arXiv 0912.20072009-12-10physics.flu-dyn

一维Burgers流中陡梯度的能量耗散和分辨

Energy dissipation and resolution of steep gradients in one-dimensional Burgers flows

Chuong V. Tran, David G. Dritschel

备注：: 6--7 journal pages, 5 figures, submitted to Physics of Fluids
DOI：: 10.1063/1.3327284

摘要：

已知具有适当形状的光滑初始波形的无粘Burgers方程的行波解在有限时间内产生激波（无限梯度）。这种奇异解的特征在于能量谱，其与波数$k$成比例为$k^{-2}$。* 在这项研究中，我们进行了一项分析，验证了上述动力学特征，并推导出$ε$和$N$的上限。发现$ε$满足$ε\le ν^{2α-1}\norm{u_*}_\infty^{2(1-α)} \norm{(-Δ)^{α/2}u_*}^2$，其中$α<1$和$u_*=u(x,t_*)$是$t=t_*$处的速度场。给定极限$ν\to0$中的$ε>0$，这意味着能谱保持不比该极限中的$k^{-2}$陡。对于临界$k^{-2}$标度，$ε$的界限减少到$ε\le\sqrt{3}k_0\norm{u_0}_\infty\norm{u_0}^2$，其中$k_0$标记惯性范围的下限，而$u_0=u(x,0)$标记惯性范围的下限。这意味着$N\le\sqrt{3}L\norm{u_0}_\infty/ν$，其中$L$是域的大小，这是一致的严格估计的自由度定义的本地李雅普诺夫指数的数量。我们证明了解析和数值的一个实例中，$k^{-2}$缩放是唯一可实现的。数值也返回$ε$和$t_*$，与从相应的极限弱解导出的解析值一致。

英文摘要：

Travelling-wave solutions of the inviscid Burgers equation having smooth initial wave profiles of suitable shapes are known to develop shocks (infinite gradients) in finite times. Such singular solutions are characterized by energy spectra that scale with the wave number $k$ as $k^{-2}$. **** In this study, we carry out an analysis which verifies the dynamical features described above and derive upper bounds for $ε$ and $N$. It is found that $ε$ satisfies $ε\le ν^{2α-1}\norm{u_*}_\infty^{2(1-α)} \norm{(-Δ)^{α/2}u_*}^2$, where $α<1$ and $u_*=u(x,t_*)$ is the velocity field at $t=t_*$. Given $ε>0$ in the limit $ν\to0$, this implies that the energy spectrum remains no steeper than $k^{-2}$ in that limit. For the critical $k^{-2}$ scaling, the bound for $ε$ reduces to $ε\le\sqrt{3}k_0\norm{u_0}_\infty\norm{u_0}^2$, where $k_0$ marks the lower end of the inertial range and $u_0=u(x,0)$. This implies $N\le\sqrt{3}L\norm{u_0}_\infty/ν$, where $L$ is the domain size, which is shown to coincide with a rigorous estimate for the number of degrees of freedom defined in terms of local Lyapunov exponents. We demonstrate both analytically and numerically an instance where the $k^{-2}$ scaling is uniquely realizable. The numerics also return $ε$ and $t_*$, consistent with analytic values derived from the corresponding limiting weak solution.

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