Combination Finite Difference & Spectral Solution to Advection-Diffusion PDE

Vorticity $\omega$ is a measure of local rotation (in a fluid). It is related to the angular velocity of a solid. Vorticity is generally a vector (valued function). However, it can be viewed as a scalar in 2D because it is always perpendicular to the plane. If fluid velocity $\vec{v}$ (in 2D) has components $u$ and $v$ \begin{equation} \vec{v}(t) = \begin{pmatrix} u(t) \\ v(t) \end{pmatrix} \end{equation} then these vector components are related to a scalar quantity called the \textbf{stream function} $\psi$ as follows \begin{align} u = -\psi_y \\ v = \psi_x \end{align} so that \begin{equation} \vec{v}(t) = \begin{pmatrix} u(t) \\ v(t) \end{pmatrix} = \begin{pmatrix} -\psi_y \\ \psi_x \end{pmatrix} \end{equation} So $\psi$ is a scalar valued function while the components of the fluid velocity vector can be derived from it. \begin{equation} \label{eq:hw4_laplacian_eqn} \nabla^2 \psi = \omega \end{equation} Vorticity-Streamfunction PDE \begin{align} \omega_t + [\psi,\omega] &= \eta \nabla^2 \omega \\ \omega_t + \psi_x \omega_y - \psi_y \omega_x &= \eta ( \omega_{xx} + \omega_{yy} ) \\ \omega_t + \psi_x \omega_y - \psi_y \omega_x &= \eta D_L \omega \\ \omega_t &= -(\psi_x \omega_y - \psi_y \omega_x) + \eta D_L \omega \end{align} where $D_L$ is the Laplacian operator $\frac{\partial^2}{\partial x^2} + \frac{\partial^2}{\partial y^2} $. With discretization in space, we can write a finite difference approximation for the linear operator $D_L$. This allows us to solve the Poisson equation by solving the linear system: \begin{equation} D_L \vec{\psi} = \vec{\omega} \end{equation} This obtains $\psi$ from $\omega$ at step n. If we use a finite difference approximation in time too we can write \begin{align} \omega_t &= -(\psi_x \omega_y - \psi_y \omega_x) + \eta D_L \omega \\ \frac{\omega^{k+1} - \omega^k }{ \Delta t} &= -(\psi_x \omega_y - \psi_y \omega_x) + \eta D_L \omega \end{align} with RHS $$ f = -(\psi_x \omega_y - \psi_y \omega_x) + \eta D_L \omega $$ The following video is made in MATLAB after integrating the system with ODE45. At each time step, the Poisson equation is solved in Fourier space (to speed things up)

Comments

Post a Comment

Popular posts from this blog

Data science & ML Video Tutorials Part II - Group & Set Theory (Groups, Rings & Fields)

Image
We continue building the knowledge required for basic probability theory and statistics (needed for data science & ML). The videos in this post are about set algebra and algebraic structures like groups, rings & fields. It includes an introduction to identity, inverse and idempotent elements. We'll see that Boolean conjunction and disjunction can be seen as multiplication and addition and investigate possible additive and multiplicative identities and inverses. This is followed by power sets and basic operations on elements of power sets (union, intersection, complement, difference). All concepts are then combined in a discussion of algebraic structures including groups, rings and fields. We end with Boolean rings and describe set algebra as an example Boolean ring. In the review session we complete an example proof from CH1 of W. Rudin's Principles Of Mathematical Analysis using the axioms of a field. Introduction to part II. The main subject of the sessions of Par...
Read more

Music Scoring With STFT-Gabor Transform

In this UW AMATH582 assignment report we analyze samples of music and attempt to reproduce guitar and bass music scores from them. In order to accomplish this we carry out a time-frequency decomposition of the signal. For the latter, we make use of the Short Time Fourier Transform (STFT) also called Gabor transform. We use low, high or bandpass filters to isolate the bass or guitar from other sound. Read a detailed explanation of the short time Fourier (Gabor) transform and its application to music scoring here . The Github project is availabe here: https://github.com/aruymgaart/AMATH/tree/master/stft_gabor_music_scoring_582HW2 . Example source code:
Read more

Guide To Walter Rudin's Principles, 1.14 (Proof Details)

Proofs - Rudin Principles 1.14 It is useful to read about Abelian groups first. In particular, review inverse and identity elements. Any book on modern or abstract algebra should cover this. In the field of reals $\mathbb{R}$, prove: 1.14.a. if $ x+y = x+z$ then $y=z$ We need to prove $y=z$ under the condition given. Rudin starts with $y=y$ (although not shown in the text) $$ \begin{array}{l c} y = y & \\ y = 0+y & additive \ \ identity\\ y = (x-x) +y & additive \ \ inverse\\ y = x - (x+y) & associative \ \ property\\ y = x - (x+z) & equation \ \ given \ \ (substitute \ \ x+y = x+z )\\ y = (x - x)+z & associative \\ y = 0 + z & additive \ \ inverse \\ y=z \end{array} $$ 1.14.b. if $ x+y = x$ then $y=0$ Note: this proof asserts uniqueness of the additive identity element. Why can we not substitute $y=0$ right away? We would get $x+0=x$ and prove $x=x$. Rudin says substitute $z=0$ in 1.14.a and follow the same path. So starting with $y=...
Read more