

In [1]:

    
%matplotlib inline 
# plots graphs within the notebook
%config InlineBackend.figure_format='svg' # not sure what this does, may be default images to svg format

from IPython.display import display,Image, Latex
from __future__ import division
from sympy.interactive import printing
printing.init_printing(use_latex='mathjax')

import time

from IPython.display import display,Image, Latex

from IPython.display import clear_output

#import SchemDraw as schem
#import SchemDraw.elements as e

import matplotlib.pyplot as plt
import numpy as np
import math
import scipy.constants as sc

import sympy as sym

from IPython.core.display import HTML
def header(text):
    raw_html = '<h4>' + str(text) + '</h4>'
    return raw_html

def box(text):
    raw_html = '<div style="border:1px dotted black;padding:2em;">'+str(text)+'</div>'
    return HTML(raw_html)

def nobox(text):
    raw_html = '<p>'+str(text)+'</p>'
    return HTML(raw_html)

def addContent(raw_html):
    global htmlContent
    htmlContent += raw_html
    
class PDF(object):
  def __init__(self, pdf, size=(200,200)):
    self.pdf = pdf
    self.size = size

  def _repr_html_(self):
    return '<iframe src={0} width={1[0]} height={1[1]}></iframe>'.format(self.pdf, self.size)

  def _repr_latex_(self):
    return r'\includegraphics[width=1.0\textwidth]{{{0}}}'.format(self.pdf)

class ListTable(list):
    """ Overridden list class which takes a 2-dimensional list of 
        the form [[1,2,3],[4,5,6]], and renders an HTML Table in 
        IPython Notebook. """
    
    def _repr_html_(self):
        html = ["<table>"]
        for row in self:
            html.append("<tr>")
            
            for col in row:
                html.append("<td>{0}</td>".format(col))
            
            html.append("</tr>")
        html.append("</table>")
        return ''.join(html)
    
font = {'family' : 'serif',
        #'color'  : 'black',
        'weight' : 'normal',
        'size'   : 12,
        }
fontlabel = {'family' : 'serif',
        #'color'  : 'black',
        'weight' : 'normal',
        'size'   : 16,
        }

from matplotlib.ticker import FormatStrFormatter
plt.rc('font', **font)

Laminar Pipe Flow

In this first tutorial, you will simulate a laminar pipe flow using the cad file CAD>PipeD0.025L2.5VOF.iges. The pipe inner diameter is 2.5cm and 250cm long. The Junior course of Fluid Mechanics has taught you that:

The entrance length for laminar flow is given by $$ \frac{L_e}{D}=0.05Re=0.05\frac{U_{b}D}{\nu} $$
The pressure gradient is constant in the fully developed region ($x > L_e$)
The velocity profile in the fully developed region is governed by the reduced streamwise momentum equation $$ 0=-\frac{dP}{dx}+\frac{\mu}{r}\frac{d}{dr}r\frac{dU}{dr} $$
The solution of this equation is $$ U(r) = \frac{R^2}{4\mu}\left(-\frac{dP}{dx}\right)\left(1-\frac{r^2}{R^2}\right)=2U_b\left(1-\frac{r^2}{R^2}\right) $$
Hence, $$ U_b=\frac{R^2}{8\mu}\left(-\frac{dP}{dx}\right) $$
Finally, the wall shear stress is proportional to the pressure drop: $$ -\frac{dP}{dx}=\frac{2\tau_w}{R} $$

Using https://www.simscale.com, Simulate a laminar flow. Verify and Validate (V & V) your simulation.



In [5]:

    
D = 2.5e-2 # m
nu = 15.e-6 #m^2/s
rho = 1.2 #kg/m^3
mu = nu/rho 

R = D/2

Re = 500.

Ub = Re * nu / D
print("Bulk velocity= %2.2f m/s" %Ub)









    



Bulk velocity= 0.30 m/s



In [15]:

    
import numpy as np

n = 30
r = np.linspace(0,R,n)

U = 2 * Ub * (1 - np.power(r,2)/R**2)
import matplotlib.pyplot as plt

plt.plot(r,U,linewidth = 2)
plt.xlabel(r"$r$ (m)", fontdict = fontlabel)
plt.ylabel(r"$U(r)$ (m/s)", fontdict = fontlabel)
plt.xlim(0,R)
plt.show()

The wall shear stress is defined: $$ \tau_w = \mu \frac{dU}{dr} $$



In [16]:

    
tauw = - mu * (U[n-1] - U[n-2])/(r[n-1] - r[n-2])
print("wall shear stress = %1.4f Pa" %tauw)









    



wall shear stress = 0.0012 Pa



In [17]:

    
mdpdx = 8 * mu *Ub / R**2
tauw_e = R * mdpdx / 2
print("Exact wall shear stress = %1.4f Pa" %tauw_e)









    



Exact wall shear stress = 0.0012 Pa



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