Homework 1
1. Consider an infinitely thin flat plate with a 1-meter chord at an angle of attack of
10 degree in a supersonic flow .The pressure and the shear stress distribution on the
upper
surfaces
are
giver
by
Pu  4  10 4 ( x  1) 2  5.4  10 4
 2  10 4 ( x  1) 2  1.73  10 5 ,  u  288x 0.2 , and  l  731x 0.2 respectively. Where x
is the distance from the leading edge in meters and P and τ are in newtons per square
meter. Calculate the normal and axial forces, the lift and drag, moments about the
leading edge, and moments about the quarter chord, all per unit span. Also calculate
the location of the center of pressure.
, Pl
Ans.
θ = 0 degree
TE
TE
LE
LE
N    ( Pu cos    u sin  )dsu   ( Pl cos    l sin  )dsl
TE
TE
LE
LE
→ N    Pu dsu   Pl dsl
→ N   113333.337 N
TE
TE
LE
LE
A   ( Pu sin    u cos )dsu   ( Pl sin    l cos )dsl
TE
TE
LE
LE
→ A    u dsu    l dsl
→ A  849.16679N
α=0 degree
L  N  cos   A sin   111464.0927N
D  N  sin   A cos  20516.39348N
TE
TE
LE
LE
   ( Pu x   u y )dsu   ( Pl x   l y )dsl
M LE
  57833.33N.m
→ M LE
→ M C / 4  57833.33  1 / 4  111464.09  85699.35N .m
xcp   M LE / L  57833.33 / 111464.0927  0.52m
2. Consider an airfoil at 12 degree angle of attack. The normal and axial force
coefficients are 1.2 and 0.03 respectively. Calculate the lift and drag coefficient.
Ans.
CN=1.2 , CA=0.03
→ CL=CN cosα – CA sinα =1.167
→ CD=CLsinα + CA cosα = 0.28
3. The shock waves on a vehicle in supersonic flight cause a component of drag
called supersonic wave drag Dw . Define the wave drag coefficient as CD,w = Dw / q S ,
where S is a suitable reference area for the body. In supersonic flight, the flow is
governed in part by its thermodynamic properties, given by the specific heats at
constant pressure ,cp, and a constant volume ,cv . Define the ratio cp/cv = γ. Using
Buckingham's pi theorem, show that CD,w= f ( M , γ).Neglect the influence of friction.
Ans.
Π1= qa . Sb. Dwc.Cpd = Ma+c L2b-a+c+2d T-2a-2c-2d
→ d=0,a=b=c=1
Π2=qa . Vb . Cpc .Cvd = Ma . L-a+b+2c+2d . T-2a-b-2c-2d. Kc+d
→ a=0 , b=0 ,c=1 ,d=-1
Π3=qa.Vb. Cpc.ad = Ma. L-a+b+2c++d. T-2a-b-2c-d. Kc
→ a=0 , b=1 , c=0 .d=-1
→ CDW = f (M,γ)
4. Consider a circular cylinder in a hypersonic flow ,with it's axis perpendicular to the
flow. Let φ be the angle measured between radial drawn to the leading edge (the
stagnation point) and to any arbitrary point on the cylinder. The pressure coefficient
distribution along the cylinder surface is given by Cp=2 cos2φ for 0 ≤ φ ≤ л/2 and
3л/2≤ φ ≤ 2 л and Cp=0 for л/2≤ φ ≤ 3 л/2. Calculate the drag coefficient for the
cylinder. Based on the projected frontal area of the cylinder.
Ans.
CP= ( P-P∞)/q∞
D= ∫ dD = ∫ P cosθ ds = CP q∞ + P∞
P  C p q   P
 /2
C D  1 / 2
3 / 2
C p cos   rd  4 / 3