2.1. Governing equations for the axisymmetric compressible jet

Numerical studies typically solve the Navier–Stokes equations in conservative form and naturally employ mass-weighted, or Favre, averaging, where density moments are absorbed in the averaging procedure. When using velocimetry data from experiments, the conventional Reynolds-averaged forms of these equations are more suitable for comparison, but result in several additional terms containing correlations of the fluctuating density. For sufficiently low Mach numbers, these additional correlations can be neglected. Morkovin (Reference Morkovin1964) suggests that for flows where the total temperature fluctuations are negligible, the magnitude of the density fluctuations can be approximated by the strong Reynolds analogy,

(2.1) \begin{equation} \frac {\sqrt {\overline {\rho ^{\prime \prime 2}}}}{\rho } \approx -\frac {\sqrt {\overline {\varTheta ^{\prime \prime }}}}{\varTheta } \approx (\gamma -1)M^2 \frac {\sqrt {\overline {u^2}}}{U} , \end{equation}

with $\rho$ the mean density, $\rho ^{\prime\prime}$ the fluctuating density, $\varTheta$ the mean temperature, $\varTheta ''$ the fluctuating temperature, $\gamma$ the specific heat ratio, $u_i$ are the components $(u,v,w)$ of the fluctuating velocities and $U_i$ the components $(U,V,W)$ of the mean velocities. This relation shows $\rho ^{\prime\prime}/\rho$ is largest in regions where high Mach number and turbulence intensity coincide. Originally derived for adiabatic flat-plate boundary layers, this has been extended to free shear flows such as mixing layers (Bradshaw Reference Bradshaw1977). For the current configuration where the stagnation temperature of the jet is approximately the ambient condition, the strong Reynolds analogy is valid, and since $\sqrt {\overline {u^2}}/U \approx 0.2$ , $\sqrt {\overline {\rho ^{\prime \prime 2}}}/\rho \,\lt 0.1$ for $M \lt 1.25$ . Per the Cauchy–Schwarz inequality, for example, $\overline {uv} \lt \sqrt {\overline {u^{2}}}\sqrt {\overline {v^{2}}}$ , terms such as $\overline {\rho ^{\prime\prime} uv}$ are an order-of-magnitude smaller than $\rho \overline {uv}$ . This reduces the governing equations to

(2.2) $${\boldsymbol{\nabla} _i \bigl (\rho U^i \bigr ) = 0,} $$

(2.3) $${\rho U^j \boldsymbol{\nabla} _{\!j} U_i = \boldsymbol{\nabla} _i \overline {p} - \boldsymbol{\nabla} _{\!j} \bigl ( \rho \overline {u_i u^j}\bigr ) + \boldsymbol{\nabla} _{\!j} \overline {\tau }^{j}_i , }$$

where $i,j,k$ are indices reserved for spatial components, $\boldsymbol{\nabla} _{\!k}$ is the covariant derivative, $\overline {p}$ the mean pressure, $\overline {\tau }^j_i$ the mean stress tensor and overline indicating a temporal mean. In this section, superscripts indicate upper indices and not exponentiation, and repeated indices imply the Einstein summation convention. In addition, to determine the scaling of high-order moments, the scaling behaviour of the RST equations is necessary. To analyse the first-order scaling effects of the RST equations, only the significant terms are considered. As the pressure–strain correlation cannot be distinguished from the pressure-gradient term using only velocimetry measurements, it is assumed that

(2.4) \begin{equation} \overline {p''\boldsymbol{\nabla} _{\!j} u_i } + \overline {p'' \boldsymbol{\nabla} _i u_{\!j} } \gg \boldsymbol{\nabla} _{\!k} \left [\overline {u_i p''} \delta ^k_j - \overline {u_{\!j} p''}\delta ^k_i \right ] . \end{equation}

Thus, the pressure–diffusion terms are neglected to isolate and characterise the Mach number scaling on the pressure–strain term, which, unlike the pressure–diffusion, has a significant role in the Mach number suppression of turbulence. Again, terms containing fluctuating-density correlations are neglected, under the assumptions of the strong Reynolds analogy, and viscous diffusion is neglected for high- ${Re}$ shear flows. This results in the first-order RST equations,

(2.5) \begin{align} \left [\boldsymbol{\nabla} _{\!k} \bigl ( \rho U^k \overline {u_i u_{\!j}} \bigr )\right ] &= - \boldsymbol{\nabla} _{\!k} \left [ \rho \overline { u_i u_{\!j} u^k} \right ] - \left [\bigl (\rho \overline {u_{\!j} u^k}\bigr ) \boldsymbol{\nabla} _{\!k} U_i + \bigl (\rho \overline {u_i u^k}\bigr )\boldsymbol{\nabla} _{\!k} U_{\!j} \right ] \nonumber \\[4pt] &\quad +\, \left [\overline {p''\boldsymbol{\nabla} _{\!j} u_i } + \overline {p'' \boldsymbol{\nabla} _i u_{\!j} } \right ] - \left [ \overline {\bigl (\tau ^{k}_j\bigr ) ^{\prime\prime} \boldsymbol{\nabla} _{\!k} u_i} + \overline {\left (\tau ^{k}_i \right ) ^{\prime\prime}\boldsymbol{\nabla} _{\!k} u_{\!j}} \right ], \end{align}

with $p''$ the fluctuating pressure and $\bigl (\tau ^{k}_i\bigr ) ^{\prime\prime}$ the fluctuating viscous stress tensor. This equation consists of the terms contributing significantly to the turbulence energetics. These are to be analysed through self-preservation, through which the effect of $M_c$ on the scaling of these terms can be determined. The exact forms, without approximation, of (2.2)–(2.5) are found in Appendix A. As in the incompressible case, the terms grouped in the square brackets correspond to the advection, diffusion, production, pressure–strain and viscous-dissipation terms, respectively.

2.2. Self-preservation of the compressible axisymmetric jet

Self-preserving flows provide insight into the dynamical processes; self-preservation implies a state of equilibrium of the turbulence energetics. A flow is considered self-preserving if, when scaled, can produce collapsing profiles. For the incompressible jet, the scales are the centreline velocity and jet width, where according to classical theory, the jet width spreading rate, $b'$ , is considered a universal constant which ‘forgets’ its initial conditions (Townsend Reference Townsend1980). George (Reference George, George and Arndt1989) challenged these ideas and instead classifies flows as fully, partially and locally self-preserving. The classification of which moments that can collapse depends on the scale determined through the similarity analysis of the governing equations. Under this classification, the incompressible jet is considered fully self-preserving when the profiles of all individual moments can collapse with the appropriate scale. Furthermore, George (Reference George, George and Arndt1989) argues that the scale depends on the individual $b'$ of the jet, which is not universal, and can differ depending on initial conditions and Reynolds number. Partially self-preserving flows then refer to flows where at least the Reynolds-shear stress collapses according to the scale $U_m^2 b'$ for the jet (George Reference George, George and Arndt1989). Lastly, locally self-preserving refers to flows where profiles are observed when normalised with a scale not determined through the governing equations.

For a compressible flow to be considered self-preserving, the thermodynamic quantities and their moments must develop either consistently with the other terms or be negligible. In the compressible mixing layer, assuming a turbulent Prandtl number $Pr_t \approx 1$ , a Reynolds analogy can be made such that the density and temperature scale with the velocity (Pantano & Sarkar Reference Pantano and Sarkar2002; Smits & Dussauge Reference Smits and Dussauge2006). After similarity analysis of the momentum equation, the main condition for self-preservation becomes $\overline {uv}/U_m^2 \propto b'$ , where $U_m$ is the velocity difference of the mixing layer. In the compressible mixing layer, $b'$ is constant as the shear layer develops, which depends on the configured $M_c$ , and satisfies this condition (Vreman et al. Reference Vreman, Sandham and Luo1996). This scaling indicates at least a partially self-preserving flow, with full self-preservation contingent on how higher moments scale.

As opposed to the mixing layer, where compressibility effects are constant and $M_c$ , $b'$ and $\rho$ do not vary with streamwise location, these quantities now change as the jet develops. Following George (Reference George, George and Arndt1989), the functional dependencies of the similarity variables $U_m(x)$ , $\rho _m(x)$ , $p_m(x)$ and $R_{ij}(x)$ are left to be determined. Defining in physical components,

(2.6) \begin{align} \eta &= r/b(x) , \end{align}

(2.7) \begin{align} U &= U_m(x) f(\eta ) , \end{align}

(2.8) \begin{align} \rho &= \rho _m(x) g(\eta ) , \end{align}

(2.9) \begin{align} p &= p_{m}(x) h(\eta ) , \end{align}

(2.10) \begin{align} \overline {u^i u^j} &= R_{ij}(x) r_{ij}(\eta ) , \end{align}

where $\eta$ is the normalised radius, and $f(\eta )$ , $g(\eta )$ , $h(\eta )$ and $r_{ij}(\eta )$ are shape functions. The density profile, $g(\eta )$ , can be considered a function of the velocity using a Crocco–Busemann relation (Smits & Dussauge Reference Smits and Dussauge2006). The momentum of the compressible jet is conserved, as in the incompressible jet. To the first order,

(2.11) \begin{equation} J_0 = 2 \pi \int _{0}^{\infty } \rho U^2 r\, {\textrm{d}}r , \end{equation}

where $J_0$ is a constant. Substituting (2.6)–(2.8) into (2.11), it can then be shown that

(2.12) \begin{equation} \frac {U_m^\prime }{U_m} = -\frac {1}{2} \frac {\rho _m^\prime}{\rho _m} - \frac {b'}{b}, \end{equation}

which provides a relationship between the density, velocity, jet thickness and their streamwise gradients. With (2.2), (2.12), the streamwise momentum equation,

(2.13) \begin{equation} \rho U \frac {\partial U }{\partial x} + \rho V \frac {\partial U}{\partial r} = \frac {\partial }{\partial x} \overline {p} - \frac {1}{r} \frac {\partial r \rho \overline {uv}}{\partial r} - \frac {\partial \rho \overline {u^2}}{\partial x} , \end{equation}

can be shown for the jet using similarity variables,

(2.14) \begin{eqnarray} \frac {1}{2}\frac {\rho _m^\prime}{\rho _m} &&\left ( \textit{gff} - \frac {\partial gf}{\partial \eta } \frac {1}{g\eta } \int _{0}^{\eta }\! gf \hat {\eta } {\textrm{d}}\hat {\eta } \right ) - \frac {b ' }{ b} \left ( \textit{gff} + \frac {\partial gf}{\partial \eta } \frac {1}{g\eta } \int _{0}^{\eta }\! gf \hat {\eta } {\textrm{d}}\hat {\eta } \right ) \nonumber \\[4pt] &&= -\frac {p_{m}b'}{b U_m^2 \rho _m} \frac {\partial h}{\partial \eta } + \frac {p_{m}'}{U_m^2 \rho _m } h - \frac {R_{12}}{b U_m^2} \frac {1}{\eta } \frac {d(g r_{12}\eta )}{{\textrm d}\eta } + \frac {R_{11}b^\prime}{b U_m^2}\eta \frac {d(g r_{11})}{{\textrm d}\eta } - \frac {(R_{11}\rho _m)'}{U_m^2 \rho _m} gr_{11} . \nonumber \\ \end{eqnarray}

In this form, the mean density effects on the jet flow is contained in the first term on the left of (2.12) and do not scale in a manner proportionally with the other terms. Therefore, self-preservation can only be approximated if the density gradients can be neglected, such that,

(2.15) \begin{equation} \frac {U_m^\prime }{U_m} \approx - \frac {b'}{b}, \end{equation}

and will be shown in § 4.2 to hold for the current jet. Using this simplification, the first-order, without the Reynolds-normal stresses, self-preservation equation is then

(2.16) \begin{equation} -\textit{ff} - \frac {1}{\eta }\frac {\partial f}{\partial \eta }\ \int _{0}^{\eta }\! f \hat {\eta } \,{\textrm{d}}\hat {\eta } = \frac {R_{12}}{U^2_m b'} \frac {1}{\eta } \frac {{\textrm d}(r_{12}\eta )}{{\textrm d}\eta } , \end{equation}

such that the Reynolds-shear stress must satisfy the scaling,

(2.17) \begin{equation} b' \propto \frac {R_{12}}{U_m^2} , \end{equation}

where a linear relationship between $b$ and $x$ is not assumed or required, and shear stress $\overline {uv}/U_m^2$ scales with the spreading rate $b'$ , and is consistent with that observed for the self-preserving compressible mixing layer (Vreman et al. Reference Vreman, Sandham and Luo1996; Menaa Reference Menaa2003). The remaining Reynolds-normal stress and pressure in (2.14) scale according to

(2.18) \begin{equation} \textit{Const.} \propto \frac {p_m}{\rho _m U_m^2 } \propto \frac {p'_m b}{\rho _m U_m^2 b^{\prime }} \propto \frac {R_{11}}{U_m^2} \propto \frac {R'_{11} b}{U_m^2 b^{\prime }}. \end{equation}

The forms of the equations and scales recover the findings for the incompressible round jet by George (Reference George, George and Arndt1989), which introduced that satisfying the condition of (2.17) is sufficient for a partially self-preserving flow, that is, the streamwise momentum equation is satisfied to the first order. For fully self-preserving flows, the normal stresses and all higher moments must behave according to the scales determined through self-preservation analysis. As the scaling and their requirements have yet to be established for compressible jets, it is unclear whether this condition can be met or its impact on describing the flow physics.

3.1. Flow conditions

Figure 1 is a schematic describing the free jet exhausting into ambient environment. A subsonic and supersonic case are investigated at Mach 0.3 and Mach 1.25, respectively. Building compressed air is supplied and controlled using an Omega PRG700 pressure regulator. The stagnation temperature, $T_0$ , and stagnation pressure, $P_0$ , are measured before flow enters a 0.5 m long stainless steel pipe of inner diameter 10.75 mm. The end of the pipe is fitted with a converging–diverging nozzle and is designed using the method of characteristics with an exit-to-throat area ratio of 1.047, with a nozzle exit diameter of 6.35 mm. Ambient laboratory conditions are also monitored, ensuring constant ambient conditions between tests. The stagnation-to-ambient nozzle pressure ratios are 1.87 and 2.70, corrected by 2 % and 5 %, respectively, to account for losses in the pipe assuming adiabatic Fanno flow for the Mach 0.3 and 1.25 cases. Both jets are considered fully turbulent, with ${Re}$ $\approx$ 36 000 and 86 000 for the Mach 0.3 and Mach 1.25 cases, respectively. The turbulence intensity exiting the converging–diverging nozzle is estimated to be approximately 2.5 % at the jet centreline at $x/d = 1$ where measurements are available.

Figure 1. Schematic of jet flow.

3.2. Particle image velocimetry system

Figure 2 shows a diagram of the experimental set-up. Planar particle image velocimetry (PIV) was employed to measure two-component velocity fields in the plane bisecting the nozzle’s centre plane. Velocity measurements were captured over a streamwise range from $x/d = 0$ to 25, using four overlapping fields-of-view (FOVs) of approximately 50 mm $\times$ 50 mm. The nozzle was mounted on rails, allowing it to slide independently of the laser and camera in the streamwise direction, ensuring consistent alignment between the light sheet and the camera across different measurements. The FOVs overlapped by 10.8 mm, covering approximately 25 % of the streamwise direction to ensure seamless data collection across the measurement region.

Figure 2. Schematic of free jet PIV experimental set-up.

Both the jet and the ambient air were seeded with ${\sim}1$ ${\rm \unicode{x03BC}}$ m di-ethyl-hexyl-sebacat (DEHS) particles, generated by a six-jet Laskin nozzle (TSI 9307-6). Although solid tracers are typically used for high-speed flows, DEHS was selected due to the need to seed the open environment of the jet flow. DEHS was preferred over solid particles due to the difficulties in collecting solid particles in an open environment while maintaining a suitable response time. Ragni et al. (Reference Ragni, Schrijer, van Oudheusden and Scarano2011) demonstrated that DEHS particles of this size have a response time of approximately $2\, {\rm \unicode{x03BC}}$ s, which is comparable to solid tracers like $\text{TiO}_{\text{2}}$ . This results in a Stokes number of ${\approx}0.1$ , and ensures that the DEHS particles closely follow the flow’s velocity fluctuations, providing accurate measurements of both mean velocity and turbulence quantities.

The PIV system consisted of a complementary metal–oxide–semiconductor (CMOS) camera (FastCAM Photron SA-5) with a 20 $\times$ 20 ${\rm \unicode{x03BC}}$ m sensor and a resolution of 1024 $\times$ 1024 pixels camera was equipped with a Nikon AF Micro-Nikkor 60 mm f/2.8D lens and positioned 20 cm perpendicular to the light sheet. This results in a spatial pixel resolution of approximately 20 pixels ${\textrm {mm}}^{-1}$ . The inter-frame time was set to 0.23 ${\rm \unicode{x03BC}}$ s, independent of frame rate and resolution. A dual-cavity Nd:YLF laser (Photonics DM20-527-DM) was used as the illumination source, operating at a repetition rate of up to 10 kHz and delivering up to 20 mJ per pulse. The light sheet was formed using two spherical lenses and a cylindrical lens with a thickness of approximately 1 mm. At the nozzle exit – where the error due to the light-sheet thickness relative to the jet is greatest – the difference between the chord length at the laser-sheet edge intersecting the nozzle and the nozzle diameter is less than 1.25 %. Hence, the potential effect of an azimuthal orientation error can be considered negligible, and the measured velocity components approximate well the two-dimensional field along the PIV centre-plane. Measurements were conducted in dual-pulse mode, with a pulse separation interval of 1–1.5 ${\rm \unicode{x03BC}}$ s, ensuring that particles moved approximately 7 pixels in the centre of the jet between image pairs. A silicon photodetector (Thorlabs DET02AFC) was used to verify the pulse separation timing to match the trigger signal.

The image pairs were processed using LaVision DaVis 10.2.0 software. Each test consisted of 2700 image pairs, with Reynolds stresses converging after approximately 750 pairs. The processing employed a multi-pass strategy, refining the vector field iteratively to improve accuracy, resulting in a 16 $\times$ 16 pixel interrogation window size with 75 % overlap. Sub-pixel accuracy when determining the particle image shift is achieved by peak fitting a Gaussian three-point estimator. This provides roughly 40 vectors across the jet’s exit diameter. The 95 % expanded uncertainty bounds were calculated to account for random sampling errors (Benedict & Gould Reference Benedict and Gould1996) and are displayed where they exceed the marker size. Only random errors are displayed for the velocity moments as they are large in magnitude, hence more conservative than a posteriori uncertainty estimation methods available from the DaVis software, which does not consider the errors associated with the high-speed nature of the current flow.

4.1. Kleinstein–Witze scaling

Methods approximating the centreline velocity development for compressible axisymmetric jets has been addressed through the Kleinstein–Witze scaling. Assuming an eddy viscosity in the form (Ferri, Libby & Zakkay Reference Ferri, Libby and Zakkay1964)

(4.1) \begin{equation} \rho \nu _t = (\kappa /4) b \rho _m U_m ,\end{equation}

where $\kappa$ is a proportionality constant, the solution of the linearised momentum equation describes the centreline velocity decay in the form,

(4.2) \begin{equation} \frac {U_m}{U_{ \textit{jet}}} = 1 - \exp \left [\frac {-1}{2 \kappa \left (\dfrac {x}{d}\right ) \left (\dfrac {\rho _{ \textit{jet}}}{\rho _{0}}\right )^{1/2}- X_c}\right ] , \end{equation}

where $X_c$ is a constant describing the jet core length (Kleinstein Reference Kleinstein1964). Figure 6 shows the centreline velocity development with fitted functions of (4.2) overlaid. The constants are found to be $\kappa = 0.0796$ and $X_c = 0.487$ for $M_{ \textit{jet}} = 0.3$ , and $\kappa = 0.0842$ and $X_c = 0.511$ for $M_{ \textit{jet}} = 1.25$ . The scaling was originally proposed with constants $\kappa = 0.074$ and $X_c = 0.7$ . Witze (Reference Witze1974) suggests $\kappa$ to have some Mach number dependence. Studies by Lau, Morris & Fisher (Reference Lau, Morris and Fisher1979), Lau (Reference Lau1981) and Wernet (Reference Wernet2016) confirm that (4.2) generally captures the effect of density, but large scatter remains in the values reported from different studies for $\kappa$ and $X_c$ . Here, the lower values of $X_c$ reflect the pipe-nozzle configuration on the jet initial condition.

Figure 6. Downstream development of $U_m/U_{ \textit{jet}}$ for Mach 0.3 () and Mach 1.25 (). Witze–Kleinstein functions for Mach 0.3 () and Mach 1.25 ().

Kleinstein (Reference Kleinstein1964) also shows that the total enthalpy, $H$ , is given in the form,

(4.3) \begin{equation} \frac {H-H_{\textrm {0}}}{H_{ \textit{jet}}-H_{\textrm {0}}} = 1 - \exp \left [\frac {-1}{2 \kappa Pr_t^{-1}\left (\dfrac {x}{d}\right ) \left (\dfrac {\rho _{ \textit{jet}}}{\rho _0}\right )^{1/2}- X_c}\right ] , \end{equation}

where $Pr_t = 0.715$ was experimentally determined by Kleinstein (Reference Kleinstein1964). This relation allows the centreline total enthalpy to be determined from the fit constants of the velocity data. This allows further derived properties like the density, temperature and Mach number to be estimated on the jet centreline, which are used in the following section to aid in the scaling of the jet.

4.2. Compressibility and spreading rate

Compressibility impacts $b'$ through two key mechanisms: the mean density ratio and the convective Mach number, $M_c$ . Where the mean density gradient affects $b'$ through continuity, $M_c$ affects turbulence mixing through the attenuation of the Reynolds stresses. These two effects have been shown to act independently (Bradbury & Riley Reference Bradbury and Riley1967; Barre, Quine & Dussauge Reference Barre, Quine and Dussauge1994). Brown & Roshko (Reference Brown and Roshko1974) compared subsonic mixing layers of different density ratios with supersonic mixing layers having the same density ratios. The behaviour of supersonic mixing layers could not be described by density variations alone, unlike the behaviour observed in compressible boundary layers. Although density variations influence the spreading rate, this effect is generally only significant at density ratios $\gt$ 7 in both subsonic (Brown & Roshko Reference Brown and Roshko1974) and supersonic mixing layers (Menaa Reference Menaa2003). Therefore, at moderate density ratios, $b'$ is mainly a function of $M_c$ in supersonic mixing layers. Figure 7 shows the spreading rate, $b'$ , for $M_{ \textit{jet}} = 0.3$ and $M_{ \textit{jet}} = 1.25$ . To reduce noise in the slope, a low-order polynomial fit of the half-width data is used to calculate the rate.

Figure 7. Downstream development of $b'$ in ${\textrm{mm}}/{\textrm{mm}}$ from curve fits of $b$ for Mach 0.3 () and Mach 1.25 (). Shaded areas show 95 % expanded uncertainty bounds in respective colours.

Figure 8. Downstream development of $M_c$ () left axis, $b'/b$ (), $U_m'/U_m$ () and $(1/2)\rho '/\rho$ () right axis for Mach 1.25. Shaded areas show 95 % expanded uncertainty bounds in respective colours.

The downstream development of $M_c$ is shown in figure 8, where the shaded area shows the 95 % expanded uncertainty bounds. For an identical specific heat ratio of the gases on both sides of the shear layer, the convective velocity is defined as

(4.4) \begin{equation} U_c = \frac {a_l U_m + a_m U_l}{a_m + a_l} , \end{equation}

where $a$ is the speed of sound, and the high-speed and low-speed sides are noted with subscripts $m$ and $l$ , respectively (Papamoschou & Roshko Reference Papamoschou and Roshko1988). In the free jet, the centreline quantity corresponds to the high-speed side and the low-speed side ambient conditions where $U_l = 0$ , where $M_c$ simplifies to the following relation,

(4.5) \begin{equation} M_c = \frac {U_m}{a_m + a_l}. \end{equation}

Here, $a_m$ is approximated using the temperature determined from the total enthalpy from (4.3) and $a_l$ from the ambient static temperature.

Figure 8 depicts the downstream development of the streamwise gradients of $U_m$ and $b$ as ratios $U_m'/U_m$ and $-b'/b$ . In the region $x/d \gt \approx$ 10, $U_m'/U_m$ and $-b'/b$ differ less than the experimental uncertainty, thus justifying the approximation of (2.15) and implies $\rho _m'/\rho$ is generally small. Satisfying (2.15) implies neglecting the first term of (2.14). This approximation extends previous observations in mixing layers (Brown & Roshko Reference Brown and Roshko1974; Menaa Reference Menaa2003) to jets, showing that at the current density ratios, the reduction in $b'$ can be attributed primarily to $M_c$ effects.

Figure 9. Shear layer thickness growth rate suppression. Current study in () compared with (a) previous experimental studies and (b) numerical studies. For the legend of used markers, see table 1. Axisymmetric data coloured in red and planar data in blue. Curve from Dimotakis (Reference Dimotakis1991) () and Langley experimental curve from Kline, Cantwell & Lilley (Reference Kline, Cantwell and Lilley1982) (). Shaded areas show 95 % expanded uncertainty bounds.

To compare the effect of compressibility on the spreading rate between the jet and the mixing layer, $\varPhi (M_c)$ defined in (1.1) is employed. Here, the spreading rate of the $M_{ \textit{jet}} = 1.25$ jet as it decays is normalised by the value from the incompressible $M_{ \textit{jet}} = 0.3$ jet from figure 7. Unlike the mixing layer, $\varPhi (M_c)$ in the jet changes as $M_c$ decays. Figure 9 compares the behaviour of $\varPhi (M_c)$ in the current jet data with earlier results for mixing layers. Similar trends between the data suggest that the local $M_c$ strongly influences $\varPhi (M_c)$ . Although the magnitude of the values observed in the present jet is similar to those of the planar measurements from Papamoschou & Roshko (Reference Papamoschou and Roshko1988), the slope of the jet data tends to be more similar to the annular mixing layer than the planar mixing layer experiments. Feng & McGuirk (Reference Feng and McGuirk2016) suggests the hypothesis where axisymmetric symmetry modifies the behaviour of $\varPhi (M_c)$ , observing that the spreading rate suppression occurs at lower $M_c$ , where the cause of this phenomenon remains unelucidated. It thus appears that while $\varPhi (M_c)$ is mainly a function of $M_c$ , geometric effects also have an influence.

5.1. Reynolds-shear stress scaling

Figure 10(a) presents the Reynolds-shear stress profiles for the Mach 0.3 jet, normalised using the classical scaling $U_m^2$ . For comparison, the incompressible LDV data from Hussein et al. (Reference Hussein, Capp and George1994) is overlaid. From axial positions beyond $x/d = 17.5$ , a good collapse of the profiles is observed and agrees with the LDV measurement from Hussein et al. (Reference Hussein, Capp and George1994). The similarity of the Reynolds stresses, like the mean profiles discussed in § 4, are observed to occur earlier upstream due to the initial conditions of the current jet. In contrast, for $M_{ \textit{jet}} = 1.25$ in figure 10(b), the collapse of the Reynolds-stress profiles is less satisfactory and the amplitude is lower in the upstream regions where $M_c$ is larger.

Figure 10. Reynolds-shear stress. Classical scaling with ${U}_m^2$ . LDV data from Hussein et al. (Reference Hussein, Capp and George1994) (). Every fourth point shown for clarity. Shaded areas show 95 % expanded uncertainty bounds in respective colours. (a) Mach 0.3. Axial positions: $x/d =17.5$ () ; 20.0 (); 22.5 (); 25.0 (); 27.5 (); (b) Mach 1.25. Axial positions: $x/d =12.5$ (); 15.5 (); 18.5 (); 21.5 (); 24.5 ().

Figure 11. Mach 1.25. Collapse of $\overline {uv}$ profiles with scaling $U_m^2 b^{\prime }$ . Same legend as figure 10(b). Profile calculated from (5.1) ().

Figure 12. Maximum Reynolds stress $\overline {uv}$ normalised by $U_m^2$ as a function of $M_c$ for mixing layers and the present jet data. Samimy & Elliott (Reference Samimy and Elliott1990) (), Goebel & Dutton (Reference Goebel and Dutton1991) (), Urban & Mungal (Reference Urban and Mungal2001) (), Debisschop, Chambers & Bonnet (Reference Debisschop, Chambers and Bonnet1994) (), Pantano & Sarkar (Reference Pantano and Sarkar2002) (), Freund et al. (Reference Freund, Lele and Moin2000) (), current study ().

In figure 11, the profiles of Reynolds stress using the self-preservation scaling, $U^2_m b'$ , collapse better, especially for the data upstream where $M_c$ is elevated. Using this scaling, the Reynolds-shear stress in the Mach 1.25 jet exhibits similarity at approximately $x/d = 12.5$ . This scaling also collapses the incompressible data. The scaled Reynolds-shear stress profile can be calculated using the mean velocity when rearranging and integrating (2.16) in the following manner:

(5.1) \begin{equation} \frac {R_{12}r_{12}}{U_m^2 b'} = \frac {1}{\eta }\int _0^{\eta }\left (-\tilde {\eta }ff - \frac {\partial f}{\partial \tilde {\eta }} \int _{0}^{\tilde {\eta }} f \hat {\eta }\,{\textrm{d}}\hat {\eta }\right ) \,{\textrm{d}}\tilde {\eta }. \end{equation}

The Reynolds-shear stress profile calculated from (5.1) is plotted and aligns with the data, validating the scaling and tacitly supporting the underlying assumptions underpinning (2.1) and (2.15). Figure 12 compares the behaviour of the max Reynolds-shear stress, $\overline {uv}/U_m^2$ , as a function of $M_c$ , providing a comparison to mixing layer data. Again, a decreasing trend in amplitude with $M_c$ is observed. Collapse of the Reynolds-shear stress indicates a form of self-preservation is observed in the compressible jet and suggests the flow is at least partially self-preserving. Self-preservation of other turbulence moments is unclear and is now investigated.

5.2. Reynolds-normal stress scaling

The proposed scaling of the streamwise Reynolds stress, $\overline {u^2}$ , derived from the self-preservation analysis of (2.14), is $R_{11} \propto U_m^2$ . This scaling suggests that assuming self-preservation implies that $b'$ and, therefore, compressibility do not affect this Reynolds-stress component. Figure 13(a) plots $\overline {u^2}/U_m^2$ for $M_{ \textit{jet}} = 1.25$ , where profiles do not collapse. Instead, the peak values of $\overline {u^2}/U_m^2$ increases progressively as the jet evolves downstream, a trend also seen in the Reynolds-shear-stress profiles of $\overline {uv}/U_m^2$ . With this observation, figure 13(b) now shows the scaling $\overline {u^2}$ normalised by $b' U_m^2$ , which collapses the profiles at $x/d \gt 15$ . Although this scaling does not have any physical basis from a self-preservation standpoint, it produces a more consistent collapse of the data across the downstream locations. This scaling suggests that factors other than those predicted by self-preservation theory influence the compressible behaviour of $\overline {u^2}$ .

In figure 14(a), the maximum magnitudes, $\overline {u^2}/U_m^2$ , at different streamwise locations, are compared at different $M_c$ for the jet and mixing layer. Two distinct trends emerge from the literature. Goebel & Dutton (Reference Goebel and Dutton1991) and Freund et al. (Reference Freund, Lele and Moin2000) (at $M_c \gt 0.4$ ) observed scaling independent with $M_c$ , consistent with the similarity assumption. However, Elliott & Samimy (Reference Elliott and Samimy1990), Samimy & Elliott (Reference Samimy and Elliott1990) and Matsuno & Lele (Reference Matsuno and Lele2020) report a suppression of $\overline {u^2}/U_m^2$ as $M_c$ increases. The current jet data trend aligns with the latter set of studies. While some scatter and asymmetry are observed in the profiles, the measurements remain within the 95 % uncertainty bounds shown in the shaded colours. Figure 14(b) examines the ratio of $\overline {uv}$ to $\overline {u^2}$ as a function of $M_c$ . This ratio remains nearly constant at approximately 0.6, consistent with findings in mixing layer studies reporting $\overline {u^2}/U_m^2$ suppression (Samimy & Elliott Reference Samimy and Elliott1990; Pantano & Sarkar Reference Pantano and Sarkar2002), and supports the evidence that $\overline {uv}$ and $\overline {u^2}$ are attenuated by compressibility proportionally.

Figure 13. Reynolds-normal stress $\overline {u^2}$ . Axial positions: $x/d$ = 15.0 (); 17.5 (); 20.0 (); 22.5 (); 25.0 (). Shaded areas show 95 % expanded uncertainty bounds in respective colours. (a) Mach 1.25. Classical scaling with $U^{2}_{m}$ ; (b) Mach 1.25. Collapse of $\overline{u^{2}}$ profiles with scaling $U^{2}_{m}b'$ .

Figure 14. Maximum Reynolds stresses as a function of $M_c$ for mixing layers and present jet data. Samimy & Elliott (Reference Samimy and Elliott1990) (), Goebel & Dutton (Reference Goebel and Dutton1991) (), Debisschop et al. (Reference Debisschop, Chambers and Bonnet1994) (), Urban & Mungal (Reference Urban and Mungal2001) (), Pantano & Sarkar (Reference Pantano and Sarkar2002) (), Matsuno & Lele (Reference Matsuno and Lele2020) (), Feng & McGuirk (Reference Feng and McGuirk2016) (), Freund et al. (Reference Freund, Lele and Moin2000) (), current study (). (a) Maximum Reynolds stress $\overline{u^{2}}$ normalised by $U^{2}_{m}$ . (b) Ratio of the maximum Reynolds stresses $\overline{uv}/\overline{u^{2}}$ .

Figure 15. Comparison of classical and self-preservation scaling of $\overline {v^2}$ . Mach 1.25 jet. Axial positions: $x/d$ = 15.0 (); 17.5 (); 20.0 (); 22.5 (); 25.0 (). Shaded areas show 95 % expanded uncertainty bounds in respective colours. (a) Classical scaling with $U^{2}_{m}$ ; (b) Collapse of $\overline{v^{2}}$ profiles with self-preservation scaling $U^{2}_{m}b^{\prime{2}}$ .

For $\overline {v^2}$ , substituting (2.6)–(2.10) yields the radial-momentum equation,

(5.2) \begin{equation} \rho U \frac {\partial V }{\partial x} + \rho V \frac {\partial V}{\partial r} = \frac {\partial \overline {p}}{\partial r}+ \frac {1}{r}\frac {\partial }{\partial r} \big(r \rho \overline {v^2}\big) , \end{equation}

and performing the similarity analysis results in the following normalised scalings:

(5.3) \begin{equation} \textit{Const.} \propto \frac {p_{m}}{\rho _m U_m^2 b^{\prime 2}} \propto \frac {R_{22}}{U_m^2 b^{\prime 2}} , \end{equation}

or $\overline {v^2} \propto U_m^2 b^{\prime 2}$ . From figure 15(a), profiles of $\overline {v^2}/U_m^2$ shows strong attenuation with $M_c$ . Instead, figure 15(b) plots the profiles using the self-preservation scaling from (5.3). Now, the profiles of $\overline {v^2}$ collapse. Figure 16 compares the behaviour of the maximum of $\overline {v^2}$ with $M_c$ with the mixing layer literature. Unlike, $\overline {u^2}$ , the behaviour of $\overline {v^2}$ shows a consensus with the literature where attenuation with $M_c$ is observed.

Figure 16. Maximum Reynolds stress $\overline {v^2}$ normalised by $U_m^2$ . Same legend as figure 14.

To further investigate the scaling observed in the Reynolds-normal stresses, the pressure term is now considered. Pressure can be estimated by integrating the radial momentum equation from $\overline {p}_\infty$ , where there is no turbulence to any location, $r$ . This yields $\overline {p}_\infty - \overline {p} = \rho \overline {v^2}$ which is substituted into the streamwise momentum equation, resulting in

(5.4) \begin{equation} \rho U \frac {\partial U }{\partial x} + \rho V \frac {\partial U}{\partial r}= \frac {1}{r} \frac {\partial r \rho \overline {uv}}{\partial r} - \frac {\partial }{\partial x} \big( \rho \overline {u^2} - \rho \overline {v^2}\big). \end{equation}

To investigate the scaling of this term, the similarity substitution $\overline {u^2} - \overline {v^2} = \varPsi (x) \psi (\eta )$ is made, yielding

(5.5) \begin{equation} \textit{Const.} \propto \frac {R_{12}}{U_m^2 b'} \propto \frac {\varPsi }{U_m^2} \propto \frac {\varPsi ' b}{U_m^2 b'} . \end{equation}

Figure 17 plots the normal stress difference, showing profiles collapse under this scaling for the compressible jet flow. The same profile is also observed for $M_{ \textit{jet}} = 0.3$ . For the $M_{ \textit{jet}} = 1.25$ case, this observation is unexpected since $\overline {u^2}$ , unlike $\overline {v^2}$ , did not obey self-preservation scaling previously. Upon closer examination, the pressure in the streamwise (2.14) and radial (5.2) momentum equations suggests different scaling. In the streamwise equation, $p_m \propto \rho _m U_m^2$ , whereas in the radial equation, $p_m \propto \rho _m U_m^2 b^{\prime 2}$ . The pressure term may involve two parts and complex dynamical interactions emphasised by compressibility. Self-preservation can be satisfied when considering the aggregate of the Reynolds-normal stresses.

Figure 17. Collapse of profiles of Reynolds-normal stress difference for the Mach 1.25 jet. Marker axial positions have the same legend as figure 15.

Figure 18 shows the streamwise development of the local-maximum scaled-Reynolds stresses for the Mach 1.25 jet. It shows that when using the correct scale determined using self-preservation arguments, the Reynolds stresses can be considered self-preserving under Mach number effects. The scaled Reynolds-shear stress, $\overline {uv}/U_m^2 b'$ , is the first to plateau, at approximately $x/d = 12.5$ , whereas the normal stresses appear to plateau further downstream, at approximately $x/d = 15$ . The relatively early state of preservation is due to the upstream condition of the jet as discussed previously. To further investigate these scaling concepts, self-preservation analysis of the Reynolds-stress-transport equations are required for higher-order turbulence moments.

Figure 18. Development of maximum scaled Reynolds stress for Mach 1.25 jet: $\overline {uv}/U_m^2 b'$ (); $\overline {u^2}/U_m^2 b'$ (); $\overline {v^2}/U_m^2 b^{\prime 2} \boldsymbol{\cdot }0.1$ (); $(\overline {u^2}-\overline {v^2})/U_m^2 \boldsymbol{\cdot }10$ (). Solid lines in respective colours represent the range where Reynolds stresses are considered self-preserving.

5.3. Reynolds-shear stress-anisotropy parameter

The anisotropy tensor, $\beta _{ij} = \overline {u_i u_{\!j}} / \overline {u_k u_k}$ , is commonly used to describe the effects of compressibility on the anisotropy of the Reynolds stresses. From the self-preservation analysis, $\beta _{ij}$ can be directly related to compressibility effects through $\varPhi (M_c)$ . The Reynolds-shear stress-anisotropy parameter, $\beta _{12}$ , can be written as

(5.6) \begin{equation} \beta _{12}=\frac {\overline {uv}}{\overline {u^2} + \overline {v^2} + \overline {w^2}}. \end{equation}

Using the self-preservation scalings, $\overline {uv} \propto U_m^2 b^{\prime }$ , $\overline {v^2} \propto U_m^2 b^{\prime 2}$ and $\overline {w^2} \propto U_m^2 b^{\prime 2}$ , and the experimentally observed scaling $\overline {u^2} \propto U_m^2 b'$ , then substituting the relations and with the relation $b' \propto \varPhi (M_c)$ , with some manipulation, the parameter $\beta _{12}$ can be written as

(5.7) \begin{equation} \beta _{12}=\frac {1}{C_1 + C_2 \varPhi (M_c)}, \end{equation}

where $C_1$ and $C_2$ are constants. Figure 19 plots $\beta _{12}$ at $\eta = 1$ , corresponding to the point of maximum shear. Here, $\overline {w^2}$ is approximated with $\overline {v^2}$ , which from symmetry is expected to scale similarly. The solid line represents (5.7) fitted to the data, with constants $C_1 = 4.4$ and $C_2 = 1.8$ , showing the suitability of the function to characterise $\beta _{12}$ . In the current jet, the parameter $\beta _{12}$ increases with $M_c$ . Like the Reynolds stresses, various behaviours of $\beta _{12}$ have been reported, also plotted are data from Freund et al. (Reference Freund, Lele and Moin2000), showing a decrease with $M_c$ , and Pantano & Sarkar (Reference Pantano and Sarkar2002), where a more constant behaviour is observed. Again, challenges are observed for turbulence models in compressible flows as there is no consensus on fundamental scaling parameters.

5.4. Optimal spreading rate scaling

Figure 19. $\beta _{12}$ scaling. Mach 1.25 jet () and fitted function () from (5.7). Freund et al. (Reference Freund, Lele and Moin2000) (); Pantano & Sarkar (Reference Pantano and Sarkar2002) ().

The spreading rate, $b'$ , plays a critical role in scaling turbulence moments. A root-mean-square-deviation (RMSD) minimisation method was developed to identify the optimal power of $b'$ for each profile within the range of self-preservation. The RMSD minimisation method provides an objective measure of profile collapse, offering a quantitative approach to compare the degree of alignment across profiles. First, to ensure comparable errors across normalised profiles, the Reynolds stresses and triple-velocity correlations were scaled and normalised to a peak magnitude of unity using the scalings $U_m^2 b^{\prime n}$ for the Reynolds stresses and $U_m^3 b^{\prime n}$ for the triple-velocity correlations. Normalising to a peak magnitude of unity ensures that the relative error remains consistent across different profiles, enabling more accurate comparisons. Since the data points are not on a fixed grid after radial normalisation, the $\eta$ axis is divided into discrete bins. The RMSD is computed for each bin and the total RMSD for the profile is obtained by summing the individual bin RMSDs in quadrature. A study was performed to ensure that the choice of bin size did not affect the final RMSD value. Figure 20 shows that the minima of the RSMD curves are close to the integer values reported for the scalings of $b'$ .

5.5. Self-preservation of the Reynolds-stress-transport equations

Figure 20. Error curves indicating ideal $b'$ scaling for Reynolds stress self-preservation scaling. Minimum $n$ () and self-preserving $n$ ().

The RST equations describe the production, transport and dissipation of each Reynolds stress. In the incompressible jet, self-preservation analysis of the RST equations to determine the scaling of higher moments were trivial as $b'$ is constant (George Reference George1995). In compressible shear flows where $b'$ scales with exponents and varies with $M_c$ , self-preservation analysis of the RST equations provides insight into how compressibility affects the turbulence energetics. Again, introducing similarity variables for the pressure-gradient, triple-velocity correlations and dissipation gives the following equations:

(5.8) \begin{align} \overline {u^i \boldsymbol{\nabla} _{\!j} p''} + \overline {u^j \boldsymbol{\nabla} _i p''} &= \varPi _{ij}(x) \pi _{ij}(\eta ) , \end{align}

(5.9) \begin{align} \overline {u_i u_{\!j} u^k} &= T_{ijk}(x) t_{ijk}(\eta ) , \end{align}

(5.10) \begin{align} - \overline {\bigl (\tau ^{jk}\bigr )' \boldsymbol{\nabla} _{\!k} u_i} - \overline {(\tau ^{ki} )'\boldsymbol{\nabla} _{\!k} u_{\!j}} & = E_{ij}(x) e_{ij}(\eta ), \end{align}

with $\pi _{ij}(\eta )$ , $t_{ijk}(\eta )$ and $e_{ij}(\eta )$ as shape functions, and $\varPi _{ij}$ , $T_{ijk}$ and $E_{ijk}$ as scaling functions. The scaling of the remaining turbulence moments can be determined using the scaling $R_{12} \propto U_m^2 b'$ from the mean-momentum equation (2.16). With these substitutions into the $\rho \overline {uv}$ balance from (2.5) for example, the balance with the scaling of leading-order terms shown in under braces after eliminating the common factor $b/U_m^3 b^{\prime 2}$ ,

(5.11) \begin{align} 0 &= - \bigg[\underbrace {\frac {\partial \rho U \overline {uv}}{\partial x} + \frac {\partial \rho V \overline {uv}}{\partial r}}_{\textstyle {{\propto \dfrac {R_{12}}{ U_m^2 b'}}} } \bigg] - \bigg[ \underbrace {\frac {1}{r} \frac {\partial r \rho \overline {uv^2}}{\partial r}}_{\textstyle {{\propto \dfrac {T_{122}}{U_m^3 b^{\prime 2}}}} } - \underbrace {\frac {\rho \overline {uw^2}}{r}}_{\textstyle {{\propto \dfrac {T_{133}}{U_m^3 b^{\prime 2}}}} } + \underbrace {\frac {\partial \rho \overline {u^2v}}{\partial x}}_{\textstyle {{\propto \dfrac {T_{112}}{U_m^3 b^{\prime }}}} } \bigg] \nonumber \\[4pt]& \quad \,\,-\, \bigg[\underbrace {\overline {uv} \frac {\partial \rho V }{\partial r}}_{\textstyle {{\propto \dfrac {R_{12}}{U_m^2 b^{\prime }}}}} + \underbrace {\overline {v^2} \frac {\partial \rho U }{\partial r}}_{\textstyle {{\propto \dfrac {R_{22}}{U_m^2 b^{\prime 2}}}} } + \underbrace { \overline {u^2} \frac {\partial \rho V}{\partial x}}_{\textstyle {{\propto \dfrac {R_{11}}{U_m^2}} } } + \underbrace { \overline {uv} \frac {\partial \rho U }{\partial x}}_{\textstyle {{\propto \dfrac {R_{12} }{U_m^2 b^\prime }}} } \bigg] + \bigg[ \underbrace {\overline {p'' \frac {\partial u}{\partial r}} + \overline {p'' \frac {\partial v}{\partial x}} }_{\textstyle {{\propto \dfrac {\varPi _{12} b}{U_m^3 b^{\prime 2}}}}} \bigg] , \end{align}

recalling that the pressure–diffusion, viscous–diffusion, viscous–dissipation terms and terms containing density correlations are neglected.

Using the remaining RST equations, a pattern emerges from the scaling of the remaining velocity correlations, where the power scaling of $b'$ depends on the order and direction of the correlation. The scaling of correlations only containing streamwise components follow

(5.12) \begin{equation} \frac {R_{11}}{U_m^2} \propto \frac {R_{111}}{U_m^3} \propto \textit{Const.} , \end{equation}

a single radial or azimuthal component follow

(5.13) \begin{equation} \frac {R_{12}}{U_m^2} \propto \frac {R_{13}}{U_m^2} \propto \frac {R_{112}}{U_m^3} \propto \frac {R_{113}}{U_m^3} \propto b', \end{equation}

two radial or azimuthal components follow

(5.14) \begin{equation} \frac {R_{22}}{U_m^2} \propto \frac {R_{33}}{U_m^2} \propto \frac {R_{122}}{U_m^3} \propto \frac {R_{123}}{U_m^3} \propto \frac {R_{133}}{U_m^3} \propto b^{\prime 2} , \end{equation}

and triple-moments of radial or azimuthal components follow

(5.15) \begin{equation} \frac {R_{222}}{U_m^3} \propto \frac {R_{223}}{U_m^3} \propto \frac {R_{233}}{U_m^3} \propto \frac {R_{333}}{U_m^3} \propto b^{\prime 3} . \end{equation}

For Cartesian flows, analogous scalings are observed, where the azimuthal component is instead the spanwise component. In the previous section where the Reynolds stresses were compared with experimental data, the scaling of $R_{12}$ and $R_{22}$ appeared consistent with (5.13) and (5.14), respectively. The scaling of $R_{11}$ was to be $\propto U_m^2b'$ and was found not to scale as per (5.12), and instead appears to be constrained by (5.5). Although not exact, these findings provide a framework for the scaling of velocity correlations in compressible shear flows. In the following section, the scalings are applied to and compared with the triple-velocity correlation jet data.

Figure 21. Similarity profiles of off-diagonal triple-velocity moments. Axial positions: $x/d$ = 15.0 (); 17.5 (); 20.0 (); 22.5 (); 25.0 (). Shaded areas show 95 % expanded uncertainty bounds in respective colours. (a) Profile collapse of $u^{2}v$ with self-preservation scaling $U^{3}_{m}b'$ ; (b) Profile collapse of $uv^{2}$ with self-preservation scaling $U^{3}_{m}b^{\prime{2}}$ .

5.6. Self-preservation scaling of the triple-velocity correlations

Figure 22. Profiles of normal triple-velocity correlations and skewness. Axial positions: $x/d$ = 15.0 (); 17.5 (); 20.0 (); 21.5 (); 25.0 (). Shaded areas show 95 % expanded uncertainty bounds in respective colours. (a) Profile collapse of $\overline{v^{3}}$ with self-preservation scaling $U^{3}_{m}b^{\prime{3}}$ ; (b) Profile collapse of $\overline{u^{3}}$ with scaling $U^{3}_{m}b^{\prime{3}/2}$ .

The compressibility scaling can be extended to the triple-velocity correlations, where profiles are also observed with the compressible jet data. Figures 21(a) and 21(b) present the triple-velocity correlations, $\overline {u^2v}$ and $\overline {uv^2}$ , respectively, normalised using their corresponding self-preservation scaling. By normalising with the appropriate power of the spreading rate, $b'$ , the profiles for $\overline {u^2v}$ and $\overline {uv^2}$ collapse onto a single curve. Demonstrating that the scaling derived from the self-preservation analysis can be successfully extended to the RST equations and applied to higher-order velocity moments.

Figure 22(a) presents the profiles of $\overline {v^3}$ , which collapse effectively when normalised using the self-preservation scaling $U_m^3 b^{\prime 3}$ . This collapse indicates that the self-preservation scaling applies well to the triple-velocity moments in the jet. Profiles of the skewness of $v$ are also shown, showing collapse and consistency of the compressible scaling of the $v$ -velocity fluctuations with $M_c$ . Rotational symmetry, where $v = w$ , also confirms the scaling for $\overline {w^2}$ and $\overline {uw^2}$ moments on the jet centreline. The agreement of the scaling for these moments further supports the applicability of the similarity analysis for higher-order velocity moments in compressible jets.

Similar to $\overline {u^2}$ , the self-preservation scaling does not yield self-similar profiles for $\overline {u^3}$ . However, as shown in figure 22(b), $\overline {u^3}$ scales with $b^{\prime 3/2}$ , consistent with the scaling found for $\overline {u^2}$ where $u \propto b^{\prime 1/2}$ . This scaling indicates that neither $\overline {u^2}$ nor $\overline {u^3}$ follows the scaling under the similarity assumption, but scale at least consistently. The skewness of $u$ also exhibits self-similar profiles across the jet for all Mach numbers. The consistent scaling behaviour, along with the skewness similarity, points to an underlying structure in the velocity field that is preserved across different flow Mach numbers.

Figure 23. Error curves indicating ideal $b'$ scaling for triple-velocity correlations self-preservation scaling. Minimum $n$ () and self-preserving $n$ ().

Figure 23 shows the error curves concerning the similarity of the triple-velocity correlation data for the Mach 1.25 jet using the method introduced in § 5.4. When compared with the error curves for Reynolds stresses shown previously in figure 20, the curves for the triple-velocity correlations exhibit a smaller depression due to increased scatter in the data. Despite this, the minimum RMSD for the profiles aligns with the self-preservation scaling for $\overline {u^2v}$ , $\overline {uv^2}$ and $\overline {v^3}$ . As discussed, $\overline {u^3}$ instead scales to the $b'$ exponent $({3}/{2})$ , such that the scale is consistent with $\overline {u^2}$ . The consistency of the scaling results between double- and triple-velocity moments increases confidence in the application of self-preservation principles in analysing compressible shear flows.

6.1. Turbulence energetics

From the velocimetry data, the advection, turbulence-transport and production terms can be calculated directly. Therefore, the pressure–strain, dissipation, molecular–diffusion and pressure–diffusion terms are lumped as the residual of the equation. For high- ${Re}$ shear flows, the effect of molecular diffusion is negligible and pressure diffusion generally small. To mitigate the effect due to random noise and to ensure differentiability, the turbulence moments are curve-fitted, with radial distributions as reported by Hussein et al. (Reference Hussein, Capp and George1994), and streamwise distributions fitted with low-order polynomials. The balances are then calculated using the fitted functions, where the fit uncertainties are propagated to indicate 95 % expanded confidence bounds in shaded areas.

Figure 24. Normalised budget of $\overline {uv}$ . Red shaded areas show 95 % expanded uncertainty bounds.

The Reynolds transport equations are presented with the common factor $b/U_m^3 b^{\prime n}$ , where $n$ is a power determined previously through self-preservation analysis. The balance of the $\overline {uv}$ transport equation is shown in figure 24, where the common factor is $b/U_m^3 b^{\prime 2}$ . The balances scale producing profiles of similar magnitude, implying they are independent of $M_c$ after normalisation. Note that $\overline {uv^2}$ is used to approximate $\overline {uw^2}$ since $w$ data are unavailable. For high- ${Re}$ incompressible jets, the Richardson–Kolmogorov–Onsager theory implies a separation of scales where dissipation, which occurs at the small scales, exhibits scale locality. While compressible turbulence superficially resembles the incompressible cascade, the exact inter-scale energy dynamics remains unclear, but analysis from Aluie (Reference Aluie2011) has shown scale locality and the existence of an inertial range in compressible flows. This suggests dissipation can be approximated as isotropic and the off-diagonal components of dissipation neglected, especially for this flow where density effects are moderate. Therefore, the residual terms in the $\overline {uv}$ transport equation approximate the pressure–strain component. With the primary source term production, the major sink term is the pressure strain, indicating turbulent kinetic energy is being redistributed to the other stress balances. Note, the shear pressure–strain component scales proportionally to the component production, agreeing with observations from Matsuno & Lele (Reference Matsuno and Lele2020). As $\overline {uv}$ is the dominating term, which spreads the jet through the streamwise momentum equation (2.13), it is unsurprising that the balance shows self-preservation.

The budget of $\overline {v^2}$ is shown in figure 25 normalised by $b/U_m^3b^{\prime 3}$ . Here, $\overline {vw^2}$ is substituted by $\overline {v^3}$ . Compared with setting $\overline {vw^2} =0$ , the magnitude of the turbulence–diffusion term changes less than 10 %. This substitution was also made by Hussein et al. (Reference Hussein, Capp and George1994) when lacking $w$ information and suggested that approximating $\overline {vw^2}$ with $\overline {v^3}$ is satisfactory for estimating the scaling behaviour of the residual of the $\overline {v^2}$ -transport equation. The balance of $\overline {v^2}$ is difficult to interpret by itself, as the major source term is the pressure–strain term and the major sink term is the dissipation (Hussein et al. Reference Hussein, Capp and George1994). With the present data, while $\overline {v^2}$ was shown to be self-similar when scaled by the self-preserving scaling $U_m^3b^{\prime 2}$ , the advection terms are not. The turbulence-transport and production terms appear self-similar, while the residual terms containing the dissipation and pressure strain is not self-preserving and balances the scaling of the advection.

Figure 25. Normalised budget of $\overline {v^2}$ . Red shaded areas show 95 % expanded uncertainty bounds.

Figure 26. Normalised budget of $\overline {u^2}$ . Red shaded areas show 95 % expanded uncertainty bounds.

In the budget of $\overline {u^2}$ , shown in (B1), the terms share a common factor of $b/U_m^3 b'$ , which has been applied to the $\overline {u^2}$ balance across various $M_c$ , as shown in figure 26. In the $\overline {u^2}$ budget, the primary source term is the production and advection terms. This turbulent kinetic energy is then transported into the $\overline {v^2}$ equation and contributes to the production of $\overline {uv}$ . Therefore, limited production of $\overline {u^2}$ due to $M_c$ is suggested to play a role in the reduced spreading rate. The production and turbulence transport terms appear to be self-preserving at all $M_c$ ; meanwhile, the advection terms increase as $M_c$ decreases. The situation in the $\overline {u^2}$ balance is similar to the $\overline {v^2}$ balance – profiles of advection are not self-similar, and a similar imbalance is observed in the residual. As the residual of the balance contains the pressure–strain term and dissipation term, the pressure strain is responsible for transferring energy to the $\overline {v^2}$ budget. Although the terms in the budgets of $\overline {u^2}$ and $\overline {v^2}$ do not conveniently collapse, a strong coupling is observed between these equations, which is already evident in the momentum equation (5.4) where the collapse of the difference $\overline {u^2} - \overline {v^2}$ collapses with $U_m^2$ , as seen previously in figure 17.

Figure 27 shows the difference between budgets (B1) and (B2) normalised by $b/U_m^3 b'$ ; remarkably, the profiles are observed to be self-preserving. The balance of the difference resembles the balance of $\overline {u^2}$ , where the turbulence production and then advection are the significant source terms. The scaling of the advection terms previously observed in the individual $\overline {u^2}$ and $\overline {v^2}$ balances appear to redistribute energy to each other such that the resulting advection of the difference is self-preserving. As a result, the residual of the difference scales is self-preserving. Closer consideration of the residual term for the difference balance, (B1)–(B2), yields further insight. An isotropic dissipation model implies that the contributor of dissipation in the residual term vanishes. In contrast, the pressure–strain terms between axial and radial directions tend to differ in sign. Hence, the residual term is dominated by the redistribution due to the pressure strain. This observation shows the importance of the pressure–strain redistribution in the coupling of $\overline {u^2}$ and $\overline {v^2}$ , and the internal dynamics leading to self-preservation and role in the attenuation of energy redistribution.

Figure 27. Normalised budget of $\overline {u^2} - \overline {v^2}$ . Red shaded areas show 95 % expanded uncertainty bounds.

Figure 28(a) plots the pressure–strain component $\varPi _{12}$ estimated from the residual of the $\overline {uv}$ balance as a function of $M_c$ . The pressure strain is normalised by its incompressible value, estimated from the furthest value available downstream. From the self-preservation analysis of the $\overline {uv}$ balance, $\varPi _{12}$ scales with $b/U_m^3 b^{\prime 2}$ . Compared with mixing layer data from Pantano & Sarkar (Reference Pantano and Sarkar2002), significant attenuation of the pressure–strain component is observed, verifying the behaviour of the shear pressure strain. From the current jet data, individual normal pressure–strain components cannot be calculated. Instead, from the balance of the difference between the $\overline {u^2}$ and $\overline {v^2}$ equations, the residual scales with $b/U_m^3 b^{\prime }$ . Again, this residual is approximately $\varPi _{11} - \varPi _{22}$ , assuming isotropic turbulence where the diagonal dissipation terms are of similar magnitude. Note that $\varPi _{11}$ and $\varPi _{22}$ are expected to have opposite signs, such that their differences increase their contribution to the $\overline {u^2}-\overline {v^2}$ transport balance. Figure 28(b) plots the normal pressure–strain terms, normalised by $b/U_m^3$ , which appears to be attenuated by $M_c$ through $b'$ , again agrees with the scaling observed in the mixing layer. Note that the off-diagonal term appears to scale to different powers of $b'$ than the diagonal terms; this difference in scaling was also exhibited in the mixing layer data of Pantano & Sarkar (Reference Pantano and Sarkar2002). This indicates that pressure modification is consistent with the self-preservation scaling framework in the current jet study and directly connected to the reduced spreading rates due to increasing $M_c$ .

Figure 28. Pressure–strain components normalised by incompressible value as a function of $M_c$ for the mixing layer (Pantano & Sarkar Reference Pantano and Sarkar2002) () and the current jet (). (a) Normalised $\Pi_{12}$ ; (b) normalised $\Pi_{12}\mbox{--}\Pi_{22}$ estimated from (B1) - (B2).