{
 "cells": [
  {
   "cell_type": "code",
   "execution_count": 204,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": [
    "%matplotlib inline\n",
    "\n",
    "import numpy as np\n",
    "from math import pi, sqrt\n",
    "import matplotlib.pyplot as plt\n",
    "from scipy.optimize import brentq\n",
    "from scipy.interpolate import Akima1DInterpolator\n",
    "from ipywidgets import interact\n",
    "\n",
    "plt.style.use('ggplot')\n",
    "plt.rcParams['font.size'] = 16.0"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# Converging Diverging Nozzle\n",
    "\n",
    "We will assume a calorically perfect, ideal gas.  We will also not include friction or heat transfer.  Thus, we will assume adiabatic flow.  \n",
    "\n",
    "## Getting Started\n",
    "Let's first set the specific heat ratio \n",
    "$$\\gamma = \\frac{c_p}{c_v}$$\n",
    "We will set a default of 1.4 for air, and define some other variables for convenience.\n"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 205,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": [
    "g = 1.4\n",
    "gp = g+1\n",
    "gm = g-1"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "For convenience, we will define the function:\n",
    "$$f(M) = M \\left[ \\frac{2}{\\gamma + 1}\\left( 1 + \\frac{\\gamma - 1}{2}M^2 \\right)\\right]^{\\frac{-(\\gamma+1)}{2(\\gamma-1)}}$$\n",
    "This is used in the area Mach relationship:\n",
    "$$\\frac{A^*}{A} = f(M)$$\n",
    "Rather than working with the less physically relevant parameter $A^*$, we often replace it with $f(M) A $.  It is important to remember that $f(1) = 1$ as should be clear from its definition.\n",
    "\n",
    "Our most important expression is conservation of mass.  Using the $f(M)$ function, conservation of mass can be expressed as:\n",
    "$$\\left( \\frac{P_0 A f(M)}{\\sqrt{T_0}}\\right)_1 = \\left( \\frac{P_0 A f(M)}{\\sqrt{T_0}}\\right)_2$$\n",
    "Because we are only concerend with adiabatic flows in this exercise, this is simplified to\n",
    "$$( P_0 A f(M))_1 = ( P_0 A f(M))_2$$\n",
    "for regions of isentropic flow this simplifes further to\n",
    "$$A_1 f(M_1) = A_2 f(M_2)$$\n",
    "Let's encapsulate $f(M)$ as a function so that we can reuse it for a given Mach number."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 206,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": [
    "def fM(M):\n",
    "    return M*(2.0/gp*(1 + gm/2.0*M**2))**(-gp/2.0/gm)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "The equation cannot be analytic inverted.  Thus, if we know the area ratio $A^*/A$ solving for the corresponding Mach number requires a root finding approach:\n",
    "$$froot(M) = \\frac{A^*}{A} - f(M) = 0$$\n",
    "Note that there are two solutions: a subsonic or a supersonic one and so we will also need to know which branch is sought.  We use brent's method for root finding."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 207,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": [
    "def areaMach(Astar_A, speed):\n",
    "    \"\"\"returns Mach number associated with isentropic flow for a given area ratio and flow regime\"\"\"\n",
    "    \n",
    "    if speed=='subsonic':\n",
    "        return brentq(lambda M: Astar_A - fM(M), 1e-6, 1.0)\n",
    "    \n",
    "    elif speed=='supersonic':\n",
    "        return brentq(lambda M: Astar_A - fM(M), 1.0, 10.0)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "We will also make heavy use of the isentropic relationships:\n",
    "$$\\frac{T_0}{T} = 1 + \\frac{\\gamma - 1}{2}M^2 $$\n",
    "$$\\frac{P_0}{P} = \\left(\\frac{T_0}{T}\\right)^{\\gamma/(\\gamma-1)} $$\n",
    "$$\\frac{\\rho_0}{\\rho} = \\left(\\frac{T_0}{T}\\right)^{1/(\\gamma-1)} $$"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 208,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": [
    "def isenT(M):\n",
    "    return 1 + gm/2.0*M**2\n",
    "\n",
    "def isenP(M):\n",
    "    return isenT(M)**(g/gm)\n",
    "\n",
    "def isenrho(M):\n",
    "    return isenT(M)**(1.0/gm)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "These expressions can also be explictly inverted in order to solve for the Mach number given a known pressure, temperature, or density ratio.  We will only need the inverse pressure expression in this exercise.\n",
    "$$ M = \\sqrt{\\frac{2}{\\gamma -1} \\left[\\left(\\frac{P_0}{P}\\right)^{(\\gamma-1)/\\gamma} -1 \\right]}$$"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 209,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": [
    "def isenM(P0_P):\n",
    "    return sqrt(2.0/gm*(P0_P**(gm/g) - 1))"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Geometry\n",
    "\n",
    "This is enough to get us started.  First we need to define the geometry that we want to solve.  We will define the radius at 5 locations along a unit length and fit an Akima spline in between.  The radii can be adjusted, but note that this simulation assumes a converging diverging nozzle so if you setup a different type of geometry the simulation may crash or the results will be nonsensical."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 221,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
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JdTNFQk5hHQQ7duxg/fr1rFixAsMweOyxx9iwYQPz5s07Y93Vq1czYcIEZsyYwcaNG1m9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\nWIuIiIS5/x8Kf8RUrw8irgAAAABJRU5ErkJggg==\n",
      "text/plain": [
       "<matplotlib.figure.Figure at 0x107012290>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "xpts = np.array([0, 0.25, 0.5, 0.75, 1.0])\n",
    "rpts = np.array([0.35, 0.15, 0.20, 0.27, 0.3])\n",
    "akima = Akima1DInterpolator(xpts, rpts)\n",
    "\n",
    "n = 200\n",
    "x = np.linspace(xpts[0], xpts[-1], n)\n",
    "r = akima(x)\n",
    "\n",
    "plt.figure()\n",
    "plt.plot(x, r)\n",
    "plt.plot(x, -r)\n",
    "plt.axis('equal')\n",
    "plt.text(1.15, 0, '$P_b$')\n",
    "plt.text(-0.4, 0, '$P_0$ (total)')\n",
    "plt.show()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "With the geometry, defined we can now define some important quantities."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 211,
   "metadata": {
    "collapsed": false
   },
   "outputs": [],
   "source": [
    "# area distribution\n",
    "A = pi*r**2\n",
    "\n",
    "# exit area\n",
    "Ae = A[-1]\n",
    "\n",
    "# area of throat, and corresponding index\n",
    "ithroat = np.argmin(A)\n",
    "Athroat = A[ithroat]"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Cases\n",
    "There are a few discrete cases that each need to be considered separately, and conditions defined for when they occur: fully subsonic flow, standing shock wave, isentropic supersonic flow.  The parameter we will use that dictates the nozzle behavior is the ratio of the back pressure (downstream of the exit) to the total pressure (upstream of the entrance) $P_b/P_0$.  Note that the back pressure is not necessarily equal to the exit pressure (it is only equal if the exit is subsonic).  We will only focus on flow in the nozzle and thus, wont discuss the under- and overexpanded cases. When $P_b/P_0 = 1$ there is no flow.  As we lower the back pressure we will move through the various cases."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Case I: fully subsonic\n",
    "If the flow is everywhere subsonic, then under our assumptions it is also isentropic throughout.  As the back pressure is lowered, the flow will accelerate, but remain subsonic, until the limiting where the throat just reaches sonic conditions.  From a mass balance (isentropic) this limiting cases occurs at:\n",
    "$$\\begin{align}\n",
    "A_{throat} &= A_e f(M_e)\\\\\n",
    "\\Rightarrow f(M_e) &= A_{throat}/A_e\n",
    "\\end{align}$$\n",
    "We can now find the critical exit Mach number at which the throat just barely reaches sonic conditions and we still have subsonic flow throughout.  If the exit Mach number was any higher, we would no longer have fully subsonic flow."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 212,
   "metadata": {
    "collapsed": false
   },
   "outputs": [],
   "source": [
    "# critical Mach number for subsonic flow throughout with sonic flow at throat\n",
    "Me_crit_sub = areaMach(Athroat/Ae, 'subsonic')"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "If subsonic at the exit, the back pressure matches the exit pressure, and because the flow is isentropic the total pressure is constant throughout the nozzle.\n",
    "$$\\frac{P_b}{P_0} = \\frac{P_e}{P_0}$$\n",
    "We already know the critical exit Mach number and using the isentropic relationship, we can compute the corresponding critical pressure ratio for subsonic flow throughout.  As long as the pressure ratio $P_b/P_0$ is above this critical value, we will have subsonic flow throughout."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 213,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": [
    "# corresponding critical pressure ratio\n",
    "Pb_P0_crit_sub = 1.0/isenP(Me_crit_sub)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "So if $$\\frac{P_b}{P_0} > \\left(\\frac{P_b}{P_0}\\right)_{crit-sub}$$ we will have subsonic flow through.  \n",
    "We can evaluate the Mach number everywhere from the (isentropic) mass balance\n",
    "$$\\begin{align}\n",
    "A\\ f(M) &= A_e f(M_e)\\\\\n",
    "f(M) &= \\frac{A_e}{A} f(M_e)\n",
    "\\end{align}$$\n",
    "where the Mach number at the exit is given by the isentropic relationship"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 214,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "image/png": 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hfBJ63duDkK5ygmM96rLxckkszCmbDW67G2Lt6PeWwNEKuOUuv46MLUQgyDu2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\nCRchhBB+9/9ZO7QUTW07zQAAAABJRU5ErkJggg==\n",
      "text/plain": [
       "<matplotlib.figure.Figure at 0x106dd6d90>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    },
    {
     "data": {
      "image/png": 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FfBIohS4rRg0dYXQ1XqW/+BROVWG59wk52hfChFocTHFxcUyYMIEJEyYAzunJ\nDxw4QGFhIYWFhezevZudO3cCznuZUlNTKSkp8UbNwgtUeAT0iEWXHTa6FK/SZ2vQq9+Hwdeg0gcZ\nXY4QogktDqYfzoEE0KlTJ0aMGMGIEc5v13a7naKiIldQHTx4kHPnzrVvtcK7EvrCEf8ez1Cveh9s\nViwzfml0KUKIS2hxMIWEhFz29dDQUAYOHMjAgQMBqKur44knnmhbdaJDqYQU59BE56x+OV6crqpE\nr/8UNWoiKj7Z6HKEEJfgtfFXgoOD6dHDvy+i+xuV0Nf5oLzE0Dq8RX+8FAA1bbbBlQghLserA4M9\n/PDD3ty9aG8JKQDoMv87nafLS9B5G1ATb0EF+LxTQphds8FUXFzMV199xfnz5z3eeZcuXVpVlDBI\nTFeI6gR+2AHC8cGbEB6BmjLT6FKEEM1o9hpT165d2blzJ//4xz/o2bMno0ePZtiwYTLagx9SSkFC\nX7/rmacPFsA/t6Nm3IWKjDa6HCFEM5pNl5iYGGbOnMnMmTMpLy9n27ZtfPjhh/Tu3ZvRo0czdOhQ\ngoKCOqJW0QFUQgp6/afoujqUn3z5cHyyDDp3QU28xehShBAt4NEnT3x8PNnZ2WRnZ1NaWkpubi4r\nVqygT58+jB49miFDhrgmDRQ+KqEv1J2HY99BnySjq2kzXbgHDvwTddt9qEsMRCyEMJdWfyVOSkoi\nKSmJ2267jcOHD5Obm8vy5ctJSkpi9OjRDBo0SELKB6mEFOfcTGXFKB8PJq01jo+XQUxX1LhJRpcj\nhGihdjlX07dvX/r2dXY1PnToENu2beOdd94hJSWFUaNGkZGRIUO/+IrYPhAc4uwAce31RlfTNvt3\nQ9E+1Oz7USGhRlcjhGghj4Lp3Llz5ObmYrPZSEtLIzU19aJ1+vXrR79+/QA4ePAgubm5LF26lNTU\nVEaPHs2AATJLqJmpoCDok+TzHSC01s5rS126o8ZkGV2OEMIDHo0u/vzzz1NVVeVaNnbsWB566KFL\nnrJLS0sjLS0NrTUHDhxg27ZtfPzxxzzzzDNtr1x4jUpMQX+Ti9bad490C76BQ4WoOx5ENTNqiRDC\nXFocTMuXL6dbt25MmzYNm81GYWEhW7duJSwsjPvuu++y2yql6N+/P/37929zwaIDxCfD5s/hZBX4\n4MyurqOlrj1QY240uhwhhIdaHEz79+9n/vz5hIeHu5YdPXqU+fPnU1JSQnJysjfqEwZQ8RfmZio/\n7JPBxN5es8J+AAAXqklEQVQdcPgg6hcPO6eNF0L4lBZ3m+vatWujUAKIjY1l7ty5bNq0qd0LEwa6\nMMCpL15ncvXE694LNfoGo8sRQrRCi4Mp7BL3gCQnJ3PmzJl2K0gYT0VEQreevjmY657tUFqEujnb\nb24QFiLQtMuNRjI8kR9K6Isu960jJq01jlU50K0nyte7ugsRwFocTOXl5ezZs4fa2tqLXvPZnlvi\nklR8Xzj2L3QTf9+mdXAvFB9ATZohR0tC+DCPuou/8MILWCwWEhMTSU9PJy0tjfT09Etus2vXLoYN\nG9YuhYqOpRKS0doBFaXQN83oclrEseo96BSDuk6uLQnhy1ocTD169GDmzJkUFhZy4MAB1qxZw5o1\nawDn7LYnTpxgwIABDBgwgJSUFIKCgvjkk08kmHxVvHMkD112GOUDwaQPfwv7dqF+dhcqVMbEE8KX\ntTiY4uLimDBhAhMmTACgurqaAwcOUFhYSGFhIbt372bnzp2Ac5r11NRUSkpKvFGz6Ajde0HYFc4u\n4z7Asfo9iIhEjZ9sdClCiDZqcTA99dRTjZ536tSJESNGMGLECADsdjtFRUWuoDp48CDnzp1rc4F5\neXnk5+cTHh5O9+7dmTFjRpPr2Ww23nnnHcLDw1FKUV1dzZ133klERIRrnePHjzN37lwcDgcAw4cP\nl1EoLkFZLBCfhC4rMbqUZumKI/BNHuqWWagrIprfQAhhai0OppBmhnUJDQ1l4MCBDBw4EIC6ujqe\neOKJNhW3Z88eVqxYwYsvvohSipdffpnVq1czefLF34qXLVtGnz59mDTJOYr06tWrWbJkCY888ohr\nnVWrVjF79mzX/FGDBw9uU33+TiX0RX+10fRDE+nPVkBoGGriVKNLEUK0A6/NSxEcHEyPHj3atI+l\nS5cyZswY14fiuHHjyMnJwW63X7RuQUFBo3utevXqRWlpqet5dXU1VquVqVOnMmXKFKZMmUJCQkKb\n6vN78X3hnBWOHzO6kkvSx4+hv9qIGjcJFd3J6HKEEO3AqxMmPfzww63etrKykpKSEhITE13LEhMT\nsVqt7Nu376L1k5OTWb58OcePHwdgy5YtjY6sVq5cyYYNG3j66adZt25dq+sKJOrCCBBmvtFWr/kQ\nlAX1k1uNLkUI0U68GkxdunRp9bZlZWUAREdHu5ZFRUUBUFFRcdH6d911F6GhocybN4/XX3+dIUOG\nMHHiRNfrgwcP5p577iEmJoZFixYxf/5817UmcQnxyaCUaYcm0qdPoresRY2eiPLFMf2EEE0y7RSz\nVqsVcIcRuEeYaOom306dOjF37lzsdnuTR0SDBg0iKyuLZ599lscee4ydO3fKkVMzVFg49Oht2hEg\n9Bf/gPo61KSmO8QIIXyTaW+Pbwikuro617KGa0uRkZEXrV9RUcG7777LK6+8wuLFi1mwYAEOh6PR\nUVODUaNGsX//fnbt2kVWVuNJ5AoKCigoKHA9z87ObnTUFmjO9r2S+sPfEh0dTWhoqGnaQtvOUb1p\nDSHXjCEy9dI3eXuLmdrCaNIWbtIWjeXk5LgeZ2RkkJGR0aLtTBtMsbGxgLPTQoOGx/Hx8Retv3Dh\nQiZOnEhERASPPvooAG+99RbXX399kz3Khg4dyvr16y9a3lTj1dTUtP6N+DhHbDz6q41Uf3+MTj17\nmaYtHBtWos/WUD/xZkNqio6ONk1bGE3awk3awi06Oprs7OxWbWvaU3mxsbGkpKRQVFTkWlZeXk5U\nVFSTwyCVlpYSExPjen733XdjtVoveS9VZWUlmZmZ7V+4n1EJzhEg+K7E0Dp+SDvq0Ws/dg6V1G+A\n0eUIIdqZaYMJYPr06eTn57ueb9y4kVmzZhEUFMTixYtZsmSJ67Xhw4c3OgVXXV3NgAEDiIiI4PTp\n0yxatMjVoeLQoUMUFxczfvz4jnszvso1NFGJsXX80K58qDyKJetWU99fJYRoHdOeygPIzMzk1KlT\nLFiwgODgYNLT013XhE6ePNnoQ+mBBx7gzTffZNmyZcTExHDixAkef/xxAIKCgiguLubZZ58lKSmJ\nkSNH8uCDDxrynnxO1+4QEWmqoYkcaz9yzhc1fJTRpQghvEBprbXRRZhdU93TA0n9/N/BeTsx//FX\nw8+f6+IDOP7jKdSse7HcOM2wOuRagpu0hZu0hVtcXFyrtzX1qTxhDiqhL3xXinbUG10K+vOP4IpI\n1JgbjS5FCOElEkyiefHJYK/FcdTYI0ddeRS9M9c5/FC4DNYqhL+SYBLNauiZV3/kkKF16C/+ARaF\nmniLoXUIIbxLgkk0Ly4RLBbqS40LJm09g96yDjVirAw/JISfk2ASzVIhodCrj7HBtHkt1J5D/eSn\nhtUghOgYEkyiRVRCX8OCSTvq0RtWQloGKrGfITUIITqOBJNomfi+6Krv0WcN6Aq7ZztUfY9Fri0J\nERAkmESLqIRk5wMD5mZyrP8UYrrB0JEd/ruFEB1Pgkm0jGtooo4dAUL/qwz270ZNmIwKNvVAJUKI\ndiLBJFqmcxdUp5gOH5pIb1gJwcGosVnNryyE8AsSTKJFlFIEJfXr0MFc9TkretsGZxfxTjHNbyCE\n8AsSTKLFghL7QcURdH3HDE2kt33h7CJ+vXR6ECKQSDCJFgtK6gd15+Hod17/XdrhQG9YBX3TUH2v\n9PrvE0KYhwSTaLGgJOc9RLojrjPt2wXHvpPhh4QIQBJMosUsfRIhKLhDuow71n8KnWJQ11zn9d8l\nhDAXCSbRYio4BHonoMuKvfp79Pf/gr07nKOIB4d49XcJIcxHgkl4RCWmwJFivDm/pN74GSiFGn+T\n136HEMK8JJiEZ5JToeY0nDzuld3r83b0tnUw7FpUTDev/A4hhLlJMAmPqKRU54OSIq/sX+/MhTM1\nWMZP8sr+hRDmJ8EkPBOfDEFB6FIvBdOmz6BHLPQf6pX9CyHMT4JJeESFhkFcItoLR0y64ggcLHB2\nerDIP00hApX87xceU8lXwpGidu8AoTetgaBg1Ogb2nW/QgjfIsEkPJeUCmdqoOr7dtultteic9ej\nrhol4+IJEeAkmITHVPKFDhDteJ1Jb98C1rPSRVwIIcEkWqFPEgQHo4sPttsu9aY1ENsH0ga12z6F\nEL5Jgkl4TAWHQFIquriwXfanyw/DoULUuJtQSrXLPoUQvkuCSbSK6tcfSorQdefbvC+9cQ0Eh6BG\nT2yHyoQQvk6CSbSKSunvnALjSNvGzdO2c+i8DahrxqAio9upOiGEL5NgEq3TLx0Afahtp/P015vB\ndg4lIz0IIS6QYBKtomK6Qbee0NZg2rIWeidAvwHtVJkQwtdJMIlWU/36t+mISVccgeIDqDE/kU4P\nQggXCSbRev36w6kqdFVlqzbXW9Y6R3oYdX07FyaE8GUSTKLVVFoGAPrAHo+31XXn0bkbYGgmKrpz\ne5cmhPBhEkyi9eKSILoz7N/t+ba7v4Yz1VjG3Nj+dQkhfJoEk2g1ZbGg+g9B79/j8YCujq3rIKYb\nZAz3UnVCCF8VbHQBl5OXl0d+fj7h4eF0796dGTNmNLmezWbjnXfeITw8HKUU1dXV3HnnnURERHi8\nL+GhAUPh681wtNzZu64F9Mkq2LsTNXkmyhLk5QKFEL7GtEdMe/bsYcWKFcydO5c5c+ZQXl7O6tWr\nm1x32bJlxMXFcfvtt3PbbbeRkJDAkiVLWrUv4RnVfwgAel/LT+fpbV+AdqDkNJ4QogmmDaalS5cy\nZswYVzficePGkZOTg91uv2jdgoICwsLCXM979epFaWlpq/YlPKN6xEL3XujClgWTdjjQW9dB+mDn\ntkII8SOmDKbKykpKSkpITEx0LUtMTMRqtbJv376L1k9OTmb58uUcP34cgC1btjB58uRW7Ut4TmUM\nh/270fba5lc+uBcqj8rRkhDikkwZTGVlZQBER7vHTouKigKgoqLiovXvuusuQkNDmTdvHq+//jpD\nhgxh4sSJrdqX8Jy6ajTU2mDvjmbX1VvXwRWRzm2EEKIJpgwmq9UKuAMEIDjY2U+jtvbib+WdOnVi\n7ty52O121q1b16Z9iVZIHwxRndDbt152NW09g96xDTVyHCo07LLrCiEClyl75TWESF1dnWtZw/Wg\nyMjIi9avqKjg3Xff5ZVXXmHx4sUsWLAAh8PBxIkTPd5XQUEBBQUFrufZ2dmNjrYCWWho6CXbwjpy\nHPYt64gKC71k6NTmrufceTuRWT8l2Mfb9HJtEWikLdykLRrLyclxPc7IyCAjI6NF25kymGJjnRfF\nq6urXcsaHsfHx1+0/sKFC5k4cSIRERE8+uijALz11ltcf/31Hu+rqcarqalpy9vxG9HR0ZdsCz0k\nE774lJrcL5s8Tae1xrHmQ0joi7V7b5SPt+nl2iLQSFu4SVu4RUdHk52d3aptTXkqLzY2lpSUFIqK\nilzLysvLiYqKIj09/aL1S0tLiYmJcT2/++67sVqtnDt3zuN9iVZKHwydu+D48hLd8A/uhe9KURNv\nkQFbhRCXZcpgApg+fTr5+fmu5xs3bmTWrFkEBQWxePHiRvcpDR8+vNHpt+rqagYMGOC6wfZy+xLt\nQwUFoX5yq7N3XhMjjjvWfwqR0ajMcQZUJ4TwJaY8lQeQmZnJqVOnWLBgAcHBwaSnp5OVlQXAyZMn\nG33rfuCBB3jzzTdZtmwZMTExnDhxgscff7xF+xLtR42/Cf3Z+zhW5hD06P9yLdffV8A3X6GybpVO\nD0KIZint6SBnAUi6lTu15Py5Y2UO+qO3UXN+i2XEGLS9FscffwtV32P59z+junbvoGq9S64luElb\nuElbuMXFxbV6W9MeMQnfpG6cht67E734TzjKD6NLi6DsMJa5z/tNKAkhvMu015iEb1Jh4Vh+8+9w\nZQZ61XtQVIia+SvUkBFGlyaE8BFyxCTanQq/AssTf4AzNRAVLb3whBAekWASXqGUguhORpchhPBB\ncipPCCGEqUgwCSGEMBUJJiGEEKYiwSSEEMJUJJiEEEKYigSTEEIIU5FgEkIIYSoSTEIIIUxFgkkI\nIYSpSDAJIYQwFQkmIYQQpiLBJIQQwlQkmIQQQpiKBJMQQghTkWASQghhKhJMQgghTEWCSQghhKlI\nMAkhhDAVCSYhhBCmIsEkhBDCVCSYhBBCmIoEkxBCCFORYBJCCGEqEkxCCCFMRYJJCCGEqUgwCSGE\nMBUJJiGEEKYiwSSEEMJUJJiEEEKYigSTEEIIU5FgEkIIYSoSTEIIIUwl2OgCmpOXl0d+fj7h4eF0\n796dGTNmNLneq6++ysaNGy9a/tJLLxEfHw/A8ePHmTt3Lg6HA4Dhw4fzzDPPeK94IYQQHjN1MO3Z\ns4cVK1bw4osvopTi5ZdfZvXq1UyePLnReufOnaOiooK7776biIgIAKxWK5999pkrlABWrVrF7Nmz\nCQoKAmDw4MEd92aEEEK0iKmDaenSpYwZMwalFADjxo3jf/7nf7jhhhsIDQ11rXfs2DGee+45wsPD\nXcs2b97Mtdde63peXV2N1Wrll7/8Zce9ASGEEB4z7TWmyspKSkpKSExMdC1LTEzEarWyb9++Rusm\nJyc3CiVwngIcNWqU6/nKlSvZsGEDTz/9NOvWrfNu8UIIIVrNtMFUVlYGQHR0tGtZVFQUABUVFZfd\n1mazcfToUZKTk13LBg8ezD333ENMTAyLFi1i/vz5rmtNQgghzMO0p/KsVivgDiOA4GBnubW1tZfd\ndseOHVx99dWNlg0aNIhBgwaRlZVFbm4uL7/8MuvWrSMrK6udKxdCCNEWpg2mhkCqq6tzLbPb7QBE\nRkZedtu8vDymT59+yddHjRrF/v372bVr10XBVFBQQEFBget5dnY2cXFxHtfvr354BBvopC3cpC3c\npC3ccnJyXI8zMjLIyMho0XamPZUXGxsLODstNGh4/MOedj9ms9moqKggJSXlsvsfOnSoq1PFD2Vk\nZJCdne36+WHDBjppCzdpCzdpCzdpC7ecnJxGn6UtDSUweTClpKRQVFTkWlZeXk5UVBTp6emX3O6b\nb75h+PDhze6/srKSzMzMdqlVCCFE+zFtMAFMnz6d/Px81/ONGzcya9YsgoKCWLx4MUuWLLlom9zc\n3Ea98cB5pLVo0SJXh4pDhw5RXFzM+PHjvfsGhBBCeMy015gAMjMzOXXqFAsWLCA4OJj09HTXNaGT\nJ09edCrObrfz3Xff0a9fv0bLLRYLxcXFPPvssyQlJTFy5EgefPDBFtXgyeGnv5O2cJO2cJO2cJO2\ncGtLWyittW7HWoQQQog2MfWpPCGEEIFHgkkIIYSpSDAJIYQwFQkmIYQQpmLqXnkdpaVzPnm6ri9q\n6fs7ceIEixcvZt++fXTu3Jlp06Zxww03dHC13tWav+vi4mKee+45li1b1gEVdhxP28Jut/Pll18S\nEhJCly5dGDhwYKMZAXxZS9vCZrPxzjvvEB4ejlKK6upq7rzzTtfUPL7u9OnTfPzxx9jtdu69997L\nruvpv5+AP2JqmPNp7ty5zJkzh/LyclavXt3mdX2RJ+/vtddeIy0tjXvvvZeYmBhee+018vLyOrhi\n72nN37Xdbue1116jvr6+g6rsGJ62xYkTJ/jLX/7CVVddxfXXX8+wYcP8JpQ8aYtly5YRFxfH7bff\nzm233UZCQkKT9176ovPnz3PgwAF27NjhGiruUlrzfyngg6mpOZ9ycnKabGxP1vVFLX1/FRUVTJky\nhVtvvZUxY8Ywb948unXrxpYtW4wo2yta83f93nvvMXbs2I4qscN40hY2m42XXnqJO+64g+7du3d0\nqV7nSVsUFBQQFhbmet6rVy9KS0s7rFZvCgkJITMz86J7RpvSmv9LAR1Mnsz55Mm6vsiT99ezZ0+G\nDBnieh4aGsqVV17pN9+KW/N3vXfvXjp16kTfvn07qswO4WlbrFixgujoaNauXcvzzz/Pq6++6pop\nwNd52hbJycksX76c48ePA7Bly5aLZt/2dQ2zgV9Kaz83AzqYPJnzqS3zQ/kCT95fw/QjP3Tq1ClG\njx7txQo7jqd/11arlS+//JKpU6fib/ere9IWdrudzz//nNTUVO644w6efPJJCgoKmD9/fscV7EWe\n/ru46667CA0NZd68ebz++usMGTKEiRMndkyxHaSpgbB/qLWfmwEdTJ7M+dSW+aF8QVve37/+9S9C\nQkK45pprvFdgB/K0Ld59911uu+22jimug3nSFkVFRdhsNsaNG4dSis6dOzN58mT27dvnF6ewPP13\n0alTJ+bOnYvdbg/YWbNb+7kS0MHkyZxPbZkfyhe09v1prfnoo4945JFHvFtgB/KkLfLy8khNTfXL\n6yngWVucOHECgPDwcNeyhvHSjh496tU6O4Kn/0cqKip49913eeWVVxg5ciQLFixg/fr1HVOsSbT2\ncyWgu4t7MudTa+eH8hWtfX8rV65k0qRJxMTEeLfADuRJW3z++ecUFBTwyiuvNFo+a9Ysfv7znzNz\n5kwvV+tdnrRFQzfompoaOnXqBEDnzp0B//jy5un/kYULFzJx4kQiIiJ49NFHAXjrrbe4/vrrmz0F\n5i9a+7kS0EdMnsz51Nr5oXxFa97fpk2bSElJaTQpoz+c1vSkLebMmcOLL77o+rn//vsBePHFF7nx\nxhs7tG5v8KQt0tLSCA4O5uDBg65lZ86cITw8vNmJO32Bp/9HSktLG31hu/vuu7FarZw7d65D6jWD\n1n5uBnQwgWdzPl1uXX/gSVusX7+ew4cPY7fb2bVrF9u3b2fRokUcO3bMiNLbXUvbIjY2lqSkJNdP\nr169AEhKSvKbo8iWtkVUVBRTp05l7dq1rk4geXl53HzzzX5zU6kn/0eGDx9OQUGB63l1dTUDBgzw\nm7YAqK+vx+FwNFrWHp+bAX0qDzyb8+ly6/qDlrbFhg0b+Otf/wrAqlWrXNsnJCRw3333dXzhXuDp\nXGD+zJO2mDVrFgALFiygW7duaK1dy/yBJ23xwAMP8Oabb7Js2TJiYmI4ceIEjz/+uFGlt7vNmzdT\nWFgIwNatW7nuuuuA9vnclPmYhBBCmErAn8oTQghhLhJMQgghTEWCSQghhKlIMAkhhDAVCSYhhBCm\nIsEkhBDCVCSYhBBCmIoEkxBCCFORYBJCCGEq/z99zFb9WmZVZAAAAABJRU5ErkJggg==\n",
      "text/plain": [
       "<matplotlib.figure.Figure at 0x106fdff90>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "Pb_P0 = 0.99\n",
    "M = np.zeros(n)\n",
    "\n",
    "if Pb_P0 > Pb_P0_crit_sub:  # subsonic\n",
    "    # back pressure equals exit pressure b.c. subsonic\n",
    "    P0_Pe = 1.0/Pb_P0  \n",
    "    \n",
    "    # find exit Mach number from isentropic relationship\n",
    "    Me = isenM(P0_Pe)\n",
    "    \n",
    "    # precompute f(Me) so we don't have to keep computing in the loop\n",
    "    fMe = fM(Me)\n",
    "    \n",
    "    # iterate across all areas solving for the subsonic Mach number\n",
    "    for i in range(n):\n",
    "        M[i] = areaMach(Ae/A[i]*fMe, 'subsonic')\n",
    "    \n",
    "    # compute pressure distribution\n",
    "    P_P0 = 1.0/isenP(M)\n",
    "\n",
    "\n",
    "plt.figure()\n",
    "plt.plot(x, M)\n",
    "plt.ylabel('M')\n",
    "\n",
    "plt.figure()\n",
    "plt.plot(x, P_P0)\n",
    "plt.ylabel('$P/P_0$')\n",
    "\n",
    "# plt.figure()\n",
    "# plt.plot(x, T_T0)\n",
    "# plt.ylabel('$T/T_0$')\n",
    "\n",
    "# plt.figure()\n",
    "# plt.plot(x, rho_rho0)\n",
    "# plt.ylabel('$\\\\rho/\\\\rho_0$')\n",
    "\n",
    "\n",
    "plt.show()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Case II: normal shock\n",
    "As the back pressure decreases, the throat becomes chocked.  However, the back pressure is not yet low enough to permit isentropic supersonic flow through the back end of the nozzle and show a shock wave must be formed.  As the back pressure is decreased the shock wave moves further back along the nozzle until it is right at the exit. As the back pressure is decreased further we have supersonic flow throughout the back end of the nozzle. We must determine the critical pressure ratio that separates the case with a shock wave from the case with isentropic supersonic flow.\n",
    "\n",
    "Computing this critical case is a little more work.  The limiting case has a shock wave right at the exit.  Thus, we need to find the Mach number behind a normal shock with isentropic supersonic flow all the way up to the exit.  From the normal shock equations we can compute the Mach number behind the shock wave as:\n",
    "$$M_2 = \\sqrt{\\frac{2 + (\\gamma-1)M_1^2}{2\\gamma M_1^2 - (\\gamma-1)}}$$"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 215,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": [
    "def shockM(M1):\n",
    "    \"\"\"computes the mach number behind a normal shock\"\"\"\n",
    "    return sqrt((2 + gm*M1**2)/(2*g*M1**2 - gm))"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Thus, with a shock wave standing right at the exit the Mach number right behind the shock is:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 216,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": [
    "# Mach number ahead of shock right before exit\n",
    "Me_up = areaMach(Athroat/Ae, 'supersonic')\n",
    "\n",
    "# Mach number at exit (downstream of shock)\n",
    "Me = shockM(Me_up)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "We now need to determine the total pressure loss across the shock wave, so we can figure out the pressure ratio across the nozzle.  Using a mass balance from the throat to exit:\n",
    "$$\n",
    "\\begin{align}\n",
    "( P_0 A f(M))_t &= ( P_0 A f(M))_e\\\\\n",
    "{P_0}_t A_t  &= {P_0}_e A_e f(M_e)\\\\\n",
    "\\Rightarrow \\frac{{P_0}_e}{{P_0}_t} f(M_e) &= \\frac{A_t}{A_e}\\\\\n",
    "\\frac{{P_0}_e}{P_e}\\frac{P_e}{{P_0}_t} f(M_e)&=\\frac{A_t}{A_e}\\\\\n",
    "\\Rightarrow \\frac{{P_0}_t}{P_e} &=\\frac{{P_0}_e}{P_e} f(M_e)\\frac{A_e}{A_t}\\\\\n",
    "\\end{align}\n",
    "$$\n",
    "Because there is a shock wave, the exit pressure is again equal to the back pressure ($P_e = P_b$).  Also, because there is no shock wave between the throat at the entrance, $P_0 = {P_0}_t$.  Thus, the critcal pressure ratio is:\n",
    "$$\\frac{P_b}{P_0} =\\frac{P_e}{{P_0}_e} \\frac{1}{f(M_e)}\\frac{A_t}{A_e}$$"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 217,
   "metadata": {
    "collapsed": false
   },
   "outputs": [],
   "source": [
    "# corresponding critical pressure ratio\n",
    "Pb_P0_crit_super = Athroat/(isenP(Me)*fM(Me)*Ae)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "As long as the pressure is above this critical value, a shock wave will exist in the throat.  This case is a bit more difficult if we want to find the location of the shock.  First we must solve for the exit Mach number using the above mass balance.  However, there is no explicit solution so we must again use a root finding technique:\n",
    "$$froot(M_e) = f(M_e) \\frac{{P_0}_e}{P_e} - \\frac{P_0}{P_b}\\frac{A_t}{A_e} = 0$$\n",
    "\n",
    "Once we know the exit Mach number we can find the total pressure loss across the shock using the equation above and repeated here:\n",
    "$$\\frac{{P_0}_e}{{P_0}_t}  = \\frac{A_t}{A_e}\\frac{1}{f(M_e)}$$\n",
    "\n",
    "If we know the total pressure loss across the shock, then we know the upstream Mach number from the normal Mach equations (and also the downstream Mach number).\n",
    "$$\\frac{{P_0}_2}{{P_0}_1} = \\left(\\frac{\\gamma +1}{2 \\gamma M_1^2 - (\\gamma - 1)}\\right)^{\\frac{1}{\\gamma-1}}   \\left( \\frac{(\\gamma + 1) M_1^2}{2 + (\\gamma - 1) M_1^2}\\right)^{\\frac{\\gamma}{\\gamma-1}}$$\n",
    "To solve this for $M_1$ requires a root finding method\n",
    "\n",
    "Using conservation of mass from just behind the shock to the exit (isentropic) allows us to compute the area of where the shock sits:\n",
    "$$\\begin{align}\n",
    "A_s f(M_s) &= A_e f(M_e) \\\\\n",
    "A_s  &= A_e \\frac{f(M_e)}{f(M_s)}\n",
    "\\end{align}$$\n",
    "\n",
    "With a known shock location we can use a mass balance from the entrance to just before the shock (subsonic solution up to throat, supersonic solution after throat).\n",
    "$$\\begin{align}\n",
    "A f(M) &= A_t \\\\\n",
    "f(M) &= \\frac{A_t}{A}\n",
    "\\end{align}$$\n",
    "and after the shock:\n",
    "$$\\begin{align}\n",
    "A f(M) &= A_e f(M_e) \\\\\n",
    "f(M) &= \\frac{A_e}{A} f(M_e)\n",
    "\\end{align}$$"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 218,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
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N65Jin86DkRDoREuJC4FwSVW4tCgN8ksIFDAS8+z2rf7zXFJb4dytcGlRGoOREChgJKYF\nwiVF4RIW6iKTEChgJGb5Z4vVh8uvFS7hoC4yCYECRmKS3XFkzCU59Ui4dPW6pPikpWIkBAoYiTn+\ncHnAHy6T/0vhEk5aKkZCoICRmGI///RIuKQcGXNRuISVMeCqi0yCExXnwZSVlTF37lx8Ph8TJkw4\n6b579+5l0qRJuEcO2wcOHMi9994biTLFY3bX57i/fwCSkv3hcqrCJeyM0RGMBM3zgKmpqeHjjz9m\n7dq15OQ0vhjh/PnzueGGG0hKSgKgf//+4S5RooDdu9sfLtbi3P0rhUukaBaZhMDzgElJSSEvL4/i\n4uJG9y0vL6eyspKbbropApVJtPBfz+UBqD6Mc/dvMKd197qkxKGlYiQEngdMvfojkpOZN28eS5cu\nZevWrXz961/n8ssvj0Bl4iVbUeYPl/IynB8/hOl+ptclJRYdwUgIomaQ3xjT6D79+/fnlltuISsr\nixkzZvDoo48GxmIk/thDB/2rIu/7AueOB3Q9Fy9oDEZCEDVHME3Rr18/+vXrR35+Pu+++y5Tp05l\n8eLF5Ofne12atDBbVYn7xIOwazvOD+/H9OnndUmJSV1kEoKYCpivuvjii/noo49Yt27dMQFTUlJC\nSUlJ4HZBQQGZmZmRLjEqpaamRn1bWF81B3//AHy6ifQf/5KUC4eE5efEQltEyona4lBqKnWQUO2k\n18XRioqKAl/n5uaSm9v0y47HbMAAnHvuuSxZsuSY7cdrhIqKikiVFdUyMzOjui1sXR3utN/CxvWY\nW35MVc65VIWp3mhvi0g6UVu4dXXYurqEaie9LhpkZmZSUFAQ9P2jZgwmGHv27CEvL8/rMqSFWGux\nf3oKPliNuX4izqDhXpckGuSXEERNwNTV1R0zYF9YWMjMmTMB/8mYM2bMYPv27QBs3ryZLVu2MHy4\n3oTihX39Bew7izHfvA5nxJVelyNwZAxGASPBiYoushUrVrBx40YA3nnnHS655BIA9u/fH5hdlpSU\nxJYtW5gyZQo9e/Zk0KBB3HbbbZ7VLC3LXTQXu2A2Ztg3MGOv97ocqedoNWUJnrE2MV49paWlXpcQ\nFaKxf9ktXop97nE4fzDO9ydjnMbPiWoJ0dgWXjnhGMzMx7GflJD0cKEHVXlDr4sGXbuGtmJG1HSR\nSWKyG9Zh/3cq5PTHmfCTiIWLNJGuaCkhUMCIZ+znn+E+8zB06Ybz/36GSUnxuiT5d8bRasoSNAWM\neMKW7ced+ktIbYUz6eeYNulelyTHoyMYCYECRiLOVlfjPvkrOFiOM+kBTMdTvC5JTkRXtJQQKGAk\noqxbh/vcY/DZZpxb78b0PMvrkuRkNE1ZQqCAkYiyf/5f+EcxZvwEzHmDvC5HGuNoDEaCp4CRiHFX\nLMQumou57Js4I8d4XY40hdF5MBI8BYxEhN28EfvSM9D3PEzBLV6XI01ltFSMBE8BI2FnD3yJO+1h\naN8JZ+JkTBMuLidRQrPIJAQKGAkrW1PjD5eqSv+5LulaBj2mqItMQqCAkbCyr86ALR/jfPdOTLcz\nvC5HmsvRNGUJ3kkD5pVXXmnWg7366qshFSPxxX1vJXbZXzFXjMNcMNjrciQYJkldZBK0kwbMkiVL\nqKqqatIDVVVVHffiX5KY7Bel2BeehF5nY67+T6/LkWBpNWUJwUmX6y8rK+PWW2+lbdu2gWXzj8da\nS1lZGTU1NS1eoMQeW+PDffYRcJJwbp2MSY6Kq0JIMIwD1mKtPel7gMjxnPQIxnEcfD4flZWV/qsN\nnuDDdV0SZNV/aQL72vPw2Rac792pZWBinXMkVNRNJkE46b+WTz75JG+++SYrV65k4MCBjBkzhlNP\nPfW4+x4+fJg777wzLEVK7LAb/oFdOg9z+VjMubqcdcwzR/4HdV3QpRSkmU56BNOpUyduvvlmnnji\nCTp27MiDDz7I448/zubNm4/Zt3Xr1owcOTJshUr0s4crcf/4JHQ5HTNO4y5xwTnyFqEeCglCkzrH\n09PTGTduHGPGjGH58uU8/fTTtG3blquuuorzzjsvsF9BQUHYCpXoZ1+bCfv34dzzMCa1ldflSEuo\nH3fRemQShGaNviYnJ3PZZZcxYsQI3n//febMmcOsWbMYM2YMQ4YMwXF0Wk2isiX/wK5YiLniGkyv\ns70uR1pKfReZrfO2DolJQSWCMYYLLriA+++/n969e/PUU08xadIk5s2b19L1SQwIdI2d1h1z1Q1e\nlyMtqf6fRh3BSBCCmj9aVVXFwoULWbBgAfv27QOgQ4cOJ5wAIPHN/uVlOPAlzr2PYFJSvS5HWlJ9\nF5nGYCQIzQqYsrIy5s+fz8KFC6msrATgggsuYOzYsZx9trpFEpH9/DPskjcwQ6/AZOd4XY60tK/O\nIhNppiYFzK5du3jjjTdYtmwZNTU1JCcnM3z4cMaOHUu3bt3CXaNEKWst7ivToXU6Zty3vS5HwiEw\ni0wBI8130oDZsmULc+fOZdWqVVhrSUtL4xvf+AajR4+mQ4cOx+w/ffp0Jk6cGLZiJcqUvA8b12Ou\nn4jJaOt1NRIOgVlkChhpvpMGzJQpUwDIyspi1KhR5Ofn06ZNm+Puu3v3blatWqWASRDWdXFf/xN0\nOhUz7Aqvy5Fw0XkwEoJGu8iSk5NJSkpi0aJFLFq06Lj7WGspLy/XWmSJ5P2/w2dbMN+7C5Oc4nU1\nEi5GS8VI8E4aMG3atOH3v/897du3b/SBDh8+zI9+9KMWK0yil7UWd/5r/mnJg4Z5XY6Ek6NBfgne\nSc+Dyc/Pb1K4gJaKSSiffAjbt2K+fhVG61PFN6MuMgneSQPm+uuvb9aDjR8/PqRiJDa4i+ZCRlvM\noOFelyLhptWUJQRa20WaxX6xE9a/hxn+Da03lgh0HoyEQAEjzWLf9V+11Awf5XElEhGaRSYhUMBI\nk1lrsauWwdkDMO07el2ORIDRasoSAgWMNN22f8GeXZg8zRxLGFpNWUKggJEms6uWQXIK5vzBXpci\nkaLVlCUEChhpEltXh31vBQy4ENMm3etyJFJ0oqWEQAEjTfPxeig/gJOnqckJRUcwEoKgrgfT0srK\nypg7dy4+n48JEyacdN/i4mJWr15NWloanTp14pprrolQlYnNrloOrdvAgAu9LkUiyWg1ZQme50cw\nNTU1fPzxx6xduxafz3fSfdevX8/s2bOZNGkSEydOZMeOHSxYsCBClSYu66vGvv93zPkX64JiiUar\nKUsIPA+YlJQU8vLy6NWrV6P7zpo1iyFDhgSmTg4bNoyioqJGg0lC9M81UHUYo+6xxKPzYCQEngdM\nvaSkk69ptWfPHrZt20aPHj0C23r06EFlZSUbNmwId3kJy9bW4P7tdWjXHs7u73U5Emka5JcQRE3A\nBE7oOoHt27cDkJmZGdiWkZEBQGlpafgKS3D2tedh6yc4192qhS0TkVZTlhBETcA0prKyEmgIFfBf\nqwagurrak5rind2wDrvkTf+qyRcO8boc8YJWU5YQRMUssqaoD5ba2trAtvqxl/T0o8/LKCkpoaSk\nJHC7oKDgqCOfRJaamtrktqj8YBW+1um0/c4P4/KiYs1pi3h3oraozUjnINC6VStSEqSt9Lo4WlFR\nUeDr3NxccnNzm3zfmAmYLl26AFBeXh7YVv91t27djtr3eI1QUVER5gpjQ2ZmZpPawrou7tq/Y3IH\ncvBwFVAV/uIirKltkQhO1Bb2sP/3frjyEFUJ0lZ6XTTIzMykoKAg6PvHTBdZly5dyM7OZtOmTYFt\nO3bsICMjg5ycHA8ri1OfbYay/TDga15XIl7SLDIJQdQETF1dHe6/DSQWFhYyc+bMwO1x48axevXq\nwO1ly5Yxfvz4RmegSfPZ9e+BcTD9LvC6FPGSzoOREERFF9mKFSvYuHEjAO+88w6XXHIJAPv37z9q\ndlleXh4HDhxg2rRpJCcnk5OTQ35+vic1xzv7wXvQKweT2dbrUsRLjs7kl+BFRcAMHTqUoUOHHrN9\n8uTJx2xToISf3V0Kn23GXPMdr0sRrxmtRSbBi5ouMokedvlfISkJc/EIr0sRr+lESwmBAkaOYn3V\n2JWLMeddhMnq4HU54rUjXWRWRzASBAWMHMWuWQmVBzGXjvK6FIkGWk1ZQqCAkaPYtxdAl26Qo3XH\nBM0ik5AoYCTAfroJtn6CuXRUo2vDSYLQeTASAgWMBNi3F0BqKw3uSwMdwUgIFDACgK08iF29DDNo\nOKZNRuN3kMRQv4K2xmAkCAoYAcC+8xb4fJjhGtyXrwhMU1YXmTSfAkawdXXYt96A3n0xPRu/sqgk\nEEddZBI8BYxg338XvvwCJ/9qr0uRaKOlYiQECpgEZ63FLpoDnbvCgDyvy5FoowuOSQgUMIlu00f+\nqclfH4tx9HKQf6NZZBICvaMkOHfhHEjPxFw80utSJBqpi0xCoIBJYHZ3KXywCjN8FKZVK6/LkWgU\nWE1ZASPNp4BJYHbxX/yrJl92pdelSLTSNGUJgQImQdmy/dh3FmMuGoFp197rciRaOboejARPAZOg\n7MI5UFuLGfUfXpci0SzQRVbnbR0SkxQwCchWlGOXLcDkDcV07up1ORLNtNilhEABk4Ds4r+Arxoz\n+lqvS5FopzEYCYECJsG4hw5il74J51+M6drD63Ik2uk8GAmBAibB+P76f3C4EufK8V6XIjHAGOMf\nh9F5MBIEBUwCsVWVVC+YDefmYbqf6XU5Eiscoy4yCYoCJoHYhXOxB8txvqmjF2kGY9RFJkFRwCQI\nW1GGXTiHlLxhmDN6e12OxBJHXWQSHAVMgrDz/wy+atLGf8/rUiTWGEdHMBIUBUwCsF/uwb49DzP4\nMpJO7+l1ORJrjMZgJDgKmARg33gZADPmeo8rkZjkOAoYCYoCJs7Znduxf1+CufRKTMdTvC5HYpFx\ntFSMBEUBE+fc2X+EVq0wo7/ldSkSq3QEI0FSwMQx++H78MFqzDfHYzLbeV2OxCpjtJqyBEUBE6ds\nbS1u0XPQ+TTMZWO8Lkdimc7klyApYOKUfXs+7NyOU3ALJiXF63IkljmapizBUcDEIVtRhv3Ly9B3\nIAz4mtflSKzTNGUJkgImDtk5s6D6MM74W/yLFYqEQkvFSJAUMHHGbt6IXfE3zIgrtRy/tAwtFSNB\nSva6AIDi4mJWr15NWloanTp14pprrjnhvnv37mXSpEm4R/6jGjhwIPfee2+kSo1qtrYG909PQVZH\nzNU3el2OxAstFSNB8jxg1q9fz+zZs3nkkUcwxjB16lQWLFjAqFGjjrv//PnzueGGG0hKSgKgf//+\nkSw3qtm/vQ6ff4pz+/2YtDZelyPxQsv1S5A8D5hZs2YxZMiQwFjBsGHDeOKJJxg5ciSpqalH7Vte\nXk5lZSU33XSTF6VGNbvrc+ybr2IuuARzbp7X5Ug8MQ5WXWQSBE/HYPbs2cO2bdvo0aNhrKBHjx5U\nVlayYcOGY/afN28eS5cu5Z577mHx4sWRLDWqWWv9XWMpqZjrbvW6HIk3mqYsQfI0YLZv3w5AZmZm\nYFtGRgYApaWlx+zfv39/brnlFrKyspgxYwaPPvpoYCwmkdm3F8AnH2K+dTMmq4PX5Ui8MQ5UHsJq\nPTJpJk8DprKyEmgIFYDkZH+vXXV19TH79+vXj/z8fKZMmcKdd97J+++/n/BHMnbXDuyfZ0K/8zFD\n870uR+KQ6XI6fPxP3F9Mwi1eiq1T0EjTeDoGUx8stbW1gW0+nw+A9PT0k9734osv5qOPPmLdunXk\n5x/9xlpSUkJJSUngdkFBwVFHSfHC1tZy8Pn/wbRKI/OHP8Np27bR+6SmpsZlWwRDbdHgZG1hf/wg\nNauWU/X6i7jPPY6ZV0Tq1TeSOuTrmGTPh3FbnF4XRysqKgp8nZubS25ubpPv6+mro0uXLoB/8L5e\n/dfdunVr9P7nnnsuS5YsOWb78RqhoqIilFKjkvv6i9gtn+Dcdi+HklOhCc8xMzMzLtsiGGqLBo22\nRb8Loe/5OOtW4c57lcPPPMLhV2diRozGDM3HZDT+z02s0OuiQWZmJgUFBUHf3/OAyc7OZtOmTfTt\n2xeAHTt2kJGRQU5OTqP337NnD3l5iTljym76CLvgz5jBIzHnD/a6HEkAxnHg/ItxBl4E/1yDu2gu\n9v9ewL7xCce3AAAPNElEQVT5CuaiEZiRY3RyrxzF8zP5x40bx+rVqwO3ly1bxvjx40lKSqKwsJCZ\nM2cCUFZWxowZMwITAzZv3syWLVsYPny4J3V7yVaU4U5/FDqeolljEnHGGMyAr5H0k1/j/GIqZtCl\n2HeX4v7iduoeux931TJsjc/rMiUKeN6BmpeXx4EDB5g2bRrJycnk5OQExlT2798fOD8mKSmJLVu2\nMGXKFHr27MmgQYO47bbbvCzdE9atwy18DCrKcO59BNNaJ1SKd0y3MzA33Y4ddxN2+V+xKxZiCx/D\ntsnAXHQpZsjXMd3P9LpM8YixNjFO0T3etOdY5M55ETuvCHPT7ThBzBpT/3IDtUWDlmoL67qwcT12\n5SLsP96F2lro0QuTNwzztaGYDp1aoNrw0uuiQdeuXUO6v+dHMNJ0dv17/nC55PKgwkUk3IzjQN/z\nMH3Pwx4sx65ahn13KfbPz2Nn/y/07ov52jD/ihOZ8TMxQI5PRzAxwu7cjvvwT6FjZ3/XWGqroB5H\n/501UFs0CHdb2N2l2PeWY1cth107/CdvnnU25rxB/o/Oof2n3JL0umgQ6hGMAiYG2PIDuL+dDNVV\nOD/7b0ynU4N+LP3xNFBbNIhUW1hrYftW7D/exa5bBTu2+b9xWnfMeXmYcwfBmX38R0Ie0euigbrI\n4pz1VeM+/Rso248z+TchhYuI14wx0CMb0yMbrroRu3c39oPV2HWrsH97HbtgNmRkYnIGQN9zMeec\nhzmli9dlS5AUMFHMui72+Sdg80acH9yLObOP1yWJtCjT6VTMyDEwcgz20EHsh2thwzrshnWw9h0s\nwCldMOecC2efi+mTi2nX3uuypYkUMFHKWot9tRC7ZqV/EcsLdDKlxDeTnoEZNBwGDfd3pe36HPvR\nOuxHH2DfWwHL/9YQOL3OgbPOwZx1jr97zcMuNTkxBUwUstZiX38Bu+RNzOVXYfLHeV2SSEQZY+C0\nbpjTusFl3/QvsLntX9jNH/lXsSh5H4qX+gOndTpk98H07I3pmQ09z4IOpwTOoRPvKGCikJ3/GnbB\nbMywb2AKvqc/FEl4JikJep2N6XU25I/zH+Hs2YXd9BFs/gi75WPsX//sPw8HID3TP9bT8yzofqZ/\nCZtTT8ekpHj7RBKMAibKuH97HTvnRf/aTjf+QOEichzGGOh8GqbzaTD4MgD/8jQ7tmE/3QyfbcZ+\nuhm7aC7U1fqPdBwHOneFrj38gdO1h/9SBKd0waS19vT5xCsFTJSw1mL/8rJ/4cALh2BuvkP9yiLN\nYFJS/VOcvzIZxtbUwO4d2M8/g9Lt2NLP/CH0j2KwLoFzNNq1h1P8gVXVrSduuw7+8DqlC7TJ0D96\nQVLARAHrutii57BvvYG5ZCTmP2/3dwmISEhMSgp0OxPT7ej10GyND3bugD07sV/shC/8n+2GdVT9\n/S3/PvU7t0qDDqdAh06YI5/pcAqmvf8zWR0wrdIi+8RihALGY7a2Fvunp7B/fwtz+VjMtd/TkYtI\nmJmUVOiR7R+n+bfvZaSmULF1kz909uyC/Xux+/bAl3uw27dC+QHgKwEE0Ko1tMuCtlnQtj2mnf8z\n7dpjjmwjI9M/NtS6TcIcESlgPGQPVeA+8zvYuB4z9gbMN8cnzAtPJFqZVmmY03vC6T2PCR84cvSz\nfy/sOxI8ZQf8oVO+H1u2H3Zux378TzjkXw3gmKVSHMcfNOkZRz5nYtIzIL2tf1tGpr9brnUbSGsD\naa2PfPi/jqWJCgoYj9hdn+M++SvY9wXme3fhXDzC65JEpAlMSqp/skDnrscNoHq2pgYqDgQCyB4s\n94fOoYNwqBwOHcQeqoADX2I//xQOVkD14Yb7n+iBk5P9gdOqNbRuEwgfk9YaWrWClFaQmgopRz5S\nWx35nAoprTD13ztym9T6/VMgOQWSUiAlGeOE3k2vgPGA/XAt7oz/hqRknJ/8GnNWX69LEpEWZlJS\njozdnOK/3YT72NqaIwFUAVWHoaoSqg5jqw7D4Ybb9d/zb6+EijJ/d56vGmqqwefzf328n9HUJ+A4\n8Mbqxvc7CQVMBNm6OuzcF/3rLXU7A+eH92ltMREJMMkp/hlt/7YcTjAd59ZaqK3xh0196NT4jrlt\nfdX+7TU1UFcDNbX++9XWhPx8FDARYvft9R+1bNqAGXYFZvyEoJfcFxFpjDGmoZuMjBPvF8YaFDBh\nZq3FFr+NfWU61LmYCT/BGTTc67JERMJOARNG9sA+3Befhg9WQ6+zcb57J+bU6LmwkohIOClgwsC6\nddjlC7Gv/wlqfP5zWy4f0yKzMkREYoUCpoXZzRtxX3oWPtsMOf1xvn0bpks3r8sSEYk4BUwLsXt2\nYefOwq5a5l864ta7MV8bqhMnRSRhKWBCZMv2Y+e9il2+EJIczKhvYUZfq9VZRSThKWCCZL/8Avu3\n/8OuXAxuHWZoPubK8ZisDl6XJiISFRQwzWQ/3YR96w3s6uWAwVx0KWb0tzCdNTtMROSrFDBNYKur\nsWtWYN9eANv+BamtMCOuxHz9akyHTl6XJyISlRQwJ2Cthc+2YIuXYv/+FlQegtO6Y66f6L/aZJt0\nr0sUEYlqCph/Y3fuwK5ejn1vBez+HJKSMedfjLl0FPTO1awwEZEmSviAsa4L2/6F/eda7LpVsGMr\nGAM5/TFXjPOHS3qm12WKiMSchAwYW1GG3bge1q/BlrwPFWVgHOiVg7nuVswFl2g2mIhIiBImYNz3\nVsAnH2I//hB2bvdvTM/E9Dsf+l+IyR2IyWjrbZEiInEkYQLGTn/UfwW4s87GXDwC06cfnNlb64OJ\niIRJwgSM87P/hh69MEkKFBGRSEiYgDFn9vG6BBGRhOJ4XYCIiMQnBYyIiISF511kxcXFrF69mrS0\nNDp16sQ111zTIvuKiIi3PD2CWb9+PbNnz2bSpElMnDiRHTt2sGDBgpD3FRER73kaMLNmzWLIkCGB\n5VeGDRtGUVERPp8vpH1FRMR7ngXMnj172LZtGz169Ahs69GjB5WVlWzYsCHofUVEJDp4FjDbt/vP\nps/MbFjnKyMjA4DS0tKg9xURkejgWcBUVlYCDUEBkJzsn3NQXV0d9L4iIhIdPJtFVh8WtbW1gW31\n4ynp6elB7wtQUlJCSUlJ4HZBQQFdu+qKk/W+eiSY6NQWDdQWDdQWDYqKigJf5+bmkpub2+T7enYE\n06VLFwDKy8sD2+q/7tatW9D7gr8RCgoKAh9fbaBEp7ZooLZooLZooLZoUFRUdNR7aXPCBTwOmOzs\nbDZt2hTYtmPHDjIyMsjJyQl6XxERiQ6eTlMeN24cq1evDtxetmwZ48ePJykpicLCQmbOnNmkfUVE\nJPp4eiZ/Xl4eBw4cYNq0aSQnJ5OTk0N+fj4A+/fvP+ryxCfbtzHNPayLZ2qLBmqLBmqLBmqLBqG2\nhbHW2haqRUREJECLXYqISFgoYEREJCwUMCIiEhYKGBERCQvPrwfTUnRdmQZNfX779u2jsLCQDRs2\n0K5dO8aOHcvIkSMjXG14BfO73rJlC/fffz8vvfRSBCqMnOa2hc/n4+233yYlJYX27dvTt29fUlNT\nI1RteDW1Laqqqnj55ZdJS0v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      "text/plain": [
       "<matplotlib.figure.Figure at 0x10748d9d0>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    },
    {
     "data": {
      "image/png": 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      "text/plain": [
       "<matplotlib.figure.Figure at 0x107a82210>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "Pb_P0 = 0.5\n",
    "\n",
    "if Pb_P0 > Pb_P0_crit_super:  # shock in nozzle\n",
    "\n",
    "    # exit Mach number\n",
    "    Me = brentq(lambda M: fM(M)*isenP(M) - Athroat/Ae/Pb_P0, 1e-6, 1.0)\n",
    "    \n",
    "    # total pressure loss across shock\n",
    "    P0e_P0t = Athroat/Ae/fM(Me)\n",
    "    \n",
    "    # Mach number ahead of shock\n",
    "    Mup = brentq(lambda M: P0e_P0t - (gp/(2*g*M**2 - gm))**(1.0/gm)*(gp*M**2/(2 + gm*M**2))**(g/gm), 1.0, 10.0)\n",
    "    \n",
    "    # Mach number after shock\n",
    "    Mdown = shockM(Mup)\n",
    "    \n",
    "    # area at shock location\n",
    "    As = Ae*fM(Me)/fM(Mdown)\n",
    "    \n",
    "    # find closest corresponding index of shock (must be aft of throat)\n",
    "    ishock = np.argwhere(A[ithroat:] >= As)[0][0]\n",
    "    ishock += ithroat\n",
    "    \n",
    "    # solution up to shock\n",
    "    for i in range(ithroat):\n",
    "        M[i] = areaMach(Athroat/A[i], 'subsonic')\n",
    "    for i in range(ithroat, ishock):\n",
    "        M[i] = areaMach(Athroat/A[i], 'supersonic')\n",
    "        \n",
    "    # solution after shock\n",
    "    fMe = fM(Me)\n",
    "    for i in range(ishock, n):\n",
    "        M[i] = areaMach(Ae/A[i]*fMe, 'subsonic')\n",
    "\n",
    "    # isentropic pressure\n",
    "    P_P0 = 1.0/isenP(M)\n",
    "    \n",
    "    # add additional pressure loss of shock\n",
    "    P_P0[ishock:] *= P0e_P0t\n",
    "\n",
    "plt.figure()\n",
    "plt.plot(x, M)\n",
    "plt.ylabel('M')\n",
    "\n",
    "plt.figure()\n",
    "plt.plot(x, P_P0)\n",
    "plt.ylabel('$P/P_0$')\n",
    "\n",
    "plt.show()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Case III: isentropic supersonic flow\n",
    "\n",
    "The last case is relatively simple.  Choked flow with isentropic flow aft of the nozzle exit.  The exit Mach number is supersonic and thus the exit mach number and the back pressure Mach number need not agree.  When the back pressure is reduced below the previous case, but is still larger than the exit pressure the nozzle is overexpanded and oblique shocks wil exist outside of the nozzle to converge the flow and increase the pressure to match the back pressure.\n",
    "\n",
    "Conversely, if the back pressure is reduced below the exit pressure then the nozzle is underexpanded and expansion waves will exist outside of the nozzle to expand the flow and decrease the pressure to match the back pressure.\n",
    "\n",
    "If they back pressure and the exit pressure match perfectly, then the nozzle is ideally expanded."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 219,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "overexpanded\n"
     ]
    },
    {
     "data": {
      "image/png": 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      "text/plain": [
       "<matplotlib.figure.Figure at 0x1070121d0>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    },
    {
     "data": {
      "image/png": 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XD/8RPtvn3fI5Ve90SSLSQQoeCRtmxCW47rgfdu/yHvNpaHC6JBHpAAWPhBVz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\n8rPevXs3P/nJT1i+fHkQKgy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      "text/plain": [
       "<matplotlib.figure.Figure at 0x107012a10>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "Pb_P0 = 0.2\n",
    "\n",
    "for i in range(ithroat):\n",
    "    M[i] = areaMach(Athroat/A[i], 'subsonic')\n",
    "for i in range(ithroat, n):\n",
    "    M[i] = areaMach(Athroat/A[i], 'supersonic')\n",
    "\n",
    "P_P0 = 1.0/isenP(M)\n",
    "\n",
    "Pe_P0 = P_P0[-1]\n",
    "\n",
    "if Pb_P0 > Pe_P0:\n",
    "    print 'overexpanded'\n",
    "elif Pb_P0 == Pe_P0:\n",
    "    print 'ideally expanded'\n",
    "else:\n",
    "    print 'underexpanded'\n",
    "\n",
    "plt.figure()\n",
    "plt.plot(x, M)\n",
    "plt.ylabel('M')\n",
    "\n",
    "plt.figure()\n",
    "plt.plot(x, P_P0)\n",
    "plt.ylabel('$P/P_0$')\n",
    "\n",
    "plt.show()\n"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Interactive\n",
    "\n",
    "Let's now combine all the cases into one simulation"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 220,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "image/png": 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pU6awa9eudu0d/dCqra295RoDidPp1JiDgMYcHJxOJ1lZWb4uQ0REeplfBPO9e/eSnJxM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\n/v73v4/D4eD555/nT3/6E5MnT+aee+7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      "text/plain": [
       "<matplotlib.figure.Figure at 0x107b2e190>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    },
    {
     "data": {
      "text/plain": [
       "None"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "# an interactive version where you can change the back pressure\n",
    "# this will not run on nbviewer.  You can view it locally if\n",
    "# you've installed Jupyter notebook or upload to http://try.jupyter.org\n",
    "\n",
    "@interact(Pb_P0=(0, 0.995, 0.005))\n",
    "def nozzle(Pb_P0=0.995):\n",
    "    \n",
    "    M = np.zeros(n)\n",
    "\n",
    "    if Pb_P0 > Pb_P0_crit_sub:  # subsonic\n",
    "\n",
    "        # back pressure equals exit pressure b.c. subsonic\n",
    "        P0_Pe = 1.0/Pb_P0  \n",
    "\n",
    "        # find exit Mach number from isentropic relationship\n",
    "        Me = isenM(P0_Pe)\n",
    "\n",
    "        # precompute f(Me) so we don't have to keep computing in the loop\n",
    "        fMe = fM(Me)\n",
    "\n",
    "        # iterate across all areas solving for the subsonic Mach number\n",
    "        for i in range(n):\n",
    "            M[i] = areaMach(Ae/A[i]*fMe, 'subsonic')\n",
    "\n",
    "        # compute pressure distribution\n",
    "        P_P0 = 1.0/isenP(M)\n",
    "\n",
    "    elif Pb_P0 > Pb_P0_crit_super:  # shock in nozzle\n",
    "\n",
    "        # exit Mach number\n",
    "        Me = brentq(lambda M: fM(M)*isenP(M) - Athroat/Ae/Pb_P0, 1e-6, 1.0)\n",
    "\n",
    "        # total pressure loss across shock\n",
    "        P0e_P0t = Athroat/Ae/fM(Me)\n",
    "\n",
    "        # Mach number ahead of shock\n",
    "        Mup = brentq(lambda M: P0e_P0t - (gp/(2*g*M**2 - gm))**(1.0/gm)*(gp*M**2/(2 + gm*M**2))**(g/gm), 1.0, 10.0)\n",
    "\n",
    "        # Mach number after shock\n",
    "        Mdown = shockM(Mup)\n",
    "\n",
    "        # area at shock location\n",
    "        As = Ae*fM(Me)/fM(Mdown)\n",
    "\n",
    "        # find closest corresponding index of shock (must be aft of throat)\n",
    "        ishock = np.argwhere(A[ithroat:] >= As)[0][0]\n",
    "        ishock += ithroat\n",
    "\n",
    "        # solution up to shock\n",
    "        for i in range(ithroat):\n",
    "            M[i] = areaMach(Athroat/A[i], 'subsonic')\n",
    "        for i in range(ithroat, ishock):\n",
    "            M[i] = areaMach(Athroat/A[i], 'supersonic')\n",
    "\n",
    "        # solution after shock\n",
    "        fMe = fM(Me)\n",
    "        for i in range(ishock, n):\n",
    "            M[i] = areaMach(Ae/A[i]*fMe, 'subsonic')\n",
    "\n",
    "        # isentropic pressure\n",
    "        P_P0 = 1.0/isenP(M)\n",
    "\n",
    "        # add additional pressure loss of shock\n",
    "        P_P0[ishock:] *= P0e_P0t\n",
    "\n",
    "    else:\n",
    "\n",
    "        for i in range(ithroat):\n",
    "            M[i] = areaMach(Athroat/A[i], 'subsonic')\n",
    "        for i in range(ithroat, n):\n",
    "            M[i] = areaMach(Athroat/A[i], 'supersonic')\n",
    "\n",
    "        P_P0 = 1.0/isenP(M)\n",
    "\n",
    "        Pe_P0 = P_P0[-1]\n",
    "\n",
    "#         if Pb_P0 > Pe_P0:\n",
    "#             print 'overexpanded'\n",
    "#         elif Pb_P0 == Pe_P0:\n",
    "#             print 'ideally expanded'\n",
    "#         else:\n",
    "#             print 'underexpanded'\n",
    "            \n",
    "    \n",
    "    plt.figure(figsize=(12,4))\n",
    "    plt.subplot(1,2,1)\n",
    "    plt.plot(x, M)\n",
    "    plt.ylim([0, 3])\n",
    "    plt.ylabel('M')\n",
    "\n",
    "    plt.subplot(1,2,2)\n",
    "    plt.plot(x, P_P0)\n",
    "    plt.plot([0.9, 1.0], [Pb_P0, Pb_P0], '--')\n",
    "    plt.ylim([0, 1])\n",
    "    plt.ylabel('$P/P_0$')\n",
    "    \n",
    "    plt.show()\n"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": []
  }
 ],
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   "language": "python",
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