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
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 Bernoull
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(
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
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 thick
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
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 o
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
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
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
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 spl
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
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.
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
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 f
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
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-