From 01f47ace6756c28deb9ea0daaee3904ffa5ce9e0 Mon Sep 17 00:00:00 2001 From: Ian Jauslin Date: Thu, 11 Jan 2018 22:48:14 +0000 Subject: Initial commit --- docs/nstrophy_doc/nstrophy_doc.tex | 146 +++++++++++++++++++++++++++++++++++++ 1 file changed, 146 insertions(+) create mode 100644 docs/nstrophy_doc/nstrophy_doc.tex (limited to 'docs/nstrophy_doc/nstrophy_doc.tex') diff --git a/docs/nstrophy_doc/nstrophy_doc.tex b/docs/nstrophy_doc/nstrophy_doc.tex new file mode 100644 index 0000000..fa5fff0 --- /dev/null +++ b/docs/nstrophy_doc/nstrophy_doc.tex @@ -0,0 +1,146 @@ +\documentclass{ian} + +\usepackage{largearray} + +\begin{document} + +\hbox{} +\hfil{\bf\LARGE +{\tt nstrophy} +} +\vfill + +\tableofcontents + +\vfill +\eject + +\setcounter{page}1 +\pagestyle{plain} + +\section{Description of the computation} +\subsection{Irreversible equation} +\indent Consider the {\it irreversible} Navier-Stokes equation in 2 dimensions +\begin{equation} + \partial_tu=\nu\Delta u+g-\nabla w-(u\cdot\nabla)u,\quad + \nabla\cdot u=0 + \label{ins} +\end{equation} +in which $g$ is the forcing term and $w$ is the pressure. +We take periodic boundary conditions, so, at every given time, $u(t,\cdot)$ is a function on the unit torus $\mathbb T^2:=\mathbb R^2/\mathbb Z^2$. We represent $u(t,\cdot)$ using its Fourier series +\begin{equation} + \hat u_k(t):=\int_{\mathbb T^2}dx\ e^{2i\pi kx}u(t,x) +\end{equation} +for $k\in\mathbb Z^2$, and rewrite~\-(\ref{ins}) as +\begin{equation} + \partial_t\hat u_k= + -4\pi^2\nu k^2\hat u_k+\hat g_k-2i\pi k\hat w_k + -2i\pi\sum_{\displaystyle\mathop{\scriptstyle p,q\in\mathbb Z^2}_{p+q=k}} + (q\cdot\hat u_p)\hat u_q + ,\quad + k\cdot\hat u_k=0 + \label{ins_k} +\end{equation} +We then reduce the equation to a scalar one, by writing +\begin{equation} + \hat u_k=\frac{2i\pi k^\perp}{|k|}\hat\varphi_k\equiv\frac{2i\pi}{|k|}(-k_y\hat\varphi_k,k_x\hat\varphi_k) +\end{equation} +in terms of which, multiplying both sides of the equation by $\frac{k^\perp}{|k|}$, +\begin{equation} + \partial_t\hat \varphi_k= + -4\pi^2\nu k^2\hat \varphi_k+\frac{k^\perp}{2i\pi|k|}\cdot\hat g_k + -\frac{2i\pi}{|k|}\sum_{\displaystyle\mathop{\scriptstyle p,q\in\mathbb Z^2}_{p+q=k}} + (q\cdot p^\perp)(k^\perp\cdot q^\perp)\hat\varphi_p\hat\varphi_q + \label{ins_k} +\end{equation} +which, since $q\cdot p^\perp=q\cdot(k^\perp-q^\perp)=q\cdot k^\perp$, is +\begin{equation} + \partial_t\hat \varphi_k= + -4\pi^2\nu k^2\hat \varphi_k+\frac{k^\perp}{2i\pi|k|}\cdot\hat g_k + +\frac{4\pi^2}{|k|}\sum_{\displaystyle\mathop{\scriptstyle p,q\in\mathbb Z^2}_{p+q=k}} + (q\cdot k^\perp)(k\cdot q)\frac{\hat\varphi_p\hat\varphi_q}{|p||q|} + . + \label{ins_k} +\end{equation} +We truncate the Fourier modes and assume that $\hat\varphi_k=0$ if $|k_1|>K_1$ or $|k_2|>K_2$. Let +\begin{equation} + \mathcal K:=\{(k_1,k_2),\ |k_1|\leqslant K_1,\ |k_2|\leqslant K_2\} + . +\end{equation} +\bigskip + +\point{\bf FFT}. We compute the last term in~\-(\ref{ins_k}) +\begin{equation} + T(\hat\varphi,k):= + \sum_{\displaystyle\mathop{\scriptstyle p,q\in\mathbb Z^2}_{p+q=k}} + \frac{\hat\varphi_p}{|p|} + (q\cdot k^\perp)(k\cdot q)\frac{\hat\varphi_q}{|q|} +\end{equation} +using a fast Fourier transform, defined as +\begin{equation} + \mathcal F(f)(n):=\sum_{m\in\mathcal N}e^{-\frac{2i\pi}{N_1}m_1n_1-\frac{2i\pi}{N_2}m_2n_2}f(m_1,m_2) +\end{equation} +where +\begin{equation} + \mathcal N:=\{(n_1,n_2),\ 0\leqslant n_1\leqslant N_1,\ 0\leqslant n_2\leqslant N_2\} +\end{equation} +for some fixed $N_1,N_2$. The transform is inverted by +\begin{equation} + \frac1{N_1N_2}\mathcal F^*(\mathcal F(f))(n)=f(n) +\end{equation} +in which $\mathcal F^*$ is defined like $\mathcal F$ but with the opposite phase. +\bigskip + +\indent The condition $p+q=k$ can be rewritten as +\begin{equation} + T(\hat\varphi,k) + = + \sum_{p,q\in\mathcal K} + \frac1{N_1N_2} + \sum_{n\in\mathcal N}e^{-\frac{2i\pi}{N_1}n_1(p_1+q_1-k_1)-\frac{2i\pi}{N_2}n_2(p_2+q_2-k_2)} + \frac{\hat\varphi_p}{|p|} + (q\cdot k^\perp)(k\cdot q)\frac{\hat\varphi_q}{|q|} +\end{equation} +provided +\begin{equation} + N_i>4K_i. +\end{equation} +Indeed, $\sum_{n_i=0}^{N_i}e^{-\frac{2i\pi}{N_i}n_im_i}$ vanishes unless $m_i=0\%N_i$ (in which $\%N_i$ means `modulo $N_i$'), and, if $p,q\in\mathcal K$, then $|p_i+q_i|\leqslant2K_i$. Therefore, +\begin{equation} + T(\hat\varphi,k) + = + \frac1{N_1N_2} + \mathcal F^*\left( + \mathcal F(|p|^{-1}\hat\varphi_p)(n) + \mathcal F((|q|^{-1}(q\cdot k^\perp)(k\cdot q)\hat\varphi_q)_q)(n) + \right)(k) +\end{equation} +which we expand +\begin{equation} + T(\hat\varphi,k) + = + \textstyle + \frac{k_x^2-k_y^2}{N_1N_2} + \mathcal F^*\left( + \mathcal F\left(\frac{\hat\varphi_p}{|p|}\right)(n) + \mathcal F\left(\frac{q_xq_y}{|q|}\hat\varphi_q\right)(n) + \right)(k) + - + \frac{k_xk_y}{N_1N_2} + \mathcal F^*\left( + \mathcal F\left(\frac{\hat\varphi_p}{|p|}\right)(n) + \mathcal F\left(\frac{q_x^2-q_y^2}{|q|}\hat\varphi_q\right)(n) + \right)(k) +\end{equation} + + +\vfill +\eject + +\begin{thebibliography}{WWW99} +\small +\IfFileExists{bibliography/bibliography.tex}{\input bibliography/bibliography.tex}{} +\end{thebibliography} + + +\end{document} -- cgit v1.2.3-70-g09d2