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湍流积分尺度修正及其对湍流耗散规律的影响

王宇杰 杨君聖 王杰 周彦宏 刘锋

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湍流积分尺度修正及其对湍流耗散规律的影响

王宇杰, 杨君聖, 王杰, 周彦宏, 刘锋

Correction of the turbulent integral length scale and its impact on turbulent dissipation law

WANG Yujie, YANG Junsheng, WANG Jie, ZHOU Yanhong, LIU Feng
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  • 湍流直接数值模拟受限于计算域尺寸,无法完全解析湍动能谱低波数区的所有波数,造成计算数据中部分大尺度信息丢失.随着湍流的演化,湍动能谱的峰值波数会向低波数迁移,使得低波数缺失现象进一步加剧,导致所计算的积分尺度和湍流耗散相关统计量偏离物理真实.本研究基于von Kármán谱模型的推广形式,充分考虑数值计算未完全解析的低波数区湍动能谱,并利用该模型对均匀各向同性自由衰减湍流的积分尺度和湍流耗散相关统计量进行修正.研究结果表明:修正后的积分尺度L显著高于未修正值,且其随时间的变化规律符合Saffmann理论预测的$L \propto t^{2 / 5}$幂律关系;修正前湍流耗散系数Cε为常数,说明此时湍流为均衡状态,而修正后耗散系数Cε的演化满足湍流非均衡耗散规律$C_{\varepsilon} \sim R e_\lambda^{-1}$.将数值计算缺失的低波数区湍动能谱引入后,湍流状态由均衡向非均衡转变,说明大尺度对湍流耗散有很强的调控作用,这与学术界普遍认为的大尺度结构是造成湍流非均衡性本质原因的结论相一致.在有限雷诺数或者受初始条件影响较大的湍流流动中,大尺度结构对流动的影响显著,湍流无法在全尺度实现均衡.
    Turbulence modeling relies critically on accurate characterization of large-scale structures, with the integral length scale L serving as a key parameter for industrial applications ranging from combustion stability optimization to wind farm design and aerodynamic load prediction. However, Direct numerical simulation (DNS) of turbulence faces inherent limitations in resolving all wavenumbers within the lowwavenumber region of the turbulent kinetic energy spectrum due to finite computational domain sizes. This unresolved low-wavenumber deficiency leads to incomplete characterization of large-scale structures and introduces systematic deviations in key statistical quantities, particularly the integral length scale L and turbulence dissipation coefficient Cε. As turbulence evolves, the spectral peak wavenumber kp migrates toward lower wavenumbers, exacerbating the loss of large-scale information and causing computed statistics to diverge from physical reality. In this study, we perform high-fidelity DNS of homogeneous isotropic decaying turbulence in a periodic cubic domain of side length 4π with 3843 grid points. DNS cases are performed by using a standard pseudospectral solver and a fourth-order Runge-Kutta time integration scheme, with a semi-implicit treatment of the viscous term. The spatial resolution kmaxη = 1.65 ensures adequate resolution of dissipative scales (η is the Kolmogorov scale). Simulations start from a fully developed field initialized with a spectrum matching Comte-Bellot and Corrsin’s experimental data and evolve within a time interval where turbulence exhibits established isotropic decay characteristics. Existing correction models, predominantly based on equilibrium turbulence assumptions, fail to capture the non-equilibrium dynamics governed by large-scale structures. Based on a generalized von Kármán spectrum model, we use a correction framework to account for unresolved low-wavenumber contributions in homogeneous isotropic decaying turbulence. DNS data reveal that the uncorrected integral scale Lm significantly underestimates the true L, with errors escalating as kL/kp increases, where kL is the minimum resolvable wavenumber in the simulation domain. After correction, L exhibits a temporal evolution following the Saffmann-predicted power-law relationship $L \propto t^{2 / 5}$, contrasting sharply with the underestimated pre-correction values. Despite the spectral correction substantially increasing the spectral integral scale L, its value remains less than the physically derived integral scale Λ computed from the velocity correlation function, primarily due to the finite domain size limiting large-scale statistics and the moderate grid resolution, though higher-resolution simulations with the same domain show L converging towards Λ. Notably, the unmodified dissipation coefficient Cε remains constant, consistent with equilibrium turbulence assumptions, whereas the corrected Cε evolves according to the non-equilibrium scaling law $C_{\varepsilon} \sim R e_\lambda^{-1}$. Further analysis confirms that the ratio L/λ shifts from Kolmogorov’s $R e_\lambda^1$ dependence to a Reynolds-number-independent plateau after correction, fundamentally altering the turbulence dissipation paradigm. This transition from equilibrium to non-equilibrium dissipation behavior underscores the dominant role of large-scale structures in regulating energy cascade dynamics. Our results demonstrate that finite Reynolds numbers or strong initial-condition effects amplify the nonequilibrium characteristics of turbulence, preventing full-scale equilibrium. These findings reconcile long-standing theoretical discrepancies and provide a paradigm for modeling scale interactions in turbulence.
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