from ae 3330, courtesy of professor brian german
TRANSCRIPT
1!
From AE 3330, courtesy of Professor Brian German! Introduction to Aerospace Vehicle Performance!
6. !Aerodynamics and the Drag Polar!
! !6.1 !Aerodynamic forces!! ! ! !Pressure and shear forces!! ! ! !Resultant aerodynamic force!! ! ! !Lift and drag!! !6.2 !Aerodynamic lift and drag coefficients!! !6.3 !Lift and drag coefficient behavior of airfoils!! ! ! !Lift curve slope, zero-lift angle!! ! ! !Stall!! ! ! !Airfoil polar!! ! ! !Effect of control surfaces!
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AE 3330!Introduction to Aerospace Vehicle Performance!
! !6.4 !Airfoils in transonic and supersonic flows!! ! ! !Critical Mach number!! ! ! !Transonic drag rise and the drag divergence !Mach number!! ! ! !Supercritical airfoils!! ! ! !Wave drag!! !6.5 !Wing terminology!! !6.6 !Flow around finite wings!! !6.7 !Aerodynamic characteristics of finite wings!! ! ! !Lift curve slope!! ! ! !Induced drag coefficient and induced drag!! ! ! !Winglets!! ! ! !Stall behavior of finite wings!
3!
AE 3310!Introduction to Aerospace Vehicle Performance!
! !6.8 !Compressibility effects on wings and airplanes!! ! ! !Wing sweep!! ! ! !Transonic drag rise and the drag divergence !Mach number!! ! ! !Area ruling!! !6.9 !Drag polar!! ! ! !Assembly of the drag polar!! ! ! !Dependency of drag polar on aircraft configuration, speed, and
! !altitude!! ! ! !Limitations of simple drag polar models!! ! ! !Example airplane drag polar buildup!
4!Aerodynamic Forces!
Aerodynamic forces arise from two sources:!!
! !• !Pressure Forces!! ! ! !- !Arise from changes in the flow velocity around the ! ! ! ! ! !body as it moves through the fluid!
!
! !• !Shear (Viscous) Forces!! ! ! !- !Arise from fluid viscosity (i.e., fluid friction) as the ! ! ! ! ! !body moves through the fluid!!Although these forces are different they are not entirely independent:!!
! !• !Viscosity effects will affect the pressure distribution!!
! !• !Pressure distribution will affect the viscous forces!
5!Pressure Forces!
s - coordinate system along airfoil surface!
Graphics source: Anderson, Aircraft Performance and Design!
p - local pressure!Force per unit area normal to the surface!
p varies along the surface, always positive
6!Why does p vary along the surface?!Example: Bernoulli’s Equation for Incompressible Flow!! !!p0 - total pressure (invariant in the flow field)!p - local pressure!V - local velocity!!Far upstream, if the vehicle is traveling at a velocity V∞ through an atmosphere of pressure p∞:!!
!!Thus, at any point on the airfoil where the local velocity is V:!! !
€
p0 = p+12ρ∞V
2
€
p0 = p∞ +12ρ∞V∞
2
€
p = p0 −12ρ∞V
2 = p∞ +12ρ∞V∞
2%
& '
(
) * −
12ρ∞V
2
7!Why does p vary along the surface?!
€
p0 = p∞ +12ρ∞V∞
2
€
p = p0 −12ρ∞V
2
Freestream!!
p∞, ρ∞, V∞!
Local V ≠ V∞!
p0 constant through flow field!
Incompressible flow!ρ∞ = constant!
Graphics source: Abbott and Doenhoff, Theory of Wing Sections !
8!Pressure Coefficient - Cp!
In practice, instead of plotting p along a surface, a nondimensional quantity Cp, known as the pressure coefficient, is plotted instead. Cp is defined by:!!!
!!!!! = freestream dynamic pressure!!On an airfoil producing lift p < p∞ and thus Cp < 0 on the upper surface. Thus, when plotting Cp it is customary to make the negative axis point up, so that the upper curve is related to the upper surface of the airfoil.!
€
Cp =p−p∞12ρ∞V∞
2=p−p∞q∞
€
q∞ =12ρ∞V∞
2
9!Pressure Coefficient - Cp!
NACA 0012 Airfoil!α = 3.93°!M∞ = 0.345!Re = 3.245 x 106!
x - distance along chordline from leading edge to trailing edge!c - airfoil chord!x/c - nondimensional distance along chord line!
Graphics source: Anderson, Introduction to Flight!
10!Shear (Viscous) Forces!
s - coordinate system along airfoil surface!
Graphics source: Anderson, Aircraft Performance and Design!
τ - shear stress!Force per unit area parallel to surface!
τ varies along the surface!
11!Viscous Flow and Boundary Layers!
Boundary Layer!Fluid velocity increases rapidly!
Viscous effects important!Boundary Layer Thickness - δ!
V!Body Surface! V = 0!y = 0
External flow!V = local free stream value!Viscous effects negligible!
y!
12!
Because of viscosity, fluid at the surface of a body has no relative motion. In other words, the fluid “sticks” to the surface. This is known as the no slip condition.!!However, as we move away from the surface the fluid velocity increases rapidly to its local free stream value.!!This region of rapidly increasing velocity is known as the boundary layer. Viscosity is usually important within the boundary layer, and negligible elsewhere.!
Viscous Flow and Boundary Layers!
13!Shear Stress at the Surface!
V!Body Surface! V = 0!y = 0
y!
€
τ = µdVdy
#
$ %
&
' ( y=0
µ - coefficient of viscosity!
14!
Boundary layers can be either laminar or turbulent!!
Boundary layers usually start as laminar, then transition to turbulent!!
As the flow progresses along the surface, the boundary layer increases in thickness!!
Turbulent boundary layers are usually thicker than laminar boundary layers!
Laminar and Turbulent Boundary Layers!
Thin flat plate!
V∞!
Laminar!Boundary!
Layer!
Turbulent!Boundary!
Layer!Transition!
δ
15!
Turbulent boundary layers have a “fuller” velocity profile. Thus, in general:!!!!!!!So,!!
τLaminar < τTurbulent!!However, all is not good news. Laminar boundary layers are more prone to separation with associated large drag increases.!
Laminar and Turbulent Boundary Layers!
Graphics Source: Anderson, Introduction to Flight!
{ }Laminar < { }Turbulent!
€
dVdy
"
# $
%
& ' y=0
€
dVdy
"
# $
%
& ' y=0
Laminar!
Turbulent!
y!
V!
δL!δT!
16!
The development of the boundary layer, including transition from laminar to turbulent and separation, depends strongly on the pressure distribution around the airfoil.!!Conversely the boundary layer, because of its thickness, affects the pressure distribution.!!Thus, pressure and shear forces influence each other. This influence is particularly important in the determination of drag.!
Pressure Distribution and Boundary Layers!
Graphics Source: Anderson, Introduction to Flight!
17!Integrating the Pressure and Shear Stress!
Note:!!s (lower case) is coordinate system along the airfoil (i.e. a distance)!!S (upper case) is an area!
Graphics source: Anderson, Aircraft Performance and Design!
n - unit vector normal to the surface (positive away from surface)!k - unit vector parallel to the surface (positive along the direction of s)!s - coordinate system along airfoil surface!dS - surface area element!
18!Integrating the Pressure and Shear!
Graphics source: Anderson, Aircraft Performance and Design!
Resultant!Aerodynamic!
Force =
19!Lift and Drag!
L: !Lift, force component perpendicular to V∞!D: !Drag, force component parallel to V∞!α: !Angle of attack, angle between V∞ and chord line!
Graphics source: Anderson, Aircraft Performance and Design!
Center of Pressure, C.P.!R acts here!No moment about C.P.!C.P. changes with α!
Chord line
20!Lift, Drag, and Moment!The C.P. moves as we change angle of attack!!
It is somewhat inconvenient to be chasing the C.P. to apply the lift and drag forces!!
A more convenient approach is to settle on a point where the lift and drag forces are applied, with a related moment!!
The quarter chord measured from the airfoil’s leading edge is a convenient location (other locations are sometimes used)!
Graphics source: Anderson, Aircraft Performance and Design!
Equivalent!Representations!
21!
Measure the location of the C.P. and the quarter chord from the airfoil’s leading edge!!
Lift and Drag applied at xcp!!
Assume angle of attack is small!!
Assume moment generated by drag force is negligible!
Lift, Drag, and Moment!
Graphics source: Anderson, Aircraft Performance and Design!
c!
x!
xcp!
€
Mc 4 =L c4− xcp
#
$ %
&
' (
22!
Note that Mc/4 is defined to be positive in a nose-up direction!!
This is consistent with our previous definition of body axes for an airplane!!
Most airfoils with positive camber have a negative Mc/4 over much of their useful angle of attack range!
Lift, Drag, and Moment!
Graphics sources: !Anderson, Aircraft Performance and Design!!Etkin, Dynamics of Flight!
23!
L = L(shape, ρ∞, V∞, S, α, µ∞, a∞)!!
D = D(shape, ρ∞, V∞, S, α, µ∞, a∞)!!
M = M(shape, ρ∞, V∞, S, c, α, µ∞, a∞)!!L! !-!Lift!D! !-!Drag!M! !-!Moment (about some specified location) ρ∞ !-!atmospheric density!V∞!-!freestream airspeed!S! !-!reference area (usually wing area)!c! !-!reference length (usually wing chord)!µ∞!-!freestream coefficient of viscosity!a∞!-!freestream speed of sound!!This is an inconvenient way of expressing the aerodynamic forces and moments. Too many variables!!
Aerodynamic Forces and Moments!
24!
A more convenient approach - define nondimensional aerodynamic coefficients:!!
!!
!where, = dynamic pressure!!
CL - lift coefficient!!
CD - drag coefficient!!
CM - moment coefficient!then,!
Aerodynamic Coefficients!
€
CL =Lq∞S
€
q∞ =12ρ∞V∞
2
€
CD =Dq∞S
€
CM =Mq∞Sc
€
L = q∞SCL
€
D = q∞SCD
€
M = q∞ScCM
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Through the method of dimensional analysis, it can be shown that:!!
CL = CL(shape, α, Re, M∞)!!
CD = CD(shape, α, Re, M∞)!!
CM = CM(shape, α, Re, M∞)!!
where: and!!This is a key observation in aeronautics! It means that vehicles of the same shape (geometric similarity), flown at the same angle of attack, Reynolds number, and Mach number have the same aerodynamic coefficients - regardless of scale. With these coefficients we can then predict lift, drag, and moment. This is why most aerodynamic data is expressed in terms of nondimensional aerodynamic coefficients - they make the data generally applicable!!
Aerodynamic Coefficients!
€
Re =ρ∞V∞c
µ∞
€
M∞ =V∞
a∞
26!
A few loose ends…!!!• !The previous statements regarding the general
applicability of aerodynamic coefficients do not apply in some situations, particularly those involving high temperatures and/or heat rates (e.g., situations related to entry vehicles). Parachutes are another exception.!
!!• !The following convention is typically (but not always)
used:!!
! ! !CL, CD, CM (upper case) - complete aircraft!!
! ! !cl, cd, cm (lower case) - two-dimensional airfoils!!
! !But be careful! Not everybody follows this convention!!
Aerodynamic Coefficients!