path: root/Problems_In_Fluid_Flow/ch3.ipynb

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   "cells": [
    {
     "cell_type": "heading",
     "level": 1,
     "metadata": {},
     "source": [
      "Chapter 3 : velocity boundary layers\n"
     ]
    },
    {
     "cell_type": "heading",
     "level": 3,
     "metadata": {},
     "source": [
      "example 3.1 page no : 50"
     ]
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "'''\n",
      "find\n",
      "fluid in boundary layer would be entirely\n",
      "reynolds \n",
      "boundary layer width\n",
      "velocity of water\n",
      "shear stress \n",
      "mean shear stress experienced\n",
      "total force experienced\n",
      "'''\n",
      "\n",
      "import math \n",
      "\n",
      "# Initialization of Variable\n",
      "rho=998.\n",
      "mu=1.002/1000\n",
      "x=48/100.\n",
      "u=19.6/100\n",
      "x1=30/100.\n",
      "b=2.6\n",
      "\n",
      "#calculation\n",
      "\n",
      "print \"fluid in boundary layer would be entirely in streamline motion \"\n",
      "Re=rho*x*u/mu\n",
      "print \"reynolds no is %.2e\"%(Re)\n",
      "\n",
      "#part 2\n",
      "Re1=rho*x1*u/mu\n",
      "delta=x1*4.64*Re1**-.5\n",
      "print \"boundary layer width in (mm): %.4f\"%(delta*1000)\n",
      "\n",
      "#part3\n",
      "y=0.5*delta           #middle of boundary layer\n",
      "ux=3/2*u*y/delta-.5*u*(y/delta)**3\n",
      "print \"velocity of water in (cm/s): %.4f\"%(ux*100)\n",
      "\n",
      "#part4\n",
      "R=0.323*rho*u**2*Re1**-0.5\n",
      "print \"shear stress at 30cm in (N/m**2): %.4f\"%R\n",
      "\n",
      "#part5\n",
      "Rms=0.646*rho*u**2*Re**-0.5\n",
      "print \"mean shear stress experienced over whole plate in (N/m**2) %.4f\"%Rms\n",
      "\n",
      "#part6\n",
      "F=Rms*x*b\n",
      "print \"total force experienced by the plate in (N) %.4f\"%F\n",
      "\n"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "output_type": "stream",
       "stream": "stdout",
       "text": [
        "fluid in boundary layer would be entirely in streamline motion \n",
        "reynolds no is 9.37e+04\n",
        "boundary layer width in (mm): 5.7520\n",
        "velocity of water in (cm/s): 8.5750\n",
        "shear stress at 30cm in (N/m**2): 0.0512\n",
        "mean shear stress experienced over whole plate in (N/m**2) 0.0809\n",
        "total force experienced by the plate in (N) 0.1010\n"
       ]
      }
     ],
     "prompt_number": 2
    },
    {
     "cell_type": "heading",
     "level": 3,
     "metadata": {},
     "source": [
      "example 3.2 page no : 52"
     ]
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "'''\n",
      "find\n",
      "% of surface over which turbulent boundary layer \n",
      "thickness of boundary layer \n",
      "velocity of air at mid point \n",
      "thickness of laminar boundary layer\n",
      "velocity at outer edge of laminar sublayer\n",
      "shearforce expericienced\n",
      "mean shearforce \n",
      "total drag force expericienced by the plate \n",
      "'''\n",
      "\n",
      "import math \n",
      "\n",
      "# Initialization of Variable\n",
      "P=102.7*1000\n",
      "M=28.8/1000\n",
      "R=8.314\n",
      "temp=273+18\n",
      "Recrit=10.**5\n",
      "u=18.4\n",
      "b=4.7#width\n",
      "x=1.3\n",
      "mu=1.827/100000\n",
      "\n",
      "#calculation\n",
      "#part1\n",
      "rho=P*M/R/temp\n",
      "xcrit=Recrit*mu/rho/u\n",
      "a=1-xcrit/1.65\n",
      "print \"%% of surface over which turbulent boundary layer exist is : %.4f\"%(a*100)\n",
      "\n",
      "#part2\n",
      "Rex=rho*u*x/mu\n",
      "thik=0.375*Rex**-.2*x\n",
      "print \"thickness of boundary layer in (cm): %.4f\"%(thik*100)\n",
      "\n",
      "y=0.5*thik\n",
      "ux=u*(y/thik)**(1./7)\n",
      "print \"velocity of air at mid point is (m/s): %.4f\"%ux\n",
      "\n",
      "#part4\n",
      "lthik=74.6*Rex**-.9*x\n",
      "print \"thickness of laminar boundary layer in (mm): %.4f\"%(lthik*1000)\n",
      "\n",
      "#part5\n",
      "ub=u*(lthik/thik)**(1./7)\n",
      "print \"velocity at outer edge of laminar sublayer in (m/s): %.4f\"%ub\n",
      "\n",
      "#part6\n",
      "R=0.0286*rho*u**2*Rex**-0.2\n",
      "print \"shearforce expericienced in (N/m**2) : %.4f\"%R\n",
      "\n",
      "#part7\n",
      "x1=1.65         #length of plate\n",
      "Rex1=rho*u*x1/mu\n",
      "Rms=0.0358*rho*u**2*Rex1**-0.2\n",
      "print \"mean shearforce in (N/m**2): %.4f\"%Rms\n",
      "\n",
      "#part8\n",
      "F=x1*Rms*b\n",
      "print \"total drag force expericienced by the plate is (N): %.4f\"%F\n",
      "\n"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "output_type": "stream",
       "stream": "stdout",
       "text": [
        "% of surface over which turbulent boundary layer exist is : 95.0776\n",
        "thickness of boundary layer in (cm): 2.7997\n",
        "velocity of air at mid point is (m/s): 16.6653\n",
        "thickness of laminar boundary layer in (mm): 0.2528\n",
        "velocity at outer edge of laminar sublayer in (m/s): 9.3924\n",
        "shearforce expericienced in (N/m**2) : 0.6798\n",
        "mean shearforce in (N/m**2): 0.8114\n",
        "total drag force expericienced by the plate is (N): 6.2921\n"
       ]
      }
     ],
     "prompt_number": 5
    },
    {
     "cell_type": "heading",
     "level": 3,
     "metadata": {},
     "source": [
      "example 3.3 page no : 55"
     ]
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "'''\n",
      "find\n",
      "pipe flow reynolds\n",
      "Water will be in streamline motion in the pipe\n",
      "velocity gradient at the pipe wall\n",
      "Sherastress at pipe wall\n",
      "new av. fluid velocity \n",
      "thickness of laminar sublayer\n",
      "thickness of buffer layer \n",
      "percentage of pipe-s core occupied by turbulent core\n",
      "velocity where sublayer and buffer layer meet \n",
      "velocity where turbulent core and buffer layer meet\n",
      "fluid velocity \n",
      "shearstress \n",
      "'''\n",
      "\n",
      "import math \n",
      "\n",
      "# Initialization of Variable\n",
      "Q=37.6/1000000\n",
      "d=3.2/100\n",
      "mu=1.002/1000\n",
      "rho=998.\n",
      "pi=3.14\n",
      "\n",
      "#calculation\n",
      "#part1\n",
      "u=4.*Q/pi/d**2.\n",
      "Re=rho*u*d/mu\n",
      "print \"pipe flow reynolds no : %.4f\"%Re\n",
      "print \"Water will be in streamline motion in the pipe\"\n",
      "\n",
      "#part2\n",
      "a=-8.*u/d\n",
      "print \"velocity gradient at the pipe wall is (s**-1): %.4f\"%a\n",
      "\n",
      "#part3\n",
      "Ro=-mu*a\n",
      "print \"Sherastress at pipe wall is N/m**2) %.2e\"%(Ro)\n",
      "\n",
      "#part4\n",
      "Q=2.10/1000\n",
      "u=4.*Q/pi/d**2\n",
      "u=round(u*1000.)/1000.\n",
      "print \"new av. fluid velocity is (m/s): %.4f\"%u\n",
      "\n",
      "Re=rho*u*d/mu\n",
      "phi=0.0396*Re**-0.25                #friction factor\n",
      "phi=round(phi*10**5)/10.**5\n",
      "delb=5*d*Re**-1*phi**-.5\n",
      "print \"thickness of laminar sublayer in (10**-6m): %.4f\"%(delb*10**6)\n",
      "\n",
      "#part5\n",
      "y=30.*d/phi**0.5/Re                 #thickness\n",
      "tbl=y-delb\n",
      "print \"thickness of buffer layer in (mm): %.4f\"%(tbl*1000)\n",
      "\n",
      "#part6\n",
      "A=pi*d**2./4                        #cross sectional area of pipe\n",
      "dc=d-2*y                            #dia of turbulent core\n",
      "Ac=pi*dc**2/4.\n",
      "p=(1-A/Ac)*100.\n",
      "print \"percentage of pipe-s core occupied by turbulent core is (%%): %.4f\"%p\n",
      "\n",
      "#part7\n",
      "uplus=5.                            #from reference\n",
      "ux=uplus*u*phi**0.5\n",
      "print \"velocity where sublayer and buffer layer meet is (m/s): %.4f\"%ux\n",
      "\n",
      "#part8\n",
      "yplus=30.                           #from reference\n",
      "ux2=u*phi**0.5*(2.5*math.log(yplus)+5.5)\n",
      "print \"velocity where turbulent core and buffer layer meet is (m/s): %.4f\"%ux2\n",
      "\n",
      "#part9\n",
      "us=u/0.81\n",
      "print \"fluid velocity along the pipe axis (m/s): %.4f\"%us\n",
      "\n",
      "#part10\n",
      "Ro=phi*rho*u**2\n",
      "print \"shearstress at pipe wall (N/m**2): %.4f\"%Ro\n"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "output_type": "stream",
       "stream": "stdout",
       "text": [
        "pipe flow reynolds no : 1490.8400\n",
        "Water will be in streamline motion in the pipe\n",
        "velocity gradient at the pipe wall is (s**-1): -11.6939\n",
        "Sherastress at pipe wall is N/m**2) 1.17e-02\n",
        "new av. fluid velocity is (m/s): 2.6120\n",
        "thickness of laminar sublayer in (10**-6m): 39.8159\n",
        "thickness of buffer layer in (mm): 0.1991\n",
        "percentage of pipe-s core occupied by turbulent core is (%): -3.0544\n",
        "velocity where sublayer and buffer layer meet is (m/s): 0.6304\n",
        "velocity where turbulent core and buffer layer meet is (m/s): 1.7655\n",
        "fluid velocity along the pipe axis (m/s): 3.2247\n",
        "shearstress at pipe wall (N/m**2): 15.8647\n"
       ]
      }
     ],
     "prompt_number": 6
    },
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     "cell_type": "code",
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