23ème Congrès Français de Mécanique

Lille, 28 au 1er Septembre 2017

Small scale statistics of turbulent fluctuations close to a stagnation point. P. D. Hucka , N. Machicoaneb and R. Volka a. Laboratoire de Physique de l’École Normale Supérieure de Lyon, Université de Lyon, 46 allée d’Italie, 69007, Lyon b. Multiphase and Cardiovascular Flow Lab (University of Washington) 4000 15th Ave NE University of Washington Box 352 600 Mechanical Engineering MEB 143 Seattle WA 98195-2600 - USA

Abstract : Experimental data measured with a 3d Shadow-Particle Tracking Velocimetry (S-PTV) setup in fully developed turbulence (Reλ = [175 − 225]) is presented. The underlying flow is of the von Kármán type and as with other similar flows, a bistability is measured. A protocol is presented to isolate each distinct stable state which reveals a stagnation point topology in the measured region of interest. Tracer particle trajectories permit the investigation of the inhomogeneity and anisotropy of the smallest scales, namely acceleration statistics. The local turbulent dissipation is shown to mimic the large scale inhomogeneity of the flow, setting the acceleration magnitude as described by the Heisenberg-Yaglom prediction. We demonstrate that this prediction is insufficient to determine the anisotropy of small scales and propose a time scale to predict anisotropy in the acceleration magnitude and the acceleration integral time.

Key Words: Turbulence; Lagrangian statistics; Acceleration measurements

1

Introduction

Inhomogeneity in fluid flows is inherent in natural [1] and industrial contexts [2] with examples in the free shear and convection of the planetary boundary layer [3] or rotation in stirred chemical reactors [4] and compression in piston engines [5]. Among the canonical flow types investigated in the literature, strain is an important mechanism that has been used to investigate the link between the imposed meanfield and the resulting anisotropy [6] [7] and theoretical formulation of the role of rapidly applied strain to turbulence succeeded in providing a mechanism to predict anisotropy [8]. Recent experimental [9] and numerical [10] investigations have demonstrated that accurate prediction is limited to larger scales when applied deformation is not sufficiently rapid to attain the small dissipative regions of the turbulence. Only recently have simulations investigated the implications of strain on the small dissipative scales of turbulence [11] demonstrating stronger root-mean-square (rms) acceleration in compressed directions than in divergent directions in asymmetric strain. Interestingly, recent study has not lead to conclusive explanations of the role such flow conditions have on dissipative scale temporal dynamics. In this contribution we build upon these concepts with an experimental Lagrangian investigation of fully developed turbulence (Reλ = [175−225]). In particular, a highly inhomogeneous and anisotropic von Kármán type flow is studied. Though often thought to belong to the free-shear category [12], recent study has pointed to the fundamental role of the stagnation point at the center of the flow [13]. By way of analogy with the characterization of spatial velocity gradients (eulerian Taylor scale, λ)

23ème Congrès Français de Mécanique

Lille, 28 au 1er Septembre 2017

[14], the lagrangian Taylor scale [15] is used to characterize high frequency motion and provides a reasonably accurate estimation of acceleration anisotropy, both in terms of temporal correlation and amplitude. The paper is organized as follows: the experimental apparatus and techniques are discussed in section 2. Statistics indicative of inhomogeneity in the velocity fluctuations and turbulent dissipation as well as anisotropy in acceleration magnitude and time scales are discussed in 3. A discussion of these results and a prediction of the maximum anistropy is presented in section 4 and finally, conclusions are given in section 5.

2

Experimental set-up

The device used in this investigation is the so-called von Kármán flow which consists of a square cylindrical enclosure, 15 cm on each side, with two counter-rotating disks of radius R = 7.1 cm driven at equal rotation frequencies by constant-current motors that are seperated by 20 cm, as depicted in figure 1(a). The advantage of this apparatus is that a strong inhomogeneous and anisotropic turbulence is created in a confined volume. Variations of this set-up have been used in a multitude of contexts, from super fluid turbulence [16] and magneto hydrodynamics [17] to Particle Tracking Velocimetry (PTV) in fluid flows [18][19]. Our experiments rely on a Shadow-Particle Tracking Velocimetry [20] where two perpendicular collimated LED beams permit the tracking of small objects over a large volume[13] approximately (6cm)3 (figure 1(b)). Trajectories are reconstructed using typical particle tracking algorithms [21] applied to films obtained with two high speed cameras (PhantomV.12, Vision Research, 1Mpix@7kHz) with a resolution 800x768 pixels, and a frame rate of fs = 12 kHz. The velocimetry and accelerometry measurements contain noise inherent to the indispensible step of particle detection. This noise propagates through, and is amplified by, the progressive differentiations of position trajectories to give the velocity and acceleration. These measurements have a signal-to-noise ratio of nearly 30 and necessitate a temporal filter with a gaussian kernel [21] which are applied to particle trajectories. A vast literature exists on the bistable nature of the von Kármán flow (e.g. [22][23]) measured primarily in round cylinder geometries. The present square cylinder manifests a unique bistability; measurements have shown the central region of the device to have a stagnation point structure with a strong converging component and two weaker diverging components. The orientation of these components may suddenly undergo a discrete 90° rotation as depicted in figure 1(c-d). Such bistability could prove problematic if typical bistable state lifetimes are shorter than the filming times (∼ 1 s). In order to determine the residence times of the bistable states Laser Doppler Velocimetry (LDV) measurements where carried out in a region slightly removed from the stagnation point so as to pick up either the strongly converging or weakly diverging velocity component. The result in figure 2(a) depicts the raw LDV signal (red) and low-pass filtered version of the signal (black) which shows alternation between a weakly diverging state (V ' 0) and and a strongly diverging state (V ' 1 m/s). For Reynolds numbers (Reλ = 600) slightly larger than those considered in the S-PTV experiments, resident times are at least an order of magnitude larger than the filming time. Thus, a collection of trajectories resulting from a single film provide a glimpse of the flow trapped in either one of the bistable states. In order to exploit the S-PTV data films are separated into two ensembles, one for each bistable state. To accomplish this we measure the kinetic energy contained in each velocity component averaged over the ensemble of trajectories measured in a single film, 2 vrms,i (t)

Np  1 X  p 2 = (vi ) − (vip )2 , Np p∈f ilm

i = x, y, z

(1)

23ème Congrès Français de Mécanique

Lille, 28 au 1er Septembre 2017

c)

a)

xy

y y z

zx z

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x

d) 0.04 0.03

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z

-0.04 -0.02

x

0

-0.02

0

0.02

0.02 0.04

0.04

z (m)

x (m)

Figure 1: Experimental apparatus. (a) The square cylindrical enclosure of the von Kármán flow consists of two counter rotating disks driven at equal rotational frequencies. (b) Particle tracking permits the study of trajectories conditioned by their proximity to a given point and the influence of the stagnation point. Here trajectories all have passed within the red hoop before exploring a larger volume. (c & d) Bistability is detected in the flow (see figure 2) and conditions the orientation of the mean flow at the stagnation point. a)

b) 0.6

vrms,i

0.55 0.5 0.45 0.4 0.35 0.3 0

c)

50 time(min)

e)

d)

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0.02

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−0.03 −0.02

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z (m)

Figure 2: Velocity statistics. (a) LDV measurement of the x velocity component (placed near the center of the red hoop in figure 1) demonstrates bistability for Reλ =?. (b) Velocity rms values over all trajectories in a single film. Symbols, #: i = x, 2: i = y, : i = z (c,d,e) Mean fields calculated in three planes (Π(xy), Π(zx) & Π(zy)) over an ensemble of trajectories in the x-dominant state. where Np indicates the number of trajectories in a film and • is an averaged quantity over a single

23ème Congrès Français de Mécanique

Lille, 28 au 1er Septembre 2017

trajectory. The result of this calculation for sequence of films separated by a data transfer time of approximately 5 minutes is given in figure 2(b). The z (axial) component remains constant while the x and y (radial) components alternate between strong and weak values, but are never equal, evidence of the bistability measured by LDV. By a simple threshold operation, two ensemble are created one where the x component is dominant and vice-versa. In the analysis to follow the ”x-dominant" state will be investigated. To understand the nature of the underlying
topology of the flow considered, the mean velocity field in 3d h~v i(x, y, z) = hvx i, hvy i, hvz i has been calculated by an Eulerian conditioning of the Lagrangian trajectory ensemble on a 123 cartesian grid, approximately 5mm in each direction. The result for three orthogonal, cartesian planes is given in figure 2(c-e) where the stagnation point topology is evident in the Π(xy) and Π(zx) planes where the strong converging direction is present. The third plane, Π(zy), demonstrates that the stagnation point has the form of two impinging jets (oriented along x) with diverging components in the y and z directions. We note that the volume over which these statistics are calculated is larger than the Eulerian integral scale L = u03 /ε = 4.8 cm, permitting an investigation of their inhomogeneity. Various Eulerian statistics are given at the geometrical center of the flow in table 1. Ω Hz 4.2 5.5 6.9

a0x m.s−2 33.5 60.0 98.8

a0y m.s−2 28.8 52.9 90.6

a0z m.s−2 24.9 45.4 76.4

a0 m.s−2 29.3 53.1 89.1

τ Lx ms 18.3 12.8 10.2

τ Ly ms 12.7 9.4 6.9

τ Lz ms 13.6 10.0 7.5

τη ms 3.2 2.1 1.5

η µm 162 131 111

εm W.kg−1 0.8 1.9 3.6

hv 0 · a0 i W.kg−1 1.5 2.8 6.4

Reλ 175 200 225

Table 1: Parameters of the flow. Ω, rotation rate of the discs; ε, dissipation rate obtained from the power consumption of the motors. Surrogate values for ε are given by hv 0 · a0 i. The rms accelerations are obtained at the geometrical center of the flow using data points located in a sphere with a 1.5 cm −6 m2 s−1 with a density radius. The kinematic viscosity of the water-UconTM mixture q is ν = 8.2 10

ρ = 1000 kg m−3 . The Taylor time scales are τLi = 2vi2 /a2i (no summation). The dissipative p time-scale and length-scaleq are τη = ν/ε and η = (ν 3 /ε)1/4 , the Taylor-based Reynolds number q being estimated as Reλ = 15u0 4 /νε with v 0 = (vx0 2 + vy0 2 + vz0 2 )/3. The large scale Reynolds number is Re = 2πR2 Ω/ν=[1.62, 2.12, 2.67]×104 .

3

Inhomogeneity and anisotropy in small scale statistics

The stagnation point in figure 2(c-e) is responsible for turbulence generation and leads to large scale inhomogeneity and anisotropy in the velocity fluctuations. Similar behavior in the smallest scales, measured by acceleration statistics, is the subject of the sections that follow. To understand the origin of large scale anisotropy, the Turbulent Kinetic Energy (TKE) budget was investigated in a previous work [13] and is presented summarily. The contracting direction (x) contributes to the " positive" production of TKE where energy in the mean flow is transformed into turbulent velocity fluctuations. The weaker dilating directions (y and z) contribution to "negative" production where energy in velocity fluctuations is redistributed to the mean flow. Overall, the former process dominates the latter resulting in an increase in the total velocity fluctuations v 02 as particles approach the stagnation point (figure 3(a)), indicating spatial inhomogeneity. Anisotropy is measured by the ratio hv 02 x i/hv 02 z i and nearly doubles as the stagnation point is reached. The TKE balance also contains turbulent dissipation which is estimated by the following scalar product: hvi0 a0i i ' −ε, (2)

23ème Congrès Français de Mécanique

Lille, 28 au 1er Septembre 2017

1.05 2 1 − h v i0 a 0i i /| h v i0 a 0i i 0 |

0.9 0.85 0.8 0.75

3.5 h v x0 2 i /h v z0 2 i

h v 0 2 i /h v 0 2 i 0

0.95

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−0.02

−0.02 0 x (m)

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0.02

Figure 3: Indication of inhomogeneity around the stagnation point at Reλ = 200 for velocity fluctuations and dissipation surrogates. (a) Overall increase in velocity fluctuations ( v 02 = (vx0 2 + vy0 2 + vz0 2 )/3) are normalized by maximum value located at the geometrical center hvi0 2 i0 . (inset): Measure of maximum anisotropy between the converging (vx0 2 ) and diverging (vz0 2 ) fluctuations. (b) Measurement of components and total dissipation as estimated by the expression in equation 2. # : hvx0 a0x i, 2 : hvx0 a0x i,
: hvz0 a0z i, +: hvi0 a0i i (summation implied over i). where summation is implied over the i indices. Normally this quantity contains contributions from pressure transport terms, though these are expected to be 50% of the total value and are neglected. Turbulent dissipation seeks to moderate the increasing strength of the velocity fluctuations and approximately mirrors the growth of v 02 . Figure 3(b) displays the various components of hvi0 a0i i as well as sum of the scalar product. The contrasting roles of "positive" (respectively "negative") production in the x (resp. z) direction are reflected in the corresponding signs of terms in figure 3(b). Clearly, the x component dominates the dissipative phenomenon and dictates the inhomogeneity of ε. The measured inhomogeneity in ε implies similar behavior in the acceleration variance following the Heisenberg-Yaglom prediction [24][25] : hai ai i = a0 ε3/2 ν −1/2 .

(3)

If this prediction from homogeneous and isotropic turbulence (HIT) is to hold, we should see a concomitant evolution in the acceleration statistics which are addressed in the following sections.

3.1

Inhomogeneity in the acceleration magnitude

We investigate acceleration statistics along trajectories confined to spherical regions with 1.5 cm radii. The various regions follow the mean flow from ~x = (±3, 0, 0)cm to ~x = (0, 0, 0)cm, ending at the stagnation point. Figure 4(a) depicts the acceleration component magnitude compensated by qa = (a0x 2 + a0y 2 + a0z 2 ), evaluated at the geometrical center. This normalization takes into account the acceleration magnitude’s dependence on the Reynolds number, and emphasizes the inhomogeneity of the acceleration field. As opposed to the increasing velocity anisotropy, the three components of the fluctuating acceleration increase in equal proportions and anisotropy is constant over the entire region of figure 4(a). Interestingly, the anisotropy a0x /a0z ' 1.68 is slightly larger than was measured in the Cornell University experiment [26], though variations are well accounted for if the x and y directions are averaged as would be the case for a statistical mixture of the x and y dominant states. The spatial profile of acceleration fluctuations attains a maximum value at the stagnation point which is coherent with the dissipation dependance of the Heisenberg-Yaglom prediction (eq. 3).

23ème Congrès Français de Mécanique

Lille, 28 au 1er Septembre 2017

3

0.45

2.8 0.4

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0.35

a0

a0i 2 =qa

2.2 0.3

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a0x2/qa

0.2

a0y2/qa a0z2/qa

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0

1.2 0.01

0.02

1

-0.02

x (m)

-0.01

0

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x (m)

Figure 4: Acceleration magnitude statistics near the stagnation point. # : Reλ = 175, 2 : Reλ = 200,
: Reλ = 225. (a) Normalized acceleration variance where qa = (a0x 2 + a0y 2 + a0z 2 ) is evaluated at (0,0,0). (b) Heisenberg-Yaglom prediction of fluid particle acceleration using local dissipation values. ; a0y ; a0z . Acceleration components: a0x Using this prediction we demonstrate the monotonic Reynolds number dependancy of these small scales by evaluating a0 in equation 3 using local values of the dissipation [27]. Figure 4(b) indicates the same hierarchy in fluctuating acceleration as figure 4(a) such that a0x > a0y > a0z . Following the increase in plateaus for a given acceleration component, the Reλ dependence of a0 is observed. However, the values obtained are slightly lower than similar Reynolds number experiments which may be attributed to the the pressure-velocity correlation term which is missing from equation 2 which effectively overestimates ε. We note that no experimental techniques have been able to measure this term and it is estimated to be about 50% of the total measured by hvi0 a0i i (summation implied). Interestingly, the normalized curves are approximately constant over nearly an integral length scale L. This is consistent with the spatially invariant Reλ deduced from hot-wire measurements in lowtemperature helium gas experiments using a similar counter rotating disk device [28]. Furthermore, constant a0 in the central region indicates that the Heisenberg-Yaglom scaling is accurate in this region and that the dissipation ε is the determining term in the acceleration magnitude. The upward turning tails of figure 4(b) may indicate that competing terms, pressure-velocity transport for instance, become important further from the stagnation point. The observed scalings for the acceleration variance indicate that K41 arguments may be valid in regions that are not necessarily homogeneous or isotropic but undergo strong turbulence production. Though the spatial evolution of the small scales may be understood in this context, neither the small scale anisotropy nor time scales are predicted from equation 3, and are the subject of the following section.

3.2

Acceleration time scales and anisotropy

We use the same trajectory sub-ensembles as in section 3.1 to calculate temporal small-scale statistics. The acceleration auto-correlation function is a useful statistical tool to determine the period over which particles "forget" the acceleration felt at a previous time, and is typically written as, Rai (τ ) =

ha0i (t)a0i (t + τ )i . ha02 i

(4)

23ème Congrès Français de Mécanique

1.6

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=a (8=")!1=2

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Lille, 28 au 1er Septembre 2017

0.02

x (m) 10 0

0.2

0

Re 6 =175 Re 6 =200 Re 6 =225

0

1

2

3

4

5

10 -1

6

= ==2

10 0

10 1

!=2

Figure 5: Acceleration correlation and spectra at the stagnation point. # : Reλ = 175, 2 : Reλ = 200,
: Reλ = 225. (a) Acceleration correlation function (x&z components) near the stagnation point for all Reynolds numbers. The color scheme is the same as 4, for clarity the y component has not been included. Inset: acceleration integral time normalized by the dissipation as estimated by hv 0 · a0 i. The black curves are averages of the three Reynolds numbers to serve as a reference. (b) Unfiltered acceleration spectra at the stagnation point normalized using the dissipation as estimated by hv 0 · a0 i. The color scheme is the same as (a) and the y component has been omitted for clarity. Figure 5(a) plots the auto-correlation function versus normalized time τ /τη where τη = (ν/ε)1/2 takes the Reynolds number dependence into account and permits a collapse of the three curves. The behavior at small times is isotropic and no difference can be made between i = x and i = z. However, at the zero-crossing t0 enough time has elapsed for the x and z components to develop significant anisotropy. To estimate the characteristic time of acceleration auto-correlation, τa , one typically takes the integral of equation 4 up to the point where Ra crosses the abscissa, t0 : Z t0
i τa = Rai τ dτ. (5) 0

The inset of figure 5(a) displays the normalized acceleration integral time for different points along the aforementioned trajectories. As stagnation point is approached from x ≶ 0 cm the integral times become increasingly anisotropic and attain a maximum at the flow’s geometric center, unlike the acceleration magnitude which maintains a constant level of anisotropy throughout particle transit. Interestingly, τax /τaz > 1 whereas a0x /a0z > 1. This observation is somewhat counter-intuitive when considering the results of homogeneous and isotropic turbulence (HIT). Early simulations [29] calculated equation 4 for a restricted range of small Reλ and found t0 ' 2.2τη . Combined with the K41 phenomenology of τη = (ν/ε)1/2 and equation 3 one would conclude that a2 ∼ 3/2 which leads to a smaller dissipative timescale (τη ) and consequently a smaller zero-crossing (t0 ). However, the opposite is observed; the strongest component (x) has the largest zero-crossing whereas the weakest component (z) has the time-scale. Obviously, the use of phenomenology based on HIT is suspect in a von Kármán flow which is both inhomogeneous and anisotropic. However, the normalization using equation 3 in the stagnation point region of the flow indicates that that ε3/2 provides a normalization that is atleast approximately correct and points to the role of anisotropy present at non dissipative scales in the inequality of ax0 and az0 . A scale by scale investigation of acceleration fluctuations sheds light on this issue, and necessitates

23ème Congrès Français de Mécanique

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the definition of the acceleration spectrum, 2 φa (ω) = π

Z

∞

Rai (τ ) cos(ωτ )dτ.

(6)

0

Figure 5(b) represents the x and z components of the acceleration spectra (eq. 6) normalized by επ −1 which has been used in the literature to account for Reλ dependence [30][18]. The anisotropy is contained in the low frequencies below ωτη ' 1 while higher frequencies in the deep dissipative region (ωτη > 1) become isotropic. Indeed, the familiar definition of the acceleration variance, Z ∞ 02 φa (ω)dω, (7) hai i = 0

leads to the conclusion that contributions to anisotropy are not necessarily confined to the dissipative range time scales and may extend to accelerative motions correlated over times within the inertial range. In the discussion to follow, we provide an argument for the role of the large scale anisotropy, embodied by the three dimensional stagnation point, in setting the hierarchy of acceleration time scales.

4

Discussion

In the previous section it was seen that anisotropy is not strictly a dissipitive scale phenomenon, and that motion in the intertial range may also contribute to its existence. In a Lagrangian scenario there is a time scale inbetween the smallest turbulence timescales τη and the largest time of persistant particle motion, the Lagrangian integral time TL , which we will refer to as the lagrangian Taylor scale τL (not to be confused with the eulerian Taylor microscale λ) [31] and is described below. Given a stationary and turbulent fluid flow, a velocity auto-correlation analogous to equation 4 is defined, Rv (τ ). At short times it may be written: Rv (τ ) = 1 −

 τ 2 , τL

(8)

where the odd ordered terms disappear under the assumption of stationnarity and 1/τL2 = −1/2 dτ 2 RL (τ ). Equation 8 is the "osculating” parabola [14][15] that describes the small time approximation of RL (τ ) and gives the Taylor scale τL at its passage through the abscissa. This time scale may be approximated by considering the following kinematic relationship, ha0i 2 iRa (τ ) = −hvi0 2 i

d2 RL (τ ). dτ 2

(9)

The Lagrangian Taylor scale τLi of the i component is then written: 2 τLi =2

hvi0 2 i ha0i 2 i

(10)

where the second order temporal derivative has been substituted following equation 8 and no summation is implied over i. By analogy with the Eulerian Taylor scale, equation 10 is the time-scale of the temporal velocity derivative and justifies its use for describing accelerative phenomena. We seek to uncover the determining factors in the anisotropy of the acceleration integral time τa . Although τL is not the same as τa , the latter is only slightly larger than the former, i.e. in the central region τai ' [1.25 − 1.45]τη and τLi ' [4 − 6]τη depending on the component observed. As such, τL falls squarely in the region of frequencies most strongly contributing to the anisotropy as seen in the spectra of figure 5(b). Anisotropy can thus be estimated: τax /τaz ' 1.13 as measured by the integral time scales in figure 5(a inset) while τLx /τLz ' 1.33 as estimated by eq.10. The Taylor timescale

23ème Congrès Français de Mécanique

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over-predicts the anisotropy by roughly 20% which is to be expected as the integral timescale (τa ) naturally includes the isotropic of the auto-correlation function (τ /τη < 1). By contrast, the Taylor scale (τL ) is most influenced by the anisotropic regions of the acceleration spectra (ωτη > 1) explaining the over-prediction of anisotropy. One may also apply the above analysis to predicting the acceleration anisotropy. To do so one may take τLi ' τa for the price of the 20% predictive error above, which gives:  a0 2 x a0z

=

 v 0 2  τ x vz0

ax

τaz

−2

,

(11)

where the left hand is 1.75 and is over-predicted (2.4) with a similar error of about 30% for all Reynolds numbers. The Taylor time scale contains information on higher frequency motions, as implied by the presence of ha0i 2 i, while at the same time containing larger scale containing a large scale motion, evident from the role of hvi0 2 i. The presence of these two terms contains information on the anisotropy at asymptotically large reynolds numbers. In a similar flow velocity anisotropy has been shown to decay slowly to just below vx0 /vz0 ' 1.5 at Reλ ' 1000 [26] while normalized acceleration amplitude decays to a0x /a0z ≤ 1.1 and may even eventually reach near isotropy. However, as shown above, time scales imply interaction of both large (velocity) and small (acceleration) scales. Consequently, the persistant anisotropy in the large scales resulting from the presence of the stagnation point inhibit isotropization of time scales at large reynolds numbers. Indeed, Lagrangian measurements spanning Reλ = [450 − 810] indicate very little evolution in τax /τaz [32].

5

Conclusion

This article presented an experimental investigation of Lagrangian data in the fully developed turbulence of a von Kármán flow . Use of LDV measurements confirmed the presence of a bistability with residence times, at minimum, an order of magnitude longer than the PTV films used. By calculating the kinetic energy contained in velocity components, averaged over all the trajectories in a single film, an ensemble of trajectories was constructed that is said to belong to either the x or y dominant state. Restricting the data analysis to the former state, the turbulent velocity fluctuations were seen to be strongly inhomogeneous close to the stagnation point. Additionally, anisotropy measured between the converging and weakest diverging direction increased monotonically. The turbulent dissipation rate was observed to be dominated by the converging direction and mirrored the evolution of the turbulent velocity fluctuations. Interestingly, the acceleration variance followed the tendency of the turbulent dissipation and velocity fluctuations to increase, though unlike the latter, acceleration anisotropy remained almost constant. The Heisenberg-Yaglom prediction held in the central region of the flow indicating that dissipation is the dominant parameter in determining acceleration magnitude despite the inhomogeneity and anisotropy of the flow. Although the verification of this classic scaling law is encouraging, it is incapable of predicting anisotropy among the acceleration components. To address this issue, the Lagrangian Taylor (τL2 = 2v 02 /a02 ) scale was proposed and addressed small scale temporal dynamics. The presence of v 0 in equation 10 suggests the presence of lower frequency contributions to accelerative motion and measurements of acceleration spectra confirm that frequencies close to ω = 2π/τL contribute most strongly to anisotropy in the acceleration magnitude. In contrast, the deep dissipative region (ωτη > 1) becomes isotropic. Similar behavior is apparent in the acceleration auto-correlation which by consequence lead to anisotropic measurements of its integral scale τa . Predictions of anisotropy in acceleration time scales and magnitudes based on the definition of τL are within 20-30% of measured values. The literature contains sparse discussion of the anisotropy of small scale staistics and this study proposes a framework in which they may be understood. The derivation of τL follows from the kinematic relationship between velocity and acceleration autocorrelation functions and is expected to not

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only hold for the fluid particle tracers studied here, but for particles whose dynamics are dominated by their inertia. Further study into the effects of inhomogeneity and anisotropy for these particle classes is of great interest for the atmospheric dispersion of pollutants [33][27] and the process of rain and ice formation [34].

References [1] Csanady G.T. Turbulent diffusion in the environment. D. Reidel Publishing Company, Dordrecht, Holland, 1973. [2] Pope S. B. Lagrangian Pdf Methods. Annual Review of Fluid Mechanics, 26:23–63, 1994. [3] Sawford B. L. Lagrangian statistical simulation of concentration mean and fluctuation fields. Journal of Climate and Applied Meteorology, 24:1152–1166, 1985. [4] Paul E.L. , Atiemo-Obeng V.A. and Kresta S.M., editors. Handbook of Industrial Mixing: Science and Practice. John Wiley and Sons, Inc., 2004. [5] Breuer S. ,Oberlack M., and Peters N. Non-isotropic length scales during the compression stroke of a motored piston engine. Flow, Turbulence and Combustion, 74(2):145–167, 2005. [6] Comte-Bellot G. and Corrsin S. The use of a contraction to improve the isotropy of gridgenerated turbulence. Journal of Fluid Mechanics, 25(04):657, 1966. [7] Tucker H. J. and Reynolds A. J. The distortion of turbulence by irrotational plane strain. Journal of Fluid Mechanics, 32:657, 1968. [8] Sagaut P. , Cambon C. Homogeneous Turbulence Dynamics. Cambridge University Press, 2008. [9] Ayyalasomayajula S. and Warhaft Z. Nonlinear interactions in strained axisymmetric highReynolds-number turbulence. 566(2006):273, 2006. [10] Clay M. P. and Yeung P. K. A numerical study of turbulence under temporally evolving axisymmetric contraction and subsequent relaxation. Journal of Fluid Mechanics, 805:460–493, 2016. [11] Lee C.-M. , Gylfason Á. , Perlekar P. and Toschi F. Inertial particle acceleration in strained turbulence. Journal of Fluid Mechanics, 785:31–53, 2015. [12] Marie L. and Daviaud L. Experimental measurement of the scale-by-scale momentum transport budget in a turbulent shear flow. Phys. Fluids, 16(2), 2004. [13] Huck P.D., Machicoane N. and Volk R. Production and dissipation of turbulent fluctuations close to a stagnation point. Submitted: Phys. Rev. Fluids., 2017. [14] Taylor G.I. Statistical Theory of Turbulence. Proc. Roy. Soc. A, 151(873):421–478, 1935. [15] S. Corrsin and M. Uberoi. Diffusion of heat from a line source in isotorpic turbulence. National Advisory Committee for aeronautics Tech Reports, 1142(1142), 1953. [16] Rousset B. et. al. Superfluid high Reynolds von Kármán experiment. Review of Scientific Instruments, 85, 2014. [17] Monchaux et. al. Generation of a magnetic field by dynamo action in a turbulent flow of liquid sodium. Phys. Rev. Lett., 98, 2007. [18] Ouellette N.T. , Xu H. , Bourgoin M. and E. Bodenschatz. Small-scale anisotropy in Lagrangian turbulence. New Journal of Physics, 8, 2006.

23ème Congrès Français de Mécanique

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[19] Mordant N. , Leveque E. and J. F. Pinton. Experimental and numerical study of the Lagrangian dynamics of high Reynolds turbulence. New Journal of Physics, 6:116, 2004. [20] Huck P.D., Machicoane N. and Volk R. A Cost-efficient Shadow Particle Tracking Velocimetry Setup Suitable for Tracking Small Objects in a Large Volume. Procedia IUTAM, 20:175–182, 2017. [21] Ouellette N.T. , Xu H. and E. Bodenschatz. A quantitative study of three-dimensional Lagrangian particle tracking algorithms. Experiments in Fluids, 40(2):301–313, 2006. [22] A. De La Torre and Burguete J. Slow dynamics in a turbulent von K??rm??n swirling flow. Physical Review Letters, 99(5):3–6, 2007. [23] Ravelet F. , Chiffaudel A. and Daviaud F. Supercritical transition to turbulence in an inertially driven von Kármán closed flow. Journal of Fluid Mechanics, 601:339–364, 2008. [24] Heisenberg W. Zur statistischen theorie der kernreaktionen. Zeitschrift fur Physik, 171:379–402, 1948. [25] Yaglom A.M. On acceleration field in a turbulent flow. Dokl. Akad.Nauk. SSSR, 69:531, 1949. [26] Voth Greg A. , La Porta A. , Crawford A. M . , Alexander J. and Bodenschatz E. Measurement of particle accelerations in fully developed turbulence. Journal of Fluid Mechanics, 469:121–160, 2002. [27] Sawford B. L. and Pinton J.F. A lagrangian view of turbulent dispersion and mixing. In Katepalli R. Sreenviasan Peter A. Davidson, Yuko Kaneda, editor, Ten Chapters in Turbulence, chapter 4, pages 132–175. Cambridge University Press, Cambridge, 2012. [28] Zocchi G. , Tabeling P. , Maurer J. and Willaime H. Mesurement of the scaling of the dissipation at high Reynolds numbers. pages 3693–3700, 1994. [29] Yeung P. K. and Pope S. B. Lagrangian statistics from direct numerical simulations of isotropic turbulence. Journal of Fluid Mechanics, 207:531, 1989. [30] Sawford B. L. and Yeung P. K. Kolmogorov similarity scaling for one-particle lagrangian statistics. Physics of Fluids, 23(9):1–5, 2011. [31] Hinze J.O. Turbulence. McGraw-Hill, 1975. [32] Volk R. , Chareyron D. and Pinton J. Mesures d’accélération lagrangienne dans un écoulement anisotrope par vélocimétrie laser Doppler étendue. 20ème Congrès Français De Mécanique, pages 1–6, 2011. [33] Toschi F. and Bodenschatz E. Lagrangian Properties of Particles in Turbulence. Annual Review of Fluid Mechanics, 41(September 2008):375–404, 2009. [34] Shaw R.A. Particle-turbulence interactions in atmospheric clouds. Annu. Rev. Fluid Mech., 37:473–491, 2003.