Kaneda’s ( J. Fluid Mech. , vol. 107, 1981, pp. 131–145) Lagrangian renormalized approximation was extended to single-time spectral closure under two assumptions: (i) Markovianization and (ii) the Lagrangian velocity response function is expressed by $G(k,\unicode[STIX]{x1D70F})=\exp (-C_{1}(k)\unicode[STIX]{x1D70F}-C_{2}(k)\unicode[STIX]{x1D70F}^{2}/2)$ . The unknown functions $C_{1}(k)$ and $C_{2}(k)$ are theoretically derived to be consistent with the exact short-time behaviour of $G(k,\unicode[STIX]{x1D70F})$ and the asymptotic short-time behaviour of assumed exponential form of $G(k,\unicode[STIX]{x1D70F})$ , i.e. the present closure is derived from the Navier–Stokes equation without introduction of any adjustable parameters and it can calculate the statistical quantities by theory. The results show that the present closure has good agreement with direct numerical simulation for single- and two-point statistics.

中文翻译：

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流体湍流中的单次马尔可夫化谱闭合

Kaneda (J. Fluid Mech. , vol. 107, 1981, pp. 131–145) 拉格朗日重归一化近似在两个假设下扩展到单时间谱闭合：(i) 马尔可夫化和 (ii) 拉格朗日速度响应函数被表示通过 $G(k,\unicode[STIX]{x1D70F})=\exp (-C_{1}(k)\unicode[STIX]{x1D70F}-C_{2}(k)\unicode[STIX]{x1D70F }^{2}/2)$ 。未知函数 $C_{1}(k)$ 和 $C_{2}(k)$ 理论上推导出来与 $G(k,\unicode[STIX]{x1D70F}) 的确切短时行为一致$ 和 $G(k,\unicode[STIX]{x1D70F})$ 的假定指数形式的渐近短时行为，即当前的闭包是从 Navier-Stokes 方程导出的，没有引入任何可调参数，它可以用理论计算统计量。