Generalised higher-order Kolmogorov scales
Generalised higher-order Kolmogorov scales
复制标题
DOI：
10.1017/jfm.2016.172
复制
发表时间：
2016-03
影响因子：
3.7
通讯作者：
J. Boschung;F. Hennig;M. Gauding;H. Pitsch;N. Peters
中科院分区：
文献类型：
--
作者：
J. Boschung;F. Hennig;M. Gauding;H. Pitsch;N. Peters
Kolmogorov引入了基于平均耗散和粘性的耗散标度，即Kolmogorov长度${it\eta}=({it\nu}^{3}/\lange{\it\varepsilon}\range)^{1/4}$和速度$u_{{it\eta}}=({it\nu}\lange{\it\varepsilon}\range)^{1/4}$。然而，在基于唯象间歇模型的文献中，已经讨论了较小尺度的存在。在这里，我们引入了偶数阶纵向结构函数的精确耗散尺度。该公式的推导是基于纵向速度梯度的偶数阶矩$(\Partial u_1}/\Partial x_{1})^{2m}$与耗散${w<sup>m}</sup>的精确关系。然后我们找到一个新的长度刻度${\it\eta}_{C，m}=({\it\nu}^{3}/\lang^\it\varepsilon}^{m/2}\rang^{2/m})^{1/4}$和$u_{C，m}=({\it\nu}\lang{\it\varepsilon}^{m/2}\Rangel^{2/m})^{1/4}$，也就是说，耗散尺度更多地依赖于耗散矩，从而取决于全概率密度函数(p.d.f)$P({\it\varepsilon})，而不是平均值的幂。在(局部)各向同性、(局部)均匀和不可压缩的假设下，本文给出的结果对于纵向偶序结构函数是精确的，并且我们发现它们在经验上也适用于混合、横向和奇数阶结构函数。我们使用从$Re_{{\it\lambda}}=88$到$Re_{{\it\lambda}}=754$的雷诺数的直接数值模拟来比较不同的标度。我们发现，作为雷诺数的函数，$P(Varepsilon)$，或者更准确地说，$Langle{varepsilon}^{m/2}\Rangel^{m/2}是一个关键参数，因为它决定了速度梯度矩的比例以及速度梯度p.d.f的比例。由于${\it\eta}_{C，m}$小于${\it\eta}$，这导致修改了DNS所需的网格点的估计。
Kolmogorov introduced dissipative scales based on the mean dissipation $\langle {\it\varepsilon}\rangle$ and the viscosity ${\it\nu}$ , namely the Kolmogorov length ${\it\eta}=({\it\nu}^{3}/\langle {\it\varepsilon}\rangle )^{1/4}$ and the velocity $u_{{\it\eta}}=({\it\nu}\langle {\it\varepsilon}\rangle )^{1/4}$ . However, the existence of smaller scales has been discussed in the literature based on phenomenological intermittency models. Here, we introduce exact dissipative scales for the even-order longitudinal structure functions. The derivation is based on exact relations between even-order moments of the longitudinal velocity gradient $(\partial u_{1}/\partial x_{1})^{2m}$ and the dissipation $\langle {\it\varepsilon}^{m}\rangle$ . We then find a new length scale ${\it\eta}_{C,m}=({\it\nu}^{3}/\langle {\it\varepsilon}^{m/2}\rangle ^{2/m})^{1/4}$ and $u_{C,m}=({\it\nu}\langle {\it\varepsilon}^{m/2}\rangle ^{2/m})^{1/4}$ , i.e. the dissipative scales depend rather on the moments of the dissipation $\langle {\it\varepsilon}^{m/2}\rangle$ and thus the full probability density function (p.d.f.) $P({\it\varepsilon})$ instead of powers of the mean $\langle {\it\varepsilon}\rangle ^{m/2}$ . The results presented here are exact for longitudinal even-ordered structure functions under the assumptions of (local) isotropy, (local) homogeneity and incompressibility, and we find them to hold empirically also for the mixed and transverse as well as odd orders. We use direct numerical simulations (DNS) with Reynolds numbers from $Re_{{\it\lambda}}=88$ up to $Re_{{\it\lambda}}=754$ to compare the different scalings. We find that indeed $P({\it\varepsilon})$ or, more precisely, the scaling of $\langle {\it\varepsilon}^{m/2}\rangle /\langle {\it\varepsilon}\rangle ^{m/2}$ as a function of the Reynolds number is a key parameter, as it determines the ratio ${\it\eta}_{C,m}/{\it\eta}$ as well as the scaling of the moments of the velocity gradient p.d.f. As ${\it\eta}_{C,m}$ is smaller than ${\it\eta}$ , this leads to a modification of the estimate of grid points required for DNS.