2.1 Power-law evolution of $\unicode[STIX]{x1D716}_{\unicode[STIX]{x1D703}}$ derived from the scale-by-scale budget

The analytical approach supporting the development of a power law expressing the longitudinal evolution of $\unicode[STIX]{x1D716}_{\unicode[STIX]{x1D703}}$ is based on a self-preservation solution of the transport equation for $\overline{(\unicode[STIX]{x1D6FF}\unicode[STIX]{x1D703})^{2}}$ , the second-order temperature structure function; $\unicode[STIX]{x1D6FF}\unicode[STIX]{x1D703}=(\unicode[STIX]{x1D703}^{+}-\unicode[STIX]{x1D703})$ , where $\unicode[STIX]{x1D703}^{+}=\unicode[STIX]{x1D703}(x+s)$ and $\unicode[STIX]{x1D703}=\unicode[STIX]{x1D703}(x)$ depends on the separations $s$ between two points along the jet axis. Starting from the temperature equation written at two independent points in space, and assuming local isotropy and local homogeneity (see Burattini et al. Reference Burattini, Antonia and Danaila2005a , for a discussion on the homogeneity condition), the transport equation for $\overline{(\unicode[STIX]{x1D6FF}\unicode[STIX]{x1D703})^{2}}$ is written as:

(2.1) $$\begin{eqnarray}-\overline{(\unicode[STIX]{x1D6FF}u)(\unicode[STIX]{x1D6FF}\unicode[STIX]{x1D703})^{2}}+2\unicode[STIX]{x1D6FC}\frac{\text{d}\overline{(\unicode[STIX]{x1D6FF}\unicode[STIX]{x1D703})^{2}}}{\text{d}s}+LST(s)=\frac{4}{3}\unicode[STIX]{x1D716}_{\unicode[STIX]{x1D703}}s,\end{eqnarray}$$

where $LST(s)$ represents the contribution of the large scales to the budget of $\overline{(\unicode[STIX]{x1D6FF}\unicode[STIX]{x1D703})^{2}}$ . Here, similar developments to those reported by Antonia et al. (Reference Antonia, Ould-Rouis, Anselmet and Zhu1997), Hill (Reference Hill1997, Reference Hill2001), Danaila et al. (Reference Danaila, Anselmet, Zhou and Antonia1999), Shivamoggi & Antonia (Reference Shivamoggi and Antonia2000) and Burattini, Antonia & Danaila (Reference Burattini, Antonia and Danaila2005b ) are used to derive the large scale term on the jet axis:

(2.2) $$\begin{eqnarray}LST(s)=-C(s)-P(s)-D(s),\end{eqnarray}$$

where

(2.3) $$\begin{eqnarray}\displaystyle & \displaystyle C(s)=\frac{1}{s^{2}}U_{0}\int _{0}^{s}y^{2}\frac{\unicode[STIX]{x2202}\overline{(\unicode[STIX]{x1D6FF}\unicode[STIX]{x1D703})^{2}}}{\unicode[STIX]{x2202}x}\,\text{d}y, & \displaystyle\end{eqnarray}$$

(2.4) $$\begin{eqnarray}\displaystyle & \displaystyle P(s)=\frac{2}{s^{2}}\frac{\text{d}\unicode[STIX]{x1D6E9}_{0}}{\text{d}x}\int _{0}^{s}y^{2}\overline{(\unicode[STIX]{x1D6FF}u)(\unicode[STIX]{x1D6FF}\unicode[STIX]{x1D703})}\,\text{d}y, & \displaystyle\end{eqnarray}$$

(2.5) $$\begin{eqnarray}\displaystyle & \displaystyle D(s)=\frac{1}{2s^{2}}\int _{0}^{s}y^{2}\frac{\unicode[STIX]{x2202}}{\unicode[STIX]{x2202}\overline{x}}\left[\overline{(u^{+}+u)(\unicode[STIX]{x1D6FF}\unicode[STIX]{x1D703})^{2}}\right]\,\text{d}y+\frac{4}{s^{2}}\int _{0}^{s}y^{2}\frac{\unicode[STIX]{x2202}}{\unicode[STIX]{x2202}r}\overline{v(\unicode[STIX]{x1D6FF}\unicode[STIX]{x1D703})^{2}}\,\text{d}y, & \displaystyle\end{eqnarray}$$

where $\overline{x}=x+s/2$ ( $\overline{x}$ being the midpoint), $\unicode[STIX]{x1D6FF}u=(u^{+}-u)$ , $u^{+}=u(x+s)$ and $u=u(x)$ . The three terms defined in (2.3) to (2.5), $C(s)$ , $P(s)$ and $D(s)$ , represent convection, production and turbulent diffusion, respectively. In the limit $s\rightarrow \infty$ , equation (2.1) yields the one-point budget of $\overline{\unicode[STIX]{x1D703}^{2}}/2$ , which is expressed (neglecting molecular diffusion) as

(2.6) $$\begin{eqnarray}-U_{0}\frac{\unicode[STIX]{x2202}\overline{\unicode[STIX]{x1D703}^{2}}/2}{\unicode[STIX]{x2202}x}-\overline{u\unicode[STIX]{x1D703}}\frac{\text{d}\unicode[STIX]{x1D6E9}_{0}}{\text{d}x}-\frac{1}{2}\frac{\unicode[STIX]{x2202}\overline{u\unicode[STIX]{x1D703}^{2}}}{\unicode[STIX]{x2202}x}-\frac{\unicode[STIX]{x2202}\overline{v\unicode[STIX]{x1D703}^{2}}}{\unicode[STIX]{x2202}r}\simeq \unicode[STIX]{x1D716}_{\unicode[STIX]{x1D703}},\end{eqnarray}$$

where $\unicode[STIX]{x1D6E9}_{0}$ is the local mean temperature excess relative to the ambient for $r=0$ . The first term on the left-hand side of this expression represents convection ( $C_{\unicode[STIX]{x1D703}}$ ), the second stands for production ( $P_{\unicode[STIX]{x1D703}}$ ) and the last two terms correspond to the longitudinal ( $D_{\unicode[STIX]{x1D703}_{x}}$ ) and radial ( $D_{\unicode[STIX]{x1D703}_{r}}$ ) contributions to the turbulent diffusion ( $D_{\unicode[STIX]{x1D703}}$ ). Darisse, Lemay & Benaïssa (Reference Darisse, Lemay and Benaïssa2014) showed that $D_{\unicode[STIX]{x1D703}_{r}}$ is more than 300 times larger than $D_{\unicode[STIX]{x1D703}_{x}}$ on the jet axis. For the one-point budget, it is thus justified to neglect the longitudinal contribution from the turbulent diffusion term (see § 2.6). However, since this term is not necessarily negligible at all scales, we have included its contribution in the definition of $D(s)$ (2.5) in the sbs budget.

Self-preservation form of the equations (2.1)–(2.5) can be obtained when the different terms are expressed using the following functional forms (see Antonia et al. Reference Antonia, Smalley, Zhou, Anselmet and Danaila2004; Burattini et al. Reference Burattini, Antonia and Danaila2005b )

(2.7) $$\begin{eqnarray}\displaystyle & \displaystyle \overline{(\unicode[STIX]{x1D6FF}\unicode[STIX]{x1D703})^{2}}=\tilde{\unicode[STIX]{x1D703}}^{2}(x)h_{\unicode[STIX]{x1D703}}(s^{\ast }), & \displaystyle\end{eqnarray}$$

(2.8) $$\begin{eqnarray}\displaystyle & \displaystyle \overline{(\unicode[STIX]{x1D6FF}u)^{2}}=\tilde{u} ^{2}(x)h_{u}(s^{\ast }), & \displaystyle\end{eqnarray}$$

(2.9) $$\begin{eqnarray}\displaystyle & \displaystyle \overline{(\unicode[STIX]{x1D6FF}u)(\unicode[STIX]{x1D6FF}\unicode[STIX]{x1D703})}=\tilde{A}_{\unicode[STIX]{x1D703}}(x)a(s^{\ast }), & \displaystyle\end{eqnarray}$$

(2.10) $$\begin{eqnarray}\displaystyle & \displaystyle \overline{(\unicode[STIX]{x1D6FF}u)(\unicode[STIX]{x1D6FF}\unicode[STIX]{x1D703})^{2}}=\tilde{B}_{\unicode[STIX]{x1D703}}(x)b_{1}(s^{\ast }), & \displaystyle\end{eqnarray}$$

(2.11) $$\begin{eqnarray}\displaystyle & \displaystyle \overline{(u^{+}+u)(\unicode[STIX]{x1D6FF}\unicode[STIX]{x1D703})^{2}}=\tilde{B}_{\unicode[STIX]{x1D703}}(x)b_{2}(s^{\ast }), & \displaystyle\end{eqnarray}$$

(2.12) $$\begin{eqnarray}\displaystyle & \displaystyle \overline{v(\unicode[STIX]{x1D6FF}\unicode[STIX]{x1D703})^{2}}=\tilde{C}_{\unicode[STIX]{x1D703}}(x)c(s^{\ast }), & \displaystyle\end{eqnarray}$$

where $s^{\ast }\equiv s/l_{\unicode[STIX]{x1D703}}$ , $l_{\unicode[STIX]{x1D703}}$ being the characteristic length scale for the temperature field. Note that the expression for $\overline{(\unicode[STIX]{x1D6FF}u)^{2}}$ is not directly related to (2.1)–(2.5), but it will be used later on, for the analysis of the mixed skewness increment. The scaling functions $\tilde{\unicode[STIX]{x1D703}}^{2}(x)$ , $\tilde{u} ^{2}(x)$ , $\tilde{A}_{\unicode[STIX]{x1D703}}(x)$ , $\tilde{B}_{\unicode[STIX]{x1D703}}(x)$ , $\tilde{C}_{\unicode[STIX]{x1D703}}(x)$ characterize the streamwise evolution of the structure functions, while the dimensionless ( $s^{\ast }$ ) functions represent the shape of these structure functions. For the mixed velocity–temperature scales, as in Antonia et al. (Reference Antonia, Smalley, Zhou, Anselmet and Danaila2004), we avoid making the a priori assumption that these functions are represented by a combination of uncoupled velocity and temperature scales. For example, $\tilde{A}_{\unicode[STIX]{x1D703}}$ is a mixed scale with dimensions of velocity times temperature, but it is not a priori considered as being represented by the product of a velocity scale with a temperature scale.

Substitution of expressions (2.7)–(2.12) (in which we drop $x$ and $s^{\ast }$ for convenience) into (2.1) and dividing by $(\unicode[STIX]{x1D716}_{\unicode[STIX]{x1D703}}s^{\ast })$ yields, after some manipulation,

(2.13) $$\begin{eqnarray}\displaystyle & & \displaystyle -\left[\frac{\tilde{B}_{\unicode[STIX]{x1D703}}}{l_{\unicode[STIX]{x1D703}}\unicode[STIX]{x1D716}_{\unicode[STIX]{x1D703}}}\right]\frac{b_{1}}{s^{\ast }}+\left[\frac{\unicode[STIX]{x1D6FC}\tilde{\unicode[STIX]{x1D703}}^{2}}{{l_{\unicode[STIX]{x1D703}}}^{2}\unicode[STIX]{x1D716}_{\unicode[STIX]{x1D703}}}\right]2\frac{\text{d}h_{\unicode[STIX]{x1D703}}}{\text{d}s^{\ast }}-\left[\frac{U_{0}}{\unicode[STIX]{x1D716}_{\unicode[STIX]{x1D703}}}\frac{\text{d}\tilde{\unicode[STIX]{x1D703}}^{2}}{\text{d}x}\right]\unicode[STIX]{x1D6E4}_{1}^{\ast }+\left[\frac{U_{0}\tilde{\unicode[STIX]{x1D703}}^{2}}{l_{\unicode[STIX]{x1D703}}\unicode[STIX]{x1D716}_{\unicode[STIX]{x1D703}}}\frac{\text{d}l_{\unicode[STIX]{x1D703}}}{\text{d}x}\right]\unicode[STIX]{x1D6E4}_{2}^{\ast }-\left[\frac{\tilde{A}_{\unicode[STIX]{x1D703}}}{\unicode[STIX]{x1D716}_{\unicode[STIX]{x1D703}}}\frac{\text{d}\unicode[STIX]{x1D6E9}_{0}}{\text{d}x}\right]\unicode[STIX]{x1D6E4}_{3}^{\ast }\nonumber\\ \displaystyle & & \displaystyle \qquad +\,\left[\frac{\tilde{B}_{\unicode[STIX]{x1D703}}}{l_{\unicode[STIX]{x1D703}}\unicode[STIX]{x1D716}_{\unicode[STIX]{x1D703}}}\frac{1}{2}\frac{\text{d}l_{\unicode[STIX]{x1D703}}}{\text{d}\overline{x}}\right]\unicode[STIX]{x1D6E4}_{4}^{\ast }-\left[\frac{1}{\unicode[STIX]{x1D716}_{\unicode[STIX]{x1D703}}}\frac{\text{d}\tilde{B}_{\unicode[STIX]{x1D703}}}{\text{d}\overline{x}}\right]\unicode[STIX]{x1D6E4}_{5}^{\ast }-\left[\frac{\tilde{C}_{\unicode[STIX]{x1D703}}}{l_{\unicode[STIX]{x1D703}}\unicode[STIX]{x1D716}_{\unicode[STIX]{x1D703}}}\right]\unicode[STIX]{x1D6E4}_{6}^{\ast }=\left[\frac{4}{3}\right],\end{eqnarray}$$

where the functions $\unicode[STIX]{x1D6E4}_{i}^{\ast }$ , which represent the shape functions of the large scale terms ( $LST$ ), are expressed as

(2.14) $$\begin{eqnarray}\displaystyle & \displaystyle \unicode[STIX]{x1D6E4}_{1}^{\ast }=\frac{1}{s^{\ast 3}}\int _{0}^{s^{\ast }}h_{\unicode[STIX]{x1D703}}y^{\ast 2}\,\text{d}y^{\ast }, & \displaystyle\end{eqnarray}$$

(2.15) $$\begin{eqnarray}\displaystyle & \displaystyle \unicode[STIX]{x1D6E4}_{2}^{\ast }=\frac{1}{s^{\ast 3}}\int _{0}^{s^{\ast }}\frac{\text{d}h_{\unicode[STIX]{x1D703}}}{\text{d}y^{\ast }}y^{\ast 3}\,\text{d}y^{\ast }, & \displaystyle\end{eqnarray}$$

(2.16) $$\begin{eqnarray}\displaystyle & \displaystyle \unicode[STIX]{x1D6E4}_{3}^{\ast }=\frac{2}{s^{\ast 2}}\int _{0}^{s^{\ast }}ay^{\ast 2}\,\text{d}y^{\ast }, & \displaystyle\end{eqnarray}$$

(2.17) $$\begin{eqnarray}\displaystyle & \displaystyle \unicode[STIX]{x1D6E4}_{4}^{\ast }=\frac{1}{s^{\ast 3}}\int _{0}^{s^{\ast }}\frac{\text{d}b_{2}}{\text{d}y^{\ast }}y^{\ast 3}\,\text{d}y^{\ast }, & \displaystyle\end{eqnarray}$$

(2.18) $$\begin{eqnarray}\displaystyle & \displaystyle \unicode[STIX]{x1D6E4}_{5}^{\ast }=\frac{1}{2s^{\ast 3}}\int _{0}^{s^{\ast }}b_{2}y^{\ast 2}\,\text{d}y^{\ast }, & \displaystyle\end{eqnarray}$$

(2.19) $$\begin{eqnarray}\displaystyle & \displaystyle \unicode[STIX]{x1D6E4}_{6}^{\ast }=\frac{4}{s^{\ast 3}}\int _{0}^{s^{\ast }}\frac{\text{d}c}{\text{d}y^{\ast }}\frac{\unicode[STIX]{x2202}y^{\ast }}{\unicode[STIX]{x2202}r^{\ast }}y^{\ast 2}\,\text{d}y^{\ast }. & \displaystyle\end{eqnarray}$$

Self-preservation requires that all terms within brackets should evolve in $x$ in exactly the same manner. Since the term on the right-hand side of (2.13) is constant, all the terms within brackets should be constant. This yields the following self-preservation constraints (where $C_{i}$ are constants):

(2.20) $$\begin{eqnarray}\displaystyle & \displaystyle \left[\frac{\tilde{B}_{\unicode[STIX]{x1D703}}}{l_{\unicode[STIX]{x1D703}}\unicode[STIX]{x1D716}_{\unicode[STIX]{x1D703}}}\right]=C_{1}, & \displaystyle\end{eqnarray}$$

(2.21) $$\begin{eqnarray}\displaystyle & \displaystyle \left[\frac{\unicode[STIX]{x1D6FC}\tilde{\unicode[STIX]{x1D703}}^{2}}{{l_{\unicode[STIX]{x1D703}}}^{2}\unicode[STIX]{x1D716}_{\unicode[STIX]{x1D703}}}\right]=C_{2}, & \displaystyle\end{eqnarray}$$

(2.22) $$\begin{eqnarray}\displaystyle & \displaystyle \left[\frac{U_{0}}{\unicode[STIX]{x1D716}_{\unicode[STIX]{x1D703}}}\frac{\text{d}\tilde{\unicode[STIX]{x1D703}}^{2}}{\text{d}x}\right]=C_{3}, & \displaystyle\end{eqnarray}$$

(2.23) $$\begin{eqnarray}\displaystyle & \displaystyle \left[\frac{U_{0}\tilde{\unicode[STIX]{x1D703}}^{2}}{l_{\unicode[STIX]{x1D703}}\unicode[STIX]{x1D716}_{\unicode[STIX]{x1D703}}}\frac{\text{d}l_{\unicode[STIX]{x1D703}}}{\text{d}x}\right]=C_{4}, & \displaystyle\end{eqnarray}$$

(2.24) $$\begin{eqnarray}\displaystyle & \displaystyle \left[\frac{\tilde{A}_{\unicode[STIX]{x1D703}}}{\unicode[STIX]{x1D716}_{\unicode[STIX]{x1D703}}}\frac{\text{d}\unicode[STIX]{x1D6E9}_{0}}{\text{d}x}\right]=C_{5}, & \displaystyle\end{eqnarray}$$

(2.25) $$\begin{eqnarray}\displaystyle & \displaystyle \left[\frac{\tilde{B}_{\unicode[STIX]{x1D703}}}{l_{\unicode[STIX]{x1D703}}\unicode[STIX]{x1D716}_{\unicode[STIX]{x1D703}}}\frac{1}{2}\frac{\text{d}l_{\unicode[STIX]{x1D703}}}{\text{d}\overline{x}}\right]=\left[\frac{\tilde{B}_{\unicode[STIX]{x1D703}}}{l_{\unicode[STIX]{x1D703}}\unicode[STIX]{x1D716}_{\unicode[STIX]{x1D703}}}\frac{\text{d}l_{\unicode[STIX]{x1D703}}}{\text{d}x}\right]=C_{6}, & \displaystyle\end{eqnarray}$$

(2.26) $$\begin{eqnarray}\displaystyle & \displaystyle \left[\frac{1}{\unicode[STIX]{x1D716}_{\unicode[STIX]{x1D703}}}\frac{\text{d}\tilde{B}_{\unicode[STIX]{x1D703}}}{\text{d}\overline{x}}\right]=\left[\frac{2}{\unicode[STIX]{x1D716}_{\unicode[STIX]{x1D703}}}\frac{\text{d}\tilde{B}_{\unicode[STIX]{x1D703}}}{\text{d}x}\right]=C_{7}, & \displaystyle\end{eqnarray}$$

(2.27) $$\begin{eqnarray}\displaystyle & \displaystyle \left[\frac{\tilde{C}_{\unicode[STIX]{x1D703}}}{l_{\unicode[STIX]{x1D703}}\unicode[STIX]{x1D716}_{\unicode[STIX]{x1D703}}}\right]=C_{8}. & \displaystyle\end{eqnarray}$$

Following the definition of the midpoint $\overline{x}$ , we can write $\text{d}l_{\unicode[STIX]{x1D703}}/\text{d}\overline{x}=\text{d}l_{\unicode[STIX]{x1D703}}/\text{d}x^{+}+\text{d}l_{\unicode[STIX]{x1D703}}/\text{d}x$ (see for example Hill Reference Hill2001; Danaila et al. Reference Danaila, Krawczynski, Thiesset and Renou2012). Along the jet axis, in the far field, it is considered that $\text{d}l_{\unicode[STIX]{x1D703}}/\text{d}x=\text{d}l_{\unicode[STIX]{x1D703}}/\text{d}x^{+}$ , and the derivative in $C_{6}$ has been written as $\text{d}l_{\unicode[STIX]{x1D703}}/\text{d}\overline{x}=2\text{d}l_{\unicode[STIX]{x1D703}}/\text{d}x$ . The same reasoning applies to the derivative of $\tilde{B}_{\unicode[STIX]{x1D703}}$ involved in $C_{7}$ .

Manipulation of $C_{i}$ constants yields the following constraints:

(2.28) $$\begin{eqnarray}\displaystyle & \displaystyle \frac{C_{6}}{C_{1}}=\frac{\text{d}l_{\unicode[STIX]{x1D703}}}{\text{d}x}, & \displaystyle\end{eqnarray}$$

(2.29) $$\begin{eqnarray}\displaystyle & \displaystyle \frac{C_{1}C_{4}}{C_{2}C_{6}}=\frac{U_{0}l_{\unicode[STIX]{x1D703}}}{\unicode[STIX]{x1D6FC}}, & \displaystyle\end{eqnarray}$$

(2.30) $$\begin{eqnarray}\displaystyle & \displaystyle \frac{C_{5}C_{6}}{C_{1}C_{4}}=\frac{\tilde{A}_{\unicode[STIX]{x1D703}}l_{\unicode[STIX]{x1D703}}}{U_{0}\tilde{\unicode[STIX]{x1D703}}^{2}}\frac{\text{d}\unicode[STIX]{x1D6E9}_{0}}{\text{d}x}, & \displaystyle\end{eqnarray}$$

(2.31) $$\begin{eqnarray}\displaystyle & \displaystyle \frac{C_{6}}{C_{4}}=\frac{\tilde{B}_{\unicode[STIX]{x1D703}}}{U_{0}\tilde{\unicode[STIX]{x1D703}}^{2}}, & \displaystyle\end{eqnarray}$$

(2.32) $$\begin{eqnarray}\displaystyle & \displaystyle \frac{C_{1}}{C_{8}}=\frac{\tilde{B}_{\unicode[STIX]{x1D703}}}{\tilde{C}_{\unicode[STIX]{x1D703}}}. & \displaystyle\end{eqnarray}$$

The two first constraints allow us to write $l_{\unicode[STIX]{x1D703}}\propto \hat{x}$ and, for a given value of $\unicode[STIX]{x1D6FC}$ , $U_{0}\propto \hat{x}^{-1}$ . It is worth mentioning that the usual power law for $U_{0}$ , viz. $U_{0}/U_{j}=B_{U}\hat{x}^{-1}$ , is more formally obtained here than the one resulting from the momentum equation.

In addition, the self-preservation analysis of the mean temperature equation presented in § A.1 strictly shows that $\unicode[STIX]{x1D6E9}_{0}$ , scales with $\hat{x}^{-1}$ (yielding $\text{d}\unicode[STIX]{x1D6E9}_{0}/\text{d}x\propto \hat{x}^{-2}$ ). This self-preservation analysis also shows, from (A 11), that $H_{0}\propto \hat{x}^{-2}$ ( $H_{0}$ represents $\overline{u\unicode[STIX]{x1D703}}$ on the jet axis). It is thus required that $\tilde{A}_{\unicode[STIX]{x1D703}}$ has to scale with $\hat{x}^{-2}$ , because $\tilde{A}_{\unicode[STIX]{x1D703}}=2H_{0}$ when $s\rightarrow \infty$ .

Using the power laws just derived, viz. $l_{\unicode[STIX]{x1D703}}\propto \hat{x}$ , $U_{0}\propto \hat{x}^{-1}$ , $\tilde{A}_{\unicode[STIX]{x1D703}}\propto \hat{x}^{-2}$ and $\text{d}\unicode[STIX]{x1D6E9}_{0}/\text{d}x\propto \hat{x}^{-2}$ into constraint (2.30) yields

(2.33) $$\begin{eqnarray}\tilde{\unicode[STIX]{x1D703}}^{2}\propto \frac{\tilde{A}_{\unicode[STIX]{x1D703}}l_{\unicode[STIX]{x1D703}}}{U_{0}}\frac{\text{d}\unicode[STIX]{x1D6E9}_{0}}{\text{d}x}\propto \hat{x}^{-2}.\end{eqnarray}$$

Finally, introducing $\text{d}\unicode[STIX]{x1D6E9}_{0}/\text{d}x\propto \hat{x}^{-2}$ and $\tilde{A}_{\unicode[STIX]{x1D703}}\propto \hat{x}^{-2}$ into expression (2.24), which defines the fifth constraint $C_{5}$ , yields the $-4$ th power law for $\unicode[STIX]{x1D716}_{\unicode[STIX]{x1D703}}$ :

(2.34) $$\begin{eqnarray}\unicode[STIX]{x1D716}_{\unicode[STIX]{x1D703}}\propto \hat{x}^{-4}.\end{eqnarray}$$

This expression is similar to the $-4$ th power law previously derived by TAD and DALL for $\unicode[STIX]{x1D716}_{k}$ . The mean dissipation rate of $\overline{\unicode[STIX]{x1D703}^{2}}/2$ , normalized by the jet exit parameters, can finally be written as

(2.35) $$\begin{eqnarray}\frac{\unicode[STIX]{x1D716}_{\unicode[STIX]{x1D703}}D}{U_{j}\unicode[STIX]{x1D6E9}_{j}^{2}}=A_{\unicode[STIX]{x1D716}_{\unicode[STIX]{x1D703}}}\hat{x}^{-4}.\end{eqnarray}$$

In the context of complete self-preservation, it should be noted that other velocity scales, like $\unicode[STIX]{x1D710}_{K}$ or $\sqrt{\overline{u^{2}}}$ could have been used, interchangeably, instead of $U_{0}$ (see Burattini et al. Reference Burattini, Antonia and Danaila2005b , and TAD), as far as the chosen velocity scales with $\hat{x}^{-1}$ . The same reasoning applies to the temperature scale. For example, the Kolmogorov temperature $\unicode[STIX]{x1D703}_{K}\equiv (\unicode[STIX]{x1D716}_{\unicode[STIX]{x1D703}}\unicode[STIX]{x1D702}/\unicode[STIX]{x1D710}_{K})^{1/2}$ , which scales with $\hat{x}^{-1}$ , could have been used, interchangeably, instead of $\unicode[STIX]{x1D6E9}_{0}$ .

Moreover, should $\tilde{A}_{\unicode[STIX]{x1D703}}$ , defined in relation (2.9), have been a priori expressed as the product of uncoupled velocity and temperature scales, the same final result would have been obtained in a simpler way, by the sole use of the sbs budget. We however prefer to avoid this assumption and use the mean temperature equation to close the problem.

2.2 Power laws for temperature variance and mixed moments

Under the assumption of local homogeneity, the temperature variance can be defined as half the temperature structure function considered for large separations, viz. $\overline{\unicode[STIX]{x1D703}^{2}}=\overline{(\unicode[STIX]{x1D6FF}\unicode[STIX]{x1D703})^{2}}/2$ (for $s\rightarrow \infty$ ). From the previous developments, it follows immediately that $\overline{\unicode[STIX]{x1D703}^{2}}\propto \unicode[STIX]{x1D6E9}_{0}^{2}\propto \hat{x}^{-2}$ . We thus introduce the parameter

(2.36) $$\begin{eqnarray}B_{I}^{2}=\frac{\overline{\unicode[STIX]{x1D703}^{2}}}{\unicode[STIX]{x1D6E9}_{0}^{2}},\end{eqnarray}$$

which must be constant along the jet axis. The term $B_{I}$ for the temperature field is comparable to the parameter $A_{I}=\overline{u^{2}}/U_{0}^{2}$ introduced by TAD for the velocity field. Darisse et al. (Reference Darisse, Lemay and Benaïssa2015) observed in their jet that $B_{I}$ was constant for $\hat{x}>20$ . For the slightly heated round jet studied by Darisse et al. (Reference Darisse, Lemay and Benaïssa2015), the longitudinal distributions of $R_{U}$ , $U_{0}$ , $\unicode[STIX]{x1D6E9}_{0}$ , $\overline{u^{2}}$ , $\overline{\unicode[STIX]{x1D703}^{2}}$ and $\unicode[STIX]{x1D716}_{\unicode[STIX]{x1D703}}$ , all normalized by the jet exit parameters, are presented in figure 1. All these quantities share the same virtual origin $x_{0}=1.65$ which has been obtained from least-square regressions performed over the whole dataset related to the self-preserving region. The log–log scale highlights the power-law behaviour of these quantities. The location corresponding to the self-preserving region depends on the quantity of interest, but it is shown that full self-preservation is clearly reached for $\hat{x}>30$ . Table 1 presents the power-law equation used for each quantity and the (measured) value of their corresponding power-law coefficients.

Figure 1. Streamwise evolution of $R_{U}/D$ , $U_{0}/U_{j}$ , $\unicode[STIX]{x1D6E9}_{0}/\unicode[STIX]{x1D6E9}_{j}$ , $\overline{u^{2}}/U_{j}^{2}$ , $\overline{\unicode[STIX]{x1D703}^{2}}/\unicode[STIX]{x1D6E9}_{j}^{2}$ and $\unicode[STIX]{x1D716}_{\unicode[STIX]{x1D703}}D/(U_{j}\unicode[STIX]{x1D6E9}_{j}^{2})$ along the jet axis (symbols) and their corresponding power laws (measurements of Darisse, Lemay & Benaïssa Reference Darisse, Lemay and Benaïssa2013a ,Reference Darisse, Lemay and Benaïssa b ; Darisse et al. Reference Darisse, Lemay and Benaïssa2015).

Table 1. Definition of the power laws shown in figure 1 and obtained from the measurements of Darisse et al. (Reference Darisse, Lemay and Benaïssa2015) ( $Re_{D}=1.4\times 10^{5}$ , $Re_{\unicode[STIX]{x1D706}}=548$ and $Pe_{\unicode[STIX]{x1D706}}=211$ ).

Recalling from expression (2.9) that $\tilde{A}_{\unicode[STIX]{x1D703}}$ has the dimensions of velocity times temperature and that $\tilde{A}_{\unicode[STIX]{x1D703}}\propto \hat{x}^{-2}$ , it immediately follows that, along the jet axis,

(2.37) $$\begin{eqnarray}\tilde{A}_{\unicode[STIX]{x1D703}}\propto U_{0}\unicode[STIX]{x1D6E9}_{0}\propto \tilde{u} \tilde{\unicode[STIX]{x1D703}}\propto \hat{x}^{-2},\end{eqnarray}$$

where $\tilde{u} \propto \hat{x}^{-1}$ is a characteristic velocity scale (see (2.8)) used by DALL in their sbs analysis of the energy equation. Relation (2.37) indicates that $\tilde{A}_{\unicode[STIX]{x1D703}}$ (a mixed velocity–temperature scale) is finally expressed as the product of uncoupled velocity and temperature scales. This also reveals that the normalized longitudinal temperature flux $\overline{u\unicode[STIX]{x1D703}}/(U_{0}\unicode[STIX]{x1D6E9}_{0})$ is constant along the jet axis.

The temperature scale is now introduced into relations (2.31) and (2.32) which leads to

(2.38) $$\begin{eqnarray}\tilde{B}_{\unicode[STIX]{x1D703}}\propto \tilde{C}_{\unicode[STIX]{x1D703}}\propto U_{0}\unicode[STIX]{x1D6E9}_{0}^{2}\propto \hat{x}^{-3}.\end{eqnarray}$$

These mixed velocity–temperature scales are both represented by the product of an uncoupled velocity scale and a temperature squared scale. The above relations also indicate that the normalized mixed moments, $\overline{u\unicode[STIX]{x1D703}^{2}}/(U_{0}\unicode[STIX]{x1D6E9}_{0}^{2})$ and $\overline{v\unicode[STIX]{x1D703}^{2}}/(U_{0}\unicode[STIX]{x1D6E9}_{0}^{2})$ , are constant along the jet axis. This also allows the mixed skewness increment to be expressed as:

(2.39) $$\begin{eqnarray}S_{\unicode[STIX]{x1D703}}=\frac{\overline{(\unicode[STIX]{x1D6FF}u)(\unicode[STIX]{x1D6FF}\unicode[STIX]{x1D703})^{2}}}{\overline{(\unicode[STIX]{x1D6FF}u)^{2}}^{1/2}\overline{(\unicode[STIX]{x1D6FF}\unicode[STIX]{x1D703})^{2}}}=\frac{\tilde{B}_{\unicode[STIX]{x1D703}}}{\tilde{u} \tilde{\unicode[STIX]{x1D703}}^{2}}\frac{b_{1}(s^{\ast })}{h_{u}(s^{\ast })h_{\unicode[STIX]{x1D703}}(s^{\ast })}=c_{S}\unicode[STIX]{x1D719}_{\unicode[STIX]{x1D703}}(s^{\ast }).\end{eqnarray}$$

The power laws for $\tilde{B}_{\unicode[STIX]{x1D703}}$ , $\tilde{u}$ and $\tilde{\unicode[STIX]{x1D703}}^{2}$ indicate that the scaling function $c_{S}$ is constant. This direct consequence of self-preservation on the jet axis was also observed for the velocity field by DALL. They have shown that $c_{q}$ , the scaling function of the mixed skewness increment of velocity and kinetic energy, was constant.

2.3 Turbulent Péclet number along the jet axis

Another outcome of this analysis pertains to the evolution of the turbulent Péclet number ( $Pe_{\unicode[STIX]{x1D706}}\equiv \sqrt{\overline{u^{2}}}\unicode[STIX]{x1D706}_{\unicode[STIX]{x1D703}}/\unicode[STIX]{x1D6FC}$ , where $\unicode[STIX]{x1D706}_{\unicode[STIX]{x1D703}}\equiv \sqrt{3\unicode[STIX]{x1D6FC}\overline{\unicode[STIX]{x1D703}^{2}}/\unicode[STIX]{x1D716}_{\unicode[STIX]{x1D703}}}$ ). Considering that $\overline{\unicode[STIX]{x1D703}^{2}}\propto \hat{x}^{-2}$ and $\unicode[STIX]{x1D716}_{\unicode[STIX]{x1D703}}\propto \hat{x}^{-4}$ , it follows immediately that $\unicode[STIX]{x1D706}_{\unicode[STIX]{x1D703}}\propto \hat{x}$ . Recalling that $\sqrt{\overline{u^{2}}}\propto \hat{x}^{-1}$ , one concludes that the turbulent Péclet number must be constant along the jet axis in the self-preserving region. This behaviour was observed by Darisse et al. (Reference Darisse, Lemay and Benaïssa2015) in their jet where $Pe_{\unicode[STIX]{x1D706}}$ was observed to be constant for $\hat{x}>30$ .

2.4 Characteristic length scales

The self-preservation analyses of the sbs budget revealed that the characteristic length scale of the temperature field evolves linearly, viz. $l_{\unicode[STIX]{x1D703}}\propto \hat{x}$ , as indicated by (2.28). Thus, any length scale evolving linearly along the jet axis would be a relevant normalization length scale, for all scales, including those corresponding to the sufficiently large values of $s$ , where the sbs budget yields the one-point $\overline{\unicode[STIX]{x1D703}^{2}}/2$ budget. It has been shown by Burattini et al. (Reference Burattini, Antonia and Danaila2005b ) and more rigorously by DALL that the Taylor and Kolmogorov microscales, viz. $\unicode[STIX]{x1D706}$ and $\unicode[STIX]{x1D702}$ , can be used interchangeably, as these quantities both evolve linearly along the jet axis. Moreover, it was shown, in the previous section, that $\unicode[STIX]{x1D706}_{\unicode[STIX]{x1D703}}$ also fulfils the requirements ensuing from the present self-preservation analysis. Finally, from § A.2, it is shown that $R_{U}$ and $R_{\unicode[STIX]{x1D6E9}}$ , the half-radii of the mean velocity and temperature profiles, respectively, evolve linearly in the streamwise direction. Thus, $\unicode[STIX]{x1D702}$ , $\unicode[STIX]{x1D706}$ , $\unicode[STIX]{x1D706}_{\unicode[STIX]{x1D703}}$ , $R_{U}$ and $R_{\unicode[STIX]{x1D6E9}}$ are all relevant quantities which could be used as an appropriate characteristic length scale. Hereafter, $R_{U}$ is used (among the other relevant options) as the local normalization length scale.

Figure 2. Second-order structure function of temperature fluctuations on the jet axis, at $x/D=30$ , normalized with Kolmogorov scales $\unicode[STIX]{x1D703}_{K}$ and $\unicode[STIX]{x1D702}$ (flow conditions similar to those of Darisse et al. Reference Darisse, Lemay and Benaïssa2013a ,Reference Darisse, Lemay and Benaïssa b , Reference Darisse, Lemay and Benaïssa2015). Symbols: measurements performed using two cold-wire probes separated by a distance $s$ in the streamwise direction; open symbols: raw data; closed symbols: signals compensated for the attenuation resulting from the wire time constant (see Lemay & Benaïssa Reference Lemay and Benaïssa2001; Lemay, Benaissa & Antonia Reference Lemay, Benaissa and Antonia2003; Darisse et al. Reference Darisse, Lemay and Benaïssa2014). Line: measurements performed using one cold-wire probe (compensated signal) and the assumption of Taylor’s hypothesis ( $s=\unicode[STIX]{x1D70F}U_{0}$ , where $\unicode[STIX]{x1D70F}$ is the time increment).

2.5 Structure functions of temperature fluctuations

Further evidence of complete self-preservation can be inferred from the analysis of the temperature structure functions. Indeed, an indicator showing the fulfilment of complete self-preservation is obtained when the structure functions measured at several positions along the jet axis all collapse over the entire range of the increment $s$ or scales, regardless of which set of scaling variables (complying with self-preservation conditions) is used for the normalization. In the present case, the second- and third-order structure functions for the temperature are considered.

Before considering the collapse of these structure functions along the jet axis, we first consider some technical aspects regarding their measurement, namely the effect of the limited time constant of the cold-wire probe and the validation of Taylor’s hypothesis used to evaluate these functions. For that purpose, figure 2 shows the distribution of $\overline{(\unicode[STIX]{x1D6FF}\unicode[STIX]{x1D703})^{2}}/\unicode[STIX]{x1D703}_{K}^{2}$ (Kolmogorov scaling) measured at a streamwise position $x/D=30$ on the jet axis. At this position, the ratio between the wire cutoff frequency and the Kolmogorov frequency is $f_{c}/f_{K}=0.7$ . The attenuation of the cold-wire signal thus needs to be compensated and this is particularly important at small scales. The compensation of the cold-wire response is performed using the numerical processing technique proposed by Lemay & Benaïssa (Reference Lemay and Benaïssa2001), Lemay et al. (Reference Lemay, Benaissa and Antonia2003). The open and closed symbols in figure 2 show respectively the uncompensated and compensated distributions of $\overline{(\unicode[STIX]{x1D6FF}\unicode[STIX]{x1D703})^{2}}/\unicode[STIX]{x1D703}_{K}^{2}$ obtained from the measurement of two distinct cold-wire probes separated by a distance $s$ along the longitudinal direction. The effect of the compensation procedure is clearly seen at small scales. The full line represents the distribution of compensated $\overline{(\unicode[STIX]{x1D6FF}\unicode[STIX]{x1D703})^{2}}/\unicode[STIX]{x1D703}_{K}^{2}$ inferred from a single cold-wire probe using Taylor’s hypothesis. The time increment $\unicode[STIX]{x1D70F}$ is converted to spatial separation $s$ using a constant convection velocity, constant for all scales, defined by the mean streamwise velocity on the jet axis ( $s=\unicode[STIX]{x1D70F}U_{0}$ ). The agreement between the full line and the closed symbols in figure 2 indicates that the use of Taylor’s hypothesis is thoroughly validated.

Hereinafter, the distributions of temperature structure functions measured at several positions along the jet axis ( $x/D=30$ to 60) are those obtained using Taylor’s hypothesis and compensated signals. For the locations $x/D=30$ , 35, 40, 45, 50, 55 and 60, the ratio of the cutoff to the Kolmogorov frequencies evolves respectively from $f_{c}/f_{K}=0.7$ , 0.9, 1.2, 1.4, 1.7, 2.1 to 2.4. Thus, the use of the compensation procedure is mandatory for $x/D=30$ , but it becomes less important as one moves downstream to $x/D=60$ . Nevertheless, the compensation technique is applied to the temperature signals for all the measurement locations.

Figure 3. Temperature structure functions $\overline{(\unicode[STIX]{x1D6FF}\unicode[STIX]{x1D703})^{2}}$ and $-\overline{(\unicode[STIX]{x1D6FF}\unicode[STIX]{x1D703})^{3}}$ on the jet axis at 7 streamwise locations extending from $x/D$ = 30 to 60, in steps of 5; normalization with $\unicode[STIX]{x1D703}_{K}$ and $\unicode[STIX]{x1D702}$ (top), $\overline{\unicode[STIX]{x1D703}^{2}}$ and $\unicode[STIX]{x1D706}$ (centre), $\unicode[STIX]{x1D6E9}_{0}$ and $R_{U}$ (bottom); flow conditions similar to those of Darisse et al. (Reference Darisse, Lemay and Benaïssa2013a ,Reference Darisse, Lemay and Benaïssa b , Reference Darisse, Lemay and Benaïssa2015).

Figure 3 shows the distribution of the normalized second- and third-order temperature structure functions measured on the jet axis in the range $30\leqslant x/D\leqslant 60$ . Three different sets of scaling variables are used for normalization, viz. Kolmogorov scales ( $\unicode[STIX]{x1D703}_{K}$ and $\unicode[STIX]{x1D702}$ ), Taylor scales ( $\overline{\unicode[STIX]{x1D703}^{2}}$ and $\unicode[STIX]{x1D706}$ ) and large scales ( $\unicode[STIX]{x1D6E9}_{0}$ and $R_{U}$ ). Regardless of the normalization, both the second- and third-order temperature structure functions present a very good collapse over the entire range of scales for all the positions in the range $x/D=30$ to 60. This indicates that complete self-preservation is satisfied for these positions on the jet axis, reinforcing confidence in the analytical development and results presented in § 2.1. Note that despite the moderately large value of $Re_{\unicode[STIX]{x1D706}}$ ( $=548$ ), $\overline{(\unicode[STIX]{x1D6FF}\unicode[STIX]{x1D703})^{2}}$ does not strictly present a power-law variation of the form $r^{\unicode[STIX]{x1D701}_{n}}$ in the scaling range. However, the figure suggests that increasing $Re_{\unicode[STIX]{x1D706}}$ would eventually lead to $\unicode[STIX]{x1D701}_{2}$ approaching the value ( $\unicode[STIX]{x1D701}_{2}=2/3$ ) predicted by Yaglom (Reference Yaglom1949). Note that nothing can be said about $-\overline{(\unicode[STIX]{x1D6FF}\unicode[STIX]{x1D703})^{3}}$ which should not be confused with $-\overline{\unicode[STIX]{x1D6FF}u(\unicode[STIX]{x1D6FF}\unicode[STIX]{x1D703})^{2}}$ for which the exponent $\unicode[STIX]{x1D701}_{3}=1$ if there is an inertial range. If $-\overline{(\unicode[STIX]{x1D6FF}\unicode[STIX]{x1D703})^{3}}$ is assumed to follow a power-law variation $r^{\unicode[STIX]{x1D701}}$ in the inertial range, its power-law exponent is yet to be determined. Interestingly, the data suggest that as $Re_{\unicode[STIX]{x1D706}}$ increases $\unicode[STIX]{x1D701}$ is likely to approach 1, a result which can be also observed in the data of Antonia & Van Atta (Reference Antonia and Van Atta1978) (see their figures 1 and 2).

2.6 One-point $\overline{\unicode[STIX]{x1D703}^{2}}/2$ budget

As previously pointed out, Darisse et al. (Reference Darisse, Lemay and Benaïssa2014) have shown that the one-point $\overline{\unicode[STIX]{x1D703}^{2}}/2$ budget on the jet axis is dominated by convection, production, the radial component of turbulent diffusion and dissipation. Their budget was evaluated in the self-preserving region since they showed that both $Re_{\unicode[STIX]{x1D706}}$ and $Pe_{\unicode[STIX]{x1D706}}$ were constant along the jet axis. Retaining only these predominant terms and recalling that outer scales $U_{0}$ , $\unicode[STIX]{x1D6E9}_{0}$ and $R_{U}$ are relevant self-preservation scales, the normalized one-point budget on the jet axis is written as (see Darisse et al. Reference Darisse, Lemay and Benaïssa2015, for details):

(2.40) $$\begin{eqnarray}\unicode[STIX]{x1D716}_{\unicode[STIX]{x1D703}}^{\ast }=\frac{\unicode[STIX]{x1D716}_{\unicode[STIX]{x1D703}}R_{U}}{U_{0}\unicode[STIX]{x1D6E9}_{0}^{2}}\simeq B_{R_{U}}\frac{\overline{\unicode[STIX]{x1D703}^{2}}}{\unicode[STIX]{x1D6E9}_{0}^{2}}+B_{R_{U}}\frac{\overline{u\unicode[STIX]{x1D703}}}{U_{0}\unicode[STIX]{x1D6E9}_{0}}-\frac{\text{d}}{\text{d}\unicode[STIX]{x1D709}}\left(\frac{\overline{v\unicode[STIX]{x1D703}^{2}}}{U_{0}\unicode[STIX]{x1D6E9}_{0}^{2}}\right)\simeq C_{\unicode[STIX]{x1D703}}^{\ast }+P_{\unicode[STIX]{x1D703}}^{\ast }+D_{\unicode[STIX]{x1D703}}^{\ast },\end{eqnarray}$$

where $\unicode[STIX]{x1D709}=r/R_{U}$ is the normalized radial coordinate. The power laws for $\unicode[STIX]{x1D716}_{\unicode[STIX]{x1D703}}$ , $R_{U}$ , $U_{0}$ and $\unicode[STIX]{x1D6E9}_{0}$ allow the dissipation term to be written as

(2.41) $$\begin{eqnarray}\unicode[STIX]{x1D716}_{\unicode[STIX]{x1D703}}^{\ast }=\frac{A_{\unicode[STIX]{x1D716}_{\unicode[STIX]{x1D703}}}B_{R_{U}}}{B_{U}B_{\unicode[STIX]{x1D6E9}}^{2}}.\end{eqnarray}$$

Introducing this expression and the definition of $B_{I}$ (2.36) into the normalized budget (2.40) and solving for $A_{\unicode[STIX]{x1D716}_{\unicode[STIX]{x1D703}}}$ yields

(2.42) $$\begin{eqnarray}A_{\unicode[STIX]{x1D716}_{\unicode[STIX]{x1D703}}}\simeq B_{U}B_{\unicode[STIX]{x1D6E9}}^{2}\left[B_{I}^{2}+\frac{\overline{u\unicode[STIX]{x1D703}}}{U_{0}\unicode[STIX]{x1D6E9}_{0}}-\frac{1}{B_{R_{U}}}\frac{\text{d}}{\text{d}\unicode[STIX]{x1D709}}\left(\frac{\overline{v\unicode[STIX]{x1D703}^{2}}}{U_{0}\unicode[STIX]{x1D6E9}_{0}^{2}}\right)\right].\end{eqnarray}$$

This expression for the prefactor of the $\unicode[STIX]{x1D716}_{\unicode[STIX]{x1D703}}$ power law (2.34) is more involved than its counterpart for $\unicode[STIX]{x1D716}_{k}$ , i.e. equation (1.2). The two mixed velocity–temperature moments accounting for the production and turbulent diffusion terms of the $\overline{\unicode[STIX]{x1D703}^{2}}/2$ budget add a level of complexity rendering this relation less attractive, from a modelling point of view, than the expression for $A_{\unicode[STIX]{x1D716}_{k}}$ . In the following section, we propose preliminary models for the longitudinal evolution of $\overline{u\unicode[STIX]{x1D703}}$ and $\overline{v\unicode[STIX]{x1D703}^{2}}$ along the jet axis. This will help us develop an ‘empirical’ expression for the prefactor $A_{\unicode[STIX]{x1D716}_{\unicode[STIX]{x1D703}}}$ that will account for the effect of the jet boundary conditions and that can be estimated from relatively simple measurements.

3.1 Longitudinal velocity–temperature correlation model

The correlation $\overline{u\unicode[STIX]{x1D703}}$ is modelled by assuming weak anisotropy of the temperature field in the self-preserving region of the jet. This hypothesis applied to the scalar flux underpins the development of the well-known algebraic second moment closure for the scalar transport (see for example Gibson & Launder Reference Gibson and Launder1976; Hanjalić & Launder Reference Hanjalić and Launder2011). In the present case, this means that the ratio $\overline{u\unicode[STIX]{x1D703}}/\sqrt{k\overline{\unicode[STIX]{x1D703}^{2}}}$ is expected to be constant. The approach can be simplified further by considering that the correlation coefficient $\unicode[STIX]{x1D70C}_{u\unicode[STIX]{x1D703}}=\overline{u\unicode[STIX]{x1D703}}/\sqrt{\overline{u^{2}}\;\overline{\unicode[STIX]{x1D703}^{2}}}$ is constant across the jet width. The data reported by Darisse et al. (Reference Darisse, Lemay and Benaïssa2015) ( $x/D=30$ ) and Chevray & Tutu (Reference Chevray and Tutu1978) ( $x/D=15$ ) indicate that $\unicode[STIX]{x1D70C}_{u\unicode[STIX]{x1D703}}(r)$ does not vary significantly ( $0.46<\unicode[STIX]{x1D70C}_{u\unicode[STIX]{x1D703}}<0.51$ ) around the centreline. Pietri, Amielh & Anselmet (Reference Pietri, Amielh and Anselmet2000) made the same observation in a coflowing jet exiting from a pipe. They also showed that, along the centreline, in the region $15\geqslant x/D\geqslant 28$ , the value of $c_{u\unicode[STIX]{x1D703}}$ ( $\equiv \unicode[STIX]{x1D70C}_{u\unicode[STIX]{x1D703}}$ at $r=0$ ) was almost constant. More precisely, $c_{u\unicode[STIX]{x1D703}}$ decreased slightly between 0.52 at $x/D=15$ and 0.49 at $x/D=28$ . Their measurements are consistent with the value of 0.48 found here. Using a combination of laser-induced fluorescence and laser Doppler anemometry (LDA) techniques in a slightly heated water jet, Lemoine et al. (Reference Lemoine, Antoine, Wolff and Lebouche1999) reported a measured value of $c_{u\unicode[STIX]{x1D703}}=0.48$ , which also confirms the validity of the present value. Finally, additional support is obtained with the results presented by Panchapakesan & Lumley (Reference Panchapakesan and Lumley1993b ) in a helium jet. From their reported longitudinal evolution of $\sqrt{\overline{u^{2}}}$ on the centreline, we can extrapolate this quantity up to the far non-buoyant region and find that, for this flow configuration too, $c_{u\unicode[STIX]{x1D703}}=0.48$ .

Based on these reported observations, the following model is adopted on the jet axis:

(3.1) $$\begin{eqnarray}\overline{u\unicode[STIX]{x1D703}}=c_{u\unicode[STIX]{x1D703}}\sqrt{\overline{u^{2}}\overline{\unicode[STIX]{x1D703}^{2}}}\quad \text{with }c_{u\unicode[STIX]{x1D703}}=\unicode[STIX]{x1D70C}_{u\unicode[STIX]{x1D703}}(r=0)=0.48.\end{eqnarray}$$

This expression is also a direct consequence of the self-preservation analysis performed on the sbs budget in § 2.1. It is shown that, along the jet axis, $\overline{u\unicode[STIX]{x1D703}}\propto \hat{x}^{-2}$ (see also expression (A 11)), $\sqrt{\overline{u^{2}}}\propto \hat{x}^{-1}$ and $\sqrt{\overline{\unicode[STIX]{x1D703}^{2}}}\propto \hat{x}^{-1}$ ; it immediately follows that $\unicode[STIX]{x1D70C}_{u\unicode[STIX]{x1D703}}$ is constant along the jet axis. Darisse et al. (Reference Darisse, Lemay and Benaïssa2015) showed that $Re_{\unicode[STIX]{x1D706}}$ and $Pe_{\unicode[STIX]{x1D706}}$ are constant along the jet axis, and concluded that complete self-preservation for the velocity and the scalar fields was satisfied. Thus, based on their measurements, $c_{u\unicode[STIX]{x1D703}}=0.48$ is adopted in (3.1) for $\overline{u\unicode[STIX]{x1D703}}$ . Introducing the previously defined coefficients terms $A_{I}$ and $B_{I}$ , the model for the normalized mixed correlation is written as

(3.2) $$\begin{eqnarray}\frac{\overline{u\unicode[STIX]{x1D703}}}{U_{0}\unicode[STIX]{x1D6E9}_{0}}=c_{u\unicode[STIX]{x1D703}}A_{I}B_{I}\quad \text{with }c_{u\unicode[STIX]{x1D703}}=0.48.\end{eqnarray}$$

3.2 Turbulent diffusion model

Using the transport equation of $\overline{u_{i}\unicode[STIX]{x1D703}^{2}}$ in which the convective transport was neglected, Dekeyser & Launder (Reference Dekeyser, Launder, Bradbury, Durst, Launder, Schmidt and Whitelaw1983) proposed a model based on a Gaussian approximation for the fourth-order moments and a return-to-isotropy representation for pressure interactions:

(3.3) $$\begin{eqnarray}\overline{u_{i}\unicode[STIX]{x1D703}^{2}}=-c_{\unicode[STIX]{x1D703}}\frac{k}{\unicode[STIX]{x1D716}_{k}}\left(\overline{u_{i}u_{j}}\frac{\unicode[STIX]{x2202}\overline{\unicode[STIX]{x1D703}^{2}}}{\unicode[STIX]{x2202}x_{j}}+2\overline{u_{i}\unicode[STIX]{x1D703}}\frac{\unicode[STIX]{x2202}\overline{u_{j}\unicode[STIX]{x1D703}}}{\unicode[STIX]{x2202}x_{j}}\right)\quad \text{with }c_{\unicode[STIX]{x1D703}}=0.11.\end{eqnarray}$$

When the last term on the right-hand side of relation (3.3) is omitted, the truncated expression represents the simple gradient-type model widely used in numerical calculations. As reported by Dekeyser & Launder (Reference Dekeyser, Launder, Bradbury, Durst, Launder, Schmidt and Whitelaw1983), this truncation is usually made to facilitate numerical solutions. They, however, mentioned that the complete form was in better agreement with their measurements. In the present case, we are interested in the radial component of (3.3) which is explicitly expressed as

(3.4) $$\begin{eqnarray}\overline{v\unicode[STIX]{x1D703}^{2}}=-c_{\unicode[STIX]{x1D703}}\frac{k}{\unicode[STIX]{x1D716}_{k}}\left[\overline{uv}\frac{\unicode[STIX]{x2202}\overline{\unicode[STIX]{x1D703}^{2}}}{\unicode[STIX]{x2202}x}+\overline{v^{2}}\frac{\unicode[STIX]{x2202}\overline{\unicode[STIX]{x1D703}^{2}}}{\unicode[STIX]{x2202}r}+2\overline{v\unicode[STIX]{x1D703}}\left(\frac{\unicode[STIX]{x2202}\overline{u\unicode[STIX]{x1D703}}}{\unicode[STIX]{x2202}x}+\frac{\unicode[STIX]{x2202}\overline{v\unicode[STIX]{x1D703}}}{\unicode[STIX]{x2202}r}\right)\right].\end{eqnarray}$$

The terms involving longitudinal derivatives are relatively small compared to the two other terms within the brackets, particularly in the mid and far fields. The measurements reported by Darisse et al. (Reference Darisse, Lemay and Benaïssa2015) indicate that the longitudinal derivative terms represent approximately $-16\,\%$ of the radial derivative contributions. To simplify the model, the longitudinal derivative terms are omitted and, to compensate for the effect of this omission, the value of $c_{\unicode[STIX]{x1D703}}$ is reduced by 16 % to 0.095. This yields a simplified model:

(3.5) $$\begin{eqnarray}\overline{v\unicode[STIX]{x1D703}^{2}}\simeq -c_{\unicode[STIX]{x1D703}}\frac{k}{\unicode[STIX]{x1D716}_{k}}\left[\overline{v^{2}}\frac{\unicode[STIX]{x2202}\overline{\unicode[STIX]{x1D703}^{2}}}{\unicode[STIX]{x2202}r}+2\overline{v\unicode[STIX]{x1D703}}\frac{\unicode[STIX]{x2202}\overline{v\unicode[STIX]{x1D703}}}{\unicode[STIX]{x2202}r}\right].\end{eqnarray}$$

Figure 4. Radial distributions of normalized velocity–temperature moments near the jet axis; right side, correlation coefficient $\unicode[STIX]{x1D70C}_{u\unicode[STIX]{x1D703}}$ : Darisse et al. (Reference Darisse, Lemay and Benaïssa2015) (black diamonds), Chevray & Tutu (Reference Chevray and Tutu1978) (blue squares) and model defined by (3.1) (red dashed line); left scale, $\overline{v\unicode[STIX]{x1D703}^{2}}/(U_{0}\unicode[STIX]{x1D6E9}_{0}^{2})$ : Darisse et al. (Reference Darisse, Lemay and Benaïssa2015) (black circles) with best fit (black line), model of Dekeyser & Launder (Reference Dekeyser, Launder, Bradbury, Durst, Launder, Schmidt and Whitelaw1983), equation (3.4), (green line) and simplified model defined by (3.5) (red dashed line).

The database of Darisse et al. (Reference Darisse, Lemay and Benaïssa2015) is used here to test the validity of these models on the jet axis. All the radial distributions of the measured quantities reported by these authors have been fitted with least-square regressions using the mathematical expressions given in Hussein, Capp & George (Reference Hussein, Capp and George1994). Near the jet axis ( $r/R_{U}\leqslant 0.5$ ), figure 4 indicates that both, relations (3.4) and (3.5) are in good agreement with the measurements of $\overline{v\unicode[STIX]{x1D703}^{2}}/(U_{0}\unicode[STIX]{x1D6E9}_{0}^{2})$ . Moreover, very close to the centreline (around $r/R_{U}\leqslant 0.2$ ), simplified and Dekeyser–Launder models are almost perfectly equivalent. As the present development focuses on the evolution of different quantities on the centreline, the simplified model (3.5) is used. The radial derivative of $\overline{v\unicode[STIX]{x1D703}^{2}}$ , which represents the dominant part of the turbulent diffusion of $\overline{\unicode[STIX]{x1D703}^{2}}/2$ on the jet axis (where $\overline{v\unicode[STIX]{x1D703}}\equiv 0$ and $\unicode[STIX]{x2202}\overline{\unicode[STIX]{x1D703}^{2}}/\unicode[STIX]{x2202}r\equiv 0$ on the axis by virtue of axisymmetry), is written as

(3.6) $$\begin{eqnarray}\frac{\unicode[STIX]{x2202}\overline{v\unicode[STIX]{x1D703}^{2}}}{\unicode[STIX]{x2202}r}\simeq -c_{\unicode[STIX]{x1D703}}\frac{k}{\unicode[STIX]{x1D716}_{k}}\left[\overline{v^{2}}\frac{\unicode[STIX]{x2202}^{2}\overline{\unicode[STIX]{x1D703}^{2}}}{\unicode[STIX]{x2202}r^{2}}+2\left(\frac{\unicode[STIX]{x2202}\overline{v\unicode[STIX]{x1D703}}}{\unicode[STIX]{x2202}r}\right)^{2}\right].\end{eqnarray}$$

The data of Darisse et al. (Reference Darisse, Lemay and Benaïssa2015) reveal that the two terms within brackets are approximately equal. Thus, we can write (3.6) as follows

(3.7) $$\begin{eqnarray}\frac{\unicode[STIX]{x2202}\overline{v\unicode[STIX]{x1D703}^{2}}}{\unicode[STIX]{x2202}r}\simeq -4c_{\unicode[STIX]{x1D703}}\frac{k}{\unicode[STIX]{x1D716}_{k}}\left(\frac{\unicode[STIX]{x2202}\overline{v\unicode[STIX]{x1D703}}}{\unicode[STIX]{x2202}r}\right)^{2}.\end{eqnarray}$$

The radial gradient of $\overline{v\unicode[STIX]{x1D703}}$ on the jet axis can be obtained directly from the mean temperature equation, after neglecting $\unicode[STIX]{x2202}\overline{u\unicode[STIX]{x1D703}}/\unicode[STIX]{x2202}x$ and the molecular diffusion term, viz.

(3.8) $$\begin{eqnarray}\frac{\unicode[STIX]{x2202}\overline{v\unicode[STIX]{x1D703}}}{\unicode[STIX]{x2202}r}\simeq -\frac{1}{2}U_{0}\frac{\text{d}\unicode[STIX]{x1D6E9}_{0}}{\text{d}x}.\end{eqnarray}$$

The combination of (3.7) and (3.8) yields, on the jet axis, a model for the turbulent diffusion of $\overline{\unicode[STIX]{x1D703}^{2}}/2$

(3.9) $$\begin{eqnarray}\frac{\unicode[STIX]{x2202}\overline{v\unicode[STIX]{x1D703}^{2}}}{\unicode[STIX]{x2202}r}\simeq -c_{\unicode[STIX]{x1D703}}\frac{k}{\unicode[STIX]{x1D716}_{k}}U_{0}^{2}\left(\frac{\text{d}\unicode[STIX]{x1D6E9}_{0}}{\text{d}x}\right)^{2}\quad \text{with }c_{\unicode[STIX]{x1D703}}=0.095.\end{eqnarray}$$

Using relation (A 17) for the streamwise gradient of $\unicode[STIX]{x1D6E9}_{0}$ , the normalized form of this model is

(3.10) $$\begin{eqnarray}\frac{\text{d}}{\text{d}\unicode[STIX]{x1D709}}\left(\frac{\overline{v\unicode[STIX]{x1D703}^{2}}}{U_{0}\unicode[STIX]{x1D6E9}_{0}^{2}}\right)\simeq -c_{\unicode[STIX]{x1D703}}B_{R_{U}}^{2}\frac{k}{U_{0}^{2}}\frac{U_{0}^{3}}{\unicode[STIX]{x1D716}_{k}R_{U}}.\end{eqnarray}$$

Introducing the notation $k/U_{0}^{2}=A_{I}^{2}({\mathcal{R}}+1/2)$ and relation (1.3), which represents $\unicode[STIX]{x1D716}_{k}^{\ast }$ , yields the final form of the normalized turbulent diffusion model:

(3.11) $$\begin{eqnarray}\frac{\text{d}}{\text{d}\unicode[STIX]{x1D709}}\left(\frac{\overline{v\unicode[STIX]{x1D703}^{2}}}{U_{0}\unicode[STIX]{x1D6E9}_{0}^{2}}\right)\simeq -c_{\unicode[STIX]{x1D703}}B_{R_{U}}\frac{({\mathcal{R}}+1/2)}{({\mathcal{R}}+2)}\quad \text{with }c_{\unicode[STIX]{x1D703}}=0.095.\end{eqnarray}$$

4.1 Mean dissipation rate of $\overline{\unicode[STIX]{x1D703}^{2}}/2$

The models for the mixed correlation (3.2) and the turbulent diffusion (3.11) introduced into relation (2.42) allow an empirical expression for the prefactor $A_{\unicode[STIX]{x1D716}_{\unicode[STIX]{x1D703}}}$ to be obtained:

(4.1) $$\begin{eqnarray}A_{\unicode[STIX]{x1D716}_{\unicode[STIX]{x1D703}}}\simeq B_{U}B_{\unicode[STIX]{x1D703}}^{2}\left[B_{I}^{2}+c_{u\unicode[STIX]{x1D703}}A_{I}B_{I}+c_{\unicode[STIX]{x1D703}}\left(\frac{{\mathcal{R}}+1/2}{{\mathcal{R}}+2}\right)\right],\end{eqnarray}$$

with $c_{u\unicode[STIX]{x1D703}}=0.48$ , $c_{\unicode[STIX]{x1D703}}=0.095$ . Using the data of Darisse et al. (Reference Darisse, Lemay and Benaïssa2015) expression (4.1) yields $A_{\unicode[STIX]{x1D716}_{\unicode[STIX]{x1D703}}}=15.02$ . This is in very good agreement with the value $A_{\unicode[STIX]{x1D716}_{\unicode[STIX]{x1D703}}}=14.58$ obtained from the measurements presented in figure 1. It is important to note that the coefficient $A_{\unicode[STIX]{x1D716}_{\unicode[STIX]{x1D703}}}=14.58$ was obtained from the power law of $\unicode[STIX]{x1D716}_{\unicode[STIX]{x1D703}}$ shown in figure 1. The values of $\unicode[STIX]{x1D716}_{\unicode[STIX]{x1D703}}$ were estimated from the integral of the dissipation spectra of temperature fluctuations, under the assumption of local isotropy and Taylor’s hypothesis. It was shown by Darisse et al. (Reference Darisse, Lemay and Benaïssa2015) that, at $\hat{x}=30$ , the isotropic estimate of $\unicode[STIX]{x1D716}_{\unicode[STIX]{x1D703}}$ on the jet centreline was in very good agreement with the value inferred from the balance of the temperature budget. Therefore, this value of 14.58 was evaluated from a dataset different than the one used to evaluate $A_{\unicode[STIX]{x1D716}_{\unicode[STIX]{x1D703}}}=15.02$ with expression (4.1). This guarantees that the validation of the empirical expression is made using an independent test.

Table 2. Estimates obtained from the simple empirical expressions compared with the results presented by Darisse et al. (Reference Darisse, Lemay and Benaïssa2015) (DLB, measurements in a heated jet), Antonia & Mi (Reference Antonia and Mi1993) (AM, measurements in a heated jet), Panchapakesan & Lumley (Reference Panchapakesan and Lumley1993b ) (PL, measurements in the non-buoyant region of a helium jet) and Ruffin et al. (Reference Ruffin, Schiestel, Anselmet, Amielh and Fulachier1994) (RAL, simulations using second-order turbulence models with a passive scalar).

After inserting (4.1) into (2.41), $\unicode[STIX]{x1D716}_{\unicode[STIX]{x1D703}}$ normalized with local outer scales, $R_{U}$ , $U_{0}$ and $\unicode[STIX]{x1D6E9}_{0}$ , can be expressed as:

(4.2) $$\begin{eqnarray}\unicode[STIX]{x1D716}_{\unicode[STIX]{x1D703}}^{\ast }\simeq B_{R_{U}}\left[B_{I}^{2}+c_{u\unicode[STIX]{x1D703}}A_{I}B_{I}+c_{\unicode[STIX]{x1D703}}\left(\frac{{\mathcal{R}}+1/2}{{\mathcal{R}}+2}\right)\right].\end{eqnarray}$$

By virtue of (2.40), expression (4.2) also defines a simple empirical expression for the dissipation of $\overline{\unicode[STIX]{x1D703}^{2}}/2$ on the jet axis. The validity of this empirical expression for $\unicode[STIX]{x1D716}_{\unicode[STIX]{x1D703}}^{\ast }$ has been tested using four datasets (see table 2). We have to point out that very few datasets in the literature provide measurements for $\unicode[STIX]{x1D716}_{\unicode[STIX]{x1D703}}^{\ast }$ in a well-established self-similar region of a slightly heated turbulent round jet. As already mentioned, Darisse et al. (Reference Darisse, Lemay and Benaïssa2015) reported such measurements (inferred from the $\overline{\unicode[STIX]{x1D703}^{2}}/2$ budget and locally isotropic estimates). In their case, the outcome of the empirical expression (4.2) differs by only $-0.3\,\%$ from their measurement. Antonia & Mi (Reference Antonia and Mi1993) presented direct measurements and locally isotropic estimates for $\unicode[STIX]{x1D716}_{\unicode[STIX]{x1D703}}^{\ast }$ (which were in agreement on the jet centreline) in a slightly heated round jet in the region $\hat{x}=15$ . The outcome of relation (4.2) differs by $+5\,\%$ from their measurement. This is in rather good agreement considering that, in their case, self-similarity is probably not completely achieved at $\hat{x}=15$ . The third dataset was provided by Panchapakesan & Lumley (Reference Panchapakesan and Lumley1993b ) which reported measurements (inferred from the budget of the variance of concentration fluctuations) in the non-buoyant region of a helium round jet. Considering that this flow field shows some analogies with the present case, it is interesting to observe that the outcome of relation (4.2) differs by only $+5.1\,\%$ from their measurement. Finally, the Reynolds-averaged Navier–Stokes (RANS) simulations reported by Ruffin et al. (Reference Ruffin, Schiestel, Anselmet, Amielh and Fulachier1994) (air jet with a passive contaminant) were used to provide a computational fluid dynamics (CFD) test case. The value of  $\unicode[STIX]{x1D716}_{\unicode[STIX]{x1D703}}^{\ast }$ calculated with expression (4.2) differs by $-12.5\,\%$ from their simulation on the jet centreline. Considering that RANS simulations of turbulent round jets are usually not in perfect concordance with the measurements (as reported by these authors), the present level of agreement is rather satisfactory.

4.2 Turbulent Péclet number

It was shown in § 2.3 that the turbulent Péclet number $Pe_{\unicode[STIX]{x1D706}}$ , which can be expressed as

(4.3) $$\begin{eqnarray}Pe_{\unicode[STIX]{x1D706}}=\sqrt{\frac{3\overline{u^{2}}\overline{\unicode[STIX]{x1D703}^{2}}}{\unicode[STIX]{x1D6FC}\unicode[STIX]{x1D716}_{\unicode[STIX]{x1D703}}}},\end{eqnarray}$$

is constant along the jet axis. The power laws for $\overline{u^{2}}$ , $\overline{\unicode[STIX]{x1D703}^{2}}$ and $\unicode[STIX]{x1D716}_{\unicode[STIX]{x1D703}}$ are now introduced in order to express $Pe_{\unicode[STIX]{x1D706}}$ as follows

(4.4) $$\begin{eqnarray}Pe_{\unicode[STIX]{x1D706}}=A_{I}B_{I}B_{U}B_{\unicode[STIX]{x1D6E9}}\sqrt{\frac{3Re_{D}Pr}{A_{\unicode[STIX]{x1D716}_{\unicode[STIX]{x1D703}}}}},\end{eqnarray}$$

where $Pr\equiv \unicode[STIX]{x1D6FC}/\unicode[STIX]{x1D708}$ , the Prandtl number, is considered to be equal to 0.71. Using (4.1), the final expression for $Pe_{\unicode[STIX]{x1D706}}$ is

(4.5) $$\begin{eqnarray}Pe_{\unicode[STIX]{x1D706}}=A_{Pe}\sqrt{Re_{D}Pr}=\left[A_{I}B_{I}\sqrt{\frac{3B_{U}}{B_{I}^{2}+c_{u\unicode[STIX]{x1D703}}A_{I}B_{I}+c_{\unicode[STIX]{x1D703}}{\displaystyle \frac{({\mathcal{R}}+1/2)}{({\mathcal{R}}+2)}}}}\right]\sqrt{Re_{D}Pr}.\end{eqnarray}$$

Since the prefactor $A_{Pe}$ (term in brackets), depends on $B_{U}$ , $A_{I}$ , $B_{I}$ and ${\mathcal{R}}$ , a set of parameters influenced by the jet boundary conditions, its value is likely to depend on the initial/boundary conditions. However, equation (4.5) gives $A_{Pe}=0.66$ and 0.71 for the Darisse et al. (Reference Darisse, Lemay and Benaïssa2015) and Antonia & Mi (Reference Antonia and Mi1993) data respectively. It appears that the boundary conditions do not severely impact this parameter. Thus, it seems reasonable to use an average – and constant – value of $A_{Pe}=0.68$ . With this assumption, relation (4.5) yields $Pe_{\unicode[STIX]{x1D706}}=211$ for Darisse et al. (Reference Darisse, Lemay and Benaïssa2015) and 79 for Antonia & Mi (Reference Antonia and Mi1993), which results are in good agreement with the measurements.

4.3 Thermal-to-mechanical time scale ratio

The thermal-to-mechanical time scale ratio is defined as

(4.6) $$\begin{eqnarray}R=\frac{\overline{\unicode[STIX]{x1D703}^{2}}/(2\unicode[STIX]{x1D716}_{\unicode[STIX]{x1D703}})}{k/\unicode[STIX]{x1D716}_{k}}.\end{eqnarray}$$

Using the different power laws derived here for the physical quantities involved in this expression allows the following simple model on the jet axis to be written as

(4.7) $$\begin{eqnarray}R\simeq \frac{B_{I}^{2}({\mathcal{R}}+2)}{\left(B_{I}^{2}+c_{u\unicode[STIX]{x1D703}}A_{I}B_{I}+c_{\unicode[STIX]{x1D703}}{\displaystyle \frac{({\mathcal{R}}+1/2)}{({\mathcal{R}}+2)}}\right)(2{\mathcal{R}}+1)}.\end{eqnarray}$$

The value reported by Darisse et al. (Reference Darisse, Lemay and Benaïssa2015), which is inferred from the budgets of $k$ and $\overline{\unicode[STIX]{x1D703}^{2}}/2$ measured in the same flow conditions, was $R=0.41$ on the jet axis. Introducing the parameters listed in table 2 for Darisse et al. (Reference Darisse, Lemay and Benaïssa2015), the simple model (4.7) gives $R\simeq 0.40$ , which is in good agreement with the measurements.