Transcript 3241 Lecture 2 - Florida Institute of Technology

MAE 3241: AERODYNAMICS AND
FLIGHT MECHANICS
Aerodynamic Force and Vector Calculus Review
January 12, 2011
Mechanical and Aerospace Engineering Department
Florida Institute of Technology
D. R. Kirk
READING ASSIGNMENT
• Fundamentals of Aerodynamics, 5th edition
– Chapter 1
• All sections except: 1.7 and 1.9
– Chapter 2
• All sections except: 2.7
WHAT CREATES AERODYNAMIC FORCES?
• Aerodynamic forces exerted by airflow comes from only two sources
• Pressure, p, distribution on surface
– Acts normal to surface
• Shear stress, tw, (friction) on surface
– Acts tangentially to surface
• Pressure and shear are in units of force per unit area (N/m2)
• Net unbalance creates an aerodynamic force
“No matter how complex the flow field, and no matter how complex the shape of
the body, the only way nature has of communicating an aerodynamic force to a
solid object or surface is through the pressure and shear stress distributions that
exist on the surface.”
“The pressure and shear stress distributions are the two hands of nature that reach
out and grab the body, exerting a force on the body – the aerodynamic force”
VECTOR CALCULUS REVIEW (SECTION 2.2)
• Natural and convenient representation to explain motion of fluids in 3D space
• Expression of field quantities
– Position
– Velocity
– Acceleration
– Vorticity
– Forces
• Directly transfer to scalar forms to solve problems
• Advantage of vector calculus/algebra:
– Definition and theorems are independent of coordinate system
– In scalar form expression are different between cartesian, cylindrical, spherical,
etc. coordinate systems
VECTOR CALCULUS REVIEW
  a   0
• The curl of the gradient of a scalar function is zero
    A   0 • The divergence of the curl of a vector is zero
 V  0
V  f
• If the curl of the velocity field is zero
– Flow is irrotational
– Velocity can be written as the gradient of a scalar function, f
 V  0
• If the divergence of the velocity field is zero
– Flow is incompressible
 f 0
• If both are true
– Laplace equation
2
SUMMARY OF VECTOR INTEGRALS
 adV   anˆdS
V
S
   AdV   A  nˆdS
V
S
   A  nˆdS   A  dC
S
C
• Gradient Theorem
– Vector equation involving a scalar
function, a
– Limits of integration such that surface
encloses the volume
– n points normal outward
• Divergence (Gauss’) Theorem
– Scalar equation
• Stokes Theorem
– Direction of n is given by right hand rule
SCRAMJET PROPULSION: X-43 MACH 10!
EXAMPLE: SHOCK ENHANCED MIXING
•
Rayleigh scattering image of flow field development following M=1.146 shock passage
over a cylinder of helium embedded in air
•
Difficult to
explain with
velocity/pressure
field description
Vorticity field
useful for insight
Shockwave
moving left to
right in pictures
Interacting with
helium cylinder in
pictures (a) – (c)
•
•
•
shockwave