tutorials/Tutorial2.ipynb

{
 "cells": [
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# noisyduck\n",
    "\n",
    "## Tutorial 2: Swirling Radial Equilibrium Flow in a Cylindrical Annulus\n",
    "\n",
    "Here we use noisyduck to analyze the case of swirling mean flow in radial equilibrium in a cylindrical annulus. We don't have analytical solutions for this case, but there are a number of authors who have reported results which we can compare against.\n",
    "\n",
    "$\\bar{v}_z = M = 0.3, \\quad \\bar{v}_\\theta = \\Gamma/r,  \\quad  \\Gamma=2,   \\quad  \\bar{p} = \\frac{1}{\\gamma} + \\frac{\\Gamma^2}{2}\\left(1 - \\frac{1}{r^2}\\right)$"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 2,
   "metadata": {},
   "outputs": [],
   "source": [
    "%matplotlib inline\n",
    "import numpy as np\n",
    "import matplotlib.pyplot as plt\n",
    "import noisyduck as nd"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "For this particular mean flow condition, authors have reported using different distributions of density. In this example, we use a homentropic profile that is consistent with Kousen and Nijbour.\n",
    "\n",
    "$\\rho = \\left(1 + \\frac{\\Gamma^2}{2}(\\gamma-1)\\left(1 - \\frac{1}{r^2}\\right)\\right)^{\\frac{1}{\\gamma-1}}$"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 17,
   "metadata": {},
   "outputs": [],
   "source": [
    "# Geometry\n",
    "res = 50 ; ri = 0.4 ; ro = 1.0\n",
    "r = np.linspace(ri,ro,res)\n",
    "\n",
    "# Define mean state\n",
    "GAMMA=0.2; gam=1.4; vr=0.; vt=GAMMA/r; vz=0.3\n",
    "p = (1./gam) + (GAMMA*GAMMA/2.)*(1. - 1./(r*r))\n",
    "\n",
    "# Homentropic density: consistent with Kousen, Nijbour\n",
    "rho = (1. + GAMMA*GAMMA*(gam-1.)*(1. - 1./(r*r))/2.)**(1./(gam-1.))\n",
    "\n",
    "# Define circumferential and temporal wavenumber\n",
    "omega=-10.; m=2"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "With the geometry and mean flow state defined, the numerical eigendecomposition is computed. The decomposition is computed twice; once without filtering and once with filtering so we can compare their eigenspectra."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 5,
   "metadata": {},
   "outputs": [],
   "source": [
    "evals_r, evecs_rl, evecs_rr = nd.annulus.numerical.decomposition(omega,m,r,rho,vr,vt,vz,p,gam,filter='None',perturb_omega=True)\n",
    "evals_f, evecs_fl, evecs_fr = nd.annulus.numerical.decomposition(omega,m,r,rho,vr,vt,vz,p,gam,filter='acoustic',alpha=0.00001,perturb_omega=True)\n",
    "\n",
    "rho_evecs = evecs_fr[0*res:1*res,:]\n",
    "vr_evecs  = evecs_fr[1*res:2*res,:]\n",
    "vt_evecs  = evecs_fr[2*res:3*res,:]\n",
    "vz_evecs  = evecs_fr[3*res:4*res,:]\n",
    "p_evecs   = evecs_fr[4*res:5*res,:]"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Here, we will compare the eigenvalues from the numerical decomposition against reported values by Nijbour.\n",
    "\n",
    "    Nijbour, \"Eigenvalues and Eigenfunctions of Ducted Swirling Flows\", 7th AIAA/CEAS Aeroacoustics Conference, AIAA 2001-2178.\n",
    "\n",
    "Data was extracted from the conference paper using the online tool WebPlotDigitizer: https://automeris.io/WebPlotDigitizer/"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 6,
   "metadata": {},
   "outputs": [],
   "source": [
    "eigenvalues_nijbour = np.array([[-3.027340779492441,  48.36867862969005  ],\n",
    "                                [-3.061040170507461,  42.82218597063621  ],\n",
    "                                [-3.062863166930221,  37.11256117455139  ],\n",
    "                                [-3.0646340777409034, 31.56606851549755  ],\n",
    "                                [-3.066509159775743,  25.693311582381725 ],\n",
    "                                [-3.0683842418105804, 19.8205546492659   ],\n",
    "                                [-3.0703634950695786, 13.621533442088094 ],\n",
    "                                [-3.008902472816523,  6.117455138662315  ],\n",
    "                                [-3.0128609793345174,-6.280587275693314  ],\n",
    "                                [-3.0471333120824085,-13.621533442088094 ],\n",
    "                                [-3.0810931311578234,-19.9836867862969   ],\n",
    "                                [-3.0830202988047404,-26.019575856443723 ],\n",
    "                                [-3.0848432952275022,-31.729200652528547 ],\n",
    "                                [-3.054737811445923, -37.438825448613386 ],\n",
    "                                [-3.056508722256604, -42.98531810766721  ],\n",
    "                                [-3.058331718679364, -48.694942903752036 ],\n",
    "                                [-9.939361930417792, -0.08156606851550663],\n",
    "                                [-12.940639069625966,-0.08156606851550663],\n",
    "                                [3.9814554386754395, -0.08156606851550663],\n",
    "                                [6.695376256044533,  -0.08156606851550663]])"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 18,
   "metadata": {},
   "outputs": [
    {
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\n",
      "text/plain": [
       "<Figure size 432x288 with 1 Axes>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "fig = plt.figure()\n",
    "# Plot raw numerical eigenvalues\n",
    "for i in range(len(evals_r)):\n",
    "    if (evals_r[i].imag > 0.):\n",
    "        plt.plot(evals_r[i].real,evals_r[i].imag, 'b^',markersize=3)\n",
    "    elif (evals_r[i].imag < 0.):\n",
    "        plt.plot(evals_r[i].real,evals_r[i].imag, 'bs',markersize=3)\n",
    "l = plt.xlabel('Eigenvalue: real')\n",
    "l = plt.ylabel('Eigenvalue: imag')"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Here we plot the filtered numerical eigenvalue spectrum against the data extracted from Nijbour. There are some subtle differences in the eigenvalues, which were found to be due to the lack of a term \n",
    "\n",
    "$\\frac{\\partial v_\\theta}{\\partial r} v_r'$\n",
    "\n",
    "that contributes to the residual of the equation for $v_\\theta$."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 19,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": 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\n",
      "text/plain": [
       "<Figure size 432x288 with 1 Axes>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "# Plot filtered numerical eigenvalues\n",
    "for i in range(len(evals_f)):\n",
    "    if (evals_f[i].imag > 0.):\n",
    "        h_up, = plt.plot(evals_f[i].real,evals_f[i].imag,   'b^',markersize=3, label='Acc. Down')\n",
    "    elif (evals_f[i].imag < 0.):\n",
    "        h_down, = plt.plot(evals_f[i].real,evals_f[i].imag, 'bs',markersize=3, label='Acc. Up'  )\n",
    "# Plot analytical eigenvalues\n",
    "h_analytical, = plt.plot(eigenvalues_nijbour[:,0],eigenvalues_nijbour[:,1], 'ko', markerfacecolor='None',markersize=7,label='Nijbour(2001)')\n",
    "l=plt.legend(handles=[h_analytical,h_up,h_down],numpoints=1)\n",
    "l = plt.xlabel('Eigenvalue: real')\n",
    "l = plt.ylabel('Eigenvalue: imag')"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 21,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": 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\n",
      "text/plain": [
       "<Figure size 288x360 with 1 Axes>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "# Plot first 4 eigenvectors of pressure, normalized by maximum value\n",
    "fig = plt.figure(figsize=(4,5))\n",
    "for i in range(4):\n",
    "    loc = np.argmax(np.abs(p_evecs[:,2*i]))\n",
    "    plt.plot(p_evecs[:,2*i].real/np.max(np.abs(p_evecs[loc,2*i].real)),r, 'ko')\n",
    "l=plt.xlim((-1,1))\n",
    "l=plt.ylim((0.4,1.0))\n",
    "l=plt.xlabel('$Re(p_{mn})$')\n",
    "l=plt.ylabel('Radius')"
   ]
  }
 ],
 "metadata": {
  "kernelspec": {
   "display_name": "Python 3",
   "language": "python",
   "name": "python3"
  },
  "language_info": {
   "codemirror_mode": {
    "name": "ipython",
    "version": 3
   },
   "file_extension": ".py",
   "mimetype": "text/x-python",
   "name": "python",
   "nbconvert_exporter": "python",
   "pygments_lexer": "ipython3",
   "version": "3.6.4"
  }
 },
 "nbformat": 4,
 "nbformat_minor": 2
}