Abstract
We investigate the effect of helicity on the scale-similar structures of homogeneous isotropic and non-mirror-symmetric turbulence based on the Lagrangian renormalised approximation (LRA), which is a self-consistent closure theory proposed by Kaneda (J. Fluid Mech., vol. 107, 1981, pp. 131–145). In this study, we focus on the time scale representing the scale-similar range. For the LRA, the Lagrangian two-time velocity correlation and response function determine the representative time scale. The LRA predicts that both the Lagrangian two-time velocity correlation and response function equation do not explicitly depend on helicity. We assume the extended scale-similar spectra and time scale by considering the helicity dissipation rate. Considering the small-scale structures, the requirements for the energy and helicity fluxes and response function equation to be scale similar, yield the conventional inertial-range power laws and provide the energy and helicity spectra $\propto k^{-5/3}$ and the time scale $\propto \varepsilon ^{-1/3} k^{-2/3}$ , where $\varepsilon$ and $k$ denote the energy dissipation rate and wavenumber, respectively. Notably, energy flux can be scale similar only when $k^H /k \ll 1$ , where $k^H = \varepsilon ^H/\varepsilon$ and $\varepsilon ^H$ denotes the helicity dissipation rate. This condition makes the energy cascade process in the scale-similar range completely independent of helicity. We also investigate the localness of the interscale interaction in the energy and helicity cascades for the LRA. We demonstrate that the helicity cascade is slightly non-local in scales compared with the energy cascade. This study provides a foundation on the modelling of non-mirror-symmetric turbulent flows.
Highlights
In three-dimensional (3-D) hydrodynamic turbulent flows, the scale-similarity law proposed by Kolmogorov (1941b) has been foundational to turbulence theories
We investigate the effect of helicity on the scale-similar structures of homogeneous isotropic and non-mirror-symmetric turbulence based on the Lagrangian renormalised approximation (LRA), which is a self-consistent closure theory proposed by Kaneda
The abridged Lagrangian history DIA (ALHDIA) predicts that the equation for the mirror-symmetric part of the two-time velocity correlation based on the generalised velocity depends on helicity
Summary
In three-dimensional (3-D) hydrodynamic turbulent flows, the scale-similarity law proposed by Kolmogorov (1941b) (which we will refer to as K41) has been foundational to turbulence theories. In the aforementioned studies on pure helicity cascade or inverse energy cascade, some modification of the nonlinear interaction or spacial forcing are essential elements for realising the maximally helical condition |EH(k)| = 2kE(k) (Biferale et al 2012, 2013; Kessar et al 2015; Sahoo et al 2015; Stepanov et al 2015; Plunian et al 2020) In this regard, it would be beneficial to investigate the statistical similarity achieved by the pure nonlinearity of the Navier–Stokes equations. Waleffe (1992) analysed the statistical properties of the Navier–Stokes equations by employing the scale-similar structure and the two detailed balance (2.7) and (2.14), which verified that the nonlinear interaction of the velocity fields comprising the same sign helical modes can trigger an inverse transfer of energy. The relationship between this tendency and the property of the closure equations is discussed in Appendix A
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