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.

中文翻译：

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基于拉格朗日闭包理论的均匀各向同性非镜对称湍流尺度相似结构

我们基于拉格朗日重整化近似 (LRA) 研究螺旋性对均匀各向同性和非镜面对称湍流的尺度相似结构的影响，这是一种由 Kaneda 提出的自洽闭包理论。J.流体机械。, 卷。107，1981，第 131-145 页）。在这项研究中，我们关注代表尺度相似范围的时间尺度。对于 LRA，拉格朗日二次速度相关性和响应函数决定了代表时间尺度。LRA 预测拉格朗日两次速度相关性和响应函数方程都没有明确依赖于螺旋度。我们通过考虑螺旋耗散率来假设扩展的尺度相似光谱和时间尺度。考虑到小尺度结构，对能量和螺旋度通量和响应函数方程的尺度相似的要求，产生了传统的惯性范围幂律，并提供了能量和螺旋度谱 $\propto k^{-5/3}$ 和时间尺度 $\propto \varepsilon ^{-1/3} k^{-2/3}$ ， 在哪里 $\伐普西隆$ 和 $k$ 分别表示能量耗散率和波数。值得注意的是，能量通量只有在 $k^H /k \ll 1$ ， 在哪里 $k^H = \varepsilon ^H/\varepsilon$ 和 $\伐普西隆 ^H$ 表示螺旋度耗散率。这种条件使得尺度相似范围内的能量级联过程完全独立于螺旋度。我们还研究了 LRA 能量和螺旋级联中尺度间相互作用的局部性。我们证明，与能量级联相比，螺旋级联在尺度上略微非局部。这项研究为非镜面对称湍流的建模奠定了基础。