We study the turbulent circulation and coherent vortical structures for flow through three separate two-dimensional planar 4:1 contractions, where the turbulence is subjected to different rates of mean strain. We use four high-speed video cameras for Lagrangian particle tracking velocimetry, to measure time-resolved velocity fields in volumes inside the contractions. We identify the coherent vortical structures and quantify their alignment with the mean strain. The intermediate strain contraction shows the strongest alignment. We also compute circulations around square loops in three perpendicular planes, to form a “circulation vector,” whose instantaneous alignment mirrors that of the coherent structures, with strong inhomogeneities between circulation components at the exit of the contractions. The circulation probability density functions show extended exponential tails for the smallest loops, with more Gaussian shapes in the inertial range.

## I. INTRODUCTION

It is our pleasure to contribute to this journal issue in honor of Professor Katepalli R. Sreenivasan's 75th birthday. This is particularly appropriate, as Sreenivasan has made seminal contributions to the two topics at hand, i.e., the statistics of circulation and the effect of contractions on turbulent flow.

Turbulent flow through contracting pipes or ducts finds extensive use across various industrial applications, such as heating and air-conditioning as well as flow in pipe networks, oil/gas separator vessels, and refining columns. The influence of the contraction on the turbulence structure and effectiveness, such as pressure drop or stirring and mixing of scalars, is therefore of practical significance.

In the early 1930s, Prandtl^{1} modeled turbulence as consisting of discrete vortex filaments aligned parallel or perpendicular to the centerline of the contraction to understand its evolution as it is subjected to the mean strain. The streamwise-aligned vortex tubes get stretched, and hence, their vorticity enhances, whereas the transversely aligned vortex tubes get compressed, and hence, their vorticity reduces. Using *Kelvin's circulation theorem* and volume conservation, he concluded that the streamwise vorticity fluctuations enhance proportional to the local contraction ratio *c*, while the transverse vorticity fluctuations should decay proportionally to $ c \u2212 0.5$. The fluctuating velocities are tangential to the cross section of the vortex tubes, and hence, their streamwise variation can be obtained through identical reasoning, to give: the streamwise velocity fluctuations decay proportional to $ c \u2212 1$ and the transverse velocity fluctuations enhance proportional to $ c 0.5$. Taylor^{2} used general vorticity distributions and Fourier analysis to arrive at variation of rms velocity, assuming inviscid conditions and rapid distortion. He showed that the streamwise velocity fluctuations should decay by a factor ranging between $ 1 / c$ and $ 2 / c$. The transverse velocity fluctuations also showed the same power-laws as above, $ c 0.5$, but the prefactors depend on the initial anisotropy of the turbulence and strength of the contraction.

Later, in the 1950s, Ribner and Tucker^{3} used spectral analysis to obtain expressions for velocity fluctuations at the contraction exit. Batchelor and Proudman^{4} derived similar expressions for ratios of kinetic energies assuming the fluid elements get distorted at a faster rate compared to their displacement, by assuming isotropic conditions. This theory would later be called the rapid distortion theory (RDT).^{5,6}

In 1978, Sreenivasan and Narasimha^{5} generalized the theory of Batchelor and Proudman^{4} for prediction of distortion of homogeneous, axisymmetric turbulence. They used Fourier coefficients and two scalar functions, the wave number and its projection on the axis of symmetry, to formulate the velocity spectral tensor. Integration obtains expressions for the ratio of energies before and after the distortion. Comparisons with experiments showed that for large $ \u2212 C$ planar and axisymmetric contractions, their longitudinal energy predictions matched with isotropic predictions. The transverse energy predictions were over-predicted by isotropic assumptions, when its initial value is smaller than for the longitudinal component.

Prandtl's conceptual formulation has been verified by experiments, but only for small to moderate contraction ratios $ C \u2264 4$. In Table I, we list the important experimental studies and summarize their main results below. Uberoi^{7} showed that Prandtl's theory for velocity fluctuations holds good for contractions with $ C < 4$ in axisymmetric contractions. He studied the evolution of free-stream turbulence in three different square cross section contractions with *C* = 4, 9, and 16. For the largest $ C = 16$, the streamwise velocity fluctuations initially decayed but then increases toward the exit. Subsequently, Uberoi and Wallis^{8} studied applications to wind tunnel design, by passed grid turbulence through a weak axisymmetric *C* = 1.25 nozzle followed by a straight section with three different grid mesh sizes, noting that the ratio of streamwise to transverse velocity fluctuations decreases in the contraction but then reverts to their initial state in the downstream straight tunnel.

Researchers . | Geometry . | Flow . | Method . | Velocity components . | Vorticity . |
---|---|---|---|---|---|

Uberoi^{7} | Axisymmetric | $ R e M$ | Hot-wire | $ u rms , v rms$ | No |

C = 4, 9, 16 | 3 700–123 00 | ||||

Reynolds and Tucker^{23} | 2-D and axisymmetric strain 2.3–6 | $ R e M$ | Hot-wire | $ u rms , v rms , w rms$ | No |

6780 | |||||

Hussain and Ramjee^{9} | Axisymmetric (four different shapes) | $ R e D$ | Hot-wire (single, X-wire) | $ u rms , v rms$, $ w rms$ | No |

C = 11 | 63 000 | ||||

Warhaft^{11} | Axisymmetric | $ R e M$ | X-wire, temperature | $ u rms , v rms$ | No |

C = 4 | 16 000 | ||||

Tan-Atichat et al.^{10} | Axisymmetric | $ R e M$ | Hot-wire | $ u rms , v rms$ | No |

C = 2–36 | 270–8130 | ||||

(ten contractions) | |||||

Nagib et al.^{12} | Axisymmetric | $ R e M$ | Hot-wire | $ u rms , v rms$ | No |

C = 9 | 370–3700 | ||||

(two contractions) | |||||

Thoroddsen and Van Atta^{13} | 2D-contraction | $ R e \lambda $ | Hot-wire | $ u rms , w rms$ | No |

C = 2.5 | 25 | ||||

Ayyalasomayajula and Warhaft^{17} | Axisymmetric | $ R e \lambda $ | Hot-wire | $ u rms$, $ v rms$ | No |

C = 4 | 35–470 | ||||

Brown et al.^{19} | Planar contraction | $ R e \lambda $ | LDA | $ u rms$, $ v rms$ | No |

(linear) C = 7.3, 11.3, and 16.7 | 51–99 | ||||

Ertunç and Durst^{20} | Axisymmetric | $ R e \lambda $ | X-config | $ u rms , v rms$ | No |

C = 3.69, 14.75 | 26–48 | Hot-wire | |||

Gylfason and Warhaft^{18} | Axisymmetric | $ R e \lambda $ | X-array probes | $ u rms , v rms$ | No |

C = 4 | 50, 190 | ||||

Mugundhan et al.^{21} | 2D-contraction | $ R e \lambda $ | Tomo PIV | $ u rms , v rms , w rms$ | Yes |

(smooth) C = 2.5 | 180–290 | and LPT | |||

Mugundhan and Thoroddsen^{22} | 2D-contraction | $ R e \lambda $ | 3-D LPT | $ u rms , v rms , w rms$ | Yes |

(smooth) C = 2.5 | 220 | ||||

Alhareth et al.^{24} | 3D-contraction | $ R e \lambda $ | 3-D LPT | $ u rms , v rms , w rms$ | Yes |

C = 16 | 192 |

Researchers . | Geometry . | Flow . | Method . | Velocity components . | Vorticity . |
---|---|---|---|---|---|

Uberoi^{7} | Axisymmetric | $ R e M$ | Hot-wire | $ u rms , v rms$ | No |

C = 4, 9, 16 | 3 700–123 00 | ||||

Reynolds and Tucker^{23} | 2-D and axisymmetric strain 2.3–6 | $ R e M$ | Hot-wire | $ u rms , v rms , w rms$ | No |

6780 | |||||

Hussain and Ramjee^{9} | Axisymmetric (four different shapes) | $ R e D$ | Hot-wire (single, X-wire) | $ u rms , v rms$, $ w rms$ | No |

C = 11 | 63 000 | ||||

Warhaft^{11} | Axisymmetric | $ R e M$ | X-wire, temperature | $ u rms , v rms$ | No |

C = 4 | 16 000 | ||||

Tan-Atichat et al.^{10} | Axisymmetric | $ R e M$ | Hot-wire | $ u rms , v rms$ | No |

C = 2–36 | 270–8130 | ||||

(ten contractions) | |||||

Nagib et al.^{12} | Axisymmetric | $ R e M$ | Hot-wire | $ u rms , v rms$ | No |

C = 9 | 370–3700 | ||||

(two contractions) | |||||

Thoroddsen and Van Atta^{13} | 2D-contraction | $ R e \lambda $ | Hot-wire | $ u rms , w rms$ | No |

C = 2.5 | 25 | ||||

Ayyalasomayajula and Warhaft^{17} | Axisymmetric | $ R e \lambda $ | Hot-wire | $ u rms$, $ v rms$ | No |

C = 4 | 35–470 | ||||

Brown et al.^{19} | Planar contraction | $ R e \lambda $ | LDA | $ u rms$, $ v rms$ | No |

(linear) C = 7.3, 11.3, and 16.7 | 51–99 | ||||

Ertunç and Durst^{20} | Axisymmetric | $ R e \lambda $ | X-config | $ u rms , v rms$ | No |

C = 3.69, 14.75 | 26–48 | Hot-wire | |||

Gylfason and Warhaft^{18} | Axisymmetric | $ R e \lambda $ | X-array probes | $ u rms , v rms$ | No |

C = 4 | 50, 190 | ||||

Mugundhan et al.^{21} | 2D-contraction | $ R e \lambda $ | Tomo PIV | $ u rms , v rms , w rms$ | Yes |

(smooth) C = 2.5 | 180–290 | and LPT | |||

Mugundhan and Thoroddsen^{22} | 2D-contraction | $ R e \lambda $ | 3-D LPT | $ u rms , v rms , w rms$ | Yes |

(smooth) C = 2.5 | 220 | ||||

Alhareth et al.^{24} | 3D-contraction | $ R e \lambda $ | 3-D LPT | $ u rms , v rms , w rms$ | Yes |

C = 16 | 192 |

Hussain and Ramjee^{9} performed experiments with axisymmetric contractions with four different shapes having the same contraction ratio *C* = 11 and found that the theory holds good for $ C <$ 4. This was also seen in the experiments of Tan-Atichat *et al.*^{10} who investigated axisymmetric contractions with *C* = 2–36, different contraction length–diameter ratios and shapes using different turbulence levels at the inlet to the contractions. They concluded that the effect of straining on turbulence depends on the incoming turbulence scales. For larger contraction ratios $ C > 4$, Prandtl's theory for streamwise velocity fluctuation does not hold.

Warhaft^{11} experimentally studied on the effects of a contraction on turbulent scalar fields generated by a heated *mandoline* of fine wires, specifically examining the decay of passive temperature fluctuations with and without uniform straining. He noted that the streamwise velocity fluctuations decays until the exit of the contraction and the return to isotropy is slow thereafter.

Nagib *et al.*^{12} performed experiments in a settling chamber, leading to an axisymmetric $ C = 9$ contraction to provide guidelines to the designers of wind tunnels, using grids and honeycomb. They conclude that it may not be possible to reach small integral scales and low fluctuating intensities simultaneously when using such larger contraction ratios.

Thoroddsen and van Atta^{13} studied stratified turbulence passing through a smooth 2-D contraction with *C* = 2.5 and showed that in the non-stratified case, the transverse turbulence $ w \u2032$ amplified over twice as much as the reduced streamwise turbulence $ u \u2032$ due to stretching of streamwise vortices. They showed how buoyancy suppressed the transverse fluctuations, to rise again through zombie turbulence.

Mydlarski and Warhaft^{14} were the first to implement an active grid to experimentally study turbulence in an axisymmetric wind tunnel contraction with $ C = 4$. Their active grid was based on the earlier design of Makita,^{15} who had established the design and reported enhanced turbulence intensity through the use of the active grid to obtain high $ R e \lambda $. Mydlarski^{16} has reviewed the design aspects and the numerous implementations of active grids in turbulence research to date. Following this work, Ayyalasomayajula and Warhaft^{17} experimentally investigated using both active and passive grids. They observed that the effect of strain on small and large scales differed and gave rise to nonlinear interactions. In the subsequent study, Gylfason and Warhaft^{18} investigated the effect of strain on passive scalars and developed a tensor model to predict the evolution of the covariance of fluctuating scalar gradients.

Brown *et al.*^{19} performed laser-doppler anemometry (LDA) measurements in a planar contraction with different converging angles. They used three angles and maintaining a constant contraction length, which results in *C* = 7.3, 11.3, and 16.7 to study the evolution of grid turbulence. They observed the streamwise velocity fluctuation to decay only initially reaching a minimum at $ C \u2248$ 1.3 followed by an increase. An initial dip before the increase was also seen with the transverse velocity, unlike previous measurements. They noted that streamwise variation of the velocity fluctuations was independent of the converging angle.

Ertunç and Durst^{20} questioned the previous hot-wire measurements of the streamwise velocity fluctuations, when measured with $ \xd7 \u2212$ wire probes. They attributed the increase observed for large $ C > 9$ contractions to be due to measurement errors. By systematic correction of these errors, they showed that streamwise velocity continues to decay until the exit in their experiments in an axisymmetric contraction with *C* = 14.75.

More recently, in 2020, Mugundhan *et al.*^{21} implemented modern experimental techniques with tomographic particle image velocimetry (Tomo PIV) and the Lagrangian particle tracking (LPT) velocimetry, to study the evolution of turbulence statistics and coherent structures in a smooth 2-D contraction with *C* = 2.5, in the facility used in this study. The alignment of the coherent vortical structures with the mean strain due to the contraction was verified and quantified. This alignment of the coherent vortices is stronger than the alignment of the local vorticity vector. The preferential alignment of the structures was reaffirmed by looking at the relative strength of circulation computed in three perpendicular planes over square loops in the subsequent work of Mugundhan and Thoroddsen.^{22}

The study of turbulent circulation gained interest in the 1990s from the pioneering works of Migdal^{25} and Sreenivasan *et al.*^{26,27} Most focus was on evaluation of the circulation probability density function (PDF) and on arriving at scaling laws with loop sizes.^{26,28–34} The circulation PDF is Gaussian for large loop sizes and, the PDF only depends on the area of the loop but not its shape, which is called the area rule. Sreenivasan *et al.*^{26} explored the circulation properties in the turbulent wake of a cylinder at $ Re \lambda \u2264$ 40, using two-dimensional particle image velocimetry (PIV). Thoroddsen^{29} attempted to use multiple hot-wires, in grid turbulence at $ Re \lambda =$ 230, with the Taylor's hypothesis of frozen turbulence, to assess the dissipation rate within large-scale rotational regions of the turbulence. These crude estimates highlight the need for more advanced experimental techniques, which have recently become available. Zhou *et al.*^{28} measured the circulation PDFs and the moments of velocity and circulation structure functions in turbulent Rayleigh–Bénard convection in a cylindrical convection cell using the PIV technique.

Iyer *et al.*^{33} performed high-resolution direct numerical simulation (DNS) of homogeneous isotropic turbulence for $ Re \lambda =$ 140–1300 and showed that in the inertial range the PDF of circulation is independent of the dimension of rectangular loops over the entire PDF and not just for the tails. In their subsequent work,^{34} they test the validity of this area rule for non-planar loops. For non-planar loops, the area taken is the minimal surface enclosed by the loop. Recently, Iyer *et al.*^{34} tested the validity of the area rule for non-planar loops.

Herein we present time-resolved volumetric measurements in three different planar contractions. With volumetric measurements, we can not only track the evolution of the velocity and vorticity vectors in space and time, but also formulate the circulation over closed contours which stay within the measurement volume. There is a second benefit of studying circulation, as for large Reynolds numbers, the vorticity vector is often not fully resolved, while the circulation is an integral quantity of the velocity and is less affected by measurement errors, especially when compared to the spatial derivatives required to calculate the vorticity. This was recently verified with experimental data by Mugundhan and Thoroddsen.^{22}

As is clear from the above literature review, in the present flow configuration, the large mean streamwise straining is expected to greatly rearrange the vorticity strengths in the streamwise vs transverse directions—an inhomogeneity that will manifest most prominently in the circulation in different planes. Herein we have therefore formulated the circulation around a point in the form of a vector, by calculating three separate circulations around squares in the three perpendicular planes centered at that point, as sketched in Fig. 3. The orientation and strength of the different components of this circulation “vector” quantify the inhomogeneity of the turbulence. However, one must keep in mind the conservation of circulation, from the *Kelvin's circulation theorem*, when the loop follows the fluid elements. Herein we keep the integrating square circuits of constant size, as we move in the streamwise direction.

## II. EXPERIMENTAL SETUP

The overall water flow facility is shown in Fig. 1(a). It is the same as used in the studies by Mugundhan *et al.*^{21} and Alhareth *et al.*,^{24} only with different contraction sections residing inside the red dashed rectangle in Fig. 1(b). The tunnel recirculates the particle suspension, where the water is pumped from a supply tank into the vertical constant-head overhead tank, from which the gravity-driven flow enters the tunnel. The water level is maintained by a centrifugal pump that generates the $ \u223c 2.5$ m head driving the flow. Flow first goes through two perforated steel plates and a metallic honeycomb to kill off any large-scale motions and make the flow uniform before entering the active grid. The active grid is used to inject turbulent fluctuations into the flow.^{15} It consists of ten rotating rods, to each of which are attached with six square flaps, with a hole in each triangular blade.^{21} Two sets of five rods are perpendicular to each other with a mesh size of $ M = 30$ mm. The rotation speed and direction are computer controlled. Herein we only show results from the Random rotation protocol, where all the rods are rotated at 210 rpm, while the direction of rotation is changed after a random time duration, which on average is $ 1 \u2009\xb1\u2009 0.5$ rotation periods. Subsequently, after leaving the grid, the turbulent flow passes through a uniform square cross section tunnel 478 mm long to enhance transverse homogeneity before the flow enters the contraction.

Three different length 2-D planar contractions were studied, with their geometries shown in Fig. 2. We call them 2-D as they contract the stream only in one of the transverse direction, i.e., the $ y \u2212$direction, while planar refers to the sidewalls being straight and not smoothly curved as in Ref. 21. Here, $ x \u2212$is the streamwise direction, while the channel remains of constant width in the $ z \u2212$direction. The contractions are identified by their length, as the short contraction (**SC**) which is 110 mm long, the intermediate contraction (**IC**) of 170 mm, and the long contraction (**LC**) at 350 mm. All three have the same total contraction ratio of *C* = 4:1. The converging angles of one of the side-walls of the three contractions are 31.5° (**SC**), 22° (**IC**), and 11° (**LC**). Photographs of the three interchangeable sections are included in the supplementary material.

The three velocity components were measured using 3-D Lagrangian particle tracking velocimetry (LPT) with the shake-the-box (STB) algorithm,^{35} as implemented in the LaVision Davis 10.2 software. Four high-speed video cameras recorded the particles inside the laser volume. The cameras are Phantom V2640, capable of capturing $ 2 k \xd7 2 k$ pixels at up to 6500 fps, even though smaller pixels areas were used near the outlet and smaller frame-rates were used in the inlet section, where the mean velocity is lower. Each camera has 288 GB of internal memory for up to 140 000 frames per experimental run. For illumination, a high-speed dual cavity pulsed laser Nd-YLF (Litron LDY 300 PIV) emitting 527 nm green light was employed. Volume optics spread the laser beam into a 25-mm-wide slice of approximately 100 mm vertical extent. Particle seeding used was fluorescent polyethylene microspheres containing Rhodamine-B, of size $ \u223c 63 \u2013 75 \u2009 \mu $m (from Cospheric). For the original volume calibration, a precision metallic plate with white equally spaced dots, on both sides, was obtained from LaVision. The dual-plane plate was held inside water of a half-filled tunnel, through a side window above the test section. Micro-stepper motor translated the plate to three different locations within the illuminated volume. Each camera thereby has six planes of dots for this initial calibration. The plate is then removed, and the tank filled with water and particles. This is followed by volume self-calibration,^{36} where particle recordings in a real experimental run are used to correct any deviations, caused, for example, by wall deformations from the increased head. This step uses about 10 000 of the brightest particles to check for triangulation disparity between the four cameras. These corrections are iterated in subsections of the volume, about five times to reach disparity down to the order of a tenth of a pixel. Using this calibration, the particles are now tracked using the STB as they move through the illuminated volume. The number of tracks varies depending on the volume size, ranging from 50 000 to 150 000 particle tracks within the volume at each instant. The conversion of the Lagrangian particle tracks onto an Eulerian spatial grid is achieved through spatial interpolation of the nearest particle tracks using information from numerous time-steps. The interpolated volume is of size $ 48 \xd7 48 \xd7 48$ voxels using overlap of 75% between adjacent velocity points. This results in a grid-spacing of $ \Delta x = 0.66$ mm. We use two different time-filter lengths, i.e., 5 time-steps for obtaining rms statistics and 11 time-steps for tracking of coherent vortical structures.

## III. CIRCULATION AND COHERENT STRUCTURES

*L*,

*A*is enclosed by the boundary curve

*L*. Herein, we use square contours of different side lengths,

*r*. We can do this calculation at the same instant over contours in any orientation. Here, we align these contours with the principal coordinates, as sketched in Fig. 3. In this way, we essentially form an instantaneous circulation vector, whose components we can compare. The individual components can be normalized by the length of this vector,

One circulation component can be calculated from a planar PIV data,^{27} but in our configuration, this would be difficult for the most interesting component, i.e., the streamwise $ \Gamma x$, as it is measured in the plane perpendicular to the flow direction.

The volumetric velocity measurements allow calculations of the full vorticity vector $ \omega = ( \omega x , \omega y , \omega z )$, at each grid point. This also allows us to search for coherent vortices, by looking at iso-contours of vorticity.

## IV. RESULTS

### A. Mean and RMS velocities

For all three contractions, the turbulent flow generated by the active grid first flows through a 478 mm long straight section before entering the contractions. This distance corresponds to 16 grid mesh sizes *M*, to allow for improved transverse homogeneity. The flow resistance differs slightly between the different piping/contractions, thereby changing the mean velocity and turbulent statistics in small but measurable ways. Therefore, Table II compares the inlet conditions entering the different contractions.

Turbulence parameter . | Short contraction . | Intermediate contraction . | Long contraction . |
---|---|---|---|

. | SC
. | IC
. | LC
. |

$C$ | 4:1 | 4:1 | 4:1 |

$H$ (mm) | 110 | 170 | 350 |

$ \u27e8 U i n \u27e9$ (m s^{−1}) | 0.35 | 0.31 | 0.34 |

$ u rms / \u27e8 U i n \u27e9$ ( $%$) | 5.56 | 6.41 | 7.15 |

$k$ (m^{2} s^{−2}) | $ 7.21 \xd7 10 \u2212 4$ | $ 6.34 \xd7 10 \u2212 4$ | $ 6.39 \xd7 10 \u2212 4$ |

$\epsilon $ (m^{2} s^{−3}) | $ 2.16 \xd7 10 \u2212 4$ | $ 1.35 \xd7 10 \u2212 4$ | $ 1.67 \xd7 10 \u2212 4$ |

$ L I$ (mm) | 24.4 | 23.1 | 25.3 |

$\lambda $ (mm) | 6.31 | 8.31 | 8.42 |

$\eta $ (mm) | 0.257 | 0.289 | 0.274 |

$ R e L$ | 668 | 592 | 654 |

$ R e \lambda $ | 172 | 213 | 217 |

$ \Delta x / \eta $ | 2.39 | 2.22 | 2.23 |

$ S x / H = 0.85$ (s^{−1}) | 28.2 | 21.2 | 13.8 |

$ S x / H = 0.85 *$ | 68 | 85 | 86 |

Turbulence parameter . | Short contraction . | Intermediate contraction . | Long contraction . |
---|---|---|---|

. | SC
. | IC
. | LC
. |

$C$ | 4:1 | 4:1 | 4:1 |

$H$ (mm) | 110 | 170 | 350 |

$ \u27e8 U i n \u27e9$ (m s^{−1}) | 0.35 | 0.31 | 0.34 |

$ u rms / \u27e8 U i n \u27e9$ ( $%$) | 5.56 | 6.41 | 7.15 |

$k$ (m^{2} s^{−2}) | $ 7.21 \xd7 10 \u2212 4$ | $ 6.34 \xd7 10 \u2212 4$ | $ 6.39 \xd7 10 \u2212 4$ |

$\epsilon $ (m^{2} s^{−3}) | $ 2.16 \xd7 10 \u2212 4$ | $ 1.35 \xd7 10 \u2212 4$ | $ 1.67 \xd7 10 \u2212 4$ |

$ L I$ (mm) | 24.4 | 23.1 | 25.3 |

$\lambda $ (mm) | 6.31 | 8.31 | 8.42 |

$\eta $ (mm) | 0.257 | 0.289 | 0.274 |

$ R e L$ | 668 | 592 | 654 |

$ R e \lambda $ | 172 | 213 | 217 |

$ \Delta x / \eta $ | 2.39 | 2.22 | 2.23 |

$ S x / H = 0.85$ (s^{−1}) | 28.2 | 21.2 | 13.8 |

$ S x / H = 0.85 *$ | 68 | 85 | 86 |

Figure 4(a) shows the mean velocity on the centerline through the contractions. The mean velocities increase approximately in accordance with the contracting cross section but are not measured all the way to the exit, where they should be slightly above the total contraction of four times the inlet velocity, owing to boundary layer formation. The optical access is limited at the exit by the transition to the outlet channel and the mandatory angles between the four cameras, which causes unavoidable shadows. The longest contraction has slightly higher centerline velocities, likely due to larger boundary layer growth on the longer walls. The corresponding mean local strain rates are shown in Fig. 4(b), normalized by the inlet velocity become, as expected, much stronger the shorter the contraction is. Here, we use the grid mesh size *M* as the length scale, which is constant in all experiments. Comparing the mean strain-rate at the half-distance through the contractions ( $ x / H = 0.5$), we see dimensional strain-rates $ S = 2 ( \u2202 \u27e8 U \u27e9 / \u2202 x )$ as 3.2, 6.0, and 12 $ s \u2212 1$ for **LC, IC**, and **SC** respectively. The largest measured strain rates near the exits are correspondingly 14, 21, and 28 $ s \u2212 1$.

In Fig. 4(c), we have included timescale considerations, i.e., how quickly is the strain imposed on the turbulence. Here, we convert the streamwise location into the elapsed time *t* it has taken a fluid element to reach that location by integrating the local mean velocity. This time is compared to the characteristic turbulent overturning time $ \tau = L I / u rms$, where $ L I$ is the streamwise integral length scale, i.e., $ t / \tau = t \u2009 u rms / L I$. The three contractions, from short to long, subject the turbulence to the strongest mean strain at 0.15, 0.3, and 0.7 times the overturning timescale $\tau $.

Figure 5 shows the rms velocities in the streamwise (a) and compressive transverse direction (b). As argued above, from conservation of mass and angular momentum, the streamwise fluctuations reduce monotonically through all three contractions, approaching the contraction ratios of $ C = 4$. The rate of decrease is however much smaller for the short contraction. On the other hand, in the compressive direction, the evolution differs between contractions, for the shortest one v_{rms} increases downstream, while for the longest one, these fluctuations decrease at first before increasing further downstream. The intermediate contraction remains constant initially before starting to rise at the end. One should also keep in mind that the *natural decay* of v_{rms} with distance from the grid and it would therefore decrease the least, from this effect, in the short contraction.

### B. Coherent structure orientation

Coherent vortical structures are ubiquitous in turbulent flows. Their shape, orientation, and coherency vary greatly between different flow geometries. Their identification is sometimes obscured by mean shear, such as in boundary layers. This has led to a number of proposed quantities or methods to extract them, such as the $ Q \u2212$^{37} or $ \Lambda \u2212$ criterion^{38} or stochastic methods.^{39} Herein we apply the simplest technique of using the iso-surfaces of vorticity magnitude, which is in accordance with Prandtl's conceptual picture. Figure 6 shows typical isosurfaces of vorticity to mark the vortical structures, at the three different measurement locations within the long contraction (**LC**). The threshold value of the vorticity magnitude $ | \omega |$ is normalized by the maximum mean strain $ S max$ in this contraction. The isosurfaces are selected by $ | \omega | / S max =$ 1.8. The surface coloring indicates the magnitude of the streamwise component of the vorticity $ \omega x$. Red and blue colors therefore indicate clockwise or counterclockwise vorticity, relative to the $ x \u2212$axis. Each row is a time-sequence of volumes, while the different rows are from experiments on different days. The orientation of the vortices is initially close to random, but become aligned with the mean strain in the middle section and even more so near the exit, where long coherent structures are seen.^{21,24} Their vorticity strength also increases from their streamwise stretching.

We now turn to characterizing the orientation of these vortices. Using an in-house MATLAB code, we use watershed algorithm to identify individual vortices. For consideration, the vortices must be above a certain minimum volume within the isocontour. The region is fit with an ellipsoid having the same moment of inertia about the principal axes, as shown schematically in Fig. 7. The orientation of the principal axis away from the streamwise $ x \u2212$direction $ \theta x$, is calculated for all identified vortices and a probability density function is constructed, as shown in Fig. 8. The probability is evaluated of the cosine of the alignment angle to account for the azimuthal freedom around the vertical axis, which requires taking $ cos ( \theta x )$ for a uniform PDF of randomly distributed $ \theta x$. Figure 8 shows relatively uniform inlet distribution of the vortex orientation, with rapidly increasing alignment with the $ x \u2212$axis $ \theta x = 0$, or $ cos ( \theta x ) = 1$, as the flow advects toward the outlet of the contraction, experiencing the strongest streamwise mean strain.

The strongly peaked distributions motivate us to find quantitative measures of the strength of the alignment, which are independent of the resolution of the PDF at this peak. We construct two parameters to quantify this:^{24} First, we fit the PDF shape approaching the peak and see where the cumulative probability reaches 99.5%, called $ P 99.5$. With 100 bins and the very large data samples of $ \u223c 10 5$ measurement volumes, the fits are converged and become independent of the number of bins and are simple interpolations, as is shown in the supplementary material. In other words, the integration to reach $ P \u2243 99.5 %$ requires no extrapolation of the data.

*Alignment Coefficient*$ C A$, defined as

Using the results in Fig. 8, we calculate these two quantitative alignment measures described above, with the results presented in Fig. 9. The alignment coefficient $ C A$ increases from $ 0.2 \u2192 0.9$ approximately linearly with the normalized distance downstream, $ x / H$, for all three contractions. The highest value of 0.93 is reached for the intermediate contraction. Keep in mind that $ x / H$ values on the abscissa coincide with $ C / 4$.

On the other hand, the difference is more pronounced in the $ P 99.5$ values. Initially, the values grow at the same rate, while for $ x / H > 0.7$, the intermediate and long contractions show stronger alignment, with the strongest peak of 40 for **IC**, while the short contraction only reaches 15 and 25 for the long contraction. This non-monotonic behavior suggests that the short contraction is too rapid to rearrange the vortices, while the longest contraction may allow start of vortex interactions and return toward isotropy through vortex breakdown. It is interesting to note that these alignment parameters are highest with the **IC**, which has its length almost equal to its inlet width.

### C. Circulation PDFs

*r*. The tails reach much larger $\Gamma $-values for the larger loop. Figure 11 shows the same PDFs near the outlet of the intermediate contraction where $ x / H = 0.85$. Here, the PDF tails of the $\Gamma $ components react differently, with the values of $ \Gamma y$, in the compressive direction, reducing greatly. $ \Gamma z$ also reduces but to lesser extent, while the stretched streamwise $ \Gamma x$ PDF grows in width. The $ \Gamma y$ distribution develops intriguing skirts at its largest values, reminiscent of scalar-gradient PDFs.

^{40}Normalizing the PDFs with their rms values $ \alpha = \Gamma i / ( \Gamma i ) rms$, we could evaluate the slope of the tails

*D*, to compare to other studies:

*et al.*,

^{34}i.e., $ P ( \alpha i ) \u223c \u2009 exp ( \u2212 D i | \alpha i | ) / \alpha i$. Similar PDF evolution is observed at the outlets for the other two contractions, as is shown in the supplementary material.

The presence of coherent structures aligned with the mean strain $ x \u2212$direction may be expected to generate large circulation values on loop sizes similar to their core size. However, despite their prominence and importance for the dynamics, they only take up a small fraction of the flow volume—for a 3-D contraction it is only $ \u223c 1 % \u2013 2 %$ at the outlet.^{24} The probability of a coherent vortex structure being captured by any one loop is therefore low.

### D. Circulation-vector orientation

As suggested in Sec. III, we can think of the circulation in different perpendicular planes as forming a *circulation vector* with a magnitude and orientation. We now quantify the relative strength of the $ \Gamma \u2212$components and show how this is affected by the contractions. Their relative strength can be expected to follow similar evolution as the orientation of the coherent vortical structures.^{22} This is indeed observed in Fig. 12 where we plot the PDF of relative strength of the three components, normalized by the length of the circulation vector $ | \Gamma |$ at each instant. Near the inlet to the contractions $ x / H = 0.2$, all the PDFs are close to uniform with $ P ( \Gamma i )$ within $ 0.5 \u2009\xb1\u2009 0.2$. On the other hand, near the outlet where $ x / H = 0.85$, the PDFs of the streamwise stretched and transverse compressed components are entirely different. The largest values, $ \Gamma i / | \Gamma | \u2243 1$, are most probable for the streamwise $ \Gamma x$ component, while $ \Gamma y$, in the compressed direction, cluster near zero for the smallest component of the circulation vector. In contrast, the other transverse component $ \Gamma z$ is mostly unaffected, but it is slightly more likely to be small. These PDF shapes are very similar to the results in Mugundhan and Thoroddsen^{22} for a weaker 2-D contraction of $ C = 2.5$. Here, our peaks in $ \Gamma x \u2248$ 1.9–2.5 are larger than 1.5 in the previous study.

## V. DISCUSSION AND CONCLUSIONS

PIV and Tomo-PIV are well suited to measure the overall velocity and vorticity fields, while hot-wires and LDV can obtain better time-series statistics at a point in the flow. The two techniques can therefore be complementary, as long as they do not clash in their paradigms. For example, the equality of rms velocity components at a fixed point in the flow does not rule out coherent structures, which are randomly places in the transverse plane. Such coherent vortical structures can in some configurations be identified in planar PIV measurements, such as the mixing layer, but are often difficult to extract from conditional point measurements. In a contraction, the stretched vortices are aligned with the mean strain and their characterization therefore require volumetric measurements, as their evolution direction is perpendicular to their cross-sectional plane. The strongest vortical structures appear randomly in time and their transverse location at the entrance to the contractions. Their orientation is also almost random, i.e., isotropic, while inside the contractions they quickly align with the mean strain, as expected by conceptual models of vortex stretching. Therefore, identifying the coherent vortical structures and quantifying their alignment requires volumetric data which we have presented herein.

We have formulated two non-dimensional quantities to characterize this degree of alignment.^{24} Interestingly, the intermediate contraction shows the strongest values of the alignment coefficients, suggesting that the shortest contraction is too sudden to allow full alignment.

We also calculate instantaneous circulations around a point in three perpendicular planes, to produce a “circulation vector,” whose alignment can also be assessed. This is done using an instantaneous measure by showing the probability of the strength of each component compared to the total length of this vector. The three contractions have similar profound effect on this orientation PDF with strong preference for the streamwise direction.

These volumetric measurements have revealed the significance of the aligned coherent vortical structures. Furthermore, our measurements in an axisymmetric contraction suggest these coherent stretched vortical structures become even more prevalent as the contraction ratio *C* increases further.^{24}

Recent study of turbulent circulation has mostly been focused on numerical simulations of periodic turbulence in a box, where the turbulent Reynolds numbers have now reached $ R e \lambda \u2243 2500$, see Yeung *et al.*^{41} Our experiments show that a new generation of experimental tools is emerging for comparison with the numerical results, especially for larger experimental devices, with inhomogeneous boundary conditions, where simulations may be more challenging.

To close this anniversary article in honor of Professor Sreenivasan, we refer to his quarter-century-old 1999 review on fluid turbulence,^{42} where he calls the analysis of coherent structures as marking the *modern era* of turbulence research. The classical era adopted the statistical approach and has persisted for almost a century since Prandtl's conceptual framework for straining turbulence, which can now be seen in action. Sreenivasan postulated that the work in large-scale structures would help in describing overall features of the flow efficiently, as well as lead to better control of the adverse and beneficial features of turbulent flow.

## SUPPLEMENTARY MATERIAL

See the supplementary material for photographs of the three different interchangeable contraction sections, the formula used for curve-fitting the PDF of alignment angles, the extent of measurements sub-volumes for the different experimental runs, examples of typical ellipsoidal fitting to coherent vortical structures, and circulation PDFs near the outlet section, for the short and long contractions, for comparison with the intermediate contraction shown in the main text.

## ACKNOWLEDGMENTS

This research was funded by King Abdullah University of Science and Technology (KAUST), under No. BAS/1/1352-01-01.

## AUTHOR DECLARATIONS

### Conflict of Interest

The authors have no conflicts to disclose.

### Author Contributions

**Abdullah A. Alhareth:** Data curation (equal); Investigation (equal); Methodology (equal); Software (equal); Validation (equal); Visualization (equal); Writing – original draft (equal); Writing – review & editing (equal). **Vivek Mugundhan:** Conceptualization (equal); Data curation (equal); Investigation (equal); Methodology (equal); Software (equal); Supervision (equal); Validation (equal); Visualization (equal); Writing – original draft (equal); Writing – review & editing (equal). **Kenneth R. Langley:** Conceptualization (equal); Investigation (equal); Software (equal); Supervision (equal); Visualization (equal). **Sigurdur T. Thoroddsen:** Conceptualization (equal); Investigation (equal); Methodology (equal); Project administration (equal); Supervision (equal); Validation (equal); Visualization (equal); Writing – original draft (equal); Writing – review & editing (equal).

## DATA AVAILABILITY

The data that support the findings of this study are available from the corresponding author upon reasonable request.

## REFERENCES

^{3}enabled by exascale computing