2.1. Numerical method and computational parameters

In this work, we use exactly the same three-dimensional data from a previous direct numerical simulation (DNS) conducted by Watanabe, Zhang & Nagata (Reference Watanabe, Zhang and Nagata2019) and Hayashi et al. (Reference Hayashi, Watanabe and Nagata2021a ) to explore the effects of the large-scale intermittecny. For the streamwise $X$ and spanwise $Z$ directions, periodic boundary conditions are imposed and uniformly distributed spacings are used, while a free-slip boundary condition and non-uniform grid spacing are adopted in the normal $Y$ direction. A fully conservative central difference scheme by Morinishi et al. (Reference Morinishi, Lund, Vasilyev and Moin1998) is adopted for spatial discretisation. Specifically, second-order schemes are applied in the vertical direction, while fourth-order schemes are used in the other two directions. For time advancement, a third-order Runge–Kutta method is used, while the Bi-CGSTAB method is implemented to solve the Poisson equation for pressure.

The inlet Reynolds number is set as $\textit{Re}_{J}= U_JH_J/\nu =40\,000$ , where $U_J$ is the initial jet velocity along the centreline, $H_J$ is the jet width and $\nu$ is the kinematic viscosity. The computational domain size of the DNS is $L_X \times L_Y \times L_Z =6H_J \times 10H_J \times 4 H_J$ . Concerning the grid mesh, a total of $ N_X \times N_Y \times N_Z = 2304 \times 3200 \times 1536$ mesh nodes are used. The simulation was run until $T_{\textit{end}}/T_{\textit{ref}} = 20$ , where $T_{\textit{ref}}=H_J/U_J$ represents the flow reference time. The computational details and the corresponding geometric parameters are presented in table 1. Here, to reduce computational costs, the initial velocity fields for the DNS are obtained from the fully developed state resulting from well-resolved large eddy simulation (LES), where the spatial resolution is approximately five times the Kolmogorov scale. Namely, LES of the temporal jet with the implicit subgrid scale model based on spatial filtering is first conducted until $T/T_{\textit{ref}}=17$ . Then, DNS is subsequently conducted by changing the spatial resolution. Note that the difference between the one-dimensional energy spectra $E_{11} (k_1)$ at $T/T_{\textit{ref}}=17$ and $20$ is mainly observed in the small-scale turbulence statistics, while $E_{11} (k_1)$ at lower wavenumbers hardly varies with time (Watanabe et al. Reference Watanabe, Zhang and Nagata2019). Watanabe et al. (Reference Watanabe, Zhang and Nagata2019) confirmed that the initial velocity fields without small-scale fluctuations effectively reproduce the small-scale turbulence characteristics in DNS at $T/T_{\textit{ref}}=18$ and will not affect the evolution of the turbulent region. For further descriptions of the numerical details and the validation of the numerical simulation, one could refer to Watanabe et al. (Reference Watanabe, Zhang and Nagata2019). Their study compared simulated datasets of temporally evolving turbulent plane jets with experimental and numerical data, validating the accuracy of the simulations (not shown herein for economy of space). The local turbulence Reynolds number $\textit{Re}_{\lambda }=u_{\textit{rms}}\lambda _C/\nu$ , based on the Taylor microscale $ \lambda _C = \langle u^{\prime 2}\rangle ^{1/2}/\langle (\partial u^{\prime }/\partial x)^{2}\rangle$ and the root-mean-square (r.m.s.) velocity $ u_{{\textit{rms}}}$ , is approximately $233$ , where the angled brackets $\langle {} \rangle$ indicate the average value of the statistical variable on the $X$ – $Z$ plane and the symmetry of the flow in the $Y$ direction is also taken into account. Hereafter, the subscript $C$ represents the variable along the jet centreline $Y/H_J=0$ .

Table 1.

Numerical parameters for the DNS of temporally evolving turbulent planar jets.

Despite using DNS data from a state-of-the-art massively parallel simulation, the local Reynolds number is still moderated in our DNS. Therefore, in this study, we overlooked the effects of small-scale intermittency. However, small-scale intermittency can, at least to some extent, affect the behaviour of the energy spectrum and structure functions, as it often leads to deviations from classical scaling laws (Sreenivasan & Antonia Reference Sreenivasan and Antonia1997).

2.2. Visualisation and identification of the TNTI

It is well known that the turbulent plane jet is an intermittent flow. The TNTI, which separates the turbulent and the non-turbulent regions, are self-similar with fractal-like properties (Sreenivasan Reference Sreenivasan1991; Mistry, Dawson & Kerstein Reference Mistry, Dawson and Kerstein2018). Before introducing the characteristics of intermittent flows, the identification of the outer edge of the TNTI is discussed. Following previous studies (da Silva et al. Reference da Silva, Hunt, Eames and Westerweel2014; Hayashi et al. Reference Hayashi, Watanabe and Nagata2021a ), the volume fraction of the turbulent region $V_T$ as a function of the vorticity threshold $| \boldsymbol{\omega } |_{\textit{th}}$ is shown in figure 1. A clear plateau region can be observed in the range of $ 1.0 \times {10^{ - 3}} \lt | \boldsymbol{\omega } |_{\textit{th}} / \langle {| \boldsymbol{\omega } |} \rangle _{C} \lt 1.0 \times {10^{ - 1}}$ , suggesting that the turbulent volume is not sensitive to the changes of the vorticity threshold $| \boldsymbol{\omega } |_{\textit{th}}$ . In this study, a vorticity threshold of $| \boldsymbol{\omega } |_{\textit{th}}$ = $0.01 \langle {| \boldsymbol{\omega } |} \rangle _{C}$ (represented by a vertical solid orange line) was selected to identify the outer edge of the TNTI. The conditional statistics of the vorticity magnitude $\langle |\boldsymbol{\omega }|\rangle _I$ and the corresponding derivative of $\langle |\boldsymbol{\omega }|\rangle _I$ are plotted in figure 1(b), where $\langle \rangle _I$ represents the conditional average operator with respect to normal distance from the TNTI in the $Y$ – $Z$ plane and $Y_I$ is the conditional distance from the outer edge of the TNTI. It can be seen that the vorticity increases rapidly near the TNTI. Following Watanabe, Nagata & da Silva (Reference Watanabe, Nagata and da Silva2017), the outer edge of the TNTI is located at the position $Y_I / \eta _C = 0$ , while the inner side is estimated to be at the location where $\langle |\boldsymbol{\omega }|\rangle _I^{\prime }$ reaches $20 \,\%$ of its maximum value. The mean thickness of the TNTI is ${\langle {{\delta _{\textit{TNTI}}}} \rangle _I} \approx 23 \eta _C$ (represented by the shaded square).

Figure 1.

(a) Normalised volume fraction of turbulent region as a function of normalised vorticity threshold $| \boldsymbol{\omega } |_{\textit{th}} / \langle {| \boldsymbol{\omega } |} \rangle _{C}$ . (b) Mean conditional distributions of the magnitudes of the vorticity $\langle |\boldsymbol{\omega }|\rangle _I$ and the corresponding derivative of $\langle |\boldsymbol{\omega }|\rangle _I$ . In panel (a), the vorticity threshold is normalised by the average value of the vorticity $\langle {| \boldsymbol{\omega } |} \rangle _{C}$ at the jet centreline. The three vertical lines indicate $| \boldsymbol{\omega } |_{\textit{th}} / \langle {| \boldsymbol{\omega } |} \rangle _{C}= 1.0 \times {10^{ - 3}}$ , $ 1.0 \times {10^{ - 2}}$ and $ 1.0 \times {10^{ - 1}}$ , respectively. In panel (b), the two vertical lines denote $Y_{_I}/\eta _{_C}=0.0$ and $Y_{_I}/\eta _{_C}={-}23.0$ .

Figure 2.

Logarithmic contours of the vorticity magnitude at a two-dimensional (2-D) $X$ – $Y$ plane. The outer edge of the TNTI is represented by the white line. Only part of the simulation domain in the $Y$ direction $ - 2 \leqslant Y/{H_J} \leqslant 2$ is visualised. The low vorticity pockets caused by large-scale engulfment mechanisms are marked by a white circle.

Figure 2 shows the contours of the vorticity magnitude field, $| \boldsymbol{\omega } |$ , in the two-dimensional (2-D) $X$ – $Y$ plane. Obviously, the solid white line, defined as the outer edge of the TNTI, completely wraps the inner high vorticity region. Thus, the highly folded outer edge of the TNTI is well defined by the vorticity threshold selected in figure 1. For the current high-Reynolds-number $\textit{Re}_J=40\,000$ case, the fine vorticity structures and small characteristic length scales can be observed in the turbulent region shown in figure 2. We also observe isolated pockets with low vorticity magnitude (e.g. $X/H_J=5.4$ , $Y/H_J=-0.7$ , marked by a white circle) in the turbulent region. The trapped pockets are caused by the large-scale inviscid engulfment process (Taveira et al. Reference Taveira, Diogo, Lopes and da Silva2013; da Silva et al. Reference da Silva, Hunt, Eames and Westerweel2014). In subsequent analyses, we excluded these isolated pockets to avoid their interference with the statistics of the structure functions.

Figure 3 visualises the outer edge of the TNTI coloured by the height of the TNTI from the jet centreline. The TNTI consists of multi-scale valleys and bulges. It is well known that significant numerical oscillations near the TNTI at high Reynolds numbers can occur, as discussed by Er, Laval & Vassilicos (Reference Er, Laval and Vassilicos2023). The three-dimensional (3-D) visualisation of the outer edge of the TNTI appears relatively smooth due to the high spatial resolution. Both figures 2 and 3 clearly show that the local instantaneous height $Y_i$ , which is the vertical distance from the centre plane with $Y/H_J=0$ , varies over a wide range. A quantitative evaluation of the vertical height of the outer edge of the TNTI is given by the distribution of the intermittency factor $\gamma$ .

Figure 3.

Top view of the outer edge of the TNTI (i.e. isosurfaces of the vorticity magnitude with $| \boldsymbol{\omega } |_{\textit{th}}$ = $0.01 \langle {| \boldsymbol{\omega } |} \rangle _{C}$ ). The colour represents the vertical height of the outer edge of the TNTI to the centreplane. Yellow indicates small height, while black represents locations far from the centreline.

Figure 4.

(a) Vertical distribution of the intermittency factors $\gamma$ . (b) Probability distribution of the vertical height of the outer edge of the TNTI from the jet centreline. The four horizontal dotted lines represent the intermittency factors $\gamma =1$ , $0.75$ , $0.50$ and $0.25$ , respectively. The orange open circles represent the Gaussian fitting curve of the TNTI vertical height. The probability distribution of the vertical height is compared with numerical simulation data measured at $\textit{Re}_{J} =4000$ in a temporally evolving turbulent plane jet (Watanabe et al. Reference Watanabe, Zhang and Nagata2019).

2.3. Intermittency

We define an intermittent function $ {H(X,Y,Z,t)}$ to mark the flow state of the fluid to describe the intermittency of the jet, where $|\boldsymbol{\omega } | \geqslant |\boldsymbol{\omega } {|_{\textit{th}}}$ 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 intermittency factor $\gamma$ at a given location represents the probability that fluid motions are located within the turbulent region with $|\boldsymbol{\omega } | \geqslant |\boldsymbol{\omega } {|_{\textit{th}}}$ , that is, $\gamma (Y,t) = \langle {H(X,Y,Z,t)} \rangle$ . In regions of fully turbulent flow, $\gamma = 1$ , whereas in purely non-turbulent regions (laminar region), $\gamma = 0$ . The vertical distribution of the intermittency factor $\gamma$ is given in figure 4(a). Here, $b_U$ denotes the jet half-width (defined as the distance from the centreplane where the mean streamwise velocity decreases to half of the centreplane value). The four horizontal dashed lines, representing intermittency factors $\gamma$ of $1.00$ , $0.75$ , $0.50$ and $0.25$ with the corresponding normalised positions $ Y/b_U$ of $0.00$ , $1.52$ , $1.70$ and $1.91$ , are subsequently selected as the normal positions for the self-similarity analysis of the energy spectrum and second-order structure function in the intermittent region. The jet flow is fully turbulent (i.e. $\gamma = 1$ ) in the normal range where $Y/b_U \leqslant 0.8$ . Laminar flow and turbulent flow appear alternately in the vertical range of $0.8 \lt Y/{b_U} \lt 2.5$ and the jet flow exhibits intermittent characteristics. A previous study (Corrsin & Kistler Reference Corrsin and Kistler1955) suggested that the intermittency factor $\gamma$ is closely related to the probability distribution function (p.d.f.) of the detected TNTI. The probability distribution of the vertical distance of the TNTI $P({Y_i,t})$ at a given position $Y_i$ can be also evaluated by the relation $ P({Y_i,t}) = - {{\partial \gamma (Y_i,t) } \mathord {\left /{\vphantom {{\partial \gamma } {\partial {Y_i}}}} \right .} {\partial {Y_i}}}$ .

Figure 5.

Vertical distributions of (a) the skewness $S_u'$ and (b) the flatness $F_u'$ of the fluctuating streamwise velocity. The two horizontal lines indicate the flatness and the skewness of a Gaussian distribution, i.e. $S_u'=0$ and $F_u'=3$ , respectively.

Figure 4(b) shows the p.d.f. of the vertical distance of the outer edge of the TNTI from the jet centreline normalised by $b_U$ . The probability distribution is different from the non-Gaussian distribution observed by da Silva & Pereira (Reference da Silva and Pereira2008) and Xie et al. (Reference Xie, Yin, Zhang and Zhou2023) in a turbulent plane jet. This may be attributed to the relatively sufficient statistical samples provided by the current larger grid number. As the Reynolds number $\textit{Re}_J$ increases, the probability distribution shows a distinct shift towards the jet centreline. As can be seen, the p.d.f. $P({Y_i}/{b_U})$ can be well approximated by a Gaussian distribution with a mean $\langle {{Y_i}/{b_U}} \rangle =1.64$ . This mean value is in good agreement with Zhou & Vassilicos (Reference Zhou and Vassilicos2017), who reports $\langle {{Y_i}/{b_U}} \rangle =1.6$ for a spatially developing axisymmetric turbulent wake. The large-scale intermittent behaviour can be assessed through the skewness $S_u'$ and the flatness $F_u'$ of the velocity fluctuations. Figure 5 indicates that the instantaneous velocity fluctuations $u'$ are nearly Gaussian-distributed at $Y/b_U \leqslant 0.8$ , where the flow is fully developed turbulent. In contrast, for $Y/b_U \gt 0.8$ , the non-Gaussian characteristics associated with the intermittency factor gradually decrease.

2.4. Conditional one-point statistics and spatial resolution

Figure 6.

Vertical distributions of (a) the kinetic energy dissipation rate, (b) the Kolmogorov length scale, (c) the Kolmogorov velocity scale and (d) the spatial resolution for a fully developed state, where $\varepsilon$ , $\eta$ and $v_\eta$ represent the traditional Reynolds average values (Reynolds Reference Reynolds1895) and $\varepsilon ^T$ , $\eta ^T$ and $v_{\eta }^T$ represent the conditional average values in the turbulent region (Kovasznay, Kibens & Blackwelder Reference Kovasznay, Kibens and Blackwelder1970).

Using the intermittency function $ {H(X,Y,Z,t)}$ , the average value of the variable $f$ in the turbulent regions is defined as $ { f(Y,t) ^T} = \langle {fH} \rangle / \langle H \rangle$ . In contrast, the corresponding average value in the laminar regions is defined as $ { {f(Y,t)} ^L} = \langle {f(1 - H)} \rangle / \langle {(1 - H)} \rangle$ (Kovasznay et al. Reference Kovasznay, Kibens and Blackwelder1970). Therefore, the traditional average value is a weighted average of two different flow states, i.e. $ f = \gamma f^T + (1 - \gamma ) f^L$ . Throughout this paper, the superscripts $T$ and $L$ represent the average of the turbulent and the laminar regions. However, as will be demonstrated later, the normalised length based on the conditional statistics is also necessary for the acquiring the similarity of the structure functions, which has been largely overlooked.

Figure 6(a–c) show the variation of the kinetic energy dissipation rate $ \varepsilon = 2 \nu \langle {{s_{\textit{ij}}}{s_{\textit{ij}}}} \rangle$ , the Kolmogorov length scale $\eta = {({\nu ^3}/\varepsilon )^{1/4}}$ and the Kolmogorov velocity scale ${v_\eta } = {(\nu \varepsilon )^{1/4}}$ with the normal position for a fully developed state, where $\varepsilon$ , $\eta$ and $v_\eta$ represent the traditional average values, while $\varepsilon ^T$ , $\eta ^T$ and $v_{\eta } ^T$ represent the conditional average values in the turbulent region, with ${s_{\textit{ij}}} = 0.5({{\partial {u_i}} \mathord {\left /{\vphantom {{\partial {u_i}} {\partial {x_j} + {{\partial {u_j}} \mathord {\left /{\vphantom {{\partial {u_j}} {\partial {x_i}}}} \right .} {\partial {x_i}}}}}} \right .} {\partial {x_j} + {{\partial {u_j}} \mathord {\left /{\vphantom {{\partial {u_j}} {\partial {x_i}}}} \right .} {\partial {x_i}}}}})$ . In the fully turbulent region with $Y/{b_U} \leqslant 1$ , the turbulence dissipation rates and the Kolmogorov length and velocity scales obtained through two different averaging methods are roughly identical. However, in the intermittent region with $1 \leqslant Y/{b_U} \leqslant 2.5$ , the averaged statistics, influenced by the large-scale laminar motions, are significantly different. We also observed that the variation of all three small-scale variables ( $\varepsilon ^T$ , $\eta ^T$ and $v_{\eta } ^T$ ) averaged in the turbulent region along the normal direction is not significant. This means that the turbulence in the intermittent region appears to have similar properties to the turbulent core region on a small scale, as shown in our subsequent analysis. The Taylor-scale Reynolds numbers $\textit{Re}_\lambda$ at four different cross-wise positions corresponding to intermittency factors $\gamma =1.00$ , $0.75$ , $0.50$ and $0.25$ are $233$ , $231$ , $178$ and $119$ , respectively. The corresponding conditional Taylor-scale Reynolds numbers $\textit{Re}_\lambda ^T = u_{C}^T\lambda ^T/\nu$ are $233$ , $232$ , $211$ and $177$ .

Since the velocity fluctuations in the laminar region are much smaller compared with those in the turbulent region, the kinetic energy dissipation rate in the laminar region can be considered negligible, that is, ${\varepsilon ^L} = 2\nu \langle {s_{\textit{ij}}}{s_{\textit{ij}}} \rangle ^L \approx 0$ . This yields $\varepsilon \approx \gamma \varepsilon ^T$ . As shown in figure 6(a), the two curves, the vertical distributions of $\varepsilon$ and $ \gamma \varepsilon ^T$ , collapse together reasonably satisfactorily, confirming the accuracy of the scaling. Similarly, the Kolmogorov length scale and the Kolmogorov velocity scale of the turbulent region exhibit the following functional relationship with the intermittency factor $\gamma$ , that is, ${\eta ^T} \approx {\gamma ^{1/4}}\eta$ and $v_\eta ^T \approx {\gamma ^{ - 1/4}}{v_\eta }$ . The vertical distribution of spatial resolution is shown in figure 6(d). The spatial resolutions $ \Delta X /\eta$ and $ \Delta Y /\eta$ are comparable to the Kolmogorov length scale $ \eta$ (as well as the conditional Kolmogorov length scale $ \eta ^T$ ), enabling a high-fidelity evaluation of fluid motions near the TNTI.

3.1. One-dimensional energy spectra in frequency domain

Figure 7.

(a) One-dimensional energy spectrum normalised by the Kolmogorov length scale $\eta$ and the averaged dissipation rate $\varepsilon$ and (b) the compensated energy spectra of the streamwise velocity fluctuations at four different cross-wise positions. (c) The same results as panel(b) presented in a linear-log scale. The shaded orange squares indicate the upper and lower limits of the inertial subrange. For comparison, the energy spectra are compared with experimental data from grid turbulence, with $\textit{Re}_\lambda$ within the range of $99$ to $ 112$ (Kitamura et al. Reference Kitamura, Nagata, Sakai, Sasoh, Terashima and Harasaki2014).

To qualitatively evaluate the turbulent characteristics of the intermittent regions, the normalised one-dimensional energy spectra ${E_{11}}({k_1})/{(\varepsilon { \nu ^5})^{1/4}}$ of the streamwise velocity fluctuation at the jet centreplane and three different positions (i.e. $\gamma =0.25$ , $0.50$ and $0.75$ ) within a wide range of intermittency are plotted in figure 7(a). The energy spectrum profile in the central plane of the jet, with $ \gamma = 1$ , shows a reasonably good agreement with experimental results of a grid turbulence (Kitamura et al. Reference Kitamura, Nagata, Sakai, Sasoh, Terashima and Harasaki2014). In the velocity signals at these locations, all energy spectrum profiles exhibit a well-defined approximate $- 5/3$ power law over a wide frequency range. Previous numerical studies by Kravchenko & Moin (Reference Kravchenko and Moin2000) and Alves Portela et al. (Reference Alves Portela, Papadakis and Vassilicos2017) have already found that a broader non-Kolmogorov $- 5/3$ power-law range can be observed in the highly intermittent region, which cannot be explained by Kolmogorov’s universal equilibrium hypothesis. However, figure 7(a) clearly shows that with the decrease of $\gamma$ , a broader $- 5/3$ power law within the inertial range has not been observed, in contrast to the fully turbulent regions where $\gamma = 1$ . One possible reason is that these selected cross-wise positions are not affected by large-scale vortex shedding (Kravchenko & Moin Reference Kravchenko and Moin2000; Alves Portela et al. Reference Alves Portela, Papadakis and Vassilicos2017).

The corresponding one-dimensional compensated energy spectra, ${E_{11}}({k_1}){\varepsilon ^{ - 2/3}}{k_1}^{5/3}$ , are plotted versus ${k_1}\eta$ in figures 7(b) and 7(c). Note that the same results as in figure 7(b) are presented in a linear-log scale in figure 7(c). At all four positions, a distinct Kolmogorov constant $C_{k1}$ is observed within the inertial range, and $C_{k1}$ decreases as $\gamma$ decreases. Kolmogorov’s phenomenology or K41 is a result proposed under sufficiently large Reynolds numbers, i.e. the K41 asymptotes can approach at high Reynolds numbers. This suggests that the current high-Reynolds-number data will be essential, as the approximate $- 5/3$ power law observed over a fairly wide frequency range allows for precise measurement of the Kolmogorov constant $C_{k1}$ , as shown in figure 7. It is well known that the inertial subrange, where viscosity is negligible, lies between the integral length scale $L_u$ and the Kolmogorov length scale. The integral length scale $L_u$ is determined by integrating the longitudinal velocity auto-correlation function $f_u(\boldsymbol{x},r)$ at the jet centreplane, that is, $L_u=\int _0^{r_{x}}f_u(\boldsymbol{x},r) {\textrm{d}}r$ , where $r_x$ is typically taken as the first zero-crossing point. The longitudinal velocity auto-correlation function is defined as

(3.1) \begin{equation} f_u(\boldsymbol{x},r)=\frac {\langle u'(\boldsymbol{x})u'(\boldsymbol{x}+r\boldsymbol{e_{1}})\rangle }{\langle u'^2(\boldsymbol{x})\rangle }, \end{equation}

where $\boldsymbol{e_{1}}$ represents the unit vector of the streamwise direction and $r$ is the separation distance between points $u^{\prime }(\boldsymbol{x}+r\boldsymbol{e_{1}})$ and $u^{\prime }(\boldsymbol{x})$ . Here, numerical integration of $f_u(\boldsymbol{x},r)$ shows that the integral length scale $L_u$ is approximately $250 \eta$ , corresponding to the angular wavenumber $k_1 = 2\pi /{L_u} \approx 0.03 \eta ^{-1}$ in spectral space. Saddoughi & Veeravalli (Reference Saddoughi and Veeravalli1994) further pointed out that the inertial subrange ends around the angular wavenumber $k_1 = 0.1 \eta ^{-1}$ . The upper and lower limits of the inertial subrange are indicated by the shaded orange squares in figure 7(c). It can be seen that an approximately clear plateau region can be observed within the selected inertial range, although there are some weak oscillations. The four constants $C_{k1}$ corresponding to the intermittency factors $\gamma =1$ , $0.75$ , $0.50$ and $0.25$ are $0.55$ , $0.52$ , $0.43$ and $0.36$ , respectively.

3.2. Conditionally averaged spectra

In experimental studies, the energy dissipation rate is sometimes determined from the one-dimensional energy spectra ${E_{11}}({k_1}) = {C_{k1}}{\varepsilon ^{2/3}}{k_1}^{ - 5/3}$ , making the estimation of the Kolmogorov constant $C_{k1}$ crucial. Following Zhou et al. (Reference Zhou, Nagata, Ito, Sakai and Hayase2023), assuming that the conditional energy spectra $E_{11}^T(k_1)$ exhibit universal behaviour in both the small-scale dissipation range and the intermediate $- 5/3$ power-law range, we can obtain the corresponding conditional one-dimensional energy spectrum expression in the turbulent region, i.e.

(3.2) \begin{equation} E_{11}^T(k_1) = \gamma ^{-1} E_{11}(k_1), \end{equation}

where $E_{11}^T({k_1}) = C_{k1}^T{({\varepsilon ^T})^{2/3}}{k_1}^{ - 5/3}$ with $C_{k1}^T$ being the conditional Kolmogorov constant, which is related to the Kolmogorov constant $C_{k1}$ by the scaling law $C_{k1} = \gamma ^{1/3} C_{k1}^{T}$ . A more detailed description of the conditional energy spectrum is included in Appendix A. The scaling law $C_{k1} = \gamma ^{1/3} C_{k1}^{T}$ has been confirmed by Zhou et al. (Reference Zhou, Nagata, Ito, Sakai and Hayase2023) in spatially intermittent wakes with strong vortex shedding. This means that a more direct method is provided to calculate the conditional Kolmogorov constant $C_{k1}^{T}$ and further determine the energy dissipation rate in the intermittent region.

Figure 8.

(a) Conditional one-dimensional energy spectrum normalised by the conditional Kolmogorov length scale $\eta ^T$ and the conditional averaged dissipation rate $\varepsilon ^T$ and (b) the corresponding conditional compensated energy spectra of the streamwise velocity fluctuations at four different cross-wise positions. In panel (a), the orange dashed line indicates the $- 5/3$ power-law and in panel (b), the two vertical solid orange lines indicate the upper and lower limits of the inertial subrange.

Figure 9.

(a) Transverse spectra ${E_{22}}({k_1})$ and (b) the corresponding conditional transverse spectra ${E_{22}^T}({k_1})$ of the normal velocity fluctuations at four different cross-wise positions. The orange dashed line indicates the −5/3 power-law range.

Figures 8(a) and 8(b) show the conditional one-dimensional energy spectra and the corresponding conditional compensated energy spectra of the streamwise velocity fluctuations. Note that the conditional energy spectrum reflects the information of the turbulent regions. The traditional Reynolds decomposition, which combines information from both laminar and turbulent flows in intermittent regions, cannot remove the influence of laminar flow signals on small-scale variables (see figure 6 a–c). As reported by Reuther & Kähler (Reference Reuther and Kähler2020), the applicability of the traditional Reynolds decomposition for flows with intermittent behaviour is questionable, and they further indicate that this method introduces artificial turbulent kinetic energy, which affects both single and multi-point statistics in intermittent regions. To minimise the influence of laminar signals, the conditional energy spectrum is normalised by the conditional average values (i.e. $\varepsilon ^T$ and $\eta ^T$ ) in intermittent regions. It can be observed from figures 8(a) and 8(b) that compared with the energy spectrum in figure 7, the conditional energy spectrum shows a good collapse in the inertial and the dissipation ranges of small and intermediate scales. The reasonably excellent collapses for different intermittency factor $\gamma$ indicate that the scaling law $C_{k1} \sim \gamma ^{1/3}$ is expected to hold. Cuypers, Maurel & Petitheans (Reference Cuypers, Maurel and Petitheans2003) suggested that vortex bursts contribute to elementary flow structures and are responsible for the $- 5/3$ scaling law. Our finding is the $- 5/3$ power law within the inertial subrange is a natural consequence of the presence of a complete turbulent region, i.e. an intrinsic feature of the high vorticity region.

The transverse spectra ${E_{22}}({k_1})= {C_{k1}^{\prime }}{\varepsilon ^{2/3}}{k_1}^{ - 5/3}$ at different cross-wise positions are shown in figure 9(a). These spectra describe the energy distribution perpendicular to the flow direction and are crucial for testing local isotropy in turbulent flows. It can be clearly seen that, similar to the longitudinal energy spectrum ${E_{11}}({k_1})$ , all transverse spectrum curves exhibit a well-defined approximate $- 5/3$ power law over a broad frequency range. For homogeneous isotropic turbulence, we have $C_{k1}^{\prime } = 4/3 C_{k1}$ (Pope Reference Pope2000). In the jet centreplane with $\gamma =1.00$ , the constant $3/4 C_{k1}^{\prime }=0.58$ is not exactly equal to $C_{k1} = 0.55$ due to the differences between isotropic turbulence and genuine turbulent flows. Based on the relationship between the energy spectra ${E_{11}}({k_1})$ and ${E_{22}}({k_1})$ , we can easily derive the corresponding conditional transverse spectra ${E_{22}^T}({k_1})$ , that is, $E_{22}^T(k_1) = \gamma ^{-1} E_{22}(k_1)$ . Here, The conditional transverse spectra $E_{22}^T({k_1}) = C_{k1}^{\prime T}{({\varepsilon ^T})^{2/3}}{k_1}^{ - 5/3}$ with $C_{k1}^{\prime T}$ being the dimensionless constant closely related to the Kolmogorov constant $C_{k1}^T$ . The conditional transverse spectra ${E_{22}^T}({k_1})$ are shown in figure 9(b). As expected, the conditional transverse spectrum exhibits good collapse in the inertial and dissipation ranges of small and intermediate scales.

Figure 10 presents the averaged Kolmogorov constant $C_{k1}$ , conditional constant $C_{k1}^T$ and $\gamma ^{1/3}C_{k1}^T$ within the inertial subrange of the spectral space as a function of intermittency factors $\gamma$ to quantitatively evaluate the relation between the Kolmogorov constants $C_{k1}$ and $C_{k1}^{T}$ . The inertial subrange is determined to be within the range of $0.03 \eta ^T$ to $0.1 \eta ^T$ . It is worth mentioning that a different choice of the average range will not change the main conclusion drawn here. It can be observed that $C_{k1}^T$ remains approximately constant across a broad range of $\gamma$ . The numerical measurement $C_{k1}$ is in fairly good agreement with the predictive model $C_{k1} = \gamma ^{1/3} C_{k1}^T$ , indicating that the scaling law $C_{k1} = \gamma ^{1/3} C_{k1}^T$ holds well in intermittent flows both with and without large-scale vortex shedding, and may be considered universal, such as in the planar jet case considered here and the wake case of Zhou et al. (Reference Zhou, Nagata, Ito, Sakai and Hayase2023).

Figure 10.

Kolmogorov constant $C_{k1}$ and conditional Kolmogorov constant $C_{k1}^T$ as a function of intermittency factors $\gamma$ within the inertial subrange of the spectral space.

4.1. Appearance of a 2/3 power-law in a highly intermittent flow

The normalised second-order structure function $ \langle \delta u^{\prime 2}\rangle (r)$ at the jet centreplane and three different positions (i.e. $\gamma =0.25$ , $0.50$ and $0.75$ ) within a wide range of intermittency are plotted in figure 11(a). Kolmogorov proposed that the smallest scales in turbulent flows exhibit local self-preservation, leading to the concept of a universal equilibrium range, where statistical properties of turbulence become independent of the specific details of the flow. Based on the above-mentioned analysis, it can be readily confirmed that the structure function has reasonably good collapses at the dissipation scale within the range of $ r/\eta \leqslant 8$ , and it follows the classical Kolmogorov scaling $\langle {\delta {u^{\prime 2}}} \rangle (r)/v_\eta ^2 \propto {(r/\eta )^2}$ . However, the situation is different at intermediate and large scales, where the structure functions exhibit obvious non-self-similar arrangements, and $ \langle \delta u^{\prime 2}\rangle (r)$ decreases as $\gamma$ decreases. This finding validates the general property of turbulence that the similarity of Kolmogorov normalised distributions is difficult to achieve across the entire range of scales. In other words, large-scale fluctuations are not universal (Pearson & Antonia Reference Pearson and Antonia2001).

Another interesting finding from figure 11(a) is that with the decrease of $\gamma$ , a broader $2/3$ power law emerges within the inertial range. The shape of the structure function at different locations hardly changes, which is consistent with the view that external intermittency of the flow mainly changes the coefficients in the scaling law of the structure function while maintaining the scaling exponent (Kuznetsov et al. Reference Kuznetsov, Praskovsky and Sabelnikov1992). We further investigate the prefactor $ C_2$ of the second-order structure function without the effects of large-scale oscillations, which are always found in wakes behind bluff bodies (Alves Portela et al. Reference Alves Portela, Papadakis and Vassilicos2017; Zhou et al. Reference Zhou, Nagata, Sakai and Watanabe2019). Zhou et al. (Reference Zhou, Nagata, Sakai and Watanabe2019) further pointed out that internal instabilities arise in the intermittent regions, which are caused by the large-scale vortex shedding. The expected $2/3$ power law at intermediate scales will be relatively weak or even disappear, and the highly intermittent upstream locations will exhibit a well-defined $1$ power law. This findings confirm that the Kolmogorov constant $C_2$ in the wake is contaminated by vortex shedding, which complicates the investigation of $C_2$ .

Figures 11(b) and 11(c) show the corresponding compensated second-order structure function. The compensated structure function profile in the jet centreplane, with $ \gamma = 1$ , shows good agreement with the numerical simulation results of isotropic turbulence at a similar Reynolds number of $\textit{Re}_\lambda = 257$ (Ishihara et al. Reference Ishihara, Gotoh and Kaneda2009). We found that, despite the presence of large-scale laminar motions in the intermittent region, a distinct plateau region appears at all considered positions, whether in the log–log scale or the linear–log scale. This indicates that large-scale anisotropy does not significantly affect the small-scale behaviour. For the centre of the jet, the intermediate eddies are completely controlled by the turbulent dissipation rate $ \varepsilon$ , and the constant $C_2$ is approximately close to 2 as predicted. Moreover, in the intermittent regions, $C_2$ decreases as $\gamma$ decreases. Gauding et al. (Reference Gauding, Bode, Brahami, Varea and Danaila2021) analysed the structure function of the intermittent regions using the random telegraphic signal. It is observed that the larger differences in the intermediate range are attributed to the fact that the morphology of the TNTI is not completely random, such as the position distribution of the TNTI satisfies the Gaussian distribution (as shown in figure 4 b).

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

The self-similar behaviour of the conditional second-order structure function is further investigated in the context of the self-similar behaviour of the conditional energy spectrum $E_{11}^T(k_1)$ . For incompressible homogeneous isotropic turbulence (HIT), one could derive the relationship between the one-dimensional conditional energy spectrum and the corresponding second-order structure function (see Appendix B). In other words, in HIT, the self-similarity of the conditional one-dimensional energy spectrum and the conditional second-order structure function are essentially interchangeable. According to Appendix B, the corresponding structure function, when properly normalised by using the conditional Kolmogorov scale and the intermittency factor $\gamma$ , develops in a self-similar manner, i.e. ${\langle {\delta {u^{\prime 2}}}\rangle }(r/{\eta ^T})/(\gamma \nu _\eta ^{T2}) \approx g(r/{\eta ^T})$ . Similar to the derivation of the Kolmogorov constant $C_{k1}$ in spectral space, making use of the relations ${\langle {\delta {u^{\prime 2}}}\rangle } (r) \gamma ^{-1} = {C_2^T}{(\varepsilon ^T r)^{2/3}}$ and ${\langle {\delta {u^{\prime 2}}}\rangle } (r) = {C_2}{(\varepsilon r)^{2/3}}$ within the $- 5/3$ power-law range yields

(4.2) \begin{equation} C_{2} = \gamma ^{1/3} C_{2}^{T}, \end{equation}

where $C_{2}^T$ is the conditional Kolmogorov constant in physical space. This means that the Kolmogorov constant $C_2$ exhibits a power-law dependence on the intermittency factor $\gamma$ .

Figure 12.

The same as figure 11, but (a) the conditional second-order structure function $\langle \delta u^{\prime 2}\rangle \gamma ^{-1}/(r\varepsilon ^T)^{2/3}$ normalised by the conditional Kolmogorov length scale $\eta ^T$ and the conditional Kolmogorov velocity scale $v_{\eta } ^T$ , and (b) the corresponding conditional compensated structure function at four different cross-wise positions.

Figure 13.

Kolmogorov constant $C_{2}$ and conditional Kolmogorov constant $C_2^T$ as a function of intermittency factors $\gamma$ within the inertial subrange of the physical space. For comparison, the profile of $\gamma ^{1/3}C_2^T$ is also plotted.

The conditional second-order structure function of the streamwise velocity fluctuations and the compensated results are shown in figure 12. Reasonable collapse of the structure function profiles for different cross-wise positions can be observed, while at the highly intermittent position, where $\gamma = 0.25$ , there remains a slight deviation compared with the other profiles. One may guess that this is attributed to the existence of the TNTI, since the turbulence within the finite thickness of the TNTI layer exhibits fundamentally different kinematic characteristics compared with the turbulent core region. As illustrated in figure 1(b), the vorticity increases rapidly within the TNTI. Figure 13 presents the averaged Kolmogorov constant $C_{2}$ , conditional constant $C_{2}^T$ and $\gamma ^{1/3} C_{2}^T$ within the inertial subrange of the physical space as a function of intermittency factors $\gamma$ to quantitatively evaluate the relation between the Kolmogorov constants $C_{2}$ and $C_{2}^{T}$ . The inertial subrange is determined to be within the range of $r/\eta ^T=60.0$ to $120.0$ , corresponding to the angular wavenumber range of $k_1 \eta ^T = 0.05$ to $0.1$ . Within this range, the one-dimensional energy spectrum exhibits a clear $- 5/3$ power law (see figure 8). It is worth mentioning that a different choice of the average range will not change the main conclusion. Distinctly different from the behaviour of the conditional constant $C_{k1}^T$ in spectral space, the value of the conditional constant $C_{2}^T$ in the physical space fails to remain roughly constant. At the cross-wise position of $\gamma = 0.25$ , the occurrence of significant small value of $C_{2}^T$ is consistent with the slight deviation of self-similarity (see figure 12), which will be discussed later. In summary, the scaling law $C_{2} = \gamma ^{1/3} C_{2}^{T}$ holds well throughout the intermittent region.

It should be noted that although this study focuses on the temporally evolving plane jet case, the correlation $C_2 \sim \gamma ^{1/3}$ also holds for the spatially evolving turbulence. The scaling $C_2 \sim \gamma ^{1/3}$ can also be observed in turbulence generated by an active grid, where energy input alternates between roughly constant and quiescent states (Chien, Blum & Voth Reference Chien, Blum and Voth2013), with the Taylor microscale Reynolds number $\textit{Re}_{\lambda }$ reaching up to 188.

4.3. Relationship between the spectra constant $C_{k1}$ and structure function constant $C_2$

Figure 14.

Kolmogorov constants $C_{k1}$ and $C_2$ as a function of the intermittency factor $\gamma$ . The profile $ 3.49 C_{k1}$ is also included for comparison.

Praskovsky & Oncley (Reference Praskovsky and Oncley1994) found that, due to the internal intermittency effect of the turbulent dissipation, the Kolmogorov constant $C_{k1}$ exhibits a weak dependence on the Reynolds number $\textit{Re}_{\lambda }$ . This dependence can be accurately characterised by the power law $C_{k1} \propto {\textit{Re}_{\lambda }}^{ - 0.10}$ . However, Sreenivasan (Reference Sreenivasan1995) measured the Kolmogorov constant $C_{k1}$ across a wide range of Reynolds numbers. They indicated that $C_{k1}$ can be considered universal, fundamentally independent of the flow type and the Reynolds number. For Reynolds numbers with $\textit{Re}_\lambda \geqslant 50$ , the average value of $C_{k1}$ is approximately 0.53, with a standard deviation of approximately 0.055 (Sreenivasan Reference Sreenivasan1995). As shown in figure 10, our numerical results show that $C_{k1}$ in the fully turbulent region is approximately 0.55, which is in good agreement with the previous findings.

The relationship between the two constants $C_{k1}$ and $C_2$ , that is, $C_2 \approx 4.02C_{k1}$ , can be directly derived from a simplified model with an infinite Reynolds number (Monin & Yaglom Reference Monin and Yaglom1975). Ni & Xia (Reference Ni and Xia2013) revisited the relationship between $C_{k1}$ and $C_2$ . Their study indicates that the widely used relationship $C_2 \approx 4.02C_{k1}$ is valid only for ${\textit{Re} _\lambda } \geqslant {10^5}$ , and the ratio $C_{k1} / C_{2}$ is observed to exhibit a distinct dependence on the local Reynolds number, that is,

(4.3) \begin{equation} C_{k1}/C_2=0.25+1.95\textit{Re}_{\lambda }^{-0.68}. \end{equation}

The local Reynolds number at $Y/b_U=0$ with $\gamma = 1$ and the returned value $C_2/C_{k1}$ by using the empirical prediction (i.e. (4.3)) is approximately $3.35$ . The empirical value $C_2/C_{k1} = 3.35$ is indeed close to our computed value of $3.49$ . An interesting finding is that the ratio $ {C_2}/{C_{k1}}$ remains constant over a wide range of intermittency with $\gamma \geqslant 0.50$ (see figure 14).

5.1. Contributions from different states to the second-order structure function

Figure 15.

Schematic diagram of the second-order structure function in five different states at $\gamma =0.50$ . For reference, the white bar in the lower right corner represents $100$ times the conditional Kolmogorov length scale in the corresponding intermittent region.

Based on the state of the two locations ( ${u^{\prime }}(\boldsymbol{x}+r\boldsymbol{e_{1}})$ and $u^{\prime }(\boldsymbol{x})$ ) and the flow field within the separation distance $ r$ , as shown in figure 15, there are five different turbulent states corresponding to different conditional second-order structure functions, i.e. $\langle \delta u^{\prime 2}\rangle _{\textit{TT}}$ , $\langle \delta u^{\prime 2}\rangle _{\textit{LL}}$ , $\langle \delta u^{\prime 2}\rangle _{\textit{TL}}$ , $\langle \delta u^{\prime 2}\rangle _{\textit{TLT}}$ and $\langle \delta u^{\prime 2}\rangle _{\textit{LTL}}$ . Take, for instance, the second-order structure function $\langle \delta u^{\prime 2}\rangle _{\textit{TT}}$ indicates that the two locations ${u^{\prime }}(\boldsymbol{x}+r\boldsymbol{e_{1}})$ and $u^{\prime }(\boldsymbol{x})$ are located in the turbulent region, and the separation distance $r$ is all in the turbulent state, while $\langle \delta u^{\prime 2}\rangle _{\textit{TLT}}$ means that there is a laminar signal between the separation distance $r$ . Therefore, the contribution of the conditional second-order structure function for analysing different states was determined to satisfy the following conditions:

(5.1)

(5.2) \begin{align} \big \langle \delta u^{\prime 2} \big \rangle _{\textit{TLT}} & \!= \big \langle H(\boldsymbol{x}+r\boldsymbol{e_{1}}) H(\boldsymbol{x})\left (u^{\prime }\left (\boldsymbol{x}+r\boldsymbol{e_{1}}\right )-u^{\prime }(\boldsymbol{x})\right )^2 \!\big \rangle \quad\!\!\!\! \text{and}\quad\!\!\!\! \int _0^rH(\boldsymbol{x}+y\boldsymbol{e}_1)\,{\textrm{d}}y \lt r+1, \end{align}

where the separation distance $r$ between points ${u^{\prime }}(\boldsymbol{x}+r\boldsymbol{e_{1}})$ and $u^{\prime }(\boldsymbol{x})$ is all in the turbulent state, which is satisfied by the condition $\int _0^rH(\boldsymbol{x}+y\boldsymbol{e}_1){\textrm{d}}y = r+1$ . Similarly, a laminar state exists when the separation distance $r$ satisfies the condition $\int _0^rH(\boldsymbol{x}+y\boldsymbol{e}_1){\textrm{d}}y \lt r+1$ , where $y$ is a dummy integration variable. The remaining structure functions are defined using a similar approach. This means that the overall structure function can be directly decomposed into the sum of the structure functions in five different states. Obviously, by definition, we obtain

(5.3) \begin{equation} \begin{aligned}\big \langle \delta u^{\prime 2}\big \rangle & = \gamma _{\textit{TT}}\big \langle \delta u^{\prime 2}\big \rangle _{\textit{TT}}+ \gamma _{LL}\big \langle \delta u^{\prime 2}\big \rangle _{LL}+ \gamma _{\textit{TL}}\big \langle \delta u^{\prime 2}\big \rangle _{\textit{TL}} \\[3pt] &\quad + \gamma _{\textit{TLT}}\big \langle \delta u^{\prime 2}\big \rangle _{\textit{TLT}}+ \gamma _{\textit{LTL}}\big \langle \delta u^{\prime 2}\big \rangle _{\textit{LTL}},\end{aligned} \end{equation}

where $\gamma _{\textit{TT}}(r)$ , $\gamma _{\textit{LL}}(r)$ , $\gamma _{TL}(r)$ , $\gamma _{TLT}(r)$ and $\gamma _{LTL}(r)$ represent the probabilities of different states at a given separation distance $r$ . This treatment is significantly different from that used by Kuznetsov et al. (Reference Kuznetsov, Praskovsky and Sabelnikov1992), where only three states are evaluated and the structure functions of the two complex states, i.e. $\langle \delta u^{\prime 2} \rangle _{\textit{TLT}}$ and $\langle \delta u^{\prime 2} \rangle _{\textit{LTL}}$ , are neglected.

Figure 16.

Schematic diagram of filtering isolated small pockets using the open-source flood fill algorithm. (a) Before filtering, (b) after filtering. The black solid line represents the selected vorticity threshold with $| \boldsymbol{\omega } |_{\textit{th}}$ = $0.14 \langle {| \boldsymbol{\omega } |} \rangle _{C}$ .

Considering the large-scale engulfment mechanism of the entrainment process near the TNTI, the isolated irrotational pockets will be wrapped in the turbulent region (Taveira et al. Reference Taveira, Diogo, Lopes and da Silva2013; da Silva et al. Reference da Silva, Hunt, Eames and Westerweel2014). Based on the definition of the conditional second-order structure function $\langle \delta u^{\prime 2} \rangle _{\textit{TT}}$ , when calculating $\langle \delta u^{\prime 2} \rangle _{\textit{TT}}$ in closed turbulent regions, some statistical points may be excluded due to contamination by the low vorticity pockets, even though they should be included. To this end, the open-source flood fill algorithm is adopted to remove the small-scale pockets, similar to the study of Heisel et al. (Reference Heisel, de Silva, Katul and Chamecki2022), which eliminated the isolated uniform momentum zones pockets in a turbulent boundary layer. Figure 16 shows a schematic diagram of filtering isolated pockets using the flood fill algorithm. It can be clearly seen that the isolated low vorticity pockets have been effectively removed and the internal turbulent region is fully preserved.

The probability of the conditional structure function under different separation distances at $\gamma =0.50$ is plotted in figure 17(a). The sum of the probabilities is equal to $1$ , which ensures the accuracy of the decomposition of the structure function. We find that $\gamma _{\textit{TT}}(r)$ and $\gamma _{\textit{LL}}(r)$ are approximately equal in the entire separation range (i.e. $\gamma _{\textit{TT}}(r) \approx \gamma _{\textit{LL}}(r)$ ) and both decrease from approximately $0.5$ to $0$ with increasing separation distance due to the intermittency of the flow. In the small-scale range of $r \leqslant 10{\eta ^T}$ , the two complex states can be almost ignored and the condition $\gamma _{\textit{TT}}(r) + \gamma _{\textit{LL}}(r) + \gamma _{TL}(r) \simeq 1$ is satisfied, which is consistent with the three flow states considered by Kuznetsov et al. (Reference Kuznetsov, Praskovsky and Sabelnikov1992). However, as $r$ increases, the two complex states begin to emerge, indicating that the flow characteristics of the intermittent region become more complex. Figure 17(b) shows the contribution of the conditional second-order structure function of five different states to the overall structure function at $\gamma = 0.50$ . For the sake of comparison, all the conditional structure functions are normalised by the conditional Kolmogorov scale of the turbulent region. For the small-scale dissipation range of $r \leqslant 10 \eta ^T$ , we have

(5.4) \begin{equation} \big \langle \delta u^{\prime 2}\big \rangle \simeq \gamma _{\textit{TT}}\big \langle \delta u^{\prime 2}\big \rangle _{\textit{TT}}, \gamma \simeq \gamma _{\textit{TT}}. \end{equation}

Within the entire range of $r \leqslant {1000} \eta ^T$ , (5.3) can be simplified as

(5.5) \begin{equation} \big \langle \delta u^{\prime 2}\big \rangle \simeq \gamma _{\textit{TT}}\big \langle \delta u^{\prime 2}\big \rangle _{\textit{TT}}+ \gamma _{\textit{TL}}\big \langle \delta u^{\prime 2}\big \rangle _{\textit{TL}} + \gamma _{\textit{TLT}}\big \langle \delta u^{\prime 2}\big \rangle _{\textit{TLT}}. \end{equation}

This means that the overall structure function $\langle \delta u^{\prime 2}\rangle (r)$ consists of the conditional structure function associated with the turbulence signal. Considering that different normal positions exhibit similar turbulence properties, it is important to note that although only the position with $\gamma =0.50$ is considered, these findings have been confirmed to be equally applicable to other positions.

Figure 17.

(a) Corresponding probability of the conditional second-order structure function under different separation distances at $\gamma = 0.50$ . (b) Contribution of the conditional compensated structure function to the overall structure function under five different turbulent states. The overall structure function (represented by black solid line) at $\gamma =0.50$ are also plotted for comparison.

5.2. Self-similarity of the second-order structure functions in turbulent regions

The conditional second-order structure function of the streamwise velocity fluctuations in the turbulent regions and the compensated results are shown in figure 18. It can be seen that, except for the cross-wise position at $\gamma =0.25$ , the profiles of the remaining conditional structure functions exhibit a good collapse at small and intermediate scales. This is consistent with the collapse of the profiles of the normalised spectra. However, compared with figure 12, the expression $\langle \delta u^{\prime 2}\rangle _{\textit{TT}} (r)\approx \gamma ^{-1} \langle \delta u^{\prime 2}\rangle (r)$ is satisfied over a relatively wide range of $\gamma$ with $\gamma \geqslant 0.50$ , albeit significant deviations are observed at the cross-wise position of $\gamma =0.25$ . This observation means that the conditional second-order structure function in the turbulent region can be indirectly calculated from the overall structure function. According to Appendix A, we have $\langle {{{ ( {{{\partial {u'}} / {\partial {x}}}} )}^2}} \rangle = \gamma {\langle {{{ ( {{{\partial {u'}} / {\partial {x}}}} )}^2}} \rangle ^T}$ (see (A4)). When the separation distance $ r$ is sufficiently small, through Taylor’s expansion and first-order finite difference, we obtain

(5.6) \begin{equation} \big \langle (u'(x+r) - u'(x))/r )^2 \big \rangle ^T = \gamma ^{-1} \big \langle (u'(x+r) - u'(x))/r )^2 \big \rangle . \end{equation}

As can be seen in figure 17(b), $\langle \delta u'^{2} \rangle _{TL} (r)\gamma _{TL} \ll \langle \delta u'^{2} \rangle _{\textit{TT}}(r) \gamma _{\textit{TT}}$ can be observed for sufficiently small $r$ , i.e. on average, the velocity increment between two points across the TNTI layer can be ignored. Further simplification of (5.6) reveals that there is an approximate functional relationship between the conditional second-order structure function $\langle \delta u'^{2} \rangle _{\textit{TT}}(r)$ , the overall structure function $\langle \delta u'^{2} \rangle (r)$ and the intermittency factor $\gamma$ , i.e. $\langle \delta u'^{2} \rangle _{\textit{TT}}(r) \approx \gamma ^{-1} \langle \delta u'^{2} \rangle (r)$ . This implies that the relationship strictly holds at sufficiently small $r$ .

Figure 18.

(a) Conditional second-order structure function $\langle \delta u^{\prime 2}\rangle _{\textit{TT}}$ of the streamwise velocity fluctuations normalised by the Kolmogorov length scale $\eta ^T$ and the Kolmogorov velocity scale $v_{\eta } ^T$ and (b) the corresponding conditional compensated structure function in the turbulent regions of four different cross-wise positions.

Gauding et al. (Reference Gauding, Bode, Brahami, Varea and Danaila2021) also explored the similarity of the conditional structure function within the jet flow with external intermittency. It is worth mentioning that in their study, the second-order structure of the turbulent flows with external intermittency range $ 0.45 \leqslant \gamma \leqslant 1$ was explored. The high intermittency case $\gamma = 0.25$ was not considered; consequently, the corresponding deviation of self-similarity in the conditional structure function (see figure 18) cannot be observed.

5.3. Second-order structure function related to the interface

It is well known that the TNTI can impose a kinematic blocking/sheltering effect, acting as a barrier to the penetration of fluctuations across the interface (da Silva et al. Reference da Silva, Hunt, Eames and Westerweel2014). The blocking/sheltering mechanisms result in a decorrelation of velocity fluctuations across the interface (Ishihara, Ogasawara & Hunt Reference Ishihara, Ogasawara and Hunt2015). In the vicinity of the TNTI, interscale energy transfers flow from small to large scales along directions aligned with the tangent plane, where stretching motions predominate. In contrast, in the directions normal to the interface, energy transfers from large to small scales, which is dominated by compressive motions (Watanabe, da Silva & Nagata Reference Watanabe, da Silva and Nagata2020; Zhou & Vassilicos Reference Zhou and Vassilicos2020).

To assess the effect of the dynamics near the TNTI on the statistics of the structure function, we calculate the conditional second-order structure function with one-point located at the outer edge of the TNTI. As shown in figure 19(a), there are two entirely distinct flow states on either side of the TNTI, characterised by different structure functions (i.e. ${\langle (u'_T - u'_{\textit{TNTI}})^2 \rangle }(r)$ and ${\langle (u'_L - u'_{\textit{TNTI}})^2 \rangle }(r)$ ). Performing data statistics on both sides of the TNTI may cause the local coordinates to cross more than one outer edge of the TNTI, due to the highly distorted nature of the TNTI. As a result, some parts of the local coordinates are excluded from the statistics, such as the data points in the dashed part of the local coordinates in figure 19(a).

The conditional structure functions along the turbulent and laminar regions on the TNTI are plotted in figures 19(b) and 19(c). It can be seen that the conditional structure functions near the TNTI are below the fully turbulent profile and there is no clear $2/3$ scaling within the intermediate range. This observation explains the slightly uncollapsed profile for the intermittent case $\gamma =0.25$ (see figure 18). Another interesting finding is that the profiles for different intermittency factors, $\gamma$ , collapse reasonably well, indicating that, regardless of the intermittency factor, the turbulence characteristics near the TNTI are similar. Our finding concerning the similar distributions of the two-point statistics somehow echoes the universality of small-scale motions across the TNTI reported by Zecchetto & da Silva (Reference Zecchetto and da Silva2021). In their study, it was demonstrated that the conditional mean profiles of various one-point statistics near the TNTI collapse together when normalised using the local Kolmogorov velocity and length scales.

Figure 19.

(a) Schematic diagram of the two different second-order structure functions at the outer edge of the TNTI. (b) ${\langle (u'_T - u'_{\textit{TNTI}})^2 \rangle }(r)$ and (c) ${\langle (u'_L - u'_{\textit{TNTI}})^2 \rangle }(r)$ represent the structure functions at the TNTI, oriented towards the turbulent and laminar regions, respectively. For comparison, the red curves corresponding to $\gamma = 1.0$ represent the fully turbulent profiles. In panel (a), the horizontal lines represent the local coordinate system defined based on the position of the outer edge of the TNTI. The locations marked with dashed lines in the local coordinate system are excluded from the conditional statistics.

To exclude the blocking effects of the TNTI, we further increase the threshold to compute the structure function in the high enstrophy region with $| \boldsymbol{\omega } |_{\textit{th}}$ = $0.14 \langle {| \boldsymbol{\omega } |} \rangle _{C}$ . Two corresponding contour lines of different thresholds (i.e. $0.01 \langle |\boldsymbol{\omega }| \rangle _C$ and $0.14 \langle |\boldsymbol{\omega }| \rangle _C$ ) are given in figure 20(a). Figure 20(b) demonstrates that the structure function in the high enstrophy region is well collapsed within the inertial and dissipation ranges. Even at positions with a departure from self-similarity, that is, the small intermittency factor $\gamma =0.25$ , a good collapse can also be observed. As the vorticity threshold is further increased, the profiles of the structure functions at different cross-wise positions remain unchanged (not shown herein). The conditional Kolmogorov constant $C_2^T$ at high vorticity threshold $| \boldsymbol{\omega } |_{\textit{th}}$ = $0.14 \langle {| \boldsymbol{\omega } |} \rangle _{C}$ for $\gamma =0.25$ is $C_2^T= 1.92$ .

Figure 20.

(a) Contour of the vorticity field. The outer edge of the detected TNTI (white lines) and the contour lines (black lines) corresponding to the high vorticity threshold $| \boldsymbol{\omega } |_{\textit{th}}$ = $0.14 \langle {| \boldsymbol{\omega } |} \rangle _{C}$ are plotted. (b) Second-order structure function of the turbulent regions under the high vorticity threshold at four different cross-wise positions.

It is well known that the TNTI can be decomposed into two different sublayers (i.e. the viscous superlayer and the turbulent sublayer) with non-uniform thickness. A typical characteristic of the continuous viscous superlayer is that pure shear motion dominates, without the presence of rigid-body rotational motion of the fluid (Hayashi et al. Reference Hayashi, Watanabe and Nagata2021b ; Xie et al. Reference Xie, Yin, Zhang and Zhou2023). Currently, the influence of the dynamical characteristics near the TNTI on the structure functions is inadequately understood. Following Van Reeuwijk & Holzner (Reference Van Reeuwijk and Holzner2014), the mean distance between two different vorticity iso-surfaces ( $| \boldsymbol{\omega } |_{\textit{th1}}$ and $| \boldsymbol{\omega } |_{\textit{th2}}$ , where $| \boldsymbol{\omega } |_{\textit{th1}} \geqslant | \boldsymbol{\omega } |_{\textit{th2}}$ ), the volume $V_T$ and the surface area $A_{\textit{iso}}$ have the following approximate functional relationship:

(5.7) \begin{equation} \frac {{{\delta }{V_T}}}{{{\langle {\delta _d}\rangle _I}}} = \frac {A_{\textit{iso}}(\left | \boldsymbol{\omega } \right |_{\textit{th1}})+A_{\textit{iso}}(\left | \boldsymbol{\omega } \right |_{\textit{th2}})}{2}, \end{equation}

where ${\delta }{V_T}={V_T}(| \boldsymbol{\omega } |_{\textit{th2}})-{V_T}(| \boldsymbol{\omega } |_{\textit{th1}})$ represents the difference in volume and $\langle {\delta _d}\rangle _I$ represents the mean distance.

The instantaneous surface area $A_{\textit{iso}}$ of the isosurface can be accurately obtained by integrating the area density (Yurtoglu, Carton & Storti Reference Yurtoglu, Carton and Storti2018), which is given by

(5.8) \begin{equation} A_{\textit{iso}}=\int {\textrm{d}}A_{\textit{iso}}=-\int _V\frac {1}{|\boldsymbol{\nabla }|\boldsymbol{\omega }||}\frac {\partial \chi (|\boldsymbol{\omega }|)}{\partial X_i}\frac {\partial |\boldsymbol{\omega }|}{\partial X_i}\, {\textrm{d}}V, \end{equation}

where the occupancy function

(5.9) \begin{equation} \left .\chi (|\boldsymbol{\omega }|)=\left \{\begin{array}{cc}1,&|\boldsymbol{\omega }|\geqslant |\boldsymbol{\omega }|_{\textit{th}},\\0,&|\boldsymbol{\omega }|\lt |\boldsymbol{\omega }|_{\textit{th}},\end{array}\right .\right . \end{equation}

and ${\textrm{d}}V$ represents the volume element. The dependence of volume and surface area on the vorticity threshold can be used to obtain information about the mean distance between two different vorticity iso-surfaces (i.e. white and black lines plotted in figure 20 a), enabling us to further quantitatively evaluate the impact of the blocking/sheltering mechanisms on the structure function.

Owing to the sharp vorticity jump across the TNTI (see figure 1 b), although the threshold increases by 14 times, the two contour lines vary within a fairly small range and the turbulent volume is only decreased by ${\delta }{V_T}/{V_T}(| \boldsymbol{\omega } |_{\textit{th2}})=5.3\, \%$ . Combining with (5.7) and (5.8), we find that the mean distance $\langle {\delta _d}\rangle _I$ between two different iso-surfaces is roughly $20 \eta _C$ , which is comparable to the mean thickness of the TNTI, ${\langle {{\delta _{\textit{TNTI}}}} \rangle _I}= 23 \eta _C$ (see figure 1 b). The comparable mean thickness indicates that the influence of large-scale inhomogeneity and anisotropy near the TNTI is significant. These results explain the departures from self-similarity in structure functions for the turbulent flows with high intermittency $\gamma = 0.25$ .