{ "cells": [ { "cell_type": "markdown", "metadata": {}, "source": [ "###### Content provided under a Creative Commons Attribution license CC-BY 4.0; code under BSD 3-Clause license. (c)2015 L.A. Barba, Pi-Yueh Chuang." ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "# Exercise: Derivation of the vortex-source panel method" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "The potential at location $(x, y)$ induced by an uniform flow, a source sheet, and a vortex sheet can be represented as" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "$$\n", "\\begin{equation}\n", "\\begin{split}\n", "\\phi(x, y) \n", "&= \\phi_{uniform\\ flow}(x, y) \\\\ \n", "&+ \\phi_{source\\ sheet}(x, y) + \\phi_{vortex\\ sheet}(x, y)\n", "\\end{split}\n", "\\end{equation}\n", "$$" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "That is" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "$$\n", "\\begin{equation}\n", "\\begin{split}\n", "\\phi(x, y) &= xU_{\\infty}\\cos(\\alpha) + yU_{\\infty}\\sin(\\alpha) \\\\\n", "&+\n", "\\frac{1}{2\\pi} \\int_{sheet} \\sigma(s)\\ln\\left[(x-\\xi(s))^2+(y-\\eta(s))^2\\right]^{\\frac{1}{2}}ds \\\\\n", "&-\n", "\\frac{1}{2\\pi} \\int_{sheet} \\gamma(s)\\tan^{-1} \\frac{y-\\eta(s)}{x-\\xi(s)}ds\n", "\\end{split}\n", "\\end{equation}\n", "$$" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "where $s$ is local coordinate on the sheet, and $\\xi(s)$ and $\\eta(s)$ are coordinate of the infinite source and vortex on the sheet. In the above equation, we assume the source sheet and the vortex sheet overlap." ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "------------------------------------------------------" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "### Q1:" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "If we discretize the sheet into $N$ panels, re-write the above equation using discretized integral. Assume $l_j$ represents the length of the panel $j$. And so that\n", "\n", "$$\n", "\\begin{equation}\n", "\\left\\{\n", "\\begin{array}{l}\n", "\\xi_j(s)=x_j-s\\sin\\beta_j \\\\\n", "\\eta_j(s)=y_j+s\\cos\\beta_j\n", "\\end{array}\n", ",\\ \\ \\ \n", "0\\le s \\le l_j\n", "\\right.\n", "\\end{equation}\n", "$$\n", "\n", "The following figure shows the panel $j$:\n", "\n", "