1. Introduction
Turbulence consists of numerous vortices. Therefore, to define—and thus, to identify—vortices is of great importance in understanding turbulence structures and mechanisms. A new vector, named Rortex and later renamed Liutex, has been introduced, whose direction is the local rotation axis and magnitude is twice the angular velocity of the rigid rotation part of fluid motion [
1,
2]. The definition of the Liutex vector is purely kinematic, i.e., based on the velocity gradient tensor
. Given that
has one real eigenvalue,
, with a corresponding real unit eigenvector
and two complex-conjugate eigenvalues
, otherwise
has three real eigenvalues and no rotational motion could be found, the rotational axis has to be in the same direction as the real eigenvector. It is argued that only compression and stretch can be found in the direction of the real eigenvector; thus, any rotational motion has to be in the plane perpendicular to the real eigenvector. Therefore, the direction of the Liutex vector is obtained by
. Both
and
are unit eigenvectors corresponding with the eigenvalue
. To uniquely define the Liutex direction, we require that
, with
as the vorticity vector. In the next step, we determine the magnitude of the Liutex vector by coordinate rotation. First, we rotate the coordinates so that the new
z-axis is in the direction of
. We then rotate the coordinates around the new
z-axis, i.e.,
, and the rotational strength is defined as twice the minimal absolute value of the off-diagonal component of the
upper left submatrix of the velocity gradient tensor. The resulting coordinate is referred to as the principal coordinate in this paper and the quantities evaluated in this coordinate are denoted by a superscript asterisk. The velocity gradient tensor in the principal coordinate had the form:
thus,
could be decomposed as:
where
,
and
represent the local rotational part, the compression/stretching part and the pure shear part of fluid motion, respectively. Here,
represents the angular velocity of the rigid rotation part of fluid motion whilst
,
and
are the residual shear along the three spatial dimensions. Different from Cauchy–Stokes decomposition, which splits the velocity gradient tensor into a symmetric part and an antisymmetric part, this decomposition extracts the pure rotation part, which would not be countervailed by the residual compression/stretching and shear. The magnitude of the Liutex vector, i.e., the rotational strength, is thus defined as twice the angular velocity of the rotational part:
Therefore, the Liutex vector in the principal and original coordinates could be written as
and
, respectively. It is easy to verify in the principal coordinate that the vorticity vector could be decomposed as the sum of the Liutex vector and the pure shear vector, i.e.,
with:
Note that the symbols
and
have been used to represent the rotation part and shear part, respectively, both in the vector form and the tensor form to highlight their correspondence between the two forms. In addition, as the vorticity vector and the Liutex vector are Galilean-invariant [
3], i.e., invariant under coordinate transformation and rotation, Equation (4) could be projected into the original coordinates as
, which is called the RS decomposition of vorticity [
4].
Although the coordinate rotation approach to calculate the Liutex vector is more intuitive, especially in prompting the systematic decomposition of fluid motion in the tensor form (Equation (2)) and vector form (Equation (4)), an explicit formula was derived that is both physically revealing and is proved to be more efficient in terms of computation efficiency [
5]. It was found, by studying the flow pattern induced by the given velocity gradient tensors, that the vorticity in the direction of the rotational axis could be viewed as twice the spatial average angular velocity; on the other hand, the imaginary part of the complex-conjugate eigenvalue could be viewed as the pseudo-time average angular velocity. Combining these two relations, the minimum angular velocity, i.e., the Liutex magnitude
, could be obtained; thus, the Liutex vector could be written as:
which is an explicit formula in the original coordinate to calculate the Liutex vector in terms of the vorticity vector
, the imaginary part of the complex-conjugate eigenvalue
and the real eigenvector of the velocity gradient tensor
. With Equation (6), we could decompose vorticity
into
and
in the original coordinate without explicitly performing a reference frame rotation.
Compared with previous vortex identification methods, the Liutex system has two distinctive features. First, the directional information is included by introducing a vector quantity
, with its direction as the local rotation axis. Intuitively, any rotational motion should have an axis to rotate about, which is generally ignored. Second, the magnitude of the Liutex vector represents the local angular speed of the rigid rotation part of fluid motion; thus, it is free from shear contamination. Six critical aspects of vortex identification, including the absolute strength, relative strength, local rotational axis, global rotation axis, vortex core size and vortex boundary, can be extracted from the flow field based on the Liutex system [
4,
6]. Multiple vortex identification strategies have been introduced based on the Liutex vector, for example, the Liutex magnitude iso-surface, objective Liutex [
7], the Liutex-
method [
8,
9] and the Liutex core line method [
10,
11]; these methods have been proven to be effective in capturing vortices in various flows [
12,
13].
Another notable feature of the Liutex vector is that the streamwise spectrum of the Liutex magnitude follows the −5/3 power law in the buffer layer of boundary-layer turbulence [
14], which reveals that turbulence has structures down to a viscous scale. To better understand the power-law spectrum of Liutex in turbulence, we used data of homogeneous isotropic turbulence (HIT) with
and a turbulent channel flow with
. The rest of the paper is organized as follows.
Section 2 introduces the numerical aspects of homogeneous isotropic turbulence. And the data from a turbulent channel of
are also used to illustrate the universality of the Liutex similarity.
Section 3 focuses on the distribution of the Liutex vector in homogeneous isotropic turbulence, with an emphasis on the power-law spectrum of Liutex. Finally, conclusions are given in
Section 4.
3. Liutex Similarity in Turbulence
Figure 1 shows the distribution of the vorticity component
and the Liutex component
on the part of the plane
. Obviously,
is more concentrated on the islands of high magnitude that are separated by regions of
whereas
is more continuously distributed across the plane. A simple calculation reveals that 40% of the whole computational volume is void of Liutex, i.e., only 60% of the domain has a Liutex magnitude larger than 0. Intuitively, the vortices are generally concentrated phenomena, which seems to agree better with the distribution of Liutex. It should be noted that the magnitude of Liutex was always smaller than the vorticity, based on Equation (6). However, this relation does not extend directly to the component because the directions of the Liutex and vorticity are generally different. For example, about 7% of the whole domain in this snapshot is equipped with a
larger than
. Nonetheless, we could claim that for homogeneous isotropic turbulence, the magnitude of the vorticity component is generally larger than that of the same Liutex component as this was true for the other 93% of the volume. It can be observed from
Figure 1b that, generally, an island with a positive
concentration (counter-clockwise rotating from the perspective of
Figure 1b) is surrounded by islands of multiple negative
concentrations (clockwise rotating) and vice versa. The probability density functions (PDFs) of
and
are shown in
Figure 2a. Note that there was only 60% volume when
was taken into account when obtaining the PDF of
. Even without taking the area of
into account, we could see that the PDF of
is significantly more concentrated near 0 than that of
; i.e., compared with
,
has a greater probability of being near zero and seldom excurses to large values. In total, 33% of the non-zero
is in the range
(nearly 60% of the whole volume if the zeros were taken into account) whereas only 19% of
is in the range
.
Despite the general coincidences of the local magnitude maximum in
Figure 1, the distributions of
and
are rather different. The term “intermittency” is often used to refer to the uneven scattering of turbulent fluctuations both in space and time, and has been argued to be a reflection and indicator of coherent structure motion [
20]. Here, we adopt the definition by Jiménez [
21]. For a quantity
with a probability density function (PDF) of
, the volume fraction of the data above a threshold
is defined as:
and the fraction of
above that threshold is defined as:
with
as the average defined by
; thus,
could be viewed as the fraction of the contribution from
greater than
to the average
. If a threshold
could be obtained such that
is relatively small whilst
is relatively large—i.e., the average
is relatively determined by a small fraction of concentrated large
—the variable
is then considered to be intermittent.
Figure 2b provides the cumulative property fraction
against the cumulative probability
of the vorticity magnitude
and the Liutex magnitude
, with or without accounting for
areas. Obviously,
is more intermittent than
. As indicated by the grey patch, we can expect the isolation of intense structures, based on
accounting for approximately 75% of the energy with a relatively small volume fraction of 20%. Ignoring the zero values of
leads to a decrease in the cumulative property fraction
to 55% for the volume fraction
of 20%, which is still higher than that of the vorticity magnitude
at 45%. The reason that the
of
approaches 1 as
approaches 0.6 is again that 40% of the
values are zero. The intermittency of the Liutex magnitude indicates that strong rotational motion is concentrated in a small volume fraction; thus, an iso-surface of the Liutex magnitude with an appropriate threshold could capture the majority of the vortical structures. Iso-surfaces of
colored by
are shown in
Figure 3 to represent the vortices in the flow. The strong vortex cores, vortex rings/arcs and longitudinal vortices, which are typically found in turbulence, are well-captured. Intermittency of the Liutex magnitude or the vortices are observed; a greater number of vortices are found in the upper half (
) than in the lower half (
) in the shown domain. Those vortices could be viewed as skeletons of the flow and served as the build-up elements of turbulence.
To study the relationship between the Liutex and vorticity inside the vortices, especially the vortex cores, we define the partial average
and
as functions of a selected threshold,
:
thus, the ratio
defines the rigidity of the partial volume
as
equals 1 if the fluid rotates similar to a rigid body and equals 0 if the fluid does not rotate at all. The calculated ratio
of the considered homogeneous isotropic turbulence is shown in
Figure 4a. At
, i.e., considering the volume with
, the ratio of the mean rotational strength over the mean vorticity magnitude equals 0.4. As
increases, the volume to obtain an average became smaller and the ratio quickly increases and saturates at around 0.85 at
; above this, the area could be considered to be the vortex core region. Note that a similar threshold has been selected in
Figure 3 to represent the vortices; thus, we could assume that the iso-surface enclosed volumes are vortex cores. According to Equation (6), the fact that the Liutex magnitude is smaller than the vorticity magnitude arise from two mechanisms: first, the non-alignment of the Liutex vector and the vorticity vector, expressed as
; and second, the residual shear in the rotational plane
. With these definitions, we could obtain
.
Figure 4b shows the similar defined partial average
and
over the vorticity against
. The conclusions are: (1) the in-plane residual shear
was generally much larger than the non-alignment shear; and (2) the inside vortex cores (
),
and
—i.e., the in-plane residual shear—represented 90% of the difference between the Liutex magnitude and the vorticity magnitude. It is understood that the role of the non-alignment shear is to change the direction of the Liutex vector whereas the in-plane residual shear provided energy for the rotation represented by the Liutex vector.
Figure 4b infers that the direction of the vortex cores is not supposed to vary much as
is small. In other words, the direction of the Liutex vector is almost the same as the direction of the vorticity vector inside the vortex cores where fluid motion behaves similar to a rigid body with in-plane residual shear.
Investigations into turbulent structures have revealed that, instead of being purely random, many organized motions exist in turbulence, i.e., coherent structures, which are often related to vortical structures [
21,
22,
23]. Xu et al. [
14] discovered that the absolute value of the Fourier coefficients of the Liutex magnitude followed a −5/3 similarity power law, which also indicates the inherent structure of turbulence. The three-dimensional energy spectrums of the velocity, vorticity and Liutex vector are shown in
Figure 5, where
,
,
and
are the wavenumber, Kolmogorov scale, turbulent kinetic energy dissipation rate and kinematic viscosity, respectively. Dotted lines with powers of 1/3, −5/3 and −10/3 are added in appropriate places to compare the numerical results and the theoretical predictions. It is shown that for ideal homogeneous isotropic turbulence, the turbulent kinetic spectrum
(red dash-dotted line) follows the −5/3 law in the inertial subrange, as predicted by Kolmogorov through a dimensional analysis. The definition of the turbulent kinetic spectrum
is given as:
where
is the turbulent kinetic energy and
,
and
are the velocity components. According to Kolmogorov’s 1941 theory (K41), the turbulent spectrum
. Similarly, the definition of the energy spectrum of the vorticity and Liutex could be given by:
where
,
and
are the vorticity components and
,
and
are the Liutex components. The energy spectrum of vorticity
or the equivalent enstrophy spectrum—which also served as the dissipation spectrum by the multiplication of the kinematic viscosity
—in homogeneous isotropic turbulence is also shown in
Figure 5. It is verified that for incompressible HIT,
and
satisfy the relation
; thus, at the same range where the turbulent kinetic spectrum
followed the −5/3 power law,
satisfied a 1/3 power law, as shown by the blue dashed line in
Figure 5. As the Liutex and vorticity had the same dimension of
(per second), it was not surprising to find that the energy spectrum of Liutex also satisfied the 1/3 power law in the inertial subrange, as shown by the black line in
Figure 5. The fact that the Liutex energy spectrum obeyed the 1/3 power law in the inertial subrange, however, was not trivial. We could arbitrarily introduce new quantities with the dimension
; for example,
and
. Yet, we could not expect all these quantities to satisfy the given power law. Therefore, the existence of the 1/3 power law for Liutex indicated that the Liutex vector might be used to signal the flow structures in homogeneous isotropic turbulence.
A distinctive feature of the energy spectrum of Liutex is that it follows the −10/3 power law in the dissipation range, which is in accordance with previous discovery of Xu
[
14], as they defined the spectrum as absolute values of Fourier coefficients. Even though the theoretical predictions of the −5/3 and 1/3 power laws match the numerical results of homogeneous isotropic turbulence well in the inertial subrange, it is the −10/3 power law of the Liutex energy spectrum in the dissipation subrange that persisted in wall-bounded turbulence where other quantities are generally significantly contaminated by residual shear. To illustrate this point, the one-dimensional energy spectrum of the velocity magnitude, vorticity magnitude, Liutex magnitude and
(the parameter in the
vortex identification method [
24]) are shown in
Figure 6 for a turbulent channel flow with
. It is shown in
Figure 6a that the turbulent kinetic spectrums only marginally match the −5/3 power law in a narrow wavenumber range. In the viscous sublayer (
), the flow there is hardly turbulent and the turbulent kinetic energy
is relatively small. Moving away from the wall surface,
quickly increases and a narrow range satisfying the −5/3 power law could be observed in the buffer layer (
). Further in the log-law layer, more energy is injected into the high wavenumbers in the log layer (
) where the turbulent fluctuations were the fiercest and the turbulent kinetic energy becomes smaller in the outer layer (
). No similarity of power law can be observed in the dissipation range. The energy of the vorticity or enstrophy, on the other hand, is relatively large even in the viscous sublayer (
), as shown in
Figure 6b, due to the inclusion of both rotational motion and residual shear. The lines of the enstrophy spectrum at various heights cross each other in the dissipation range; thus, they do not obey any power-law similarity.
As shown in
Figure 6c, the fluctuating energy of Liutex is negligible in the viscous sublayer because the Liutex vector is free from shear contamination. This is in the same order in the dissipation range as that of
, which is already near the center line of the channel. High energy in the dissipation range (a high wavenumber) is observed in the log layer (
) and, more importantly and independent of height, all the Liutex energy spectrums—except for
in the viscous layer, where the flow could hardly be viewed as turbulence—follow the −10/3 power law in the dissipation range as in the homogeneous isotropic case. The energy spectra of
, on the other hand, are irregular in the dissipation range, as shown in
Figure 6d, and obey no power-law similarity. The zigzags of the spectral lines in
Figure 6 are due to insufficient averages as only the spanwise average for one snapshot was taken. However, the general shape of the spectrum would remain the same with the average in time over many snapshots. In summary, among the 1/3, −5/3 and −10/3 power laws shown in
Figure 5 for homogeneous isotropic turbulence, only the −10/3 power law of the Liutex energy spectrum in the dissipation range extends to wall-bounded turbulence.
From the discussion above, we observed that the energy spectrum of Liutex follows the −10/3 power-law spectrum in the dissipation scale range. We further hypothesized that this power law is determined by the turbulence dissipation rate
and the kinematic viscosity
. Through a dimensional analysis, we obtained:
where
is a non-dimensional parameter. Unlike the −5/3 power-law spectrum of turbulent kinetic energy—which only exists for homogeneous isotropic turbulence and only marginally matches could be found in wall-bounded turbulence, especially with mid-to-low Reynolds numbers—the Liutex energy spectrum in the dissipation range exists even in wall-bounded turbulence of various heights; thus, this could lay a theoretical foundation for new turbulence models.