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Proc Math Phys Eng Sci. 2016 April; 472(2188): 20150817.
PMCID: PMC4892279

On the axisymmetric stability of heated supersonic round jets

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 Mj≥2, whereas the subsonic instability waves, in addition, require the core-to-ambient flow temperature ratio Tj/To>1. We show, for moderately heated jets at Tj/To>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.

Keywords: hydrodynamic stability, aeroacoustics, supersonic heated jets

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 [13]. 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 [68]. Thus, in spite of their basic turbulent nature, methods based on the linear and quasi-linear stability theories [911], 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 (MjMo)>2 [3], where Mj and Mo 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 (MjMo), 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 Mj, 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 Mj [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 Tj exceeds the ambient To, such that κT=Tj/To>1, a situation common in many practical applications, it remains to be seen what effect the temperature ratio κT has on the growth rates of acoustically coupled instability modes. At the higher levels of jet compressibility (MjMo)>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 κT, 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 [2224], but as Mj is raised the region of absolute instability disappears even at moderate subsonic Mach numbers Mj>0.5, for negligible-thickness shear layers subjected to moderate to low heating κT<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<Mj<5 and 1<κT<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].

Figure 1.
Schematic of the free jet. (Online version in colour.)

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/Uj, where f is the frequency in Hz, D the nozzle diameter and Uj 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/Rj=0.1, where Rj 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≈0.05Rj [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 Mj, 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 tanh velocity profiles until Mj=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:

[u¯zT¯ρ¯]={UjTjρjfor r<Rj,UoToρofor r>Rj.
2.1

The core jet mean temperature Tj is typically much larger than the ambient flow temperature To, which is included by the parameter κT=Tj/To, whereas the mean densities of the two streams are related via κρ=ρj/ρo. Although, κT and κρ 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 κρ=1/κT2. The variables are non-dimensionalized by Rj, cj and ρjcj2 for the length, velocity and pressure variables, respectively, with cj being the speed of sound in the core jet. This yields the following non-dimensional numbers: Mach numbers Mj=Uj/cj and Mo=Uo/cj, for the respective streams and the Helmholtz number ω=ω*Rj/cj, where ω* is the dimensional angular frequency. The Strouhal number St can then be related to the Helmholtz number via ω=2πMjSt, which will be used here to select physically relevant frequencies. In this work, the core jet is always supersonic (Mj>1) and usually hot (κT≥1), whereas the ambient is subsonic (Mo<1).

(b) Governing equations and boundary conditions

Advected wave equations, written in terms of a velocity potential ϕ(r,θ,z) such that u=[nabla]ϕ, for both streams yield

(t+Mjz)2ϕΔϕ=0,r<1
2.2a

and

κT2(t+Moz)2ϕΔϕ=0,r>1
2.2b

are used to study the acoustic and vorticity waves, whereas a homentropic assumption coupled with the neglect of dissipation, heat conduction and any heat source drops out the entropy perturbations. Here, for convenience, the non-dimensional lengths and time are expressed by their corresponding dimensional symbols. In forming (2.2), the pressure in each stream is

pk=(ϕt+Mkϕz),
2.3

for k[equivalent]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

(t+Mjz)ξ(θ,z)=ϕr(1,θ,z),<z<
2.4

and

(t+Moz)ξ(θ,z)=ϕr(1+,θ,z),<z<,
2.5

where ξ(θ,z) is the radial displacement of the vortex sheet from r=1, whereas the dynamic pressure continuity gives

κρ(t+Mjz)ϕ(1,θ,z,t)=(t+Moz)ϕ(1+,θ,z,t),<z<.
2.6

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^(r,ζ)exp{i(mθωt)}=+q(r,θ,z,t)exp(iωζz)dz,
2.7

where 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

1rr(rϕ^r)+(ω2λk2m2r2)ϕ^=0forkj,o,
2.8

at r<1 and r>1, respectively. Here, the radial wavenumbers are defined as λk=λk+λk with

λj±=[1ζ(Mj±1)]1/2
2.9a

and

λo±=[κTζ(κTMo±1)]1/2.
2.9b

The location of the principal branch cuts are determined via a causality criterion [29,30], discussed in detail elsewhere [31,3]. Basically, for harmonic time dependence of exp(−iωt), where ω in general can be complex, causality requires its imaginary part ωi≥0 or, 0≤δπ/2, where ω=|ω|exp(). This translates to Im [λk]>0 as ζ for the branch cuts, which is ensured by finite branch points at

ζj±=1/(Mj±1)
2.10a

and

ζo±=κT/(κTMo±1).
2.10b

This choice of branch cuts also defines the domain of regularity (analyticity) of the Fourier transform as

Rj±:±Im(ζζj±)<Re(ζζj±)tanδforr<1
2.11a

and

Ro±:±Im(ζζo±)<Re(ζζo±)tanδforr>1.
2.11b

The boundary conditions are similarly transformed using (2.8) to yield

iω(1ζMk)ξ^(ζ)=ϕ^r(1,ζ),
2.12

for k[equivalent]j,o, respectively, and

κρ(1ζMj)ϕ^(1,ζ)=(1ζMo)ϕ^(1+,ζ).
2.13

Bessel equations of (2.9) have solutions of the form

ϕ^(r,ζ)={Aj(ζ)Jm(λjωr)forr<1,Ao(ζ)Hm(1)(λoωr)forr>1,
2.14

ensuring regular behaviour of ϕ^ as r0 and at infinity. From equations (2.13)–(2.15) on eliminating Aj, Ao, ϕ^, ξ^ the following dispersion relation is obtained

D(ω,ζ)=ω{κρ(1ζMj)2λjJm(λjω)Jm(λjω)(1ζMo)2λoHm(1)(λoω)Hm(1)(λoω)},
2.15

which is equivalent to that obtained by Tam & Hu [2] by ensuring the present equation to yield identical instability modes (e.g. the case in figure 7 of Tam & Hu).

Figure 7.
(ah) Same as in figure 5 but as a function of κT at Mj=2.5 and different Strouhal numbers St. (Online version in colour.)

Poles and zeros of D(ω,ζ) for real ω yields the complex spatial modes ωζ=ωζr+iωζi of the free jet via (2.8). The spatial wavenumber is given by ωζr, whereas the growth rate is ωζi, with ζi<0 indicating unstable modes. The corresponding phase speed is then c=1/ζr. The poles of (2.16) represent a continuous spectrum of neutral modes (figure 2a), 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 Rj [32] and as Mj or κT 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].

Figure 2.
The zeros (open circles) and poles (multiplication) of the dispersion relation (2.16) with the common parameters Mo=0.2 and m=0 at (a) Mj=2.5, κT=1, ω=8; (b) Mj=2.5, κT=3, ω=8, (c) Mj=4.5, κT=3, ω ...

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, Mj and κT, at different operating conditions. Of course, a generalized measure of compressibility like the convective Mach number Mv [35] can include the effects of Mj and κT simultaneously for a given Mo, because Mv=(MjMo)/(1+1/κT), but here we separately focus on Mj and κT 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 (Mj>1), both the branch points corresponding to the core jet ζj± are positive (see (2.11a)), which requires the corresponding branch cut to lie on the positive real axis, joining ζj+ and ζj. The branch points for the outer flow ζo± depend upon κT and for κT>1/Mo, all the four branch points are positive, yielding a second branch cut along the positive real axis connecting ζo+ and ζo. In this work, we have set a low-subsonic ambient flow at Mo=0.2 and restricted κT<3, sufficient for most practical applications where our model is expected to hold, yielding a ζo that is still negative for the cases considered. This makes the respective cuts from ζo± to terminate at ±, as shown in figure 2.

Figure 2a shows the lowest-azimuthal-order modal spectrum of our reference cold supersonic case at Mj=2.5 and κT=1, for a given frequency as indicated in the figure caption. The vertical shaded strip satisfies ζo+>ζj, which via a procedure derived from the vortex sheet model for axisymmetric jets yields [2,3]

MjMo>1+κTκT,
3.1

equivalent to Mv>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 κT>1 (see (3.2)). However, for the cold supersonic jet of figure 2a, (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 2a. 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, Mj or κT 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 2a, with new unstable modes indexed when they first appear, for example, in figure 2b as κT is increased. Following this scheme, the K–H mode in figure 2a 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 2a 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 κT=3 keeping other parameters of figure 2a fixed, the modal spectrum changes significantly as evident from figure 2b. The vertical strip, still satisfying (3.1), widens as ζo+ increases with κT (see (2.11b)) and some of the otherwise cut-off neutral modes (zeros) of figure 2a are now cut on and unstable, appearing inside the shaded strip in addition to the unstable modes of figure 2a: 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 2a (filled square) is no longer the unstable mode with the highest phase speed in figure 2b but continuing our indexing scheme, we label this mode as per its phase speed in the former (reference) case. Moreover, because ζo+>1, any mode ζn such that 1<Re(ζn)<ζo+ and Im(ζn)<0 is unstable with a subsonic phase speed, two of which are the (0,5) and (0,6) modes, respectively, in figure 2b. In fact, it follows from (2.11b), that subsonic instability modes exist if

κT>11Mo,
3.2

along with (3.1), which together define the required κT and Mj, respectively, for a given ambient flow Mo. If the ambient is static (Mo=0), subsonic instability modes are predicted in any heated jet (κT>1) of the requisite supersonic Mach number Mj>1+1/κT.

Figure 2c,d shows the changes in the spectrum of figure 2b as one of the parameters is raised from its reference value to Mj=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 2c as ζj decreases. The high sensitivity of the unstable modal growth rates to Mj is apparent in figure 2c, where in spite of a larger number of unstable modes their respective growth rates are largely diminished compared with figure 2b.

(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 ζj defines a boundary (the left edge of the vertical shaded strips in figure 2) in terms of the phase speed 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>(Mj−1) (see (2.11)a), a supersonic instability mode if (Mj−1)>c>Mo+1/κT>1 and a subsonic instability mode if 1>c>Mo+1/κT, where the last limit corresponds to 1/ζo+ (see (2.11)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 and44 track the three unstable modes: the (0,1), (0,2) and (0,3), respectively, for the reference case corresponding to figure 2a. Figure 3 shows the phase speed c and the growth rate −αi as a function of Strouhal numbers St, where the thicker grey lines in figure 3a are c=Mj−1 and c=Mo+1/κT, corresponding to 1/ζj and 1/ζo+ (see (2.11)), respectively, the upper and lower phase speed bounds for the acoustically coupled (supersonic) instability modes, defined in §3b. As St is raised, phase speeds of all the tracked modes asymptotically approach the upper bound such that c(Mj1), where beyond St>2 the individual phase speeds are almost indistinguishable. The (0,1) mode with a phase speed c>(Mj−1) at all St, particularly for St<1, yields cMj as St0, standard characteristic of K–H modes (irrespective of κT). However, this mode switches to an acoustically coupled instability mode with changing St and Mj, as we shall see in figure 4g. 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 3a. 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 2d), not shown here.

Figure 3.
Variation of (a) phase speed and (b) spatial growth rate with frequency St, shown for (filled squares) (0,1), (filled triangles) (0,2) and (filled right triangles) (0,3) instability modes of figure 2a at Mj=2.5, Mo=0.2, κT=1, m=0. The ...
Figure 4.
(ah) Same as in figure 3 but as a function of Mj at different Strouhal numbers St, as labelled. (Online version in colour.)

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 3b, 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 ([proportional, variant]ω), as seen in figure 3b. Note that for a finite-thickness shear layer model, the sharper modal cut-offs for the supersonic instability modes of figure 3b 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 Mj) 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 3a, which in this case intersect at the cut-on Mach number Mc=Mo+1+1/κT (see §3b) for the acoustically coupled instability modes to propagate, independent of St and modal orders (m,n).

As Mj<Mc, only the (0,1) mode, a K–H instability wave propagates with c>Mj−1, seen from figure 4a,c,e and g. As Mj 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 4b,d,f and h, show a rising trend with increased St (compare also with figure 3b), whereas the corresponding phase speeds are asymptotic to the c=Mj−1 line progressively earlier in Mj, so that at St=0.3, for example, the (0,1) mode ceases to be distinctly of K–H type beyond Mj>2.7 (figure 4e). Interestingly, at St=0.4, this (0,1) mode switches over to an acoustically coupled supersonic instability mode near Mj≈2.6 (figure 4g), where instead the (0,2) mode is the one with the highest phase speed and asymptotic to the c=Mj−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 Mj>2.8 for St≤0.3 (figure 4b,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 Mj>3.0, which by this point has itself switched to a supersonic instability mode (figure 4g,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 Mj0 [36], it may not be as straightforward for the highly compressible supersonic jets, considered here. Because as expected, increased compressibility suppresses hydrodynamic growth [18,19] as apparent from the growth rate curves of figure 4, but an increase in κT with other parameters held constant boosts the speed of sound leading to an effective drop in Mj, 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 2b at κT=3. For heated jets, the lower phase-speed limit for acoustically coupled instability modes drops below the sonic c=1 line (figure 5a), 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.

Figure 5.
Variation of (a) phase speed and (b) spatial growth rate with frequency ω, shown for (filled squares) (0,1), (filled triangles) (0,2), (filled right-side triangles) (0,3), (filled down-side triangle) (0,4), (filled left-side triangle) (0,5) and ...

In figure 5a, over the entire range of St, the (0,1) mode is an acoustically coupled instability, unlike in the cold jet case of figure 3a, 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 5b 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 cMj1, enter a decaying phase.

The effect of jet compressibility (Mj) appears in figure 6 where the (0,2) mode is the K–H instability at lower St<0.3 (figure 6a,c,e), whereas the (0,1) mode clearly takes this role at the higher frequencies (figure 6g). Once the acoustically coupled instabilities are cut on at subsonic phase speeds c=Mo+1/κT, quite expectedly, as in figure 5, their growth rates show rapid rise with increased Mj (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 Mj, because frequently this mode is of subsonic phase speed, especially at the higher radial orders. In figure 6f, for example, all of the (0,3)−(0,6) instability modes have already started their decaying phases once supersonic c>1.

Figure 6.
(ah) Same as in figure 5 but as a function of Mj at different Strouhal numbers St. (Online version in colour.)

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 Mj>2.8, whereas the same limit for figure 6 is Mj>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 2c), 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 Mj<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 7a,c,e,g, appear to be almost independent of κT, whereas that of the K–H mode drops gradually with increasing κT 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 Mv at the vortex sheet, which changes slightly with the relative change of core jet temperature compared with the ambient even if Mj 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 κT, and hence non-radiating. For example, at St=0.1, the (0,1) and (0,3) modes are cut on (figure 7a) 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 7b).

The insensitivity of the phase speed is also reflected in the corresponding growth rate curves (figure 7b,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 κT is raised beyond κT>1.4, the growth rate of the most unstable mode present at any κT seems to be −αi≈0.6. Note that at St=0.4, the (0,1) mode recovers its role as a K–H mode at the lowest κT values (figure 7g) similar to what has been observed at the lowest Mj (figure 6g).

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: κT>1.16 for St=0.3 and κT>1.11 for St=0.4 compared with κT≈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 −αi|max, the maximum −αi over all perturbation frequencies St. In figures 8 and 9, we track its variation with respect to Mj and κT, respectively.

Figure 8.
Variation of (a) the maximum growth rate −αi|max with Mj and the corresponding (b) Strouhal number St|max and (c) phase speed c|max, shown for (filled squares) (0,1), (filled upside triangles) (0,2), (right-side filled triangles) (0,3) ...
Figure 9.
(ac) Same as in figure 8 but as a function of κT for Mj=2.5. (Online version in colour.)

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 2b, on considering only the first four modes from (0,1) to (0,4), figure 8a shows each of these modes to support finite maximum growths except at specific Mj, where their growth rates seem unbounded. Except the (0,2) mode, which is unbounded at Mj=2.26, the remaining modes seem to show this behaviour near Mj≈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 Mj 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 Mj is lowered from these limiting-growth points, the maximum growth rates drop sharply, with the corresponding St|max (figure 8b) signifying the approach of the cut-off Mach number (Mc≈1.78 in figure 8), when all the respective phase speeds converge to c=1/κT+Mo. 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 (Stmax<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=Mj−1 line has a near-constant St|max at these higher Mach numbers.

Figure 9a shows the variation of maximum growth rate curves as a function of temperature ratio κT. As κT1, the vortex-sheet model predicts unbounded growth for the corresponding K–H mode (0,1) (see also figure 3b), but as κT is raised this mode gets morphed into an acoustically coupled mode with finite instability beyond κT>1.48, from which point it is tracked in figure 9. Also, for the cold (κT=1) jet, we note from figure 9c that the maximum phase speed for all the unstable modes converges to c|max=0.5(Mj+Mo), while the corresponding maximum growth rates drop to magnitudes typically lower than heated jets (κT<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 κT≈1.52, except the (0,2) mode which is singular near κT~1.31. As expected, the decay rates of −αi|max, as κT is raised beyond these points, are slower than when Mj is increased (see figure 8 but also compare figures 6 and 7) which points to their decreased sensitivity with κT. The maximum phase speed c|max, which for a cold jet is always c|max≥1, drops to subsonic speeds as κT is raised (figure 9c) 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, κT>2.26 in figure 9c) 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 9a,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 Mj) are more hydrodynamically unstable when heated (investigated here up to κT=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 Mj, 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. κT≈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.

Acknowledgements

This work does not acknowledge anyone in particular.

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.

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