3.1. Mean flow field

Figure 5 presents the mean flow field for each case, showing mean velocity streamlines and a colour contour of the magnitude of the mean velocity vector, $|\langle {\boldsymbol{u}}\rangle |$ . The streamlines indicate that opposing flows approach from both sides along the $x$ direction and expand vertically near the test-section centre. The stagnation point is identified by the minimum value of $|\langle {\boldsymbol{u}}\rangle |$ , close to zero, and is marked by red crosses. In all cases, the stagnation point is located near the geometric centre of the test section, $(x,y)=(0,0)$ , owing to the symmetric configuration of the test section and jet arrays. The streamline patterns imply $\partial \langle u \rangle /\partial x \lt 0$ and $\partial \langle v \rangle /\partial y \gt 0$ , consistent with axisymmetric expansion described by (1.3). The mean velocity remains below 45 m s⁻¹ in all cases, implying that the mean flow Mach number is less than 0.3.

Figure 5.

Averaged velocity fields coloured by the mean velocity magnitude, $|\langle \boldsymbol{u}\rangle |=\sqrt {\langle u\rangle ^2+\langle v\rangle ^2}$ , on the $x$ – $y$ plane at $z=0$ . Black lines with arrows indicate streamlines. Red crosses denote the minimum value of $|\langle \boldsymbol{u}\rangle |$ . Panels show results for different test section geometries: ( $a$ ) case 1, ( $b$ ) case 2, ( $c$ ) case 3, ( $d$ ) case 4, ( $e$ ) case 5 and ( $f$ ) case 6.

Figure 6 visualises the mean flow on the $x$ – $z$ plane at $y=0$ and on the $x$ – $y$ plane at $z/M=1$ . Owing to the geometry of the test section and jet arrays, the $x$ – $z$ plane at $y=0$ is equivalent to the $x$ – $y$ plane at $z=0$ . The mean velocity pattern of axisymmetric expansion is visible on the $x$ – $z$ plane as well as on the off-centre $x$ – $y$ plane.

Figure 6.

Averaged velocity fields coloured by the mean velocity magnitude in case 1: ( $a$ ) $|\langle \boldsymbol{u}\rangle |=\sqrt {\langle u\rangle ^2+\langle w\rangle ^2}$ on the $x$ – $z$ plane at $y=0$ ; ( $b$ ) $|\langle \boldsymbol{u}\rangle |=\sqrt {\langle u\rangle ^2+\langle v\rangle ^2}$ on the $x$ – $y$ plane at $z/M=1$ . Black lines with arrows indicate streamlines. Red crosses denote the minimum value of $|\langle \boldsymbol{u}\rangle |$ .

The mean velocity gradients associated with axisymmetric strain are given by the diagonal components of ${\unicode{x1D63C}}$ in (1.1), namely $\mathsf{A}_{\textit{xx}} = \partial \langle u \rangle /\partial x$ , $\mathsf{A}_{\textit{yy}} = \partial \langle v \rangle /\partial y$ and $\mathsf{A}_{zz} = \partial \langle w \rangle /\partial z$ . Since central-difference approximations introduce significant errors due to statistical uncertainty in the mean velocity distribution, the gradients are instead obtained by fitting $\langle u \rangle$ , $\langle v \rangle$ and $\langle w \rangle$ with bicubic functions $(x,y)$ or $(x,z)$ . This choice is motivated by the observation that a third-order polynomial adequately captures one-dimensional profiles of mean velocities along either $x$ , $y$ or $z$ axis. Higher-order functions were also tested but offered no improvement in accuracy and instead introduced unnecessary complexity and the risk of over-fitting.

Figures 7 and 8 show colour contours of $|\mathsf{A}_{\textit{xx}}|$ and $\mathsf{A}_{\textit{yy}}$ , normalised by their respective maxima $|\mathsf{A}_{\textit{xx}}|_{\textit{max}}$ and $(\mathsf{A}_{\textit{yy}})_{\textit{max}}$ . In all cases, $|\mathsf{A}_{\textit{xx}}|$ and $\mathsf{A}_{\textit{yy}}$ decrease with increasing $|x|$ , indicating that the mean strain strengthens as the flow enters the gap between the inner ducts and approaches the test-section centre. At $x \approx 0$ , both $|\mathsf{A}_{\textit{xx}}|$ and $\mathsf{A}_{\textit{yy}}$ increase with $|y|$ in cases 1–3, but the dependence on $x$ is stronger than on $y$ .

Figure 7.

Distributions of the mean velocity gradient magnitude $|\mathsf{A}_{\textit{xx}}|$ normalised by its maximum value $|\mathsf{A}_{\textit{xx}}|_{\textit{max}}$ . Panels show results for different test-section geometries: ( $a$ ) case 1, ( $b$ ) case 2, ( $c$ ) case 3, ( $d$ ) case 4, ( $e$ ) case 5 and ( $f$ ) case 6.

Figure 8.

Same as figure 7, but for $\mathsf{A}_{\textit{yy}}$ normalised by $(\mathsf{A}_{\textit{yy}})_{\textit{max}}$ : ( $a$ ) case 1, ( $b$ ) case 2, ( $c$ ) case 3, ( $d$ ) case 4, ( $e$ ) case 5 and ( $f$ ) case 6.

Table 1 includes the maximum values of $|\mathsf{A}_{\textit{xx}}|$ in each case. The mean velocity in the $x$ direction decreases from that in the inner duct to nearly zero at the stagnation point, and when $D_E$ is large this decrease occurs over a longer distance. Consequently, the mean velocity gradient tends to be smaller for larger $D_E$ . For fixed $D_N$ , $|\mathsf{A}_{\textit{xx}}|_{\textit{max}}$ decreases with increasing $D_E$ , but variations in $D_N$ also affect $|\mathsf{A}_{\textit{xx}}|_{\textit{max}}$ , indicating that it is not determined solely by $D_E$ .

A key feature of axisymmetric expansion is the ratio of the diagonal velocity gradients, satisfying $\mathsf{A}_{\textit{yy}} = \mathsf{A}_{zz} = -\mathsf{A}_{\textit{xx}}/2 \gt 0$ . Figure 9 shows the distribution of $-\mathsf{A}_{\textit{yy}}/\mathsf{A}_{\textit{xx}}$ on the $x$ – $y$ plane at $z=0$ . Deviations from $-\mathsf{A}_{\textit{yy}}/\mathsf{A}_{\textit{xx}}=0.5$ are observed only near the edges of the measurement region. In most regions close to the test-section centre, $-\mathsf{A}_{\textit{yy}}/\mathsf{A}_{\textit{xx}}$ is close to $0.5$ , lying within the range 0.4–0.6. Figure 10( $a$ ) presents the distribution of $-\mathsf{A}_{zz}/\mathsf{A}_{\textit{xx}}$ measured on the $x$ – $z$ plane at $y=0$ in case 1. As with $-\mathsf{A}_{\textit{yy}}/\mathsf{A}_{\textit{xx}}$ , $-\mathsf{A}_{zz}/\mathsf{A}_{\textit{xx}}$ is close to 0.5 except near the edges of the measurement region. Taken together, $-\mathsf{A}_{\textit{yy}}/\mathsf{A}_{\textit{xx}}\approx -\mathsf{A}_{zz}/\mathsf{A}_{\textit{xx}}\approx 0.5$ provides direct evidence of the formation of axisymmetric expansion and suggests that the continuity equation for the mean velocity in an incompressible flow is satisfied without compressibility effects. This is also consistent with the subsonic mean velocity, as low as about 45 m s⁻¹. Figure 10( $b$ ) shows the distribution of $-\mathsf{A}_{\textit{yy}}/\mathsf{A}_{\textit{xx}}$ on the $x$ – $y$ plane at $z/M=1$ in case 1. Even on this off-centre plane, $-\mathsf{A}_{\textit{yy}}/\mathsf{A}_{\textit{xx}}\approx 0.5$ holds except near the edges of the measurement region. These results demonstrate that axisymmetric strain is broadly established in the gap between the inner ducts of the two jet generators.

Figure 9.

Distributions of the ratio of mean velocity gradients, $-\mathsf{A}_{\textit{yy}}/\mathsf{A}_{\textit{xx}}$ . Panels show results for different test-section geometries: ( $a$ ) case 1, ( $b$ ) case 2, ( $c$ ) case 3, ( $d$ ) case 4, ( $e$ ) case 5 and ( $f$ ) case 6.

Figure 10.

Distributions of ( $a$ ) $-\mathsf{A}_{zz}/\mathsf{A}_{\textit{xx}}$ on the $x$ – $z$ plane at $y=0$ and ( $b$ ) $-\mathsf{A}_{\textit{yy}}/\mathsf{A}_{\textit{xx}}$ on the $x$ – $y$ plane at $z/M=1$ in case 1.

Table 1 lists the maximum values of $|\mathsf{A}_{\textit{xx}}|$ and $\mathsf{A}_{\textit{yy}}$ , denoted by $|\mathsf{A}_{\textit{xx}}|_{\textit{max}}$ and $(\mathsf{A}_{\textit{yy}})_{\textit{max}}$ , together with spatially averaged values of $|\mathsf{A}_{\textit{xx}}|$ and $\mathsf{A}_{\textit{yy}}$ . The ratios $(\mathsf{A}_{\textit{yy}})_{\textit{max}}/|\mathsf{A}_{\textit{xx}}|_{\textit{max}}$ and $\mathsf{A}_{\textit{yy}}/|\mathsf{A}_{\textit{xx}}|$ are also shown, both close to 1/2. Using the incompressible continuity equation, $\mathsf{A}_{zz}$ is estimated as $\mathsf{A}_{zz} = -\mathsf{A}_{\textit{xx}} - \mathsf{A}_{\textit{yy}}$ , which yields $\mathsf{A}_{\textit{yy}} \approx \mathsf{A}_{zz}$ when $\mathsf{A}_{\textit{yy}} \approx -\mathsf{A}_{\textit{xx}}/2$ . Thus, the condition for axisymmetric expansion is satisfied for both maximum and average values. Moreover, the relation $\mathsf{A}_{\textit{yy}} \approx \mathsf{A}_{zz}$ indicates that the flow is statistically axisymmetric in the $y$ and $z$ directions.

In summary, all cases successfully produce a mean strain characterised by compression in the $x$ direction and expansion in the $y$ and $z$ directions. Axisymmetric expansion, with $\mathsf{A}_{\textit{xx}} : \mathsf{A}_{\textit{yy}} : \mathsf{A}_{zz} = -2 : 1 : 1$ , is broadly observed near the test-section centre. This condition is uniformly satisfied for $|x/M| \leqslant 3$ and $|y/M| \leqslant 1.5$ . Although $-\mathsf{A}_{\textit{yy}}/\mathsf{A}_{\textit{xx}}$ remains approximately constant, both $-\mathsf{A}_{\textit{xx}}$ and $\mathsf{A}_{\textit{yy}}$ exhibit spatial variations, such that the strain rate intensifies as the flow approaches the test-section centre in the $x$ direction. Similar non-uniformity of axisymmetric strain has often been reported in wind tunnel experiments with variable cross-sections (Ayyalasomayajula & Warhaft Reference Ayyalasomayajula and Warhaft2006). The relation $\mathsf{A}_{\textit{yy}} = \mathsf{A}_{zz}$ , implied by the continuity equation, indicates statistical axisymmetry in the $y$ and $z$ directions, consistent with the symmetry of the test section and jet generators. Importantly, $-\mathsf{A}_{\textit{xx}}/2 = \mathsf{A}_{\textit{yy}} = \mathsf{A}_{zz}$ is satisfied not only along the symmetry axis ( $y=0$ ) but also for $y \neq 0$ and $z \neq 0$ , demonstrating that the mean strain remains axisymmetric throughout the central region of the test section. This mean strain characterisation provides the basis for interpreting the turbulence statistics discussed below.

3.2. One-point statistics of velocity fluctuations

One-point statistics of velocity fluctuations are evaluated to characterise the turbulence properties under axisymmetric expansion. Table 1 reports the spatially averaged values of $u_{\textit{rms}}$ and $v_{\textit{rms}}$ for all cases. Cases 1–3 exhibit lower values of $u_{\textit{rms}}$ and $v_{\textit{rms}}$ than cases 4–6. The turbulence generated by jet interaction decays within the inner duct. Since cases 1–3 employ longer inner ducts, the turbulence decays more before reaching the test-section centre, leading to smaller r.m.s. velocity fluctuations in the region where the mean strain forms. The present experiments test three values of the distance $D_E$ between the ends of the two inner ducts. However, the r.m.s. velocity fluctuations are less sensitive to $D_E$ than to $D_N$ . In all cases, $u_{\textit{rms}}$ exceeds $v_{\textit{rms}}$ , indicating anisotropy. Turbulence generated by the single jet array exhibits $u_{\textit{rms}}/v_{\textit{rms}} \approx 1.08$ (Mori et al. Reference Mori, Watanabe and Nagata2024). In the present experiments, the nearly HIT generated by jet interaction experiences axisymmetric expansion near the test-section centre, resulting in anisotropic velocity fluctuations. Table 1 also includes $u_{\textit{rms}}/v_{\textit{rms}}$ values evaluated from spatial averages of $u_{\textit{rms}}$ and $v_{\textit{rms}}$ . This ratio exceeds 1.4, much larger than the value of 1.08 for turbulence without axisymmetric expansion. Direct numerical simulations of initially isotropic turbulence subject to axisymmetric expansion have also shown that $u_{\textit{rms}}/v_{\textit{rms}} \gt 1$ (Lee Reference Lee1989), even though the turbulent Reynolds number, estimated in § 3.4, is much larger in the present experiments. Thus, the observed increase in $u_{\textit{rms}}/v_{\textit{rms}}$ due to axisymmetric expansion is consistent with previous numerical results.

Figures 11 and 12 show the distributions of $u_{\textit{rms}}$ and $v_{\textit{rms}}$ on the $x$ – $y$ plane at $z=0$ . The colour contours indicate relative deviations from the spatially averaged value in the FOV, $F_{{avg}}$ , defined as $\Delta (F)=(F-F_{{avg}})/F_{{avg}}$ for $F=u_{\textit{rms}}$ or $v_{\textit{rms}}$ . In figure 11, $u_{\textit{rms}}$ increases towards $x=0$ from both sides, whereas $v_{\textit{rms}}$ shows little dependence on $x$ for cases 1–3 and decreases to the centre for cases 4–6. Turbulence begins to be influenced by the mean strain near the ends of the inner ducts. As it is advected by the mean flow towards the test-section centre, it is exposed to the mean strain for a longer time. Additionally, the mean strain intensifies towards the test-section centre. This combination of strain intensification and accumulation of strain effects likely contributes to the increase in $u_{\textit{rms}}$ near the test-section centre. In contrast, variations in $v_{\textit{rms}}$ in the same region are less pronounced. Results of DNS studies of turbulence subject to axisymmetric strain at low Reynolds numbers have shown that the strain exerts stronger influence on r.m.s. velocity fluctuations in the symmetry axis direction, here the $x$ direction, than in the lateral ( $y$ and $z$ ) directions (Lee Reference Lee1989). These numerical findings qualitatively agree with the greater changes in $u_{\textit{rms}}$ observed toward the test-section centre.

Figure 11.

Relative difference of r.m.s. velocity fluctuations in the $x$ direction, $u_{\textit{rms}}$ , from the spatially averaged value $(u_{\textit{rms}})_{{avg}}$ , defined as $\Delta (u_{\textit{rms}}) = [u_{\textit{rms}} - (u_{\textit{rms}})_{{avg}}]/(u_{\textit{rms}})_{{avg}}$ , on the $x$ – $y$ plane at $z=0$ . Here the subscript ‘avg’ denotes spatial averages within the FOV. Panels show results for different test-section geometries: ( $a$ ) case 1, ( $b$ ) case 2, ( $c$ ) case 3, ( $d$ ) case 4, ( $e$ ) case 5 and ( $f$ ) case 6.

Figure 12.

Same as figure 11, but for r.m.s. velocity fluctuations in the $y$ direction, $v_{\textit{rms}}$ : ( $a$ ) case 1, ( $b$ ) case 2, ( $c$ ) case 3, ( $d$ ) case 4, ( $e$ ) case 5 and ( $f$ ) case 6. The relative difference is defined as $\Delta (v_{\textit{rms}}) = [v_{\textit{rms}} - (v_{\textit{rms}})_{{avg}}]/(v_{\textit{rms}})_{{avg}}$ .

Under axisymmetric expansion with $\mathsf{A}_{\textit{xx}}=-2\mathsf{A}_{\textit{yy}}=-2\mathsf{A}_{zz}\lt 0$ , the production term of turbulent kinetic energy of the $\alpha$ -directional velocity component is written as $P_\alpha =-\langle {u^{\prime}_\alpha }^2\rangle \mathsf{A}_{\alpha \alpha }$ with $\alpha =x,\ y$ or $z$ , where no summation is applied over $\alpha$ (Nieuwstadt et al. Reference Nieuwstadt, Westerweel and Boersma2016). This gives $P_x\gt 0$ and $P_y=P_z\lt 0$ , indicating an increase in the streamwise and a decrease in the transverse velocity fluctuations. Moreover, when turbulence is initially isotropic with $\langle {u'}^2\rangle =\langle {v'}^2\rangle =\langle {w'}^2\rangle$ , the relation $\mathsf{A}_{\textit{xx}}:\mathsf{A}_{\textit{yy}}:\mathsf{A}_{zz}=-2:1:1$ yields $P_x=2|P_y|=2|P_z|$ , implying that the strain induces greater changes to the velocity fluctuations in the $x$ direction than in the transverse ( $y$ and $z$ ) directions. Consistently, a significant increase in $u_{\textit{rms}}$ towards the test-section centre is observed in figure 11. These explanations are also consistent with DNS and RDT results (Lee Reference Lee1989). For axisymmetric contraction, the signs of $P_\alpha$ are reversed; transverse velocity fluctuations are amplified and longitudinal fluctuations attenuated, resulting in a decrease in $u_{\textit{rms}}/v_{\textit{rms}}$ (Lee Reference Lee1989; Ayyalasomayajula & Warhaft Reference Ayyalasomayajula and Warhaft2006).

Figure 13 compares $v_{\textit{rms}}$ and $w_{\textit{rms}}$ along the $x$ direction at the test-section centre, $(y,z)=(0,0)$ , for case 1. These components exhibit similar r.m.s. levels, as indicated by $v_{\textit{rms}}/w_{\textit{rms}}\approx 1$ , which is required for statistical axisymmetry. This also follows from the geometry of the test section and jet arrays, which are identical in the $y$ and $z$ directions.

Figure 13.

The $x$ -directional distributions of r.m.s. velocity fluctuations in the $y$ and $z$ directions, $v_{\textit{rms}}$ and $w_{\textit{rms}}$ , and their ratio $v_{\textit{rms}}/w_{\textit{rms}}$ at $(y,z)=(0,0)$ in case 1.

Statistical axisymmetry in the present set-up requires that, at a given $x$ , velocity statistics depend solely on $r_{yz}=\sqrt {y^2+z^2}$ and are independent of the azimuthal angle $\theta =\tan ^{-1}(y/z)$ . Table 2 compares $u_{\textit{rms}}$ at the same $r_{yz}$ but at different $(y,z)$ locations at $x=0$ on three measurement planes: the $x$ – $y$ planes at $z/M=0$ and 1 and the $x$ – $z$ plane at $y=0$ . For $r_{yz}/M=1$ , two points are presented because the locations at this $r_{yz}$ on the $x$ – $y$ plane at $z/M=1$ and the $x$ – $z$ plane at $y=0$ are identical. Although the square cross-section and square nozzle arrangement could, in principle, introduce $\theta$ dependence, $u_{\textit{rms}}$ at the same $r_{yz}$ shows no significant variation across $(y,z)$ locations. Therefore, the square geometries do not introduce $\theta$ dependence near the test-section centre.

Table 2.

Radial variations of $u_{\textit{rms}}$ at different azimuthal locations at $x=0$ in case 1.

Large-scale anisotropy is often characterised using the normalised anisotropy tensor of Reynolds stress (Pope Reference Pope2000), defined as

(3.1) \begin{align} \mathsf{b}_{\textit{ij}} = \frac {\langle u^{\prime}_i u^{\prime}_{\!j}\rangle }{\langle u^{\prime}_k u^{\prime}_k\rangle } - \frac {1}{3} \delta _{\textit{ij}}, \end{align}

where the summation convention is applied to repeated indices, $i,j,k=x,y$ or $z$ , and $\delta _{\textit{ij}}$ is the Kronecker delta. Second and third invariants of $\mathsf{b}_{\textit{ij}}$ , denoted by $B_2$ and $B_3$ , are given by $B_2 = (\mathsf{b}_{\textit{ij}} \mathsf{b}_{ji}/6)^{1/2}$ and $B_3 = (\mathsf{b}_{\textit{ij}} \mathsf{b}_{jk} \mathsf{b}_{ki}/6)^{1/3}$ . Anisotropy states are commonly visualised using a plot of $(B_2, B_3)$ , known as the Lumley triangle. The PIV measurements of $(u,v)$ on the $x$ – $y$ plane provide $\mathsf{b}_{\textit{xx}}$ , $\mathsf{b}_{\textit{yy}}$ and $\mathsf{b}_{xy}=\mathsf{b}_{yx}$ . Based on the assumption of axisymmetry at $(y,z)=(0,0)$ , the following conditions are adopted in the analysis of $B_2$ and $B_3$ : $\mathsf{b}_{\textit{yy}}=\mathsf{b}_{zz}$ and $\mathsf{b}_{\textit{ij}}=0$ for $i\neq j$ . The present PIV yields $|\langle u'v'\rangle |/u_{\textit{rms}}v_{\textit{rms}}\lt 0.1$ at $(y,z)=(0,0)$ for all cases, indicating that $\mathsf{b}_{\textit{ij}}$ with $i\neq j$ is sufficiently small. These simplifications for axisymmetric expansion yield $B_2=B_3$ . Accordingly, this study examines the spatial distribution of $B_2$ . The omission of $\mathsf{b}_{12}$ and $\mathsf{b}_{21}$ has negligible effect, as their inclusion hardly alters $B_2$ values. For isotropic turbulence, $B_2=0$ . As the flow departs from isotropy under axisymmetric expansion, $B_2=B_3$ increases, with the theoretical upper bound of $1/3$ attained when $\langle u^{\prime 2}\rangle \gg \langle v^{\prime 2}\rangle =\langle w^{\prime 2}\rangle$ .

Figure 14 shows the $x$ -directional distribution of $B_2$ at $(y,z)=(0,0)$ . Nearly HIT generated by the single jet array exhibits $B_2 \approx 0.03$ (Mori et al. Reference Mori, Watanabe and Nagata2024). In contrast, the present experimental set-up yields higher $B_2$ values, indicating that the flow becomes anisotropic. For all cases, $B_2$ increases from both sides towards the test-section centre, demonstrating that anisotropy intensifies as the fluid subjected to the mean strain is advected with the mean flow. This increasing trend of $B_2$ is observed consistently across all test-section geometries.

Figure 14.

The $x$ -directional distribution of the invariant $B_2$ of the normalised anisotropy tensor $\mathsf{b}_{\textit{ij}}$ along the centreline, $(y,z)=(0,0)$ .

Figure 15 presents the profiles of skewness and flatness of $u$ , defined as $S_u=\langle {u'}^3\rangle /\langle {u'}^2\rangle ^{3/2}$ and $F_u=\langle {u'}^4\rangle /\langle {u'}^2\rangle ^{2}$ . For Gaussian velocity fluctuations, $S_u=0$ and $F_u=3$ . Deviations from these values typically indicate large-scale intermittency, i.e. intermittency of energy-containing motions (Veeravalli & Warhaft Reference Veeravalli and Warhaft1989; Davidson Reference Davidson2015). This differs from small-scale intermittency, which refers to the burstiness of small-scale motions, typically evident in velocity-gradient statistics. Large-scale intermittency is observed in turbulent free-shear flows (e.g. jets and wakes) and other inhomogeneous turbulent flows, where the presence and absence of large coherent structures alternate at a fixed point. In such cases, the magnitudes of skewness and flatness can reach values as large as $1$ and $5$ , respectively (Kang & Meneveau Reference Kang and Meneveau2008; Matsushima, Nagata & Watanabe Reference Matsushima, Nagata and Watanabe2021; Nakamura et al. Reference Nakamura, Matsushima, Zheng, Nagata and Watanabe2022, Reference Nakamura, Watanabe and Nagata2023). In the present measurements, $|S_u|$ remains below 0.3 and $F_u$ is close to 3 near the test-section centre, showing that large-scale intermittency is negligible in this region. While $F_u$ tends to exceed 3 as $|x|$ increases, suggesting slight intermittency in the flow entering the gap region from the inner duct ends, the deviation remains moderate and far smaller than that in flows with pronounced large-scale intermittency.

Figure 15.

Distributions of ( $a$ ) skewness $S_u$ and ( $b$ ) flatness $F_u$ of $x$ -directional velocity fluctuations along the centreline, $(y,z)=(0,0)$ .

3.3. Two-point statistics and energy spectra

We examine two-point statistics and energy spectra from PIV on the $x$ – $y$ plane at $z=0$ to evaluate integral length scales and the scale dependence of velocity fluctuations, using autocorrelation functions, energy spectra and structure functions. For evaluating these quantities, spatial averages in the $x$ and $y$ directions are taken over $|x|\lt 2M$ and $|y|\lt M$ , respectively, thereby excluding corner regions where the mean velocity gradients depart from axisymmetric strain.

Large-scale properties are examined using the longitudinal autocorrelation functions of velocity fluctuations at two points separated by a distance $r$ , defined as

(3.2) \begin{align} f_{u}(r,x,y) &= \frac {\langle u'(x - r/2, y) u'(x + r/2, y)\rangle }{\langle u^{\prime 2}(x, y)\rangle }, \end{align}

(3.3) \begin{align} f_{v}(r,x,y) &= \frac {\langle v'(x, y - r/2) v'(x, y + r/2)\rangle }{\langle v^{\prime 2}(x, y)\rangle }. \end{align}

These functions are used to estimate the integral length scale, which characterises large-scale motions. The symmetric forms of the autocorrelation functions, i.e. correlations between $x+r/2$ and $x-r/2$ rather than between $x$ and $x+r$ , are used to evaluate turbulent properties at a given point because the flow is statistically inhomogeneous. Here, $f_u$ is evaluated at $x=0$ and $f_v$ at $y=0$ with spatial averaging applied in the $y$ and $x$ directions, respectively.

Figure 16 presents $f_u$ and $f_v$ for cases 1 and 4, representing $D_N=930$ and $725$ mm. Both $f_u$ and $f_v$ decrease from unity as $r$ increases, indicating decorrelation of velocity fluctuations between two spatially separated points. The measurable range of $r$ is limited by the PIV FOV, and although $f_u$ and $f_v$ are expected to decay to zero, this decay cannot be fully captured. It is known that the tails of autocorrelation functions at large $r$ are well approximated by an exponential form $a_{\!f}\exp (-b_{\!f}r)$ (Morikawa et al. Reference Morikawa, Urano, Sanada and Saito2008; Bewley, Chang & Bodenschatz Reference Bewley, Chang and Bodenschatz2012; Griffin et al. Reference Griffin, Wei, Bodenschatz and Bewley2019; Mori et al. Reference Mori, Watanabe and Nagata2024). For the measured $r$ range where $f_u$ and $f_v$ fall below 0.3, a least-squares method is used to determine $a_{\!f}$ and $b_{\!f}$ . The resulting fits, $f_{u,{\textit{exp}}}$ and $f_{v,{\textit{exp}}}$ , are also shown in figure 16, confirming that the exponential function accurately approximates the measured autocorrelation of both $u$ and $v$ .

Figure 16.

Longitudinal autocorrelation functions of $u$ and $v$ , denoted by $f_u$ and $f_v$ , respectively, for ( $a$ ) case 1 and ( $b$ ) case 4. Dashed lines indicate exponential fits that approximate the correlations at large separation distances.

The integral scales are computed by integrating the autocorrelation functions with respect to $r$ , defined as $L_u=\int f_u(r)\,\textrm{d}r$ and $L_v=\int f_v(r)\,\textrm{d}r$ . The integration is performed from $r=0$ to the position where the correlation function becomes zero. Since the PIV measurements do not always capture correlations at large $r$ , the integral scales are estimated with the aid of the exponential approximations $f_{u,{\textit{exp}}}$ and $f_{v,{\textit{exp}}}$ , following previous studies (Morikawa et al. Reference Morikawa, Urano, Sanada and Saito2008; Bewley et al. Reference Bewley, Chang and Bodenschatz2012; Griffin et al. Reference Griffin, Wei, Bodenschatz and Bewley2019; Mori et al. Reference Mori, Watanabe and Nagata2024):

(3.4) \begin{align} L_{u} = \int _{0}^{r_{1}} f_{u}(r)\,\textrm{d}r + \int _{r_{1}}^{r_{2}} f_{u,{\textit{exp}}}(r)\,\textrm{d}r, \end{align}

where $r_1$ is defined by the condition $f_u(r_1)=0.3$ and $r_{2}$ is the position at which $f_{u,{\textit{exp}}}(r)$ reaches $1 \times 10^{-5}$ . The integral scale in the $y$ direction, $L_v$ , is computed in the same manner using $f_v$ and $f_{v,{\textit{exp}}}$ as in (3.4). Table 1 lists $L_u$ , $L_v$ and their ratio $L_u/L_v$ for each case. Nearly HIT generated by the single jet array yields $L_u/L_v\approx 1.03$ (Mori et al. Reference Mori, Watanabe and Nagata2024). In contrast, turbulence subject to axisymmetric expansion in the present experiments exhibits larger values of $L_u/L_v$ , indicating increased anisotropy. For the case of the single jet array, both $L_u$ and $L_v$ are about 0.022 m in the measurement region, comparable to that of the present set-up. Relative to this reference, $L_u$ increases and $L_v$ tends to decrease in the current experiments, with the deviations from the reference being more pronounced for $L_u$ than for $L_v$ . Although the case of the single jet array does not provide exact turbulence statistics prior to the onset of mean strain in the OMJA experiments, it is noteworthy that the greater changes observed in $L_u$ are consistent with DNS results at low Reynolds numbers of Lee (Reference Lee1989), where $L_u$ departs more strongly from the isotropic state than $L_v$ under axisymmetric expansion. The increase in $L_u$ and the decrease in $L_v$ under axisymmetric expansion are also predicted by RDT, which attributes these trends to anisotropic redistribution of energy in three-dimensional wavenumber space and the consequent changes in the one-dimensional spectra (Lee Reference Lee1989).

The scale dependence of velocity fluctuations is examined using longitudinal energy spectra. The Fourier transform of $u$ in the $x$ direction is denoted by $\hat {u}(k_x,y)$ , where $k_x$ is the streamwise wavenumber. With $\hat {u}^{*}$ representing the complex conjugate of $\hat {u}$ , the energy spectrum of $u$ is defined as $E_u(k_x,y) = \textrm{Re}[\langle \hat {u}\hat {u}^{*}\rangle ]$ , where $\textrm{Re}$ denotes the real part. As with the autocorrelation functions, a spatial average in the $y$ direction is applied to obtain $E_u(k_x)$ . Similarly, the energy spectrum of $v$ , denoted by $E_v(k_y)$ , is computed from the Fourier transform of $v$ in the $y$ direction, where $k_y$ is the vertical wavenumber.

Figures 17( $a$ ) and 18( $a$ ) show $E_u$ and $E_v$ , respectively, together with reference results for nearly HIT generated by the single jet array (Watanabe et al. Reference Watanabe, Inagaki, Mori, Ishizawa and Nagata2025). The accessible wavenumber range is determined by the spatial resolution and the size of the measurement region. As discussed below, the measured range lies within the inertial subrange. In high-Reynolds-number turbulence, Kolmogorov scaling is observed in this subrange, with $E_u \sim k_x^{-5/3}$ and $E_v \sim k_y^{-5/3}$ . This scaling is more clearly seen in the compensated spectra, $k_x^{5/3}E_u$ and $k_y^{5/3}E_v$ , shown in figures 17( $b$ ) and 18( $b$ ). The $-5/3$ law appears as a plateau in the compensated spectra and is evident in the single-jet-array results. In contrast, the present experiments show that $k_x^{5/3}E_u$ and $k_y^{5/3}E_v$ decrease with increasing $k_x$ and $k_y$ , indicating a faster decay than predicted by the $-5/3$ law.

Figure 17.

( $a$ ) Longitudinal energy spectra of $x$ -directional velocity fluctuations, $E_u(k_x)$ , where $k_x$ is the $x$ -directional wavenumber. ( $b$ ) Compensated spectra $k_x^{5/3}E_u$ . Results for nearly HIT generated by the single jet array, evaluated 605 mm downstream of the nozzles, are shown for comparison (Watanabe et al. Reference Watanabe, Inagaki, Mori, Ishizawa and Nagata2025).

Figure 18.

( $a$ ) Longitudinal energy spectra of $y$ -directional velocity fluctuations, $E_v(k_y)$ , where $k_y$ is the $y$ -directional wavenumber. ( $b$ ) Compensated spectra $k_y^{5/3}E_v$ . Results for nearly HIT generated by the single jet array, evaluated 605 mm downstream of the nozzles, are shown for comparison (Watanabe et al. Reference Watanabe, Inagaki, Mori, Ishizawa and Nagata2025).

Previous studies of HIT subject to axisymmetric contraction, based on DNS and wind tunnel experiments, reported deviations in the opposite direction, where the spectra decay more slowly than the $-5/3$ law (Ayyalasomayajula & Warhaft Reference Ayyalasomayajula and Warhaft2006; Clay & Yeung Reference Clay and Yeung2016). In contrast, the present results demonstrate that axisymmetric expansion leads to faster decay. This deviation from the $-5/3$ law under axisymmetric expansion could not be assessed in previous DNS because the Reynolds numbers were too low (Lee Reference Lee1989). The mean velocity gradient tensors for axisymmetric contraction and expansion can be expressed in the same form with opposite signs using a single strain-rate parameter $S\gt 0$ , as shown in (1.2) and (1.3). These forms indicate that contraction and expansion exert opposite effects on turbulence. In this context, the faster decay in $E_u$ and $E_v$ observed here under axisymmetric expansion aligns with earlier findings for axisymmetric contraction.

Figure 19 shows $E_u/E_v$ calculated at identical wavenumber values $k = k_x = k_y$ , providing a measure of anisotropy at each scale. The result for the single jet array is also included for comparison (Watanabe et al. Reference Watanabe, Inagaki, Mori, Ishizawa and Nagata2025). For the single jet array, $E_u/E_v$ remains close to 1.05 across all wavenumbers, indicating approximate isotropy at all scales. In contrast, the present results show $E_u/E_v \gt 1.2$ over most wavenumbers. As $E_u/E_v$ decreases with increasing wavenumber, the velocity fluctuations are more anisotropic at larger scales, where the influence of mean strain is more pronounced. Generally, smaller-scale turbulent motions have shorter characteristic time scales. Since the time scale of the mean strain is determined by the mean velocity gradients, it acts relatively slowly on such small-scale motions. Consequently, the direct effect of mean strain is weaker at small scales, which explains why the greatest deviation from $E_u/E_v \approx 1$ appears at larger scales.

Figure 19.

Ratios of energy spectra of $u$ and $v$ , $E_u/E_v$ . Results for nearly HIT generated by the single jet array, evaluated 605 mm downstream of the nozzles, are shown for comparison (Watanabe et al. Reference Watanabe, Inagaki, Mori, Ishizawa and Nagata2025).

The energy spectra describe the scale-by-scale distribution of turbulent kinetic energy in wavenumber space. A real-space equivalent is the second-order structure function, defined using the velocity difference between two points (Davidson Reference Davidson2015). The longitudinal structure function of $u$ is given by $\langle (\Delta u)^2 \rangle$ , where $\Delta u = u'(x + r/2, y) - u'(x - r/2, y)$ . In the inertial subrange, the scaling $\langle (\Delta u)^2 \rangle \sim r^{2/3}$ corresponds to the spectral scaling $E_u \sim k_x^{-5/3}$ . More generally, Kolmogorov’s second similarity hypothesis predicts that the $n$ th-order longitudinal structure function scales as $r^{n/3}$ (Kolmogorov Reference Kolmogorov1991a , Reference Kolmogorovb ; Tsinober Reference Tsinober2009; Davidson Reference Davidson2015). Previous studies have shown that the scaling exponents deviate from $n/3$ as $n$ increases, a phenomenon attributed to small-scale intermittency. The deviation becomes pronounced for $n \geqslant 5$ (Arneodo et al. Reference Arneodo1996; Saw et al. Reference Saw, Debue, Kuzzay, Daviaud and Dubrulle2018; Iyer, Sreenivasan & Yeung Reference Iyer, Sreenivasan and Yeung2020). The present results for the energy spectra deviating from the $k_x^{-5/3}$ scaling imply that the structure functions deviate from the $n/3$ scaling even for small $n$ , with the departures arising from mechanisms other than small-scale intermittency.

We evaluate the $n$ th-order longitudinal structure function of $u'$ as $\langle |\Delta u|^n \rangle = \langle |u'(x + r/2, y) - u'(x - r/2, y)|^n \rangle$ at $x = 0$ , using ensemble averages combined with spatial averages in the $y$ direction. Similarly, the longitudinal structure function of $v'$ is computed at $y=0$ as $\langle |\Delta v|^n \rangle = \langle |v'(x, y + r/2) - v'(x, y - r/2)|^n \rangle$ , using ensemble averaging together with spatial averaging in the $x$ direction. Figure 20( $a$ ) shows the third- and fifth-order structure functions together with the power laws $r^{n/3}$ . We set the maximum separation to $r/M=0.75$ so that $r\leqslant L_u/3$ . Below this bound the structure functions exhibit clear power laws and the $-5/3$ spectral slope is observed at the corresponding wavenumbers for the single-jet-array case (Watanabe et al. Reference Watanabe, Mori, Ishizawa and Nagata2024). In contrast, for turbulence subject to axisymmetric expansion, both the third- and fifth-order structure functions follow power-law behaviour but with exponents that deviate substantially from $n/3$ .

Figure 20.

( $a$ ) Third- and fifth-order structure functions for case 4. Solid lines indicate the power laws predicted by Kolmogorov’s second similarity hypothesis. ( $b$ ) Power-law exponents $p$ for the $n$ th-order structure functions. Present results are compared with DNS of isotropic turbulence (Iyer et al. Reference Iyer, Sreenivasan and Yeung2020) and experimental data for various turbulent flows, including jets, wakes, mixing layers, duct flows and grid turbulence (Arneodo et al. Reference Arneodo1996), as well as nearly HIT generated by the same single jet array used in this study (Watanabe et al. Reference Watanabe, Mori, Ishizawa and Nagata2024).

Figure 20( $b$ ) shows the power-law exponents $p$ of the $n$ th-order longitudinal structure functions of $u$ and $v$ , obtained by least-squares fitting. The present results are compared with reference data from DNS of isotropic turbulence (Iyer et al. Reference Iyer, Sreenivasan and Yeung2020) and from experiments of various flows (Arneodo et al. Reference Arneodo1996). Results for nearly HIT generated by the single jet array at $\textit{Re}_\lambda \approx 600$ (Watanabe et al. Reference Watanabe, Mori, Ishizawa and Nagata2024), obtained with the same PIV set-up, are also shown. In Watanabe et al. (Reference Watanabe, Mori, Ishizawa and Nagata2024), the available range of $r$ , set by the spatial resolution and FOV of the PIV, is similar to that of the present measurements. Although the fitting range spans less than one decade, the exponents for the single jet array agree well with DNS and experimental data, confirming that the limited $r$ range does not compromise exponent evaluation. For these reference cases, the exponents follow $p=n/3$ predicted by Kolmogorov’s second similarity hypothesis for $n \leqslant 4$ , but become smaller than $n/3$ as $n$ increases, reflecting small-scale intermittency. In contrast, the present results for turbulence subject to axisymmetric expansion yield exponents larger than $n/3$ , in clear departure from other turbulent flows. The departure is similar for $u$ and $v$ . Since the single-jet-array results reproduce established turbulence data, the observed deviation is most likely due to mean strain effects. The energy spectra exhibit a faster decay than $k_x^{-5/3}$ , consistent with an exponent for $n=2$ exceeding $2/3$ ; indeed, $p(n=2)=1.1$ for $\langle |\Delta u|^n \rangle$ and 0.98 for $\langle |\Delta v|^n \rangle$ is obtained. Previous comparisons across grid turbulence, jets, mixing layers and cylinder wakes have shown that mean shear exerts little influence on the scaling of structure functions (Arneodo et al. Reference Arneodo1996). Consistently, numerous experiments and numerical simulations have reported the persistence of the $-5/3$ energy spectral law in canonical turbulent shear flows (Uberoi & Freymuth Reference Uberoi and Freymuth1970; Fiscaletti et al. Reference Fiscaletti, Elsinga, Attili, Bisetti and Buxton2016; Sadeghi, Oberlack & Gauding Reference Sadeghi, Oberlack and Gauding2018; Takamure et al. Reference Takamure, Sakai, Ito, Iwano and Hayase2019; Watanabe, Zhang & Nagata Reference Watanabe, Zhang and Nagata2019). Taken together with reports of non- $-5/3$ spectra for turbulence under axisymmetric contraction (Ayyalasomayajula & Warhaft Reference Ayyalasomayajula and Warhaft2006), the present findings for axisymmetric expansion suggest that mean axisymmetric strain exerts a stronger influence on structure–function scaling than mean shear.

The turbulent kinetic energy distribution across scales can be examined using second-order structure functions, which estimate the kinetic energy contained in scales below a given separation distance (Davidson Reference Davidson2015). The transport equation for the second-order structure function is often utilised to assess the scale-dependent energy budget (Deissler Reference Deissler1992; Valente & Vassilicos Reference Valente and Vassilicos2015; Wang et al. Reference Wang, Yurikusa, Iwano, Sakai, Ito, Zhou and Hattori2023). This scale-by-scale energy-transport equation, originating from the Kármán–Howarth equation, contains two inviscid terms that describe interscale energy transfers: a nonlinear transfer due to turbulent fluctuations and a linear transfer due to mean velocity gradients. For the longitudinal structure functions of $u$ and $v$ , which are related to the longitudinal energy spectra in figures 17 and 18, the relevant nonlinear transfer terms measured on the $x$ – $y$ plane at $z=0$ are

(3.5) \begin{align} \varPi _{u'}(x,y,r)&=-\frac {\partial \left\langle [u'(x^+,y) - u'(x^-,y)]^3\right\rangle }{\partial r}, \end{align}

(3.6) \begin{align} \varPi _{v'}(x,y,r)&=-\frac {\partial \left\langle [v'(x,y^+) - v'(x,y^-)]^3\right\rangle }{\partial r}, \end{align}

with $x^+=x+r/2$ , $x^-=x-r/2$ , $y^+=y+r/2$ and $y^-=y-r/2$ , as derived in Valente & Vassilicos (Reference Valente and Vassilicos2015). The corresponding linear transfer terms are

(3.7) \begin{align} \varPi _{\langle u\rangle }(x,y,r)&= -\frac {\partial [\langle u\rangle (x^+,y) - \langle u\rangle (x^-,y)] \left\langle [u'(x^+,y) - u'(x^-,y)]^2\right\rangle }{\partial r}, \end{align}

(3.8) \begin{align} \varPi _{\langle v\rangle }(x,y,r)&= -\frac {\partial [\langle v\rangle (x,y^+) - \langle v\rangle (x,y^-)] \left\langle [v'(x,y^+) - v'(x,y^-)]^2\right\rangle }{\partial r}. \end{align}

Figure 21 shows these terms at $(x,y)=(0,0)$ normalised by $u_{\textit{rms}}^{3}/L_u$ or $v_{\textit{rms}}^{3}/L_v$ . A second-order central difference is used for $\partial /\partial r$ . Because small errors in the structure functions can be amplified by differentiation, we focus on the overall trends rather than precise quantification. The maximum separation for evaluation is $r=1.8M$ , close to the integral scales. Positive and negative values indicate transfers to smaller and larger scales (forward and backward transfers), respectively. Thus, $\varPi _{\langle u\rangle }\gt 0$ and $\varPi _{\langle v\rangle }\lt 0$ show that the mean velocity gradients associated with axisymmetric expansion have opposite effects: compression in the $x$ direction gives $\langle u\rangle (x^+,y)-\langle u\rangle (x^-,y)\lt 0$ and hence a forward transfer with $\varPi _{\langle u\rangle }\gt 0$ , whereas expansion in the $y$ direction yields $\langle v\rangle (x,y^+)-\langle v\rangle (x,y^-)\gt 0$ and thus a backward transfer with $\varPi _{\langle v\rangle }\lt 0$ . At large scales, the linear transfer terms exceed the nonlinear ones. Even at $r/L_u\approx 10^{-1}$ and $r/L_v\approx 10^{-1}$ they are of comparable magnitude. Thus, the mean velocity gradients of axisymmetric expansion alter the energy transfer across scales. The nonlinear terms also differ substantially between components. The $x$ -component $\varPi _{u'}$ is positive, indicating forward transfer that diminishes as $r$ approaches the integral scale, consistent with other turbulent flows (Valente & Vassilicos Reference Valente and Vassilicos2015). In general, the nonlinear term scales as $u_{\textit{rms}}^{3}/L_u$ (or $v_{\textit{rms}}^{3}/L_v$ ), and the normalised quantity $\varPi _{u'}/(u_{\textit{rms}}^{3}/L_u)$ is close to 1. By contrast, the $y$ -component $\varPi _{v'}$ becomes negative at large scales, indicating that axisymmetric expansion modifies the nonlinear transfer. Deviations from the $-5/3$ law are observed in the longitudinal energy spectra of both $u$ and $v$ . Since axisymmetric expansion affects the interscale transfers of these components differently, the spectral changes are unlikely to be explained solely by the interscale transfers.

Figure 21.

Linear and nonlinear longitudinal interscale energy transfers defined by (3.5)–(3.8). ( $a$ ) Energy transfer of the $x$ -directional velocity across scales in the $x$ direction; ( $b$ ) energy transfer of the $y$ -directional velocity across scales in the $y$ direction. The separation distance and energy-transfer terms are normalised by the integral scales and r.m.s. velocity fluctuations.

3.4. Summary of turbulence statistics

Table 1 summarises velocity statistics obtained for all experimental conditions with different test-section geometries. The dissipation scaling is used to provide an order-of-magnitude estimate of the turbulent kinetic energy dissipation rate per unit mass, $\varepsilon = A\mathcal{U}^3/\mathcal{L}$ , where $\mathcal{U}$ and $\mathcal{L}$ denote the large-scale velocity and length scales. The present analysis relies on this dissipation scaling because the PIV does not resolve the smallest turbulent scales. The non-dimensional constant $A$ varies among flows and may depend on position in the present set-up, but $A=O(10^0)$ holds in most cases, including nearly HIT generated by the single jet array (Mori et al. Reference Mori, Watanabe and Nagata2024). We therefore take $A=1$ for the present estimates. This approximation is used only to evaluate the orders of the Taylor microscale $\lambda$ , the Kolmogorov scale $\eta$ and the turbulent Reynolds number $\textit{Re}_\lambda$ ; accurate values of these quantities would require direct measurements of $\varepsilon$ . The velocity and length scales are defined as $\mathcal{U}=\sqrt {2k_T/3}$ and $\mathcal{L}=(L_u+L_v+L_w)/3$ , where the turbulent kinetic energy is $k_T=(\langle u^{\prime 2}\rangle +\langle v^{\prime 2}\rangle +\langle w^{\prime 2}\rangle )/2$ and $L_w$ is the integral scale in the $z$ direction. With statistical equivalence in the $y$ and $z$ directions, we assume $\langle w^{\prime 2}\rangle =\langle v^{\prime 2}\rangle$ and $L_w=L_v$ . The Taylor microscale and Kolmogorov scale are given by $\lambda =\sqrt {10(\mu /\rho )k_T/\varepsilon }$ and $\eta =(\mu /\rho )^{3/4}\varepsilon ^{-1/4}$ , respectively. The turbulent Reynolds number is $\textit{Re}_\lambda =\rho \sqrt {2k_T/3}\,\lambda /\mu$ . The resulting values of $k_T$ , $\varepsilon$ , $\lambda$ , $\eta$ and $\textit{Re}_\lambda$ are listed in table 1.

The Kolmogorov scales are $\eta /M \approx 1 \times 10^{-3}$ for cases 1–3 and $0.7\times 10^{-3}$ for cases 4–6. The larger $\eta$ in the former cases reflects the longer inner ducts with larger $D_N$ , as turbulence decay increases $\eta$ . Turbulence decay also enlarges the Taylor microscale, giving $\lambda$ values greater in cases 1–3 than in cases 4–6. Consistently, $\textit{Re}_\lambda$ is about 600 for cases 1–3 and 700 for cases 4–6. These values are sufficiently large to allow observation of the inertial subrange with the $-5/3$ law, as this scaling and its structure–function equivalent have been successfully observed in turbulence generated by the single jet array at comparable $\textit{Re}_\lambda$ (Mori et al. Reference Mori, Watanabe and Nagata2024; Watanabe et al. Reference Watanabe, Mori, Ishizawa and Nagata2024). The observed deviation from the $-5/3$ law is therefore unlikely to originate from low-Reynolds-number effects. Consistently, turbulence subject to axisymmetric contraction at similar $\textit{Re}_\lambda$ also departs from the $-5/3$ scaling, as discussed above (Ayyalasomayajula & Warhaft Reference Ayyalasomayajula and Warhaft2006).

An important parameter describing the influence of mean strain on turbulence is the ratio of the strain time scale to the turbulence time scale. From (1.3), the strain rate $S$ is evaluated from the experimental data as $S = (|\mathsf{A}_{\textit{xx}}| + 2\mathsf{A}_{\textit{yy}} + 2\mathsf{A}_{zz})/3,$ using the continuity relation $\mathsf{A}_{zz} = -\mathsf{A}_{\textit{xx}} - \mathsf{A}_{\textit{yy}}$ . The characteristic time scale of large-scale motion is defined as $T_L = {\mathcal L}/{\mathcal U}$ . The normalised strain rate is expressed as $ST_L$ , representing the ratio of $T_L$ to the strain time scale $1/S$ . Table 1 lists $ST_L$ values for all cases. In all experiments, $ST_L$ is close to unity, indicating that the mean strain acts on a time scale comparable to that of large-scale turbulent motions. Comparisons among cases suggest that the distance between the two opposing jet arrays, $D_N$ , has a stronger influence on $ST_L$ , through turbulence decay effects on $T_L$ , than the gap length between the inner-duct exits, $D_E$ , which affects $S$ . Systematic experiments varying $D_N$ would therefore be valuable for further investigations of turbulence under axisymmetric strain.

Turbulence generated by the OMJA facility exhibits $ST_L \approx 1$ while the necessary condition for the RDT assumption to be valid is $ST_L \gg 1$ . Nonetheless, several findings are qualitatively consistent with RDT. Detailed comparisons with RDT require knowledge of the total strain (distortion) experienced along mean flow trajectories. When the turbulence intensity is small, this can be inferred from spatial variations of the mean velocity using Taylor’s frozen-turbulence hypothesis, as in wind tunnel studies of axisymmetric contraction. In the present set-up, however, the turbulence intensity is very high near the stagnation region, owing to the small mean velocity there and to the jet-interaction turbulence, so this approach is not suitable.

Rapid distortion theory for axisymmetric turbulence has been presented in previous studies (Batchelor Reference Batchelor1946; Sreenivasan & Narasimha Reference Sreenivasan and Narasimha1978; Hunt & Carruthers Reference Hunt and Carruthers1990). Based on these frameworks, temporal evolutions of $u_{\textit{rms}}/v_{\textit{rms}}$ and $L_u/L_v$ under axisymmetric expansion have been derived (Pope Reference Pope1985; Lee Reference Lee1989). For initially HIT, RDT predicts that axisymmetric expansion induces anisotropy, increasing both $u_{\textit{rms}}/v_{\textit{rms}}$ and $L_u/L_v$ above unity. The present measurements likewise show $u_{\textit{rms}}/v_{\textit{rms}}\gt 1$ and $L_u/L_v\gt 1$ . However, RDT predicts an upper bound $u_{\textit{rms}}/v_{\textit{rms}}=\sqrt {2}$ , whereas our results exceed this limit (Pope Reference Pope2000). Consistently, other experiments and DNS on turbulence with axisymmetric expansion at low Reynolds numbers report greater anisotropy than RDT when the strain time scale is comparable to turbulence time scales (Liu et al. Reference Liu, Katz and Meneveau1999; Choi & Lumley Reference Choi and Lumley2001). In addition, RDT predicts that axisymmetric strain does not modify the $-5/3$ slope of the energy spectra at high Reynolds numbers, as shown in Appendix A. This contrasts with the present results in figures 17 and 18 and with previous findings for axisymmetric contraction (Ayyalasomayajula & Warhaft Reference Ayyalasomayajula and Warhaft2006).