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.

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, Prandtl1 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 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 1 and the transverse velocity fluctuations enhance proportional to c 0.5. Taylor2 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 Tucker3 used spectral analysis to obtain expressions for velocity fluctuations at the contraction exit. Batchelor and Proudman4 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 Narasimha5 generalized the theory of Batchelor and Proudman4 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 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 4. In Table I, we list the important experimental studies and summarize their main results below. Uberoi7 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 Wallis8 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.

TABLE I.

Summary of important experimental studies on the evolution of grid turbulence in axisymmetric and planar contractions, arranged in chronological order. The table highlights the geometry, Reynolds number based on the mesh size R e M, Taylor Reynolds number R e λ, measurement methodology adopted, and quantities measured in each study.

Researchers Geometry Flow Method Velocity components Vorticity
Uberoi7   Axisymmetric  R e M  Hot-wire  u rms , v rms  No 
  C = 4, 9, 16  3 700–123 00       
Reynolds and Tucker23   2-D and axisymmetric strain 2.3–6  R e M  Hot-wire  u rms , v rms , w rms  No 
    6780       
Hussain and Ramjee9   Axisymmetric (four different shapes)  R e D  Hot-wire (single, X-wire)  u rms , v rms, w rms  No 
  C = 11  63 000       
Warhaft11   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 Atta13   2D-contraction  R e λ  Hot-wire  u rms , w rms  No 
  C = 2.5  25       
Ayyalasomayajula and Warhaft17   Axisymmetric  R e λ  Hot-wire  u rms, v rms  No 
  C = 4  35–470       
Brown et al.19   Planar contraction  R e λ  LDA  u rms, v rms  No 
  (linear) C = 7.3, 11.3, and 16.7  51–99       
Ertunç and Durst20   Axisymmetric  R e λ  X-config  u rms , v rms  No 
  C = 3.69, 14.75  26–48  Hot-wire     
Gylfason and Warhaft18   Axisymmetric  R e λ  X-array probes  u rms , v rms  No 
  C = 4  50, 190       
Mugundhan et al.21   2D-contraction  R e λ  Tomo PIV  u rms , v rms , w rms  Yes 
  (smooth) C = 2.5  180–290  and LPT     
Mugundhan and Thoroddsen22   2D-contraction  R e λ  3-D LPT  u rms , v rms , w rms  Yes 
  (smooth) C = 2.5  220       
Alhareth et al.24   3D-contraction  R e λ  3-D LPT  u rms , v rms , w rms  Yes 
  C = 16  192       
Researchers Geometry Flow Method Velocity components Vorticity
Uberoi7   Axisymmetric  R e M  Hot-wire  u rms , v rms  No 
  C = 4, 9, 16  3 700–123 00       
Reynolds and Tucker23   2-D and axisymmetric strain 2.3–6  R e M  Hot-wire  u rms , v rms , w rms  No 
    6780       
Hussain and Ramjee9   Axisymmetric (four different shapes)  R e D  Hot-wire (single, X-wire)  u rms , v rms, w rms  No 
  C = 11  63 000       
Warhaft11   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 Atta13   2D-contraction  R e λ  Hot-wire  u rms , w rms  No 
  C = 2.5  25       
Ayyalasomayajula and Warhaft17   Axisymmetric  R e λ  Hot-wire  u rms, v rms  No 
  C = 4  35–470       
Brown et al.19   Planar contraction  R e λ  LDA  u rms, v rms  No 
  (linear) C = 7.3, 11.3, and 16.7  51–99       
Ertunç and Durst20   Axisymmetric  R e λ  X-config  u rms , v rms  No 
  C = 3.69, 14.75  26–48  Hot-wire     
Gylfason and Warhaft18   Axisymmetric  R e λ  X-array probes  u rms , v rms  No 
  C = 4  50, 190       
Mugundhan et al.21   2D-contraction  R e λ  Tomo PIV  u rms , v rms , w rms  Yes 
  (smooth) C = 2.5  180–290  and LPT     
Mugundhan and Thoroddsen22   2D-contraction  R e λ  3-D LPT  u rms , v rms , w rms  Yes 
  (smooth) C = 2.5  220       
Alhareth et al.24   3D-contraction  R e λ  3-D LPT  u rms , v rms , w rms  Yes 
  C = 16  192       

Hussain and Ramjee9 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.

Warhaft11 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 Atta13 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 amplified over twice as much as the reduced streamwise turbulence u due to stretching of streamwise vortices. They showed how buoyancy suppressed the transverse fluctuations, to rise again through zombie turbulence.

Mydlarski and Warhaft14 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 λ. Mydlarski16 has reviewed the design aspects and the numerous implementations of active grids in turbulence research to date. Following this work, Ayyalasomayajula and Warhaft17 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 Warhaft18 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  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 Durst20 questioned the previous hot-wire measurements of the streamwise velocity fluctuations, when measured with × 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 Migdal25 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 λ  40, using two-dimensional particle image velocimetry (PIV). Thoroddsen29 attempted to use multiple hot-wires, in grid turbulence at Re λ = 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 λ = 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.

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 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  ±  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.

FIG. 1.

(a) Schematic of the gravity-driven water tunnel with interchangeable contraction sections of different streamwise lengths. The details of each of the three contraction used are shown in the next figure. The flow is closed loop, with water pumped into a constant-head overflow tank with meshes to arrest any bubbles, which directs the water into a flow-control section, with perforated plates and honeycomb. The laminar flow is then passed through an active grid section to generate turbulence, which travels through a 478-mm-long straight section before entering the contraction which varies in streamwise length H. (b) Close-up schematic of the main test section including the flow straighteners and active grid upstream of the contraction. The interchangeable test section containing the contractions is marked by the dashed red rectangle, with details in the next figure. Photographs of the test sections are shown in the supplementary material.

FIG. 1.

(a) Schematic of the gravity-driven water tunnel with interchangeable contraction sections of different streamwise lengths. The details of each of the three contraction used are shown in the next figure. The flow is closed loop, with water pumped into a constant-head overflow tank with meshes to arrest any bubbles, which directs the water into a flow-control section, with perforated plates and honeycomb. The laminar flow is then passed through an active grid section to generate turbulence, which travels through a 478-mm-long straight section before entering the contraction which varies in streamwise length H. (b) Close-up schematic of the main test section including the flow straighteners and active grid upstream of the contraction. The interchangeable test section containing the contractions is marked by the dashed red rectangle, with details in the next figure. Photographs of the test sections are shown in the supplementary material.

Close modal

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 direction, while planar refers to the sidewalls being straight and not smoothly curved as in Ref. 21. Here, x is the streamwise direction, while the channel remains of constant width in the z 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.

FIG. 2.

Sketches showing the dimensions of the three different 2-D planar contractions. They all have the same 4:1 contraction ratio but are characterized by their different streamwise lengths, which in turn controls the strength of the mean streamwise strain-rate. The three different lengths marked by the red arrows are (a) short contraction (SC) 110 mm long, (b) intermediate contraction (IC) 170 mm, and (c) long contraction (LC) 350 mm. The separate measurement volumes, within each contraction, are indicated by the blue squares. Restricted optical access owing to the support structure prevents measurements near the outlet of the longest contraction. Experimental runs for each square are obtained on separated days, after the necessary realignment of the optics. The abbreviations SOC and EOC indicate the start and end of the contractions. All dimensions are in millimeters. Photographs of the three test sections are shown in supplementary material.

FIG. 2.

Sketches showing the dimensions of the three different 2-D planar contractions. They all have the same 4:1 contraction ratio but are characterized by their different streamwise lengths, which in turn controls the strength of the mean streamwise strain-rate. The three different lengths marked by the red arrows are (a) short contraction (SC) 110 mm long, (b) intermediate contraction (IC) 170 mm, and (c) long contraction (LC) 350 mm. The separate measurement volumes, within each contraction, are indicated by the blue squares. Restricted optical access owing to the support structure prevents measurements near the outlet of the longest contraction. Experimental runs for each square are obtained on separated days, after the necessary realignment of the optics. The abbreviations SOC and EOC indicate the start and end of the contractions. All dimensions are in millimeters. Photographs of the three test sections are shown in supplementary material.

Close modal

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 × 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 63 75 μm (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 × 48 × 48 voxels using overlap of 75% between adjacent velocity points. This results in a grid-spacing of Δ 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.

Table I showed that earlier turbulence experiments in contraction geometries have focused on single-point measurements, primarily using hot-wires or Laser Doppler Velocimetry (LDV) probes. One of the advantages of having volumetric measurements is that it allows us to calculate circulation, by integrating the tangential velocity around a closed contour L,
Γ i = L u · d l = A ω · d A ,
(1)
where the area 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,
| Γ | = Γ x 2 + Γ y 2 + Γ z 2 .
(2)
FIG. 3.

The mutually perpendicular square loops used to compute circulation components, centered about point P in a 5  × 5  × 5 Eulerian grid. The loops shown on the left, middle, and the right sides are used to calculate Γ x, Γ y, and Γ z respectively. The × marks and the arrows represent the velocity locations on the grid and integration direction. The stretching and compression directions are indicated by the bold arrows. The colored plane indicate qualitative deformation of a fluid element which initially matches with the square loop indicated in the same color.

FIG. 3.

The mutually perpendicular square loops used to compute circulation components, centered about point P in a 5  × 5  × 5 Eulerian grid. The loops shown on the left, middle, and the right sides are used to calculate Γ x, Γ y, and Γ z respectively. The × marks and the arrows represent the velocity locations on the grid and integration direction. The stretching and compression directions are indicated by the bold arrows. The colored plane indicate qualitative deformation of a fluid element which initially matches with the square loop indicated in the same color.

Close modal

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 Γ x, as it is measured in the plane perpendicular to the flow direction.

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

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.

TABLE II.

Turbulence quantities slightly inside the three different contractions, at x = 5 mm. Here, the symbols denote the following: U i n —mean inlet velocity, u rms—streamwise rms velocity, k—turbulent kinetic energy, ε—dissipation rate, L I—streamwise integral length scale, and λ—Taylor microscale. The Taylor microscale is computed by two-point spatial correlation function f ( r ) = [ u ( x ) u ( x + e x r ) ] / u 2 . Kolmogorov length scale is η = ( ν 3 / ε ) 1 / 4. Reynolds numbers based on L I and λ are denoted as R e L, R e λ respectively. The velocity-grid measurement resolution, Δ x = 0.66 mm. S * = S k / ε characterizes the maximum mean strain measured, in terms of a timescale ratio between the mean strain S = 2 ( U / x ) and the turbulent straining.

Turbulence parameter Short contraction Intermediate contraction Long contraction
SC IC LC
C  4:1  4:1  4:1 
H (mm)  110  170  350 
U i n (m s−1 0.35  0.31  0.34 
u rms / U i n ( % 5.56  6.41  7.15 
k (m2 s−2 7.21 × 10 4  6.34 × 10 4  6.39 × 10 4 
ε (m2 s−3 2.16 × 10 4  1.35 × 10 4  1.67 × 10 4 
L I (mm)  24.4  23.1  25.3 
λ (mm)  6.31  8.31  8.42 
η (mm)  0.257  0.289  0.274 
R e L  668  592  654 
R e λ  172  213  217 
Δ x / η  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 
U i n (m s−1 0.35  0.31  0.34 
u rms / U i n ( % 5.56  6.41  7.15 
k (m2 s−2 7.21 × 10 4  6.34 × 10 4  6.39 × 10 4 
ε (m2 s−3 2.16 × 10 4  1.35 × 10 4  1.67 × 10 4 
L I (mm)  24.4  23.1  25.3 
λ (mm)  6.31  8.31  8.42 
η (mm)  0.257  0.289  0.274 
R e L  668  592  654 
R e λ  172  213  217 
Δ x / η  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 ( U / x ) as 3.2, 6.0, and 12  s 1 for LC, IC, and SC respectively. The largest measured strain rates near the exits are correspondingly 14, 21, and 28  s 1.

FIG. 4.

(a) Streamwise evolution of the mean centerline velocity U normalized by the mean velocity U i n at the inlet to the contractions. (b) Streamwise variation of the mean strain rate, U / x, normalized by U i n and the mesh size, M. (c) How rapidly is the mean strain-rate imposed on the turbulence? The mean strain-rate is normalized by U i n and the integral scale L I, shown vs non-dimensional time t u rms / L I. This timescale is the ratio of advection time t to the eddy turn-over timescale, τ = L I / u rms. All the evolutions are shown along the centerline of the contraction.

FIG. 4.

(a) Streamwise evolution of the mean centerline velocity U normalized by the mean velocity U i n at the inlet to the contractions. (b) Streamwise variation of the mean strain rate, U / x, normalized by U i n and the mesh size, M. (c) How rapidly is the mean strain-rate imposed on the turbulence? The mean strain-rate is normalized by U i n and the integral scale L I, shown vs non-dimensional time t u rms / L I. This timescale is the ratio of advection time t to the eddy turn-over timescale, τ = L I / u rms. All the evolutions are shown along the centerline of the contraction.

Close modal

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 τ = L I / u rms, where L I is the streamwise integral length scale, i.e., t / τ = t 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 τ.

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 vrms 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 vrms with distance from the grid and it would therefore decrease the least, from this effect, in the short contraction.

FIG. 5.

Streamwise variation of the streamwise rms velocity u rms (a) and the transverse rms velocity vrms (b) along the centerline of the contraction. The rms velocities are normalized by the corresponding values at the inlet to the contractions, at x = 0. The streamwise location x is normalized by the contraction streamwise length H—for SC ( H = 110 mm), for IC ( H = 170 mm) and for LC ( H = 350 mm).

FIG. 5.

Streamwise variation of the streamwise rms velocity u rms (a) and the transverse rms velocity vrms (b) along the centerline of the contraction. The rms velocities are normalized by the corresponding values at the inlet to the contractions, at x = 0. The streamwise location x is normalized by the contraction streamwise length H—for SC ( H = 110 mm), for IC ( H = 170 mm) and for LC ( H = 350 mm).

Close modal

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 37 or Λ criterion38 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 | ω | is normalized by the maximum mean strain S max in this contraction. The isosurfaces are selected by | ω | / S max = 1.8. The surface coloring indicates the magnitude of the streamwise component of the vorticity ω x. Red and blue colors therefore indicate clockwise or counterclockwise vorticity, relative to the x 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.

FIG. 6.

Typical time evolution of the coherent structures visualized by iso-surfaces of vorticity magnitude, | ω | = 25 s−1, normalized by the maximum mean strain rate, | ω | / S max, in the long contraction LC. The maximum mean strain S max = 2 ( U / x ) max is measured as 13.8 s−1. The iso-surfaces are colored based on the magnitude and rotation direction of ω x. The top, middle, and bottom rows show time sequences of vortical structures in regions LCP1 ( x = [ 10 , 96 ] mm), LCP2 ( x = [ 122 , 226 ] mm), and LCP3 ( x = [ 231 , 331 ] mm) respectively, as marked in Fig. 2. Note that the measurements in each region correspond to different experimental runs. Videos are captured at frame rates of 1200 fps for LCP1 and LCP2 and 1800 fps for LCP3, but the volumes shown are separated by 30 ms. The depth of the volumes into the board is 23 mm.

FIG. 6.

Typical time evolution of the coherent structures visualized by iso-surfaces of vorticity magnitude, | ω | = 25 s−1, normalized by the maximum mean strain rate, | ω | / S max, in the long contraction LC. The maximum mean strain S max = 2 ( U / x ) max is measured as 13.8 s−1. The iso-surfaces are colored based on the magnitude and rotation direction of ω x. The top, middle, and bottom rows show time sequences of vortical structures in regions LCP1 ( x = [ 10 , 96 ] mm), LCP2 ( x = [ 122 , 226 ] mm), and LCP3 ( x = [ 231 , 331 ] mm) respectively, as marked in Fig. 2. Note that the measurements in each region correspond to different experimental runs. Videos are captured at frame rates of 1200 fps for LCP1 and LCP2 and 1800 fps for LCP3, but the volumes shown are separated by 30 ms. The depth of the volumes into the board is 23 mm.

Close modal

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 direction θ 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 ( θ x ) for a uniform PDF of randomly distributed θ x. Figure 8 shows relatively uniform inlet distribution of the vortex orientation, with rapidly increasing alignment with the x axis θ x = 0, or cos ( θ x ) = 1, as the flow advects toward the outlet of the contraction, experiencing the strongest streamwise mean strain.

FIG. 7.

Illustration to show the angle θ X Struct between the principal axis of the ellipsoid fitted to a coherent structure and the streamwise x axis.

FIG. 7.

Illustration to show the angle θ X Struct between the principal axis of the ellipsoid fitted to a coherent structure and the streamwise x axis.

Close modal
FIG. 8.

Streamwise evolution of the probability density function of the cosine of θ X Struct, the angle between the principal axis of the equivalent ellipsoid fitted to the structure and the streamwise x axis. The PDF is computed for 6–9 equal sub-regions within the measurement regions shown in Fig. 2. Here sub-region R1 is closest to the inlet of the contraction, while R6 or R9 are the closest to the exit of the contraction. The x extents of the sub regions R1–R6 is given in the supplementary material. The | ω | threshold used for structure identification in: SC is | ω | = 25 s−1 for R1–R3 and | ω | = 30 s−1 for R4–R6; In the IC it is | ω | = 25 s−1 for R1–R6; In LC it is | ω | = 25 s−1 for all R1–R9. The PDFs are computed with 100 bins. Note that the extent of the vertical axes are different.

FIG. 8.

Streamwise evolution of the probability density function of the cosine of θ X Struct, the angle between the principal axis of the equivalent ellipsoid fitted to the structure and the streamwise x axis. The PDF is computed for 6–9 equal sub-regions within the measurement regions shown in Fig. 2. Here sub-region R1 is closest to the inlet of the contraction, while R6 or R9 are the closest to the exit of the contraction. The x extents of the sub regions R1–R6 is given in the supplementary material. The | ω | threshold used for structure identification in: SC is | ω | = 25 s−1 for R1–R3 and | ω | = 30 s−1 for R4–R6; In the IC it is | ω | = 25 s−1 for R1–R6; In LC it is | ω | = 25 s−1 for all R1–R9. The PDFs are computed with 100 bins. Note that the extent of the vertical axes are different.

Close modal

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 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 99.5 % requires no extrapolation of the data.

The second approach is to calculate the moment-of-inertia of the distribution about cos ( θ ) = 0.5, to formulate an Alignment Coefficient C A, defined as
C A = 0 π / 2 2 [ cos θ 0.5 ] P ( cos θ ) d θ .
C A takes a value of zero for uniform orientation of vortices and a value of unity for vortices which are all perfectly aligned with the x axis. We have included a factor of 2 in the formula, to confine possible values of the orientation coefficient in the range C A [ 1 , 1 ]. Perfect transverse alignment would give a value of −1.

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 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.

FIG. 9.

The streamwise evolution of the orientation coefficient C A (a) and the PDF peak value corresponding to the cumulative probability of 99.5% (b). These quantities are calculated for the PDFs corresponding to sub-regions R1–R6 (or R1–R9 in the case of long contraction) shown in Fig. 8. Table I in the supplementary material shows the streamwise extent of each subregion.

FIG. 9.

The streamwise evolution of the orientation coefficient C A (a) and the PDF peak value corresponding to the cumulative probability of 99.5% (b). These quantities are calculated for the PDFs corresponding to sub-regions R1–R6 (or R1–R9 in the case of long contraction) shown in Fig. 8. Table I in the supplementary material shows the streamwise extent of each subregion.

Close modal

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.

Our formulation of circulations in perpendicular planes was introduced in Sec. III. Volumetric measurements of 3-D velocity fields allow us to form a large number of possible planar loops within the volume. These loops can differ in orientation, size, and their center location. However, at each x location, we limit results to loops evaluated around 3 × 3 grid of points around the centerline. To evaluate the changes in the circulation statistics, we have calculated the PDFs of the three circulation components near the entrance to the contractions as well as near their outlets. Figure 10 shows the Γ PDFs near the inlet for the intermediate contraction, but all three contractions are expected to be quite similar, as the inlet conditions are similar for all three contractions. All three components of Γ are also quite similar, suggesting isotropy. The comparison is for two different loop sizes—the smallest one r = 5 η approaching an estimate of the vorticity and a much larger one r = 36 η, on the larger energy-containing scales. These two PDFs are quite different, with exponential tails for the smaller loop, while the larger loop takes closer to a Gaussian form. For a meaningful comparison, the axes are normalized by the mean inlet velocity and the width of the square loop r. The tails reach much larger Γ-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 Γ components react differently, with the values of Γ y, in the compressive direction, reducing greatly. Γ z also reduces but to lesser extent, while the stretched streamwise Γ x PDF grows in width. The Γ y distribution develops intriguing skirts at its largest values, reminiscent of scalar-gradient PDFs.40 Normalizing the PDFs with their rms values α = Γ i / ( Γ i ) rms, we could evaluate the slope of the tails D, to compare to other studies:
P ( α i ) exp ( D i | α i | ) ,
or one could consider the functional form used in Iyer et al.,34 i.e., P ( α i ) exp ( D i | α i | ) / α i. Similar PDF evolution is observed at the outlets for the other two contractions, as is shown in the supplementary material.
FIG. 10.

The inlet PDF of circulation components Γ x, Γ y and Γ z computed in mutually perpendicular planes, as seen in Fig. 3, by integrating over square loops of dimension r / η = 5 (red) and 36 (green). The loops are centered in the streamwise plane x / H = 0.2, close to the inlet of the intermediate contraction (IC). Both the x and y axes are normalized using the inlet mean velocity, U i n and the loop size, r, such that the unit area under the PDF is preserved.

FIG. 10.

The inlet PDF of circulation components Γ x, Γ y and Γ z computed in mutually perpendicular planes, as seen in Fig. 3, by integrating over square loops of dimension r / η = 5 (red) and 36 (green). The loops are centered in the streamwise plane x / H = 0.2, close to the inlet of the intermediate contraction (IC). Both the x and y axes are normalized using the inlet mean velocity, U i n and the loop size, r, such that the unit area under the PDF is preserved.

Close modal
FIG. 11.

The PDF of circulation components Γ x (a), Γ y (b), and Γ z (c), computed in mutually perpendicular planes by integrating over square loops of dimensions r / η = 5, 36, and 50. The loops are centered in the streamwise location at x / H = 0.85, close to the exit of the intermediate contraction (IC). The normalization is same as that used in Fig. 10.

FIG. 11.

The PDF of circulation components Γ x (a), Γ y (b), and Γ z (c), computed in mutually perpendicular planes by integrating over square loops of dimensions r / η = 5, 36, and 50. The loops are centered in the streamwise location at x / H = 0.85, close to the exit of the intermediate contraction (IC). The normalization is same as that used in Fig. 10.

Close modal

The presence of coherent structures aligned with the mean strain x 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 1 % 2 % at the outlet.24 The probability of a coherent vortex structure being captured by any one loop is therefore low.

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 Γ 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 | Γ | at each instant. Near the inlet to the contractions x / H = 0.2, all the PDFs are close to uniform with P ( Γ i ) within 0.5  ±  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, Γ i / | Γ | 1, are most probable for the streamwise Γ x component, while Γ y, in the compressed direction, cluster near zero for the smallest component of the circulation vector. In contrast, the other transverse component Γ 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 Thoroddsen22 for a weaker 2-D contraction of C = 2.5. Here, our peaks in Γ x  1.9–2.5 are larger than 1.5 in the previous study.

FIG. 12.

Comparison of the PDFs of the relative strengths of the circulation components Γ x / | Γ | (a), Γ y / | Γ | (b), and Γ z / | Γ | (c), between inlet and outlet of the contractions. The circulations are measured with loop sizes r 36 η. The components are normalized with the instantaneous circulation strength is | Γ | = ( Γ x 2 + Γ y 2 + Γ z 2 ) 0.5. The red and black curves correspond to circulation computed in loops centered along the centerline and at the streamwise locations x / H = 0.2 and 0.85 respectively. The PDFs are computed using 100 bins.

FIG. 12.

Comparison of the PDFs of the relative strengths of the circulation components Γ x / | Γ | (a), Γ y / | Γ | (b), and Γ z / | Γ | (c), between inlet and outlet of the contractions. The circulations are measured with loop sizes r 36 η. The components are normalized with the instantaneous circulation strength is | Γ | = ( Γ x 2 + Γ y 2 + Γ z 2 ) 0.5. The red and black curves correspond to circulation computed in loops centered along the centerline and at the streamwise locations x / H = 0.2 and 0.85 respectively. The PDFs are computed using 100 bins.

Close modal

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 λ 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.

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.

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

The authors have no conflicts to disclose.

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).

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

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