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'hijof L: ux mwMf +o C}.
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C H = :w
32' M 1C.
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3f(M JJ )
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Face (a): EX/pasxm UAW, Smre PW Is meIaJ away
{W Prisoff (See Andrrmmy : NH
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 fuel. Relate weight
change to time change:
dW
dt
dt
fu
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 (angle between flight path and chord line)
aircraft weight
l
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 + surface idS
dt volume
Note :
ds
= ( xx dx + yx dy + zx
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 ) +
t
x j
2
i
ij =
x j
,
q=k
x j
+
1
( e + v 2 )v j =
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 Potential
Incompressible Invicid
Incompressible Bounda
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 ( x, y ) and a rotated system
( x, y ) as shown below:
y
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 is in + xi
direction and the stress is in + x j directi
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 +
j + k
2 y z
z x
x y
Normal strain rates:
dLx
u
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
vv
(Fi + Gj ) nds
ci i
U
n +1
i
Update formula for cel
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 judgement. Here are some of the issues: (1) Complex geometry:
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-D, Incompressible, Irrotational Inviscid
Vortex lattice
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
M1 > 1
1 2
Rearward Mach line waves
Just as we saw with
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
( )2 : downstream flow
v2 , M 2
condition
x
: angle of
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 U = u cos i + U sin j
1
y u ( x) = y c ( x) + ( x)
2
1
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 you saw, for small perturbations to a uniform flow, the
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 through the atmosphere and these wave propagate away
from
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 thickness scales with 1 ):
Re
M
ML
shock
1
Upstream Mach:
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 the source. Suppose that the flow velocity were u and th
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 manner.
Mass
dl
( x)dx + u l u 0 = 0
dt 0
l
d xs
( + d )dx
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 = ( + d )(u + du )( A + dA)
= uA + duA + Adu + udA + H .
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-G:
On the airfoil
surface:
C p (M ) =
C p ( M = 0)
2
1
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 (?) and bcs= 2 2 2 (1 M ) 2 + 2 = 0 y x
BC: v( x,0) = V d
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 estimate drag tare:
Remove model and run tests to find dr
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 for this looks like:
images
images
images
images
actual
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 far downstream:
z
h
b
So, to satisfy bc:
h
b
h
b
Image
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
=
V b
1 V2 S 2
Single Horseshoe Vortex Wing Model
Induc
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, ,.)
In many aerodynamic applications,
C L = C L ( 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. the boundary layer edge pressure. Note: since
have been
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
x
C
Changes in y direction occur over a distance
1
1