2 Equations for the trace of structure functions and dissipation source terms at the fourth-order level

With the procedure used by Hill (Reference Hill2001), we have derived equations for successive source terms appearing in the equations for structure functions up to the sixth order under the assumptions of homogeneity and isotropy. The idea is that the source terms determine the solution of the structure function equations, i.e. the structure functions themselves. In the following, we present the equations used in the next sections. The entire set of successive equations (up to the sixth order) is documented in the archive material available from http://arxiv.org/abs/1504.07490. While it is possible to carry out the same analysis at higher orders, both the number of equations and the number of terms to be analysed increase very quickly; for that reason we consider only the fourth order here. We will look at the fourth-order equations in the viscous range in § 3, while the inertial-range assumptions are used in § 4. We have also performed DNS in a cubic box for Taylor-based Reynolds numbers up to 529 and for some parts of the analysis up to $Re_{\unicode[STIX]{x1D706}}=754$ . This is briefly described in appendix A.

The two points at which velocity fluctuations are sampled are denoted as $\boldsymbol{x}=(x_{1},x_{2},x_{3})$ and $\boldsymbol{x}^{\prime }=(x_{1}^{\prime },x_{2}^{\prime },x_{3}^{\prime })$ , with $x_{1}$ pointing in the longitudinal direction and $x_{2}$ and $x_{3}$ in the transverse direction. Structure functions are defined as $S_{p,q}=\langle (\unicode[STIX]{x0394}u_{1})^{p}(\unicode[STIX]{x0394}u_{2})^{q}\rangle$ , with $\unicode[STIX]{x0394}u_{1}=u_{1}-u_{1}^{\prime }$ being the longitudinal and $\unicode[STIX]{x0394}u_{2}=u_{2}-u_{2}^{\prime }$ the transverse velocity difference, with $u_{1}$ and $u_{2}$ taken at $x_{1}$ and $u_{1}^{\prime }$ and $u_{2}^{\prime }$ at $x_{1}^{\prime }$ . For the $n$ th-order moment, $n=p+q$ in these equations. Hence, at the fourth order, $S_{4,0}$ is the longitudinal, $S_{2,2}$ is the mixed and $S_{0,4}$ is the transverse structure function. In the following, Einstein’s summation convention is used, which implies summation over indices appearing twice. For instance, $\langle \unicode[STIX]{x1D700}_{ij}^{2}\rangle =\langle \unicode[STIX]{x1D700}_{ij}\unicode[STIX]{x1D700}_{ij}\rangle$ , where the sum is over $i$ and $j$ . The equations for $S_{4,0}$ , $S_{2,2}$ and $S_{0,4}$ are given in appendix B. In this paper, we examine the trace of the structure function tensor

(2.1) $$\begin{eqnarray}\mathit{S}_{[4]}=\langle \unicode[STIX]{x0394}u_{i}\unicode[STIX]{x0394}u_{i}\unicode[STIX]{x0394}u_{j}\unicode[STIX]{x0394}u_{j}\rangle ,\end{eqnarray}$$

and the same analysis can be carried over to the individual equations. As

(2.2) $$\begin{eqnarray}\left.\begin{array}{@{}c@{}}\langle (\unicode[STIX]{x0394}u_{1})^{2}(\unicode[STIX]{x0394}u_{2})^{2}\rangle =\langle (\unicode[STIX]{x0394}u_{1})^{2}(\unicode[STIX]{x0394}u_{3})^{2}\rangle ,\\ \langle (\unicode[STIX]{x0394}u_{2})^{4}\rangle =3\langle (\unicode[STIX]{x0394}u_{2})^{2}(\unicode[STIX]{x0394}u_{3})^{2}\rangle =\langle (\unicode[STIX]{x0394}u_{3})^{4}\rangle ,\end{array}\right\}\end{eqnarray}$$

cf. Hill (Reference Hill2001), the trace amounts to the sum

(2.3) $$\begin{eqnarray}\mathit{S}_{[4]}=S_{4,0}+4S_{2,2}+{\textstyle \frac{8}{3}}S_{0,4}.\end{eqnarray}$$

Consequently, the transport equation for $\mathit{S}_{[4]}$ is derived by adding the individual structure function equations (B 1)–(B 3) as given in appendix B multiplied by the corresponding prefactor. This results in

(2.4) $$\begin{eqnarray}\frac{\unicode[STIX]{x2202}\mathit{S}_{[4]}}{\unicode[STIX]{x2202}t}+\frac{\unicode[STIX]{x2202}}{\unicode[STIX]{x2202}r_{n}}\langle \unicode[STIX]{x0394}u_{n}(\unicode[STIX]{x0394}u_{i})^{2}(\unicode[STIX]{x0394}u_{j})^{2}\rangle =2\unicode[STIX]{x1D708}\frac{\unicode[STIX]{x2202}^{2}\mathit{S}_{[4]}}{\unicode[STIX]{x2202}r_{n}^{2}}-\langle T_{[4]}\rangle -\langle E_{[4]}\rangle .\end{eqnarray}$$

Here, the second term on the left-hand side is the transport term containing the fifth-order longitudinal, mixed and transverse structure functions $S_{5,0}$ , $S_{3,2}$ and $S_{1,4}$ respectively. The first term on the right-hand side of (2.4) is the viscous term containing the fourth-order structure functions $S_{4,0}$ , $S_{2,2}$ and $S_{0,4}$ . The remaining two terms in the equation are the pressure source term and the dissipative source term in that order. Equation (2.4) is closely related to equation (4.3) of Falkovich et al. (Reference Falkovich, Fouxon and Oz2010) (also a fourth-order equation), which they derived from the kinetic energy equation. From this, they found a new relation for the pressure–velocity correlation in the inertial range, similar to Kolmogorov’s $4/5$ law stemming from the second order.

In explicit notation, the transport term is written as

(2.5) $$\begin{eqnarray}\displaystyle \frac{\unicode[STIX]{x2202}}{\unicode[STIX]{x2202}r_{n}}\langle \unicode[STIX]{x0394}u_{n}(\unicode[STIX]{x0394}u_{i})^{2}(\unicode[STIX]{x0394}u_{j})^{2}\rangle & = & \displaystyle \frac{\unicode[STIX]{x2202}S_{5,0}}{\unicode[STIX]{x2202}r}+4\frac{\unicode[STIX]{x2202}S_{3,2}}{\unicode[STIX]{x2202}r}+\frac{8}{3}\frac{\unicode[STIX]{x2202}S_{1,4}}{\unicode[STIX]{x2202}r}+\frac{2}{r}S_{5,0}+\frac{8}{r}S_{3,2}-\frac{16}{3r}S_{1,4}\nonumber\\ \displaystyle & = & \displaystyle \frac{\unicode[STIX]{x2202}\mathit{S}_{[5]}}{\unicode[STIX]{x2202}r}+\frac{2}{r}\mathit{S}_{[5]},\end{eqnarray}$$

where $\mathit{S}_{[5]}$ is the trace of the general fifth-order structure function tensor,

(2.6) $$\begin{eqnarray}\mathit{S}_{[5]}=\langle \unicode[STIX]{x0394}u_{1}(\unicode[STIX]{x0394}u_{i})^{2}(\unicode[STIX]{x0394}u_{j})^{2}\rangle =S_{5,0}+4S_{3,2}+{\textstyle \frac{8}{3}}S_{1,4}.\end{eqnarray}$$

The pressure source term is given by

(2.7) $$\begin{eqnarray}\displaystyle \langle T_{[4]}\rangle & = & \displaystyle 4\langle (\unicode[STIX]{x0394}u_{i})^{2}\unicode[STIX]{x0394}u_{j}\unicode[STIX]{x0394}P_{j}\rangle \nonumber\\ \displaystyle & = & \displaystyle 4\langle (\unicode[STIX]{x0394}u_{1})^{3}\unicode[STIX]{x0394}P_{1}\rangle +8\langle (\unicode[STIX]{x0394}u_{2})^{2}\unicode[STIX]{x0394}u_{1}\unicode[STIX]{x0394}P_{1}+(\unicode[STIX]{x0394}u_{1})^{2}\unicode[STIX]{x0394}u_{2}\unicode[STIX]{x0394}P_{2}\rangle \nonumber\\ \displaystyle & & \displaystyle +\,{\textstyle \frac{32}{3}}\langle (\unicode[STIX]{x0394}u_{2})^{3}\unicode[STIX]{x0394}P_{2}\rangle ,\end{eqnarray}$$

the dissipative source term by

(2.8) $$\begin{eqnarray}\displaystyle \langle E_{[4]}\rangle & = & \displaystyle \langle 2(\unicode[STIX]{x0394}u_{i})^{2}(\unicode[STIX]{x1D700}_{jj}+\unicode[STIX]{x1D700}_{jj}^{\prime })+4\unicode[STIX]{x0394}u_{i}\unicode[STIX]{x0394}u_{j}(\unicode[STIX]{x1D700}_{ij}+\unicode[STIX]{x1D700}_{ij}^{\prime })\rangle \nonumber\\ \displaystyle & = & \displaystyle 6\langle (\unicode[STIX]{x0394}u_{1})^{2}(\unicode[STIX]{x1D700}_{11}+\unicode[STIX]{x1D700}_{11}^{\prime })\rangle +4\langle (\unicode[STIX]{x0394}u_{2})^{2}(\unicode[STIX]{x1D700}_{11}+\unicode[STIX]{x1D700}_{11}^{\prime })+(\unicode[STIX]{x0394}u_{1})^{2}(\unicode[STIX]{x1D700}_{22}+\unicode[STIX]{x1D700}_{22}^{\prime })\rangle \nonumber\\ \displaystyle & & \displaystyle +\,16\langle \unicode[STIX]{x0394}u_{1}\unicode[STIX]{x0394}u_{2}(\unicode[STIX]{x1D700}_{12}+\unicode[STIX]{x1D700}_{12}^{\prime })+(\unicode[STIX]{x0394}u_{2})^{2}(\unicode[STIX]{x1D700}_{22}+\unicode[STIX]{x1D700}_{22}^{\prime })\rangle\end{eqnarray}$$

and the viscous term by

(2.9) $$\begin{eqnarray}\displaystyle 2\unicode[STIX]{x1D708}\frac{\unicode[STIX]{x2202}^{2}\mathit{S}_{[4]}}{\unicode[STIX]{x2202}r_{n}^{2}} & = & \displaystyle 2\unicode[STIX]{x1D708}\left[\frac{\unicode[STIX]{x2202}^{2}S_{4,0}}{\unicode[STIX]{x2202}r^{2}}+\frac{2}{r}\frac{\unicode[STIX]{x2202}S_{4,0}}{\unicode[STIX]{x2202}r}+4\frac{\unicode[STIX]{x2202}^{2}S_{2,2}}{\unicode[STIX]{x2202}r^{2}}+\frac{8}{r}\frac{\unicode[STIX]{x2202}S_{2,2}}{\unicode[STIX]{x2202}r}+\frac{8}{3}\frac{\unicode[STIX]{x2202}^{2}S_{0,4}}{\unicode[STIX]{x2202}r^{2}}+\frac{16}{3r}\frac{\unicode[STIX]{x2202}S_{0,4}}{\unicode[STIX]{x2202}r}\right]\nonumber\\ \displaystyle & = & \displaystyle 2\unicode[STIX]{x1D708}\left[\frac{\unicode[STIX]{x2202}^{2}\mathit{S}_{[4]}}{\unicode[STIX]{x2202}r^{2}}+\frac{2}{r}\frac{\unicode[STIX]{x2202}\mathit{S}_{[4]}}{\unicode[STIX]{x2202}r}\right].\end{eqnarray}$$

In (2.4)–(2.9), the definitions

(2.10) $$\begin{eqnarray}\displaystyle & \displaystyle \unicode[STIX]{x0394}P_{i}=\frac{\unicode[STIX]{x2202}p}{\unicode[STIX]{x2202}x_{i}}-\frac{\unicode[STIX]{x2202}p^{\prime }}{\unicode[STIX]{x2202}x_{i}^{\prime }}, & \displaystyle\end{eqnarray}$$

(2.11) $$\begin{eqnarray}\displaystyle & \displaystyle \unicode[STIX]{x1D700}_{ij}=2\unicode[STIX]{x1D708}\frac{\unicode[STIX]{x2202}u_{i}}{\unicode[STIX]{x2202}x_{k}}\frac{\unicode[STIX]{x2202}u_{j}}{\unicode[STIX]{x2202}x_{k}} & \displaystyle\end{eqnarray}$$

have been used.

The balance of the different terms in (2.4) is shown in figure 1(a) for the case $R0$ ( $\mathit{Re}_{\unicode[STIX]{x1D706}}=88$ ) and in figure 1(b) for the case $R5$ ( $\mathit{Re}_{\unicode[STIX]{x1D706}}=529$ ) of our DNS. We will need this balance to estimate which terms we may neglect when we integrate (2.4) in the viscous range in § 3 and in the inertial range in § 4 below. The terms have been normalized with $(\unicode[STIX]{x1D708}^{2}\langle \unicode[STIX]{x1D700}^{2}\rangle ^{3})^{1/4}$ and the separation distance $r$ with the Kolmogorov scale $\unicode[STIX]{x1D702}$ . It is seen that in the viscous range for values of $r/\unicode[STIX]{x1D702}$ up to approximately 5, the transport term and the pressure source term are an order of magnitude smaller than the viscous terms and the dissipation source term for the six data sets $R0$ – $R5$ ( $R1$ – $R4$ are not shown here). These terms therefore balance to leading order, which will lead to exact relations in the viscous range, as will be shown in § 3. As expected, the inertial range increases with increasing Reynolds number: beginning at $r/\unicode[STIX]{x1D702}=30$ , for the case $R0$ ( $Re_{\unicode[STIX]{x1D706}}=88$ ) only a very small inertial range is identifiable, whereas for $R5$ ( $Re_{\unicode[STIX]{x1D706}}=529$ ) it extends to $r/\unicode[STIX]{x1D702}\approx 300$ . In the inertial range and for all data sets, the pressure source term is smaller by a factor of four on average and the viscous term is much smaller. This order-of-magnitude estimate will be used in § 4.

Figure 1. Balance of the different terms in (2.4) for the cases $R0$ ( $\mathit{Re}_{\unicode[STIX]{x1D706}}=88$ ) (a) and $R5$ ( $\mathit{Re}_{\unicode[STIX]{x1D706}}=529$ ) (b): ○, transport term; ▫, dissipative source term; ▿, pressure source term; ▵, viscous term.

The dissipation term $\langle E_{[4]}\rangle$ defined in (2.8) is a correlation between squared velocity increments $\unicode[STIX]{x0394}u_{1}$ and $\unicode[STIX]{x0394}u_{2}$ and the instantaneous dissipations defined by (2.11) at $x_{1}$ and likewise at $x_{1}^{\prime }$ . Since it balances the transport term in the inertial range and the viscous term in the viscous range, $\langle E_{[4]}\rangle$ is of particular interest and will be further examined in the following. Furthermore, one might expect the dissipative source term to contain dissipative fluctuations. The equation for the dissipative source term,

(2.12) $$\begin{eqnarray}\langle E_{[4]}\rangle =\langle E_{4,0}\rangle +4\langle E_{2,2}\rangle +{\textstyle \frac{8}{3}}\langle E_{0,4}\rangle ,\end{eqnarray}$$

reads with the equations for $\langle E_{4,0}\rangle$ , $\langle E_{2,2}\rangle$ and $\langle E_{0,4}\rangle$ , cf. (B 10), (B 12) and (B 14) given in appendix B,

(2.13) $$\begin{eqnarray}\displaystyle \frac{\langle \unicode[STIX]{x2202}E_{[4]}\rangle }{\unicode[STIX]{x2202}t}+\frac{\unicode[STIX]{x2202}\langle \unicode[STIX]{x0394}u_{n}E_{[4]}\rangle }{\unicode[STIX]{x2202}r_{n}} & = & \displaystyle \unicode[STIX]{x1D708}\frac{\unicode[STIX]{x2202}^{2}}{\unicode[STIX]{x2202}r_{n}^{2}}\langle E_{[4]}\rangle \nonumber\\ \displaystyle & & \displaystyle -\,\underbrace{(F_{[4]}+Q_{[4]}+P_{[4]}+T_{[4]}+D_{[4]}+\unicode[STIX]{x1D700}_{[4]}^{2})}_{\sum \mathop{P}_{[4]}^{E}},\end{eqnarray}$$

where $\sum P_{[4]}^{E}$ is the trace of the sum of source terms in the dissipative source term equation, where the transport term is defined as

(2.14) $$\begin{eqnarray}\displaystyle \frac{\unicode[STIX]{x2202}\langle \unicode[STIX]{x0394}u_{n}E_{[4]}\rangle }{\unicode[STIX]{x2202}r_{n}} & = & \displaystyle \frac{\unicode[STIX]{x2202}\langle \unicode[STIX]{x0394}u_{1}E_{4,0}\rangle }{\unicode[STIX]{x2202}r}+4\frac{\unicode[STIX]{x2202}\langle \unicode[STIX]{x0394}u_{1}E_{2,2}\rangle }{\unicode[STIX]{x2202}r}+\frac{8}{3}\frac{\unicode[STIX]{x2202}\langle \unicode[STIX]{x0394}u_{1}E_{0,4}\rangle }{\unicode[STIX]{x2202}r}\nonumber\\ \displaystyle & & \displaystyle +\,\frac{2}{r}\langle \unicode[STIX]{x0394}u_{1}E_{4,0}\rangle +\frac{8}{r}\langle \unicode[STIX]{x0394}u_{1}E_{2,2}\rangle +\frac{16}{3r}\langle \unicode[STIX]{x0394}u_{1}E_{0,4}\rangle \nonumber\\ \displaystyle & = & \displaystyle \frac{\unicode[STIX]{x2202}\langle \unicode[STIX]{x0394}u_{1}E_{[4]}\rangle }{\unicode[STIX]{x2202}r}+\frac{2}{r}\langle \unicode[STIX]{x0394}u_{1}E_{[4]}\rangle ,\end{eqnarray}$$

with

(2.15) $$\begin{eqnarray}\unicode[STIX]{x0394}u_{1}E_{[4]}=\unicode[STIX]{x0394}u_{1}E_{4,0}+4\unicode[STIX]{x0394}u_{1}E_{2,2}+{\textstyle \frac{8}{3}}\unicode[STIX]{x0394}u_{1}E_{0,4},\end{eqnarray}$$

the viscous term

(2.16) $$\begin{eqnarray}2\unicode[STIX]{x1D708}\frac{\unicode[STIX]{x2202}^{2}\langle E_{[4]}\rangle }{\unicode[STIX]{x2202}r_{n}^{2}}=2\unicode[STIX]{x1D708}\left[\frac{\unicode[STIX]{x2202}^{2}\langle E_{[4]}\rangle }{\unicode[STIX]{x2202}r^{2}}+\frac{2}{r}\frac{\unicode[STIX]{x2202}\langle E_{[4]}\rangle }{\unicode[STIX]{x2202}r}\right],\end{eqnarray}$$

the $F$ -term

(2.17) $$\begin{eqnarray}F_{[4]}=8\unicode[STIX]{x1D708}\langle (\unicode[STIX]{x0394}u_{i})^{2}(A_{jj}+A_{jj}^{\prime })+\unicode[STIX]{x0394}u_{i}\unicode[STIX]{x0394}u_{j}(A_{ij}+A_{ij}^{\prime }+A_{ji}+A_{ji}^{\prime })\rangle ,\end{eqnarray}$$

the $Q$ -term

(2.18) $$\begin{eqnarray}Q_{[4]}=4\unicode[STIX]{x1D708}\langle (\unicode[STIX]{x0394}u_{i})^{2}(\unicode[STIX]{x1D712}_{jj}+\unicode[STIX]{x1D712}_{jj}^{\prime })+2\unicode[STIX]{x0394}u_{i}\unicode[STIX]{x0394}u_{j}(\unicode[STIX]{x1D712}_{ij}+\unicode[STIX]{x1D712}_{ij}^{\prime })\rangle ,\end{eqnarray}$$

the $P$ -term

(2.19) $$\begin{eqnarray}P_{[4]}=4\langle \unicode[STIX]{x0394}u_{i}\unicode[STIX]{x0394}P_{i}(\unicode[STIX]{x1D700}_{jj}+\unicode[STIX]{x1D700}_{jj}^{\prime })+(\unicode[STIX]{x0394}u_{i}\unicode[STIX]{x0394}P_{j}+\unicode[STIX]{x0394}u_{j}\unicode[STIX]{x0394}P_{i})(\unicode[STIX]{x1D700}_{ij}+\unicode[STIX]{x1D700}_{ij}^{\prime })\rangle ,\end{eqnarray}$$

the $T$ -term

(2.20) $$\begin{eqnarray}T_{[4]}=8\unicode[STIX]{x1D708}\langle (\unicode[STIX]{x0394}u_{i})^{2}(P_{jj}+P_{jj}^{\prime })+\unicode[STIX]{x0394}u_{i}\unicode[STIX]{x0394}u_{j}(P_{ij}+P_{ij}^{\prime }+P_{ji}+P_{ji}^{\prime })\rangle ,\end{eqnarray}$$

the $D$ -term

(2.21) $$\begin{eqnarray}\displaystyle D_{[4]} & = & \displaystyle 4\unicode[STIX]{x1D708}\left\langle \left(\frac{\unicode[STIX]{x2202}\left(\unicode[STIX]{x0394}u_{i}\right)^{2}}{\unicode[STIX]{x2202}x_{n}}\frac{\unicode[STIX]{x2202}\unicode[STIX]{x1D700}_{jj}+\unicode[STIX]{x1D700}_{jj}^{\prime }}{\unicode[STIX]{x2202}x_{n}}+\frac{\unicode[STIX]{x2202}\left(\unicode[STIX]{x0394}u_{i}\right)^{2}}{\unicode[STIX]{x2202}x_{n}^{\prime }}\frac{\unicode[STIX]{x2202}\unicode[STIX]{x1D700}_{jj}^{\prime }+\unicode[STIX]{x1D700}_{jj}^{\prime }}{\unicode[STIX]{x2202}x_{n}^{\prime }}\right)\right.\nonumber\\ \displaystyle & & \displaystyle \left.+\,2\left(\frac{\unicode[STIX]{x2202}\unicode[STIX]{x0394}u_{i}\unicode[STIX]{x0394}u_{j}}{\unicode[STIX]{x2202}x_{n}}\frac{\unicode[STIX]{x2202}\unicode[STIX]{x1D700}_{ij}+\unicode[STIX]{x1D700}_{ij}^{\prime }}{\unicode[STIX]{x2202}x_{n}}+\frac{\unicode[STIX]{x2202}\unicode[STIX]{x0394}u_{i}\unicode[STIX]{x0394}u_{j}}{\unicode[STIX]{x2202}x_{n}^{\prime }}\frac{\unicode[STIX]{x2202}\unicode[STIX]{x1D700}_{ij}^{\prime }+\unicode[STIX]{x1D700}_{ij}^{\prime }}{\unicode[STIX]{x2202}x_{n}^{\prime }}\right)\right\rangle\end{eqnarray}$$

and the $\unicode[STIX]{x1D700}^{2}$ -term

(2.22) $$\begin{eqnarray}\displaystyle \unicode[STIX]{x1D700}_{[4]}^{2} & = & \displaystyle \langle 2(\unicode[STIX]{x1D700}_{ii}+\unicode[STIX]{x1D700}_{ii}^{\prime })(\unicode[STIX]{x1D700}_{jj}+\unicode[STIX]{x1D700}_{jj}^{\prime })+4(\unicode[STIX]{x1D700}_{ij}+\unicode[STIX]{x1D700}_{ij}^{\prime })^{2}\rangle \nonumber\\ \displaystyle & = & \displaystyle 2\langle (\unicode[STIX]{x1D700}+\unicode[STIX]{x1D700}^{\prime })^{2}\rangle +4\langle (\unicode[STIX]{x1D700}_{ij}+\unicode[STIX]{x1D700}_{ij}^{\prime })^{2}\rangle .\end{eqnarray}$$

In our notation, $\unicode[STIX]{x1D700}_{ii}=\unicode[STIX]{x1D700}$ is twice the pseudo-dissipation. For better readability, the definitions

(2.23) $$\begin{eqnarray}\displaystyle & \displaystyle A_{ij}=\frac{\unicode[STIX]{x2202}u_{n}}{\unicode[STIX]{x2202}x_{m}}\frac{\unicode[STIX]{x2202}u_{i}}{\unicode[STIX]{x2202}x_{n}}\frac{\unicode[STIX]{x2202}u_{j}}{\unicode[STIX]{x2202}x_{m}}, & \displaystyle\end{eqnarray}$$

(2.24) $$\begin{eqnarray}\displaystyle & \displaystyle P_{ij}=\frac{\unicode[STIX]{x2202}u_{i}}{\unicode[STIX]{x2202}x_{m}}\frac{\unicode[STIX]{x2202}^{2}p}{\unicode[STIX]{x2202}x_{j}\unicode[STIX]{x2202}x_{m}} & \displaystyle\end{eqnarray}$$

and

(2.25) $$\begin{eqnarray}\unicode[STIX]{x1D712}_{ij}=2\unicode[STIX]{x1D708}\left[\frac{\unicode[STIX]{x2202}^{2}u_{i}}{\unicode[STIX]{x2202}x_{n}\unicode[STIX]{x2202}x_{m}}\frac{\unicode[STIX]{x2202}^{2}u_{j}}{\unicode[STIX]{x2202}x_{n}\unicode[STIX]{x2202}x_{m}}\right]\end{eqnarray}$$

have been used in (2.17)–(2.20).

For the stationary case, the transport term on the left-hand side in (2.13) balances the viscous term and the sum of source terms $\sum P_{[4]}^{E}$ . The source terms are defined in (2.17) to (2.22) and are shown in figure 2 for the cases $R1$ ( $Re_{\unicode[STIX]{x1D706}}=119$ ) and $R4$ ( $Re_{\unicode[STIX]{x1D706}}=331$ ) together with the transport term and the viscous term. Negative terms are denoted with a minus sign, meaning that we have changed their sign in order to be able to show them in a log–log plot. As seen in figure 2, the largest ones are the $F$ -term and the $Q$ -term, which nearly balance each other. The $F$ -term is a correlation between the velocity increments squared and the triple product of velocity gradients $(A_{ij}+A_{ij}^{\prime })$ . The $Q$ -term is a correlation between the velocity increments and $(\unicode[STIX]{x1D712}_{ij}+\unicode[STIX]{x1D712}_{ij}^{\prime })$ . The term $\unicode[STIX]{x1D712}_{ij}$ describes the dissipation of velocity gradients squared, while $A_{ij}$ describes their production by stretching. The next two source terms, called the $T$ -term and the $P$ -term, containing pressure derivatives, are relatively small and will not be discussed here in detail. Finally, there are the $D$ -term and the $\unicode[STIX]{x1D700}^{2}$ -term, which balance each other in the inertial range, but strongly diverge from each other in the viscous range. In the following, the focus will be on the $\unicode[STIX]{x1D700}^{2}$ -term, which is the sum of the second-order dissipation parameters.

Figure 2. Balance of the different terms in (2.13) for the cases $R1$ ( $Re_{\unicode[STIX]{x1D706}}=119$ ) (a) and $R4$ ( $Re_{\unicode[STIX]{x1D706}}=331$ ) (b): grey ○, transport term; grey ▫, $F$ -term; grey ▿, $P$ -term; grey ▵, $T$ -term; ○, $Q$ -term; ▫, $D$ -term; ▿, $\unicode[STIX]{x1D700}^{2}$ -term; ▵, viscous term.

The $\unicode[STIX]{x1D700}^{2}$ -term is defined as $2\langle (\unicode[STIX]{x1D700}_{ii}+\unicode[STIX]{x1D700}_{ii}^{\prime })^{2}\rangle +4\langle (\unicode[STIX]{x1D700}_{ij}+\unicode[STIX]{x1D700}_{ij}^{\prime })^{2}\rangle$ and is a two-point quantity depending on  $r$ . Noticeably, Hill (Reference Hill2002) showed that the sum of the two-point dissipation $\langle \unicode[STIX]{x1D700}+\unicode[STIX]{x1D700}^{\prime }\rangle$ also appears in the trace of the second-order structure function equations, which then reduces to $2\langle \unicode[STIX]{x1D700}\rangle$ independent of $r$ for homogeneous flows.

Since $\unicode[STIX]{x1D700}_{ij}$ becomes equal to $\unicode[STIX]{x1D700}_{ij}^{\prime }$ at $r\rightarrow 0$ , $\unicode[STIX]{x1D700}_{[4]}^{2}$ approaches the value $8\langle \unicode[STIX]{x1D700}_{ii}^{2}\rangle +16\langle \unicode[STIX]{x1D700}_{ij}^{2}\rangle$ and balances the viscous term there, as seen in figure 2. The quantity $\langle \unicode[STIX]{x1D700}_{ii}^{2}\rangle =\langle \unicode[STIX]{x1D700}^{2}\rangle$ is the second-order moment of the dissipation distribution of $\unicode[STIX]{x1D700}$ . It is a sum of dissipation parameters that appear in the set of successive equations presented in the archive. It is worth noting that the $\unicode[STIX]{x1D700}^{2}$ -term in figure 2 starts as a constant for small values of $r$ , then decreases slightly and becomes constant again at the large scales. In the latter range, $2\langle (\unicode[STIX]{x1D700}_{ii}+\unicode[STIX]{x1D700}_{ii}^{\prime })^{2}\rangle +4\langle (\unicode[STIX]{x1D700}_{ij}+\unicode[STIX]{x1D700}_{ij}^{\prime })^{2}\rangle$ approaches the asymptotic limit $4\langle \unicode[STIX]{x1D700}_{ii}^{2}\rangle +8\langle \unicode[STIX]{x1D700}_{ij}^{2}\rangle$ ; i.e. it differs from the value for $r\rightarrow 0$ only by a constant factor of $2$ . That is, the $\unicode[STIX]{x1D700}^{2}$ -term links the remaining terms of (2.13) at the very large and very small scales, similarly to $\langle \unicode[STIX]{x1D700}\rangle$ at the second order. In the following, we are mostly interested in the influence of the $\unicode[STIX]{x1D700}^{2}$ -term, since this term is closest to the classical picture of dissipative fluctuations in turbulent flows in the spirit of Kolmogorov.

3 Relations between the trace of fourth-order structure functions $\mathit{S}_{[4]}$ and $\langle \unicode[STIX]{x1D700}_{[4]}^{2}\rangle$ in the viscous range

First, we examine the equations outlined above in the viscous range, before we look at the inertial range in the next section. We find that for $r\rightarrow 0$ , $\mathit{S}_{[4]}$ is exactly determined by $\unicode[STIX]{x1D700}_{[4]}^{2}$ , $\unicode[STIX]{x1D708}$ and $r$ .

In the viscous range, the transport term of the fourth-order structure function (2.4) as well as the pressure source terms are proportional to $r^{4}$ for $r\rightarrow 0$ , while the viscous and the dissipative source terms are proportional to $r^{2}$ . Therefore, to leading order the transport and pressure source terms may be neglected in the viscous range and (2.4) simplifies to

(3.1) $$\begin{eqnarray}2\unicode[STIX]{x1D708}\left(\frac{\text{d}^{2}\mathit{S}_{[4]}}{\text{d}r^{2}}+\frac{2}{r}\frac{\text{d}\mathit{S}_{[4]}}{\text{d}r}\right)=\langle E_{[4]}\rangle .\end{eqnarray}$$

Similarly, from the equation for $\langle E_{[4]}\rangle$ , the viscous and $\unicode[STIX]{x1D700}^{2}$ -term balance,

(3.2) $$\begin{eqnarray}2\unicode[STIX]{x1D708}\left(\frac{\text{d}^{2}\langle E_{[4]}\rangle }{\text{d}r^{2}}+\frac{2}{r}\frac{\text{d}\langle E_{[4]}\rangle }{\text{d}r}\right)=\unicode[STIX]{x1D700}_{[4]}^{2},\end{eqnarray}$$

where $\unicode[STIX]{x1D700}_{[4]}^{2}$ is constant in the viscous range. Solving (3.2) and inserting the result into (3.1) then gives

(3.3a,b ) $$\begin{eqnarray}\mathit{S}_{[4]}=\frac{\unicode[STIX]{x1D700}_{[4]}^{2}}{480\unicode[STIX]{x1D708}^{2}}r^{4},\quad \langle E_{[4]}\rangle =\frac{\unicode[STIX]{x1D700}_{[4]}^{2}}{12\unicode[STIX]{x1D708}}r^{2}.\end{eqnarray}$$

Since $\mathit{S}_{[4]}$ is the sum of $S_{4,0}$ , $S_{2,2}$ and $S_{0,4}$ , all of which are also proportional to $r^{4}$ for $r\rightarrow 0$ , the individual structure functions are also determined by $\unicode[STIX]{x1D700}_{[4]}^{2}$ in the viscous range. In particular, by expanding the structure functions as

(3.4) $$\begin{eqnarray}\displaystyle & \displaystyle S_{4,0}=F_{1}r^{4}+\cdots =\left\langle \left(\frac{\unicode[STIX]{x2202}u_{1}}{\unicode[STIX]{x2202}x_{1}}\right)^{4}\right\rangle r^{4}+\cdots \,, & \displaystyle\end{eqnarray}$$

(3.5) $$\begin{eqnarray}\displaystyle & \displaystyle S_{2,2}=F_{2}r^{4}+\cdots =\left\langle \left(\frac{\unicode[STIX]{x2202}u_{1}}{\unicode[STIX]{x2202}x_{1}}\right)^{2}\left(\frac{\unicode[STIX]{x2202}u_{2}}{\unicode[STIX]{x2202}x_{1}}\right)^{2}\right\rangle r^{4}+\cdots \,, & \displaystyle\end{eqnarray}$$

(3.6) $$\begin{eqnarray}\displaystyle & \displaystyle S_{0,4}=F_{3}r^{4}+\cdots =\left\langle \left(\frac{\unicode[STIX]{x2202}u_{2}}{\unicode[STIX]{x2202}x_{1}}\right)^{4}\right\rangle r^{4}+\cdots \,, & \displaystyle\end{eqnarray}$$

one can relate $\unicode[STIX]{x1D700}_{[4]}^{2}/\unicode[STIX]{x1D708}^{2}$ to the sum of $F_{1}$ , $F_{2}$ and $F_{3}$ . Siggia (Reference Siggia1981) has identified the four invariants of the fourth-order velocity gradient tensor as

(3.7a-d ) $$\begin{eqnarray}\mathit{I}_{1}\equiv \langle s^{4}\rangle ,\quad \mathit{I}_{2}\equiv \langle s^{2}\unicode[STIX]{x1D714}^{2}\rangle ,\quad \mathit{I}_{3}\equiv \langle \unicode[STIX]{x1D714}_{i}\mathit{s}_{ij}\unicode[STIX]{x1D714}_{k}s_{kj}\rangle ,\quad \mathit{I}_{4}\equiv \langle \unicode[STIX]{x1D714}^{4}\rangle ,\end{eqnarray}$$

where $\mathit{s}_{ij}=(\unicode[STIX]{x2202}u_{i}/\unicode[STIX]{x2202}x_{j}+\unicode[STIX]{x2202}u_{j}/\unicode[STIX]{x2202}x_{i})/2$ is the rate of strain tensor and the vorticity $\unicode[STIX]{x1D714}_{i}=\unicode[STIX]{x1D716}_{ijk}(\unicode[STIX]{x2202}u_{k}/\unicode[STIX]{x2202}x_{j})$ , where $\unicode[STIX]{x1D716}_{ijk}$ is the Levi-Civita symbol. Siggia then derived

(3.8) $$\begin{eqnarray}\displaystyle & F_{1}=4\mathit{I}_{1}/105, & \displaystyle\end{eqnarray}$$

(3.9) $$\begin{eqnarray}\displaystyle & F_{2}=\mathit{I}_{1}/105+\mathit{I}_{2}/70-\mathit{I}_{3}/105, & \displaystyle\end{eqnarray}$$

(3.10) $$\begin{eqnarray}\displaystyle & F_{3}=3\mathit{I}_{1}/140+11\mathit{I}_{2}/140-3\mathit{I}_{3}/35+\mathit{I}_{4}/80, & \displaystyle\end{eqnarray}$$

resulting in

(3.11) $$\begin{eqnarray}\frac{\unicode[STIX]{x1D700}_{[4]}^{2}}{480\unicode[STIX]{x1D708}^{2}}=F_{1}+4F_{2}+\frac{8}{3}F_{3}=\frac{2}{15}\mathit{I}_{1}+\frac{4}{15}\mathit{I}_{2}-\frac{4}{15}\mathit{I}_{3}+\frac{1}{30}\mathit{I}_{4}.\end{eqnarray}$$

Since

(3.12) $$\begin{eqnarray}\frac{\langle \unicode[STIX]{x1D700}_{11}^{2}\rangle }{\unicode[STIX]{x1D708}^{2}}=\frac{\langle \unicode[STIX]{x1D700}_{22}^{2}\rangle }{\unicode[STIX]{x1D708}^{2}}=\frac{\langle \unicode[STIX]{x1D700}_{11}\unicode[STIX]{x1D700}_{22}\rangle +2\langle \unicode[STIX]{x1D700}_{12}^{2}\rangle }{\unicode[STIX]{x1D708}^{2}}=\frac{8}{15}\mathit{I}_{1}+\frac{16}{15}\mathit{I}_{2}-\frac{16}{15}\mathit{I}_{3}+\frac{2}{15}\mathit{I}_{4},\end{eqnarray}$$

cf. (C 20) in appendix C, the trace $\unicode[STIX]{x1D700}_{[4]}^{2}$ is related in the viscous range to the dissipation parameters

(3.13) $$\begin{eqnarray}\unicode[STIX]{x1D700}_{[4]}^{2}=120\langle \unicode[STIX]{x1D700}_{11}^{2}\rangle =120\langle \unicode[STIX]{x1D700}_{22}^{2}\rangle =120\langle \unicode[STIX]{x1D700}_{11}\unicode[STIX]{x1D700}_{22}\rangle +360\langle \unicode[STIX]{x1D700}_{12}^{2}\rangle .\end{eqnarray}$$

A similar analysis can be carried out for higher even orders in the viscous range. However, there are more intermediate equations linking the higher-order moments of the dissipation to the higher-order dissipative source term and ultimately to the structure function; cf. the archive for the sixth-order equations. For instance, one then finds for the sixth order $\mathit{S}_{[6]}\sim r^{6}\unicode[STIX]{x1D700}_{[6]}^{3}/\unicode[STIX]{x1D708}^{3}$ in the viscous range, with two intermediate equations.

4 Relation between dissipation fluctuations and inertial-range scaling exponents

In the inertial range, Kolmogorov (Reference Kolmogorov1962) included dissipation fluctuations by using a locally averaged dissipation $\unicode[STIX]{x1D700}_{r}$ as the structure function scaling parameter instead of only the mean value $\langle \unicode[STIX]{x1D700}\rangle$ . One would therefore think that $\unicode[STIX]{x1D700}_{r}$ must appear in the system of equations. For that reason, we examine in this section a connection between the $\unicode[STIX]{x1D700}^{2}$ -term and the second moment of $\unicode[STIX]{x1D700}_{r}$ . Therefore, in order to analyse the effect of the dissipation parameter $\unicode[STIX]{x1D700}_{[4]}^{2}$ on the inertial-range scaling exponent $\unicode[STIX]{x1D701}_{[5]}$ , we must consider the dissipative source term equation. In a first step, we will integrate (2.4) in the inertial range in order to calculate the fifth-order scaling exponent $\unicode[STIX]{x1D701}_{[5]}$ implicitly defined by assuming a power law $\mathit{S}_{[5]}\sim r^{\unicode[STIX]{x1D701}_{[5]}}$ . The idea is that because there are equations for the structure functions as well as equations for their source terms, the scaling exponent has to be contained in the system of equations.

For some time, there has been an expectation that the connection between the Navier–Stokes equations and the RSH would be discovered one day or, as Kaneda & Morishita (Reference Kaneda, Morishita, Davidson, Kaneda and Sreenivasan2013) proffered, ‘the link between these models and the Navier–Stokes dynamics governing the fluid motion appears still to be missing’. This would require that a correlation exists between $\unicode[STIX]{x1D700}_{r}$ and the terms representing dissipation fluctuations in the Navier–Stokes equations. In the two-point equations derived systematically from the Navier–Stokes equations in the archive material, from which we derived (2.13) shown above, we have identified the terms describing dissipation fluctuations as $\unicode[STIX]{x1D700}^{2}$ -terms. They contain moments of the sum of components of the dissipation at two points, for instance the moments $\langle (\unicode[STIX]{x1D700}+\unicode[STIX]{x1D700}^{\prime })^{m}\rangle$ . An integral expression such as

(4.1) $$\begin{eqnarray}\widetilde{\unicode[STIX]{x1D700}}_{r}=\frac{1}{V}\int \unicode[STIX]{x1D700}(\boldsymbol{x})\,\text{d}\boldsymbol{x},\end{eqnarray}$$

where $V\sim O(r^{3})$ is a volume of dimension $r^{3}$ , or the corresponding one-dimensional expression

(4.2) $$\begin{eqnarray}\unicode[STIX]{x1D700}_{r}=\frac{1}{r}\int \unicode[STIX]{x1D700}(x)\,\text{d}x,\end{eqnarray}$$

where the integral is over any length $r$ , does not appear directly in these equations.

Since $\unicode[STIX]{x1D700}_{r}$ exhibits a power-law scaling in the inertial range, dimensional analysis suggests that this could be related to the power-law scaling of the structure functions, as Kolmogorov (Reference Kolmogorov1962) had predicted. Stolovitzky, Kailasnath & Sreenivasan (Reference Stolovitzky, Kailasnath and Sreenivasan1992) have performed a test of the relation between $\unicode[STIX]{x0394}u(r)$ and $(r\unicode[STIX]{x1D700}_{r})^{1/3}$ by investigating the statistical properties of $\unicode[STIX]{x0394}u(r)/(r\unicode[STIX]{x1D700}_{r})^{1/3}$ . They found that the distribution function of $\unicode[STIX]{x0394}u(r)/(r\unicode[STIX]{x1D700}_{r})^{1/3}$ is independent of $r$ and $\unicode[STIX]{x1D700}_{r}$ in the inertial range and that it is approximately universal. Additional work addressing tests of the RSH is referenced by Sreenivasan & Antonia (Reference Sreenivasan and Antonia1997).

On the other hand, properties such as inertial-range scaling exponents should, as a matter of principle, result from the two-point equations. If a power-law scaling can be assumed for the source terms in the structure function equations, the scaling exponents would follow from equations like (4.7) below, to which both the pressure source term and the dissipation source term contribute. For even-order scaling exponents resulting from the odd-order structure function equations, Yakhot (Reference Yakhot2003) has argued that the dissipation source terms can be neglected and has proposed an algebraic closure relating the pressure source terms of arbitrary order to the structure functions. A similar closure could conceivably be developed for the odd-order scaling exponents in the even-order structure function equations. In these equations, the closure would be between the dissipation source terms and structure functions, while the pressure source terms can be neglected. The two-point equations show that dissipation fluctuations are represented by the dissipation parameters appearing in the successive source term equations and not by the source terms in the structure function equations. If one wanted to reconcile the RSH with the two-point equations, one would have to develop a closure between the moments of $\unicode[STIX]{x1D700}_{r}$ and the dominating source terms in the structure function equations.

It can be shown that the second moment of $\unicode[STIX]{x1D700}_{r}$ is related to the dissipation correlation $\langle \unicode[STIX]{x1D700}\unicode[STIX]{x1D700}^{\prime }\rangle$ under the assumption of homogeneity by

(4.3) $$\begin{eqnarray}\langle \unicode[STIX]{x1D700}\unicode[STIX]{x1D700}^{\prime }\rangle =\frac{1}{2}\frac{\text{d}^{2}}{\text{d}r^{2}}[r^{2}\langle \unicode[STIX]{x1D700}_{r}^{2}\rangle ],\end{eqnarray}$$

cf. Nelkin (Reference Nelkin1994), and similar relations can be found for the higher moments of $\unicode[STIX]{x1D700}_{r}$ . We only discuss the second moment, because we examine here the fourth-order structure function equations. In particular, we only look at the trace (2.13), but our findings carry over to the individual equations for longitudinal, mixed and transverse dissipative source terms as well. If one assumes a power law for $\langle \unicode[STIX]{x1D700}_{r}^{2}\rangle \sim r^{\unicode[STIX]{x1D6FE}}$ , where the scaling exponent and the prefactor are independent of $r$ , as one is apt to do and which is at the very core of the RSH and similar theories, one also finds by virtue of (4.3) that $\langle \unicode[STIX]{x1D700}\unicode[STIX]{x1D700}^{\prime }\rangle \sim r^{\unicode[STIX]{x1D6FE}}$ and hence has the same $r$ -scaling. The first term of the $\unicode[STIX]{x1D700}^{2}$ -term in (2.22) can be written as the sum of constants and correlations; i.e.

(4.4) $$\begin{eqnarray}\langle (\unicode[STIX]{x1D700}+\unicode[STIX]{x1D700}^{\prime })^{2}\rangle =2\langle \unicode[STIX]{x1D700}^{2}\rangle +2\langle \unicode[STIX]{x1D700}\unicode[STIX]{x1D700}^{\prime }\rangle .\end{eqnarray}$$

In other words, the $r$ -dependence of $\unicode[STIX]{x1D700}_{r}$ is manifested in the $\unicode[STIX]{x1D700}^{2}$ -terms in the dissipative source terms by virtue of (4.3) and (4.4).

We begin with the integration of (2.4) to link the structure function exponent to the source terms. We will neglect the diffusive and unsteady terms in (2.4) and (2.13) and perform an order-of-magnitude estimate of the source terms. If we assume in (2.4) a power-law scaling for the source terms of the form

(4.5) $$\begin{eqnarray}\displaystyle & \langle T_{[4]}\rangle & \displaystyle =A_{[4]}^{T}r^{\unicode[STIX]{x1D709}_{[4]}^{T}},\end{eqnarray}$$

(4.6) $$\begin{eqnarray}\displaystyle & \langle E_{[4]}\rangle & \displaystyle =A_{[4]}^{E}r^{\unicode[STIX]{x1D709}_{[4]}^{E}},\end{eqnarray}$$

which is consistent with figure 1, the trace of the fifth-order structure functions $\mathit{S}_{[5]}$ can be determined from (2.4) by integration. Of course, (4.5) and (4.6) are approximations. However, without these assumptions, a power law of the form $\mathit{S}_{[5]}=C_{[5]}r^{\unicode[STIX]{x1D701}_{[5]}}$ is not compatible with (2.4). In other words, only if the source terms follow a power law in the inertial range, do the structure functions as well. The range of integration will be divided into two parts. The first part ranges from $r=0$ to $r^{\ast }\approx 30\unicode[STIX]{x1D702}$ , where the power law in the inertial range starts to be valid. The second part ranges from there on to the value $r$ of interest in the inertial range. The integration then yields

(4.7) $$\begin{eqnarray}\displaystyle -\!\mathit{S}_{[5]} & = & \displaystyle \frac{1}{r^{2}}\int _{0}^{r^{\ast }}r^{2}(\langle T_{[4]}\rangle +\langle E_{[4]}\rangle )\,\text{d}r-\frac{1}{r^{2}}\left\{\frac{A_{[4]}^{T}r^{3+\unicode[STIX]{x1D709}_{[4]}^{T}}}{3+\unicode[STIX]{x1D709}_{[4]}^{T}}+\frac{A_{[4]}^{E}r^{3+\unicode[STIX]{x1D709}_{[4]}^{E}}}{3+\unicode[STIX]{x1D709}_{[4]}^{E}}\right\}_{r=r^{\ast }}\nonumber\\ \displaystyle & & \displaystyle +\,\frac{A_{[4]}^{T}r^{1+\unicode[STIX]{x1D709}_{[4]}^{T}}}{3+\unicode[STIX]{x1D709}_{[4]}^{T}}+\frac{A_{[4]}^{E}r^{1+\unicode[STIX]{x1D709}_{[4]}^{E}}}{3+\unicode[STIX]{x1D709}_{[4]}^{E}}.\end{eqnarray}$$

Since the diffusion terms have been neglected from the very beginning, the first integral is incomplete. The purpose here is to show that this and the second term on the right-hand side of (4.7) can be neglected for large $r\gg r^{\ast }$ because of the term  $r^{-2}$ in front of them. Then, only the last two terms on the right-hand side of (4.7) remain, and we find the trace of the structure functions to be given as a weighted sum of the two power laws (4.5) and (4.6). More specifically, there are several possible contributions to the inertial range that have been neglected: the integral from 0 to $r$ over the viscous term might not be negligible in the inertial range at $r$ ; the integral of the dissipative and pressure source terms over 0 to $r^{\ast }$ might not be negligible in the inertial range at some $r$ ; the value of the transport term at $r=r^{\ast }$ might not be small compared with its inertial-range value. Additionally, because $\unicode[STIX]{x1D702}$ is not the correct dissipative length scale for the fourth-order equations, $r^{\ast }/\unicode[STIX]{x1D702}$ is dependent on the Reynolds number, and this dependence is required to be small. While the importance of some of these contributions may already be estimated from the balance figure 1, it is more advantageous to look at the integrated balance, in the spirit of (4.7) and the present section. For that reason, the numerical integration of (2.4) over $r$ is presented in figure 3. The integrated balances for the data sets $R0$ ( $Re_{\unicode[STIX]{x1D706}}=88$ ) and $R5$ ( $Re_{\unicode[STIX]{x1D706}}=529$ ) normalized by $\langle \unicode[STIX]{x1D700}^{2}\rangle$ and $\unicode[STIX]{x1D708}$ are plotted over $r/\unicode[STIX]{x1D702}$ . Indeed, the integrated viscous terms are negligible compared with all other integrated terms in the inertial range after $r/\unicode[STIX]{x1D702}=30$ . Moreover, the integrated dissipative source term, pressure source term and transport term are much smaller in the viscous range than in the inertial range. This holds for all data sets and not only the cases $R0$ and $R5$ shown in figure 3(a). Last, as shown in figure 3(b), both the integrated dissipative source terms and the integrated viscous terms cross over to the inertial range at approximately the same value of $r/\unicode[STIX]{x1D702}$ for the data sets analysed here, i.e. the true viscous cutoff length scale depends only weakly on the Reynolds number. Therefore, the simplifications with regard to the integration carried out in (4.7) as described above seem to be justifiable.

Figure 3. (a) Terms of (2.4) numerically integrated over $r$ for cases $R0$ (grey) and $R5$ (black), where ○ is the transport term, ▫ is the dissipative source term, ▿ is the pressure source term and ▵ is the viscous term. The integrated dissipative source term ▫ and integrated viscous terms $\triangle$ for all cases $R0$ – $R5$ (from light grey to black) are shown in (b).

The sum of two power laws with constant prefactor and exponent does not give a power law with constant prefactor and exponent. Only if $\unicode[STIX]{x1D709}_{[4]}^{T}=\unicode[STIX]{x1D709}_{[4]}^{E}$ or if one of the two terms is much smaller than the other one will the scaling of $\mathit{S}_{[5]}$ result in an (approximate) power law. We will explore the second possibility by approximating the scaling exponent $\unicode[STIX]{x1D701}_{[5]}$ by neglecting the pressure source term (cf. figure 1, where the pressure source term is smaller than the dissipative source terms by a factor of four, and also figure 3), resulting in

(4.8) $$\begin{eqnarray}\unicode[STIX]{x1D701}_{[5]}=1+\unicode[STIX]{x1D709}_{[4]}^{E}.\end{eqnarray}$$

This relationship between $\unicode[STIX]{x1D701}_{[5]}$ and $\unicode[STIX]{x1D709}_{[4]}^{E}$ is consistent with fusion rules, cf. L’vov & Procaccia (Reference L’vov and Procaccia1996a ) and Benzi et al. (Reference Benzi, Biferale and Toschi1998), where the same relation for the fifth-order structure function is provided for the case when one of the two separation distances is in the dissipative and the other in the inertial range. Here, (4.8) follows directly from neglecting the pressure source term. We have calculated the scaling exponent $\unicode[STIX]{x1D709}_{[4]}^{E}$ of $\langle E_{[4]}\rangle$ already introduced in (4.6) for all of our DNS cases by using

(4.9) $$\begin{eqnarray}\unicode[STIX]{x1D709}_{[4]}^{E}=\frac{r}{\langle E_{[4]}\rangle }\frac{\unicode[STIX]{x2202}\langle E_{[4]}\rangle }{\unicode[STIX]{x2202}r}.\end{eqnarray}$$

As seen from figure 4, $\unicode[STIX]{x1D709}_{[4]}^{E}=0.56$ in the inertial range for our data sets $R1$ – $R6$ . The trace of the fifth-order structure functions $\mathit{S}_{[5]}$ is shown in figure 5 in compensated form; i.e. divided by $(r/\unicode[STIX]{x1D702})^{1.56}$ and normalized by $\unicode[STIX]{x1D708}$ and $\langle \unicode[STIX]{x1D700}^{2}\rangle$ . Since $\langle \unicode[STIX]{x1D700}^{2}\rangle$ represents the dissipation parameters at the fourth-order level, $\langle \unicode[STIX]{x1D700}^{2}\rangle$ provides indeed a better scaling than $\langle \unicode[STIX]{x1D700}\rangle$ for quantities at that level. We plot the scaling exponents $\unicode[STIX]{x1D701}_{[5]}$ in figure 5, as computed by

(4.10) $$\begin{eqnarray}\unicode[STIX]{x1D701}_{[5]}=\frac{r}{\mathit{S}_{[5]}}\frac{\unicode[STIX]{x2202}\mathit{S}_{[5]}}{\unicode[STIX]{x2202}r},\end{eqnarray}$$

where again implicitly a power law for $\mathit{S}_{[5]}$ with constant prefactors and exponents is assumed. The dashed black horizontal line indicates the value $\unicode[STIX]{x1D701}_{[5]}=1.56$ .

Figure 4. Scaling exponent $\unicode[STIX]{x1D709}_{[4]}^{E}$ for the cases $R1$ – $R6$ with $\mathit{Re}_{\unicode[STIX]{x1D706}}$ ranging from 119 to 754 (higher Reynolds numbers indicated by darker shading). The dashed black horizontal line indicates $\unicode[STIX]{x1D709}_{[4]}^{E}=0.56$ .

Figure 5. Compensated structure function $\mathit{S}_{[5]}$ in the inertial range for the cases $R0$ – $R6$ with $Re_{\unicode[STIX]{x1D706}}$ ranging from 88 to 754 (a) and scaling exponent $\unicode[STIX]{x1D701}_{[5]}$ (b). Higher Reynolds numbers are indicated by darker shading. The horizontal dashed black line in (b) indicates $\unicode[STIX]{x1D701}_{[5]}=1.56$ .

In the next step, a relation between $\unicode[STIX]{x1D709}_{[4]}^{E}$ and the $\unicode[STIX]{x1D700}^{2}$ -term is needed. Therefore, we need to look at the transport equation of the dissipative source term $\langle E_{[4]}\rangle$ in more detail. However, under the assumption of stationarity, there is no term containing $\langle E_{[4]}\rangle$ in its transport equation when the inertial-range assumptions are invoked, as both the unsteady and the viscous terms are neglected. That is, information is lost by averaging, and, hence, additional assumptions and closures are needed. The situation is similar to the second-order structure function equations, where one finds in the inertial range a solution for the third-order structure functions (i.e. the $4/5$ law), but all connections to the second order are lost. If one is interested in the solution of the second order in the inertial range, one has to make additional assumptions such as Kolmogorov’s constant-skewness assumption.

That is, we would now need a relation between $\langle \unicode[STIX]{x0394}u_{1}E_{[4]}\rangle$ and $\langle E_{[4]}\rangle$ , e.g. by employing a gradient flux ansatz. In the following, we rather use a slightly different approach. In analogy to K41 and K62 theory, one might postulate that fluctuations of the dissipation represent the relevant scaling parameter in the inertial range. Since the transport equation for $\langle E_{[4]}\rangle$ contains the $\unicode[STIX]{x1D700}^{2}$ -term, which represents dissipation fluctuations, it seems reasonable to scale the dissipative source term with $\unicode[STIX]{x1D700}_{[4]}^{2}$ and  $r$ . It should be mentioned that the $r$ -dependence of the $\unicode[STIX]{x1D700}^{2}$ -term is cancelled out by the $D$ -term in the sum of the source terms. However, neither of the other terms combined with only $r$ is sufficient to provide a scaling of the dissipative source term without an additional parameter like $\unicode[STIX]{x1D6FF}_{[4]}^{E}$ , and the same holds for the cancellations of $D_{[4]}+\unicode[STIX]{x1D700}_{[4]}^{2}$ and $F_{[4]}+Q_{[4]}$ . This indicates that some combination of the source terms in (3.1) is needed to scale the dissipative source term and ultimately $\mathit{S}_{[5]}$ . Consequently, not only the dissipation fluctuations characterized by the $\unicode[STIX]{x1D700}^{2}$ -term are relevant, but also the other source terms in (3.1). However, the necessary combination of source terms cannot be determined by scaling arguments alone, because there are only two dimensions (m) and (s) and more than two scaling quantities. Furthermore, all terms are needed if the prefactor is also of interest.

In the following, we proceed to look only at the dissipation fluctuations, because they remain a quantity of high interest. We generalize the scaling of $\langle E_{[4]}\rangle$ with $\unicode[STIX]{x1D700}_{[4]}^{2}$ and $r$ by adding a prefactor $(r/r^{\ast })^{\unicode[STIX]{x1D6FF}_{[4]}^{E}}$ , i.e. we use the ansatz

(4.11) $$\begin{eqnarray}\langle E_{[4]}\rangle =c_{[4]}^{E}(\unicode[STIX]{x1D700}_{[4]}^{2})^{5/6}r^{2/3+\unicode[STIX]{x1D6FF}_{[4]}^{E}}.\end{eqnarray}$$

Here, $r^{\ast \unicode[STIX]{x1D6FF}_{[4]}^{E}}$ is contained in $c_{[4]}^{E}$ . Of course, this is an ad hoc ansatz and only one of many possibilities. We do not want to claim that this is the best or only way to close the equations; rather, we use it for its simplicity. We use this closure for analytical purposes only, i.e. do not want to make predictions regarding higher orders, other data sets or flows. Because there are other source terms in (2.13) that contribute to the balance, one cannot expect that $\unicode[STIX]{x1D6FF}_{[4]}^{E}$ vanishes. Indeed, from our DNS, we find that $\unicode[STIX]{x1D6FF}_{[4]}^{E}=-0.09$ . The numerical values of $\unicode[STIX]{x1D6FF}_{[4]}^{E}$ for $R1$ – $R6$ are shown in figure 6(a), where

(4.12) $$\begin{eqnarray}\unicode[STIX]{x1D6FF}_{[4]}^{E}=r(\unicode[STIX]{x1D700}_{[4]}^{2})^{5/6}\langle E_{[4]}\rangle ^{-1}\frac{\unicode[STIX]{x2202}}{\unicode[STIX]{x2202}r}(\langle E_{[4]}\rangle (\unicode[STIX]{x1D700}_{[4]}^{2})^{-5/6})-\frac{2}{3}\end{eqnarray}$$

and the dashed horizontal black line corresponds to $\unicode[STIX]{x1D6FF}_{[4]}^{E}=-0.09$ . This value is empirically determined and not claimed to be universal or to carry over to other flows. Moreover, its exact value is not important in the present context. We will now insert (4.11) into (4.7) and integrate. That is, we now examine the scaling of the dissipative source term $\langle E_{[4]}\rangle$ compared with the scaling of the $\unicode[STIX]{x1D700}^{2}$ -term, $\unicode[STIX]{x1D700}_{[4]}^{2}$ , which contains $\langle \unicode[STIX]{x1D700}_{r}^{2}\rangle$ . Using the ansatz (4.11), we find that this ratio should scale as $r^{2/3+\unicode[STIX]{x1D6FF}_{[4]}^{E}}$ in the inertial range, under the assumption that we may approximate the terms by power laws in the inertial range. The ratio $\langle E_{[4]}\rangle /(\unicode[STIX]{x1D700}_{[4]}^{2}r^{2/3+\unicode[STIX]{x1D6FF}_{[4]}^{E}})$ is shown in figure 6(b) for the cases  $R1$ – $R6$ . We find a scaling range in the inertial range. In short, we have now replaced the dissipative source terms with $\unicode[STIX]{x1D700}_{[4]}^{2}r^{2/3+\unicode[STIX]{x1D6FF}_{[4]}^{E}}$ , which has the same $r$ -scaling in the inertial range. If the $\unicode[STIX]{x1D700}^{2}$ -term has a power-law scaling in the inertial range, we can now integrate to obtain for the structure function trace

(4.13) $$\begin{eqnarray}\mathit{S}_{[5]}\sim r^{5/3+\unicode[STIX]{x1D6FF}_{[4]}^{E}}(\unicode[STIX]{x1D700}_{[4]}^{2})^{5/6}.\end{eqnarray}$$

This should be compared with the RSH, for which

(4.14) $$\begin{eqnarray}\mathit{S}_{[5]}\sim r^{5/3}\langle \unicode[STIX]{x1D700}_{r}^{5/3}\rangle .\end{eqnarray}$$

That is, from the system of equations we would rather have a contribution from $\langle \unicode[STIX]{x1D700}_{r}^{2}\rangle$ , which is contained in $\unicode[STIX]{x1D700}_{[4]}^{2}$ , instead of $\langle \unicode[STIX]{x1D700}_{r}^{5/3}\rangle$ . As the RSH gives satisfactory results when compared with experimental data, the differences have to be contained in the parameter $\unicode[STIX]{x1D6FF}_{[4]}^{E}$ , which contains empirically the influence of the other source terms in (3.1) on the dissipative source term $\langle E_{[4]}\rangle$ . It should be noted that (4.14) implies $\langle E_{[4]}\rangle \sim r^{2/3}\langle \unicode[STIX]{x1D700}_{r}^{5/3}\rangle$ , if a power law for $\unicode[STIX]{x1D700}_{r}$ can be assumed. However, there is no term containing $\langle \unicode[STIX]{x1D700}_{r}^{5/3}\rangle$ in either the fourth- or the fifth-order equations.

Figure 6. The term $\unicode[STIX]{x1D6FF}_{[4]}^{E}$ as evaluated by (4.12) for $R1$ – $R6$ (a). Plot of the ratio $\langle E_{[4]}\rangle /(\unicode[STIX]{x1D700}_{[4]}^{2}r^{2/3-\unicode[STIX]{x1D6FF}_{[4]}^{E}})$ with $\unicode[STIX]{x1D6FF}_{[4]}^{E}=-0.09$ (b). Higher Reynolds numbers are indicated by darker shading.

We may conclude that the RSH assumption of using $\unicode[STIX]{x1D700}_{r}$ instead of the K41 ansatz using $\langle \unicode[STIX]{x1D700}\rangle$ is compatible with the results of our approach here, in the sense that $\unicode[STIX]{x1D700}_{r}$ appears in the $\unicode[STIX]{x1D700}^{2}$ -term. However, we find in the system of equations a contribution to the fifth-order structure functions by $\langle \unicode[STIX]{x1D700}_{r}^{2}\rangle$ ; i.e. the connection to $\langle \unicode[STIX]{x1D700}_{r}^{n/3}\rangle$ is still missing. We expect similar results at higher orders, which should then be related to higher moments of $\unicode[STIX]{x1D700}_{r}$ . However, the pressure source terms also might be important at higher orders, which one would then need to close as well. Furthermore, we find that the $r$ -scaling of the $\unicode[STIX]{x1D700}^{2}$ -term is cancelled out by the $D$ -term in the full system of equations. We are left with the fact that while we have found all higher moments of $\unicode[STIX]{x1D700}_{r}$ in the dissipative source term equations and further equations derived therefrom, we cannot say why $n$ th-order structure functions should be determined by $\langle \unicode[STIX]{x1D700}_{r}^{n/3}\rangle$ , i.e. by the RSH or other similar theories. While the numerical values obtained in either way agree fairly well (at least for the fourth order we examined here), such a connection cannot be obtained from the Navier–Stokes equations without additional empirical closures.

The analysis can be carried to higher orders. It can easily be shown that the third successive equation for dissipation source terms in the trace of the eighth-order structure function equations contains a term $\langle (\unicode[STIX]{x1D700}_{ii}+\unicode[STIX]{x1D700}_{ii}^{\prime })(\unicode[STIX]{x1D700}_{jj}+\unicode[STIX]{x1D700}_{jj}^{\prime })(\unicode[STIX]{x1D700}_{kk}+\unicode[STIX]{x1D700}_{kk}^{\prime })(\unicode[STIX]{x1D700}_{ll}+\unicode[STIX]{x1D700}_{ll}^{\prime })\rangle$ , which generates a dissipation parameter $\langle \unicode[STIX]{x1D700}^{4}\rangle$ . As one continues further, one will find all moments of the dissipation distribution function in the system of averaged equations. On the basis of the equations at the sixth-order structure function level, for instance, one could perform similar integrations to those for the fourth-order level. For instance, in the archive material, we have derived an equation for the dissipation source term $\langle E_{6,0}\rangle =15\langle (\unicode[STIX]{x0394}u_{1})^{4}(\unicode[STIX]{x1D700}_{11}+\unicode[STIX]{x1D700}_{11}^{\prime })\rangle$ which appears in the sixth-order longitudinal structure function equation. In the equation for $\langle E_{6,0}\rangle$ , a source term $\langle F_{6,0}\rangle =15\langle (\unicode[STIX]{x0394}u_{1})^{2}(\unicode[STIX]{x1D700}_{11}+\unicode[STIX]{x1D700}_{11}^{\prime })^{2}\rangle$ appears. In the equation for $\langle F_{6,0}\rangle$ , the term $15\langle (\unicode[STIX]{x1D700}_{11}+\unicode[STIX]{x1D700}_{11}^{\prime })^{3}\rangle$ appears, which generates the new dissipation parameter $\langle \unicode[STIX]{x1D700}_{11}^{3}\rangle$ , and combined with the mixed and transverse equations, one finds an $\unicode[STIX]{x1D700}_{[6]}^{3}$ -term which contains $\langle (\unicode[STIX]{x1D700}+\unicode[STIX]{x1D700}^{\prime })^{3}\rangle$ , i.e. one would expect to find a contribution by $\langle \unicode[STIX]{x1D700}_{r}^{3}\rangle$ in the sixth-order equations. In order to calculate seventh-order inertial-range scaling coefficients from the sixth-order structure function equations and their dissipation source terms, the trace of the structure function equations and two successive source term equations would have to be integrated. An ansatz similar to (4.11) would provide at leading order the exponent $7/3$ . The deviation from this K41 scaling exponent would then involve more empirical terms. Similar arguments will hold for all other uneven-order scaling exponents. Because there are no dissipation parameters in the uneven-order source term equations, even-order scaling exponents cannot be determined in this way.