论文标题
关于随机强迫Navier-Stokes方程的动荡法
On the zeroth law of turbulence for the stochastically forced Navier-Stokes equations
论文作者
论文摘要
我们认为在没有边界的情况下，遭受白色时间（彩色空间）强迫的三维随机迫使Navier-Stokes方程。时间平均耗能率的平均值的上限和下限，$ \ mathbb {e} [\ langle \ varepsilon \ rangle] $是直接从方程式得出的。首先，我们表明，对于弱（mar）的解决方案（martingale）解决方案，\ [\ mathbb {e} [\ langle \ langle \ varepsilon \ rangle] \ leq g^2 +（2 + \ frac \ frac {1} {1} {1} {1} {1} {re} {re freac g^$ is $ is $ g^3}随机力量提供的费率，$ u $是根平方的速度，$ l $是应用强迫函数中最长的长度比例，而$ re $ $是雷诺数。在能量平等的额外假设下，如果随机力给出的能量速率主导流动的确定性行为，那么我们还会得出一个下限。我们获得，\ [\ [\ frac {1} {3} g^2 - \ frac {1} {3}（2+ \ frac {1} {1} {re}）\ frac {u^3} {u^3} {l} {l} \ leq \ leq \ mathb {e} \ frac {1} {re}）\ frac {u^3} {l} \，。 g^2。\]此外，我们还获得了模型耗散率的方差估计。
We consider three-dimensional stochastically forced Navier-Stokes equations subjected to white-in-time (colored-in-space) forcing in the absence of boundaries. Upper and lower bounds of the mean value of the time-averaged energy dissipation rate, $\mathbb{E} [\langle\varepsilon \rangle] $, are derived directly from the equations. First, we show that for a weak (martingale) solution to the stochastically forced Navier-Stokes equations, \[ \mathbb{E} [\langle\varepsilon \rangle] \leq G^2 + (2+ \frac{1}{Re})\frac{U^3}{L},\] where $G^2$ is the total energy rate supplied by the random force, $U$ is the root-mean-square velocity, $L$ is the longest length scale in the applied forcing function, and $Re$ is the Reynolds number. Under an additional assumption of energy equality, we also derive a lower bound if the energy rate given by the random force dominates the deterministic behavior of the flow in the sense that $G^2 > 2 F U$, where $F$ is the amplitude of the deterministic force. We obtain, \[\frac{1}{3} G^2 - \frac{1}{3} (2+ \frac{1}{Re})\frac{U^3}{L} \leq \mathbb{E} [\langle\varepsilon \rangle] \leq G^2 + (2+ \frac{1}{Re})\frac{U^3}{L}\,.\] In particular, under such assumptions, we obtain the zeroth law of turbulence in the absence of the deterministic force as, \[\mathbb{E} [\langle\varepsilon \rangle] = \frac{1}{2} G^2.\] Besides, we also obtain variance estimates of the dissipation rate for the model.
