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Brequet Range Equation
V
t=tf
W=Wf
t=ti
W=Wi
Range
tf
Range = Vdt
ti
In level flight at const speed:
L = W , Lift = Weight
T = D , Thrust = Drag
The aircraft weight changes during flight due to use of
Equations of Aircraft Motion
Force Diagram Conventions
line
Chord
T
V
T
Flight path
M
D
Horizontal
W
Definitions
V
W
L
D
M
T
T
flight speed
angle between horizontal & flight path
angle of attack (angl
Compressible Equations
Conservation of mass
d
dv + surface u ndS = 0
dt volume
t
D
Dt
+ ( u )
=0
+ u
=0
u
= 0,
( incompressible )
Conservation of Momentum
d
udv + surface uu ndS = surface pn idS + s
Compressible Viscous Equations
Also known as the compressible Navier-Stokes equations:
Mass:
(v j ) = 0
+
t x j
Momentum:
( i )
p
+
( i j ) = x + ( ij ) , i = 1, 2,3
xj
t
x j
i
Energy:
1
( e + v2 )
Compressible Viscous Note: Other effects could have been included in this diagram. e.g. steady vs. unsteady flows
Compressible Inviscid
Incompressible Viscous
Compressible Boundary Layer
Compressible
Coordination Transformations for Strain & Stress Rates
To keep the presentation as simple as possible, we will look at purely two-dimensional
stress-strain rates. Given an original coordinate system (
Stress-Strain Relationship for a Newtonian Fluid
First, the notation for the viscous stresses are:
y
yy
yz
x y
xz
z y
zz
yx
xx
x
zx
z
ij = stress acting on the fluid element with a face whose normal
Kinematics of a Fluid Element
Rotation
Convection
Shear Strain
Compression/Dilation
(Normal strains)
Convection: u
i
Rotation rate:
=
1
1
u =
2
2 x
u
j
k
y
v
z
w
= vorticity
=
v u
1 w v
u w
i +
Solution Convergence
Recall for our triangular grid finite volume scheme, the basic iterative scheme
looked like:
Rin residual of cell
Ai
U in +1 U in
n
n
+ n i + bci + cai = 0
ab
t
Approximation of
v
Structured vs. Unstructured Grids The choice of whether to use a structured or an unstructured mesh is very problem specific (as well as company/lab specific). The answer is one of engineering judgeme
Computational Methods for the Euler Equations
Before discussing the Euler Equations and computational methods for them, lets
look at what weve learned so far:
Method
2-D panel
Assumptions/Flow type
2-
Prandtl-Meyer Expansion Waves When a supersonic flow is turned around a corner, an expansion fan occurs producing a higher speed, lower pressure, etc. in an isentropic process. fan Forward Mach line
M
Oblique Shock Waves
Heres a quick refresher on oblique shock waves. We start with the oblique shock
as shown below:
w2 , M t 2
(1)
(2)
( )1 : upstream flow
u2 , M n 2
y
condition
u1 , M n1
w1 , M t1
(
Implications of Linearized Supersonic Flow on Airfoil Lift & Drag
To begin, we will divide the airfoil geometry into camber and thickness
distributions:
y
yc ( x )
( x)
yu ( x)
x
yl ( x )
1
i
v
v
v
U
Linearized Compressible Potential Flow Governing Equation
Recall the 2-D full potential eqn is:
1
1
2
1 2 ( x ) 2 xx + 1 2 ( y ) 2 yy 2 x y xy = 0
a
a
a
1
2
Where a 2 = a o
( x ) 2 + ( y ) 2
[
]
As
Propagation of Disturbances By a Moving Object
Consider an object moving at speed V :
Vt1
t3
t2
V t =0
t1
Suppose that the atmospheric speed of sound is a . The body emits sound
waves as it travels th
Normal Shock Waves
In our quasi-1D flows, shocks can occur from a supersonic-to-subsonic state.
These shocks are discontinuous in our inviscid flow model (recall that shocks are
very thin and their th
Waves in 1-D Compressible Flow
Imagine we have a steady 1-D compressible flow. Then suppose a small
disturbance occurs at a location x = xo . This disturbance will cause waves to
propagate away from t
Derivation of Sound Wave Properties
Sound wave
propagating to
right
+ d
du
p + dp
a
u = 0 ( x) =
p
Similar for u ( x), p( x)
xs
l
Assume:
Sound wave creates small disturbances in an isentropic manne
Behavior of Isentropic Flow in Quasi-1D
Recall cons. of mass:
uA = const.
Consider a perturbation in the area
R = L + d
L =
p R = p L + dp
pL = p
u R = u L + du
uL = u
AR = AL + dA
AL = A
xL
xR
uA =
Critical Mach Number
We can estimate the freestream Mach number at which the flow first accelerates
above M > 1 (locally) using the Prandtl-Glauert scaling and isentropic
relationships.
Recall from P-
Subsonic Small Disturbance Potential Flow v vv 1. V = (V + u )i + vj where | u2 + v2 | < 1 2 V small disturbances are assumed
vv 2. ui + vj = perturbation potential u = v= x y
3. small-disturbance (?)
Drag Tare Due to Mount
Force balance will measure drag which is due to exposed portion of mount:
extra drag
due to
exposed
mount
Forces
measured
here
fairing
mechanism
to adjust
Two techniques to est
Three-Dimensional Wall Effects
In a freestream, recall that a lifting body can e modeled by a horseshoe vortex:
V
Consider a rectangular cross-section tunnel:
Flow is into page
wing
The image system f
Ground Effect Using Single Vortex Model
b
h
What is the boundary condition at ground ( z = 0 ) and does a single horseshoe
vortex satisfy it?
B.C.: solid wall
vv
u n = 0
w = 0 at z = 0!
Consider from
Single Horseshoe Vortex Wing Model
S
b b
trailing vortices
bound vortex
Lift due to a horseshoe vortex Kutta-Joukowsky Theorem
b 2
L = V dy = V b
b 2
1 V2 S 2 2 b 2 CL = V b S
CL = 2 A V b
CL =
L
=
Similarity in Wind Tunnel Testing
In terms of non-dimensional force and moment coefficients, these depend on
numerous non-dimensional input parameters.
C L = C L ( M , Re, ,.)
C D = C D ( M , Re, ,.)
Solutions of the Laminar Boundary Layer Equations The boundary layer equations for incompressible steady flow, i.e.,
u v + =0 x y
u
dp u u 2u + v = e + 2 x y dx y
p = 0, we set p = p e ( x) , y i.e. t
Laminar Boundary Layer Order of Magnitude Analysis
( x)
u
x
y
c
Assumptions (in addition to incompressible, steady & 2-D)
Changes in x direction occur over a distance c
1
1
~
or, we write
= O( )
x C