# CHAPTER05:INCOMPRESSIBLE FLOW OVER FINITE WINGS¶

## Example E01 : Pg 182¶

In [1]:
# All the quantities are expressed in SI units
from math import pi,sqrt
AR = 8.;                # Aspect ratio of the wing
alpha = 5.*pi/180.;             # Angle of attack experienced by the wing
a0 = 2.*pi             # airfoil lift curve slope
alpha_L0 = 0;          # zero lift angle of attack is zero since airfoil is symmetric
# from fig. 5.20, for AR = 8 and taper ratio of 0.8
delta = 0.055;
tow = delta;           # given assumption
# thus the lift curve slope for wing is given by
a = a0/(1.+(a0/pi/AR/(1.+tow)));
# thus C_l can be calculated as
C_l = a*alpha;
# from eq.(5.61)
C_Di = C_l**2./pi/AR*(1.+delta);
print"Cl =",round(C_l,2)
print"CDi =",round(C_Di,2)

Cl = 0.44
CDi = 0.01


## Example E02 : Pg 185¶

In [2]:
# All the quantities are expressed in SI units
from math import sqrt,pi
CDi1 = 0.01;                        # induced drag coefficient for first wing
delta = 0.055;                      # induced drag factor for both wings
tow = delta;
alpha_L0 = -2.*pi/180.;              # zero lift angle of attack
alpha = 3.4*pi/180.;                # angle of attack
AR1 = 6.;                            # Aspect ratio of the first wing
AR2 = 10.;                           # Aspect ratio of the second wing

# from eq.(5.61), lift coefficient can be calculated as
C_l1 = sqrt(pi*AR1*CDi1/(1.+delta));

# the lift slope for the first wing can be calculated as
a1 = C_l1/(alpha-alpha_L0);

# the airfoil lift coefficient can be given as
a0 = a1/(1.-(a1/pi/AR1*(1.+tow)));

# thus the list coefficient for the second wing which has the same airfoil is given by
a2 = a0/(1.+(a0/pi/AR2*(1.+tow)));
C_l2 = a2*(alpha-alpha_L0);
CDi2 = C_l2**2./pi/AR2*(1.+delta);

print"The induced drag coefficient of the second wing is CD,i =",CDi2

The induced drag coefficient of the second wing is CD,i = 0.00741411360464


## Example E03 : Pg 189¶

In [4]:
# all the quantities are expressed in SI units
from math import pi
a0 = 0.1*180./pi;                    # airfoil lift curve slope
AR = 7.96;                   # Wing aspect ratio
alpha_L0 = -2.*pi/180.;               # zero lift angle of attack
tow = 0.04;                  # lift efficiency factor
C_l = 0.21;                  # lift coefficient of the wing

# the lift curve slope of the wing is given by
a = a0/(1+(a0/pi/AR/(1.+tow)));

# thus angle of attack can be calculated as
alpha = C_l/a + alpha_L0;

print"alpha =",alpha*180./pi,"degrees\n"

alpha = 0.562642629213 degrees



## Example E04 : Pg 191¶

In [5]:
# All the qunatities are expressed in SI units
from math import pi,sqrt
alpha_L0 = -1.*pi/180.;                 # zero lift angle of attack
alpha1 = 7.*pi/180.;                    # reference angle of attack
C_l1 = 0.9;                            # wing lift coefficient at alpha1
alpha2 = 4.*pi/180.;
AR = 7.61;                             # aspect ratio of the wing
taper = 0.45;                          # taper ratio of the wing
delta = 0.01;                          # delta as calculated from fig. 5.20
tow = delta;
# the lift curve slope of the wing/airfoil can be calculated as
a0 = C_l1/(alpha1-alpha_L0);
e = 1./(1.+delta);
# from eq. (5.70)
a = a0/(1.+(a0/pi/AR/(1.+tow)));
# lift coefficient at alpha2 is given as
C_l2 = a*(alpha2 - alpha_L0);
# from eq.(5.42), the induced angle of attack can be calculated as
alpha_i = C_l2/pi/AR;
# which gives the effective angle of attack as
alpha_eff = alpha2 - alpha_i;
# Thus the airfoil lift coefficient is given as
c_l = a0*(alpha_eff-alpha_L0);
c_d = 0.0065;                         # section drag coefficient for calculated c_l as seen from fig. 5.2b
# Thus the wing drag coefficient can be calculated as
C_D = c_d + ((C_l2**2.)/pi/e/AR);
print"The drag coefficient of the wing is C_D =",C_D

The drag coefficient of the wing is C_D = 0.014827553741