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

Falkner-Skan Flows
For the family of flows, we assume that the edge velocity, u e ( x) is of the
following form:
u e ( x) = Kx m
K = arbitrary constant
The pressure can be calculated from the Bernoulli in the outer, inviscid flow:
12
u e = const.
2
dp
du

Method of Assumed Profiles
Here are the basic steps:
1. Assume some basic boundary velocity profile for u ( x, y ) . For example, this is a
crude approach but illustrates the ideas:
y
, 0 y < ( x)
u( x, y )
= ( x)
ue ( x )
1, y ( x )
where ( x) is the s

Correlation Methods for Integral Boundary Layers
We will look at one particularly well-known and easy method due to Thwaites in
1949.
First, start by slightly re-writing the integral b.l. equation. We had:
w
d
du e
=
+ (2 + H )
2
u e dx
e u e dx
Multipl

Integral Boundary Layer Equations
Displacement Thickness
The displacement thickness * is defined as:
* = 1
0
u
u
dy = 1 dy
e ue
ue
0
compressible
flow
incompressible
flow
The displacement thickness has at least two useful interpretations:
Interpret

Viscous Flow: Stress Strain Relationship Objective: Discuss assumptions which lead to the stress-strain relationship for a Newtonian, linear viscous fluid:
ij =
ui u j u + + k x j xi ij xk uk u v w = + + = iV xk x y z
where = dynamic viscosity coeffici

Problem #1
Assume:
Incompressible
1
2-D flow Vz = 0, = 0
z
Steady
=0
t
Parallel Vr = 0
r1
r0
a) Conservation of mass for a 2-D flow is:
1
1
( r Vr ) +
(V ) = 0
r r
r
=0
(V ) = 0 V does not depend on
V = V ( r )
b) -mometum equation is:
V
VV
1 p
2 Vr

Trefftz Plane Analysis of Induced Drag
Consider an inviscid, incompressible potential flow around a body (say a wing).
We define a control volume surrounding the body as follows
Trefftz plane
ST (part of S )
body
z
Sbody
y
8
S
8
x, V
Upstream flow is V an

Force Calculations for Lifting Line
Recall:
N
( y ) = ( ) = 2bV An sin n
n =1
b
y = cos
2
The local two-dimensional lift distribution is given by Kutta-Joukowsky:
L ( y ) = V ( y )
N
L ( ) = 2bV2 An sin n
n =1
To calculate the total wing lift, we integr

Prandtls Lifting Line Introduction Assumptions: 3-D steady potential flow (inviscid & irrotational) Incompressible High aspect ratio wing Low sweep Small crossflow (along span) flow looks like 2-D flow locally with adjusted Outputs: Total lift and induced

Important Concepts in Thin Airfoil Theory
1. This airfoil theory can be viewed as a panel method with vortex solutions
taking the limits of infinite number of panels & zero thickness & zero camber
cfw_ lim vortex panel = thin airfoil theory
lim
thickness

Effect of Turbulent Fluctuations on Mean Flow: Reynolds-Averaging
In a turbulent flow, we can define the mean, steady flow as:
T
1
u ( x, y , z ) = lim u ( x, y , z , t )dt
T T
0
This allows us to split the flow properties into a mean and a fluctuating pa

Poiseuille Flow Through a Duct in 2-D
y
y = +h
x
y = h
Assumptions:
u v
=
=0
x x
Velocity is independent of x,
Incompressible flow
Constant viscosity,
Steady
Pressure gradient along length of pipe is non-zero, i.e.
p
0
x
Boundary conditions:
u ( y = h )

Assume steady
=0
t
L
Assume > 1
h
V
=0
x
Assume 2-D
w = 0, = 0
z
y= h
y
pL
y=-h
Incompressible N-S equations:
1.
u v
+
=0
x y
2u 2u
u
u
u
1 p
+u +v
=
+ 2 + 2
t
x
y
y
x
x
2v 2v
v
v
v
1 p
3.
+u +v
=
+ 2 + 2
t
x
y
y
y
x
2.
BCs
v ( x, h ) = 0
u

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

z
Thin Airfoil Theory Summary
(x) = thickness
z(x) = camber line
x
c
Replace airfoil with camber line (assume small
c
)
z
z(x) = camber line
x
c
Distribute vortices of strength ( x) along chord line for 0 x c .
z
(x)dx
x
c
Determine ( x) by satisfying flo

Solution
2
y
V = 100 mph
8
3
=10 miles
1
u n = 0 at control pt #1:
The velocity at control pt #1 is the sum of the freestream + 3 point vortices
velocities at that point:
u1 = Vi +
1
2
3
i
i+
j
2
2
2
2
2
2
The normal at control pt #1 is:
n1 = i
u1 n1

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