12th International Symposium on Turbulence and Shear Flow Phenomena (TSFP12)
Osaka, Japan (Online), July 19-22, 2022

SCALE-DEPENDENT GEOMETRIC STATISTICS OF THE LAGRANGIANAND EULERIAN ACCELERATIONS IN HOMOGENEOUS TURBULENTSHEAR FLOWFrank G. Jacobitz
Department of Mechanical Engineering, University of San Diego
5998 Alcalá Park, San Diego, CA 92110, USAjacobitz@sandiego.edu

Kai Schneider
Institut de Mathématiques de Marseille (I2M), Aix-Marseille Université, CNRS
39 rue Joliot-Curie, 13453 Marseille cedex 13, France
kai.schneider@univ-amu.fr

ABSTRACT

Tennekes, 1975) acceleration fluctuations and their different
contributions have been studied in Pinsky et al. (2000) andTsinober et al. (2001) for isotropic turbulence. Their work
is based on the prediction of Tennekes (1975), which statesthat the Lagrangian acceleration must be small, justified by
the consideration of Eulerian and Lagrangian time scales.
Homogeneous turbulent shear flow has been investigatedas a prototypical example of turbulence due to the importanceof shear production in the geophysical environment and inmany engineering applications. In homogeneous flows, thestatistical properties of turbulence do not change in the spatial
directions (x, y, and z), but they evolve in time t. While thissimplification has been used extensively as the basis of numerical simulations, homogeneous turbulent shear flow was firststudied experimentally. Rose (1966) and Champagne et al.
(1970) are credited with the first experimental studies of homogeneous turbulent shear flow. Rohr et al. (1988) established that the eventual evolution of homogeneous turbulent
shear flow follows an exponential growth law. Using direct numerical simulations, Jacobitz et al. (1997) confirmed the exponential evolution and also investigated the impact of buoyancy
forces on the flow. This prototypical flow has been studied
extensively including the effects of buoyancy and rotation as
well as using a variety of statistical methods (e.g., Salhi et al.,2014).
Motivated by our recent work in Jacobitz & Schneider(2021), in which we studied the Lagrangian and Eulerian acceleration properties of fluid particles in homogeneous turbulence with uniform shear and uniform stable stratification, we
generalize this approach here to investigate scale dependent
geometrical statistics. A wavelet-based scale-dependent decomposition of the Lagrangian and Eulerian accelerations is
performed and the alignment properties are analyzed at different scales of the turbulent motion (e.g., Farge & Schneider,2015). The aim of this work is to address the applicability of
Taylor’s hypothesis at small scales in homogeneous turbulent
shear flow.
The remainder of the manuscript first presents the numerical approach based on direct numerical simulations. Then,
Lagrangian and Eulerian accelerations are introduced, theircorresponding orthogonal wavelet decomposition is presented,

The alignment properties of different vector-valued flow
quantities, including the Lagrangian, Eulerian, and convective
accelerations in homogeneous turbulent shear flow, are quantified through the introduction of scale-dependent geometricalstatistics. The vector fields are decomposed into an orthogonal wavelet series and the angles of the scale-wise contributions of different vector-valued flow quantities can be determined. This approach allows us to revisit the random Taylor
hypothesis by examining the cancellation properties of Eulerian and convective accelerations at different flow scales. The
results for homogeneous turbulent shear flow, computed by direct numerical simulation, show that Taylor’s hypothesis holds
at small scales of the flow as reflected by the anti-alignment of
the Eulerian acceleration and the convective term. Such antialignment, however, is not observed at the largest scales of the
turbulent motion, indicating that Taylor’s hypothesis does not
generally hold for homogeneous turbulent shear flow.

INTRODUCTION
Taylors frozen-ow hypothesis (Taylor, 1938) supposesthat small-scale eddies in turbulent ows move downstream at
a constant speed with little distortion. Temporal and spatial
fluctuations in turbulent flows can thus be related, and, forexample, temporal spectra in experimental measurements canbe converted into spatial spectra. Hot-wire measurements inturbulent flows often implicitly assume its validity. Theoretically, it is also of utmost importance to understand convection
in turbulent flows and the related temporal to spatial intermittency. Moreover, both play an important role in two-point closure turbulence models for space-time correlations in Eulerianand Lagrangian reference frames (He et al., 2017). However,
Taylor’s hypothesis has many limitations, such as the requirements of weak shear rates and low turbulence intensities. Discussions on its validity have a long history and are still subject
to some controversy in the recent literature (e.g., Moin, 2009;
Del Alamo & Jiménez, 2009).
Motivated by the random Taylor hypothesis or sweeping
decorrelation hypothesis, stating that ‘small eddies in turbulent flow being swept past a stationary Eulerian observer,’ (cf.

112th International Symposium on Turbulence and Shear Flow Phenomena (TSFP12)
Osaka, Japan (Online), July 19-22, 2022
1.4

0.6

1.2

0.4

1

K/K0

0.2
0.8
0
0.6
−0.2
0.4
−0.4

0.2
0

γ
P/(SK)−ε/(SK)

−0.6
0

2

4

6

8

10

0

St

2

4

6

8

10

St

Figure 1. Evolution of the normalized turbulent kinetic energy K/K0 with normalized time St (left) as well as the growth rate of the
turbulent kinetic energy γ , the normalized production rate P/(SK), and the normalized dissipation rate ε /(SK) (right).

For defining scale-dependent alignment statistics of theabove quantities, we use a three-dimensional orthogonal
vector-valued wavelet decomposition. To this end we consider
a generic vector field a = (a1 , a2 , a3 ) at a fixed time instant anddecompose each component aα (xx) into an orthogonal wavelet
series,and the scale-dependent alignment of Eulerian and convective accelerations are considered. Results illustrate our findings, considering the direct numerical simulation data of turbulent homogeneous shear flow, and conclusions complete themanuscript.

aα (xx) = ∑ aeλα ψλ (xx),NUMERICAL APPROACH

where the wavelet coefficients are given by the scalar productaeα = haα , ψλ i (e.g., Farge & Schneider, 2015). The wavelets
ψλ with the multi-index λ = ( j, i , d) are well localized in scale
L0 2− j (where L0 corresponds to the size of the computational
domain), around position L0 i/2 j , and orientated in one of theseven directions d = 1, ..., 7, respectively. Reconstructing thethree components aα at scale L0 2− j by summing only over the
position i and direction d indices in eq. (2) yields the vector
field a j at scale index j. By construction we have a = ∑ j a j ,where the a j are mutually orthogonal.
The scale-dependent statistical moments of the flow
fields, including scale-dependent flatness, and scale-dependent
pdfs, can thus be computed from a j using classical statisticalestimators. For instance, the q-th order moment of a j (xx) canbe defined by,
Mq [aa j ] = h(aa j )q i,

The Lagrangian and Eulerian accelerations aL = aE + aC ,where aC denotes the convective contribution, are defined as∂u∂u+ u · ∇uu and a E =
,
∂t
∂t

(3)

and by construction the mean value vanishes with haa j i =0. The moments are thus central moments. These scaledependent moments are directly related to the q-th order structure functions (Schneider et al., 2004), where the increment
size is ∝ 2− j .
The scale-dependent flatness, which measures the intermittency of a j at scale 2− j , is defined by

SCALE-DEPENDENT STATISTICS

aL =

(2)

λ

The present study is based on a direct numerical simulation of homogeneous turbulent shear flow using the incompressible form of the Navier-Stokes equations for the fluctuating components of velocity and pressure with an imposedconstant vertical shear rate S = ∂ U/∂ y. The equations of
motion are transformed into a frame of reference moving
with the mean velocity (Rogallo, 1981). This transformation enables the application of periodic boundary conditions
for the fluctuating components of velocity and pressure. A
spectral collocation method is used for the spatial discretization and the solution is advanced in time with a fourth-order
RungeKutta scheme. The simulations are performed using
5123 grid points. The initial conditions are taken from a separate simulation of isotropic turbulence, which was allowed todevelop for approximately one eddy turnover time. The initialvalue of the Taylor-microscale Reynolds number is Reλ = 89
and SK/ε = 2 for the shear number. The corresponding values
at St = 10 are Reλ = 157 and SK/ε = 5.22. The results areanalyzed at St = 10.

(1)Fl[aa j ] =respectively. Both accelerations are computed as a volume average at a fixed time, which is an appropriate choice for homogeneous flows. The effect of shear is considered as an external
force. The pressure-gradient term is given by a P = ∇ (p/ρ0 ),where ρ0 is the ambient density.

M4 [aa j ](M2 [aa j ])2.

(4)

For a Gaussian distribution the flatness equals three at all
scales.

2

12th International Symposium on Turbulence and Shear Flow Phenomena (TSFP12)
Osaka, Japan (Online), July 19-22, 2022
2

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1

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aC/aCrms

aC/aCrms

1

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0
0.6
−0.5

0.2
0
0.15
−0.5

0.4

−1
−1.5
−2
−2 −1.5 −1 −0.5 0 0.5
aE/aErms

1

1.5

0.2

−1.5

0

−2

0.05
0
−2 −1.5 −1 −0.5 0 0.5
aE/aErms

0.6

1.5

0.1

−1

2

2

0.5

1

1

1.5

2

2

1.4

1.5

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1
0.4

0.5
0

aC/aCrms

aC/aCrms

0.25

0.5

0.3

−0.5

1

0.5
0.8
0
0.6
−0.5

0.2

−1

0.4

−1
0.1

−1.5
−2

0
−2 −1.5 −1 −0.5 0 0.5
aE/aErms

1

1.5

0.2

−1.5
−2

2

0
−2 −1.5 −1 −0.5 0 0.5
aE/aErms

1

1.5

2

Figure 2. Joint probability distribution functions (pdfs) of the the Eulerian acceleration a E and the convective acceleration aC at
nondimensional time St = 10 for the total fields (top, left) and for the decomposed fields at scales j = 2 (top, right), j = 3 (center, left),j = 4 (center, right), j = 6 (bottom, left), and j = 8 (bottom, right).

To understand the magnitude of the Eulerian and Lagrangian accelerations, we statistically assess, following Tsinober et al. (2001), the alignment properties of a E , aC , andits sum, corresponding to the Lagrangian acceleration aL =a E + aC . When the vectors of the Eulerian acceleration a E andthe convective acceleration aC are anti-parallel, then the magnitude of the Lagrangian acceleration a L , is small compared tothose of the Eulerian and convective contributions, sinceof alignment yields a measure for how well Taylor’s approximation holds at different scales of motion, as detailed below.

RESULTS AND DISCUSSION
Flow Evolution
Fig. 1 (left) provides the evolution of the turbulent kinetic
energy K = 1/2huu, u i normalized by its initial value K0 . The
turbulent kinetic energy initially decays due to the isotropic
initial conditions, starts to grow at about St = 2, and eventually
grows exponentially starting at about St = 4. The evolution
equation for K can be written in the following nondimensionalform:

haaL , a L i = haaE + aC , a E + aC i = haaE , a E i + haaC , aC i
+2 cos(aaE , aC ) ||aaE || ||aaC ||. (5)Applying the wavelet decomposition to the different accelerations in eq. (2), we can study, in addition to the total, also
the scale-dependent alignment properties of the Eulerian andjjconvective accelerations, i.e., cos(aaE , aC ) and check its implications for the Lagrangian acceleration.
jj
We can directly conclude that, if a E and aC are antijaligned and the cosine term is negative, the norm of a L becomes minimal. This means that for perfect anti-alignment
Taylor’s hypothesis would hold and temporal variations thus
correspond to the spatial variations. Quantifying the departure

γ=

1 dK
ε
P
=
−
SK dt
SK SK

(6)

Here, γ is the growth rate of the turbulent kinetic energy,
P/(SK) its normalized production rate, and ε /(SK) the normalized dissipation rate. The evolution of γ , P/(SK), and−ε /(SK) is shown in Fig. 1 (right) and the three terms areapproximately constant for St ≥ 4. Once the normalized production and dissipation rates have reaches constant values, resulting in a constant growth rate γ , eq. (6) can be integrated to3

12th International Symposium on Turbulence and Shear Flow Phenomena (TSFP12)
Osaka, Japan (Online), July 19-22, 2022
Table 3. Scale-dependent flatness of the Eulerian acceleration FlaE as a function of scale j and nondimensional time St
or Taylor-microscale Reynolds number Reλ .

Table 1. Pearson correlation coefficient between the Eulerian acceleration a E and the convective acceleration aC as afunction of scale j and nondimensional time St or Taylormicroscale Reynolds number Reλ .

St

St = 4

St = 7

St = 10Reλ103.43136.40156.90total-0.79688-0.79152-0.79612r( j = 0)-0.20819-0.48831

0.22471r( j = 1)-0.01186-0.17431-0.23597r( j = 2)-0.23966-0.34748-0.36848r( j = 3)-0.52569-0.55758-0.58615r( j = 4)-0.67770-0.68890-0.69352r( j = 5)-0.75734-0.75537-0.75556

r( j = 6)-0.81143-0.80741-0.80288r( j = 7)-0.86320-0.86073-0.85560r( j = 8)-0.89679-0.90279-0.90486

St

St = 4

St = 7

St = 10Reλ103.43136.40156.90

total

10.10

11.96

14.41

Fl( j = 0)

3.71

5.51

5.22

Fl( j = 1)

3.54

3.47

4.09

Fl( j = 2)

4.35

4.36

4.72

Fl( j = 3)

4.70

5.43

5.53

Fl( j = 4)

5.86

6.01

6.55

Fl( j = 5)

7.30

7.70

8.22

Fl( j = 6)

9.69

11.35

12.45

Fl( j = 7)

16.63

22.57

25.92

Fl( j = 8)

32.88

47.88

56.43

Scale-Dependent Statistics
Fig. 2 provides the joint probability distribution functions(pdfs) of the Eulerian acceleration a E and the convective acceleration aC as a function of scale j at nondimensional time
St = 10. While the joint pdf of the total fields (top, left) showssome negative correlation, the joint pdfs clearly become more
and more negatively correlated as the scale j increases and thescale of the motion decreases.
Table 1 shows the Pearson correlation coefficient r between the Eulerian acceleration a E and the convective acceleration aC as a function of scale j and nondimensional time
St. For all times, a negative value of r = −0.8 is found forthe total fields. For a given time St, the value of r decreases
with increasing scale index j or decreasing scale of the turbulent motion. For the smaller scales of the turbulent motion
with scale index j ≥ 4, the results obtained for r at different
times St remain approximately the same. For the larger scalesof the turbulent motion with scale indices 1 ≤ j ≤ 3, the results for r decrease with time St and a stronger anti-alignment
is observed for later times. Only the results for j = 0 show adifferent trend, but only few wavelet modes contribute to thisvalue for r. Table 1 also gives the Taylor-microscale Reynolds
number corresponding to the different times in the flow’s evolution with later times corresponding to higher Reynolds numbers due to the exponential evolution of the flow.
Tables 2 and 3 provide the flatness values for the Lagrangian and Eulerian accelerations, respectively. Both thevalues for the total accelerations and for their scale dependence
are shown. The results are presented at three nondimensionaltimes St = 4, St = 7, and St = 10, corresponding to three different Reynolds numbers Reλ = 103.43, Reλ = 136.40, andReλ = 156.90. Except for the flatness values at the largest
scale of motion ( j = 0), the flatness values increase with decreasing scale of motion (or increasing j). For a given scale,the flatness values of both accelerations increase with increasing nondimensional time St and increasing Reynolds number
Reλ . While the flatness values of the total Lagragian acceleration are always larger than those of the corresponding total
Eulerian acceleration, this result does not hold at all scales ofobtainK = K ∗ eγ St(7)In the following, the flow is analyzed in the exponential growth
regime with St ≥ 4.

Table 2. Scale-dependent flatness of the Lagrangian acceleration FlaL as a function of scale j and nondimensional time St
or Taylor-microscale Reynolds number Reλ .

St

St = 4

St = 7

St = 10Reλ103.43136.40156.90

total

13.47

20.77

27.81

Fl( j = 0)

3.77

6.35

6.72

Fl( j = 1)

3.26

3.63

3.75

Fl( j = 2)

4.30

4.55

4.63

Fl( j = 3)

4.79

5.38

5.93

Fl( j = 4)

5.91

6.29

7.25

Fl( j = 5)

8.17

10.35

12.44

Fl( j = 6)

16.55

25.42

30.54

Fl( j = 7)

45.81

107.90111.72Fl( j = 8)

82.07

298.47327.364

12th International Symposium on Turbulence and Shear Flow Phenomena (TSFP12)
Osaka, Japan (Online), July 19-22, 2022
10

10

total
j=2
j=4
j=6
j=8

8

total
j=2
j=4
j=6
j=8

8

pdf

6

pdf

6

4

4

2

2

0

0
−1

−0.5

0
cos(aE, aC)

10

0.5

1

−1

0
cos(aL, aC)

10

total
j=2
j=4
j=6
j=8

8

−0.5

1

total
j=2
j=4
j=6
j=8

8

pdf

6

pdf

6

0.5

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4

2

2

0

0
−1

−0.5

0
cos(aL, aE)

10

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1

−1

0
cos(aL, aP)

10

total
j=2
j=4
j=6
j=8

8

−0.5

1

total
j=2
j=4
j=6
j=8

8

pdf

6

pdf

6

0.5

4

4

2

2

0

0
−1

−0.5

0
cos(aE, aP)

0.5

1

−1

−0.5

0
cos(aC, aP)

0.5

1

Figure 3. Probability distribution functions (pdfs) of the cosine of the angle between the Eulerian acceleration and the advection term
(top, left), Lagrangian acceleration and the advection term (top, right), Lagrangian and Eulerian accelerations (center, left), Lagrangian
acceleration and the pressure-gradient term (center, right), Eulerian acceleration and the pressure-gradient term (bottom, left), as wellas the advection term and the pressure gradient term (bottom, right) at time St = 10. Note that the pdfs are shown for the total flow
fields (dashed lines), and the flow fields at the scales j = 2 (large scale), 4, 6 and 8 (small scale).

motion. For the larger scales of the motion (0 ≤ j ≤ 4), theflatness values of the two accelerations are similar. For thesmaller scales of the motion (5 ≤ j ≤ 8), the flatness valuesof the Lagrangian acceleration are in some cases substantiallylarger than the corresponding values for the Eulerian accelera-

tion.
These observations are consistent with previous work by
Yoshimatsu et al. (2009) considering homogeneous isotropic
turbulence at a Reynolds number Reλ = 732. It was observedthat the flatness values of the Lagrangian and Eulerian ac-

5

12th International Symposium on Turbulence and Shear Flow Phenomena (TSFP12)
Osaka, Japan (Online), July 19-22, 2022
ACKNOWLEDGEMENTS

celerations ‘increase with scale [index] for the turbulent flow,
but the flatness of the Lagrangian acceleration is one order ofmagnitude larger than the flatness of the Eulerian acceleration,
which shows the extreme intermittency of the former.’ In a
comparison to random fields with phase randomization, it was
found that ‘the flatness remains almost constant, around 5 forthe Lagrangian acceleration and around 6 for the Eulerian acceleration, which confirms that the latter yields a Laplace distribution whose flatness is 6. This proves that the random fields
are non intermittent, as no scale dependence can be observed,
even if their pdfs of acceleration are strongly non-Gaussian.’Fig. 3 shows the pdf of the cosine of the angle between different vector-valued quantities for the total and scaledependent fields for j = 2, 4, 6, and 8 (from large to small
scales). The following observations can be made. The Eulerian and convective accelerations have the tendency of being strongly anti-aligned, reflected in a peak at −1 in the pdf
of cos(aaE , a c ) (top, left). The peak is more strongly pronounced at small scales, and weak or no alignment is foundat large scales. The reason for this observation is that the advection term is essentially a small scale quantity. This finding
supports that Taylor’s hypothesis holds at small scales in homogeneous turbulent shear flow. In contrast, for the pdf ofcos(aaL , aC ) (top, right), we observe a slight probability foralignment, which becomes weaker at small scales. Consequently, the Langrangian and Eulerian accelerations are somewhat aligned (center, left), corresponding to a peak at +1 in
the pdf of cos(aaL , a E ). This behavior becomes stronger forlarge scales. The reason is again that the convective term is asmall scale quantity and thus supports Taylor’s hypothesis.
In the pdf for cos(aaL , a p ) we find likewise strong
anti-alignment, which is most pronounced at intermediate
scales and weakens at the largest and smallest scales (center, right). The alignment of the Lagrangian accelerationand the pressure-gradient confirms the results in Jacobitz &Schneider (2021) that the pressure-gradient is the dominantterm contributing to the Lagrangian acceleration. The pdf ofcos(aaE , aP ) shows some anti-alignment, which is stronger atlarge scales, and no alignment is observed at small scales (bottom, left). A similar weak anti-alignment behavior is found forthe pdf of cos(aC , a p ) (bottom, right).

KS acknowledges support from Agence Nationale de laRecherche, project CM2E (ANR-20-CE46-0010-01), and theFrench Research Federation for Fusion Studies within theframework of the European Fusion Development Agreement(EFDA). FGJ acknowledges support from the University Professor award and the Saber3 cluster computer at the Universityof San Diego.

REFERENCES
Champagne, F. H., Harris, V. G. & Corrsin, S. 1970 Experiments on nearly homogeneous turbulent shear flow. J. Fluid
Mech. 41 (1), 81–139.
Del Alamo, J. C. & Jiménez, J. 2009 Estimation of turbulent
convection velocities and corrections to Taylor’s approximation. J. Fluid Mech. 640, 5–26.
Farge, M. & Schneider, K. 2015 Wavelet transforms and theirapplications to mhd and plasma turbulence: a review. J.
Plasma Phys. 81 (6), 435810602.
He, G., Jin, G. & Yang, Y. 2017 Space-time correlations
and dynamic coupling in turbulent flows. Annu. Rev. Fluid
Mech. 49, 51–70.
Jacobitz, F. G., Sarkar, S. & Van Atta, C. W. 1997 Direct numerical simulations of the turbulence evolution in a uniformly sheared and stably stratified flow. J. Fluid Mech.
342, 231–261.
Jacobitz, F. G. & Schneider, K. 2021 Lagrangian and Eulerian
accelerations in turbulent stratified shear flows. Phys. Rev.
Fluids 6, 074609.
Lin, C. C. 1953 On Taylors hypothesis and the acceleration
terms in the Navier-Stokes equation. Quarterly of Applied
Mathematics 10 (4), 295–306.
Moin, P. 2009 Revisiting Taylor’s hypothesis. J. Fluid Mech.
640, 1–4.
Pinsky, M., Khain, A. & Tsinober, A. 2000 Accelerations inisotropic and homogeneous turbulence and Taylors hypothesis. Phys. Fluids 12 (12), 3195–3204.
Rogallo, R. S. 1981 Numerical experiments in homogeneous
turbulence. Technical Report TM 81315. NASA Ames Research Center, Moffett Field, CA, United States.
Rohr, J. J., Itsweire, E. C., Helland, K. N. & Van Atta, C. W.
1988 An investigation of the growth of turbulence in a
uniform-mean-shear flow. J. Fluid Mech. 187, 1–33.
Rose, W. G. 1966 Results of an attempt to generate a homogeneous turbulent shear flow. J. Fluid Mech. 25, 97–120.
Salhi, A., Jacobitz, F. G., Schneider, K. & Cambon, C. 2014
Nonlinear dynamics and anisotropic structure of rotating
sheared turbulence. Phys. Rev. E 89, 013020.
Schneider, K., Farge, M. & Kevlahan, N. 2004 Spatial intermittency in two–dimensional turbulence: a wavelet approach. In Woods Hole Mathematics, Perspectives in Mathematics and Physics, pp. 302–328. World Scientific.
Taylor, G. I. 1938 The spectrum of turbulence. Proceedingsof the Royal Society of London. Series A-Mathematical andPhysical Sciences 164 (919), 476–490.
Tennekes, H. 1975 Eulerian and Lagrangian time microscales
in isotropic turbulence. J. Fluid Mech. 67 (3), 561–567.
Tsinober, A., Vedula, P. & Yeung, P.K. 2001 Random Taylor
hypothesis and the behavior of local and convective accelerations in isotropic turbulence. Phys. Fluids 13 (7), 1974–
1984.
Yoshimatsu, K., Okamoto, N., Schneider, K., Kaneda, Y. &Farge, M. 2009 Intermittency and scale-dependent statistics
in fully developed turbulence. Phys. Rev. E 79 (2), 026303.

CONCLUSIONS
We performed direct numerical simulation in order to revisit Taylor’s hypothesis in homogeneous turbulent shear flow.
We found strong anti-correlation of the Eulerian acceleration
with the convective acceleration based on the joint pdf and thePearson correlation coefficient. The scale dependence of theresults was studied using an orthogonal wavelet decomposition. We observed that the anti-correlation of the Eulerian acceleration and the convective acceleration is strongest for thesmallest scales of the turbulent pdf the Eulerian accelerationand the convective acceleration. The pdf shows again strong
anti-alignment of the terms, indicating a cancellation between
the two accelerations, which is again strongest at small scales.
However, this result does not hold for the largest scales of the
motion. Hence, the results confirm that Taylor’s hypothesis
holds at small scales and not at large scales of the turbulent
motion, which is in agreement with Lin (1953), who wrotethat ‘there is no general extension of Taylor’s hypothesis tothe case of shear flow.’6