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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",
"\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",
"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",
"\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
},
{
"cell_type": "code",
"collapsed": false,
"input": [],
"language": "python",
"metadata": {},
"outputs": []
}
],
"metadata": {}
}
]
}