3.1 Turbulence parameters

The important turbulent parameters computed at two different downstream locations are tabulated in table 1. Point A is located 125 mm ($x=-113~\text{mm}$) downstream of the active grid still inside the straight section and point B is located at 30 mm ($x=30~\text{mm}$) inside the start of the contraction. These points lie on the centreline of the tunnel. The mean streamwise velocity in the straight inlet section varied between 0.27 and $0.28~\text{m}~\text{s}^{-1}$ in our experiments. Turbulence intensities after the active grid are initially high, then decay along the flow direction, as expected. Higher values of fluctuations at point A are observed in modes S1, S3 and R, with the highest of 16.9 % obtained in the random mode. This point is located on the centreline of the tunnel at 125 mm ($x=-113~\text{mm}$, $x_{AG}/M=4.2$) from the active grid. In modes S2 and S4, where shafts rotate in the same direction, lower turbulence intensities are obtained. This trend is also noted for the turbulent Reynolds number based on the Taylor microscale, $Re_{\unicode[STIX]{x1D706}}$, at both of the points. The highest value of $Re_{\unicode[STIX]{x1D706}}=292$ is achieved at point A in the random mode, which then reduces to 231 at point B. The value of $Re_{\unicode[STIX]{x1D706}}$ initially reduces in the straight section, due to the decay of the velocity fluctuations, while the Taylor microscale remains approximately a constant. There is some increase in $\unicode[STIX]{x1D706}$, as the flow enters the contraction, which results in higher values of $Re_{\unicode[STIX]{x1D706}}$.

Table 1. Flow parameters at location $x=-113~\text{mm}$ ($x_{AG}/M=4.2$) upstream of the start of the contraction (top), at $x=30~\text{mm}$ ($x_{AG}/M=8.9$), which is slightly inside the contraction (middle), and at $x=30~\text{mm}$ ($x_{AG}/M=16.9$) with the extensional part (bottom).

Turbulence kinetic energy is computed as $k=(u_{rms}^{2}+v_{rms}^{2}+w_{rms}^{2})/2$, based on the root mean square of the fluctuating velocity components. The dissipation rate is approximated by,

(3.1)$$\begin{eqnarray}\unicode[STIX]{x1D700}=15\unicode[STIX]{x1D708}\left\langle \!\left(\frac{\unicode[STIX]{x2202}u}{\unicode[STIX]{x2202}x}\right)^{2}\right\rangle .\end{eqnarray}$$

The notation is the following: the instantaneous velocity $U=\langle U\rangle +u$, $V=\langle V\rangle +v$, $W=\langle W\rangle +w$, where $\langle U\rangle$, $\langle V\rangle$ and $\langle W\rangle$ are the time averaged velocities and, $u$, $v$ and $w$ denote the fluctuations. Equation (3.1) assumes homogeneity and local isotropy. Sirivat & Warhaft (Reference Sirivat and Warhaft1983) showed for grid turbulence that the estimate of $\unicode[STIX]{x1D700}$ using (3.1) agrees well with that computed by two other approaches: differentiation of the energy decay and integration of the velocity spectrum. Brown et al. (Reference Brown, Parsheh and Aidun2006) also use (3.1), showing that the dissipation rate agrees well with other estimates even though their flow is not isotropic. We also compare the isotropic estimate of $\unicode[STIX]{x1D700}$ to a more complete expression of $\langle s_{ij}s_{ij}\rangle$ in our high-magnification experiments, which are described in the supplementary material § S7.

The length scales associated with the turbulence are computed using the spatial auto-correlation of the streamwise velocity (Pope Reference Pope2000). For homogeneous turbulence, the correlation of streamwise velocities at two points axially separated by a distance $r$ is given by,

(3.2)$$\begin{eqnarray}f(r)=\frac{\langle u(x)u(x+\boldsymbol{e}_{\boldsymbol{x}}r)\rangle }{\langle u^{2}\rangle }.\end{eqnarray}$$

The integral length scale $L$ and Taylor microscale $\unicode[STIX]{x1D706}$, are computed using the spatial correlation function $f$ as

(3.3a,b)$$\begin{eqnarray}L=\int _{0}^{\infty }f(r)\,\text{d}r,\quad \unicode[STIX]{x1D706}=\left(-{\textstyle \frac{1}{2}}f^{\prime \prime }(0)\right)^{-1/2}.\end{eqnarray}$$

The variation of the correlation $f$ is shown for modes S1 and R in the supplementary figure S16. In both cases the curve reaches zero, where we stop the integration. The Reynolds numbers based on these length scales, presented in table 1, are defined as, $Re_{L}=k^{1/2}L/\unicode[STIX]{x1D708}$ and $Re_{\unicode[STIX]{x1D706}}=k^{1/2}\unicode[STIX]{x1D706}/\unicode[STIX]{x1D708}$. In case of isotropic flows, the Taylor microscale is often estimated as $\sqrt{u^{2}/\left\langle \left(\unicode[STIX]{x2202}u/\unicode[STIX]{x2202}x\right)^{2}\right\rangle }$ (Pope Reference Pope2000), which would give slightly smaller values for $Re_{\unicode[STIX]{x1D706}}$. If we apply this estimate to our data, the largest of our $Re_{\unicode[STIX]{x1D706}}$, which occurs for the random mode, reduces from 292 to 244. The smaller Kolmogorov length scale is computed as $\unicode[STIX]{x1D702}=(\unicode[STIX]{x1D708}^{3}/\unicode[STIX]{x1D700})^{1/4}$. The velocity grid resolution including the 75 % overlap is given by $\unicode[STIX]{x1D6FF}$ and is compared in table 1.

The turbulence-to-mean-strain time ratio is defined as $S^{\ast }=Sk/\unicode[STIX]{x1D700}$ (Lee & Reynolds Reference Lee and Reynolds1985; Pope Reference Pope2000; Ayyalasomayajula & Warhaft Reference Ayyalasomayajula and Warhaft2006), where $S$ is the mean strain rate. Here, we use the dominant extensional strain rate ($2\unicode[STIX]{x2202}\langle U\rangle /\unicode[STIX]{x2202}x$) for the mean strain rate $S$. The value of $S^{\ast }$ should be very large to apply rapid distortion theory. (Ayyalasomayajula & Warhaft Reference Ayyalasomayajula and Warhaft2006). For their flows, $S^{\ast }$ varied between 10 and 100. However, as they study an axisymmetric contraction they use a different prefactor ($\sqrt{3}$) to calculate the mean strain.

Figure 7. (a) Streamwise evolution of the mean streamwise velocity $\langle U\rangle$, on the centreline, for different oscillation modes of the active grid: synchronous modes S1–S4 and random mode R. The streamwise velocity is normalized by the area averaged mean inlet velocity in the straight section, $\langle U_{in}\rangle$. Vertical black lines mark the start and end of the contraction. Transverse profiles of mean streamwise velocity $\langle U\rangle (y)$ at different $x$ locations for mode S1 (b) and mode R (c). Locations $x=-90~\text{mm}$ and $x=-30~\text{mm}$ are before the entrance to the contraction. They correspond to $x_{AG}/M\approx 5$, 7, 9 and 11, where $x_{AG}$ is the distance from the bottom shafts of the active grid. The streamwise velocities in (b) and (c) are normalized by the local area averaged velocity at the corresponding $x$-location, $\langle U_{lm}\rangle$.

Figure 8. (a) Streamwise variation of the normalized mean strain $\unicode[STIX]{x2202}\langle U\rangle /\unicode[STIX]{x2202}x$ for different modes. (b) Streamwise variation of $S^{\ast }$ for grid-rotation modes S1 and R. Vertical black lines mark the start and end of the contraction. The vertical red line marks the maximum curvature location.

3.2 Mean velocity

Figure 7(a) shows the mean streamwise velocity $\langle U\rangle$ on the centreline of the tunnel. The values have been time averaged over 4.2 s (this corresponds to ${\approx}8\unicode[STIX]{x1D70F}$, where $\unicode[STIX]{x1D70F}$ is the eddy turnover time computed based on the integral length scale and the fluctuation at the start of contraction), and comparison between four realizations confirms good convergence. Velocity is normalized by the corresponding mean inlet velocity $\langle U_{in}\rangle$ in the straight channel averaged over the $y$ direction. The mean velocity in our experiments varies in the range $0.27{-}0.28~\text{m}~\text{s}^{-1}$. The flow accelerates inside the contraction and the measured mean streamwise values match closely the local cross-sectional area ratio, the outlet speed reaching 2.5 times the inlet velocity. There are minor differences seen between the different grid-rotation modes in the initial straight section and into the first part of the contraction (figure 7(a)). This is due to the flow not being entirely uniform in the transverse direction, both from the periodicity of the grid and the nature of the contracting geometry. We quantify the errors in the mean and root mean square (r.m.s.) quantities and discuss the convergence of our results in the supplementary material sections § S3 and S4 respectively.

The transverse profile of $\langle U\rangle$ for modes S1 and R is shown in figures 7(b) and 7(c) respectively. The profiles are shown at four locations, $x=-90$, $-30$, $30$ and $90$ mm, corresponding to $x_{AG}/M\approx 5$, 7, 9 and 11, where $x_{AG}$ is the distance from the bottom shafts of the active grid. Velocity profiles have been normalized by the local mean velocity $\langle U_{lm}\rangle$ averaged over the transverse $y$ direction. For mode S1 a strong signature of the shaft rotation can be seen in the velocity profile at $x=-90~\text{mm}$. The non-uniformity reduces with the vigorous turbulent transport of momentum in the transverse direction, as we move further downstream. However, the flow still remains slightly non-uniform for mode S1 when it enters the contraction. These profiles are much more uniform in the case of the random mode. We present additional information how the active grid rotation influences the homogeneity of the flow for the different rotation modes in the supplementary material § S5.

An important parameter in strained turbulence is the strain-rate parameter, which is defined above. The streamwise variations of the mean strain rate and the strain-rate parameter $S^{\ast }$ are shown in figures 8(a) and 8(b) respectively. We plot $S^{\ast }$ for the modes S1 and R which have the highest turbulent fluctuations. The maximum value of $S^{\ast }$ is approximately 25.

Figure 9. Streamwise variation of velocity and vorticity r.m.s. along the centreline. (a) Streamwise velocity $u_{rms}$; (b) transverse velocity $v_{rms}$; and (c) spanwise velocity $w_{rms}$ for different modes of the active grid, i.e. the four synchronous modes S1–S4 and the random mode R. Streamwise evolution of the vorticity r.m.s. of (d) streamwise vorticity $\unicode[STIX]{x1D714}_{x,rms}$; (e) transverse vorticity $\unicode[STIX]{x1D714}_{y,rms}$; and (f) spanwise vorticity $\unicode[STIX]{x1D714}_{z,rms}$, for different modes of the active grid. Here, $\langle U_{in}\rangle$ is the area averaged mean inlet velocity in the straight section. The vertical black lines represent the start and end of the contraction. The vertical red line marks the maximum curvature location.

3.3 Fluctuating velocity statistics

Figure 9(a–c) shows the streamwise variation of the r.m.s. velocities $u_{rms}$, $v_{rms}$ and $w_{rms}$ on the centre of the tunnel. They are normalized by the mean velocity $\langle U_{in}\rangle$ in the straight section. There is a slight discontinuity seen in the profiles within the contraction, as measurements at positions P2 and P3 are from different experimental realizations taken on different days. However, the trends seen for $u_{rms}$ and $v_{rms}$ in the contraction are as expected from the literature. All r.m.s. fluctuations decay rapidly in the straight section, but in the contraction, $u_{rms}$ continues to decay whereas there is an increase in the transverse components $v_{rms}$ and $w_{rms}$. In wind-tunnel experiments with planar contraction of $C=2.5$, the same as the current study, Thoroddsen & Van Atta (Reference Thoroddsen and Van Atta1995a) also observed decay in the $u_{rms}$ and increase in $v_{rms}$. These trends are similar to that proposed in Prandtl’s theory for an axisymmetric contraction (Prandtl Reference Prandtl1933). According to his theory, streamwise velocity fluctuations vary as inverse of $C$ and lateral fluctuations grow proportionally to $C^{1/2}$. This behaviour has been verified experimentally by Uberoi (Reference Uberoi1956) in his experiments with an axisymmetric square contraction and he concludes Prandtl’s theory holds good when $C<$4, which is true in our case. The decay in $u_{rms}$ in our current results is proportional to $1/C$, but the increase in $v_{rms}$ is not proportional to $C^{1/2}$, its increase is weaker.

The spanwise $w_{rms}$ relatively has higher values in region P1 and shows a decay upstream of the contraction. In the contraction, similar to the transverse $v_{rms}$, $w_{rms}$ also increases, reaching a clear local maximum at the contraction’s maximum curvature location (MCL). This local maximum, which occurs at $x\approx 142~\text{mm}$, could be due to the delayed reaction to the strongest mean strain, which is observed at $x\approx 120~\text{mm}$, as seen in figure 8(a). After this local maximum at the MCL, there is a dip in the r.m.s. followed by an increase further downstream. The increase towards the downstream could be due to the tendency of the turbulence to return to isotropy. This local maximum is also seen in experiments with the longer distance between the grid and SOC, while it is less prominent. This variation of $w_{rms}$ for a planar contraction has not been reported before. However, it should also to be noted that $w_{rms}$ has the highest noise, as discussed in the supplemental material subsection § S3. These variations in r.m.s. velocities along the contraction are similar for all grid-oscillation modes.

We see a mismatch in the r.m.s. curves between the different measurement regimes. This is most likely caused by two effects: these regimes are necessarily run on different days, due to the major adjustments needed for the optical system and camera arrangement/calibration; and also the laser volumes are not perfectly aligned in the depth direction, which introduces minor differences due to the transverse inhomogeneity arising from the grid-rod locations.

The random mode (R) shows the largest level of fluctuations for all components of the velocity. This is also seen in table 1 with higher values of the turbulent intensity, turbulent kinetic energy and $Re_{\unicode[STIX]{x1D706}}$. Modes S2, S4 show lower values of velocity fluctuations for $u_{rms}$ (figure 9(a)).

As seen in figure 4, shafts on the top plane are oriented along $y$, and their flaps’ rotation displaces flow in the $z$, $x$ directions; whereas those on the bottom plane are along $z$, and their flaps’ rotation displaces flow in the $x$, $y$ directions. As we measure the statistics at the centre of the tunnel at a sufficiently long distance from the grid, the flow is more complex and hence corroboration of $v_{rms}$ and $w_{rms}$ with the protocol is difficult. However, the trends of all the fluctuations in the contraction are qualitatively similar for all modes.

3.4 Vorticity statistics

In figure 9(d–f) we present the streamwise variation of the r.m.s. vorticity components, $\unicode[STIX]{x1D714}_{x,rms}$, $\unicode[STIX]{x1D714}_{y,rms}$ and $\unicode[STIX]{x1D714}_{z,rms}$. All components of the vorticity decay upstream of the contraction owing to diffusion and dissipation (Tennekes & Lumley Reference Tennekes and Lumley1972). The $\unicode[STIX]{x1D714}_{x,rms}$ and $\unicode[STIX]{x1D714}_{y,rms}$ exhibit similar trends with downstream location. Both exhibit local maxima at the MCL similar to that seen for $w_{rms}$ in figure 9(c). Keep in mind that the gradient in $w_{rms}$ contributes to these two vorticity components. When we view the vorticity field as consisting of large discrete entities, vortex tubes which are aligned with the $x$-direction, they undergo stretching by the mean strain and, thus, there is an increase in $\unicode[STIX]{x1D714}_{x,rms}$ by conservation of angular momentum. The $\unicode[STIX]{x1D714}_{x,rms}$ reaches a prominent local maximum at the MCL, which could be due to the delayed response to the strongest mean strain at $x\approx 120~\text{mm}$. After this local maximum, $\unicode[STIX]{x1D714}_{x,rms}$ decreases locally until $x\approx 170~\text{mm}$ after which it increases again until the exit. This increasing trend qualitatively agrees with Prandtl’s theory for an axisymmetric contraction. According to the theory, streamwise vorticity increases as $C$ and the transverse vorticity decreases as $C^{-0.5}$. In our case, the transverse component $\unicode[STIX]{x1D714}_{y,rms}$ does not show a monotonic decrease as per the theory. Initially, there is a slight decrease in its value due to the compression of vortex tubes aligned with $y$. But then, there is a subsequent increase observed due to the variation of $w_{rms}$. The increase in $\unicode[STIX]{x1D714}_{y,rms}$ is at a lower rate than $\unicode[STIX]{x1D714}_{x,rms}$. The spanwise component $\unicode[STIX]{x1D714}_{z,rms}$, which is parallel to contracting walls, undergoes minimum variations along the contraction. This is justified as any vortex tube aligned with the $z$ direction ‘feels’ minimum changes due to the contraction. To the best of our knowledge, there has been no earlier experimental measurement of vorticity inside a contraction. However, the gradients of velocity depend on the numerical grid used for the velocity computation, which in turn depends on the particle density used. Here, we use the grid based on the grid-dependence study detailed in the supplementary material § S2. This can lead to fine structures being under-resolved, which can underestimate their gradients and in turn the vorticity strength, while capturing the spatial structure of the vorticity field.

3.5 Coherent vortical structures

Identification and visualization of coherent vortical structures is one approach to better understand the underlying dynamics in turbulent flows. Many definitions of coherent structures exist in the literature, and there is no uniquely accepted best one. One conceptual definition, given by Hussain et al. (Reference Hussain1986), is ‘a coherent structure is a connected turbulent fluid mass with instantaneously phase correlated vorticity over its spatial extent’. There are also many methods for identification of such vortical structures based on: (i) some intuitive characteristics of the vorticity where the pressure has a minimum or (ii) the characteristics of the velocity gradient tensor. In the first approach, the simplest and most intuitive method uses the vorticity magnitude as a criterion. In the second approach, based on the velocity gradient tensor, parameters based on this tensor are used as its criteria. For example, the discriminant $\unicode[STIX]{x1D6E5}$ (Chong et al. Reference Chong, Perry and Cantwell1990), the second invariant of velocity gradient tensor $\unicode[STIX]{x1D64C}$ (Hunt et al. Reference Hunt, Wray and Moin1988) or the second eigenvalue $\unicode[STIX]{x1D706}_{2}$ (Jeong & Hussain Reference Jeong and Hussain1995). The $\unicode[STIX]{x1D706}_{2}$ parameter uses the eigenvalues of the real symmetric matrix $(\unicode[STIX]{x1D64E}^{2}+\unicode[STIX]{x1D734}^{2})$, where $\unicode[STIX]{x1D64E}$ is the strain-rate tensor and $\unicode[STIX]{x1D734}$ is the rotation-rate tensor (Pope Reference Pope2000). A vortex is identified as the connected region where two of the three real eigenvalues are negative. This implies that the second eigenvalue $\unicode[STIX]{x1D706}_{2}<0$, when they are arranged such that $\unicode[STIX]{x1D706}_{1}>\unicode[STIX]{x1D706}_{2}>\unicode[STIX]{x1D706}_{3}$. The eigenvalues of the strain-rate tensor in the descending order ($\unicode[STIX]{x1D706}_{S}>\unicode[STIX]{x1D706}_{I}>\unicode[STIX]{x1D706}_{C}$) will give the extensional (stretching), intermediate and compressive strains respectively. The corresponding eigenvectors give the principal directions.

In figure 10 we compare the instantaneous isovorticity (blue) and $\unicode[STIX]{x1D706}_{2}$ structures (orange) in region P2. The characteristic larger structures are captured by both criteria, although there are differences in the visualization of smaller structures. Due to ease of physical interpretation, we choose the vorticity magnitude in our further analysis. This structure analysis is presented only for mode S1, as a larger number of realizations were made for this mode.

Figure 10. Comparison of the instantaneous isosurfaces of $|\unicode[STIX]{x1D74E}|=30~\text{s}^{-1}$ (blue) and $\unicode[STIX]{x1D706}_{2}=-130$ (orange) in position P2. The coordinates ($x$, $y$, $z$) indicate the principal directions but do not coincide with the actual origin.

Figure 11. Downstream changes in the appearance of vorticity isosurfaces for a constant value of $|\unicode[STIX]{x1D74E}|=35~\text{s}^{-1}$, for the three different streamwise measurement regions: (a) P1, (b) P2 and (c) P3, for grid mode S1. The region between the planes marked in red and green represents the overlap between P2 and P3. The plane marked in blue in (c) corresponds to the end of the contraction. The sizes ($x$, $y$, $z$) of the measurement regions in mm are P1 (92.0, 55.9, 23.6), P2 (95.1, 54.8, 17.6) and P3 (97.8, 43.2, 18.2). The coordinates ($x$, $y$, $z$) indicate the principal directions only and do not coincide with the actual origin.

Figure 12. Time evolution of isosurfaces of $|\unicode[STIX]{x1D74E}|$ for measurement regions (a) P1 with $|\unicode[STIX]{x1D74E}|=60~\text{s}^{-1}$; (b) P2 with $|\unicode[STIX]{x1D74E}|=35~\text{s}^{-1}$; and (c) P3 with $|\unicode[STIX]{x1D74E}|=35~\text{s}^{-1}$, for grid-oscillation mode S1. The sequences are spaced by 20 video frames, which corresponds to a time interval of 20 ms for P1 and P2, and 15 ms for P3. The region between the planes marked in red and green represents the overlap between measurement volumes P2 and P3. The plane marked blue in (c) corresponds to the end of the contraction. Dimensions of the region are the same as in figure 11.

The representative change in the spatial structure in the three measurement regions is shown in figure 11. It presents isosurfaces of $|\unicode[STIX]{x1D74E}|=35~\text{s}^{-1}$. The vorticity is strong in P1, immediately below the grid, which then decays as the flow enters the contraction, as was observed in the r.m.s. vorticity presented. Big interconnected structures are seen in region P1, which appear as blobs of comparable dimensions in all directions. As the flow enters the contraction, the structures coalesce and break down, forming long tubular structures under the straining action of the mean flow and exit the contraction as long tubes.

Time evolution of structures within the three measurement zones is shown in figure 12. Note that the results presented at different positions do not correspond to the experiment done on the same day, thus we do not track the same structures through the entire contraction. For better visualization, isosurfaces are shown for a higher vorticity ($|\unicode[STIX]{x1D74E}|=60~\text{s}^{-1}$) for P1, which exhibits strong vorticity, versus $|\unicode[STIX]{x1D74E}|=35~\text{s}^{-1}$ for regions P2 and P3. Smaller structures are seen in P1 as compared to figure 11(a), due to the difference in the vorticity magnitude. Due to the active grid motion we expect the production of larger structures than for passive grids. Under the action of mean strain these structures are stretched, forming long tubular structures. The evolution of two such structures in the mid position P2 is shown in figure 12(b). Stretching continues as the flow moves through the contraction, forming longer structures which get more aligned with the centreline of the tunnel as they exit the contraction. A representative long structure observed in the lower part of the contraction is shown in figure 12(c). We quantify this preferential orientation by looking at the pointwise and structure-based statistics in the following sections.

3.6 Vorticity orientation with respect to the principal strain rates

Vorticity and strain-rate interdependence is important to understand the dynamics of turbulent flows. In the vorticity evolution equation, the vorticity intensification terms, due to stretching and tilting, include the local strain-rate tensor. The orientation of $\unicode[STIX]{x1D74E}$ with respect to the largest strain is thus an important quantity that governs redistribution of vorticity across its components. We specifically look at the PDFs of orientation of $\unicode[STIX]{x1D74E}$ with respect to the principal directions of the strain-rate tensor. We present the probability density function for the cosine of the angles between $\unicode[STIX]{x1D74E}$ and the three principal eigenvectors. Using cos $\unicode[STIX]{x1D703}$ rather than $\unicode[STIX]{x1D703}$ is mandated by the non-uniform probability of the angular locations on a sphere i.e. for a random orientation it is much more likely to end near the equator than at the North Pole. Using the PDF (cos $\unicode[STIX]{x1D703}$) corrects for this and a random distribution gives a uniform probability.

We use the notation that $\unicode[STIX]{x1D703}_{S}$, $\unicode[STIX]{x1D703}_{I}$, $\unicode[STIX]{x1D703}_{C}$ represent the angle between $\unicode[STIX]{x1D74E}$ and the direction of the principal stretching, intermediate and compressive stresses respectively. The PDF is obtained at four characteristic $x$-locations along the tunnel which are: $x=-90~\text{mm}$ (close to the active grid), 30 mm (in the contraction), 142 mm (corresponds to the maximum curvature of contraction) and 190 mm (after the exit of contraction). The PDF at a given $x$-plane is obtained using data from $11\times 11$ grid points around the channel centreline, sampled over the entire set of approximately $4\times 5500$ frames captured, corresponding to the total number of samples of $0.67\times 10^{6}$. The PDFs at a particular $x$-location are averaged over those obtained from all of the independent realizations (we have three realizations in P1; four each in P2 and P3). To account for all directions of the vorticity vector, PDFs are plotted for the full range of cos $\unicode[STIX]{x1D703}\in [-1,1]$. We note that for perfect spanwise grid homogeneity the distribution should be symmetric about zero. There is a strong tendency for $\unicode[STIX]{x1D74E}$ to align with the intermediate strain, i.e. perpendicular to the other two principal directions at $x=-90~\text{mm}$ (figure 13). This overall alignment persists downstream through the contraction.

Figure 13. PDF of the cosine of the angle between $\unicode[STIX]{x1D74E}$ and the principal strain-rate eigenvectors at $x=-90~\text{mm}$ (a), 30 mm (b), 142 mm (c), 190 mm (d). The legend denotes the angle between the vorticity vector and the $\unicode[STIX]{x1D703}_{S}$: stretching vector;$\unicode[STIX]{x1D703}_{I}$: intermediate vector; and $\unicode[STIX]{x1D703}_{C}$: compression vector. The PDFs are obtained using data from $11\times 11$ grid points in constant $x$-planes around the channel centreline. The distributions are calculated using 50 bins.

This behaviour of $\unicode[STIX]{x1D74E}$ aligning with the intermediate eigenvector has been observed in both simulations (Ashurst et al. Reference Ashurst, Kerstein, Kerr and Gibson1987; Nomura & Post Reference Nomura and Post1998; Hamlington et al. Reference Hamlington, Schumacher and Dahm2008) and in experiments (Tsinober et al. Reference Tsinober, Kit and Dracos1992; Su & Dahm Reference Su and Dahm1996; Mullin & Dahm Reference Mullin and Dahm2006) of different turbulent flows. This tendency of vorticity alignment was first reported by Ashurst et al. (Reference Ashurst, Kerstein, Kerr and Gibson1987) using simulated data for isotropic and shear flow turbulence. Later, this was verified with measurements in grid flow, in the outer region of a turbulent boundary layer by Tsinober et al. (Reference Tsinober, Kit and Dracos1992) and for coflowing jets by Mullin & Dahm (Reference Mullin and Dahm2006). All of these studies report that $\unicode[STIX]{x1D74E}$ tends to align with the intermediate strain perpendicular to the compressive strain direction. Furthermore, it has weak or no correlation with the stretching direction. This can be understood for flow fields which are locally two-dimensional, such as vortices or plane shear layers, where the principal stretching and compression are in planes perpendicular to the vorticity vector, requiring that the vorticity be parallel to the remaining, i.e. intermediate strain direction.

In our study we reach essentially the same conclusion. However, we observe vorticity to be perpendicular to both the stretching and compression strain directions, with nearly similar probability distributions. This is similar to the observations made by Su & Dahm (Reference Su and Dahm1996) in the case of an axisymmetric jet. It is perhaps intuitive to expect $\unicode[STIX]{x1D74E}$ to align with the direction of stretching as it amplifies the stretched vorticity component. However, vorticity being the derivative of the velocity, its local value favours contributions from the smaller scales. On the other hand, we could also expect alignment with the stretching direction due to the imposed contraction to appear at larger scales based on the instantaneous coherent structures. This behaviour was a mystery for a couple of decades before Hamlington et al. (Reference Hamlington, Schumacher and Dahm2008) gave an explanation. Indeed Hamlington et al. (Reference Hamlington, Schumacher and Dahm2008), based on their analysis of DNS data of homogeneous isotropic turbulence, showed vorticity tends to align with the stretching direction of the background strain. However, with respect to the local strain rate, $\unicode[STIX]{x1D74E}$ showed preferred alignment with the intermediate strain direction. They separate local and background strain rates based on the radius of integration used for the computation of the strain rate. It is computed from a specific form of the Biot–Savart integral, that expresses the velocity as an integral of the vorticity. Strain rate induced at a location by the vorticity within this radius (${\approx}12\unicode[STIX]{x1D702}$, where $\unicode[STIX]{x1D702}$ is the Kolmogorov scale) is called local and beyond that is the background strain rate.

Figure 14. Joint PDF of $|\unicode[STIX]{x1D74E}|$ and cos $\unicode[STIX]{x1D703}$. Rows from top correspond to cos $\unicode[STIX]{x1D703}_{C}$, cos $\unicode[STIX]{x1D703}_{I}$ and cos $\unicode[STIX]{x1D703}_{S}$ respectively. Columns from left correspond to $x=-90~\text{mm}$, 30 mm, 142 mm and 190 mm respectively. Samples obtained and bins used are the same as mentioned in figure 13 for cos $\unicode[STIX]{x1D703}$. A total of 50 bins are used in $|\unicode[STIX]{x1D74E}|$.

Figure 15. The alignment of the vorticity vector with respect to the centreline of the contraction. PDF of the cosine of the angle between $\unicode[STIX]{x1D74E}$ and the $-x$-axis at four different streamwise locations. The distribution is calculated using 50 bins. For a perfectly spanwise homogeneous turbulent field, this distribution should be symmetric about zero.

While the PDFs in figure 13 show no qualitative change with $x$ in the general alignment of $\unicode[STIX]{x1D74E}$ with the principal strains, there are quantitative changes. The magnitude of the peak probability changes. We see $\unicode[STIX]{x1D74E}$ to deviate from alignment with the intermediate strain through the contraction, but it then tends back to its pre-contraction behaviour when it leaves the contraction. The PDF values for the first bin at $\cos \unicode[STIX]{x1D703}=\pm 1$ are 3, 1.6, 2.2 and 2.4 for locations $x=-90$, 30, 142 and 190 mm respectively. As far as the stretching and compressive directions, the contraction reduces the PDF value at $\cos \unicode[STIX]{x1D703}=0$, but this tends to its pre-contraction behaviour at the exit.

The effect of the contraction on the orientation observed from the PDFs is more clear in the joint PDFs (JPDF) plotted between $|\unicode[STIX]{x1D74E}|$ and the cosine of the angles (figure 14). The red regions in the middle row of figure 14, which shows alignment with intermediate strain, become wider as we move through the contraction. This indicates slight loss of alignment. The red/yellow regions in the top and bottom rows of figure 14 show a slightly preferential perpendicular alignment of $\unicode[STIX]{x1D74E}$ with the compressive and stretching strains, as they are concentrated in the middle, corresponding to cos $\unicode[STIX]{x1D703}=0$.

Figure 16. Joint PDF of $|\unicode[STIX]{x1D74E}|$ and cos $\unicode[STIX]{x1D703}_{-X}$, where $\unicode[STIX]{x1D703}_{-X}$ is the angle between the vorticity and the $-x$-axis. Plots correspond to downstream locations at $x=-90~\text{mm}$ (a), 30 mm (b), 142 mm (c), 190 mm (d). Samples obtained and bins used are the same as mentioned in figure 13.

Figure 17. Normalized PDF of $|\unicode[STIX]{x1D74E}|$ at different values of cos $\unicode[STIX]{x1D703}_{-X}$. The PDF is extracted along horizontal lines in the JPDF of figure 16(d) at cos $\unicode[STIX]{x1D703}_{-X}=0.98$, 0.5 and 0. The PDFs are normalized by the integral under each curve.

3.7 Orientation of vorticity vectors

To investigate how the contraction affects the alignment of vortical structures at the different turbulent length scales, we examine both pointwise and coherent-structure-based statistics, to quantify the average vertical orientation of $\unicode[STIX]{x1D74E}$. First, we present the PDFs of the cosine of the angle ($\unicode[STIX]{x1D703}_{-X}$) between $\unicode[STIX]{x1D74E}$ and the $-x$ direction. PDFs at the same four characteristic locations described in the previous section are shown in figure 15. The range of values for $\cos \unicode[STIX]{x1D703}_{-X}$ varies between $[-1,1]$ going between vertically up to down, which includes all possible orientations of the local $\unicode[STIX]{x1D74E}$. The pointwise $\unicode[STIX]{x1D74E}$ does indeed show significantly increased alignment with $x$ as it passes through the contraction. The vorticity vectors close to the grid are closest to random orientation. As we move along the contraction, the rising values of the PDF at cos $\unicode[STIX]{x1D703}_{-X}=\pm 1$ indicate a strong tendency of $\unicode[STIX]{x1D74E}$ to align with the $x$-direction. This was also qualitatively observed with the instantaneous structures presented above in figure 12, where longer structures aligned along the mean flow. Figure 16 shows the JPDF between cos $\unicode[STIX]{x1D703}_{-X}$ and $|\unicode[STIX]{x1D74E}|$ at the same four locations. Sampling for this is done in the same way as employed for the PDFs. Contours show the JPDF after averaging over all of the independent realizations. At $x=-90~\text{mm}$ (figure 16(a)), the probability density is nearly random over the range of vorticity magnitudes. As we move along the tunnel, the distribution tends to concentrate close to $\cos \unicode[STIX]{x1D703}_{-X}=-1$ and 1 and this bias towards alignment with the vertical direction is more probable at higher vorticity magnitudes. This can be seen clearly from figure 17, wherein we show the conditional PDFs for three different values of $\cos \unicode[STIX]{x1D703}_{-X}=0.98$, 0.5 and 0. These PDFs are extracted along the corresponding horizontal lines in the contour plot shown in figure 16(d). The values are normalized by the total conditional probability in each case. For the fully vertically aligned condition $\cos \unicode[STIX]{x1D703}_{-X}=1$, the normalized PDF has a larger contribution at higher vorticity.

In both the PDFs and JPDFs, we observe the distribution to be nearly symmetric (except at the point near to the grid) which also indicates that this distribution is independent of the up or down orientation of the vorticity. This is expected as the horizontal symmetries of the grids should not generate $\unicode[STIX]{x1D714}_{x}$ with preferential clockwise or counterclockwise rotation, but the measurement volume is not perfectly aligned with the grids. At locations in proximity to the grid, the orientation could also be slightly biased in $\unicode[STIX]{x1D714}_{y}$ and $\unicode[STIX]{x1D714}_{z}$ by the rotation protocol used. Figure 17 suggests that stronger vorticity is most aligned in the vertical direction, i.e. in the direction of the large-scale strain.

Figure 18. Orientation of instantaneous coherent vortical structures visualized based on average vorticity magnitude and their total size. Vorticity weighted centroid of each structure is shown with a dot, together with the major principal axis of the equivalent ellipsoid with the length corresponding to the major axis length. (a) P1 with $|\unicode[STIX]{x1D74E}|=70~\text{s}^{-1}$, (b) P2 with $|\unicode[STIX]{x1D74E}|=35~\text{s}^{-1}$ and (c) P3 with $|\unicode[STIX]{x1D74E}|=35~\text{s}^{-1}$. In all the regions the size criterion used is ${>}100$ voxels. The $x$-coordinate shown here is local to each measurement region, centred at the middle of the region.

Figure 19. PDFs of the orientation of coherent vortical structures for different values of $|\unicode[STIX]{x1D74E}|$ and structure size, for regions close to the grid (a) $x=-23$ to $-115~\text{mm}$ (P1), and near the exit of the contraction (b) $x=103$ to 198 mm (P3). Grid-rotation mode is S1. PDF calculated with 20 bins. The 100 voxel size corresponds to ${\approx}24~\text{mm}^{3}$.

Figure 20. Streamwise evolution of the PDF of orientation of coherent vortical structures versus downstream location regions P1–P3 for modes (a) S1, and (b) R. Structures are identified using the criteria: size of 100 voxels, $|\unicode[STIX]{x1D74E}|=70~\text{s}^{-1}$ for P1; $|\unicode[STIX]{x1D74E}|=35~\text{s}^{-1}$ for P2 and P3. PDFs are plotted with 50 bins.

Figure 21. Streamwise evolution of the vertical orientation PDF of coherent vortical structures in sub-regions R1–R10, for grid-oscillation modes (a) S1, and (b) R. Structures are identified using criteria: size of 100 voxels, $|\unicode[STIX]{x1D74E}|=70~\text{s}^{-1}$ for P1; $|\unicode[STIX]{x1D74E}|=35~\text{s}^{-1}$ for P2 and P3. PDFs are plotted with 50 bins. The mid-$x$ locations (in mm) for these sub-regions are: $-100$ (R1), $-69$ (R2), $-38$ (R3), 46 (R4), 77 (R5), 108 (R6), 120 (R7), 151 (R8), 182 (R9).

3.8 Orientation of coherent vortical structures

The analysis of vorticity orientation is now extended to large-scale coherent vortical structures. The coherent structures are identified based on the vorticity magnitude and their size. The structure identification and tagging is done using the 3-D watershed function in MATLAB. Voxels with $|\unicode[STIX]{x1D74E}|$ higher than a particular cutoff value are flagged. A ‘turbulent structure’ here corresponds to a group of connected flagged voxels. These connected regions are then sequentially numbered using the watershed function. Numbered structures are then filtered based on their size in voxels. A large cutoff value results in relatively smaller structures, corresponding to regions close to the vortex core. Only big structures with sizes greater than a particular value are considered for the statistics. Using the regionprops3 function in MATLAB, we extract the principal axis lengths and directions of an equivalent ellipsoid, which has the same normalized central moments as the structure. Filtered structures identified at a representative time instant in measurement regions P1, P2 and P3 are shown in figure 18. The $x$–$y$ projections show the orientation becoming more aligned with $x$ as we move through the contraction from P1 to P3, with most of the structures in P3 nearly aligned with $x$. In figure 19 we present the PDFs of the $\cos (\unicode[STIX]{x1D703}_{-X\text{-}Struct})$, where $\unicode[STIX]{x1D703}_{-X\text{-}Struct}$ is the angle between the major axis direction of the coherent structure and the $-x$-axis. The PDF of absolute value of $\cos (\unicode[STIX]{x1D703}_{-X\text{-}Struct})$ is shown as we consider only the orientation of the principal axis.

We note that the structure shape and quantity depend on the chosen vorticity magnitude and size criterion and are therefore quite subjective. Thus we quantify the sensitivity of the results on the specific choice of these thresholds. Based on the visualization of isosurfaces of $|\unicode[STIX]{x1D74E}|$, we see that the coherent structures are distinct and clear for $|\unicode[STIX]{x1D74E}|$ in the range $60{-}80~\text{s}^{-1}$ inside P1, and $30{-}40~\text{s}^{-1}$ inside P2 and P3. The PDFs of $|\!\cos (\unicode[STIX]{x1D703}_{-X\text{-}Struct})|$, obtained for P1 and P3 for different size criteria in the above ranges of $|\unicode[STIX]{x1D74E}|$, are shown in figure 19. The maximum size in this study is chosen to ensure that large structures are identified at all the time realizations. Sizes of 50, 100, 200 and 300 voxels correspond to ${\approx}12$, 24, 48 and $72~\text{mm}^{3}$. Figure 19 shows that the cutoff values chosen have a minimal effect on the PDFs. In P1 (figure 19(a)), the alignment distribution is random. In P3 (figure 19(b)), the PDF shows strong preference towards cos $\unicode[STIX]{x1D703}_{-X\text{-}Struct}=1$ for both values of $|\unicode[STIX]{x1D74E}|$ and especially for the largest structures. This indicates that larger structures tend to align more as compared to the smaller ones. Thus for all the further analysis of structures, we use a cutoff value of $|\unicode[STIX]{x1D74E}|=70~\text{s}^{-1}$ for P1, $|\unicode[STIX]{x1D74E}|=35~\text{s}^{-1}$ for P2 and P3. In all cases only structures of size greater than 100 voxels (${\approx}24~\text{mm}^{3}$) are considered. For each realization, the PDF is computed using structures that satisfy the above criteria, over all of the 5499 frames. In each measurement region, the average of the PDFs over all realization is presented in figure 20(a) (average is over three realizations in P1, and four in P2 and P3). With the chosen cutoff values, we have the average of approximately 100 000 big structures in each measurement volume. It is clear that there is a strong tendency of structures to align with $x$ near the exit of the contraction in P3, with almost a random distribution at the inlet in P1. The random mode, which has a totally different grid-oscillation protocol compared to the rest, exhibits an identical trend in the PDFs (figure 20(b)). This confirms the structures tend to align in $x$-direction in a contraction irrespective of the protocol of the grid.

The evolution of the orientation along the contraction is looked at more closely by dividing each measurement region into three equal sub-regions, P1 to R1–R3, P2 to R4–R6 and P3 to R7–R9 (R1 is closest to the grid and R9 is the last one at the outlet of the contraction). The PDF of the structure orientation in each of these sub-regions for modes S1 (figure 21(a)) and R (figure 21(b)) shows a clear systematic transition from a random orientation at R1 to a preference for alignment in the vertical at the outlet of the contraction at R8. Due to the overlap of P2 and P3, sub-regions R6 and R7 also overlap and thus show nearly identical PDFs. However, in region R9, which is in the straight section downstream the contraction, the PDF shows a slightly reduced peak tendency to align with $x$. This deorientation immediately downstream of the contraction could be physical but we will require measurements further downstream to confirm this behaviour and investigate the underlying reasons, such as the return to isotropy. This reduction in the peak PDF value persists after we increase the distance after the grid to the entrance of the contraction, as shown in the supplementary figure S12 and discussed in supplementary material § S6.

3.9 Increasing the distance between the grid and contraction

There is a trade-off between large $Re_{\unicode[STIX]{x1D706}}$ and transverse homogeneity of the inlet flow, which requires a long distance between the active grid and the entrance of the contraction. To address whether this inhomogeneity has significant effect on the vortical-orientation results, we have performed an additional experiment where this distance was doubled from $x/M=8$ to 16, as shown in figure 2(a). Thormann & Meneveau (Reference Thormann and Meneveau2014) conclude that the traverse spatial homogeneity was found to be good beyond $x/M\approx 15$ in their study on decay of turbulence behind active fractal grids. This was done for the primary synchronous mode S1, which showed the large inhomogeneity in figures 7(b) and S9(a). While the value of $Re_{\unicode[STIX]{x1D706}}$ is more than cut in half, the orientation results are essentially the same, as will now be shown. The bottom part of table 1 lists the conditions at the inlet of the contraction with the additional length. Figure 22(a) presents the inlet inhomogeneity of the mean flow, showing that the maximum has reduced from $\pm 6\,\%$ in figure 7(b), to $\pm 3\,\%$. Figure 22(b) shows the orientation of the vorticity vector with respect to the principal rates of strain. Compared to figure 13(c) the results are quite similar. Similarly, figure 22(c) shows that the orientation of the pointwise vorticity vector with the vertical centreline is the same as in figure 15. Finally, figure 22(d) also shows that the orientation of the large-scale coherent structures is essentially the same as in figure 20(a). Here the vorticity level of the isocontour must be adjusted to a smaller value to account for the diffusion of vorticity further downstream from the grid.

Figure 22. Results obtained from experiments for which the length of the straight section between the active grid and the inlet of the contraction is doubled from $x/M=8$ to 16. All results presented are for mode S1. (a) Transverse profiles of $\langle U\rangle (y)$ normalized by $\langle U_{lm}\rangle$ at different $x$-locations are presented in comparison to figure 7(b). The profiles marked by dotted lines are obtained from experiments with the extension added. (b) PDF of cosine of the angle between $\unicode[STIX]{x1D74E}$ and the principal strain-rate eigenvectors at $x=142~\text{mm}$, as compared to those in figure 13(c). (c) The alignment PDF of $\unicode[STIX]{x1D74E}$ with respect to the tunnel centreline at different $x$-locations, as compared to figure 15. (d) Streamwise evolution of PDF of the orientation of coherent vortical structures in regions $\text{P}2^{\ast }$ ($x=3{-}109~\text{mm}$) and $\text{P}3^{\ast }$ ($x=86{-}192~\text{mm}$). Here, $\text{P}2^{\ast }$ starts 20 mm above the previous region P2 and is closer to the inlet of contraction. Structures are identified using the two criteria of size ${>}100$ voxels and $|\unicode[STIX]{x1D74E}|=20~\text{s}^{-1}$ (refer to figure 20(a) for comparison). PDFs are evaluated with 50 bins.

3.10 Experiment at larger optical magnification

Our experiments are focused on capturing the larger-scale coherent vortices in the turbulence and cannot resolve all the small scales and the velocity gradients accurately. Thus, in order to evaluate the effect of the spatial resolution, we repeat an experiment using a higher optical zoom (${\sim}2\times$). The measurements were made in a region $30\times 50~\text{mm}$ approximately centred on the location of the maximum contraction curvature. We use these data to assess the accuracy of the vorticity and the dissipation rate, as described in the supplementary material § S7. The PDF of the vorticity magnitudes is slightly shifted to higher values, as shown in supplementary figure S15. Similarly, the dissipation shows values up to 2.8 times higher than for the less zoomed imaging. However, this results in only 25 % smaller estimates of the Kolmogorov scale due to its weak dependence on $\unicode[STIX]{x1D700}^{-1/4}$.