# A self-induced mechanism of large-scale helical structures in compressible
turbulent flows

Zheng Yan  Institute of Applied Physics and Computational Mathematics, Beijing
100094, China  Jianchun Wang  Department of Mechanics and Aerospace
Engineering, Southern University of Science and Technology, Shenzhen,
Guangdong 518055, China  Lifeng Wang<EMAIL_ADDRESS>Institute of
Applied Physics and Computational Mathematics, Beijing 100094, China  Center
for Applied Physics and Technology, HEDPS, Peking University, Beijing 100871,
China  Zhu Lei  Institute of Applied Physics and Computational Mathematics,
Beijing 100094, China  Junfeng Wu  Institute of Applied Physics and
Computational Mathematics, Beijing 100094, China  Junyi Duan  LHD, Institute
of Mechanics, Chinese Academy of Sciences, Beijing 100190, China  School of
Engineering Science, University of Chinese Academy of Sciences, Beijing
100049, China  Fulin Tong  State Key Laboratory of Aerodynamics, China
Aerodynamics Research and Development Center, Mianyang 621000, China  Xinliang
Li  LHD, Institute of Mechanics, Chinese Academy of Sciences, Beijing 100190,
China  School of Engineering Science, University of Chinese Academy of
Sciences, Beijing 100049, China  Changping Yu<EMAIL_ADDRESS>LHD, Institute
of Mechanics, Chinese Academy of Sciences, Beijing 100190, China  School of
Engineering Science, University of Chinese Academy of Sciences, Beijing
100049, China

###### Abstract

A novel self-sustaining mechanism is proposed for large-scale helical
structures in compressible turbulent flows. The existence of two channels of
subgrid-scale and viscosity terms for large-scale helicity evolution is
confirmed for the first time, through selecting a physical definition of the
large-scale helicity in compressible turbulence. Under the influence of the
fluid element expansion, it is found that the helicity is generated at small
scales via the second-channel viscosity, and the inverse cross-scale helicity
transfers at inertial scales through the second-channel helicity flux.
Together, they form a self-induced mechanism, which provides a physical
insight into the long-period characteristic of large-scale helical structures
in the evolution of compressible flow systems.

††preprint: APS/123-QED

##  I Introduction

Large-scale helical motions widely exist in natural phenomena and practical
applications, such as tornadoes in the atmosphere, Langmuir circulations in
the ocean, and Taylor-Görtler vortices in turbomachinery (Sanada, 1993;
Moffatt and Tsinober, 1992; Yan  _et al._ , 2020a). Nevertheless, these large-
scale helical structures have a long-period characteristic, which might
originate from an external forcing and intrinsic viscosity. The pseudoscalar
of helicity indicates that the intrinsic viscosity might play a self-induced
role to decelerate the decay of helicity in some specific circumstances (Agoua
_et al._ , 2021). The existence of large-scale helical structures has an
important influence on flow system evolution (Teitelbaum and Mininni, 2009).
This weakens the nonlinearity strength of the Navier-Stokes equations and
leads to a smaller viscosity dissipation rate of kinetic energy (Bos and
Rubinstein, 2013; Linkmann, 2018) and an inverse energy cascade process of
kinetic energy (Biferale  _et al._ , 2012; Sahoo  _et al._ , 2017). In
engineering flows, large-scale helical structures can strengthen the mixing
efficiency in a jet-stirred tubular reactor, decrease aircraft drag, and lead
to high heat flux in hypersonic vehicles (Moffatt and Tsinober, 1992; Yu  _et
al._ , 2022). Hence, an induced mechanism should be found to develop
appropriate flow control strategies.

Large-scale helical motions can be characterized by high helicity (Sanada,
1993), and the investigation of large-scale helical structures can be carried
out from the perspective of helicity. Helicity is defined as the integral of
the scalar product of velocity $\textbf{{u}}(\textbf{{x}},t)$ and vorticity
$\bm{\omega}(\textbf{{x}},t)$ over volume. Consistent with Helmholtz and
Kelvin theorems, if we assume that the flows are inviscid and barotropic, the
conservation law of helicity is present (Moffatt and Tsinober, 1992). Helicity
measures the twisting, linking and writhing of vortex lines, and it is
conservative even in viscous flows (Moffatt, 1969, 2014; Scheeler  _et al._ ,
2014, 2017). According to the Noether theorem, the conservation law determines
the evolution of a three-dimensional turbulent system (Pouquet and Mininni,
2010). It is worth exploring the statistical characteristics of helicity
evolution, which is one of the only two inviscid quadratic invariants in
three-dimensional flows.

From the perspective of its governing equation, the external sources of
helicity generation might be rotating dynamo or buoyancy (Agoua  _et al._ ,
2021; Moffatt, 1978; Hide, 1976; Frisch  _et al._ , 1987; Marino  _et al._ ,
2013), and the self-induced mechanism might stem from the viscosity term
(Melander and Hussain, 1993). The role of viscosity in helicity evolution
differs from kinetic energy evolution. From the perspective of governing
equations, the viscosity plays a dissipation role statistically in the
transfer of kinetic energy into internal energy. Nevertheless, viscosity makes
a bridge for helicity transferring between different chiralities.
Specifically, if viscosity plays a dissipation role for left-chirality
helicity, it must play a generating role for right-chirality helicity, and
vice versa. Hence, the kinetic energy governing equation must be accompanied
by an internal energy governing equation to describe the energy path
stringently. However, the helicity governing equation does not need to couple
other equations. Besides, viscosity also has other effects (Moffatt and
Tsinober, 1992), such as mediating the reconnection process of vortex lines,
causing vorticity to diffuse away from the vortex centerline (Kida and
Takaoka, 1988; Irvine, 2018; Yao and Hussain, 2020; Yao  _et al._ , 2021). In
compressible flows, the role of viscosity is more complex, and compressibility
plays an essential role in helicity generation (Reshetnyak, 2012; Serre,
2018). The coupling of the solenoidal and compressible modes leads to a
theoretical challenge for exploring the intrinsic mechanism of compressible
turbulent flows (John and Donzis, 2019). For the existence of large-scale
helical structures, there must exist a bridge to transfer helicity from small
scales to large scales (Kivotides and Leonard, 2021; Wu and Bos, 2022).

The unresolved long-period characteristics of the large-scale helical
structures in compressible turbulent flows motivate us to comprehensively
explore the dynamic evolutions of large-scale helicity. Except for the case-
by-case external forcing, we focus on the universal statistical
characteristics of helicity evolution at middle and small scales. The rest of
the paper is organized as follows. In $\S$ 2, we exhibit the details of DNS of
compressible helical turbulent flows. In $\S$ 3, we derive the governing
equation of large-scale helicity in compressible turbulent flows, and confirm
the existence of dual channel of subgrid-scale (SGS) and viscosity terms of
helicity transfer. Next in $\S$ 4, we propose a self-induced mechanism based
on the statistical properties of the two channels of SGS and viscosity terms.
In $\S$ 5, we summarize the major conclusions and discussions.

##  II Direct numerical simulations of compressible homogeneous and isotropic
turbulence flows

We carry out direct numerical simulations of compressible homogeneous and
isotropic turbulent flows in a cubic box with a length of $2\pi$ with a mesh
resolution of $2048^{3}$. The three-dimensional dimensionless Navier-Stokes
equations are numerically solved by a hybrid finite difference method, and it
combines an eighth-order compact finite difference scheme in smooth regions
and a seventh-order weighted essential nonoscillatory scheme in shock regions
(Wang  _et al._ , 2013; Yan  _et al._ , 2020b). The dimensionless Navier-
Stokes equations of ideal gas are

$\frac{\partial\rho}{\partial t}+\frac{\partial(\rho u_{j})}{\partial
x_{j}}=0,$ (1a) $\frac{\partial(\rho u_{i})}{\partial t}+\frac{\partial(\rho
u_{i}u_{j})}{\partial x_{j}}=-\frac{\partial p}{\partial
x_{i}}+\frac{1}{Re}\frac{\partial\sigma_{ij}}{\partial x_{j}}+F_{i},$ (1b)
$\frac{\partial\mathcal{E}}{\partial
t}+\frac{\partial[(\mathcal{E}+p)u_{j}]}{\partial
x_{j}}=\frac{1}{\alpha}\frac{\partial}{\partial x_{j}}(\kappa\frac{\partial
T}{\partial x_{j}})+\frac{1}{Re}\frac{\partial(\sigma_{ij}u_{i})}{\partial
x_{j}}-\Lambda+F_{j}u_{j},$ (1c)

where ${\rho}$ is the density, u is the velocity, $p$ is the pressure, $T$ is
the temperature, $\Lambda$ is a cooling-function for sustaining a steady state
statistically (Wang  _et al._ , 2010), and F is the large-scale force composed
of multi-parameters controlling kinetic energy and helicity (Yan  _et al._ ,
2020b). The multi-parameters include the energy inputting rates of compressive
and solenoidal components and helicity inputting rate. Hence, the specific
external forcing can be constructed as

$F_{i}=\sqrt{\rho}\times(\pi_{1}u_{i}^{C}+\pi_{2}u_{i}^{S}+\pi_{3}\omega_{i}),$
(2)

where the superscript $C$ and $S$ denote the compressible and solenoidal
component of velocity, and $\pi_{1}$, $\pi_{2}$ and $\pi_{3}$ are three
indeterminate dimensional parameters. The external forcing is fixed within the
lowest two wave number shells to avoid the pollution of external forcing on
the statistical results at middle and small scales. Similar to our previous
work (Yan  _et al._ , 2020b), the injection rates of kinetic energy and
helicity are fixed at 0.30 and -0.37, respectively. To obtain the highly
compressible turbulent flows, the injection rate of the kinetic energy is
divided into the solenoidal and compressible modes equally. More details can
refer to our previous work (Yan  _et al._ , 2020b).

The equation of state is

$p=\rho T/\gamma M^{2}.$ (3)

Besides, the viscous stress $\sigma_{ij}$ and total energy per unit volume
$\mathcal{E}$ are defined as

$\sigma_{ij}=\mu(\frac{\partial u_{i}}{\partial x_{j}}+\frac{\partial
u_{j}}{\partial x_{i}})-\frac{2}{3}\mu\frac{\partial u_{k}}{\partial
x_{k}}\delta_{ij},\mathcal{E}=\frac{p}{\gamma-1}+\frac{1}{2}\rho u_{i}u_{i}.$
(4)

Some one-point statistical parameters are defined to describe the
characteristics of the present compressible helical turbulence (Samtaney  _et
al._ , 2001; Wang  _et al._ , 2012a). The integral length scale $L_{f}$ is

$L_{f}=\frac{3\pi}{2(u^{\prime})^{2}}\int_{0}^{\infty}\frac{E(k)}{k}dk.$ (5)

The Taylor microscale $\lambda$ and the Kolmogorov length scale $\eta$ are

$\lambda=\frac{u^{\prime}}{\langle[(\partial u_{1}/\partial
x_{1})^{2}+(\partial u_{2}/\partial x_{2})^{2}+(\partial u_{3}/\partial
x_{3})^{2}]/3\rangle^{1/2}},$ (6)
$\eta=[\langle\mu/Re\rho\rangle^{3}/\epsilon]^{1/4}.$ (7)

Here, $u^{\prime}$ is the root mean square (rms) of velocity vector u which is
defined as

$u^{\prime}=\sqrt{\langle u_{1}^{2}+u_{2}^{2}+u_{3}^{2}\rangle/3}.$ (8)

$E(k)$ is the spectrum of kinetic energy per unit mass, $\epsilon$ is the
ensemble-averaged viscous dissipation rate of kinetic energy per unit volume
which is defined as

$\epsilon=\langle\sigma_{ij}S_{ij}/Re\rangle.$ (9)

The strain-rate tensor is

$S_{ij}=(1/2)(\partial u_{i}/\partial x_{j}+\partial u_{j}/\partial x_{i}).$
(10)

Hence, the Taylor microscale Reynolds number $Re_{\lambda}$ and the turbulent
Mach number $M_{t}$ are defined as

$Re_{\lambda}=\frac{u^{\prime}\lambda\langle\rho\rangle}{\langle\mu\rangle},\qquad
M_{t}=M\frac{\sqrt{3}u^{\prime}}{\langle\sqrt{T}\rangle}.$ (11)

Table 1: Characteristic parameters of steady numerical simulations. Here, $\Delta x$ is the mesh spacing, $\theta^{\prime}$ is the rms of the velocity divergence and $\omega^{\prime}$ is the rms of the vorticity magnitude. Resolution | $Re_{\lambda}$ | $M_{t}$ | $\epsilon$ | $\delta$ | $k_{max}\eta$ | $\eta/\Delta x$ | $L_{f}/\eta$ | $\lambda/\eta$ | $\tau_{\eta}$ | $T_{0}$ | $\theta^{\prime}$ | $\omega^{\prime}$ | $S_{3}$
---|---|---|---|---|---|---|---|---|---|---|---|---|---
$2048^{3}$ | 514 | 0.67 | 0.30 | -0.37 | 3.23 | 1.03 | 477 | 33.96 | 0.032 | 0.94 | 22.22 | 20.47 | -30.76

Figure 1: (a) Isosurface of the velocity divergence with threshold
$\theta=-3\theta^{\prime}$ rendered in yellow and $\theta=0.5\theta^{\prime}$
rendered in red. (b) Energy spectra of different mesh resolutions, and the
energy spectrum of a mesh resolution with $1024^{3}$ is obtained from our
previous work (Yan  _et al._ , 2020b).

The mean viscous dissipation rate of helicity is

$\delta=4\left\langle S_{ij}R_{ij}-S_{ii}R_{jj}/3\right\rangle/Re.$ (12)

And
$R_{ij}=\left(1/2\right)\left[\partial\left(\omega_{i}/\rho\right)/\partial
x_{j}+\partial\left(\omega_{j}/\rho\right)/\partial x_{i}\right]$.
$\tau_{\eta}$ is the Kolmogorov time scale, which is defined as

$\tau_{\eta}=\left(\left\langle\mu/\rho\right\rangle/\epsilon\right)^{1/2}.$
(13)

$T_{0}$ is the large eddy turn-over time, which is defined as
$T_{0}=L_{f}/u^{\prime}$. The skewness of the velocity gradient $S_{3}$ is
defined as

$S_{3}=\frac{\left[\left\langle\left(\partial u_{1}/\partial
x_{1}\right)^{3}+\left(\partial u_{2}/\partial x_{2}\right)^{3}+\left(\partial
u_{3}/\partial
x_{3}\right)^{3}\right\rangle\right]/3}{\left[\left\langle\left(\partial
u_{1}/\partial x_{1}\right)^{2}+\left(\partial u_{2}/\partial
x_{2}\right)^{2}+\left(\partial u_{3}/\partial
x_{3}\right)^{2}\right\rangle/3\right]^{3/2}}.$ (14)

The characteristic parameters of the flow fields are listed in table 1. To
obtain the stationary status of fully developed helical turbulent flows, the
computational physical time exceeds five times large eddy turnover time. We
show the three-dimensional shocklets structures represented by the local
velocity divergence of the present mesh resolution in figure 1(a). Due to a
strong compressibility, the shocklets exhibit large-scale sheet structures,
and the expansion structures show a small-scale random distribution. The
energy spectra with different mesh resolutions are shown in figure 1(b). In
contrast to our previous result (Yan  _et al._ , 2020b), the present energy
spectrum with a higher Reynolds number and a higher mesh resolution shows more
small-scale structures, which is beneficial to our statistical analysis.

##  III Theoretical derivation

To study the helicity evolution at different scales simultaneously, we need to
derive the governing equation of large-scale helicity in compressible flows.
The first undetermined issue is the definition of large-scale helicity because
its governing equation should satisfy the conservation law of helicity cascade
and inviscid criteria in compressible flows (Zhao and Aluie, 2018). This means
that no other source terms exist other than the large-scale pressure, SGS and
viscosity terms (Yan  _et al._ , 2019). The definition based on the Favre
filter is appropriate, and it is defined as
$\widetilde{H}=\widetilde{\textbf{{u}}}\cdot\widetilde{\bm{\omega}}$. In
contrast, the definition based on the direct filter (
$\overline{H}=\overline{\textbf{{u}}}\cdot\overline{\bm{\omega}}$) might not
be appropriate because of existing source terms, and the specific derivation
is provided in Appendix A. Here,
$\widetilde{\textbf{{u}}}(\textbf{x})=\overline{\rho\textbf{{u}}(\textbf{{x}})}/\overline{\rho}$
is the Favre-filtered velocity, and
$\widetilde{\bm{\omega}}=\nabla\times\widetilde{\textbf{{u}}}$.
$\overline{\cdot}$ denotes a filter operation, which is defined as
$\overline{\textbf{f}}(\textbf{x})=\int{{\rm
d}^{3}\textbf{{r}}G_{l}(\textbf{{r}})\textbf{f}(\textbf{{x}}+\textbf{{r}})}$
with kernel $G_{l}(\textbf{{r}})$.

Making a filter operation on equation 1b and dividing by $\overline{\rho}$, we
can obtain the following large-scale velocity equation as

$\frac{\partial\widetilde{u}_{i}}{\partial
t}+\widetilde{u}_{j}\frac{\partial\widetilde{u}_{i}}{\partial
x_{j}}=-\frac{1}{\overline{\rho}}\frac{\partial\overline{\rho}\widetilde{\tau}_{ij}}{\partial
x_{j}}-\frac{1}{\overline{\rho}}\frac{\partial\overline{p}}{\partial
x_{i}}+\frac{1}{Re}\frac{1}{\overline{\rho}}\frac{\partial\overline{\sigma}_{ij}}{\partial
x_{j}}+\frac{\overline{F}_{i}}{\overline{\rho}},$ (15)

where Favre-filtered SGS stress is
$\widetilde{\tau}_{ij}=\widetilde{u_{i}u_{j}}-\widetilde{u}_{i}\widetilde{u}_{j}$.
Making a curl operation on equation 15, we can obtain the following Favre-
filtered vorticity governing equation as

$\begin{split}\frac{\partial\widetilde{\omega}_{i}}{\partial
t}+\widetilde{u}_{j}\frac{\partial\widetilde{\omega}_{i}}{\partial
x_{j}}&+\widetilde{\omega}_{i}\widetilde{\theta}=\widetilde{\omega}_{j}\frac{\partial\widetilde{u}_{i}}{\partial
x_{j}}-\epsilon_{ijk}\frac{\partial}{\partial
x_{j}}\left(\frac{1}{\overline{\rho}}\frac{\partial\widetilde{\tau}_{kl}}{\partial
x_{l}}\right)\\\ &-\epsilon_{ijk}\frac{\partial}{\partial
x_{j}}\left(\frac{1}{\overline{\rho}}\frac{\partial\overline{p}}{\partial
x_{k}}\right)+\frac{1}{Re}\epsilon_{ijk}\frac{\partial}{\partial
x_{j}}\left(\frac{1}{\overline{\rho}}\frac{\partial\overline{\sigma}_{kl}}{\partial
x_{l}}\right)\\\ &+\epsilon_{ijk}\frac{\partial}{\partial
x_{j}}\left(\frac{\overline{F}_{k}}{\overline{\rho}}\right).\end{split}$ (16)

Hence, combining equation 15 and 16, and we can get the following governing
equations of large-scale helicity $\widetilde{H}$ with a Favre filter,

$\frac{\partial\widetilde{H}}{\partial
t}+\frac{\partial\widetilde{J}_{j}}{\partial
x_{j}}=-\Pi^{H1}_{l}-\Pi^{H2}_{l}+\Phi_{l}^{1}+\Phi_{l}^{2}+D_{l}^{1}+D_{l}^{2}+\mathcal{F}_{l}^{1}+\mathcal{F}_{l}^{2},$
(17)

where $\widetilde{J}_{j}$ is the spatial transport term, which is defined as

$\begin{split}\widetilde{J}_{j}=&\widetilde{u}_{j}\widetilde{H}-\left(1/2\right)\widetilde{u}_{i}\widetilde{u}_{i}\widetilde{\omega}_{j}+\widetilde{\omega}_{i}\widetilde{\tau}_{ij}+\epsilon_{ikl}\frac{\partial\overline{\rho}\widetilde{\tau}_{lj}}{\partial
x_{k}}\frac{\widetilde{u}_{i}}{\overline{\rho}}\\\
&+\overline{\rho}\widetilde{\tau}_{ij}\epsilon_{ikl}\widetilde{u}_{k}\frac{\partial}{\partial
x_{l}}\left(\frac{1}{\overline{\rho}}\right)+\overline{p}\frac{\widetilde{\omega}_{j}}{\overline{\rho}}+\overline{p}\epsilon_{jkl}\frac{\partial\overline{\rho}}{\partial
x_{k}}\frac{\widetilde{u}_{l}}{\overline{\rho}^{2}}\\\
&-\overline{\sigma}_{ij}\frac{\widetilde{\omega}_{i}}{\overline{\rho}}+\overline{\sigma}_{ij}\epsilon_{ikl}\frac{\widetilde{u}_{k}}{\overline{\rho}^{2}}\frac{\partial\overline{\rho}}{\partial
x_{l}}-\frac{\widetilde{u}_{i}}{\overline{\rho}}\epsilon_{ikl}\frac{\partial\overline{\sigma}_{lj}}{\partial
x_{k}}.\end{split}$ (18a) The other terms are defined as
$\Pi^{H1}_{l}=-\overline{\rho}\widetilde{\tau}_{ij}\frac{\partial}{\partial
x_{j}}\left(\frac{\widetilde{\omega}_{i}}{\overline{\rho}}\right),\quad\Pi^{H2}_{l}=\Pi^{H21}_{l}+\Pi^{H22}_{l},$
(18b)
$\Pi^{H21}_{l}=-\epsilon_{ikl}\frac{\partial\overline{\rho}\widetilde{\tau}_{lj}}{\partial
x_{k}}\frac{\partial\left(\widetilde{u}_{i}/\overline{\rho}\right)}{\partial
x_{j}},$ (18c)
$\Pi^{H22}_{l}=-\overline{\rho}\widetilde{\tau}_{ij}\frac{\partial}{\partial
x_{j}}\left(\epsilon_{ikl}\widetilde{u}_{k}\frac{\partial\left(1/\overline{\rho}\right)}{\partial
x_{l}}\right),$ (18d) $\Phi^{1}_{l}=\overline{p}\frac{\partial}{\partial
x_{j}}\left(\frac{\widetilde{\omega}_{j}}{\overline{\rho}}\right),\quad\Phi^{2}_{l}=\overline{p}\frac{\partial}{\partial
x_{j}}\left(\epsilon_{jkl}\frac{\partial\overline{\rho}}{\partial
x_{k}}\frac{\widetilde{u}_{l}}{\overline{\rho}^{2}}\right),$ (18e)
$D^{1}_{l}=-\overline{\sigma}_{ij}\frac{\partial}{\partial
x_{j}}\left(\frac{\widetilde{\omega}_{i}}{\overline{\rho}}\right),\quad
D^{2}_{l}=D^{21}_{l}+D^{22}_{l},$ (18f)
$D^{21}_{l}=-\epsilon_{ikl}\frac{\partial\overline{\sigma}_{lj}}{\partial
x_{k}}\frac{\partial}{\partial
x_{j}}\left(\frac{\widetilde{u}_{i}}{\overline{\rho}}\right),\quad
D^{22}_{l}=\overline{\sigma}_{ij}\frac{\partial}{\partial
x_{j}}\left(\epsilon_{ikl}\frac{\widetilde{u}_{k}}{\overline{\rho}^{2}}\frac{\partial\overline{\rho}}{\partial
x_{l}}\right).$ (18g)
$\mathcal{F}_{l}^{1}=\widetilde{\omega}_{i}\frac{\overline{F}_{i}}{\overline{\rho}},\qquad\mathcal{F}_{l}^{2}=\widetilde{u}_{i}\epsilon_{ijk}\frac{\partial}{\partial
x_{j}}\left(\frac{\overline{F}_{k}}{\overline{\rho}}\right).$ (18h)

The above governing equation of the large-scale helicity describes the
evolution of helical structures in compressible turbulent flows via different
physical processes, such as the spatial transportation process
($\widetilde{J}_{j}$), the SGS transfer ($\Pi^{H1}_{l}$ and $\Pi^{H2}_{l}$),
pressure process ($\Phi^{1}_{l}$ and $\Phi^{2}_{l}$), viscosity process
($D^{1}_{l}$ and $D^{2}_{l}$) and external forcing ($\mathcal{F}^{1}_{l}$ and
$\mathcal{F}^{2}_{l}$). For the different physical processes, all of them
consist of two expressions. To distinguish these two expressions, we name the
expression from the momentum equation as the first channel, and name the
expression from the vorticity equation as the second channel. These two
channels describe the large-scale helicity evolutions from different
perspectives, and we investigate their statistical characteristics
subsequently.

Here, for the SGS terms, if we assume that the filtered density is a constant,
the first and second SGS channels are recovered to the first and second
channels in incompressible flows(Yan  _et al._ , 2020a). Notably, the second
term of the second-channel helicity flux depends on the density gradient, and
it is only present in variable-density flows. In addition, their tensor
geometries are more complex than that in incompressible form, and the specific
illustration is provided in Appendix B. For the above two pressure terms, we
take the difference and obtain the following relation

$\begin{split}\Theta&\equiv\overline{p}\frac{\partial}{\partial
x_{j}}\left(\epsilon_{jkl}\frac{1}{\overline{\rho}}\frac{\partial\widetilde{u}_{l}}{\partial
x_{k}}+\epsilon_{jkl}\frac{\partial}{\partial
x_{k}}\left(\frac{1}{\overline{\rho}}\right)\widetilde{u}_{l}\right)\\\
&\equiv\overline{p}\frac{\partial}{\partial
x_{j}}\left(\epsilon_{jkl}\frac{\partial}{\partial
x_{k}}\left(\frac{1}{\overline{\rho}}\widetilde{u}_{l}\right)\right)\overset{\nabla\cdot\nabla\times\textit{{f}}\equiv
0}{\equiv}0,\end{split}$ (19)

where f is a vector. Hence, the first and second channels of the pressure
terms are the same, which means that $\Phi^{1}_{l}=\Phi^{2}_{l}$. Regarding
the statistical characteristics of the pressure terms of helicity transfer in
compressible turbulent flows, we performed a detailed analysis in our previous
paper (Yan  _et al._ , 2019). The viscosity terms under a constant-density
hypothesis can be simplified as

$D^{1}_{l}=-2\nu\overline{S}_{ij}\overline{R}_{ij},\qquad
D^{2}_{l}=-2\nu\overline{R}_{ij}\overline{S}_{ij}.$ (20)

Under the hypothesis of constant density, the Favre filter can be reduced to
other direct filters, and we replace the symbol $\widetilde{\cdot}$ with the
symbol $\overline{\cdot}$ in the above derivation. Here,
$\overline{S}_{ij}=\left(\partial\overline{u}_{i}/\partial
x_{j}+\partial\overline{u}_{j}/\partial x_{i}\right)/2$ is the symmetric
component of the velocity gradient tensor, and
$\overline{R}_{ij}=\left(\partial\overline{\omega}_{i}/\partial
x_{j}+\partial\overline{\omega}_{j}/\partial x_{i}\right)/2$ is the symmetric
component of the vorticity gradient tensor. We can conclude that
$D^{1}_{l}=D^{1}_{2}$ in incompressible turbulent flows. Hence, the dual-
channel characteristics of the viscosity term do not exist in incompressible
turbulent flows, and it is consistent with our previous work (Yan  _et al._ ,
2020a). However, the equality relationship of the dual channels of the
viscosity term in compressible turbulent flows does not exist. Therefore, we
conclude that the dual-channel characteristics of viscosity terms only exist
in compressible turbulent flows.

$\mathcal{F}_{l}^{1}$ and $\mathcal{F}_{l}^{2}$ are the external forcing
terms. In our numerical simulations, we construct an external forcing
consisting of velocity and vorticity within the first two wave number shells
to sustain a stationary highly helical turbulent flow. Because we focus on the
statistical characteristics of helicity transfer at middle and small scales,
the statistical analysis of the external forcing will not be carried out.
However, in practical applications, the unknown external forcing might have
the dual-channel characteristic.

##  IV A self-induced mechanism of large-scale helical structures

The conditional averaging method is used to evaluate the local compression or
expansion effects on helicity transfer, and the sufficient sample points in
our numerical simulations verify its accuracy (Block  _et al._ , 2006; Wang
_et al._ , 2012b, 2013). Before evaluating the compression or expansion
effects, we need to investigate the scale distributions of the velocity
divergence. The power spectrum of the velocity divergence $\theta$ is defined
as $\int_{0}^{\infty}E^{\theta}(k)dk=\left\langle\theta^{2}\right\rangle$. In
figure 2(a), we show the power spectrum of the velocity divergence. The power
spectrum density indicates the slight discrepancy of the fluctuations of the
velocity divergence at different length scales, especially at large and middle
scales. However, both the compression and expansion motions are involved in
the power spectrum of the velocity divergence. In order to access the
compression and expansion motions separately, we exhibit the probability
distribution functions (PDF) of the velocity divergence at corresponding
length scales in figure 2(b) marked up in figure 2(a). The PDF of the velocity
divergence indicates that the compression motion is dominant at all scales.
With the decrease of the length scale, the dominant degree increases, which is
associated with the shocklet structures. For the PDF of the expansion motion
shown in the inset of figure 2(b), weak expansion motions are more
concentrated at middle and large length scales. Hence, the degree of
compression and expansion motions is high at middle and small scales in our
numerical simulations. This is beneficial for accessing the compression and
expansion effects on the SGS transfer at middle length scales and on the
viscosity dissipation at small scales.

Figure 2: (a) Power spectrum of the velocity divergence $\theta$. (b) PDF of
the velocity divergence at different length scales, and the line color
corresponds to the length scales marked up in (a). The inset shows the local
enlarged PDF of the velocity divergence to highlight the distribution of the
dilatation at different length scales.

In figure 3(a), we show the ensemble averages of the first and second terms of
the second-channel helicity flux conditioned in compression and expansion
regions. Flows in the compression regions dominate the first and second terms
of the second-channel helicity flux, which is consistent with the dominant
forward helicity cascade process. However, the sign of the second term in the
expansion regions is positive, and it corresponds to an inverse helicity
cascade process. The expansion effects on the second term are further
evaluated through ensemble averages conditioned in local velocity divergence,
and the statistical results are exhibited in the inset in figure 3(a).
Compression and expansion have opposite effects on the second term of the
second-channel helicity flux. With an increase in expansion, the magnitude of
the second term is larger, which indicates a stronger inverse helicity cascade
process. Similarly, with an increase in compression, the magnitude of the
second term also increases, and it corresponds to a stronger forward helicity
cascade process. Hence, we can conclude that the second term of the second-
channel helicity flux in expansion regions serves as an intrinsic mechanism
for the inverse helicity cascade process, which might help to sustain large-
scale helical structures. The statistical results of the first- and second-
channel helicity fluxes conditioned in compression and expansion regions are
consistent with the whole forward helicity cascade process, and we do not show
them for the sake of simplicity.

Figure 3: (a) The ensemble averages of the first and second terms of the
second-channel helicity flux conditioned in the compression $\theta_{l}<0$ and
expansion regions $\theta_{l}>0$. The inset shows the ensemble average of the
second term of the second-channel helicity flux conditioned in local velocity
divergence with filter widths $l/\eta$=32. Here,
$\theta_{l}=\partial\overline{u}_{j}/\partial x_{j}$. The light green region
corresponds to the inverse cross-scale transfer phenomenon. (b) The ensemble
averages of the first and second terms of the second-channel viscosity term
conditioned in compression and expansion regions. The inset shows the ensemble
average of the second term of the second-channel viscosity term conditioned in
local velocity divergence with filter widths $l/\eta$=4. The light pink region
corresponds to the source for the generation of helicity.

The second term of the second-channel helicity flux in expansion regions
provides an inverse cross-scale transfer mechanism at medium scales to sustain
large-scale helical structures. However, at small scales, the viscosity terms
are dominant. Hence, we also employ the above conditional averaging method to
evaluate the compression and expansion effects on the statistical
characteristics of the involved viscosity terms. In figure 3(b), we exhibit
the ensemble averages of the first and second terms of the second-channel
viscosity term conditioned in the compression and expansion regions. With an
increase in the lengths of the scales, their magnitudes decrease rapidly. This
indicates that the selected definition of large-scale helicity is appropriate
because it satisfies the ”inviscid criterion” proposed by Aluie (2011); Zhao
and Aluie (2018). Similar to the conclusions for SGS terms, only the sign of
the second term of the second-channel viscosity term in the expansion regions
is the same as the total helicity. This means that the second term serves as a
source for helicity generation at small scales in expansion regions. The inset
in figure 3(b) shows the strength of the compression and expansion effects on
the second term of the second-channel viscosity term. When the expansion is
strong, the magnitude of the second term of the second-channel viscosity term
is very large. This validates the expansion effects on the source role of the
second term of the second-channel viscosity term.

In the above statistical analysis, we found that the second term of the
second-channel helicity flux and the second term of the second-channel
viscosity term in expansion regions serve as a self-induced mechanism for
large-scale helicity. Both originate from the variable-density effects in the
vorticity equation, which is neglected in incompressible turbulent flows. In
fully developed helical turbulent flows, the helicity spectrum obeys a power-
law solution as follows:

$H(k)\sim C_{H}\delta\varepsilon^{-1/3}k^{-5/3},$ (21)

where $C_{H}$ is the Kolmogorov constant of the helicity spectrum (Chen  _et
al._ , 2003; Alexakis and Biferale, 2018; Milanese  _et al._ , 2021). In
figure 4, we show the normalized helicity spectrum in the present numerical
simulations with a high Reynold number. Due to the limitation of the
computational cost, the present Reynolds number is still not large to exhibit
an apparent inertial subrange. Even the present Reynolds number is larger than
our previous work (Yan  _et al._ , 2019, 2020b). In addition, we schematically
summarize the proposed induced mechanism in figure 4. At small scales, the
second term of the second-channel viscosity in expansion regions
($D_{\textit{l}}^{22}|\theta_{\textit{l}}>0$) serves as a source role for
helicity generation. At medium scales, the second term of the second-channel
helicity flux in expansion regions
($\Pi_{\textit{l}}^{H22}|\theta_{\textit{l}}>0$) serves as an inverse cross-
scale transfer mechanism. It transfers helicity generated or deposited at
small scales to large scales, and the large-scale helical structures can be
sustained to some extent.

Figure 4: Normalized helicity spectrum.

##  V Conclusions and discussions

To summarize, we propose a self-induced mechanism for large-scale helical
structures in compressible turbulent flows. The self-induction of large-scale
helical structures originates from the intrinsic mechanism of its own system.
The proposed self-induced mechanism provides an alternative theoretical
foundation from the perspective of the intrinsic mechanism at inertial and
small scales. In contrast to the dominant dissipation process, the self-
induced mechanism is composed of the generation of helicity via the viscous
process at small scales and the inverse cascade of helicity via scale
interaction at inertial scales, which is associated with the expansion process
of fluid elements. A high-order and high-resolution DNS with $2048^{3}$
verifies the accuracy of the present statistical analysis.

To comprehensively investigate the statistical characteristics of helicity
transfer in compressible turbulent flows, we select a suitable definition for
large-scale helicity in compressible turbulent flows to represent the
conservative character of helicity transfer, getting rid of pressure and
viscosity terms. Similar to inviscid criteria for large-scale kinetic energy
in compressible turbulent flows (Aluie, 2011; Aluie  _et al._ , 2012; Aluie,
2013; Zhao and Aluie, 2018), large-scale helicity should involve the Favre-
filtered velocity and vorticity, which is defined as
$\widetilde{H}=\widetilde{\textbf{{u}}}\cdot\widetilde{\bm{\omega}}$. Its
governing equation 17 indicates that only SGS transfer exists in the inertial
subrange, without considering pressure and viscosity terms. Based on a simple
algebraic operation, we can prove that the two point-to-point pressure terms
originating from momentum and vorticity governing equations are the same in
compressible turbulent flows, and the dual-channel characteristics of SGS and
viscosity terms are highlighted. Through the conditional averaging method, the
expansion effect on the statistical characteristics of the SGS and viscosity
terms are investigated, and the proposed self-induced mechanism is based on
the above statistical analysis.

In engineering flows, this self-induced mechanism provides an explanation of
high skin-friction and heat flux induced by near-wall helical structures in
highspeed aircraft. This self-induced mechanism opens the way to a more
comprehensive physical understanding of long-period characteristic of large-
scale helical structures, widely existing in the atmosphere, ocean and
turbomachinery.

##  VI Acknowledgements

This work was supported by the National Natural Science Foundation of China
(NSFC Grants No.12302281 and No. 12072349), and the National Key Research and
Development Program of China (2019YFA0405300 and 2020YFA0711800). Z. Y. is
also supported by the China Postdoctoral Science Foundation (No. 2022M710459).
L. F. W. is also supported by NSFC Grant No. 11975053.

##  Appendix A Governing equation of the large-scale helicity
$\overline{H}=\overline{\textbf{{u}}}\cdot\overline{\bm{\omega}}$ in
compressible turbulent flows

Making filter operations on the governing equations of velocity and vorticity,
the following governing equations of filtered velocity and vorticity could be
obtained,

$\frac{\partial\overline{u}_{i}}{\partial
t}+\overline{u}_{j}\frac{\partial\overline{u}_{i}}{\partial
x_{j}}+\overline{u}_{i}\overline{\theta}=-\frac{\partial\overline{\tau}_{ij}}{\partial
x_{j}}-\overline{u_{i}\theta}-\overline{\frac{1}{\rho}\frac{\partial
p}{\partial
x_{i}}}+\frac{1}{Re}\overline{\frac{1}{\rho}\frac{\partial\sigma_{ij}}{\partial
x_{j}}},$ (22) $\begin{split}\frac{\partial\overline{\omega}_{i}}{\partial
t}+\overline{u}_{j}\frac{\partial\overline{\omega}_{i}}{\partial
x_{j}}+\overline{\omega}_{i}\overline{\theta}=&\overline{\omega}_{j}\frac{\partial\overline{u}_{i}}{\partial
x_{j}}-\frac{\partial\overline{\gamma}_{ij}}{\partial
x_{j}}-\overline{\epsilon_{ijk}\frac{\partial}{\partial
x_{j}}\left(\frac{1}{\rho}\frac{\partial p}{\partial x_{k}}\right)}\\\
&+\frac{1}{Re}\overline{\epsilon_{ijk}\frac{\partial}{\partial
x_{j}}\left(\frac{1}{\rho}\frac{\partial\sigma_{kl}}{\partial
x_{l}}\right)},\end{split}$ (23)

where $\overline{\tau}_{ij}$ is the subgrid-scale stress defined as
$\overline{\tau}_{ij}=\overline{u_{i}u_{j}}-\overline{u}_{i}\overline{u}_{j}$,
and $\overline{\gamma}_{ij}$ is the subgrid-scale vortex stretching stress
defined as
$\overline{\gamma}_{ij}=\overline{u_{j}\omega_{i}}-\overline{u_{i}\omega_{j}}-\left(\overline{u}_{j}\overline{\omega}_{i}-\overline{u}_{i}\overline{\omega}_{j}\right)$.

According to the definition of $\overline{H}$, we can get the following
governing equation of large-scale helicity $\overline{H}$ through combining 22
and 23 as

$\frac{\partial\overline{H}}{\partial
t}+\frac{\partial\overline{J}_{j}}{\partial
x_{j}}=-\overline{H}\overline{\theta}-\Pi^{\overline{H}1}-\Pi^{\overline{H}2}+\overline{u_{i}\theta}\overline{\omega}_{i}+\overline{\Phi}+\overline{D}.$
(24)

Here, the spatial transport term $\overline{J}_{j}$, and the first and second
channels of helicity cascade $\Pi^{\overline{H}1}$ and $\Pi^{\overline{H}2}$
are defined as

$\overline{J}_{j}=\overline{u}_{j}\overline{H}-\left(1/2\right)\overline{u}_{i}\overline{u}_{i}\overline{\omega}_{j}+\overline{\omega}_{i}\overline{\tau}_{ij}+\overline{u}_{i}\overline{\gamma}_{ij},$
(25)
$\Pi^{\overline{H}1}=\overline{\tau}_{ij}\frac{\partial\overline{\omega}_{i}}{\partial
x_{j}},\quad\Pi^{\overline{H}2}=\overline{\gamma}_{ij}\frac{\partial\overline{u}_{i}}{\partial
x_{j}}.$ (26)

$\overline{\Phi}$ referring to pressure term and $\overline{D}$ referring to
viscous term are defined as

$\overline{\Phi}=-\overline{\frac{1}{\rho}\frac{\partial p}{\partial
x_{i}}}\overline{\omega}_{i}+\overline{\frac{\epsilon_{ijk}}{\rho^{2}}\frac{\partial\rho}{\partial
x_{j}}\frac{\partial p}{\partial x_{j}}}\overline{u}_{i},$ (27)
$\overline{D}=\frac{1}{Re}\overline{\frac{1}{\rho}\frac{\partial\sigma_{ij}}{\partial
x_{j}}}\overline{\omega}_{i}+\frac{1}{Re}\overline{\epsilon_{ijk}\frac{\partial}{\partial
x_{j}}\left(\frac{1}{\rho}\frac{\partial\sigma_{kl}}{\partial
x_{l}}\right)}\overline{u}_{i}.$ (28)

The first term on the right hand side (r.h.s.) of equation 24 only includes
large-scale flow field, and it act as a resource term at all scales similar to
vortex stretching term of enstrophy governing equation in three-dimensional
turbulent flows. The presence of large-scale source term originating from
velocity field breaks the conservative characteristic of helicity cascade,
except pressure and viscosity terms in compressible turbulent flows. The other
terms on the r.h.s. of equation 24 involve large- and small-scale flow field.
Especially, the two expressions of the inter-scale helicity are the same as
those in incompressible turbulent flows.

##  Appendix B Tensor geometry of the first term of the second-channel
helicity cascade

For the convenience of tensor analysis, we introduce a density-weighted
velocity $\textbf{{v}}=\sqrt{\rho}\textbf{{u}}$ and a filtered density-
weighted velocity
$\textbf{{w}}=\sqrt{\overline{\rho}}\widetilde{\textbf{{u}}}$. This weighted
velocity form is suitable to investigate the spectral properties of physical
variables in compressible flows. The SGS stress can be rewritten
as$\overline{\rho}\widetilde{\tau}_{ij}\equiv\overline{\sqrt{\rho}u_{i}\cdot\sqrt{\rho}u_{j}}-\sqrt{\overline{\rho}}\widetilde{\textit{u}}_{i}\cdot\sqrt{\overline{\rho}}\widetilde{\textit{u}}_{j}\equiv\overline{v_{i}v_{j}}-w_{i}w_{j}$.
Using above definitions, the tensor operation of the first term of the SGS
stress involved in the first term can be written as

$\begin{split}\epsilon_{ijk}\frac{\partial}{\partial
x_{j}}&\left[\frac{\partial}{\partial
x_{m}}\left(\overline{v_{k}v_{m}}\right)\right]\equiv\epsilon_{ijk}\frac{\partial}{\partial
x_{j}}\left(\overline{v_{m}\frac{\partial v_{k}}{\partial
x_{m}}}+\overline{v_{k}\frac{\partial v_{m}}{\partial x_{m}}}\right)\\\
&\equiv\overline{v_{j}\frac{\partial\omega_{i}}{\partial
x_{j}}}-\overline{\omega_{j}\frac{\partial u_{i}}{\partial
x_{j}}}+2\overline{\omega_{i}\theta}+\overline{\epsilon_{ijk}\frac{\partial\theta}{\partial
x_{j}}v_{k}}\\\ &\equiv\frac{\partial}{\partial
x_{j}}\left(\overline{\omega_{i}v_{j}}-\overline{v_{i}\omega_{j}}\right)+\overline{\omega_{i}\theta}+\overline{\epsilon_{ijk}\frac{\partial\theta}{\partial
x_{j}}v_{k}},\end{split}$ (29)

where $\bm{\omega}=\nabla\times\textbf{{v}}$, and
$\theta=\nabla\cdot\textbf{{v}}$. Similarly, the tensor operation of the
second term of the SGS stress involved in the first term can be written as

$\begin{split}\epsilon_{ijk}\frac{\partial}{\partial
x_{j}}&\left[\frac{\partial}{\partial
x_{m}}\left(w_{k}w_{m}\right)\right]\equiv\epsilon_{ijk}\frac{\partial}{\partial
x_{j}}\left(w_{m}\frac{\partial w_{k}}{\partial x_{m}}+w_{k}\frac{\partial
w_{m}}{\partial x_{m}}\right)\\\ &\equiv
w_{j}\frac{\partial\varpi_{i}}{\partial x_{j}}-\varpi_{j}\frac{\partial
w_{i}}{\partial
x_{j}}+2\varpi_{i}\vartheta+\epsilon_{ijk}\frac{\partial\vartheta}{\partial
x_{j}}w_{k}\\\ &\equiv\frac{\partial}{\partial
x_{j}}\left(\varpi_{i}w_{j}-w_{i}\varpi_{j}\right)+\varpi_{i}\vartheta+\epsilon_{ijk}\frac{\partial\vartheta}{\partial
x_{j}}w_{k},\end{split}$

where $\bm{\varpi}=\nabla\times\textbf{{w}}$, and
$\vartheta=\nabla\cdot\textbf{{w}}$. Hence, the SGS term involved in the first
term can be gotten

$\begin{split}\epsilon_{ijk}\frac{\partial}{\partial
x_{j}}\left[\frac{\partial}{\partial
x_{m}}\left(\overline{v_{k}v_{m}}-w_{k}w_{m}\right)\right]&=\frac{\partial}{\partial
x_{j}}\left[\left(\overline{\omega_{i}v_{j}}-\varpi_{i}w_{j}\right)-\left(\overline{v_{i}\omega_{j}}-w_{i}\varpi_{j}\right)\right]+\overline{\omega_{i}\theta}-\varpi_{i}\vartheta+\overline{\epsilon_{ijk}\frac{\partial\theta}{\partial
x_{j}}v_{k}}-\epsilon_{ijk}\frac{\partial\vartheta}{\partial
x_{j}}w_{k}\end{split}$ (30)

We define
$\Upsilon_{ij}=\left(\overline{\omega_{i}v_{j}}-\varpi_{i}w_{j}\right)-\left(\overline{v_{i}\omega_{j}}-w_{i}\varpi_{j}\right)$.
It is an antisymmetric tensor, which is similar to the form in incompressible
turbulent flows (Yan  _et al._ , 2020a).

## References

  * Sanada (1993) T. Sanada, “Helicity production in the transition to chaotic flow simulated by navier-stokes equation,” Phys. Rev. Lett. 70, 3035–3038 (1993).
  * Moffatt and Tsinober (1992) H. K. Moffatt and A. Tsinober, “Helicity in laminar and turbulence flow,” Annu. Rev. Fluid Mech. 24, 281–312 (1992).
  * Yan  _et al._ (2020a) Z. Yan, X. L. Li, C. P. Yu, J. C. Wang, and S. Y. Chen, “Dual channels of helicity cascade in turbulent flows,” J. Fluid Mech. 894, R2 (2020a).
  * Agoua  _et al._ (2021) W. Agoua, B. Favier, A. Delache, A. Briard, and W. J. T. Bos, “Spontaneous generation and reversal of helicity in anisotropic turbulence,” Phys. Rev. E 103, L061101 (2021).
  * Teitelbaum and Mininni (2009) T. Teitelbaum and P. D. Mininni, “Effect of helicity and rotation on the free decay of turbulent flows,” Phys. Rev. Lett. 103, 014501 (2009).
  * Bos and Rubinstein (2013) W. J. T. Bos and R. Rubinstein, “On the strength of the nonlinearity in isotropic turbulence,” J. Fluid Mech. 733, 158–170 (2013).
  * Linkmann (2018) M. Linkmann, “Effects of helicity on dissipation in homogeneous box turbulence,” J. Fluid Mech. 856, 79–102 (2018).
  * Biferale  _et al._ (2012) L. Biferale, S. Musacchio, and F. Toschi, “Inverse energy cascade in three-dimensional isotropic turbulence.” Phys. Rev. Lett. 108, 164501 (2012).
  * Sahoo  _et al._ (2017) G. Sahoo, A. Alexakis, and L. Biferale, “Discontinuous transition from direct to inverse cascade in three-dimensional turbulence,” Phys. Rev. Lett. 118, 164501 (2017).
  * Yu  _et al._ (2022) C. P. Yu, R. N. Hu, Z. Yan, and X. L. Li, “Helicity distributions and transfer in turbulent channel flows with streamwise rotation,” J. Fluid Mech. 940, A18 (2022).
  * Moffatt (1969) H. K. Moffatt, “Degree of knottedness of tangled vortex lines,” J. Fluid Mech. 35, 117–129 (1969).
  * Moffatt (2014) H. K. Moffatt, “Helicity and singular structures in fluid dynamics.” Proc. Natl. Acad. Sci. U.S.A. 111, 3663–3670 (2014).
  * Scheeler  _et al._ (2014) M. W. Scheeler, D. Klecknera, D. Promentb, G. L. Kindlmann, and W. T. M. Irvine, “Helicity conservation by flow across scales in reconnecting vortex links and knots,” Proc. Natl. Acad. Sci. U.S.A. 111, 15350–15355 (2014).
  * Scheeler  _et al._ (2017) M. W. Scheeler, W. M. van Rees, H. Kedia, D. Kleckner, and W. T. M. Irvine, “Complete measurement of helicity and its dynamics in vortex tubes,” Science 357, 487–491 (2017).
  * Pouquet and Mininni (2010) A. Pouquet and P. D. Mininni, “The interplay between helicity and rotation in turbulence: implications for scaling laws and small-scale dynamics,” Philos. Trans. R. Soc. London, Ser. A 368, 1635–1662 (2010).
  * Moffatt (1978) H. K. Moffatt,  _Field Generation in Electrically Conducting Fluids_ (Cambridge University Press, 1978).
  * Hide (1976) R. Hide, “A note on helicity.” Geophys. Astrophys. Fluid Dyn. 7, 157–161 (1976).
  * Frisch  _et al._ (1987) U. Frisch, Z. She, and P. Sulem, “Large-scale flow driven by the anisotropic kinetic alpha effect.” Physica D 28, 382–392 (1987).
  * Marino  _et al._ (2013) R. Marino, P. D. Mininni, D. Rosenberg, and A. Pouquet, “Emergence of helicity in rotating stratified turbulence,” Phys. Rev. E 87, 033016 (2013).
  * Melander and Hussain (1993) M. V. Melander and F. Hussain, “Polarized vorticity dynamics on a vortex column,” Phys. Fluids 5, 1992–2003 (1993).
  * Kida and Takaoka (1988) S. Kida and M. Takaoka, “Reconnection of vortex tubes,” Fluid Dyn. Res. 3, 257–261 (1988).
  * Irvine (2018) W. T. M. Irvine, “Moreau’s hydrodynamic helicity and the life of vortex knots and links,” C. R. Méc. 346, 170–174 (2018).
  * Yao and Hussain (2020) J. Yao and F. Hussain, “A physical model of turbulence cascade via vortex reconnection sequence and avalanche,” J. Fluid Mech. 883, A51 (2020).
  * Yao  _et al._ (2021) J. Yao, Y. Yang, and F. Hussain, “Dynamics of a trefoil knotted vortex,” J. Fluid Mech. 923, A19 (2021).
  * Reshetnyak (2012) M. Yu. Reshetnyak, “Effect of compressibility on the generation of hydrodynamic helicity in the earth’s liquid core,” Geomagn. Aeron. 52, 398–402 (2012).
  * Serre (2018) D. Serre, “Helicity and other conservation laws in perfect fluid motion,” C. R. Mec. 346, 175–183 (2018).
  * John and Donzis (2019) P. J. John and D. A. Donzis, “Solenoidal scaling laws for compressible mixing,” Phys. Rev. Lett. 123, 224501 (2019).
  * Kivotides and Leonard (2021) D. Kivotides and A. Leonard, “Helicity spectra and topological dynamics of vortex links at high reynolds numbers,” J. Fluid Mech. 911, A25 (2021).
  * Wu and Bos (2022) T. Wu and W. J. T. Bos, “Cascades of enstrophy and helicity in turbulence without vortex stretching,” Phys. Rev. Fluids 7, 094601 (2022).
  * Wang  _et al._ (2013) J. Wang, Y. Yang, Y. Shi, Z. Xiao, X. T. He, and S. Chen, “Cascade of kinetic energy in three-dimensional compressible turbulence,” Phys. Rev. Lett. 110, 214505 (2013).
  * Yan  _et al._ (2020b) Z. Yan, X. L. Li, C. P. Yu, and J. C. Wang, “Cross-chirality transfer of kinetic energy and helicity in compressible helical turbulence,” Phys. Rev. Fluids 5, 084604 (2020b).
  * Wang  _et al._ (2010) J. Wang, L.-P. Wang, Z. Xiao, Y. Shi, and S. Chen, “A hybrid numerical simulation of isotropic compressible turbulence,” J. Comput. Phys. 229, 5257–5259 (2010).
  * Samtaney  _et al._ (2001) R. Samtaney, D. I. Pullin, and B. Kosović, “Direct numerical simulation of decaying compressible turbulence and shocklet statistics.” Phys. Fluids 13, 1415–1430 (2001).
  * Wang  _et al._ (2012a) J. Wang, Y. Shi, L. P. Wang, Z. Xiao, X. T. He, and S. Chen, “Effect of compressibility on the small-scale structures in isotropic turbulence.” J. Fluid Mech. 713, 588–631 (2012a).
  * Zhao and Aluie (2018) D. Zhao and H. Aluie, “Inviscid criterion for decomposing scales,” Phys. Rev. Fluids 3, 054603 (2018).
  * Yan  _et al._ (2019) Z. Yan, X. Li, J. Wang, and C. Yu, “Effect of pressure on joint cascade of kinetic energy and helicity in compressible helical turbulence,” Phys. Rev. E 99, 033114 (2019).
  * Block  _et al._ (2006) D. Block, I. Teliban, F. Greiner, and A. Piel, “Prospects and limitations of conditional averaging.” Phys. Scr. T122, 25–33 (2006).
  * Wang  _et al._ (2012b) J. Wang, Y. Shi, L. Wang, Z. Xiao, X. T. He, and S. Chen, “Scaling and statistics in three-dimensional compressible turbulence,” Phys. Rev. Lett. 108, 214505 (2012b).
  * Aluie (2011) H. Aluie, “Compressible turbulence: the cascade and its locality,” Phys. Rev. Lett. 106, 174502 (2011).
  * Chen  _et al._ (2003) Q. Chen, S. Chen, G. L. Eyink, and D. D. Holm, “Intermittency in the joint cascade of energy and helicity.” Phys. Rev. Lett. 90, 214503 (2003).
  * Alexakis and Biferale (2018) A. Alexakis and L. Biferale, “Cascades and transitions in turbulent flows.” Phys. Rep. 767, 1–101 (2018).
  * Milanese  _et al._ (2021) L. M. Milanese, N. F. Loureiro, and S. Boldyrev, “Dynamic phase alignment in navier-stokes turbulence,” Phys. Rev. Lett. 127, 274501 (2021).
  * Aluie  _et al._ (2012) H. Aluie, S. Li, and H. Li, “Conservative cascade of kinetic energy in compressible turbulence,” Astrophys. J. 751, L29 (2012).
  * Aluie (2013) H. Aluie, “Scale decomposition in compressible turbulence.” Physica D 247, 54–65 (2013).
  * Dolling (2001) D. S. Dolling, “Fifty years of shock-wave/boundary-layer interaction research: What next?” AIAA J. 39, 1517–1531 (2001).
  * Duan  _et al._ (2021) J. Y. Duan, X. Li, X. L. Li, and H. W. Liu, “Direct numerical simulation of a supersonic turbulent boundary layer over a compression–decompression corner,” Phys. Fluids 33, 065111 (2021).
  * Li  _et al._ (2010) X. Li, D. Fu, Y.Ma, and X. Liang, “Direct numerical simulation of shock/turbulent boundary layer interaction in a supersonic compression ramp,” Sci. China Phys. Mech. Astron. 53, 1651–1658 (2010).
  * Pirozzoli  _et al._ (2004) S. Pirozzoli, F. Grasso, and T. B. Gatski, “Direct numerical simulation and analysis of a spatially evolving supersonic turbulent boundary layer at M=2.25,” Phys. Fluids 16, 530–545 (2004).
  * Wu and Martin (2007) M. Wu and M. P. Martin, “Direct numerical simulation of supersonic turbulent boundary layer over a compression ramp,” AIAA J. 45, 879–889 (2007).
  * Wu and Moin (2009) X. Wu and P. Moin, “Direct numerical simulation of turbulence in a nominally zero-pressure-gradient flat-plate boundary layer,” J. Fluid Mech. 630, 5–41 (2009).
  * Yan  _et al._ (2022) Z. Yan, X. L. Li, and C. P. Yu, “Helicity budget in turbulent channel flows with streamwise rotation,” Phys. Fluids 34, 065105 (2022).