all the unstable modes are subsonic except the (0,2) mode, which consequently yields a peak radiation for such jets from a near-neutral instability state (figure 9*a*,*c*).

The vortex-sheet model seems to better model the peak growths of unstable modes in high-speed heated jets than in cold, low-speed cases, where it predicts unbounded growth for the K–H mode. However, the presence of unbounded growth is not completely eliminated by the elevated compressibility and heating which now appears at specific compressibility/heating levels when the growth of supersonic instability modes tend to the limiting maximum growth of the incompressible K–H mode. It would be interesting to consider finite-thickness shear layer models to investigate how such unbounded growth regions of figures 8 and 9 get modified in more practical flows.

## 4. Summary and conclusion

The main findings of this work are summarized below:

(i) Cold supersonic jets of fixed compressibility (constant

*M*_{j}) are more hydrodynamically unstable when heated (investigated here up to*κ*_{T}=3), with the maximum instability state largely unchanged beyond moderate heating levels. In contrast, increased compressibility has a more pronounced impact on jet instability which is increasingly stabilized.(ii) At higher

*M*_{j}, the acoustically coupled modes govern jet stability, where the role of purely hydrodynamic K–H wave is minimal, itself being converted to an acoustically coupled mode at moderate to low core jet heating (e.g.*κ*_{T}≈1.48 in figure 8).(iii) Increased heating of supersonic jets raises the potential for subsonic instability modes, all of which may turn into supersonic modes if, in addition, jet speed is also raised. Subsonic instability modes, in general, are more unstable than the supersonic ones but have no role on its radiated sound. Hence, although supersonic heated jets are usually more hydrodynamically unstable, it rarely translates into increased sound levels at the far-field over equivalent cold jets.

(iv) Likewise for the K–H mode in cold, subsonic jets, a vortex sheet model also predicts unbounded growths for the acoustically coupled supersonic instability modes in hot, supersonic jets that may occur over narrow ranges of core jet Mach number or temperature, which actually points to the basic hydrodynamic nature of these modes driven by acoustic reflections.

In conclusion, it has been shown that the dynamics of heated supersonic jets that support acoustically coupled instability modes are not governed by K–H instability, rather their near-field is dominated by the usually more unstable subsonic instability modes with the supersonic instability modes contributing to the primary far-field sound.

## Authors' contributions

The author was responsible for all aspects of the manuscript.

## Competing interests

The author declare that there are no competing interests.

## Funding

Partial financial support for this work was provided by the ISRO–IISc Space Technology Cell via grant no. ISTC/MAE/AS/296.

## Acknowledgements

This work does not acknowledge anyone in particular.

- Received November 30, 2015.
- Accepted March 22, 2016.

- © 2016 The Author(s)

Published by the Royal Society. All rights reserved.