Chapter 3 : velocity boundary layers¶

example 3.1 page no : 50¶

In :
import math

# Initialization of Variable
rho=998.
mu=1.002/1000
x=48/100.
u=19.6/100
x1=30/100.
b=2.6

#calculation

print "fluid in boundary layer would be entirely in streamline motion "
Re=rho*x*u/mu
print "reynolds no is %.2e"%(Re)

#part 2
Re1=rho*x1*u/mu
delta=x1*4.64*Re1**-.5
print "boundary layer width in (mm): %.4f"%(delta*1000)

#part3
y=0.5*delta           #middle of boundary layer
ux=3/2*u*y/delta-.5*u*(y/delta)**3
print "velocity of water in (cm/s): %.4f"%(ux*100)

#part4
R=0.323*rho*u**2*Re1**-0.5
print "shear stress at 30cm in (N/m**2): %.4f"%R

#part5
Rms=0.646*rho*u**2*Re**-0.5
print "mean shear stress experienced over whole plate in (N/m**2) %.4f"%Rms

#part6
F=Rms*x*b
print "total force experienced by the plate in (N) %.4f"%F
fluid in boundary layer would be entirely in streamline motion
reynolds no is 9.37e+04
boundary layer width in (mm): 5.7520
velocity of water in (cm/s): 8.5750
shear stress at 30cm in (N/m**2): 0.0512
mean shear stress experienced over whole plate in (N/m**2) 0.0809
total force experienced by the plate in (N) 0.1010

example 3.2 page no : 52¶

In :
import math

# Initialization of Variable
P=102.7*1000
M=28.8/1000
R=8.314
temp=273+18
Recrit=10.**5
u=18.4
b=4.7#width
x=1.3
mu=1.827/100000

#calculation
#part1
rho=P*M/R/temp
xcrit=Recrit*mu/rho/u
a=1-xcrit/1.65
print "%% of surface over which turbulent boundary layer exist is : %.4f"%(a*100)

#part2
Rex=rho*u*x/mu
thik=0.375*Rex**-.2*x
print "thickness of boundary layer in (cm): %.4f"%(thik*100)

y=0.5*thik
ux=u*(y/thik)**(1./7)
print "velocity of air at mid point is (m/s): %.4f"%ux

#part4
lthik=74.6*Rex**-.9*x
print "thickness of laminar boundary layer in (mm): %.4f"%(lthik*1000)

#part5
ub=u*(lthik/thik)**(1./7)
print "velocity at outer edge of laminar sublayer in (m/s): %.4f"%ub

#part6
R=0.0286*rho*u**2*Rex**-0.2
print "shearforce expericienced in (N/m**2) : %.4f"%R

#part7
x1=1.65         #length of plate
Rex1=rho*u*x1/mu
Rms=0.0358*rho*u**2*Rex1**-0.2
print "mean shearforce in (N/m**2): %.4f"%Rms

#part8
F=x1*Rms*b
print "total drag force expericienced by the plate is (N): %.4f"%F
% of surface over which turbulent boundary layer exist is : 95.0776
thickness of boundary layer in (cm): 2.7997
velocity of air at mid point is (m/s): 16.6653
thickness of laminar boundary layer in (mm): 0.2528
velocity at outer edge of laminar sublayer in (m/s): 9.3924
shearforce expericienced in (N/m**2) : 0.6798
mean shearforce in (N/m**2): 0.8114
total drag force expericienced by the plate is (N): 6.2921

example 3.3 page no : 55¶

In :
import math

# Initialization of Variable
Q=37.6/1000000
d=3.2/100
mu=1.002/1000
rho=998.
pi=3.14

#calculation
#part1
u=4.*Q/pi/d**2.
Re=rho*u*d/mu
print "pipe flow reynolds no : %.4f"%Re
print "Water will be in streamline motion in the pipe"

#part2
a=-8.*u/d
print "velocity gradient at the pipe wall is (s**-1): %.4f"%a

#part3
Ro=-mu*a
print "Sherastress at pipe wall is N/m**2) %.2e"%(Ro)

#part4
Q=2.10/1000
u=4.*Q/pi/d**2
u=round(u*1000.)/1000.
print "new av. fluid velocity is (m/s): %.4f"%u

Re=rho*u*d/mu
phi=0.0396*Re**-0.25                #friction factor
phi=round(phi*10**5)/10.**5
delb=5*d*Re**-1*phi**-.5
print "thickness of laminar sublayer in (10**-6m): %.4f"%(delb*10**6)

#part5
y=30.*d/phi**0.5/Re                 #thickness
tbl=y-delb
print "thickness of buffer layer in (mm): %.4f"%(tbl*1000)

#part6
A=pi*d**2./4                        #cross sectional area of pipe
dc=d-2*y                            #dia of turbulent core
Ac=pi*dc**2/4.
p=(1-A/Ac)*100.
print "percentage of pipe-s core occupied by turbulent core is (%%): %.4f"%p

#part7
uplus=5.                            #from reference
ux=uplus*u*phi**0.5
print "velocity where sublayer and buffer layer meet is (m/s): %.4f"%ux

#part8
yplus=30.                           #from reference
ux2=u*phi**0.5*(2.5*math.log(yplus)+5.5)
print "velocity where turbulent core and buffer layer meet is (m/s): %.4f"%ux2

#part9
us=u/0.81
print "fluid velocity along the pipe axis (m/s): %.4f"%us

#part10
Ro=phi*rho*u**2
print "shearstress at pipe wall (N/m**2): %.4f"%Ro
pipe flow reynolds no : 1490.8400
Water will be in streamline motion in the pipe
velocity gradient at the pipe wall is (s**-1): -11.6939
Sherastress at pipe wall is N/m**2) 1.17e-02
new av. fluid velocity is (m/s): 2.6120
thickness of laminar sublayer in (10**-6m): 39.8159
thickness of buffer layer in (mm): 0.1991
percentage of pipe-s core occupied by turbulent core is (%): -3.0544
velocity where sublayer and buffer layer meet is (m/s): 0.6304
velocity where turbulent core and buffer layer meet is (m/s): 1.7655
fluid velocity along the pipe axis (m/s): 3.2247
shearstress at pipe wall (N/m**2): 15.8647
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