3.1. Theoretical analysis assuming self-similarity

In this subsection, a theoretical analysis on the mean SDR is developed, using the assumption that flow variables remain self-similar in the unsteady jet. The assumptions of self-similarity will be thoroughly checked by DNS data in §§ 3.3 and 3.4. Furthermore, obtained theoretical predictions will be validated in § 3.5.

The analysis starts from the transport equation for the mean SDR in the case of an incompressible jet with constant mass diffusivity $\mathscr {D}$ :

(3.1) \begin{equation} \begin{gathered} \frac {\partial \,\overline {\chi }}{\partial t} + \frac {\partial \overline {u_j}\,\, \overline {\chi }}{\partial x_j} = \mathscr {D} \frac {\partial ^2 \overline {\chi }}{\partial x_j \partial x_j} -\frac {\partial \overline {u^{\prime}_j \chi^{\prime}}}{\partial x_j} -4 \mathscr {D} \overline {\frac {\partial \xi }{\partial x_j} \frac {\partial u_j}{\partial x_i} \frac {\partial \xi }{\partial x_i}} -4 \mathscr {D}^2 \overline {\frac {\partial ^2 \xi }{\partial x_j \partial x_i}\frac {\partial ^2 \xi }{\partial x_j \partial x_i}}. \end{gathered} \end{equation}

Next, assuming that all involved variables are self-similar, the scaled ensemble-averaged statistics can be represented by a scaled radius $\eta =r/(x-x_0)$ , where $x_0$ represents the jet virtual origin. Flow variables can be normalised by centreline values as

(3.2) \begin{equation} \begin{gathered} \overline {\chi } = \overline {\chi }_c g_{\chi }(\eta ),\quad\overline {u} = \overline {u}_c g_u(\eta ),\quad\overline {v} = \overline {u}_c g_v(\eta ), \\[5pt] \overline {u^{\prime}\chi^{\prime}} = \overline {u}_c\,\overline {\chi }_c\, g_{u^{\prime}\chi^{\prime}}(\eta ),\quad\overline {v ^{\prime}\chi^{\prime}} = \overline {u}_c\,\overline {\chi }_c\, g_{v ^{\prime}\chi^{\prime}}(\eta ), \\[5pt] 4 \mathscr {D}^2\, \overline {\frac {\partial ^2 \xi }{\partial x_j\, \partial x_k}\frac {\partial ^2 \xi }{\partial x_j\, \partial x_k}} =\frac {\overline {u}_c\, \overline {\chi }_c}{r_{1/2}} h(\eta ), \\[5pt] 4 \mathscr {D}\, \overline {\frac {\partial \xi }{\partial x_j} \frac {\partial u_j}{\partial x_k} \frac {\partial \xi }{\partial x_k}} = \frac {\overline {u}_c\, \overline {\chi }_c}{r_{1/2}} l(\eta ), \end{gathered} \end{equation}

where $g_\chi$ , $g_u$ , $g_v$ , $g_{u^{\prime}\chi^{\prime}}$ , $g_{v ^{\prime}\chi^{\prime}}$ , $h$ and $l$ are dimensionless shape functions characterising the self-similar profiles, respectively. Note that $r_{1/2}$ is the so-called half-radius, i.e. the radial location where axial velocity is half of the centreline axial velocity. Centreline variables $\overline {u}_c$ and $\overline {\chi }_c$ are functions of $\eta$ and $t$ . In previous studies of the stopping jet, $\overline {u}_c$ and $r_{1/2}$ are characterised as (Shin et al. Reference Shin, Aspden and Richardson2017; Pope Reference Pope2000)

(3.3) \begin{align} \overline {u}_c &= C_u \frac {x-x_0}{t-t_0}, \notag\\ r_{1/2} &= S (x-x_0), \end{align}

where $t_0$ is a time shift, and $C_u$ and $S$ are some constants.

Substituting the relations in (3.2) and (3.3) into (3.1) results in

(3.4) \begin{align} \frac {g_{\chi }}{C_u}\frac {t-t_0}{\overline {\chi }_c} \frac {\partial \overline {\chi }_c}{\partial t} + \left(g_u g_{\chi } + g_{u^{\prime}\chi^{\prime}}\right) \frac {x-x_0}{\overline {\chi }_c} \frac {\partial \overline {\chi }_c}{\partial x}&= \eta g_u \frac {{\rm d} g_{\chi }}{{\rm d} \eta }-g_v\frac {{\rm d} g_{\chi }}{{\rm d} \eta }-\frac {1}{\eta } \frac {{\rm d}\left(\eta g_{v ^{\prime}\chi^{\prime}}\right)}{{\rm d}\eta }\nonumber \\ &\quad +\eta \frac {{\rm d} g_{u^{\prime}\chi^{\prime}}}{{\rm d} \eta }-g_{u^{\prime}\chi^{\prime}}-\frac {1}{S}(h+l). \end{align}

The terms on the right-hand side of (3.4) are functions of $\eta$ only, while the terms on the left-hand side are functions of $x$ and $t$ . In order for equality to hold, $\overline {\chi }_c$ only accepts a power-law form as

(3.5) \begin{equation} \overline {\chi }_c(x,t) = C_X (x-x_0)^a (t-t_0)^b, \end{equation}

where $C_X$ is a constant. The power-law exponents $a$ and $b$ cannot be determined any further in this analysis. In Shin et al. (Reference Shin, Aspden and Richardson2017), power-law exponents of $\overline {u}_c$ for a stopping jet were determined explicitly using a similar approach. However, the difference is that velocity is nonlinear in its governing equation (momentum equation), while $\overline {\chi }$ is linear in its governing equation (see (3.1)).

It will be shown in § 3.2 that present DNS data indicate $a=1$ and $b=-2$ as a good fit for $\overline {\chi }_c$ evolution, giving

(3.6) \begin{equation} \overline {\chi }_c(x,t) = \frac {C_{\overline {\chi }_c}}{U_0} \frac {x-x_0}{(t-t_0)^{2}}, \end{equation}

where a division by $U_0$ is included to match dimensions. Then substitution of (3.6) into (3.4) gives

(3.7) \begin{align} -\frac {2}{C_u} g_{\chi } + g_u g_{\chi } - \eta g_u \frac {{\rm d} g_{\chi }}{{\rm d} \eta } + g_v \frac {{\rm d} g_{\chi }}{{\rm d} \eta } &= - \left [ \frac {g_{v^{\prime }\chi ^{\prime }}}{\eta } + \frac {{\rm d} g_{v^{\prime }\chi ^{\prime }}}{{\rm d} \eta } \right ] - \left [ 2 g_{u^{\prime }\chi ^{\prime }} - \eta \frac {{\rm d} g_{u^{\prime }\chi ^{\prime }}}{{\rm d} \eta } \right ] \nonumber \\ &\quad - \frac {1}{S}(h+l). \end{align}

Note that $C_{\overline {\chi }_c}$ does not appear in (3.7), due to linear dependency of $\overline {\chi }$ in the SDR transport equation (see (3.1)). At $\eta =0$ , the following relationships hold due to definitions and symmetry:

(3.8) \begin{align} g_u(0) = 1, \quad g_v(0) = 0, \quad g_{\chi }(0) = 1, \quad g_{v ^{\prime} \chi^{\prime}} = 0. \end{align}

With the above relationships, evaluating (3.7) at $\eta =0$ gives

(3.9) \begin{equation} \left [\frac {{\rm d} g_{v ^{\prime}\chi^{\prime}}}{{\rm d}\eta } \right ]_{\eta =0} = \frac {1}{C_u} - \frac {1}{2} - g_{u^{\prime}\chi^{\prime}}(0) - \frac {1}{2S}\big [h(0)+l(0)\big ]. \end{equation}

Next, (3.7) can be integrated to obtain an explicit expression for $g_{v^{\prime }\chi ^{\prime }}$ as

(3.10) \begin{align} g_{v^{\prime }\chi ^{\prime }}={}&\frac {2}{\eta C_u} \int _{0}^\eta \eta ^{\prime } g_{\chi }\, {\rm d}\eta ^{\prime } - \frac {4}{\eta } \int _{0}^\eta \eta ^{\prime } g_u g_{\chi }\, {\rm d}\eta ^{\prime } + \frac {3 g_{\chi }}{\eta } \int _{0}^\eta \eta ^{\prime } g_u\, {\rm d}\eta ^{\prime }\nonumber \\ &{}+ \eta g_{u^{\prime }{\chi }^{\prime }}-\frac {4}{\eta } \int _{0}^\eta \eta ^{\prime } g_{u^{\prime }{\chi }^{\prime }}\, {\rm d}\eta ^{\prime } - \frac {1}{\eta S} \int _{0}^\eta \eta ^{\prime } (h+l)\, {\rm d}\eta ^{\prime }. \end{align}

The two explicit expressions of (3.9) and (3.10) are compared with simulation data in § 3.5.

3.2. Centreline SDR

In this subsection, centreline SDR ( $\overline {\chi }_c$ ) of the stopping jet is characterised using DNS. As shown in (3.2), centreline characteristics are the starting point of the self-similar analysis. The theoretical analysis in § 3.1 has not reached a conclusion on spatial and temporal dependencies. This will be solved in this subsection (see the paragraph after (3.5)).

Figure 5. Centreline SDR variation with axial distance at multiple time instances.

Figure 5 shows the spatial and temporal evolution of $\overline {\chi }_c$ after stopping. Thin coloured lines represent ensemble-averaged $\overline {\chi }_c$ – note that ensemble averaging was done over 10 realisations. Profiles are normalised by the characteristic jet time $\tau = D/U_0$ , with $D$ and $U_0$ as the source diameter and velocity, respectively. The jet virtual origin $x_0 = 2.39D$ remains the same as for the steady-state jet (Shin et al. Reference Shin, Sandberg and Richardson2017b ). For reference, the steady-state $\overline {\chi }_c$ is added as a thick black line. Overall, $\overline {\chi }_c$ decreases from its steady-state profile after stopping.

In the inlet vicinity ( $x/D \approx 0$ ), centreline SDR momentarily increases immediately after stopping, then slowly decreases again to zero. The region corresponds to the potential core so that $\overline {\chi }_c$ is zero in the steady-state jet (see figure 4 a). After the jet is stopped, the potential core collapses inwards, and SDR in the nozzle vicinity is no longer zero (see figure 4 b). As time advances, SDR approaches to zero, as seen in figure 4(c,d).

Further downstream, after a peak, $\overline {\chi }_c$ evolves in the same way as for the steady-state profile. Behaviour is also similar to the observations of decelerating diesel jets (Musculus Reference Musculus2009). After stopping, the decelerating wave propagates downstream with a certain speed. At locations where the decelerating wave has not arrived, $\overline {\chi }_c$ remains the same as in the steady-state jet. Similarity with the steady-state profile can also be seen in the instantaneous images of figure 4 – the instantaneous SDR characteristics after $x/D\gt 25$ remain similar for $t/\tau = 0{-}50$ .

Figure 6. (a) The parabolic regime and (b) the inverse of $\overline {\chi }_c$ slope behaviour near the nozzle.

Long-term behaviour of the intermediate region between the nozzle inlet and the peak of $\overline {\chi }_c$ (e.g. $1\lt (x-x_0)/D\lt 7$ at $t/\tau = 5$ ) will be analysed further.

Figures 6(a) and 6(b) replotted centreline SDR in two different ways to investigate spatial and temporal dependencies on the intermediate region. Figure 6(a) shows $\overline {\chi }_c$ in the log-log scale, with the $x$ -axis being the axial distance with $1\lt (x-x_0)/D\lt 12$ . For reference, $y=x$ is added as a thick black line. As time advances, $\overline {\chi }_c$ becomes parallel to the black line, indicating that $\overline {\chi }_c$ becomes linear in $x$ . Similarly, figure 6(b) shows $\overline {\chi }_c$ in the log-log scale with the $x$ -axis being the time with $5\lt (t-\tau _0)/\tau \lt 56$ . For reference, $y=x^{-1}$ and $y=x^{-2}$ are added as thick blue and black lines. In the long time and in the far field, $\overline {\chi }_c$ becomes parallel to the black line, meaning that $\overline {\chi }_c$ asymptotes to a $[(t-t_0)/\tau ]^{-2}$ dependency. The analysis indicates that exponents in (3.5) are $a=1$ and $b=-2$ , respectively. The long term behaviour of $\overline {\chi }_c$ in the intermediate regime can be expressed as

(3.11) \begin{equation} \overline {\chi }_c = \frac {C_{\overline {\chi }_c}}{U_0} \frac {x-x_0}{(t - t_0)^2}, \end{equation}

where $C_{\overline {\chi }_c}$ is a dimensionless coefficient whose value will be evaluated, and $U_0$ is included to match the dimension.

Figure 7. Scaled centreline SDR with time and axial distance at different axial locations.

Figure 8. (a) False colours of $\overline {\chi }_c U_0 (t - t_0)^2/(x - x_0)$ , and (b) deceleration wave speed ( $\overline {u}_{wave}$ ) for $\overline {\chi }_c$ , along with evolution of centreline axial velocity $\overline {u}_c$ .

Figure 7 shows scaled $\overline {\chi }_c$ over time to reconfirm spatial and temporal dependencies – the centreline SDR is scaled by $(x - x_0)/U_0(t - t_0)^2$ . Again, the scaled $\overline {\chi }_c$ asymptotes to a constant in the long term. Scatter around the asymptotic value remains low, compared to the magnitude of early ( $t/\tau \lt 14$ ) scaled profiles. The asymptotic constant $C_{\overline {\chi }_c}$ is $0.00105$ , as shown by the dashed line.

Next, the decelerating wave for the SDR is analysed. To quantify the decelerating wave, figure 8(a) shows the false colour of $ \overline {\chi }_cU_0 (t - t_0)^2/(x - x_0)$ , using log scales. The false colour displays a boundary separating two regions, one in red and the other in white. The boundary is not distinctive, but rather smooth with some width. Still, the boundary is clearly parallel to $y=x^2$ . Hence this is treated as a characteristic line and drawn on the figure with a functional form as

(3.12) \begin{equation} \frac {t-t_0}{\tau } = \frac {1}{2C_{wave,SDR}} \left ( \frac {x-x_0}{D} \right )^2. \end{equation}

As the boundary is not very distinctive, three candidates for $1/(2C_{wave,SDR})$ are selected, representing high, medium and low bounds with values $0.061$ , $0.053$ and $0.046$ , respectively. Hence $C_{wave,SDR}$ takes values 8.2, 9.43 and 10.87, respectively. By the method of characteristics, the characteristic line in (3.12) should be a solution of the characteristic equation

(3.13) \begin{equation} \frac {{\rm d}x}{{\rm d}t} = u_{wave,SDR}. \end{equation}

With (3.12), $u_{wave,SDR}$ satisfies

(3.14) \begin{equation} \frac {u_{wave,SDR}}{U_0} = \frac {C_{wave,SDR}}{(x-x_0)/D}. \end{equation}

The $1/x$ dependency of $u_{wave,SDR}$ indicates that the SDR deceleration wave has a behaviour similar to that of the deceleration wave for axial velocity ( $\overline {u}_c$ ) in the stopping jet (Shin et al. Reference Shin, Aspden and Richardson2017), which is

(3.15) \begin{equation} \frac {u_{wave,velocity}}{U_0} = \frac {7.71}{(x-x_0)/D}. \end{equation}

For comparison, figure 8(b) shows the decelerating wave speeds for centreline SDR and axial velocity. This indicates that overall, the deceleration wave speed for SDR is slightly higher than for axial velocity. A quantitative comparison between the SDR wave speed and centreline axial velocity indicates that the former is between 1.06 and 1.41 times higher for the high and low bounds in figure 8(b), respectively.

Figure 8(a) can also be used to set the ranges, where the long-term asymptotic behaviour is observed. The horizontal black dashed line is the line $(t-\tau _0)/\tau =36$ , which crosses the decelerating wave line (black solid line) at $(x-x_0)/D=26$ . Considering $x_0=2.39D$ and $t_0=14\tau$ , the ranges $14\lt x/D\lt 28$ and $50\lt t/\tau \lt 69$ can be used to obtain statistics for long-term behaviour analysis.

3.3. Self-similarity of the mean SDR in a stopping jet

This subsection analyses radial profiles of the mean SDR and its axial, radial and azimuthal components. As shown in § 3.2, scaled centreline SDR $ \overline {\chi }_c U_0 (t - t_0)^2/(x - x_0)$ asymptotes to a constant value, after some transient time. Based on this behaviour, the current subsection focuses on identifying new self-similar states in the stopping jet. In the first subsubsection, stationarity and homogeneity of normalised radial profiles of the SDR are investigated. The second subsubsection presents new self-similar states, which are different from their steady-state counterparts.

3.3.1. Stationarity and homogeneity of the SDR radial profiles

Stationarity of SDR and its components is first checked at selected axial locations $(x-x_0)/D = 14$ and $28$ . Figure 9 shows radial profiles of axial, radial, azimuthal and the total SDR components, at different times after the jet is stopped. Profiles are normalised by centreline SDR ( $\overline {\chi }_c$ ). On each plot, the blue thick line represents averaged profile over $t/\tau = 50{-}69$ to be termed hereafter ‘long-term profiles’. As seen in figure 8(a), the decelerating wave passes by at approximately $t/\tau = 24$ and $35$ for $(x-x_0)/D=14$ and $28$ , respectively. Hence the time range $t/\tau = 50{-}69$ falls into the interval where long-term behaviour can be observed at both locations.

Figure 9. Normalised radial profiles of transient axial SDR component at (a) $(x-x_0)/D = 14$ and (e) $(x-x_0)/D = 28$ , radial SDR component at (b) $(x-x_0)/D = 14$ and (f) $(x-x_0)/D=28$ , azimuthal SDR component at (c) $(x-x_0)/D = 14$ and (g) $(x-x_0)/D = 28$ , total SDR at (d) $(x-x_0)/D = 14$ and (h) $(x-x_0)/D = 28$ .

Overall, transient profiles increase over time. As time advances, profiles start near the steady-state profiles (black lines), and slowly move to long-term profiles (blue lines), then settle down at the long-term profiles.

Figures 9(a) and 9(e) show the normalised axial SDR components. The long-term profiles are higher than their steady-state counterparts. At both axial locations, long-term profiles slightly increase with $\eta$ , from the centreline towards a peak, and then decay to zero further away from the centreline. Figures 9(b) and 9( f) show the normalised radial SDR components. The long-term profiles show the highest increase in magnitude, compared to the steady-state profiles. The increase is mostly visible at $(x-x_0)/D=28$ , which also shows the highest peak dissipation at $\eta \approx 0.09$ , approximately 2.5 times higher than the centreline value. Figures 9(c) and 9(g) show the azimuthal SDR component. Long-term profiles resemble those of the axial component, which is also the case for the steady-state jet. At $(x-x_0)/D=14$ , the long-term profile is relatively flat, whereas at $(x-x_0)/D=28$ , there is a distinct peak in dissipation at $\eta \approx 0.07$ .

Finally, figures 9(d) and 9(h) show the total SDR long-term profiles. Again, the long-term profiles are higher than the steady-state profiles, with the largest contribution from the radial component. Also, on all figures, the instantaneous profiles of $t/\tau =50, 60, 69$ are all around the long-term profiles, which confirms stationarity in the inspected time range. Furthermore, total SDR goes to zero at the same $\eta$ location for both steady-state and long-term profiles. This indicates that the jet spreading angle does not change while the jet is decelerating.

Figure 10. Temporally averaged profiles of normalised SDR components (a) axial, (b) radial and (c) azimuthal as well as (d) total SDR at $(x-x_0)/D = 14{-}28$ ; and self-similar profiles averaged over $t/\tau = 50{-}69$ and $(x-x_0)/D = 14{-}28$ for SDR components (e) axial, (f) radial and (g) azimuthal as well as (h) total SDR.

3.4. Self-similarity of the mean SDR transport equation in a stopping jet

In previous subsections, the mean SDR in the stopping jet reaches new self-similar states after a transient time. These profiles are termed decelerating self-similar profiles. In this subsection, analysis is extended to the terms in the mean SDR transport equation (3.1). For ease of reading, this equation is rearranged so that all the terms are moved to the right-hand side, as follows:

(3.16) \begin{equation} \begin{gathered} 0 = - \underbrace {\frac {\partial \overline {\chi }}{\partial t}}_{\text {I}} - \underbrace {\frac {\partial \overline {u_j}\, \overline {\chi }}{\partial x_j}}_{\text {II}} \underbrace {{}+ \mathscr {D} \frac {\partial ^2 \overline {\chi }}{\partial x_j\, \partial x_j}}_{\text {III}} \underbrace {{}-\frac {\partial \overline {u^{\prime}_j \chi^{\prime}}}{\partial x_j}}_{\text {IV}} \underbrace {{}-4 \mathscr {D} \overline {\frac {\partial \xi }{\partial x_j} \frac {\partial u_j}{\partial x_i} \frac {\partial \xi }{\partial x_i}}}_{\text {V}} -\underbrace {4 \mathscr {D}^2 \overline {\frac {\partial ^2 \xi }{\partial x_j\, \partial x_i}\frac {\partial ^2 \xi }{\partial x_j\, \partial x_i}}}_{\text {VI}}. \end{gathered} \end{equation}

In addition, all the terms are defined by Roman numerals, which are used hereafter. Note that some definitions contain a minus sign and some do not. The inclusion or exclusion of a minus sign is consistent with previous studies (Mantel & Borghi Reference Mantel and Borghi1994; Mura & Borghi Reference Mura and Borghi2003; Chakraborty et al. Reference Chakraborty, Champion, Mura and Swaminathan2011).

A previous analysis of the mean SDR equation shows that as for the steady-state case (Aparece-Scutariu & Shin Reference Aparece-Scutariu and Shin2022), mean SDR convection (term II), mean SDR diffusion (term III) and turbulent transport (term IV) remain negligible. Hence the analysis focuses on the remaining significant terms (I, V, VI). Significance will be reconfirmed by evaluating balance in the last subsection. The analysis will be conducted in the same manner as the SDR analysis in the previous subsection. Stationarity and homogeneity for radial profiles are first analysed. Then decelerating self-similar profiles are presented.

Figure 11. Transient behaviour of radial profiles for dominant budget terms in the mean SDR transport equation: term I at (a) $(x-x_0)/D = 19$ and (d) $(x-x_0)/D = 28$ ; term V at (b) $(x-x_0)/D = 19$ and (e) $(x-x_0)/D = 28$ ; and term VI at (c) $(x-x_0)/D = 19$ and (f) $(x-x_0)/D = 28$ .

Figure 11 shows normalised radial profiles of significant terms in (3.16) at axial locations $(x-x_0)/D = 19$ and $28$ . Terms are normalised by $(\overline {u}_c\, \overline {\chi }_c)/r_{1/2}$ , as in § 3.1. In each plot, profiles are shown in time increments of $10\tau$ after stopping. Blue thick lines represent averaged profiles over $t/\tau = 50{-}69$ . The time interval is considered based on the scaled $\overline {\chi }_c$ , which displays an asymptotic trend, beyond $t/\tau \gt 40$ , at both axial locations (see figure 6 c).

Figures 11(a) and 11(d) show the normalised temporal term of SDR (term I in (3.16)). At all times, the value remains negative, indicating that the mean SDR decreases over time. At $t/\tau \gt 50$ , radial profiles remain close to the long-term profiles. While profiles of $x/D=19$ and $28$ are not identical, both show the same trend. Long-term profiles start at approximately $-0.8$ at the centreline, remain on a plateau, except for any spurious oscillations, and then, at approximately $\eta =0.09$ , the value goes to zero. As term I includes a derivative of SDR (i.e. higher-order statistics), spurious oscillations are unavoidable.

Figures 11(b) and 11(e) show terms associated with construction by the stretch of scalar field due to local curvature (term V in (3.16)) at the two axial locations $(x-x_0)/D = 19$ and $28$ . Again, the profiles are not exactly identical at the two locations, but they show similar trends. Over time, profiles transition from the steady-state profile to the long-term profile. After $t/\tau =50$ , the profile remains close to the long-term profile. The long-term profile starts from approximately 6.5 at the centreline, and increases with $\eta$ . Then it peaks at $\eta \approx 0.05$ , followed by a decay to zero.

Figures 11(c) and 11(f) show the destruction term (term VI in (3.16)) at the two axial locations. At both locations, profiles transition again from the steady-state profile to the long-term profile, until $t/\tau =50$ . The long-term profile starts from 6.3 at the centreline, and decreases with $\eta$ , reaching a negative peak at approximately $\eta =0.07$ , and then goes to zero. Overall, figure 11 illustrates that all dominant terms reach the long-term profiles at the two inspected locations after stopping, especially after $t/\tau \gt 50$ .

Figure 12. Temporally averaged dominant terms of the mean SDR transport equation for (a) term I, (b) term V and (c) term VI; and normalised radial profiles of (d) term I, (e) term V and (f) term VI, self-similar over $t/\tau=50{-}69$ and $x/D=19{-}28$ .

Next, homogeneity of the long-term profiles is inspected. Figure 12(a–c) show the normalised radial profiles for the dominant terms in the axial range $(x-x_0)/D = 19{-}28$ . Profiles are first averaged over the time interval $t/\tau = 50{-}69$ and then are normalised by $(\overline {u}_c\, \overline {\chi }_c)/r_{1/2}$ . These centreline variables are also averaged over the time interval $t/\tau = 50{-}69$ . Overall, variations among locations are small.

Then figure 12(d–f) show spatially averaged long-term profiles for the three dominant terms, denoted hereafter as decelerating self-similar profiles. The blue shaded region indicates one standard deviation of the averaged profile, and the black line indicates its corresponding radially averaged steady-state profile.

Term I (the temporal term) starts from a negative value near the jet centreline, peaking at $\eta \approx 0.07$ , followed by a monotonic increase to 0 until $\eta \approx 0.25$ . As for terms V and VI (figure 12 d–f), radial profiles are homogeneous in the interval $(x-x_0)/D = 19{-}28$ . Term V (production term) has an off-centreline peak, followed by a decrease at outer radii. Term VI (destruction term) shows the same trend as the production term. In terms of order of magnitude, terms V and VI are approximately 5 times larger than term I. Hence terms V and VI balance the equation at leading order, although term I has a non-negligible influence on this balance.

Figure 13. Balance of temporally and spatially averaged budget terms of the mean SDR transport equation (3.16).

Finally, figure 13 shows the balance of the mean SDR transport equation. The balance term considers all terms in (3.16), including the significant terms I, V and VI, as well as the negligible ones, II, III and IV. Balance is zero at the centreline, but becomes negative away from the jet axis. Note that the three significant terms still have a certain amount of uncertainty, as shown in figure 12(d–f). Hence variation of the balance term would be attributed to statistical uncertainty of the dominant terms. Still, the relatively small magnitude balance term validates the computation of the three significant terms, as they are introduced in the first part of this subsection. Overall, the biggest contributions are due to terms V and VI in (3.16). These are followed by the temporal term I, which is approximately $20\,\%$ in magnitude of either term V or term VI. Equation (3.16) is considered to be balanced, given the fact that the balance is overall smaller than relevant budget terms (i.e. terms I, V and VI).

3.5. Prediction of the self-similar profile of $\overline {v ^{\prime}\chi^{\prime}}$ in a stopping jet

In § 3.1, the self-similar radial profile of $\overline {v ^{\prime}\chi^{\prime}}$ and its derivative at $\eta =0$ are given in (3.9) and (3.10). Derivation is based on the assumption of self-similarity of the other involved terms. Hence agreement of the derivation with actual DNS profiles can serve as indirect evidence of self-similarity occurring in the stopping jet.

Figure 14. Normalised decelerating self-similar profiles over $t/\tau = 50{-}69$ and $(x-x_0)/D = 14{-}28$ for $\overline {v ^{\prime} \chi^{\prime}}$ .

For ease of reading, (3.9) and (3.10) are rewritten below:

(3.17) \begin{align} &\left [\frac {{\rm d} g_{v ^{\prime}\chi^{\prime}}}{{\rm d}\eta } \right ]_{\eta =0} = \frac {1}{C_u} - \frac {1}{2} - g_{u^{\prime}\chi^{\prime}}(0) - \frac {1}{2S}\big [h(0)+l(0)\big ], \end{align}

(3.18) \begin{align} g_{v^{\prime }\chi ^{\prime }}&=\frac {2}{\eta C_u} \int _{0}^\eta \eta ^{\prime } g_{\chi }\, {\rm d}\eta ^{\prime } - \frac {4}{\eta } \int _{0}^\eta \eta ^{\prime } g_u g_{\chi }\, {\rm d}\eta ^{\prime } + \frac {3 g_{\chi }}{\eta } \int _{0}^\eta \eta ^{\prime } g_u\, {\rm d}\eta ^{\prime }\nonumber \\[5pt] &\quad + \eta g_{u^{\prime }{\chi }^{\prime }}-\frac {4}{\eta } \int _{0}^\eta \eta ^{\prime } g_{u^{\prime }{\chi }^{\prime }}\, {\rm d}\eta ^{\prime } - \frac {1}{\eta S} \int _{0}^\eta \eta ^{\prime } (h+l)\, {\rm d}\eta ^{\prime }. \end{align}

The derivations require profiles of

(3.19) \begin{align}\overline {u},\ \overline {\chi },\ \overline {u^{\prime}\chi^{\prime}},\ 4 \mathscr {D}^2 \overline {\frac {\partial ^2 \xi }{\partial x_j \partial x_i}\frac {\partial ^2 \xi }{\partial x_j \partial x_i}},\quad 4 \mathscr {D} \overline {\frac {\partial \xi }{\partial x_j} \frac {\partial u_j}{\partial x_i} \frac {\partial \xi }{\partial x_i}}.\end{align}

The decelerating self-similar profile of $\overline {u^{\prime}\chi^{\prime}}$ is presented in Appendix A, while all the other profiles are presented in previous sections.

Figure 14 shows the decelerating self-similar profile of $\overline {v ^{\prime}\chi^{\prime}}$ along with various predictions. Again, the shaded region indicates one standard deviation of the mean profile. The light green dot-dashed straight line is the predicted slope, using (3.9). The values $C_u=0.441$ (Shin et al. Reference Shin, Aspden and Richardson2017), $S=0.085$ (i.e. jet spreading rate), $g_{\overline {u^{\prime}\chi^{\prime}}}(0)=-0.092$ , $h(0)=-5.405$ and $l(0)=5.642$ are assigned to the equation. The predicted slope (0.468) is close to the actual average slope of the averaged decelerating self-similar profile until $\eta \approx 0.11$ (0.353).

Predicted radial profiles based on (3.18) are plotted as dashed and dot-dashed lines. The purple dashed line includes all terms in (3.18), while the green dot-dashed line includes only the first three terms on the right-hand side. Prediction with only three terms shows better agreement with the actual averaged profile. If all terms are included, the prediction seems to be shifted rightwards to some extent. Note that the first three terms in (3.18) have a high level of confidence, as these are related to first-order statistics, while the remaining terms have a lower confidence level, due to their nature of higher-order statistics. Hence including the remaining terms would lead to undesirable uncertainty, due to their lower confidence level with regard to statistics.

Overall, proximity of slope and radial profile predictions increases the validity of self-similar behaviour of the SDR in the stopping jet.

3.6. Scaling of turbulent modelling for terms in the $\overline {\chi }$ transport equation

In this subsection, the existing turbulence models for the SDR are compared with DNS data, and additional modifications are proposed. For brevity, spatial and temporal virtual origins ( $x_0$ and $t_0$ ) are omitted in this subsection.

Three algebraic models are found in the literature for SDR modelling. Libby & Bray (Reference Libby and Bray1980) proposed a model for mean SDR, relating turbulent energy dissipation ( $\epsilon$ ), turbulent kinetic energy ( $k$ ) and mixture fraction variance ( $\overline {\xi ^{\prime 2}}$ ), denoted the LB model:

(3.20) \begin{equation} \begin{gathered} \overline {\chi } = C_{LB} \frac {\epsilon }{k} \overline {\xi ^{\prime 2}}. \end{gathered} \end{equation}

Mantel & Borghi (Reference Mantel and Borghi1994) proposed algebraic models for the two dominant terms in the SDR transport equation (3.16). Mantel & Borghi (Reference Mantel and Borghi1994) argued that as every term in (3.1) depends on the turbulent Reynolds number, in the large turbulent Reynolds number limit, the following models for dominant terms associated with scalar–turbulence interaction (term V in (3.16)) and effects of local curvature (term VI in (3.16)) can be developed. The model for term V, denoted the MB5 model, is

(3.21) \begin{equation} \begin{gathered} -4 \mathscr {D} \overline {\frac {\partial \xi }{\partial x_j}\frac {\partial \xi }{\partial x_i}\frac {\partial u_j}{\partial x_i}} = 2 \alpha \frac {\epsilon }{k} \overline {\chi }, \end{gathered} \end{equation}

and the model for term VI, denoted the MB6 model, is

(3.22) \begin{equation} \begin{gathered} 4 \mathscr {D}^2 \overline {\frac {\partial ^2 \xi }{\partial x_j \partial x_i} \frac {\partial ^2 \xi }{\partial x_j \partial x_i}} = 2 \beta \frac {\overline {\chi }^2}{\overline {\xi ^{\prime 2}}}, \end{gathered} \end{equation}

where $\alpha$ and $\beta$ are dimensionless constants.

As shown in Aparece-Scutariu & Shin (Reference Aparece-Scutariu and Shin2022) and in previous subsections, both steady-state and stopping jets show self-similar characteristics. Henceforth, the three models in (3.20)–(3.22) should be self-similar as well. At first glance, the models would seem to be self-similar, as constituent variables ( $\epsilon$ , $k$ , $\overline {\xi ^{\prime 2}}$ , $\overline {\chi }$ ) are self-similar. Still, caution is necessary, as self-similarity requires normalisation by certain centreline values, which can be different from case to case. An alternative way to check self-similarity is to check if the $x$ and $t$ scalings of the models are the same as the targeted variables.

In the steady-state jet, $\overline {\chi }$ scales as $1/x^4$ (Aparece-Scutariu & Shin Reference Aparece-Scutariu and Shin2022). The constituent variables in the LB model are $\epsilon$ , $k$ and $\overline {\xi ^{\prime 2}}$ , which scale as $1/x^4$ , $1/x^2$ and $1/x^2$ , respectively. Then the collective scaling of the LB model without $C_{LB}$ is $1/x^4$ , which agrees with the scaling of $\overline {\chi }$ for the steady-state jet. The same analysis is done for the other two models. Table 1 summarises $x$ and $t$ scalings for each variable for the steady-state jet. The first two rows are the measured scaling from Aparece-Scutariu & Shin (Reference Aparece-Scutariu and Shin2022). The next two rows show the scalings of the three algebraic models, without the modelling parameters. The three models agree with the $x$ scalings of their target variables in the steady-state jet, so comparisons can be made by self-similar variables.

Table 1. Spatial and temporal scalings of flow variables for the steady-state jet.

Table 2. Spatial and temporal scalings of flow variables for the stopping jet.

Figure 15. Comparison between the actual data and the algebraic models in the steady-state jet corresponding to (a) SDR and (b) terms V and VI from (3.1).

Figure 15 shows self-similar profiles of $\overline {\chi }$ , term V, and term VI over the scaled radius. Note that the three profiles are normalised by $\overline {\chi }_c$ , $(\overline {u_c}\,\overline {\chi _c})/r_{1/2}$ and $(\overline {u_c}\,\overline {\chi _c})/r_{1/2}$ , respectively. In addition, the three algebraic models with optimal parameters are included. Radial profiles of the models agree well with actual data. The optimal parameters for the steady-state jet are

(3.23) \begin{equation} C_{LB} = 2, \quad \alpha = 9, \quad \beta = 0.1. \end{equation}

Note that previous works reported similar values of $C_{LB}$ : 1.0 from Jones (Reference Jones1994), 2.0 from Janicka & Peters (Reference Janicka and Peters1982), 2.0 from Overholt & Pope (Reference Overholt and Pope1996), and 3.0 from Juneja & Pope (Reference Juneja and Pope1996).

Next, the same scaling analysis is conducted for the stopping jet. Table 2 summarises the scalings of flow variables and the models. Again, the first two rows are the measured scalings from Shin et al. (Reference Shin, Aspden and Richardson2017, Reference Shin, Aspden, Aparece-Scutariu and Richardson2023) and this work, and the next two rows show scalings of the three algebraic models without modelling parameters. The scaling of the MB5 model agrees with that of term V. However, the LB model without $C_{LB}$ , and the MB6 model without $\beta$ , do not match the targeted scalings.

Table 3. Spatial and temporal scaling of the turbulent Reynolds number and its related variables.

A plausible fix for the scaling mismatch is to impose scalings on the modelling parameters. The necessary scalings are $t/x$ for $C_{LB}$ , and $x/t$ for $\beta$ . Mantel & Borghi (Reference Mantel and Borghi1994) applied a $Re_t^{1/2}$ scaling to $\beta$ in the context of decaying homogeneous isotropic turbulence flow in order to match with their order of magnitude analysis. Hence a scaling with $Re_t$ is tested for the model parameters $C_{LB}$ and $\beta$ .

Table 3 shows the scalings of the turbulent Reynolds number ( $Re_t$ ) and its related variables, which are based on (Pope Reference Pope2000)

(3.24) \begin{equation} \begin{gathered} Re_t=\frac {l_t u^{\prime}}{\nu }, \quad \text {where} \ l_t = \frac {k^{3/2}}{\epsilon } \ \text {and} \ u^{\prime} = k^{1/2}. \end{gathered} \end{equation}

Given table 3, two types of modifications are proposed, as follows. The first modification, denoted Type 1, is

(3.25) \begin{equation} \begin{gathered} C_{LB} = \frac {C_{LB,1}}{D} \frac {l_t}{Re_t}, \quad \beta = \beta _1 D \frac { Re_t}{l_t}. \end{gathered} \end{equation}

The second modification, denoted Type 2, is

(3.26) \begin{equation} \begin{gathered} C_{LB} = C_{LB,2} \left (\frac {U_0}{D} \frac {l_t}{Re_t\, u^{\prime}}\right )^{1/2}, \quad \beta = \beta _2 \left (\frac {D}{U_0} \frac {Re_t\, u^{\prime}}{l_t} \right )^{1/2}. \end{gathered} \end{equation}

Note that $l_t$ and $u^{\prime}$ have dimensions of length and velocity, so additional length/velocity scales ( $D$ and $U_0$ ) are added to keep the new parameters dimensionless. Type 1 has simpler expressions for $C_{LB}$ and $\beta$ . Type 2 includes a $Re_t^{1/2}$ scaling, which appears in the scaling of the two dominant terms in the SDR transport equation (Mantel & Borghi Reference Mantel and Borghi1994; Mura & Borghi Reference Mura and Borghi2003).

By using the two types of modification, radial profiles can be predicted with self-similar variables. Figures 16 and 17 show a comparison of the normalised radial profiles for the models, over the scaled radius. The resulting optimal parameters are

(3.27) \begin{equation} \begin{gathered} \alpha = 42, \\ \end{gathered} \end{equation}

parameters for Type 1

(3.28) \begin{equation} \begin{gathered} C_{LB,1} = 70, \quad \beta _1 = 0.1, \end{gathered} \end{equation}

and parameters for Type 2

(3.29) \begin{equation} \begin{gathered} C_{LB,2} = 3, \quad \beta _2 = 25. \end{gathered} \end{equation}

For LB models, Type 2 modification seems to fit better with actual data. For the MB5 model, which does not need a modification, there is very good agreement with the actual data. For the MB6 model, both modifications produce results that fit within one standard deviation.

Figure 16. Comparison between the actual data and the algebraic models using Type 1 modification (3.25) in the stopping jet corresponding to (a) the SDR and (b) terms V and VI from (3.1).

Figure 17. Comparison between the actual data and the algebraic models using Type 2 modification (3.26) in the stopping jet corresponding to (a) the SDR and (b) terms V and VI from (3.1).