{ "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)2014 Lorena A. Barba, Olivier Mesnard. Thanks: NSF for support via CAREER award #1149784." ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "# Infinite row of vortices" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "The objective of this assignment is to visualize the streamlines around an infinite row of vortices. First, you will consider the case of a finite number of vortices, obtained by simple superposition. By adding more and more vortices, you should be able to see how the flow pattern approaches that of an infinite row of vortices. But there will always be some differences (pay attention to what these may be).\n", "\n", "It's possible to derive an analytical expression for the infinite case, and the derivation is provided below. With this analytical expression, you can visualize the streamlines for the infinite case. Observe and think: how are the streamlines different from one case to the other?\n", "\n", "\n", "In this notebook, there is no Python code. Your job is to study the theory (and follow the mathematics on your own handwritten notes), to think how you could implement it in an efficient manner and finally to code it and visualize the results." ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "## Vortex flow (from previous lesson)" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "You might not suspect it, but the vortex has a very important role in classical aerodynamic theory. You'll discover some of its uses in this assignment.\n", "\n", "First, a little review of the basics. As seen in a previous lesson, a vortex of strength $\\Gamma$ has a stream-function:\n", "\n", "$$\\psi\\left(r,\\theta\\right) = \\frac{\\Gamma}{2\\pi}\\ln r$$\n", "\n", "and a velocity potential\n", "\n", "$$\\phi\\left(r,\\theta\\right) = -\\frac{\\Gamma}{2\\pi}\\theta$$\n", "\n", "We can now derive the velocity components in a polar coordinate system, as follows:\n", "\n", "$$u_r\\left(r,\\theta\\right) = 0$$\n", "\n", "$$u_\\theta\\left(r,\\theta\\right) = -\\frac{\\Gamma}{2\\pi r}$$" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "In a Cartesian coordinate system, the velocity components at $\\left(x,y\\right)$ around a vortex of strength $\\Gamma$ located at $\\left(x_\\text{vortex},y_\\text{vortex}\\right)$, are given by\n", "\n", "$$u\\left(x,y\\right) = +\\frac{\\Gamma}{2\\pi}\\frac{y-y_\\text{vortex}}{(x-x_\\text{vortex})^2+(y-y_\\text{vortex})^2}$$\n", "\n", "$$v\\left(x,y\\right) = -\\frac{\\Gamma}{2\\pi}\\frac{x-x_\\text{vortex}}{(x-x_\\text{vortex})^2+(y-y_\\text{vortex})^2}$$\n", "\n", "and the stream-function is written as\n", "\n", "$$\\psi\\left(x,y\\right) = \\frac{\\Gamma}{4\\pi}\\ln\\left((x-x_\\text{vortex})^2+(y-y_\\text{vortex})^2\\right)$$" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "## Superposition of many vortices" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "In this assignment, we consider a useful example to illustrate the concept of a *vortex sheet*: an infinite row of vortices of equal strength $\\Gamma$ (same sign and magnitude) evenly spaced by a distance $a$. But let's start with a finite row of vortices first.\n", "\n", "\n", "The stream-function $\\psi_i$ of the $i^{th}$ vortex at a distance $r_i$ is given by:\n", "\n", "$$\\psi_i = \\frac{\\Gamma}{2\\pi}\\ln r_i$$\n", "\n", "Applying the principle of superposition, the stream-function of $N$ vortices is, then\n", "\n", "$$\\psi = \\frac{\\Gamma}{2\\pi} \\sum_{i=1}^N \\ln r_i$$\n", "\n", "And the velocity field (in Cartesian coordinates) of the row of vortices is\n", "\n", "$$u\\left(x,y\\right) = + \\frac{\\Gamma}{2\\pi} \\sum_{i=1}^N \\frac{y-y_i}{(x-x_i)^2+(y-y_i)^2}$$\n", "\n", "$$v\\left(x,y\\right) = - \\frac{\\Gamma}{2\\pi} \\sum_{i=1}^N \\frac{x-x_i}{(x-x_i)^2+(y-y_i)^2}$$\n", "\n", "where $\\left(x_i,y_i\\right)$ are the Cartesian coordinates of the $i^{\\text{th}}$ vortex.\n", "\n", "Here is a diagram of the situation:\n", "\n", "