2.1. Numerical details and computational parameters

In this study, the three-dimensional data from prior DNS studies of the temporally evolving turbulent jet performed by Watanabe et al. (Reference Watanabe, Zhang and Nagata2019b ) and Hayashi et al. (Reference Hayashi, Watanabe and Nagata2021) are used to investigate the effects of external intermittency on the second- and third-order structure functions. In the streamwise $X$ and spanwise $Z$ directions, periodic boundary conditions are imposed and uniformly distributed spacings are used. In the normal $Y$ direction, a free-slip boundary condition is applied and the grid is stretched using a mapping function (Watanabe et al. Reference Watanabe, Zhang and Nagata2019b ). The spatial derivatives are discretised using the fully conservative central difference scheme proposed by Morinishi et al. (Reference Morinishi, Lund, Vasilyev and Moin1998). A second-order scheme is applied in the normal $Y$ direction, while fourth-order difference schemes are applied in the streamwise $X$ and spanwise $Z$ directions. A third-order Runge–Kutta method is employed for the time advancement and the Bi-CGSTAB method is adopted to solve the pressure Poisson equation. Nagarajan, Lele & Ferziger (Reference Nagarajan, Lele and Ferziger2003) and Suzuki et al. (Reference Suzuki, Nagata, Sakai, Hayase, Hasegawa and Ushijima2013) quantified the errors in the first- and second-order derivatives, and higher-order statistics of the fourth-order and second-order fully conservative finite difference schemes using Fourier analysis. The results indicate that each term in the Reynolds-stress transport equation shows good agreement with the corresponding results obtained using spectral methods. For a more detailed examination of the errors associated with the finite difference schemes adopted, please refer to Appendix A of Watanabe et al. (Reference Watanabe, Riley, Nagata, Onishi and Matsuda2018a ).

Table 1.

Numerical parameters for the DNS of temporally evolving turbulent plane jets. The turbulence statistics at $T/T_{\textit{ref}}=20.0$ , including $\Delta X/\eta$ , $\Delta Y/\eta$ , $\Delta Z/\eta$ and $ \textit{Re}_\lambda$ , are computed at the centre plane of the jet.

Table 1. Long description

The table presents numerical parameters for the direct numerical simulation (DNS) of temporally evolving turbulent plane jets. It includes columns for Reynolds number (ReH), number of samples (Ns), initial momentum thickness to half-velocity thickness ratio (θ0/H), and various grid dimensions and spacings. The table has two rows of data, each representing different simulation conditions. The first row shows ReH of , Ns of 6, θ0/H of 1/100, and grid dimensions LX/H of 6, LY/H of 10, and LZ/H of 4, with corresponding grid points NX of 864, NY of 1200, and NZ of 576. The spacings ΔX/η, ΔY/η, and ΔZ/η are 1.5, 1.2, and 1.5 respectively, with a final Reynolds number Reλ of 100. The second row shows ReH of , Ns of 1, θ0/H of 1/100, and grid dimensions LX/H of 6, LY/H of 10, and LZ/H of 4, with corresponding grid points NX of 4608, NY of 5800, and NZ of 3072. The spacings ΔX/η, ΔY/η, and ΔZ/η are 1.7, 1.5, and 1.7 respectively, with a final Reynolds number Reλ of 353. The turbulence statistics at x=6H, including u, v, and w, are computed at the center plane of the jet.

The initial streamwise velocity component of a turbulent jet is given by the hyperbolic tangent function (da Silva & Pereira Reference da Silva and Pereira2008; Hayashi, Watanabe & KNagata Reference Hayashi, Watanabe and Nagata2021), while the normal and spanwise velocity components are set to zero. Considering that the Reynolds number significantly affects the behaviour of structure functions (Pearson & Antonia Reference Pearson and Antonia2001), the present study uses two different cases to investigate the effects of external intermittency on the longitudinal and transverse structure functions. Although high-fidelity and high-resolution three-dimensional DNS data are obtained using state-of-the-art massively parallel computations, the Reynolds number remains limited. Therefore, the influence of small-scale intermittency and the related anomalous scaling is not considered in the present study. The global Reynolds number based on the initial streamwise velocity $U_{\textit{in}}$ along the centreline and the jet width $H$ is defined as $ \textit{Re}_H=U_{\textit{in}} H / \nu$ , and takes the values $10^4$ and $10^5$ . The corresponding mesh numbers are $N_X \times N_Y \times N_Z = 864 \times 1200 \times 576$ and $N_X \times N_Y \times N_Z = 4608 \times 5800 \times 3072$ , respectively. The numerical parameters of DNS are summarised in table 1 and a schematic of the DNS of the turbulent plane jet is shown in figure 1. It is well known that the computational cost of temporally evolving flows is significantly lower than that of spatially evolving flows at the same Reynolds number. In other words, temporally evolving flows often provide a more feasible approach for accessing data at higher Reynolds numbers. Furthermore, for the temporally evolving jet, the ensemble averages are taken over the two homogeneous directions (i.e. the streamwise and spanwise directions), which enables the statistical results to converge readily. Another motivation for selecting the jet configuration arises from our previous studies showing that the second-order structure function in the developing region of spatially evolving wakes is significantly affected by large-scale strong vortex shedding. The observed power-law exponent is $1$ , which is significantly larger than the classical value of $2/3$ . Therefore, it is a reasonable choice to use a temporally evolving plane jet for this investigation. However, a significant limitation of temporally evolving flows is the limited number of spatial samples available in a single instantaneous field. This makes it difficult to obtain statistically converged results, particularly for higher-order statistics such as the third-order structure function. For this reason, the DNS is performed $N_S$ times for each Reynolds number $ \textit{Re}_H$ using different initial velocity perturbations, where the values of $N_S$ are listed in table 1. Note that the strength of the random initial perturbations is consistent and the root mean square (r.m.s.) magnitude of the perturbations is equal to $2.5\,\%$ of the initial mean streamwise velocity at the centre plane. The mean streamwise velocity and the r.m.s. velocity components in all three directions were examined, which showed reasonably good agreement with previous numerical simulations and experimental data. For further description of the numerical validation, one can refer to Watanabe et al. (Reference Watanabe, Zhang and Nagata2019b ) and Hayashi et al. (Reference Hayashi, Watanabe and Nagata2021).

Since the thickness of the TNTI is comparable to the Kolmogorov scale (Watanabe et al. Reference Watanabe, Zhang and Nagata2018b ; Zhang, Watanabe & Nagata Reference Zhang, Watanabe and Nagata2018), high spatial resolution is required for accurate numerical studies of external intermittency. In the centre plane of the jet, the normalised spatial resolutions in the $X$ and $Y$ directions are $ \mathrm{\Delta } X/\eta = 1.5,\, \Delta Y/\eta = 1.2$ for $ Re_H = 10^4$ and $ \Delta X/\eta = 1.7,\, \Delta Y/\eta = 1.5$ for $ Re_H = 10^5$ at $T/T_{\textit{ref}}=20.0$ , where $\eta =(\nu ^{3}/\varepsilon )^{1/4}$ represents the Kolmogorov length scale and $\varepsilon =2\nu \langle s_{\textit{ij}}s_{\textit{ij}}\rangle$ is the kinetic energy dissipation rate. Note that the normalised spatial resolutions in the $X$ and $Z$ directions are identical, i.e. $\Delta X/\eta = \Delta Z/\eta$ . The local Reynolds numbers, $ Re_\lambda$ , is calculated using the Taylor microscale $ \lambda = u_{\textit{rms}} / \sqrt {\langle (\partial u^{\prime }/\partial x)^{2}\rangle }$ and the r.m.s. velocity $ u_{\textit{rms}}$ of the streamwise component along the centre plane of the jet. For $ \textit{Re}_H = 10^4$ and $10^5$ , these values remain approximately constant after reaching the fully developed state at $100$ and $353$ , respectively. Here, the angled brackets $\langle \rangle$ denote the ensemble average of statistical quantities over the $X$ – $Z$ plane, taking into account the flow symmetry in the $Y$ direction. The spatial resolution and turbulent Reynolds number are also presented in table 1.

2.2. Large-scale and small-scale intermittency of the turbulent plane jet

Figure 1.

Schematic of the DNS of a turbulent plane jet. (a) Two-dimensional initial velocity field in the $X$ – $Y$ plane; (b) three-dimensional computational domain together with a randomly chosen vorticity field in the $X$ – $Y$ plane. The initial jet parameters are also shown, where $H$ denotes the initial jet width and $U_{\textit{in}}$ denotes the initial centreline velocity of the jet.

Figure 1. Long description

A schematic of the DNS of a turbulent plane jet. The left part shows a two-dimensional initial velocity field in the plane, with labels indicating the initial jet width and the initial centerline velocity of the jet. The right part displays a three-dimensional computational domain with a randomly chosen vorticity field in the plane. The jet flow is indicated with arrows, and the ambient flow is also labeled.

Figure 2.

Logarithmic contours of the vorticity magnitude $|\boldsymbol{\omega }|$ at the cross-wise position $Y/b_U = 1.6$ . The solid white line indicates the outer edge of the TNTI, which is defined by the vorticity threshold $\omega _{{th}} = 0.01 \langle |\boldsymbol{\omega }|\rangle _{C}$ . (a) $ \textit{Re}_H=10^4$ ; (b) $ \textit{Re}_H=10^5$ . Note that only a part of the computational domain in the $X$ – $Z$ plane is plotted. For reference, the white bar in the lower left corner represents $100$ and $500$ times the Kolmogorov length scale of the jet centre plane.

Figure 2. Long description

A heat map displays the vorticity magnitude at a cross-wise position, with logarithmic contours representing different levels of vorticity. The solid white line marks the outer edge of the turbulent/non-turbulent interface (TNTI), defined by a specific vorticity threshold. The heat map is divided into two parts: (a) and (b). Each part shows a section of the computational domain in the plane. For scale, a white bar in the lower left corner represents a specific length relative to the Kolmogorov length scale of the jet center plane. The color scale at the top ranges from blue to red, indicating varying magnitudes of vorticity, with blue representing lower values and red representing higher values.

Before exploring the second- and third-order structure functions of the intermittent region, the identification of the outer edge of the TNTI layer is briefly discussed. Following previous studies on TNTI, an isosurface of vorticity threshold $|\boldsymbol{\omega }| = |\boldsymbol{\omega }|_{\textit{th}}$ is used to detect the outer edge of the TNTI. The specific value of $|\boldsymbol{\omega }|_{\textit{th}}$ is determined by the sensitivity of the detected turbulent volume fraction $V_T$ to $|\boldsymbol{\omega }|_{\textit{th}}$ (not shown herein for economy of space). Consistent with previous studies by Hayashi et al. (Reference Hayashi, Watanabe and Nagata2021) and Xie et al. (Reference Xie, Xiong, Zheng, Nagata, Watanabe and Zhou2025b ), the vorticity threshold $|\boldsymbol{\omega }|_{\textit{th}}=0.01 \langle |\boldsymbol{\omega }|\rangle _{C}$ was chosen to identify the outer edge of the TNTI, where the subscript $C$ denotes the value at the centre plane of the jet.

Figure 2 visualises the colour contour of the vorticity magnitude $|\boldsymbol{\omega }|$ and the outer edge of the TNTI on an $X$ – $Z$ plane at the cross-wise position $Y/b_U = 1.6$ . Here, $b_U$ indicates the jet half-width, defined as the distance from the centreplane where the mean streamwise velocity falls to half of its value at the centreplane. The highly contorted outer edge of the TNTI can be characterised by the multiscale turbulent motions occurring under the TNTI layer over a wide range of length scales. Therefore, the turbulent region at $ \textit{Re}_H=10^5$ exhibits finer vorticity structures and smaller characteristic length scales compared with that at $ \textit{Re}_H=10^4$ . It should be mentioned that the outer edge of TNTI appears to be relatively smooth, similar to previous observations in turbulent jets (da Silva et al. Reference da Silva, Hunt, Eames and Westerweel2014; Watanabe et al. Reference Watanabe, Silva and Nagata2019a ). This implies that the vorticity threshold selected in the present DNS effectively distinguishes between the turbulent and non-turbulent regions.

As can be seen from figure 2, the temporally evolving plane turbulent jet is a typical intermittent flow and exhibits distinct external intermittency near the TNTI. Once the outer edge of the TNTI is detected, conditional statistics related to the intermittency of the flow can be computed. Figure 3(a) shows the vertical distribution of the intermittent factor $\gamma$ at two different Reynolds numbers. Here, the intermittent factor $\gamma$ is defined as the probability that a fluid point is in a turbulent region at a fixed cross-wise position (Zhou et al. Reference Zhou, Nagata, Sakai, Watanabe, Ito and Hayase2020; Hayashi et al. Reference Hayashi, Watanabe and Nagata2021). By definition, $\gamma =1$ for a fully turbulent region, whereas $\gamma =0$ for a non-turbulent region. It can be seen that the vertical distribution of the intermittency factor $\gamma$ is almost independent of the Reynolds number when normalised by $b_U$ . Specifically, for the flow region with $Y/b_{U} \leqslant 0.8$ , the flow is always turbulent, while it is intermittent in the region with $0.8 \leqslant Y/b_{U} \leqslant 2.5$ , as shown in figure 3(a).

Figure 3.

(a) Vertical distribution of the intermittency factors $\gamma$ ; (b) vertical distributions of the turbulent Reynolds number for a fully developed state. In panel (a), the vertical dashed line separates the fully turbulent region from the intermittent region. In panel (b), $ \textit{Re}_{\lambda }$ denotes the traditional averaged value, while $ \textit{Re}_{\lambda }^{T}$ denotes the conditionally averaged value in the turbulent region.

Figure 3. Long description

The image contains two line graphs. The first graph (a) shows the vertical distribution of the intermittency factors for two different Reynolds numbers, 10 to the power of 4 and 10 to the power of 5. The x-axis represents the normalized vertical distance Y divided by bU, ranging from 0 to 2.5. The y-axis represents the intermittency factor gamma, ranging from -0.25 to 1.25. The graph is divided into a fully turbulent region and an intermittent region by a vertical dashed line at approximately Y/bU equals 1. The second graph (b) shows the vertical distributions of the turbulent Reynolds number for a fully developed state. The x-axis is the same as in the first graph, representing the normalized vertical distance Y divided by bU. The y-axis represents the turbulent Reynolds number on a logarithmic scale, ranging from 10 to the power of 1 to 10 to the power of 3. Two lines are plotted: one for the traditional averaged value and another for the conditionally averaged value in the turbulent region. The lines show variations in the turbulent Reynolds number across the vertical distance. All values are approximated.

Reuther & Kähler (Reference Reuther and Kähler2020) revealed that the traditional averaging methods combine information from both laminar and turbulent flows in the intermittent region, which affects the statistics of small-scale turbulence. Therefore, the applicability of the traditional averaging method in intermittent regions is questionable. For this reason, a conditional averaging method based on the intermittency factor $\gamma$ is employed to analyse the turbulence characteristics within intermittent regions (Kovasznay, Kibens & Blackwelder Reference Kovasznay, Kibens and Blackwelder1970). In particular, for any statistical variable $f$ , the traditional average value can be expressed as a weighted average of two different flow states, that is, $f=\gamma f^T+(1-\gamma )f^L$ . Here, the superscripts $T$ and $L$ denote the ensemble averages in the turbulent and laminar regions, respectively. The vertical distributions of local Reynolds number $ \textit{Re}_\lambda$ and the corresponding conditional local Reynolds number $ \textit{Re}_\lambda ^T$ are shown in figure 3(b). Note that $ \textit{Re}_\lambda ^T$ is calculated by $ \textit{Re}_\lambda ^T=u_{\textit{rms}}^T \lambda ^T / \nu$ , where $u_{\textit{rms}}^T$ and $ \lambda ^T = u_{\textit{rms}}^T/\sqrt {{((\partial u^{\prime }/\partial x)^{2})}^T}$ denote the r.m.s. velocity and Taylor microscale in the turbulent region, respectively. For the fully turbulent region with $\gamma =1$ (corresponding to $Y/b_{U} \leqslant 0.8$ ), the local Reynolds number increases continuously along the normal direction. The maximum value of $ \textit{Re}_\lambda$ occurs at approximately $Y/b_{U} = 1.0$ . Considering that both the mean shear $\partial U /\partial Y$ and the r.m.s. velocity fluctuations (Hayashi et al. Reference Hayashi, Watanabe and Nagata2021; Xie et al. Reference Xie, Zhang, Xiong and Zhou2024) tend to reach their maximum at this location, this finding is expected. For the intermittent region with $1.0 \leqslant Y/b_{U} \leqslant 2.0$ , both $ \textit{Re}_\lambda$ and $ \textit{Re}_\lambda ^T$ decrease monotonically, with $ \textit{Re}_\lambda ^T$ being significantly larger than $ \textit{Re}_\lambda$ , as it should be. Note that a higher conditional local Reynolds number $ \textit{Re}_\lambda ^T$ in the intermittent region is observed over a wide range of intermittency compared with the fully turbulent jet centre plane. This results from the fact that in the intermittent region with lower $\gamma$ values, although $u_{\textit{rms}}^T$ is relatively smaller, $\lambda ^T$ increases significantly and dominates the overall trend.

The large-scale intermittency of the flow in the intermittent region was evaluated by the skewness $S(u^{\prime })=\langle u^{\prime 3}\rangle /\langle u^{\prime 2}\rangle ^{3/2}$ and flatness $F(u^{\prime })=\langle u^{\prime 4}\rangle /\langle u^{\prime 2}\rangle ^{2}$ of the velocity fluctuations (Xie et al. Reference Xie, Xiong, Zheng, Nagata, Watanabe and Zhou2025b ). The results show that large-scale motions in the intermittent region are highly intermittent, with $F(u^{\prime })$ up to 9 and $S(u^{\prime })$ around 2, deviating significantly from the Gaussian values ( $F(u^{\prime })=3.0$ , $S(u^{\prime })=0.0$ ). In this study, the skewness $S(\partial u^{\prime }/\partial X)$ and flatness $F(\partial u^{\prime }/\partial X)$ of the velocity derivative are further investigated to explore small-scale intermittency in the intermittent region. The skewness and flatness of the velocity derivative as a function of the cross-wise position are shown in figure 4. It can be observed that $S(\partial u^{\prime }/\partial X)$ is almost independent of $ \textit{Re}_H$ , whereas $F(\partial u^{\prime }/\partial X)$ increases with increasing $ \textit{Re}_H$ in the fully turbulent region. On the centreplane of the jet, the skewness $S(\partial u^{\prime }/\partial X)$ is nearly identical at the two Reynolds numbers ( $ \textit{Re}=10^4$ and $10^5$ ), with a value of approximately $-0.57$ , while $F(\partial u^{\prime }/\partial X)$ are $5.51$ and $7.52$ , respectively. These values are consistent with the previous research results on small-scale intermittency at comparable local Reynolds numbers (Van Atta & Antonia Reference Van Atta and Antonia1980; Kerr Reference Kerr1985; Kitamura et al. Reference Kitamura, Nagata, Sakai, Sasoh, Terashima, Saito and Harasaki2014). One interesting finding is that the skewness $S^{T}(\partial u^{\prime }/\partial X)$ and flatness $F^{T}(\partial u^{\prime }/\partial X)$ of the conditional velocity derivative remain approximately constant in the region with $ Y/b_{U} \leqslant 1.5$ . This observation supports the argument of Zhou et al. (Reference Zhou, Nagata, Ito, Sakai and Hayase2023) that the turbulent motions in the intermittent region exhibit characteristics similar to those in fully turbulent flows.

Figure 4.

Vertical distributions of (a) the skewness, $S(\partial u^{\prime }/\partial X)$ and $S^{T}(\partial u^{\prime }/\partial X)$ , and (b) the flatness, $F(\partial u^{\prime }/\partial X)$ and $F^{T}(\partial u^{\prime }/\partial X)$ , of the streamwise velocity derivative for two different Reynolds numbers. The vertical dashed line separates the fully turbulent region from the intermittent region.

Figure 4. Long description

The line graph displays vertical distributions of skewness and flatness of the streamwise velocity derivative for two different Reynolds numbers. The x-axis represents the normalized vertical distance Y divided by bU, ranging from 0 to 2.5. The y-axis in panel (a) represents the skewness values, ranging from -3 to 4, while in panel (b) it represents the flatness values, ranging from 0 to 60. The graph includes four data lines for each Reynolds number, with different colors representing different variables. The vertical dashed line separates the fully turbulent region from the intermittent region. All values are approximated.

3.1. Self-similarity of the conditional second-order structure functions

Townsend (Reference Townsend1951) proposed that turbulence in most shear flows reaches a self-similar state after the flow becomes fully developed. In this state, the profiles of certain turbulence statistics exhibit reasonably good collapses when normalised by a set of characteristic length and velocity scales. The recent work by Xie et al. (Reference Xie, Xiong, Zheng, Nagata, Watanabe and Zhou2025b ) investigated the self-similar behaviour of the conditional second-order structure function in the intermittent region, following the self-similarity observed in the conditional energy spectrum (Zhou et al. Reference Zhou, Nagata, Ito, Sakai and Hayase2023). The results indicate that the distribution of the conditional longitudinal structure function exhibits self-similarity at small and intermediate scales when properly normalised using the conditional Kolmogorov scale. The presence of self-similarity implies that the conditional longitudinal structure function can be described using a single similarity variable (Xie et al. Reference Xie, Xiong, Zheng, Nagata, Watanabe and Zhou2025b ), i.e.

(3.1) \begin{align} \big\langle \delta u^{\prime 2}\big\rangle (\zeta )/\left(\gamma \nu _{\eta }^{T2} \right)=g(\zeta ), \end{align}

where $\zeta = r/\eta ^T$ represents the single similarity variable and $\nu _{\eta }^{T}$ denotes the conditional Kolmogorov velocity scale. Note that the difference between the standard and conditional structure functions lies in the choice of normalisation parameters. Therefore, their overall shapes are expected to be the similar. The self-similar conditional second-order longitudinal structure function further allows the derivation of the scaling law for the Kolmogorov constant $C_2^{\textit{LSF}}$ in the intermittent region, namely

(3.2) \begin{align} C_2^{\textit{LSF}} = \gamma ^{1/3} C_2^{\textit{LSF},\,\textit{Cond}}, \end{align}

where $C_2^{\textit{LSF}}$ and $C_2^{\textit{LSF},\,\textit{Cond}}$ are the Kolmogorov constant and the conditional Kolmogorov constant, respectively. Here, $C_2^{\textit{LSF}}$ and $C_2^{\textit{LSF},\,\textit{Cond}}$ are determined by the structure function and conditional structure function, respectively. A more detailed derivation of the conditional second-order structure function and the scaling law for the Kolmogorov constant $C_2^{\textit{LSF}}$ can be found in Appendix B of Xie et al. (Reference Xie, Xiong, Zheng, Nagata, Watanabe and Zhou2025b ). According to the local isotropic assumption, the conditional transverse second-order structure function (TSF) will exhibit self-similar behaviour. This self-similar behaviour is accompanied by a power-law scaling between the prefactor $C_2^{\textit{TSF}}$ of $\langle \delta v^{\prime 2}\rangle (r)$ and the intermittency factor $\gamma$ , that is, $C_2^{\textit{TSF}} = \gamma ^{1/3} C_2^{\textit{TSF},\,\textit{Cond}}$ . In Appendix A, a simplified two-state model effectively establishes the connection between the Kolmogorov constants $C_2^{\textit{LSF}}$ and $C_2^{\textit{TSF}}$ and $\gamma$ .

Figure 5.

(a) Second-order longitudinal structure function, normalised by the Kolmogorov length scale $\eta$ and the averaged dissipation rate $\varepsilon$ , and (b) the corresponding conditional longitudinal structure function, normalised by the conditional Kolmogorov length scale $\eta ^{T}$ and the conditional energy dissipation rate $\varepsilon ^{T}$ , of the streamwise velocity fluctuations at four different cross-wise positions for $ \textit{Re}_H= 10^4$ . The horizontal dashed line shows the asymptotic value of the Kolmogorov constant $C_2^{\textit{LSF}}$ as the Reynolds number approaches infinity. For comparison, the distribution of structure functions is compared with DNS data for isotropic turbulence with $ \textit{Re}_\lambda = 257$ (Ishihara, Gotoh & Kaneda Reference Ishihara, Gotoh and Kaneda2009). The vertical arrow indicates the location corresponding to the integral length scale $L$ at the jet centre plane.

Figure 5. Long description

The two line graphs present the second-order longitudinal structure function and the corresponding conditional longitudinal structure function of the streamwise velocity fluctuations at four different cross-wise positions. The x-axis represents the normalized separation distance, while the y-axis represents the normalized structure function. The graphs include data for four different values of gamma (1.00, 0.75, 0.50, and 0.25), each represented by a different colored line. Additionally, the distribution of structure functions is compared with DNS data for isotropic turbulence with a specific Reynolds number. The horizontal dashed line indicates the asymptotic value of the Kolmogorov constant as the Reynolds number approaches infinity. The vertical arrow marks the location corresponding to the integral length scale at the jet center plane. All values are approximated.

Figure 6.

Same as figure 5, but for $ \textit{Re}_H= 10^5$ .

Figure 6. Long description

The line graph presents the second-order longitudinal structure function for various values of gamma. The x-axis represents the normalized separation distance, while the y-axis shows the normalized structure function. Different colored lines represent different gamma values: red for gamma equals one, blue for gamma equals zero point seven five, green for gamma equals zero point five, and purple for gamma equals zero point two five. Orange triangles represent data from Ishihara et al. Two thousand nine. A horizontal dashed line indicates a reference level. A red arrow marks a specific point on the x-axis. All values are approximated.

In this section, we further investigate the Reynolds number dependence of the self-similarity of the structure function in the intermittent region. Figures 5 and 6 show the normalised longitudinal structure function and the corresponding conditional longitudinal structure function at four different cross-wise positions (i.e. $\gamma = 0.25, 0.50, 0.75$ and $1.00$ ) for two different Reynolds numbers. The statistical convergence of the ensemble average is discussed in Appendix B. The vertical arrow indicates the location corresponding to the integral length scale $L$ at the jet centre plane, where $L$ is determined by the integral of the velocity auto-correlation function. Numerical integration of the velocity auto-correlation function shows that $L$ is approximately $112\eta$ and $638\eta$ for the two different Reynolds numbers ( $ \textit{Re}_H = 10^4$ and $10^5$ ), respectively. Furthermore, the jet half-width $b_U$ is approximately $200\eta$ and $1062\eta$ , respectively.

For the jet centre plane with $\gamma =1.0$ , the distribution of structure functions shows good agreement with the results of isotropic turbulence at similar Reynolds numbers. Kolmogorov’s universal equilibrium theory suggests that the statistical properties of turbulence become independent of the flow type and specific location for the small-scale dissipation range, and follow the classical Kolmogorov scaling law: $\langle \delta u'^2 \rangle (r)/v_\eta ^2 \propto ( r/\eta )^2$ . The good collapse of the structure functions at $r/\eta \leqslant 8$ in figures 5(a) and 6(a) shows good agreement with the theory, except for the highly intermittent region of $\gamma =0.25$ at the lower Reynolds number $ \textit{Re}_H=10^4$ . The present results provide insights into the statistical properties of small-scale turbulence in the intermittent region. It is worth noting that the conditional structure functions are identical to the normal structure functions, with the only difference arising from the normalisation parameters. Therefore, the general profile of the conditional structure function in the intermittent region is unaffected by patch size.

Figures 5(b) and 6(b) show the corresponding conditional longitudinal structure functions for two different Reynolds numbers. It can be observed that the conditional longitudinal structure function profiles at different cross-wise positions reasonably collapse at small and intermediate scales. A previous study (Xie et al. Reference Xie, Xiong, Zheng, Nagata, Watanabe and Zhou2025b ) has shown that the distribution of the conditional second-order longitudinal structure function deviates from self-similarity in the highly intermittent region of $\gamma =0.25$ at the lower Reynolds number $ \textit{Re}_H=40000$ . This slight deviation has been attributed to the blocking/sheltering mechanisms of TNTI (Hunt & Durbin Reference Hunt and Durbin1999; da Silva et al. Reference da Silva, Hunt, Eames and Westerweel2014), since the turbulence near the TNTI layer is characterised by the large-scale inhomogeneity and anisotropy (da Silva et al. Reference da Silva, Hunt, Eames and Westerweel2014; Zecchetto & da Silva Reference Zecchetto and da Silva2021). This result is consistent with the current findings for the lower Reynolds number, as shown in figure 5(b). For a higher Reynolds number $ \textit{Re}_H=10^5$ , however, the slight deviation from the self-similarity of the conditional structure function is not observed (see figure 6 b). This behaviour may be related to the turbulence near the TNTI layer tending to become more isotropic at the higher $ \textit{Re}_H$ , as demonstrated by the analysis of the anisotropy tensors of Reynolds stress and vorticity (Zhang et al. Reference Zhang, Watanabe and Nagata2023). Finally, it should also be mentioned that the reasonable collapse of the conditional structure function in the intermediate range implies that the scaling law of the Kolmogorov constant $C_2^{\textit{LSF}}$ (i.e. $C_2^{\textit{LSF}}\sim \gamma ^{1/3}$ ) is independent of the Reynolds number within the currently investigated range. Therefore, the scaling law of Kolmogorov constant $C_2^{\textit{LSF}}$ derived from the K41 theoretical framework may be considered robust, at least for the two simulations considered.

Figure 7.

(a) Second-order transverse structure function, normalised by the Kolmogorov length scale $\eta$ and the averaged dissipation rate $\varepsilon$ , and (b) the corresponding conditional transverse structure function, normalised by the conditional Kolmogorov length scale $\eta ^{T}$ and the conditional energy dissipation rate $\varepsilon ^{T}$ , of the normal velocity fluctuations at four different cross-wise positions for $ \textit{Re}_H= 10^4$ . The horizontal dashed line shows the asymptotic value of the Kolmogorov constant $C_2^{\textit{TSF}}$ as the Reynolds number approaches infinity, i.e. $\langle \delta v^{\prime 2}\rangle (r)/{(\varepsilon r)}^{2/3}=2.67$ (Pope Reference Pope2000). The vertical arrow indicates the location corresponding to the integral length scale $L$ at the jet centre plane.

Figure 7. Long description

The image contains two line graphs labeled (a) and (b). Graph (a) shows the second-order transverse structure function normalized by the Kolmogorov length scale and the averaged dissipation rate. Graph (b) shows the corresponding conditional transverse structure function normalized by the conditional Kolmogorov length scale and the conditional energy dissipation rate. Both graphs plot the data for four different cross-wise positions. The x-axis represents the normalized distance r divided by the Kolmogorov length scale, while the y-axis represents the normalized structure function. The horizontal dashed line in both graphs indicates the asymptotic value of the Kolmogorov constant as the Reynolds number approaches infinity. The vertical arrow in both graphs marks the location corresponding to the integral length scale at the jet center plane. The data lines are color-coded for different values of gamma: red for gamma equals 1.00, blue for gamma equals 0.75, green for gamma equals 0.50, and purple for gamma equals 0.25. All values are approximated.

The longitudinal structure function is related to the energy dissipation, while the transverse structure function reflects the characteristics of the enstrophy field. Numerous studies (Dhruva, Tsuji & Sreenivasan Reference Dhruva, Tsuji and Sreenivasan1997; Shen & Warhaft Reference Shen and Warhaft2002; Grauer, Homann & Pinton Reference Grauer, Homann and Pinton2012; Iyer, Sreenivasan & Yeung Reference Iyer, Sreenivasan and Yeung2017) have shown that the longitudinal and transverse structure functions are different. One notable difference is that the scaling exponent of the transverse structure function is always smaller than that of the longitudinal structure function, although the K41 theory predicts identical scaling exponents for both. It is not surprising that different physical fields exhibit distinct degrees of intermittency. So far, universal relations between the longitudinal and transverse structure functions have been established in the fully turbulent region. To assess the local isotropy of turbulence, further analysis has been conducted on the transverse structure function. The normalised second-order transverse structure function, and the corresponding conditional transverse structure function are plotted in figures 7 and 8 for two different Reynolds numbers. It can be seen from figures 7(a) and 8(a) that the shape of the structure function appears to remain almost unchanged across four different cross-wise positions. This observation somewhat echoes the argument by Kuznetsov, Praskovsky & Sabelnikov (Reference Kuznetsov, Praskovsky and Sabelnikov1992) that the external intermittency induced by TNTI primarily affects the Kolmogorov constant without altering the scaling exponent of the structure function. For a higher Reynolds number $ \textit{Re}_H$ , the Kolmogorov constant $C_2^{\textit{TSF}}$ in the fully turbulent region approaches the theoretical value of infinite Reynolds number. Moreover, in the intermittent region, the Kolmogorov constant $C_2^{\textit{TSF}}$ decreases as the intermittent factor $\gamma$ decreases.

Figure 8.

Same as figure 7, but for $ \textit{Re}_H= 10^5$ .

Figure 8. Long description

The line graph presents the normalized second-order structure function of longitudinal velocity increments as a function of the separation distance. The x-axis represents the separation distance normalized by the Kolmogorov length scale, while the y-axis shows the normalized structure function. The graph includes multiple data lines corresponding to different values of the parameter gamma, specifically gamma equals 1.00, 0.75, 0.50, and 0.25. Each line is color-coded: red for gamma equals 1.00, blue for gamma equals 0.75, green for gamma equals 0.50, and purple for gamma equals 0.25. A horizontal dashed line is present, indicating a reference value. An arrow on the x-axis marks a specific point of interest. All values are approximated.

The corresponding conditional transverse structure functions for the two different Reynolds numbers are shown in figures 7(b) and 8(b). For four different cross-wise positions, profiles of the conditional transverse structure function exhibit self-similarity at small and intermediate scales. For the higher Reynolds number case in figure 8(b), a more satisfactory collapse is observed. This means that the turbulent properties in the intermittent region are similar to those of the fully turbulent flow. This idea was first proposed by Zhou et al. (Reference Zhou, Nagata, Ito, Sakai and Hayase2023) in spatially evolving wakes to explain the presence of a non-Kolmogorov – $5/3$ spectra (Zhou et al. Reference Zhou, Nagata, Sakai and Watanabe2019) in the highly intermittent region. It is worth noting that, although the turbulence statistics related to structure functions are shown only in log–log scales, we confirm that the reported findings are not sensitive to different coordinate scales (e.g. linear–log scale). Therefore, our conclusions are expected to be robust, especially for the current higher Reynolds number (not shown herein for economy of space).

Another probably interesting finding is that, compared with the fully turbulent region with $\gamma = 1.0$ , the intermittent region exhibits significant mean shear, which is expected to affect the behaviour of the structure functions, at least to some extent (Thiesset, Antonia & Djenidi Reference Thiesset, Antonia and Djenidi2014; Sadeghi, Lavoie & Pollard Reference Sadeghi, Lavoie and Pollard2015). Here, the mean shear effect on the structure functions is further investigated through the Corrsin scale (Corrsin Reference Corrsin1958). The Corrsin scale is obtained by equating the eddy-turnover time scale to the mean-shear time scale, yielding $L_c=(\varepsilon /S^3)^{1/2}$ , where $S=\partial U/\partial Y$ . The Corrsin scale separates the region dominated by mean shear ( $r \gt L_c$ ) from the inertial region ( $r \lt L_c$ ). Therefore, the onset of shear-affected scales may be characterised by $r = L_c$ , as reported by Rosa (Reference Rosa2026). Based on theoretical analysis, mean shear effects do not have a significant influence on the behaviour of the energy spectrum over the corresponding wavenumber range when $kL_c \gg 8^{-1/2}$ , where $k$ represents the angular wavenumber. However, the theoretical criterion $kL_c \gg 8^{-1/2}$ does not provide a clearly defined numerical threshold. This criterion remains somewhat ambiguous when used to quantitatively characterise the influence of mean shear on the structure functions. According to the experimental findings of Saddoughi & Veeravalli (Reference Saddoughi and Veeravalli1994) at high Reynolds numbers, the influence of mean shear on the energy spectrum can be considered negligible when $kL_c\gt 3$ . Correspondingly, the mean shear does not affect the behaviour of the structure functions when the separation scale $r \lt 2\pi L_c/3$ in physical space. Saddoughi & Veeravalli (Reference Saddoughi and Veeravalli1994) further concluded that the inertial subrange remains unaffected by mean shear only when the condition $S_c^{*} = S(\nu /\varepsilon )^{1/2} \ll 1$ is satisfied, where $S_c^{*}$ denotes the ratio of the Kolmogorov to mean-shear time scales. It should be noted that these conclusions apply only at higher Reynolds numbers.

To quantitatively assess the mean shear effects, the Corrsin scale $L_c$ and the related shear parameters for three intermittent regions for the higher Reynolds number case of $ \textit{Re}_H=10^5$ are summarised in table 2. In this study, the inertial subrange is determined to be within the range of $r/\eta ^T$ = $60.0$ to $100.0$ . Table 2 implies that the scale-dependent shear parameter $2\pi L_c^T/3$ ranges from $238\eta ^T$ to $307\eta ^T$ , which is significantly larger than the separation scale r considered in the structure function analysis. Furthermore, the conditional ratio $S^T(\nu /\varepsilon ^T)^{1/2}$ remains below 0.043 in three intermittent regions, which is much smaller than 1. These quantitative results confirm that the mean shear effect is not strong enough to reach the inertial range. As shown in figures 6(b) and 8(b), differences in the structure function profiles can only be found at large scales, which suggests that the influence of mean shear on the structure functions at small and intermediate scales can be negligible. Conversely, for lower Reynolds numbers, the mean shear effect may extend into the inertial range of the structure functions to some extent.

Table 2.

Turbulence statistics associated with mean shear at three intermittent regions.

Table 2. Long description

The table presents turbulence statistics associated with mean shear at three intermittent regions. It includes columns for gamma, Lc, LcT, 2πLc/3, 2πLcT/3, S(v/ε)^1/2, and ST(v/ε)^T^1/2. The rows provide specific values for these parameters at different gamma levels of 0.75, 0.50, and 0.25. The values are given in terms of eta and include both numerical and symbolic representations. The table highlights variations in turbulence statistics across different regions and shear parameters.

The scaling law $C_2^{\textit{LSF}} = \gamma ^{1/3} C_2^{\textit{LSF},\,\textit{Cond}}$ for $\langle \delta u^{\prime 2}\rangle (r)$ has been evaluated and shows fairly good agreement with the intermittency factor $\gamma$ (Xie et al. Reference Xie, Xiong, Zheng, Nagata, Watanabe and Zhou2025b ). To evaluate the scaling law $C_2^{\textit{TSF}} = \gamma ^{1/3} C_2^{\textit{TSF},\,\textit{Cond}}$ for $\langle \delta v^{\prime 2}\rangle (r)$ , figure 9 presents the Kolmogorov constant $C_2^{\textit{TSF}}$ , and conditional Kolmogorov constant $C_2^{\textit{TSF},\,\textit{Cond}}$ and $\gamma ^{1/3} C_2^{\textit{TSF},\,\textit{Cond}}$ versus the intermittency factor $\gamma$ . The self-similar behaviour of the conditional transverse structure functions at intermediate scales implies that $C_2^{\textit{TSF},\,\textit{Cond}}$ is approximately constant, i.e. $C_2^{\textit{LSF}} \sim \gamma ^{1/3}$ . The inertial subrange is selected to lie within the range $r/\eta ^T = 60{-}100$ , where the second-order structure function exhibits a distinct plateau region (see figures 7 and 8). It can be observed from figure 9 that the scaling law $C_2^{\textit{TSF}}\sim \gamma ^{1/3}$ holds roughly throughout the intermittent region. Moreover, the scaling law for the Kolmogorov constant $C_2^{\textit{TSF}}$ is influenced by the Reynolds number. For a higher Reynolds number $ \textit{Re}_H=10^5$ , the scaling law $C_2^{\textit{TSF}}\sim \gamma ^{1/3}$ becomes more consistent. In summary, these findings confirm that the scaling law of the Kolmogorov constant for $\langle \delta v^{\prime 2}\rangle (r)$ can be considered robust, particularly at higher Reynolds numbers.

Figure 9.

Kolmogorov constant $C_2^{\textit{TSF}}$ and conditional Kolmogorov constant $C_2^{\textit{TSF},\,\textit{Cond}}$ as a function of intermittency factors $\gamma$ within the inertial subrange for two different Reynolds numbers. For comparison, the profile of $\gamma ^{1/3} C_2^{\textit{TSF},\,\textit{Cond}}$ is also plotted. (a) $ \textit{Re}_H=10^4$ , (b) $ \textit{Re}_H=10^5$ . The horizontal dashed line shows the asymptotic value of the Kolmogorov constant $C_2^{\textit{TSF}}$ as the Reynolds number approaches infinity, i.e. $\langle \delta v^{\prime 2}\rangle (r)/{(\varepsilon r)}^{2/3}=2.67$ (Pope Reference Pope2000).

Figure 9. Long description

The two line graph presents the Kolmogorov constant and conditional Kolmogorov constant as a function of intermittency factors within the inertial subrange for two different Reynolds numbers. The x axis represents the intermittency factor, ranging from 0.25 to 1.00. The y axis represents the Kolmogorov constant and conditional Kolmogorov constant, ranging from 0 to 4. The red line with square markers represents the conditional Kolmogorov constant, while the blue line with triangle markers represents the Kolmogorov constant. Additionally, a purple line with circle markers represents the product of the intermittency factor raised to the power of one-third and the conditional Kolmogorov constant. The horizontal dashed line indicates the asymptotic value of the Kolmogorov constant as the Reynolds number approaches infinity. All values are approximated.

Figure 10.

(a) Fourth- and sixth-order structure functions and (b) the corresponding conditional structure functions at four different cross-wise positions for $ \textit{Re}_H= 10^4$ . The horizontal dashed lines indicate the asymptotic values of the Kolmogorov constants $C_4^{\textit{LSF}}$ and $C_6^{\textit{LSF}}$ (18.18 and 413.33, respectively) as the Reynolds number approaches infinity. These asymptotic values are determined based on an empirical formula for the structure function (Tang et al. Reference Tang, Antonia, Djenidi, Danaila and Zhou2017).

Figure 10. Long description

The line graph presents fourth- and sixth-order structure functions and their corresponding conditional structure functions at four different cross-wise positions. The x-axis represents the normalized separation distance, while the y-axis represents the normalized structure functions. The graph includes data lines for different values of n, specifically n equals 4 and n equals 6. Each data line is color-coded to represent different values of gamma, with red for gamma equals 1.00, cyan for gamma equals 0.75, green for gamma equals 0.50, and purple for gamma equals 0.25. The horizontal dashed lines indicate the asymptotic values of the Kolmogorov constants as the Reynolds number approaches infinity. All values are approximated.

3.2. Self-similarity of the conditional higher-order structure functions

Figure 10 presents the higher even-order (fourth- and sixth-order) structure functions and the corresponding conditional structure functions for $ \textit{Re}_H=10^4$ . The self-similarity of higher-order structure functions differs significantly from that of the second-order structure function discussed already. The profiles of the conditional structure functions exhibit a gradual departure from self-similarity and Kolmogorov’s universal equilibrium theory from the jet centre to the edge. Even within the small-scale dissipative range, the conditional fourth- and sixth-order structure functions exhibit clear non-self-similar profiles, with the deviations being more pronounced for the sixth-order case. This finding is consistent with the standard paradigm of higher-order statistics, which are not universal and are highly sensitive to factors such as external intermittency, the FRN effect and intrinsic characteristics of TNTI. A similar observation was also reported in Gauding et al. (Reference Gauding, Bode, Brahami, Varea and Danaila2021). Moreover, they employed the extended self-similarity (ESS) framework to investigate the relationship between the scaling exponent of the inertia range of the structure function and external intermittency. Their results indicate that the application of the ESS framework does not improve the relative scaling exponent of the inertia range of higher-order structure functions in intermittent regions. Note that the results at the higher Reynolds number $ \textit{Re}_H=10^5$ are consistent with those at $ \textit{Re}_H=10^4$ . For a higher Reynolds number, the Kolmogorov constants in the fully turbulent region $C_4^{\textit{LSF}}$ and $C_6^{\textit{LSF}}$ are closer to their theoretical values at infinite Reynolds number (see figure 11).

Figure 11.

Same as figure 10, but for $ \textit{Re}_H= 10^5$ .

Figure 11. Long description

The line graph consists of two panels labeled (a) and (b). Panel (a) plots the normalized second-order structure function against the normalized separation distance, while panel (b) plots the normalized third-order structure function against the normalized separation distance. Each panel contains multiple lines representing different values of n (4 and 6) and gamma (0.25, 0.50, 0.75, and 1.00). The x-axis in both panels is labeled with the normalized separation distance, and the y-axis is labeled with the respective normalized structure function. The lines show how these structure functions vary with the separation distance for different values of n and gamma. The data points on the lines indicate the values of the structure functions at specific separation distances. All values are approximated.

Figure 12.

Same as figures 10(b) and 11(b), but for the vorticity threshold $|\boldsymbol{\omega }|_{\textit{th}}=0.14 \langle |\boldsymbol{\omega }|\rangle _{C}$ . (a) $ \textit{Re}_H= 10^4$ ; (b) $ \textit{Re}_H= 10^5$ .

Figure 12. Long description

The line graph presents the vorticity threshold for different values of n and gamma. The x-axis represents the normalized distance r divided by the Kolmogorov length scale, ranging from to . The y-axis represents the normalized structure function, ranging from to . The graph includes four data lines, each corresponding to a different value of gamma: 1.00, 0.75, 0.50, and 0.25. Each data line shows the behavior of the structure function for two different values of n: 4 and 6. The dashed horizontal lines indicate the values of n. The data lines show how the structure function varies with the normalized distance for each value of gamma and n. All values are approximated.

Owing to the blocking/sheltering mechanisms of TNTI (Hunt & Durbin Reference Hunt and Durbin1999; da Silva et al. Reference da Silva, Hunt, Eames and Westerweel2014), the turbulence near the TNTI layer can significantly affect the self-similarity of $\langle \delta u^{\prime 2}\rangle (r)$ (Xie et al. Reference Xie, Xiong, Zheng, Nagata, Watanabe and Zhou2025b ). Their analysis further shows that the conditional second-order structure functions recover self-similarity once the turbulent region within a distance approximately equal to the mean TNTI thickness from its outer edge is excluded. In this study, to eliminate the effect of the TNTI blocking/sheltering mechanism on higher-order structure functions, we computed the conditional structure functions in the high-enstrophy region with $|\boldsymbol{\omega }|_{\textit{th}}=0.14 \langle |\boldsymbol{\omega }|\rangle _{C}$ , presented in figure 12. Here, the high-vorticity threshold is consistent with the value reported by Xie et al. (Reference Xie, Xiong, Zheng, Nagata, Watanabe and Zhou2025b ), who found that the average distance between two different iso-surfaces (i.e. $|\boldsymbol{\omega }|_{\textit{th}}=0.14 \langle |\boldsymbol{\omega }|\rangle _{C}$ and $0.01 \langle |\boldsymbol{\omega }|\rangle _{C}$ ) is comparable to the mean thickness of the TNTI. Note that the characteristic scales used for normalisation (i.e. $\varepsilon ^{T}$ , $\eta ^T$ and $\gamma$ ) are also determined using the modified threshold to mitigate the effects of TNTI. Compared with figures 10(b) and 11(b), the profiles of the conditional structure functions exhibit a satisfactory collapse at small and intermediate scales in the high-enstrophy region, except for the sixth-order structure function at $ \textit{Re}_H=10^5$ . This may be related to the high sensitivity of higher-order structure functions to the external intermittency. In summary, our findings suggest that the large-scale inhomogeneity and anisotropy near the TNTI significantly affect the behaviour of higher-order structure functions, leading to their deviation from self-similarity.

4.1. Approach to the Kolmogorov $4/5$ law in the presence of external intermittency

The self-similar behaviour of the conditional third-order structure function is further investigated in the context of the self-similar behaviour of the conditional second-order structure function. The third-order structure function, in contrast, is directly related to the energy flux in the inertial range through the Kolmogorov $4/5$ law and its distribution provides insights into the energy cascade within intermittent regions. The normalised third-order structure functions $ -\langle \delta u^{\prime 3}\rangle (r)/ ( r \varepsilon )$ at four different cross-wise positions are plotted (see figures 13 a and 14 a). For the lower local Reynolds number $ \textit{Re}_\lambda =100$ , the third-order structure function distribution in the central plane of the jet shows good agreement with DNS results for grid turbulence at Reynolds number $ \textit{Re}_\lambda = 100$ (Antonia & Burattini Reference Antonia and Burattini2006). It is worth noting that they actually plotted $ -\langle \delta u^{\prime 3}\rangle (r)/ ( r \varepsilon )$ versus $r/\lambda$ , instead of $ -\langle \delta u^{\prime 3}\rangle (r)/ ( r \varepsilon )$ versus $r/\eta$ . According to the local isotropic assumption, the Taylor microscale $\lambda$ and the Kolmogorov length scale $\eta$ satisfy $\lambda /\eta =15^{1/4}(Re_\lambda )^{1/2}$ . For a local Reynolds number of $ \textit{Re}_\lambda = 100$ , the value of $\lambda /\eta$ estimated from this formula is approximately 20. In other words, Antonia & Burattini (Reference Antonia and Burattini2006) actually plotted $ -\langle \delta u^{\prime 3}\rangle (r)/ ( r \varepsilon )$ versus $r/(20\eta )$ , applying a scaling factor of $ \textit{SF}=20$ . For comparison, after removing the scaling factor $ \textit{SF}$ , the distribution of the structure function collapses together at almost all scales. However, for two different Reynolds numbers, the expected Kolmogorov $4/5$ law is not achieved, although it is closer at higher Reynolds numbers (see figure 14 a).

Figure 13.

(a) Third-order structure function, normalised by the Kolmogorov length scale $\eta$ and the averaged dissipation rate $\varepsilon$ , and (b) the corresponding conditional third-order structure function, normalised by the Kolmogorov length scale $\eta ^{T}$ and the averaged dissipation rate $\varepsilon ^{T}$ , of the streamwise velocity fluctuations at four different cross-wise positions for $ \textit{Re}_H= 10^4$ . For comparison, the distribution of structure functions is compared with DNS data for grid turbulence with $ \textit{Re}_\lambda = 100$ (Antonia & Burattini Reference Antonia and Burattini2006).

Figure 13. Long description

The image contains two line graphs. The first graph (a) shows the third-order structure function normalized by the Kolmogorov length scale and the averaged dissipation rate. The second graph (b) shows the corresponding conditional third-order structure function normalized by the Kolmogorov length scale and the averaged dissipation rate of the streamwise velocity fluctuations at four different cross-wise positions for Reynolds number 100. The data is compared with DNS data for grid turbulence with Reynolds number 100. The x-axis represents the normalized separation distance, and the y-axis represents the normalized third-order structure function. The lines are color-coded for different values of gamma: 1.00, 0.75, 0.50, and 0.25. All values are approximated.

The maximum values of the structure function $ -\langle \delta u^{\prime 3}\rangle (r)/ ( r \varepsilon )$ on the central plane are $0.52$ and $0.65$ , and reveal a significant Reynolds number dependence, as reported by Pearson & Antonia (Reference Pearson and Antonia2001). Recent work by Tang, Antonia & Djenidi (Reference Tang, Antonia and Djenidi2023) shows that the Reynolds number effect may differ for odd and even orders. Specifically, $-\langle \delta u^{\prime 3}\rangle (r)/ ( r \varepsilon )$ requires a higher Reynolds number than $\langle \delta u^{\prime 2}\rangle (r)/ ( r \varepsilon )^{2/3}$ to establish a discernible inertial range. In the intermittent region, the maximum value $ -\langle \delta u^{\prime 3}\rangle (r)/ ( r \varepsilon )$ decreases with decreasing $\gamma$ (see figure 13 a). Highly intermittent regions are associated with larger deviations from the Kolmogorov $4/5$ law. However, the two-state model analysis in Appendix A suggests that this result cannot be attributed to external intermittency, as it does not affect the Kolmogorov constant in the inertial range for $-\langle \delta u^{\prime 3}\rangle (r)$ . Instead, one may guess that this is attributed to large-scale motions in the flow. Compared with the fully turbulent region with $\gamma =1.0$ , the intermittent region exhibits significant mean shear, which is expected to affect the behaviour of $-\langle \delta u^{\prime 3}\rangle (r)$ in the inertial range.

Figures 13(b) and 14(b) show the corresponding conditional third-order structure functions for two different Reynolds numbers. For the centre plane of the jet where $\lambda ^T/\eta ^T =\lambda /\eta$ and $ \textit{Re}_\lambda ^{T}=Re_\lambda$ , the DNS result for grid turbulence is also included. The expected self-similarity is not observed at small and intermediate scales. Even for higher Reynolds number, the structure function exhibits self-similarity only at small scales where $r/\eta ^T \leqslant 7$ . One interesting finding is that the range of self-similarity extends to increasingly larger values of $r/\eta ^T$ at a higher Reynolds number $ \textit{Re}_H$ .

Figure 14.

Same as figure 13, but for $ \textit{Re}_H= 10^5$ . For comparison, the distribution of structure functions is compared with DNS data for isotropic turbulence with $ \textit{Re}_\lambda = 471$ (Ishihara et al. Reference Ishihara, Gotoh and Kaneda2009).

Figure 14. Long description

The line graph compares the distribution of structure functions with DNS data for isotropic turbulence. The x-axis represents the normalized separation distance, while the y-axis represents the normalized third-order structure function. Four different data series are plotted, each corresponding to a different cross-wise position. The data series are color-coded: red for gamma equals one point zero zero, blue for gamma equals zero point seven five, green for gamma equals zero point five zero, and orange for Reynolds number lambda equals four seven one. The graph shows how the structure functions vary with the separation distance for different positions and conditions. All values are approximated.

4.2. Contribution of the unsteady term

The transport equation for $\langle \delta u^{{\prime }2}\rangle (r)$ , i.e. the general form of the Kármán–Howarth equation (Kolmogorov Reference Kolmogorov1941c ), is used to further investigate the deviation of the third-order structure function from self-similarity at intermediate scales:

(4.1) \begin{align} -\big\langle \delta u^{\prime 3}\big\rangle (r)=\frac {4}{5}r \varepsilon -6\nu \frac {\partial }{\partial r}\big\langle \delta u^{\prime 2}\big\rangle (r)-I_u(r), \end{align}

where the second term on the right-hand side denotes the viscous term. The last term represents the unsteady term $I_u(r)$ , which reflects the effect of large-scale motion on the structure function (Danaila et al. Reference Danaila, Anselmet, Zhou and Antonia1999; Hill Reference Hill2001; Kaneda, Yoshino & Ishihara Reference Kaneda, Yoshino and Ishihara2008). For the temporally evolving plane jet, $I_u(r)$ can be expressed as (Danaila et al. Reference Danaila, Anselmet, Zhou and Antonia2001; Kaneda et al. Reference Kaneda, Yoshino and Ishihara2008)

(4.2) \begin{align} I_u(r)=-\frac {3}{r^4}\int _0^rs^4\left [\frac {\partial \big\langle \delta u^{\prime 2}\big\rangle (s)}{\partial t}\right ]\mathrm{d}s-\frac {6}{r^4}\int _0^rs^4\left [\frac {\partial \big\langle \delta v^{\prime }\delta u^{\prime 2}\big\rangle (s)}{\partial Y}\right ]\mathrm{d}s, \end{align}

where $s$ is a dummy variable. A detailed description of the self-similarity analysis of the third-order structure function is provided in Appendix C. Note that the self-similarity analysis based on the Kármán–Howarth equation neglects the contribution of $I_u(r)$ , since this term can be considered negligible in the inertial range under the assumption of infinite Reynolds number. However, for the current finite Reynolds number case, the unsteady term is expected to have a significant effect on $-\langle \delta u^{\prime 3}\rangle (r)/ ( r \varepsilon ^T )$ (Antonia & Burattini Reference Antonia and Burattini2006; Tang et al. Reference Tang, Antonia and Djenidi2023), especially in the intermittent region. Therefore, it is important to consider the contribution of the unsteady term, as shall be discussed later.

After dividing (4.1) by $r \varepsilon$ , the normalised form of the equation is given by

(4.3) \begin{align} -T_{r}^* = 4/5 - V_{r}^* - I_u^*(r), \end{align}

where the superscript $*$ represents the normalised variable. In particular, (4.3) can be written as $-T_{\lambda ^T}^*=4/5 - V_{\lambda ^T}^* - I_u^*(\lambda ^T)$ when $r = \lambda ^T$ . It should be noted that $r = \lambda ^T$ represents a scale located near the lower end of the inertial range. Tang et al. (Reference Tang, Antonia, Djenidi, Danaila and Zhou2017) have investigated the effect of $I_u^*(\lambda ^T)$ on the Kolmogorov $4/5$ law in different flow types (e.g. grid turbulence, circular and square cylinder wakes, jet axis and channel centreline, etc.). These studies show that $I_u^*(\lambda ^T)$ can significantly influence the behaviour of scales within the inertial range. To date, the influence of the dynamical characteristics of the unsteady term $I_u^*(\lambda ^T)$ on the third-order structure function in the intermittent region still remains elusive.

Figure 15.

Variation of each term in the generalised transport equation with respect to the separation distance $r$ on the central plane of the jet at two different Reynolds numbers. Only the scale range where the deviation from local isotropy is less than $20\,\%$ is shown. The left and right vertical arrows indicate the positions of $r = \lambda ^T$ at $ \textit{Re}_{\lambda }^T=100$ and $353$ , respectively. For comparison, each term of the transport equation is compared with DNS data from decaying grid turbulence at the same Taylor Reynolds number of $ \textit{Re}_{\lambda }^T=100$ (Antonia & Burattini Reference Antonia and Burattini2006). Note that each term in the transport equation from the results of Antonia & Burattini (Reference Antonia and Burattini2006) has been rescaled by removing the scaling factor $ \textit{SF} = 20$ , which is determined from the relation $\lambda ^T/\eta ^T=15^{1/4}(Re_\lambda ^{T})^{1/2}$ .

Figure 15. Long description

The line graph displays the variation of each term in the generalised transport equation with respect to the separation distance on the central plane of the jet at two different Reynolds numbers. The x-axis represents the separation distance normalized by the Taylor microscale, while the y-axis represents the normalized terms of the transport equation. The graph includes multiple data lines representing different terms and conditions. The orange and brown lines correspond to the HIT case with a Reynolds number of 100, showing the normalized temperature structure function, radial velocity structure function, and longitudinal velocity increment. The red, blue, and green lines correspond to the Jet case with a Reynolds number of 100, showing similar terms. The purple, dark blue, and gray lines correspond to the Jet case with a Reynolds number of 353, showing the same terms. The left and right vertical arrows indicate the positions of the Taylor microscale at different Reynolds numbers. The graph also includes comparison data from decaying grid turbulence at the same Taylor Reynolds number. All values are approximated.

Considering the significant effect of the Reynolds number on the third-order structure function $-\langle \delta u^{\prime 3}\rangle (r)$ (see figures 13 and 14), the contribution of the unsteady term $I_u^*(r)$ is evaluated at three different cross-wise positions with $\gamma =1.0$ , $0.75$ and $0.5$ . Each term of the generalised transport equation on the jet central plane at two different Reynolds numbers is plotted in figure 15. Note that only the scale range where the deviation from local isotropy is less than $20\,\%$ is shown, i.e. $r/\eta ^T \lt 500$ (see figure 23). Each term of the equation in the fully turbulent region shows reasonably good agreement with DNS results for grid turbulence at a similar Taylor Reynolds number $ \textit{Re}_{\lambda }^T=100$ across almost all scales (Antonia & Burattini Reference Antonia and Burattini2006). This suggests that the scale-by-scale energy budget may exhibit similar statistical characteristics even for different types of turbulent flows. Tang et al. (Reference Tang, Antonia, Djenidi, Danaila and Zhou2017) found that the value of $I_u^*(\lambda ^T)$ at $r = \lambda ^T$ varies significantly across different flows. Specifically, $I_u^*(\lambda ^T)$ is smallest in stationary forced periodic box turbulence and largest along the centreline of the channel flow. With increasing $ \textit{Re}_{\lambda }^T$ , the unsteady term $I_u^*(r)$ gradually decreases, while the third-order structure function $T_{r}^*$ gradually increases. In contrast, the viscous term $V_{r}^*$ appears to be independent of the local Reynolds number $ \textit{Re}_{\lambda }^T$ . The unsteady term $I_u^*(r)$ is negligible when $r/\eta ^T\lt 8$ , indicating that large-scale motions have little effect on small-scale motions, as reported by Danaila et al. (Reference Danaila, Anselmet, Zhou and Antonia1999).

The examination of local isotropy at each scale is provided in Appendix C. The local isotropy approximately satisfied at $r = \lambda ^T$ enables us to make a rough estimate of the contribution of $I_u(r)$ in the intermittent region at the same scale. Table 3 lists the values of each term $T_{\lambda ^T}^*$ , $V_{\lambda ^T}^*$ and $I_u^*({\lambda ^T})$ in the transport equation at $r = \lambda ^T$ for three different cross-wise positions with $\gamma =1.0$ , $0.75$ and $0.5$ at two Reynolds numbers ( $ \textit{Re}_H=10^4$ and $10^5$ ). As expected, the contribution of the unsteady term $I_u^*({\lambda ^T})$ in the intermittent region is larger than that in the jet central plane at a fixed separation distance $r = \lambda ^T$ . According to § 3.1, the conditional longitudinal second-order structure function exhibits self-similarity at small and intermediate scales, indicating that the viscous term $V_{r}^*$ also exhibits self-similarity over these scales. For two different cross-wise positions (i.e. $\gamma =1.0$ and $0.5$ ) of $ \textit{Re}_H=10^4$ , it is also worth noting that the two positions have approximately the same conditional local Reynolds number $ \textit{Re}_\lambda ^T$ . This indicates that the significant deviation of $T_{\lambda ^T}^*$ at two different cross-wise positions is unlikely to be caused by Reynolds number effects. Instead, this deviation is attributed to the contribution of $I_u^*({\lambda ^T})$ . For the lower Reynolds number, the values of $I_u^*({\lambda ^T})$ at $\gamma =1.0$ and $0.5$ are 0.127 and 0.450, respectively, while for the higher Reynolds number, the values at the same positions are 0.081 and 0.344. This means that the contribution of $I_u^*({\lambda ^T})$ decreases as $ \textit{Re}_\lambda ^T$ increases and increases as $\gamma$ decreases. In summary, the larger contribution of $I_u^*({\lambda ^T})$ at $\gamma =0.5$ compared with that at $\gamma =1.0$ indicates that large-scale motions are responsible for the significant deviation from self-similarity of the third-order structure function within the inertial range (see figures 13 b and 14 b). Nevertheless, as the local Reynolds number increases, the contribution of $I_u^*({r})$ in the inertial range gradually decreases and, thus, the self-similar behaviour of the third-order structure function is expected to be recovered at sufficiently high Reynolds numbers.

Table 3.

Values of each term $T_{\lambda ^T}^*$ , $V_{\lambda ^T}^*$ and $I_u^*({\lambda ^T})$ in the transport equation at $r = \lambda ^T$ for there different cross-wise positions with $\gamma =1.0$ , $0.75$ and $0.5$ at two different Reynolds numbers ( $ \textit{Re}_H=10^4$ and $10^5$ ). The conditional local Reynolds numbers $ \textit{Re}_\lambda ^T$ at two cross-wise positions are also listed.

Table 3. Long description

The table presents values of terms in the transport equation for three different cross-wise positions with varying conditions and at two different Reynolds numbers. The table has 2 rows and 10 columns. The columns are labeled as ReH, gamma, ReTA, T*AT, V*AT, I*u(λT), ReH, gamma, ReTλ, T*AT, V*AT, and I*u(λT). The rows provide data for different Reynolds numbers and cross-wise positions. Notable trends include the variation in the contribution of the unsteady term in the intermittent region compared to the jet central plane at a fixed separation distance. The table also highlights the self-similarity of the viscous term over small and intermediate scales. The conditional local Reynolds numbers at two cross-wise positions are also listed, showing that the deviation in certain values is not due to Reynolds number effects but rather the contribution of other factors.

5.1. Comparison between the empirical prediction and numerical data at $\gamma =1.0$

To further investigate the FRN effect of the structure functions, an empirical formula for $\langle \delta u^{\prime n}\rangle (r)$ is used to indirectly estimate the behaviour of the structure function at high Reynolds numbers (Dhruva Reference Dhruva2000; Kurien & Sreenivasan Reference Kurien and Sreenivasan2000), that is,

(5.1) \begin{align} \frac {\langle \delta u^{\prime n}\rangle \big(r/\eta ^T \big)}{ \big(v_{\eta }^{T} \big)^n}=\frac {1}{15^{n/2}}F_{n}\frac {\big(r/\eta ^T \big)^{n}\left (1+D_{n}{r/L^T}\right )^{2C_{n}-n}}{\left (1+B_{n}{(r/\eta ^T)}^{2}\right )^{C_{n}}}, \end{align}

where $B_n$ , $C_n$ , $D_n$ and $F_n$ ( $=\langle (\partial u^{\prime }/\partial x)^n \rangle /\langle (\partial u^{\prime }/\partial x)^2\rangle ^{n/2}$ ) are constants that satisfy the asymptotic case for infinite Reynolds number. Following Antonia & Burattini (Reference Antonia and Burattini2006) and Tang et al. (Reference Tang, Antonia and Djenidi2023), the values of the empirical parameters $B_n$ and $D_n$ are determined by trial and error until (5.1) fits the data of $\langle \delta u^{\prime n}\rangle (r/\eta ^T)$ at two different $ \textit{Re}_\lambda ^T$ . In particular, substituting the isotropic relations ( $L^T/\eta ^T=C_\varepsilon ^T 15^{-3/4}(Re_\lambda ^T)^{3/2}$ and $(u_{\textit{rms}}^T)^{2}/(v_\eta ^T)^2=Re_\lambda ^T/15^{1/2}$ ) into (5.1) yields $C_n=n/3$ , since the prefactor of the structure function is independent of the Reynolds number. Here, $C_\varepsilon ^T$ and $u_{\textit{rms}}^T$ are the conditional non-dimensional dissipation rate and r.m.s. of the conditional velocity fluctuations, respectively. Clearly, the term $D_{n}{r/L^T}$ is associated with the large-scale motions of the flow, which involves the conditional integral length scale $L^T$ , as well as the constraint constant $D_n$ . However, for the intermittent region with $\gamma =0.5$ , $L^T$ cannot be directly determined from the velocity auto-correlation function due to the interference of laminar signals in the intermittent regions. As a result, the conditional non-dimensional dissipation rate $C_\varepsilon ^T=L^T \varepsilon ^T /(u^{\prime T})^3$ cannot be calculated directly. Following the approach of Gauding et al. (Reference Gauding, Bode, Brahami, Varea and Danaila2021) for defining conditional structure functions in the intermittent region, we define the conditional velocity auto-correlation function $f_u^{TT} (\boldsymbol{x}, r)$ in a similar manner, i.e.

(5.2) \begin{align} f_u^{TT}(\boldsymbol{x},r)=\frac {\langle H(\boldsymbol{x})H(\boldsymbol{x}+r\boldsymbol{e}_1)u^{\prime }(\boldsymbol{x})u^{\prime }(\boldsymbol{x}+r\boldsymbol{e}_1)\rangle }{\langle (u^{\prime T})^2(x)\rangle }. \end{align}

The formula defines an intermittent function $H(X,Y,Z)$ to mark the flow state of the fluid to describe the intermittency of the flow, where $|\boldsymbol{\omega }|\geq |\boldsymbol{\omega }|_{\textit{th}}$ is marked as 1 to indicate the turbulent region and $|\boldsymbol{\omega }| \lt |\boldsymbol{\omega }|_{\textit{th}}$ is marked as 0 to indicate the laminar region. The advantage of this formula is that it ensures the state of the two locations ( $u^{\prime }(\boldsymbol{x}+r\boldsymbol{e}_1)$ and $u^{\prime }(\boldsymbol{x})$ ) is restricted to the turbulent portion of the jet. It is worth noting that the jet centre plane lies in a fully turbulent region, where the intermittent function $H(X,Y,Z)$ is always equal to 1. Consequently, the conditional velocity auto-correlation function is identical to the normal velocity auto-correlation function. In contrast, for the intermittent region with $\gamma =0.5$ , the conditional velocity auto-correlation function can exclude influences of the laminar motions on the conditional integral length scale $L^T$ . Based on (5.2), $L^T$ can be directly obtained by integrating the conditional velocity auto-correlation function $f_u^{TT} (\boldsymbol{x},r)$ and the conditional non-dimensional dissipation rate is evaluated by $C_\varepsilon ^T=L^T \varepsilon ^T /(u^{\prime T})^3$ . Numerical integration of $f_u^{TT} (\boldsymbol{x},r)$ shows that the integral length scale $L^T$ in the jet centre plane is approximately $112\eta ^T$ , corresponding to the conditional non-dimensional dissipation rate $C_\varepsilon ^T=0.86$ . For the intermittent region with $\gamma =0.5$ , $L^T$ is approximately $144 \eta ^T$ and the corresponding conditional non-dimensional dissipation rate is $C_\varepsilon ^T=0.92$ . As expected, the values of the conditional non-dimensional dissipation rate are approximately the same. The slight differences may be related to the intrinsic characteristics of TNTI. Watanabe et al. (Reference Watanabe, Silva and Nagata2019a ) investigated the non-dimensional dissipation rate $C_\varepsilon$ near the TNTI. The results indicate that $C_\varepsilon$ is significantly higher in the TNTI sublayer (i.e. the turbulent sublayer) than in the turbulent core region.

According to the local isotropic assumption, the relationship between $\langle \delta u^{\prime 2}\rangle$ and $\langle \delta v^{\prime 2}\rangle$ satisfies (C1). For $n=2$ , combining with (5.1) and (C1), a good approximation for $\langle \delta v^{\prime 2}\rangle (r)$ can be written as

(5.3) \begin{align} \big\langle \delta v^{\prime 2}\big\rangle =\frac {F_{2}(r/\eta )^2(1+D_2r/L)^{(2C_2-2)}}{15(1+B_2(r/\eta )^2)^{C_2}}\left (2+\frac {(2C_2-2)D_2r/L}{2(1+D_2r/L)}-\frac {C_2B_2(r/\eta )^2}{1+B_2(r/\eta )^2}\right ). \end{align}

It should be emphasised that the parameters $B_n$ and $D_n$ can be adjusted in different regions since (5.1) is mainly used to effectively fit the measured data of $\langle \delta u^{\prime n}\rangle (r)$ in a specific flow region (Antonia et al. Reference Antonia, Tang, Djenidi and Zhou2019b ). This approach enables us to extrapolate the existing results to the Reynolds number range that cannot be directly obtained by experiments or numerical simulations. Dhruva (Reference Dhruva2000) conducted detailed tests on the empirical formula and the results showed that the formula provides a good agreement with atmospheric surface layer data across the entire range of $r$ . Antonia & Burattini (Reference Antonia and Burattini2006) investigated the FRN effect on third-order structure functions in forced and decaying turbulence based on empirical formula.

Figure 16.

Distributions of structure functions in the fully turbulent region with $\gamma =1.0$ , based on (5.1), at $ \textit{Re}_\lambda ^T = 100$ , $353$ , $10^3$ , $10^4$ and $10^5$ . (a) $\langle \delta u^{\prime 2}\rangle (r)/(r \varepsilon ^T)^{2/3}$ ; (b) $\langle \delta v^{\prime 2}\rangle (r)/(r \varepsilon ^T)^{2/3}$ ; (c) $-\langle \delta u^{\prime 3}\rangle (r)/(r \varepsilon ^T)$ . The solid lines of the same colour represent the corresponding DNS results for the turbulent plane jet. The horizontal dashed line shows the asymptotic value of the Kolmogorov constant as the Reynolds number approaches infinity.

Figure 16. Long description

The line graph displays the distributions of structure functions in the fully turbulent region with r over eta T at different Reynolds numbers. The x-axis represents r over eta T on a logarithmic scale ranging from to . The y-axis represents the normalized structure functions. The graph includes multiple data lines for different Reynolds numbers: 100, 353, , , and . Each data line is color-coded and marked with different symbols. The solid lines of the same color represent the corresponding DNS results for the turbulent plane jet. The horizontal dashed line shows the asymptotic value of the Kolmogorov constant as the Reynolds number approaches infinity. The graph is divided into three subplots: (a) for the second-order structure function, (b) for the second-order structure function with different scaling, and (c) for the third-order structure function. All values are approximated.

Figure 16 presents a comparison between the numerical results of the structure function and the empirical formula in the jet centre plane. The parameters $C_\varepsilon$ , $B_n$ , $C_n$ , $D_n$ and $F_n$ used to fit the structure functions are summarised in table 4. Here, the values of $B_2$ , $C_2$ , $D_2$ and $F_2$ differ slightly from the grid turbulence data reported by Tang et al. (Reference Tang, Antonia and Djenidi2023). In their study, $B_2=0.0056$ was used, whereas we used $B_2=0.006$ to achieve a better fit to the data; this value is consistent with that reported by Antonia et al. (Reference Antonia, Tang, Djenidi and Zhou2019b ). The numerical results show reasonably good agreement with the empirical formula at the small and intermediate scales for two different local Reynolds numbers $ \textit{Re}_\lambda ^T$ . This confirms that the empirical formula effectively describes the available data for $\langle \delta u^{\prime 2}\rangle (r)$ , $\langle \delta v^{\prime 2}\rangle (r)$ and $-\langle \delta u^{\prime 3}\rangle (r)$ . The good agreement makes it possible to extrapolate the existing results to higher local Reynolds numbers $ \textit{Re}_\lambda ^T$ . It should be pointed out that the agreement between the model and data at large scales is poor, which is likely due to the difficulty of achieving statistical convergence at those scales. Specifically, the distribution of $\langle \delta v^{\prime 2}\rangle (r)$ at $ \textit{Re}_\lambda ^T=353$ exhibits an unexpected oscillation at approximately $r/\eta =1500$ . Although we performed repeated simulations to mitigate this issue as much as possible, the available data gradually decreases with increasing $r$ , possibly leading to the emergence of an unexpected oscillation.

Table 4.

Values of parameters $B_n$ , $C_n$ , $D_n$ and $F_n$ in (5.1) within the jet centre plane. Note that the superscripts $u$ and $v$ denote the fitting parameters for the longitudinal and transverse structure functions, respectively.

Table 4. Long description

The table presents values of parameters B, C, D, and F in a jet center plane, with data for different conditions. It includes values for parameters B, C, D, and F, with superscripts u and t denoting fitting parameters for longitudinal and transverse structure functions, respectively. The table has 2 rows and 13 columns, with the first row listing the parameters and the second row providing their corresponding values. Notable values include as 0.006, as 0.667, as 1.09, and as 1.0 for the first set, and as 0.006, as 0.667, as 0.553, and as 1.0 for the second set. Other parameters such as , , , and are also listed with values 0.014, 1.84, and 0.53 respectively.

Figure 17.

Range of normalised separation distance $r/\eta ^T$ in the jet centre plane over which the empirical distributions of $\langle \delta u^{\prime 2}\rangle$ , $\langle \delta v^{\prime 2}\rangle$ and $-\langle \delta u^{\prime 3}\rangle$ deviate from $2.00$ , $2.66$ and $0.798$ , respectively, by less than $2.5\,\%$ . The separation distance between the curves approximately represents the upper and lower limits of the inertial range at a fixed conditional local Reynolds number $ \textit{Re}_\lambda ^T$ . The vertical arrows indicate the approximate values of $ \textit{Re}_\lambda$ required to achieve an inertial range of two decades in extent for $\langle \delta u^{\prime 2}\rangle (r)/(r \varepsilon ^T)^{2/3}$ (red arrow), $\langle \delta v^{\prime 2}\rangle (r)/(r \varepsilon ^T)^{2/3}$ (blue arrow) and $-\langle \delta u^{\prime 3}\rangle (r)/(r \varepsilon ^T)$ (green arrow).

Figure 17. Long description

The line graph presents the relationship between normalized separation distance (r/ηT) and Reynolds number (ReTλ) for different structure functions. The x-axis represents the Reynolds number (ReTλ) ranging from to , while the y-axis represents the normalized separation distance (r/ηT) ranging from to . Three data lines are plotted: a red line for δu'² /((rεT)^(2/3)), a blue line for δv'² /((rεT)^(2/3)), and a green line for − δu'³ /(rεT). The graph shows that the scaling exponent of the transverse structure function is smaller than that of the longitudinal structure function. Vertical arrows indicate the approximate values of Reynolds number required to achieve an inertial range of two decades in extent for each structure function. All values are approximated.

The extrapolated results in figure 16 also show that when $ \textit{Re}_\lambda ^T$ tends to infinity, the Kolmogorov constants for $\langle \delta u^{\prime 2}\rangle (r)$ , $\langle \delta v^{\prime 2}\rangle (r)$ and $-\langle \delta u^{\prime 3}\rangle (r)$ are 2.00, 2.676 and 0.798, respectively, which are consistent with the theoretically predicted values. For $ \textit{Re}_\lambda ^T = 10^4$ , the distribution of $\langle \delta u^{\prime 2}\rangle (r)$ exhibits a discernible plateau over more than one decade in extent, whereas a similar discernible plateau in the distribution of $-\langle \delta u^{\prime 3}\rangle (r)$ is only observed at $ \textit{Re}_\lambda ^T = 10^5$ . This suggests that FRN effects may differ between the second- and third-order structure functions. To satisfy the K41 theoretical prediction, i.e. $\langle \delta u^{\prime n}\rangle (r) \sim r^{n/3}$ , the third-order structure function may require a much higher $ \textit{Re}_{\lambda }^{T}$ .

The approximate value of $ \textit{Re}_\lambda ^T$ required to establish the inertial range for the structure functions $\langle \delta u^{\prime n}\rangle (r)$ is estimated from the range of normalised separation distances $r/\eta ^T$ in the jet centre plane. As shown in figure 17, this range corresponds to where the normalised structure functions $\langle \delta u^{\prime 2}\rangle (r)/(r \varepsilon ^T)^{2/3}$ , $\langle \delta v^{\prime 2}\rangle (r)/(r \varepsilon ^T)^{2/3}$ and $-\langle \delta u^{\prime 3}\rangle (r)/(r \varepsilon ^T)$ deviate from their theoretical predictions by less than $2.5\,\%$ . The vertical arrows indicate the approximate values of $ \textit{Re}_\lambda ^T$ required to achieve an inertial range of two decades in extent. The corresponding Reynolds numbers $ \textit{Re}_\lambda ^T$ for $\langle \delta u^{\prime 2}\rangle (r)/(r \varepsilon ^T)^{2/3}$ , $\langle \delta v^{\prime 2}\rangle (r)/(r \varepsilon ^T)^{2/3}$ and $-\langle \delta u^{\prime 3}\rangle (r)/(r \varepsilon ^T)$ are approximately $9 \times 10^3$ , $8 \times 10^3$ and $4 \times 10^4$ , respectively. As $ \textit{Re}_\lambda ^T$ increases, the inertial range continues to expand. This observation somewhat echoes the argument by Antonia et al. (Reference Antonia, Tang, Djenidi and Zhou2019b ) that FRN effects should be properly considered before determining whether modifications to K41 are necessary.

5.2. Finite Reynolds number effect on structure functions in the intermittent region

Figure 18.

Distributions of conditional structure functions in the intermittent region with $\gamma =0.5$ , based on (5.1), at $ \textit{Re}_\lambda ^T = 114$ , $220$ , $459$ , $10^3$ , $10^4$ and $10^5$ . (a) $\langle \delta u^{\prime 2}\rangle (r)/(r \varepsilon ^T)^{2/3}$ ; (b) $\langle \delta v^{\prime 2}\rangle (r)/(r \varepsilon ^T)^{2/3}$ ; (c) $-\langle \delta u^{\prime 3}\rangle (r)/(r \varepsilon ^T)$ . The solid lines of the same colour represent the corresponding DNS results for the turbulent plane jet. Noted that the distribution of the structure functions for the global Reynolds number $ \textit{Re}_H = 4 \times 10^4$ is plotted, with the data taken from Watanabe et al. (Reference Watanabe, Zhang and Nagata2019b ).

Figure 18. Long description

The line graph displays distributions of conditional structure functions in the intermittent region. The x-axis represents the ratio of distance to Kolmogorov length scale, while the y-axis represents normalized structure functions. The graph includes multiple data series for different Reynolds numbers, with solid lines representing DNS results for a turbulent plane jet. The data points are plotted for various Reynolds numbers, showing how the structure functions vary with distance. All values are approximated.

A comparison between the numerical results of the conditional structure function and the empirical formula at $\gamma = 0.5$ is presented in figure 18. The corresponding parameters $C_\varepsilon$ , $B_n$ , $C_n$ , $D_n$ and $F_n$ used to fit the structure functions are summarised in table 5. For the conditional second-order structure function, although large-scale fluctuations are not universal, the differences between the curves of the conditional second-order structure functions at large scales are not pronounced (see figures 5 b, 6 b, 7 b and 8 b). Therefore, the constant $D_2$ requires only slight adjustments. Conversely, for the conditional third-order structure function, significant differences are observed at large scales (see figures 13 b and 14 b). This means that the term $D_3 r/L^T$ , related to large-scale fluctuations, needs to be adjusted to accurately fit the behaviour of the structure function at large scales. Therefore, the value of the constant $D_3$ differs considerably between $\gamma =1$ and $\gamma =0.5$ . To enhance the reliability of the statistical results, the structure function profiles of the temporally evolving plane jet for the global Reynolds number $ \textit{Re}_H = 4 \times 10^4$ from Watanabe et al. (Reference Watanabe, Zhang and Nagata2019b ) are also shown. The corresponding conditional local Reynolds number $ \textit{Re}_\lambda ^T$ is approximately $220$ at $\gamma =0.5$ . Except for the conditional longitudinal structure function $\langle \delta u^{\prime 2}\rangle (r)\gamma ^{-1}/(r \varepsilon ^T)^{2/3}$ , the numerical results collapse reasonably well with the empirical formula at small and intermediate scales. This means that the empirical formula (5.1) seems to be difficult to describe the behaviour of $\langle \delta u^{\prime 2}\rangle (r)\gamma ^{-1}/(r \varepsilon ^T)^{2/3}$ in the intermittent region. This deviation may result from the influence of the intrinsic characteristics of the TNTI on the inertial range of the longitudinal structure functions (see figure 12). The extrapolation results in figures 18(b) and 18(c) indicate that the conditional structure functions $\langle \delta v^{\prime 2}\rangle (r)\gamma ^{-1}/(r \varepsilon ^T)^{2/3}$ and $-\langle \delta u^{\prime 3}\rangle (r)\gamma ^{-1}/(r \varepsilon ^T)$ may satisfy the classical K41 theory predictions even in the highly inhomogeneous intermittent region when $ \textit{Re}_\lambda ^T$ is sufficiently large.

Table 5.

Same as table 4, but for $ \gamma = 0.5$ .

Table 5. Long description

The table presents a comparison between the numerical results of the conditional structure function and the empirical formula at specific scales. It includes parameters such as C subscript s, B subscript 2 superscript u, C subscript 2 superscript u, D subscript 2 superscript u, F subscript 2 superscript u, B subscript 2 superscript d, C subscript 2 superscript d, D subscript 2 superscript d, F subscript 2 superscript d, B subscript 3 superscript u, C subscript 3 superscript u, D subscript 3 superscript u, and F subscript 3 superscript u. The table has 2 rows and 14 columns. Notable values include 0.92 for C subscript s, 0.006 for B subscript 2 superscript u, 0.667 for C subscript 2 superscript u, 0.94 for D subscript 2 superscript u, 1.0 for F subscript 2 superscript u, 0.006 for B subscript 2 superscript d, 0.667 for C subscript 2 superscript d, 0.836 for D subscript 2 superscript d, 1.0 for F subscript 2 superscript d, 0.0114 for B subscript 3 superscript u, 1.0 for C subscript 3 superscript u, 19.7 for D subscript 3 superscript u, and 0.53 for F subscript 3 superscript u. The table highlights the differences and adjustments needed for the constants in the conditional structure functions at large scales.

Figure 19.

Same as figure 17, but for $ \gamma = 0.5$ . Note that the extrapolation results of the conditional longitudinal structure function are not shown, since (5.1) fails to adequately describe its behaviour.

Figure 19. Long description

The line graph presents the influence of large-scale motions on small scales in various grid-generated turbulence. The x-axis represents the Reynolds number (Re subscript lambda) on a logarithmic scale ranging from to . The y-axis represents the ratio of r to eta subscript T on a logarithmic scale ranging from to . Two data lines are plotted: a blue line representing the second-order structure function and a green line representing the third-order structure function. The blue line shows a steep increase as the Reynolds number increases, while the green line also increases but at a slower rate. Arrows indicate the direction of the data trends. All values are approximated.

According to § 3, due to the effect of external intermittency, the Kolmogorov constant $C_2^{\textit{TSF}}$ for $\langle \delta v^{\prime 2}\rangle (r)$ exhibits a power-law dependence on the intermittency factor $\gamma$ , i.e. $C_2^{\textit{TSF}}\sim \gamma ^{1/3}$ , while the Kolmogorov constant $C_3^{\textit{LSF}}$ is unaffected by the external intermittency (see Appendix A). This means that the Kolmogorov constant $C_3^{\textit{LSF}}$ in the intermittent region may approach $4/5$ at infinite Reynolds number, as shown in figure 18(c). These results clearly reflect FRN effects on the structure functions.

Figure 19 presents the range of normalised separation distances $r/\eta ^T$ at $\gamma =0.5$ over which $\langle \delta v^{\prime 2}\rangle (r)\gamma ^{-1}/(r \varepsilon ^T)^{2/3}$ and $-\langle \delta u^{\prime 3}\rangle (r)\gamma ^{-1}/(r \varepsilon ^T)$ depart from 2.66 and 0.798, respectively, by no more than $2.5\,\%$ . Note that the extrapolation results of the conditional longitudinal structure function are not shown here, since (5.1) fails to adequately describe its behaviour (see figure 18 a). The vertical arrows indicate the approximate values of $ \textit{Re}_\lambda ^T$ required to achieve an inertial range of two decades in extent. For $\langle \delta v^{\prime 2}\rangle (r)\gamma ^{-1}/(r \varepsilon ^T)^{2/3}$ and $-\langle \delta u^{\prime 3}\rangle (r)\gamma ^{-1}/(r \varepsilon ^T)$ , the values of $ \textit{Re}_\lambda ^T$ corresponding to the positions of the arrows need to reach approximately $10^4$ and $10^5$ , respectively. Compared with figure 17, the values of $ \textit{Re}_\lambda ^T$ required in the intermittent region with $\gamma = 0.5$ are significantly higher than those obtained in the jet centre plane. This is expected since the contribution of unsteady terms $I_u^*({\lambda ^T})$ is greater in the intermittent region than in the jet centre plane.

Figure 20.

Variation of $T_{\lambda ^T}^*$ with $ \textit{Re}_\lambda ^T$ at two different cross-wise positions with $\gamma = 0.5$ and $1.0$ , based on (5.1). The horizontal dashed line indicates $T_{\lambda ^T}^*=0.8$ . The red () and blue () solid circles correspond to the data at the two different Reynolds numbers considered in the present study. Purple () and green () squares correspond to the grid turbulence data from Zhou & Antonia (Reference Zhou and Antonia2000) and Tang et al. (Reference Tang, Antonia and Djenidi2023), respectively. The orange triangle () corresponds to the data from the intermittent region with $\gamma =0.5$ in the temporally evolving turbulent plane jet at $ \textit{Re}_H=4 \times 10^4$ (Watanabe et al. Reference Watanabe, Zhang and Nagata2019b ).

Figure 20. Long description

The line graph displays the variation of T lambda T star with Re T lambda for different values of gamma and theoretical fits. The x-axis represents Re T lambda on a logarithmic scale ranging from to . The y-axis represents T lambda T star ranging from 0 to 1. The red solid line corresponds to gamma equals 1.00, and the blue solid line corresponds to gamma equals 0.50. The green dashed line represents the theoretical fit T lambda T star equals 4 over 5 minus 9.7 times Re T lambda to the power of negative 0.764. The purple dashed line represents the theoretical fit T lambda T star equals 4 over 5 minus 9.7 times Re T lambda to the power of negative 0.553. The red and blue solid circles correspond to data at two different Reynolds numbers. Purple and green squares correspond to grid turbulence data from Zhou & Antonia (2000) and Tang et al. (2023), respectively. The orange triangle corresponds to data from the intermittent region with in the temporally evolving turbulent plane jet at (Watanabe et al. 2019b). All values are approximated.

It is helpful to examine the self-similarity of the third-order structure function using (5.1) at high local Reynolds numbers. Figure 20 shows the variation of the conditional third-order structure function $T_{\lambda ^T}^*$ with $ \textit{Re}_\lambda ^T$ at $r = \lambda ^T$ for $\gamma =1.0$ and $0.5$ , based on (5.1). The conditional Taylor microscale $\lambda ^T$ is determined based on the assumption of local isotropy (i.e. $\lambda ^T/\eta ^T=15^{1/4}(Re_\lambda ^{T})^{1/2}$ ), since the turbulence in the intermittent region also satisfies local isotropy at small scales (see figure 23 in Appendix C). For the fully turbulent region with $\gamma =1.0$ , the extrapolated results are compared with data obtained by Zhou & Antonia (Reference Zhou and Antonia2000) and Tang et al. (Reference Tang, Antonia and Djenidi2023) in grid turbulence. It can be observed that the results from grid turbulence show a good agreement with the extrapolated results. There seem to be no earlier works on the conditional third-order structure function for the intermittent region with $\gamma =0.5$ . We have supplemented an additional DNS result from Watanabe et al. (Reference Watanabe, Zhang and Nagata2019b ). Nevertheless, a good agreement can still be observed. For two different cross-wise positions, the contribution of the unsteady term at $r = \lambda ^T$ becomes negligible only when the local Reynolds number reaches approximately $10^4$ and $10^5$ , respectively. This means that when $ \textit{Re}_\lambda ^T$ is greater than $10^5$ , the conditional third-order structure functions at different positions are expected to recover self-similarity in the range of $r \lt \lambda ^T$ . Correspondingly, a higher local Reynolds number is required for the conditional third-order structure function to exhibit self-similarity at small and intermediate scales. It should be noted that the maximum local Reynolds number $ \textit{Re}_\lambda ^T$ achieved in the current experimental measurements is approximately $5000$ (Tabeling et al. Reference Tabeling, Zocchi, Belin, Maurer and Willaime1996; Belin et al. Reference Belin, Maurer, Tabeling and Willaime1997), whereas the maximum $ \textit{Re}_\lambda ^T$ obtained in DNS is much smaller, approximately $2500$ (Yeung et al. Reference Yeung, Ravikumar, Uma-Vaideswaran, Dotson, Sreenivasan, Pope, Meneveau and Nichols2025). The value of $ \textit{Re}_\lambda ^T = 10^5$ is far beyond the local Reynolds number range currently achievable through experiments or numerical simulations.

A simple algebraic model is used to quantify, albeit approximately, the dependence of $T_{\lambda ^T}^*$ on $ \textit{Re}_\lambda ^T$ at $r = \lambda ^T$ . We use the following expression for $T_{\lambda ^T}^*$ as a function of $ \textit{Re}_\lambda ^T$ , namely, $T_{\lambda ^T}^* = 4/5 - C (Re_\lambda ^{T})^{D}$ , where $C$ and $D$ are constants determined by nonlinear least squares fitting. The numerical results show reasonably good agreement with the empirical formula indicating that the empirical parameters are independent of $ \textit{Re}_\lambda ^{T}$ at a fixed cross-wise position. For the positions where $\gamma =1.0$ and $0.5$ , the expression for $T_{\lambda ^T}^*$ is given by

(5.4) \begin{align} T_{\lambda ^T}^* = \frac {4}{5} - 9.7\,\big(Re_\lambda ^{T}\big)^{-0.764} \quad \text{and} \quad T_{\lambda ^T}^* = \frac {4}{5} - 9.7\,\big(Re_\lambda ^{T}\big)^{-0.553}. \end{align}

As shown in figure 20, the selected constant $C$ and $D$ fit the data reasonably well for the fully turbulent region with $\gamma = 1.0$ . For the intermittent regions with $\gamma = 0.5$ , the simplified curve deviates moderately from (5.1) and the three data points, particularly in the range $ \textit{Re}_\lambda ^T \lt 350$ . This deviation may be attributed to the blocking/sheltering mechanisms of TNTI (da Silva et al. Reference da Silva, Hunt, Eames and Westerweel2014). The turbulence near the TNTI layer is characterised by large-scale inhomogeneity and anisotropy at low Reynolds numbers (Zhang et al. Reference Zhang, Watanabe and Nagata2023), which implies that the simplified equation cannot adequately capture the characteristics of the third-order structure function within the intermittent region. In contrast, at higher Reynolds numbers, the deviation is small with a maximum error of less than $10\,\%$ relative to (5.1). Note that the smaller value of $T_{\lambda ^T}^*$ in the intermittent region at $\gamma = 0.5$ is primarily due to a larger contribution from the unsteady term at $r = \lambda ^T$ compared with the fully turbulent region (see table 3). Therefore, a smaller intermittent factor $\gamma$ corresponds to a larger constant $D$ , which stems from the larger unsteady term $I_u({\lambda ^T})$ in the intermittent region. To our knowledge, this should be the first time reasonable evidence has supported the possibility that the conditional structure functions in the intermittent region may satisfy K41 predictions.