Contents
Page
Subject(s)
1 Properties of Air & Ideal Gas Law
2 Properties of Air at Various Altitudes
3 Isentropic Flow, Compressibility & Bulk Modulus, and Speed of Sound
4 Stagnation Properties
5 Bernoulli, Mach Number & Compressibility, and the Doppler Effect
6 Fluid Mechanics
7 Deformation of a Fluid Element
8 Angular Velocity, Vorticity, and Circulation
9 Stream Function, Velocity Potential Function, and various Polar Coordinates conversions
10 Conservation Laws
11 Conservation Laws & the Jet Engine Equation
12 Words (and their translation to math) & the Euler Equation
13 The Navier-Stokes Equation
14 Shear Forces and External Flow [Reynolds Number]
15 External Flow cont. [Displacement Thickness & Momentum Thickness]
16 Coefficients [Drag & Pressure coefficients] & Parameter chart [laminar and turbulent]
17 Coefficients cont. [Pressure Coefficient]
18 Flow Separation and Reversal & Wings (general)
19 Wings - Rectangular
20 Wings - Tapered & Swept
21 Swept Supersonic Graphs
22 Swept Supersonic Graphs Cont.
23 Wings - Delta & Supersonic Swept Graphs Cont.
24 Delta Supersonic Graph & Steady Level Flight
25 Steady Level Flight - Lift & Drag Conditions (Thrust and Power Required)
26 Steady Level Flight - Vortices
Page
Subject(s)
27 Study Set 1
28 Study Set 1 Cont.
29 Study Set 1 Cont.
30 Study Set 2
31 Study Set 2 Cont.
32 Study Set 2 Cont.
33 Study Set 3
34 Study Set 3 Cont.
35 Study Set 3 Cont.
36 Final Review 1
37 Final Review 1 Cont.
38 Final Review 2
39 Final Review 2 Cont. & Final Review 3
40 Final Review 3 Cont.
Page 1
Air and Ideal Gas Law
• Assumed to be an ideal gas
- Assumptions
1. No intermolecular forces
2. Collisions are perfectly elastic
3. Point mass with negligible volume
note: high temperatures and low pressures make gasses behave more like ideal gases
Ideal Gas Equation:
Pν=nRUT
We use:
Where:
P∀=mRT
P=ρRT
Pν=RT
P = pressure = N/(m^2)
∀ = volume = (m^3)
m = mass = kg
RU = universal gas constant = 8.31 J/(mol * k)
R = specific gas constant = RU / MW = 287 J/(kg * k)
Properties of air
• Density (ρ) = 1.225 kg / m^3
• Specific gas constant (R) = c - c = 287 J / (kg*k)
• Specific heat at constant pressure (c ):
γR
cp =
= 1004.5 J / (kg*k)
γ−1
P
V
P
• Specific heat at constant volume (c ):
V
cv =
• Specific heat ratio (γ) = c / c = 1.4
P
V
R
= 717.5 J / (kg*k)
γ−1
Page 2
Properties of Air at Various Altitudes
Page 3
Isentropic flow
• ds = 0
• Requires the flow to be adiabatic (no heat transfer) and reversible (no frictional losses)
• Polytropic flow equation:
Pν n = constant
- Cases:
1. n = 0; isobaric (constant pressure)
2. n = 1; isothermal (constant temperature)
3. n = γ; isentropic (constant entropy) [This is the case that will be important for us]
4. n = ∞; isochoric (constant volume)
• Isentropic flow for ideal gasses:
P1ν1γ = P2 ν2γ
γ− 1
P1 γ
ρ1 γ− 1 T1
=
=
( P2 )
( ρ2 )
T2
Compressibility & Bulk Modulus
Compressibility (τ)
τ= −
1 δν
1 δρ
m2
=
=
ν δP
ρ δP
N
(τ)isoth ermal =
1
P
(τ)isentopic =
1
γP
Bulk Modulus (κ)
κ =
1
N
= 2
τ
m
Speed of Sound
a=
κ
=
ρ
RT γ
Page 4
Stagnation Properties
Inlet
Outlet
Tin, Pin, vin
Tou t , Pou t , vou t
Stagnation Enthalpy (h 0)
h +
v2
= h0
2
note: stagnation enthalpy is constant
Stagnation Temperature (T0)
• Incompressible:
T0 = T +
• Compressible:
v2
2cp
T0
γ−1 2
= 1+
M
T
2
note: other properties can be found by substituting for T0 /T for the desired
isentropic property (as shown in the stagnation pressure section)
γ− 1
T0
P0 γ
ρ0 γ− 1
=
=
(P)
(ρ)
T
Stagnation Pressure (P0)
• Incompressible:
ρv 2
ρv 2
; P = static pressure,
P0 = P +
= dynamic pressure
2
2
• Compressible:
γ
P0
γ − 1 2 γ− 1
= 1+
M
(
)
P
2
note: stagnation properties are conserved in isentropic processes
Page 5
Bernoulli
Pressure
P+ ρ
v2
+ ρg z = constant
2
Energy
P v2
+
+ g z = constant
ρ
2
Head
P
v2
+
+ z = constant
ρg
2g
Mach Number and Compressibility
The following equation relates the Mach number and the compressibility of the flow:
dρ
M2
≈
ρ
2
If this ratio is <.05 (or, in other words, if M < 0.3), then the flow is considered incompressible.
For incompressible flows, use Bernoulli as usual. For compressible flows, use the following:
γ
P0
γ − 1 2 γ− 1
= 1+
M
(
)
P
2
To use this equation, the flow must be isentropic, adiabatic, and reversible.
Doppler Effect
at
vt
Mach Angle (μ):
sin(μ) =
a
1
=
v
M
Page 6
Fluid Mechanics
In this course, we work using the Eularian approach to fluid flow analysis (N = N(t, x, y, z))
and as a result we may have to convert from Lagrangian equations (N = N(t)); to do this, we
use this equation, expressed here in terms of an unspecified characteristic N:
where:
DN
δN
=
+ ( v ⃗ ∙ ∇ ⃗) N
Dt
δt
d
d
d
v ⃗ ∙ ∇⃗ =
u +
v+
w
dx
dy
dz
In this system, velocity is given as a function of the three dimensions:
v ⃗ = u ı ̂ + v 𝚥 ̂ + w k̂
Say N = v, the local derivative in the above equation can now be written:
where:
dN
d v⃗
=
= a = ax ı ̂ + a y 𝚥 ̂ + az k̂
dt
dt
ax ⃗ =
δu
δu
δu
δu
+ u
+ v
+ w
δt
δx
δy
δz
a y⃗ =
δv
δv
δv
δv
+ u
+ v
+ w
δt
δx
δy
δz
az ⃗ =
δw
δw
δw
δw
+ u
+ v
+ w
δt
δx
δy
δz
In these equations, a steady flow results in the d/dt term being equal to 0; in a uniform flow, the
d/dx, d/dy, and d/dz terms are all equal to 0.
Streamline
- Velocity is tangent to the streamline at any point. If they converge, the flow is accelerating;
if they diverge, it is decelerating.
- Equations:
dy
v
=
dx
u
u
v
w
=
=
dx
dy
dz
Pathline
- Follows the path of an individual molecule
Streakline
- Uses the injection of a visible fluid to show how particles move from a point
Page 7
Deformation of a Fluid Element
Rate of linear strain
δu
t·x =
δx
δv
t·y =
δy
δw
t·z =
δz
Volume Dilatation Rate (VDR)
VDR =
δ ∀/ ∀
δu
δv δw
=
+
+
= ∇⃗ ∙ v ⃗
δt
δx δy
δz
note: if VDR = 0, the flow is incompressible; if VDR > 0, the flow is expanding; if
VDR < 0, the flow is shrinking.
Rate of angular deformation (shear strain)
δu
δv
γ· xy =
+
δy δx
δv δw
γ· yz =
+
δz
δy
δu
δw
γ· xz =
+
δz
δx
Page 8
Angular Velocity, Vorticity, and Circulation
Vorticity ( ζ )⃗
The vorticity of the fluid at any point is given by the curl of the velocity vector:
𝚥̂
ı̂
k̂
δ
δ
δ
δw δv
δ w δu
δ v δu
ζ = curl(v) = ∇ ⃗ × v ⃗ =
=
−
ı̂−
−
𝚥̂+
−
k̂
( δy
( δx
( δx δy )
δ x δ y δz
δz )
δz )
u
v
w
⃗ = 0, the flow is “irrotational”
note: if ( ∇ ⃗ × v )
Angular Velocity (ω)
The angular velocity is half of this value:
ω⃗ =
1
1
⃗
ζ = ( ∇⃗ × v )
2
2
Circulation (c)
Vorticity examines the rotational characteristics of the flow on the microscopic level, whereas
circulation (c) examines the rotational characteristics of flow on the macroscopic level.
c=
Or equivalently:
c=
This ends up simplifying to:
∮
∮
v ⃗ ∙ d ℓ ⃗ = | v | | d r | cos(θ )
v ⃗∙ ℓ ⃗=
∫A (
⃗ dA =
∇⃗ × v )
∫A
ζd A
c = Aζ
note: in this equation, A is the area enclosed by the path.
Page 9
Stream Function and Velocity Potential Function
Velocity Potential Function (ϕ)
• In order to define a velocity potential function, the flow must be irrotational!
⃗
➡ ( ∇ × v ) MUST = 0
⃗
v ⃗ = ∇ ⃗ϕ =
δϕ
δx
u =
δϕ
δϕ
δϕ ̂
ı̂+
𝚥̂+
k
δx
δy
δz
v=
δϕ
δy
w=
δϕ
δz
note: Velocity potential lines are perpendicular to streamlines
Stream Function (Ψ)
• In order to define a stream function, the flow must be 2-D AND incompressible
• The Cauchy-Riemann equations allow us to convert between the stream and velocity
potential functions (provided that both exist)
➡ Cauchy-Riemann equations:
u =
δϕ
δΨ
=
δx
δy
v=
δϕ
δΨ
= −
δy
δx
Polar Coordinates
Del operator in polar coordinates:
∇⃗ =
δ
δ
δ ̂
δ
1 δvθ
δ
ı̂+
𝚥̂+
k=
er̂ +
eθ̂ +
eẑ
δx
δy
δz
δr
r δθ
δz
Divergence in polar coordinates:
∇⃗ ∙ v ⃗ =
1 δ
1 δvθ δvz
(r vr ) +
+
r δr
r δθ
δz
Curl in polar coordinates:
δvz δvr
δ(r vθ ) δvr
1 δvz δvθ
−
er̂ −
−
eθ̂ +
−
ê
( r δθ
( δr
( δr
δz )
δz )
δθ ) z
Laplacian in polar coordinates:
∇2 P =
1 δ δP
1 δ2P δ2P
+ 2 2 +
r δr ( δr ) r δθ
δz 2
Cauchy-Riemann equations in polar:
vr =
1 δΨ
δϕ
=
r δθ
δr
vθ = −
δΨ
1 δϕ
=
δr
r δθ
vz =
δϕ
δz
Page 10
Conservation Laws
Conservation equation stated in terms of some extensive property (N), some intensive property
(n = N/m), and a unit vector (n):̂
dN
δN
=
+
n∙ ρ ( v ⃗ ∙ n)̂ d A
∫A
dt
δt
total change
amount in & out
amount generated
v2
v1
n1̂
n2̂
Conservation of Mass
• For the conservation of mass, the extensive property (N) is the mass (m):
- Steady flow
dN
dm
=
= 0; since the mass of the system is always the same
dt
dt
δN
= 0; if the system is steady
δt
m
ρ v ⃗ ∙ n)̂ d A = 0
n=
= 1, thus the steady flow equation simplifies to:
∫A (
m
- Uniform flow
m· = constant = ρ1 A1v1 = ρ2 A2 v2
- Unsteady flow
δm
δ
δρ
=
=
ρd ∀ =
d∀
∫∀ δt
δt
δt ∫∀
Thus, the conservation of mass reduces to:
0=
δρ
d∀+
ρ v ⃗ ∙ n)̂ d A
∫∀ δt
∫A (
note: v ⃗ is the relative velocity of fluid with respect to the control volume.
⃗ ntrolvolu me , since v co
⃗ ntrolvolu me = 0, it is v ⃗
v o⃗ bser ver = v f⃗ lu id + v co
Page 11
After applying the Gauss divergence theorem, we are left with the continuity equation:
Dρ
⃗ ρ= 0
+ ( ∇⃗ ∙ v )
Dt
For incompressible flow systems:
δu
δv δw
Dρ
+
+
= 0 ; thus, this reduces to: ∇ ⃗ ∙ v ⃗ = 0 =
δx δy
δz
Dt
Conservation of Momentum
• For the conservation of momentum, the extensive property (N) is the mass multiplied by the
velocity (mv):
d (mv)
δ(mv)
mv
=
+
∙ ρ ( v ⃗ ∙ n)̂ d A
∫A m
dt
δt
δ(ρv)
ΣFnet =
d∀+
ρv v ⃗ ∙ n)̂ d A
∫∀ δt
∫S (
note: ΣFnet includes body forces (
shear [
∫S
∫∀
g ρd ∀ ) and surface forces (which includes
τd A] and pressure [−
∫S
P∙dA =
the flow is inviscid, the shear forces are 0.
• For steady and Inviscid flow:
ΣFnet =
∫∀
⃗
(ρg − ∇ P) d ∀ =
Eventually:
ΣFnet =
note:
∫S
∫S
∫∀ (
∇ ⃗ ∙ P) d ∀ ] forces) if
δ(ρ v ⃗ )
d∀+
ρv v ⃗ ∙ n)̂ d A
∫∀ δt
∫S (
ρv ( v ⃗ ∙ n)̂ d A
ΣFx =
∫S
·
ρu ( v ⃗ ∙ n)̂ d A = mu
ΣFy =
∫S
·
ρv ( v ⃗ ∙ n)̂ d A = mv
ΣFz =
∫S
·
ρw ( v ⃗ ∙ n)̂ d A = mw
ρv ( v ⃗ ∙ n)̂ d A = ρ u (y)bd y for a rectangle and = ρ u (r)2π rd r for a circle
∫
∫
Patm
• The jet engine equation:
ΣFx = m· (v2 − v1) + P2,g ag e A2 − P1,g ag e A1
note: Pg ag e = Pabs − Patm
P1
Fx
P2
Patm
Page 12
Words
Incompressible
ρ = constant; M < .3; ∇ ⃗ ∙ v ⃗ = 0;
Steady
δ
= 0
δt
Uniform
d
d
d
= 0;
= 0;
= 0;
dx
dy
dz
Fully Developed
d
= 0
dx
Euler Equation
This is the Euler equation for inviscid flow:
− ∇P + ρg = ρ
Dv⃗
Dt
note: Here, the term ρg represents the body forces.
The following are expansions of the inviscid Euler equation in the coordinate directions:
Inviscid Euler in x direction:
−
δP
δu
δu
δu
δu
+ ρg x = ρ
+ u
+ v
+ w
[ δt
δx
δx
δy
δz ]
Inviscid Euler in x direction:
−
δP
δv
δv
δv
δv
+ ρg y = ρ
+ u
+ v
+ w
[ δt
δy
δx
δy
δz ]
Inviscid Euler in x direction:
−
δP
δw
δw
δw
δw
+ ρg z = ρ
+ u
+ v
+ w
[ δt
δz
δx
δy
δz ]
note: be careful to ensure you are applying gravity in the correct plane
Page 13
Navier-Stokes Equation
Viscous Flow
Fully Developed Flow
Boundary Layer
Inviscid Flow
Boundary Layer
Viscous Flow
The form of the Navier-Stokes momentum equation shown is called the “convective form” and
is expressed:
ρ
Dv⃗
1
⃗ + μ ∇ 2⃗ v ⃗
= − ∇ ⃗ P + ρg +
μ + λ ∇ ⃗( ∇ ⃗ ∙ v )
(3
)
Dt
note: here the D/Dt term is the material derivative
The simplified Navier-Stokes equation shown below is applicable only for flows that are both
incompressible AND viscous:
dv
= − ∇P + ρg + μ ∇2 v ⃗
Dt
The following are expansions of the μ ∇2 v ⃗ term in the coordinate directions:
ρ
in x: μ
δ 2u
δ 2u
;
+
( δx 2 δy 2 )
in y: μ
δ 2v δ 2v
;
+
( δx 2 δy 2 )
in z: μ
δ2w δ2w
+
δy 2 )
( δx 2
The following is a comprehensive Navier-Stokes equation solved for velocity in the x direction,
which is applicable in all cases of viscous, incompressible, AND fully developed flow (as
shown in the diagram above):
Where:
y
H2
δP
y
y
u (y) =
+
−
1−
⨆H
2μ ( δx ) ( H ) (
H)
⨆
= relative velocity of the top plate to the bottom plate
H = the total distance between the plates
μ = the viscosity of the fluid
Page 14
Shear Forces
The total shear is made up of the laminar component and the turbulent component:
τtotal = τlaminar + τtu rbu lent
note: in inviscid flow regions, the shear is 0
Laminar Shear
τxy = μ γ· xy = μ
τyz = μ γ· yz = μ
τxz = μ γ· xz = μ
δu
δv
+
( δy δx )
δv δw
+
( δz
δy )
δu
δw
+
( δz
δx )
Turbulent Shear
τtu rbu lent,xx = − ρu ′u ′
τtu rbu lent,xy = − ρu ′v′ = μt
δu¯ δ v̄
+
( δy δx )
τtu rbu lent,xz = − ρu ′w′
note: the term μt is the eddy viscosity, barred terms are average values
External Flow
Laminar Flow
u∞
Turbulent Flow
r
Boundary Laye
r
Boundary Laye
x (critical)
In external flow, as the flow develops the flow changes from laminar to turbulent; in the case of
external flow, the Reynolds Number at which the flow changes from laminar to turbulent is:
Rexcritical = 5 * 105
at any x: Rex =
ρu ∞ x
u x
= ∞
μ
ϑ
Page 15
Streamline Height (h )
h =
γ
u (y)
dy
∫0 u ∞
Displacement Thickness (γ ⋆)
As the boundary layer grows, a phenomenon occurs wherein the plate over which the fluid is
flowing appears to grow from the viewpoint of the fluid; the thickness of this ‘apparent wall’ is
called the “displacement thickness (γ ⋆),” and it is given by:
⋆
γ =
∫0 (
γ
1−
u (y)
dy
u∞ )
where: γ = height of the boundary layer at some point along the plate
u (y) = the velocity profile at that point
u ∞ = the original flow
u (y)/u ∞ is often given as a function
Momentum Thickness (θ ⋆)
Outside the boundary layer, the momentum equation is the Euler equation, because the shear
forces are 0:
− ∇P + ρg = ρ
Dv⃗
Dt
note: from this, we can make the following assumptions and be left with the
Bernoulli equation:
- on the same streamline
- inviscid flow
- steady flow
- incompressible flow
Inside the boundary layer shear forces must be considered, in this course we will focus on the
laminar portion of shear force; this allows us to simplify the Navier-Stokes equation to:
ρ
dv
= − ∇P + ρg + μ ∇2 v ⃗
Dt
The momentum thickness equation inside the boundary layer is given by:
⋆
θ =
u (y)
u (y)
1−
dy
∫0 u ∞ (
u∞ )
γ
Page 16
Airfoils & their Coefficients
Coefficients are non-dimensionalized so that they are scalable.
Terminology
Leading
Edge
Trailing
Edge
Here, the solid line is the “chord line” (c) and the dashed line is the “mean camber line”; the
mean camber line is the line over which a cambered airfoil is symmetric, thus a symmetrical
airfoil is one whose mean camber line is a straight line between the leading edge and the
trailing edge.
NACA Airfoils
NACA airfoils are standardized airfoils that can be described using a four digit code. The first
digit gives the maximum camber of the airfoil as a percentage of the chord length, the second
digit gives the location of the max camber divided by 10, and the final two digits give the
maximum thickness of the airfoil as a percentage of the total chord.
Skin Friction Coefficient (cf)
cf =
Here τw is the wall shear stress
dynamic pressure.
(
τw
= 2
2
(1/2)ρ∞v∞
τw = μ
d θ⋆
dx
⋆
du
2 dθ
|y= 0 = ρ∞u ∞
and the denominator is the
dy
dx )
Drag Coefficient (cd )
1 L
D
;
cd =
cf d x =
2 Lb
L ∫0
(1/2)ρ∞v∞
D = − Fnet
note: the denominator is dynamic pressure multiplied by area; this makes the
coefficient of drag a dimensionless quantity
Thin Airfoil Theory
This airfoil theory declares that under certain assumptions, a thin enough airfoil can be
assumed to have very similar flow characteristics to that of a flat plate. It requires flow to be
2D, inviscid, AND incompressible.
Page 17
d cℓ
; approximately 2π/radian or .11/degree
dα
Lift coefficient for a symmetric airfoil = cℓ = α 2π
Lift curve slope of an airfoil = a 0 = tan(α) =
Lift coefficient for a cambered airfoil = cℓ = 2π (α − αcℓ = 0)
note: alpha MUST be in RADIANS
Definitions of Various Parameters in Laminar or Turbulent Flow
Parameter
Laminar
(Blasius exact solution)
Turbulent
(Prandtl approximation)
Rex
γ
0.16
= 7
x
Rex
Displacement Thickness
γ⋆
1.72
=
x
Rex
γ⋆
0.02
= 7
x
Rex
Momentum Thickness
θ⋆
0.664
=
x
Rex
θ⋆
0.016
= 7
x
Rex
γ
=
x
Boundary Layer Thickness
γ
γ
⋆
θ
⋆
Shape Factor
H ( > 1)
H=
Local Skin Friction Coefficient
γ⋆
= 2.59
θ⋆
cf =
cf
Wall Shear Stress
5
0.664
Drag Coefficient
cd =
cd
μρU 3
x
1.328
ReL
γ⋆
= 1.25
θ⋆
cf =
Rex
τw = 0.332
τw
H=
0.027
7
τw = 0.0135
cd =
Rex
7
μρ 6U 13
x
0.031
7
ReL
Pressure Coefficient (cp)
1
2
v2 2
P − P∞
cp =
= 1−
2
( v∞ )
(1/2)ρv∞
cp = 1 at the stagnation point
note: this equation uses Bernoulli assumptions
Page 18
Flow Separation
Flow separation occurs as a pressure gradient becomes more positive (adverse) and eventually
leads to flow reversal. In the diagram below, the area below the plate becomes a region of high
pressure while the region above the plate has low pressure; this pressure difference leads to
the creation of the lift force. As α → 0∘ the skin drag increases (until at 0º it accounts for 100%
of drag), and as α → 90∘ pressure drag increases (until at 90º it accounts for 100% of drag).
u∞
L
α
Wings
A wing is an airfoil with some finite span (b).
Lift coefficient for a cambered wing = cL = a (α − αcℓ = 0), where α is geometric angle of
attack and αCℓ = 0 is the “zero-lift angle of attack”.
note: alpha MUST be in RADIANS, a is the lift curve slope of the wing
Due to the three dimensional nature of wings, they produce a drag component called induced
drag, it can be given by:
cDi =
Di
cL2
=
q∞S πe A R
because of this induced drag, the angle of attack (that is, the geometric angle of attack) is
reduced:
αg eometric = αindu ced + αef fective
Page 19
Wings produce a downward stream of air at their trailing edge called downwash (w∞). The
induced angle of attack can be found using the downwash and the free-stream velocity:
αindu ced =
w∞
u∞
Rectangular
Shape
Planform Area (s)
Aspect Ratio (AR)
bc
b2
S
c
b
Lift Curve Slope (a) for Straight Wing
Low Aspect Ratio (AR < 4)
High Aspect Ratio (AR > 4)
a0
a0
1 + (a0 /(πe A R))
Incompressible
a + [a0 /(π A R)]2 + a0 /(π A R)
a0
a0
Subsonic Compressible
2 + [a /(π A R)]2 + a /(π A R)
1 − M∞
0
0
Supersonic Compressible
4
2 − 1
M∞
Here, e =
1−
1
, and δ can be found by
1+ δ
using the graph shown to the right:
1
2A R
2 − 1
M∞
2 + a /(π e A R)
1 − M∞
0
4
2 − 1
M∞
Page 20
Tapered
Shape
cT
Planform Area (s)
cR
Aspect Ratio (AR)
cT + cR
b
( 2 )
2b
cT + cR
In a tapered wing, the taper ratio (λ) is given by: cT /cR, thus rectangular wings have a taper
ratio of 1 and a delta wing has a taper ratio of 0.
2 1 + λ + λ2
The mean aerodynamic chord (c̄) is given by: c̄ =
cR
3
1+ λ
Swept
Shape
Planform Area (s)
cR
Aspect Ratio (AR)
cT + cR
b
( 2 )
2b
cT + cR
cT
For swept wings, the flow velocity over the wing (u ) is given by: u = V∞ cos(Λ), where Λ is
the sweep angle measured from the leading edge of the wing.
Lift Curve Slope (a) for Swept Wing
Incompressible
a 0 cos(Λ)
1 + [(cos(Λ))/(π A R)]2 + (a 0 cos(Λ))/(π A R)
Subsonic Compressible
a 0 cos(Λ)
2 cos2(Λ) + [(cos(Λ))/(π A R)]2 + (a cos(Λ))/(π A R)
1 − M∞
0
Page 21
note: in these graphs, λ is the wing’s taper ratio, β =
2
M∞
−1
Page 22
Page 23
Delta
To find lift coefficient for a low speed delta wing:
Shape
s ℓ
s
Planform Area (s)
Aspect Ratio (AR)
cR
b
2
2b
cR
2πα
α
cL ≈CN cos(α) = (s /ℓ )
+ 4.9
cos(α)
( s /ℓ ) ]
[ s /ℓ
note: α is in radians and (s /ℓ ) = (A R /4)
2
1.7
Page 24
Steady-Level Flight
Steady-Level flight conditions require the forces on the flying body to be balanced.
In the figure, lift (L) is equal to weight (W)
and thrust (T) is equal to drag (D); These
expressions can be rewritten in a way that
solves for various flight characteristics.
To find the equivalent skin friction coefficient
(cfe), use the dashed line in the graph on the
right, then plug in the value to the following
equation to calculate the zero-lift drag (cD,0):
cD,0 =
Swet
S
cfe
D
L
T
W
Page 25
Lift Conditions
L= W
Cruise speed: v∞ =
2mg
cL ρ∞S
Stall speed: v∞ =
2mg
(cL )ma x ρ∞S
-or-
2
cL ((1/2)ρ∞v∞
)S = mg
Takeoff speed: vtakeof f ≈1.2vstall
Drag Conditions
To minimize the required thrust you must maximize the list coefficient, which we can see from
the equation below:
W
(cL /cD )ma x
(Treq′d )min =
After expanding the denominator, we are left with the following equation:
cL
cL
=
cD
cD,0 + KcL2
We can differentiate this equation and set it equal to zero to find the max lift coefficient, after
differentiating and solving for cL we are left with the following equation:
cL =
CD,0
K
K = k1 + k 2 +
;
1
πe A R
To minimize the power required, however, we must maximize a performance characteristic
stated mathematically as cL(3/2) which can be explained by the following substitutions:
c
W
Preq′d = Treq′d u ∞ =
u∞ = DW
L /D
cL
From this, it can be seen that:
(Preq′d )min ∝
2W
=
ScL ρ∞
2W 3cD2
ScL3 ρ∞
1
cL(3/2) /cD
Using the same process described above for the minimum thrust required, we arrive at this
equation for minimum power required:
cL =
3CD,0
K
;
K = k1 + k 2 +
1
πe A R
A general form of the derivative completed above is shown below as a function of exponent (n):
cLn
d
= 0 ⇒ ncLn− 1cD,0 + KncLn+ 1 − 2KcLn+ 1 = 0 ;
2
d cL ( cD,0 + KcL )
K = k1 + k 2 +
1
πe A R
Page 26
Vortices
The strength of vertices at the wingtips is given by:
Γ = vθ 2π r
The strength of vortices formed over the wing is given by:
L=
b/2
∫− b/2
ρ∞u ∞Γ(y)d y
The downwash created by the wings can be estimated as:
w∞ =
Γ0
− 2b
The following equations apply only to the elliptical lift distribution case:
Γ(y) = Γ0
2y
1−
(b)
π
ρ∞v∞Γ0 b
4
c
αi ≈ L
πAR
L=
2