2.1. Static density/temperature ratio effect in low-speed shear layers/jets

In incompressible flows, free shear layers are known to be susceptible to instabilities excited by velocity and density ratios across the shear layer (Michalke Reference Michalke1984; Morris Reference Morris2010). Instability waves grow faster when density decreases on the higher-speed side of the shear layer. For a laminar boundary layer at nozzle exit, such instabilities help trigger the transition to turbulence. Even for fully turbulent shear layers they influence the development of the near nozzle large-scale eddies which dominate mixing. The effect of static density ratio $s = {\rho _2}/{\rho _i}$ across a turbulent shear layer with velocity ratio $r = {U_2}/{U_1}$ has been studied experimentally (using different gases to control s) by Brown & Rosko (Reference Brown and Rosko1974) (subscript 1 is the faster (jet) stream and 2 is the slower (stagnant ambient) stream). Measurements identified the dependence of shear layer width growth rate $(\delta ^{\prime})$ on r and s. Note that, in what follows, although results were expressed in terms of s, to place these in the current hot jet context, they are here presented in terms of static temperature ratio, since: $s = {\rho _a}/{\rho _j} = {T_{s,j}}/{T_a} = t$ (ignoring pressure effects). The vorticity thickness $\delta _{\omega} = ({U_1} - {U_2})/{(\partial U/\partial z)_{max}}$ was used to characterise shear layer width (z is the coordinate across the shear layer) and the growth rate relation was

(2.1)\begin{equation}\delta ^{\prime}_\omega = 0.0825\frac{{(1 - r)}}{{(1 + r{t^{1/2}})}}(1 + {t^{1/2}}).\end{equation}

Dimotakis (Reference Dimotakis1986) proposed a modified version of (2.1) taking into account the asymmetric entrainment into the mixing region from high-speed and low-speed sides, although only ~15 % difference to (2.1) resulted and only at extreme values of r and t. For a jet in stagnant surroundings (U ₂ = 0, i.e. r = 0) with an expected maximum t of ~3, the difference was only ~6 %. When applied to a hot jet in stagnant surroundings (2.1) indicates shear layer width growth is modified by the square root of jet/ambient static temperature ratio

(2.2)\begin{equation}\delta ^{\prime}_\omega = 0.0825(1 + {t^{1/2}}).\end{equation}

This implies a significant although fairly weak effect; the growth rate increases by only 37 % for a static temperature ratio increase of 300 %. The LES study of McMullan, Coats & Gao (Reference McMullan, Coats and Gao2011) did not compare directly with the Brown & Roshko data, but the predicted results may be deduced as in very good agreement with (2.1).

A similar square root density ratio influence on the spreading rate of a variable density jet in stagnant surroundings – this time for the developed jet not just the initial shear layer – had been suggested earlier by Thring & Newby (Reference Thring and Newby1953). These authors introduced the concept of an ‘effective diameter’ $({D_{eff}} = D{({\rho _j}/{\rho _a})^{1/2}})$; when D_eff was used as the governing length scale axial decay of measured jet velocity collapsed for all temperature ratios studied (t = 0.196–7.14). Ricou & Spalding (Reference Ricou and Spalding1961) performed direct measurements of entrainment for jets with different density ratios and confirmed this effect. Jet entrainment and centreline velocity decay rate both increased when jet density was less than ambient density but all data collapsed onto a single line when scaled by t ^1/2. Other variable density jet experiments have observed a similar relationship (Pitts Reference Pitts1991; Amielh et al. Reference Amielh, Djeridane, Anselmet and Fulachier1996).

2.2. Compressibility effects in high-speed shear layers

The jet studies described above were restricted to low Mach number. However, measurements by Zaman (Reference Zaman1998) showed that a t ^1/2 parameter was equally successful in collapsing the downstream asymptotic spreading rate behaviour of compressible jets as long as shock-free conditions existed. The same was not true, however, for the initial shear layer development. Whilst the Brown & Rosko (Reference Brown and Rosko1974) t ^1/2-effect still applies in the shear layer region close to nozzle exit, Papamoschou & Roshko (Reference Papamoschou and Roshko1988) carried out measurements for high-speed shear layers demonstrating an additional, compressibility-related effect. A significant reduction in shear layer growth rate $\delta ^{\prime}_\omega$ was observed. This was best correlated by the convective Mach number (M_C) – the Mach number in a frame of reference moving with the speed (U_C) of dominant shear layer instability waves (or other disturbances such as turbulent structures). The convective Mach numbers in a 2-stream turbulent shear layer are defined as

(2.3a,b)\begin{equation}{M_{{C_1}}} = \frac{{{U_1} - {U_C}}}{{{a_1}}}\quad {M_{{C_2}}} = \frac{{{U_C} - {U_2}}}{{{a_2}}}.\end{equation}

For mixing where both streams are pressure matched and have the same γ, a speed of sound weighted average provided the optimum estimation of U_C

(2.4a,b)\begin{equation}{U_C} = \frac{{{a_2}{U_1} + {a_1}{U_2}}}{{{a_1} + {a_2}}}\quad {M_{{C_1}}} = {M_{{C_2}}} = {M_C} = \frac{{{U_1} - {U_2}}}{{{a_1} + {a_2}}}.\end{equation}

Here, U and a are axial velocity and speed of sound in each flow stream. The relation between M_c and M_j which results for a jet discharged into a stagnant ambient is thus

(2.5)\begin{equation}{M_c} = \frac{{{U_j}}}{{{a_j} + {a_a}}} = \frac{{{M_j}}}{{(1 + {a_a}/{a_j})}} = \frac{{{M_j}}}{{[1 + {{({T_{s,j}}/{T_a})}^{ - 1/2}}]}} = \frac{{{M_j}}}{{[1 + {t^{ - 1/2}}]}}.\end{equation}

Note the static temperature ratio appears in this relation (due to the presence of both jet and ambient speed of sound) indicating that the compressibility effect and the t ^1/2-effect are interlinked.

Many planar shear layer experiments have been carried out to confirm the original Papamoschou & Roshko (Reference Papamoschou and Roshko1988) data. Barone, Oberkampf & Blottner (Reference Barone, Oberkampf and Blottner2006) collated 11 data sets to deduce what is accepted as the classical curve demonstrating the growth rate reduction (note – relative to the incompressible growth rate at the same value of r and t), shown via the dotted line curve fitted to the measured data in figure 1. Direct numerical simulation (DNS) predictions (Pantano & Sarkar Reference Pantano and Sarkar2002) have shown this effect is caused by decrease of pressure fluctuation magnitude with increasing Mach number. This leads to decorrelation of pressure and strain rate fluctuations, inhibited energy transfer from streamwise to cross-stream fluctuations and thus reduced turbulent shear stress and shear layer growth rate.

Figure 1. Compressibility-affected annular shear layer growth (Feng & McGuirk Reference Feng and McGuirk2016).

While the first 2–3D axial distance of the shear layer bordering a round jet will behave like a planar flow, annular effects grow downstream. Feng & McGuirk (Reference Feng and McGuirk2016) conducted measurements to investigate this, indicating similar but stronger suppression of growth rate with M_c was observed in annular shear layers, as also present in the data of Lau, Morris & Fisher (Reference Lau, Morris and Fisher1979) when plotted in this format. Note, Barone et al. (Reference Barone, Oberkampf and Blottner2006) also pointed out that, although compressibility effects in shear layers were well correlated by M_C alone, this may not be universal, and the total temperature ratio may also be influential. Of the 29 experimental data points considered by Barone et al. (Reference Barone, Oberkampf and Blottner2006) to establish their recommended curve there were only 5 high M_c points with high-speed total temperature greater than low speed (Goebel & Dutton Reference Goebel and Dutton1991) of direct relevance in the present context. These 5 points appear as ‘outliers’ to the recommended curve (the 5 points at M_c = 0.4–1.0 lying well beneath the curve). Given also evidence in figure 1 that unheated compressible annular and planar shear layers in the jet near-field context display different behaviours, hot jet annular shear layers appear worthy of further study.

2.3. Near-field experimental data for hot high-speed jets

The first study of potential core length and centreline velocity decay in the near field of compressible high temperature jets was provided by Witze (Reference Witze1972), analysing eight sets of measurements of (mainly) heated air jets (0.07 < M_j < 0.97 and 0.64 < t < 2.9). Five data sets for supersonic jets were also included (1.4 < M_j < 2.6, fully expanded) but all were cooled flows (0.4 < t < 0.6) not relevant to the present context. An empirical curve fit to centreline velocity decay was extrapolated backwards to intersect ${U_{cl}}/{U_j} = 1.0$ to establish L_p; two relations for L_p(M_j, t) were proposed

(2.6)\begin{gather}\textrm{subsonic}\quad \frac{{{L_p}}}{D} = 4.375{(1.0 - 0.16{M_j})^{ - 1.0}}{(t)^{ - 0.28}},\end{gather}

(2.7)\begin{gather}\textrm{supersonic}\quad \frac{{{L_p}}}{D} = 5.56{(M_j^2 - 1.0)^{0.15}}{(t)^{ - 0.5}}.\end{gather}

Lau (Reference Lau1980, Reference Lau1981) cast doubt on the correctness of different power laws for t in (2.6) and (2.7). In Lau's study of centreline velocity decay a factor of t ^−0.2 for both sub- and supersonic Mach number jets was observed. Lau conducted measurements for 0.5 < M_j < 1.67 (fully expanded) and t = 1.0, 1.5, and 2.32 and applied the same definition for L_p as Witze (Reference Witze1972). Lau's empirical curve fit to measured L_p data did not explicitly contain t but did include a t-dependent value for the parameter C

(2.8)\begin{equation}\frac{{{L_p}}}{D} = C + 1.1M_j^2.\end{equation}

The parameter C represents the potential core length for (effectively) incompressible Mach numbers (although this will be dependent on experimental facility details, see § 2.4). A value of C = 4.2 was suggested for t = 1.0 and C = 3.2 for t = 1.5 and 2.32. No rationale was put forward for these values. On the basis of the observations described in § 2.1, the shear layer growth rate should increase by ~11 % between t = 1.0 and 1.5 and a further 13 % between t = 1.5 and 2.32. Whist L_p cannot be accurately predicted based solely on shear layer growth values (this is discussed further in § 4.3) the values proposed for C do not seem justified. Despite the different functional forms chosen, L_p from (2.6) and (2.8) do show similar trend with M_j for subsonic conditions, increasing at approximately the same gradient but with different responses to t (Witze's equation produced values ~7 % larger at t = 1.0 rising to ~21 % at t = 1.5). Complete disagreement occurs for supersonic M_j with the Witze equation also showing a discontinuous behaviour.

Lau's data included detailed mean velocity profiles within the first 10D of jet development, producing evidence of a linear spread of the shear layer bordering the potential core. The results for vorticity thickness growth rate $(\delta ^{\prime}_\omega)$ supported the following empirical fit for the influence of M_j and t

(2.9)\begin{equation}\delta ^{\prime}_\omega = 0.177\{ (1 - 0.294M_j^2)[1 + 0.5(M_j^2 - 1){(t - 1.4)^2}]\} .\end{equation}

This matched measurement data very well but no arguments were put forward to explain the flow physics underpinning the influence of either M_j or t (no reference was made to the work described in §§ 2.1 or 2.2). Two strange and unexplained results in Lau's data were that the supersonic data (M_j = 1.67) displayed an opposite trend for $\delta ^{\prime}_\omega$ variation with t than all subsonic data, and for the highest t = 2.32 jet the same values of $\delta ^{\prime}_\omega$ were obtained at all values of M_j.

Only two other experimental investigations have appeared on hot high-speed jet near-field flows. The first, Seiner et al. (Reference Seiner, Ponton, Jansen and Lagen1992), was aimed at the military aerospace application, concentrating on an axisymmetric supersonic jet (M_j = 2.0, fully expanded) at total temperatures between 313 K and 1370 K (NTR = 1.1–4.9, t = 0.62–2.73). Mainly acoustic measurements were made but useful supersonic L_p data were also presented (note the definition of L_p now used intersection with ${U_{cl}}/{U_j} = 0.99$ rather than 1.0 as adopted by Witze and Lau). The second was a comprehensive NASA aerodynamic study of the civil aerospace high subsonic M_j application aimed at creation of a benchmark computational fluid dynamics (CFD) validation data set (Bridges & Wernet Reference Bridges and Wernet2010). Particle image velocimetry (PIV) instrumentation was used with substantial detail provided for mean velocity and turbulence variables over the first 25D of jet development. Jets were considered with 0.38 < M_j < 1.0 and 0.84 < t < 2.7.

In Witze's, Lau's, and Seiner's studies no nozzle exit profile measurements were provided, in spite of the strong possibility of this influencing L_p. Nozzle exit data were reported for the first time by Bridges & Wernet (Reference Bridges and Wernet2010) and are discussed further in § 2.4. Four PIV data sets were obtained for each of 7 M_j/t test points. To establish a ‘consensus’ data set, point-by-point averaging was undertaken including weighting by a ‘quality’ metric (the number of valid PIV velocity vectors relative to the number of image pairs available). Axisymmetry was assumed and the two ‘halves’ of the flow field in each PIV image were averaged. The consensus data set was compared against measurements in the literature; generally, agreement was very good for low M_j/t conditions, somewhat worse (e.g. ±10 % variation) at M_j/t = 0.99/0.84 and difficult to estimate accurately for t = 2.7, with only one other data set available.

Values of $\delta ^{\prime}_\omega$ and L_p were extracted from the measurements. However, $\delta ^{\prime}_\omega$ was evaluated at just one axial location (measuring local shear layer width and dividing by distance from nozzle exit). Unfortunately, this was done at x = L_p, where the shear layer would already have begun its transition to a merged jet. Direct comparison with Lau's data is thus difficult, although Bridges & Wernet described this comparison as showing: ‘general trends on change with M_j and t in agreement between the two data sets’. Closer evaluation shows that, although both Lau and Bridges & Wernet results showed $\delta ^{\prime}_\omega$ decreasing with M_j, the slopes were quite different. Further, whilst the Bridges & Wernet data fell on a single line independent of t, the Lau data displayed significant sensitivity to t. On the basis of this comparison, the quantitative effect of t seems inconclusive.

Bridges & Wernet also adopted a different approach for L_p making direct comparison with earlier data problematic. L_p was selected for each M_j/t data set such that the axial turbulence peak value on the centreline was shifted to the same x/L_p location for all jet conditions. One benefit of this approach was that it clearly identified the dominant parameter in the flow region downstream of L_p. Comparison of axial mean velocity and turbulence root-mean-square (r.m.s.) contours in this region revealed that axial and radial penetration of $U/{U_j}$ (or $u^{\prime}/{U_j}$) contours was controlled solely by t and was independent of M_j. This is strong evidence that downstream of L_p it is the ${t^{1/2}}$ effect of § 2.1 which is influential rather than the compressibility effect of § 2.2. However, the substantially different approaches adopted in evaluating $\delta ^{\prime}_\omega$ and L_p makes it difficult for detailed comparison with the Bridges & Wernet data in the current work.

2.4. Nozzle exit profile effects

Details of exit profiles will influence initial flow development downstream of the nozzle trailing edge. The target engineering scenarios are at high Reynolds number: $R{e_j} = {\rho _j}{U_j}D/{\mu _j} = O(107)$ and it is important that laboratory-based experiments are conducted which are representative of this. The turbine efflux entering the nozzle, containing a core region mixing out with multiple blade wakes and wall boundary layers, is highly unlikely to produce a laminar nozzle exit boundary layer. Finally, internal nozzle acceleration introduces the possibility of re-laminarised boundary layers with low momentum thickness Reynolds number $(R{e_\theta } = {\rho _j}{U_j}\theta /{\mu _j}$, where $\theta$ is the momentum thickness at inlet to the nozzle) (Narasimha & Sreenivasan Reference Narasimha and Sreenivasan1973; Piomelli & Yuan Reference Piomelli and Yuan2013). These facts set a challenge for small-scale laboratory tests of near-field jets intended to be representative of engine operating conditions. It is essential to remember the main interest is in the mixing layer/jet not the internal flow. Careful control and monitoring of nozzle exit conditions would seem wise but has rarely been applied to date.

The importance of nozzle exit conditions for hot jets was illustrated by Lepicovsky (Reference Lepicovsky1990, Reference Lepicovsky1992). Intended to explore the effect of heat on subsonic jets, the measurements merely demonstrated strong facility-dependent effects. Jet heating – affecting density and viscosity such as to reduce the Reynolds number – had a large influence on exit boundary layer characteristics (e.g. the precise location of laminar/turbulent transition in the mixing layer). If the initial mixing layer were laminar, no dependence on jet temperature could be identified; for turbulent conditions, the effect of temperature level could be seen, although substantial scatter persisted. The recommendation was Re_j >  1.0 × 10⁶ should be used in laboratory-based test programmes to avoid laminar/turbulent transition in the external mixing layer. Birch (Reference Birch2006) also carried out a careful study of cold and hot round jet subsonic experiments with particular relevance to civil aerospace jet noise. An important conclusion was that at the high Reynolds numbers of practical relevance: ‘the laminar viscosity has little influence on jet mixing. The Reynolds No. enters the problem because the thickness of the initial wall boundary layer depends on the Reynolds No$\ldots \ldots$.it is the characteristics of the initial boundary layer that is the controlling factor, not the Reynolds No’. It was thus concluded that: ‘a minimum requirement for a jet to be only weakly dependent on initial conditions is that the mixing layer becomes fully developed within the potential core’. A minimum Re_j = 4 × 10⁵ was recommended.

None of the studies described in § 2.3 have quite met this constraint; only the lower 2 temperature ratios of the Seiner data, 10 of the 12 Lau test points and 4 of the 7 Bridges & Wernet test points had Re_j values above the recommended limit. Bridges & Wernet (Reference Bridges and Wernet2010) were the first to monitor nozzle exit conditions, applying hot-wire measurements to characterise the boundary layer. The selected nozzle geometry produces a shape factor (H ₁₂) of ~2.2 and a peak axial turbulence intensity of ~5 %, representing more a disturbed laminar than a turbulent boundary layer. In spite of providing this data, LES predictions of the Bridges & Wernet measurements (e.g. Naqavi et al. Reference Naqavi, Wang, Tucker, Mahak and Strange2016) have usually ignored this important information, and inaccuracies in comparisons with near-field measurements are probably caused by this omission.

The unheated data for high subsonic and supersonic jets presented by Trumper, Behrouzi & McGuirk (Reference Trumper, Behrouzi and McGuirk2018) did meet the minimum Re_j requirement. The approach adopted introduced a short parallel extension at nozzle exit to allow boundary layer recovery to a representative turbulent state, whilst still maintaining a high rate of internal nozzle acceleration. Most importantly, when NPR was lowered from 2.32 to 1.5 the jet Reynolds number decreased by ~70 % (whilst still remaining above 1 × 10⁶) but the exit boundary layer shape factor H ₁₂ remained within a fully turbulent range (1.3–1.45), demonstrating the robustness provided by this nozzle modification. In addition, both nozzle inlet and exit profiles were measured, to aid in validation of LES studies which simulate both internal nozzle as well as jet flow (Bres et al. Reference Bres, Jordan, Jaunet, Le Rallic, Cavalieri, Towne, Lele, Colonius and Schmidt2018; Wang & McGuirk Reference Wang and McGuirk2020). The cold flow data of Feng & McGuirk (Reference Feng and McGuirk2016) for compressible annular shear layer development mentioned in § 2.2 were taken with this nozzle design, and it was adopted for the current measurement programme (more details are provided below).

2.5. Summary

Past research on the fundamental fluid mechanics of planar shear layers has created a solid understanding of how static temperature ratio and compressibility influence turbulent mixing rates. To date, however, this seems to have had only a minor impact on our understanding of the near-field aerodynamics of hot high Mach number jets, as required to address important engineering design challenges related to jet noise and IR signature reduction. A review of experimental investigations in this area revealed just four studies with little coherent agreement on how $\delta^{\prime}_{\omega }$ and L_p depend on jet Mach number and temperature. Feng & McGuirk (Reference Feng and McGuirk2016) have also provided evidence that compressibility effects in unheated annular shear layers are different to that in planar shear layers, suggesting that further measurements on hot annular shear layer would be useful.

Perfect matching of small-scale laboratory experiments to industrially relevant conditions is particularly difficult, especially if high Re, a representative range of M_j and t conditions and appropriate nozzle exit conditions are all essential. The experiments reported here have thus been undertaken to ensure retention of representative conditions and to complement and extend the small number of existing studies of hot, high-speed jets. Improperly expanded supersonic jets, not previously considered, are also addressed. These new data, added to existing information, are analysed to establish physically based explanations for the influence of jet Mach number and static temperature ratio (t) on $\delta^{\prime}_{\omega}$ and L_p. To illustrate the benefits of the insight gained from this analysis a new empirical correlation for potential core length is introduced. The experiments were conducted under the following carefully controlled conditions:

1. (i) nozzle internal acceleration similar to aerospace propulsion nozzle practice;

2. (ii) nozzle exit Reynolds numbers above Re_j = 4 × 10⁵ in all tests;

3. (iii) NPR, NTR test points to cover representative M_j and t values (including shocks);

4. (iv) emphasis placed on examination of M_j and t effects on $\delta^{\prime}_{\omega}$ and L_p.

3.1. Experimental facility

Experiments were performed in the Loughborough University High Pressure Nozzle Test Facility (HPNTF); a photo and diagram of the system layout are in figure 2. High pressure compressor air (15 bar abs) is stored in air receivers with a volume of 110 m³ after treatment in a desiccant drier to a dew point of −40 °C. The receivers serve as pulsation dampers as well as an HP air reservoir for system operation in ‘blow-down’ mode when the desired air mass flow rate exceeds the maximum continuous supply rate (1 kg s⁻¹, with typical blow down times ~30 mins). The air total temperature is constant and equal to the ambient temperature due to the large surface area of, and long residence time in, the receivers. A control valve external to the test cell regulates the air pressure down to ~5 bar; a supply pipeline (G-150 mm diameter) then transfers the air into the HPNTF test cell within which a globe valve (E) is available to isolate the rig if required. The flow is split into two streams, one to feed a primary nozzle via a central delivery pipe (A) and the other to supply a larger diameter co-axial secondary nozzle (not used in the present study) via a branched delivery pipe (F). Mass flow and pressure in the two streams are set using separate valves (D for primary flow, H for secondary). These are computer controlled pneumatic valves automatically adjusted to maintain constant total pressure in each stream. The air supply total pressure was measured via a single probe mounted on the pipe centreline ~1.3 m upstream of the nozzle; the automatic rig pneumatic control valve held this constant to a set NPR to an accuracy of ±1 %. Primary and secondary stream delivery pipes are each fed via an initial plenum followed by a contraction (area ratios: 4/1 for primary, 11/1 for secondary). The facility was able to produce heated jets using a combustor (C) located downstream of the primary control valve. A carefully machined groove on the outside of pipe A allowed for attachment of the test nozzle using grub screws distributed equally around the circumference. The jet plume leaving the nozzle exit was available for measurement over a distance of ~1.5 m before entering a detuner for noise attenuation/exhaust.

Figure 2. High pressure nozzle test facility.

In measurements presented below jet total temperatures up to 900 K were achieved using a single can combustor (similar to those employed in the Rolls-Royce Tay engine) located in a combustor test rig (C) and positioned in the red section downstream of the control valve (figure 2a). The engine combustion system has 10 cans arranged circumferentially. So that the combustor environment is as similar as possible to the engine configuration, the single can test rig (figure 3) contains upstream and downstream transition pieces, changing the flow cross-section upstream of the combustor from a round pipe to a 36° annular sector and at combustor exit from an annular sector back to a round pipe. The pipe section following the combustor has an internal liner comprising a transition duct to convert the flow area gradually back to a circular shape over a distance of ~1.5 m. The combustor section consists of several elements: [1] main combustor housing, [2] an annular (36° sector) inlet (including aerodynamic-conditioning – vanes designed to present airflow to the combustor – as far as possible – as would occur in an engine), [3] the fuel injector, [4] a transply combustor can (constructed with a porous metal surface to effect wall cooling), [5] an exit transition nozzle (also made of transply) held within a carrier [6], a locator pin [7] to secure the combustor axially and an ignitor plug [8]. Photos of the main components – fuel injector, combustor and exit transition nozzle are shown in figure 3. The fuel injector – located centrally in the hemispherical head of the combustor – is a double swirler, air-spray design using liquid kerosene as fuel. The ignition system is a Vibro-Meter ignitor plug operating on 240 volt ac mains power, releasing a 16 Joules discharge at 1 Hz frequency to ignite the atomised fuel/air mixture.

Figure 3. Combustor arrangement within the test rig and its components left: [3] fuel injector, middle: [5] can combustor, right: combustor exit nozzle.

For the present tests a simple conical convergent nozzle of exit diameter D = 48 mm was employed, as used in Feng & McGuirk (Reference Feng and McGuirk2016) (see figure 4a). The area ratio and length of the baseline geometry were chosen to be similar to the propulsion nozzle of the BAE Systems Hawk jet trainer aircraft. A short (34.1 mm) parallel extension was added to the baseline nozzle geometry to minimise vena contracta effects and allow boundary layer recovery (final lip thickness was 1 mm).

Figure 4. (a) Axisymmetric nozzle geometry and (b) nozzle dimensions (mm).

Velocity profiles at nozzle inflow, outflow and in the near field of an un-heated jet plume from this nozzle were documented in Trumper et al. (Reference Trumper, Behrouzi and McGuirk2018) for a range of NPRs from low subsonic to moderately under-expanded (1.3–2.4). Figures 5 and 6 demonstrate that inflow and outflow profiles meet the turbulent requirements discussed in § 2. Figure 5(a) shows measured inlet boundary layer shape factor (H ₁₂) and figure 5(b) the inlet profile in (u⁺, y ⁺) wall coordinates. The shape factor varies very little with NPR (1.33 ± 0.001), as expected for a fully turbulent boundary layer; the mean velocity profile is also in close agreement with an equilibrium log law over the whole range of NPR tested. At nozzle exit, figure 6(a) shows the mean axial velocity profile collapses well over a range of NPRs for both subsonic and super-critical NPRs; figure 6(b) indicates this is also true for the turbulent Reynolds stresses (only $\overline{u^{\prime}v}$, a dash indicates a turbulence fluctuation and an over-bar a time-average, shown at 2 NPRs but a similar level of collapse was observed for other stresses and at other conditions). The data in figures 5 and 6 indicate this nozzle geometry is a good choice to study hot jet near-field development.

Figure 5. Nozzle inlet: (a) boundary layer shape factor and (b) velocity profile, log-law format.

Figure 6. Nozzle exit profiles: (a) mean velocity and (b) Reynolds shear stress, NPR = 1.55, 1.89.

The nozzle size and the NPR/NRT test conditions chosen allowed jet Reynolds numbers always to remain above 4.0 × 10⁵. For NPR = 2.32 the range of Re_j covered varied from 1.76 × 10⁶ at NTR=1.0 to 4.5 × 10⁵ at NTR = 3.01. Finally, the coordinate system used to present the data is shown in figure 4(a); the x-axis is in the jet direction, the y-axis horizontal and the z-axis vertical. Measurements of axial and radial velocity were carried out in both vertical x–z and horizontal y–z planes; these indicated excellent axisymmetry, so in most cases only x–z data are presented below.

3.2. Instrumentation

Standard (black and white) and colour schlieren systems (to guide test NPR selection) and a 2-component laser doppler anemometry (LDA) system for mean velocity and turbulence were employed to visualise and quantify flow development. A mineral-insulated K-type thermocouple was installed in a small (1.8 mm outside diameter) passively aspirated tube designed to decelerate the flow over the measuring junction to a fraction of the approach velocity and minimise recovery factor effects. This allowed the thermocouple to measure a time-averaged temperature close to the local flow total temperature (calibration against a platinum resistance thermometer showed the error for the hot test conditions chosen was typically ±1 %). Colour schlieren uses an orange–green–blue slide (horizontally orientated) instead of a knife edge; orange indicates flow expansion regions, blue indicates compression and green corresponds to undeflected light. The Z-type schlieren arrangement consisted of a mercury vapour lamp, 2-concave mirrors of 10 inch diameter, 2-plane mirrors of 12 inch diameter, a knife-edge unit containing the orange–blue–green slide and a Sony digital camera. Schlieren pictures were taken of the flow downstream of nozzle exit up to x/D = 4. The LDA system (figure 7) was a Dantec 2-component fibre optic system: a 5 W argon/ion laser, a beam transmitter and projector, and a high-speed signal processor (maximum analysable frequency 80 MHz equivalent to ~800 m s⁻¹). For the present measurements, the beam projector had a focal length of 310 mm, beam spacing and diameter were 38 mm and 1.35 mm, resulting in an LDA measurement volume with spatial dimensions of 0.15 mm (vertical and horizontal) and 2.3 mm (longitudinal (along optical axis)). Traversing of the LDA probe was achieved using a three-axis Dantec traverse with a positional accuracy of 0.005 mm. For low NPR jets, liquid droplet seeding was possible (0.3 μm, density 920 kg m⁻³), but for high NPR and hot jets solid alumina oxide particles were necessary (0.3 μm, 3960 kg m⁻³). Data rates were typically 7 ~ 10 KHz and a sample population of 20–50 K validated readings was used to evaluate time-averaged statistics. The same LDA configuration was employed by Feng & McGuirk (Reference Feng and McGuirk2016) and Trumper et al. (Reference Trumper, Behrouzi and McGuirk2018), who have provided assessment of mean and turbulent velocity measurement uncertainty. For mean velocity in regions of low (< 2 %) turbulence an accuracy of ±0.05 % of the true mean was estimated, rising to ±5 % in regions where data rates were low and turbulence levels high; measured turbulent stresses were estimated to lie within ±5 % of the true value.

Figure 7. LDA system and traversing table.

3.3. Initial exploratory measurements

Schlieren imaging of unheated jets was employed to guide selection of the optimum NPR for hot tests; a range of NPRs from low subcritical (1.8) to under-expanded (~6.0) were examined. Section 2 had indicated that data for jets containing embedded shocks have never been considered. It was therefore decided the focus of testing should be on a supersonic jet with moderate under-expansion. Jet size and NPR should require a mass flow rate allowing LDA measurement in reasonable time with continuous rig operation over as large a range of NTR as possible. Figure 8 indicates the inviscid shock cell patterns observed in under-expanded jets within the NPR range 2.0–3.0. The development of the jet shear layer is clearly visible, with the core jet flow containing repeated oblique shock waves reflecting (imperfectly and becoming weaker) from the shear layer/ambient pressure boundary. Expansion and compression regions grow longer as NPR increases; at NPR = 2.0 the first shock diamond length is less than 0.5D increasing to ~0.75D at NPR 2.32 and ~1.2 at NPR = 3.0.

Figure 8. (a) Black and white schlieren for NPR = 2.0, NTR = 1.0. (b) Colour schlieren for (left) NPR = 2.32, (right) NPR = 3.0, both at NTR = 1.0.

The ability of seed particles to follow the high spatial gradient velocity change in the vicinity of shocks influences measurement accuracy. Velocity slip error was investigated by assuming a Stokes drag law and calculating the particle relaxation time ($\tau_{p}$) when experiencing a step change in velocity – $\tau_{p} = \rho_{p}d_{p}^{2}/18\mu $ ($\rho_{p}$, d _p and $\mu$ are particle density, diameter and air viscosity and $\tau_{p}$ is the time for the particle velocity lag (difference between particle and fluid velocity) to reduce by a factor e ⁻¹).

The relaxation length is the distance moved in this time. Relaxation times for solid seeding were 1.3 − 0.6 μs (NTR = 1.0–3.01), with relaxation lengths 1.1–0.5 mm. Values for liquid particles were ~1/4th of these values, but liquid seeding was not viable in hot jet testing due to evaporation. To investigate the impact of a longer relaxation time for all-important shear layer measurements, LDA data were taken using both seeding types at 2 locations in the first shock cell of an NPR = 3.0, NTR = 1.0 jet (liquid seeding SNR was just acceptable at this NPR, at higher NPR condensation of water vapour in the entrained ambient air made LDA signal processing difficult).

In figure 9 the first location (x/D = 0.5) is in the accelerating expansion region immediately after nozzle exit, and the second (x/D = 1.0) in the following compression region; results for time-averaged axial velocity and turbulence r.m.s. are shown. Mean velocity results quantify the extent that liquid seeding follows the acceleration and deceleration process better than solid seeding. The maximum velocity difference between the two stations was 110 m s⁻¹ for liquid but only 95 m s⁻¹ for solid seeding. Differences were ~12 % in the inviscid core but only ~1 % in the shear layer. For turbulence (figure 7b) only x/D = 1.0 is shown for clarity (other results were similar), the difference was less than 1 % at x/D = 1.0. The conclusion drawn was that solid seeding had acceptable performance for the measurements undertaken here.

Figure 9. Seeding particle tests at NPR = 3.0, NTR = 1.0. (a) x/D = 0.5 and 1.0 mean axial velocity, (b) x/D = 1.0 axial turbulence r.m.s.

LDA measurements in hot jets were conducted first at NPR = 1.89 (M_j = 1.0) to establish optimum practice for hot testing and explore test time variation with NTR. For stable and controllable hot starting, the rig was initially operated with HP air and fuel flow rates close to the optimum point of the known combustor stability loop (furthest removed from rich and lean blow-off boundaries). For a 60 mm diameter nozzle this corresponded to an operating condition of NPR = 1.75, NTR = 2.45 (T_t,j = 706 K), requiring only marginal adjustment for other nozzle sizes. Once stable, NPR and fuel flow rate were gradually adjusted to achieve the desired test point. Initial measurements (figure 8) at NTR = 1.0 and 3.0 confirmed the expected behaviour, faster turbulent mixing and reduced potential core length for increased jet heating. This figure also demonstrates how L_p was deduced in the current work (following the Witze (Reference Witze1972) and Lau (Reference Lau1980) practice) by backwards extrapolation of the centreline velocity decay rate to intersect $U_{cl}/U_{j} = 1.0$. For the conditions shown in figure 10 L_p reduced by 13 % and the centreline velocity decay rate increased by 36 % ($\partial {U^{\ast}}/\partial {x^{\ast}}$, $U^{\ast} = {U_{cl}}/{U_j}$ and $x^{\ast} = x/D$).

Figure 10. Mean centreline axial velocity at NPR = 1.89 for NTR = 1.0 and 3.0.

Based on these preliminary measurements (in particular the time taken), a jet at NPR = 2.32 was chosen for detailed testing. A complete set of near-field mean velocity, temperature and turbulence measurements was targeted for NTR = 2.0 (nominal) and centreline profiles of the same variables at 5 (nominal) NTRs: 1.0, 1.5, 2.0, 2.5 and 3.0. The literature review above showed existing work predominantly covered subsonic and fully expanded jets. Hence, in the present work emphasis was placed on jet flows relevant to high-speed military aircraft, i.e. supersonic jets containing embedded shocks and covering as large a temperature ratio range as was feasible in the experimental facility used. This ensured a sufficiently wide and representative range of M_j and t were included to capture important compressibility and temperature ratio effects.