University of California, Los Angeles
Department of Civil & Env. Engineering
E. Taciroglu
cee130
Elementary Structural Mechanics
Quiz 2
Winter 2011
Instructions
1.
2.
3.
4.
There are three problems. Solve all of them.
All work must be written on the sheet
University of California, Los Angeles
Department of Civil & Env. Engineering
E. Taciroglu
cee130
Elementary Structural Mechanics
Quiz 2
Winter 2013
Instructions
1.
2.
3.
4.
There are three problems. Solve all of them.
All work must be written on the sheet
cee130: Elementary Structural Mechanics
University of California, Los Angeles
Prof. E. Taciroglu
Winter 2013
Homework 2 (due on Thursday, January 24)
Problem 1 . S olve Problem 1 from
Problem 3 (RSM4.14). Determine the state
Homework Series No. 2
University of California, Los Angeles
Department of Civil & Env. Engineering
E. Taciroglu
cee130
Elementary Structural Mechanics
Quiz 1(Solution)
Winter 2013
Instructions
1.
2.
3.
4.
There are three problems. Solve all of them.
All work must be written on
University of California, Los Angeles
Department of Civil & Env. Engineering
E. Taciroglu
cee130
Elementary Structural Mechanics
Quiz 4
Winter 2011
Instructions
1.
2.
3.
4.
There are three problems. Solve all of them.
All work must be written on the sheet
5.
Since the expansion process is isentropic, the isentropic flow relations can be used throughout the
expansion fan. The isentropic relations may also be used in a limited region within a non-centered
compression fan where the Mach waves do not intersect
Example:
Calculate the lift and drag coefficients (per unit depth into the page) for a flat-plate airfoil with a chord
length of 1 m. The plate is at an angle of attack of 6 degree with respect to the incoming flow which has a
Mach number of 2.5. Clearly
Example:
A wind tunnel nozzle is designed to yield a parallel uniform flow of air with a Mach number of 3.0. The
stagnation pressure of the air supply reservoir is 7000 kPa, and the nozzle exhausts into the atmosphere
(100 kPa). Calculate the flow angle a
3.
Recall that there is a maximum possible angle through which the flow can be turned using an oblique
shock. This maximum angle (max) decreases as the incoming Mach number decreases. Thus, when the
flow passes through the initial oblique shock, there exi
Example:
A uniform supersonic flow at Mach 2.0, static pressure 10 psia, and temperature 400 R expands around a
10 corner. Determine the downstream Mach number, pressure, temperature, and the fan angle.
fan angle
10
C. Wassgren
Chapter 11: Gas Dynamics
60
Prandtl-Meyer Expansion Fans
Now lets consider expanding a steady supersonic flow around a gradual, outward-turning corner as shown
in the figure below.
Ma1
1
Mach waves
2< 1
Ma2 > Ma1
The gradual curve can be approximated by a series of very small, discr
Notes:
1. The angle, , is positive for counter-clockwise (compressive) rotations and negative for clockwise
(expansive) rotations. The convention, however, is to drop the negative sign when reporting PrandtlMeyer angles.
The Prandtl-Meyer angle is plotted
For a perfect gas:
V 2 RT Ma 2
1
RT0 Ma 1
Ma 2
2
1
2
1
dV 1
d Ma
Ma 2
1
V
2
Ma
(12.275)
Substituting (12.275) into equation (12.274) gives:
Ma 2 1 d Ma
d
1 2 Ma
1
Ma
2
Note: Since Ma > 1, d > 0 d(Ma) < 0 and d < 0 d(Ma) > 0.
Integrating equat
Example:
A uniform supersonic air flow traveling at Mach 2.0 passes over a wedge. An oblique shock, making an
angle of 40 with the flow direction, is attached to the wedge for these flow conditions. If the static
pressure and temperature in the uniform fl
14. Expansion Waves
Recall the piston experiment in our previous discussion regarding the formation of shock waves. Now lets
consider what happens when we move the piston toward the left (as shown in the figure below) so that an
expansion (or rarefaction)
Now lets analyze the steady, 2D flow through a single Mach wave as shown in the figure below. Our
analysis will be for a compression Mach wave (d > 0 in the figure below) but the analysis will also hold
for an expansion Mach wave (d < 0). Note that the up
Example:
An aircraft is to cruise at a Mach number of 3. The stagnation pressure in the flow ahead of the aircraft is
400 kPa. Compare the stagnation pressure recovery ratio for three possible intake scenarios.
a. An intake that involves a normal shock in
Example:
Consider a compression corner with a deflection angle of 28. How does the pressure ratio across the
oblique shock change if the incoming Mach number is doubled from 3 to 6?
Ma1, p1
Ma2, p2
28
C. Wassgren
Chapter 11: Gas Dynamics
592
Last Updated:
Example:
Air having an initial Mach number of 3.0, a free stream static pressure of 101.3 kPa, and a free stream static
temperature of 300 K is deflected by a wedge by an angle of 15. Assuming that a weak oblique shock
occurs, calculate:
a. the downstream
15. Reflection and Interaction of Oblique Shock Waves
In this set of notes well consider the interaction between oblique shocks and solid boundaries, free
surfaces, and with other oblique shocks.
Reflection with a Solid Boundary
When an oblique shock inte
Oblique Shock Reflection from a Free Surface
When an oblique shock intersects a free surface, the reflection must be an expansion fan so that the flow
pressure remains equal to the free surface pressure as shown in the figure below.
free surface
p0
free s
Example:
Air flowing with a Mach number of 2.5 with a pressure of 60 kPa (abs) and a temperature of 253 K passes
over a wedge which turns the flow through an angle of 4 leading to the generation of an oblique shock
wave.
a. If this oblique shock wave impi
Notes:
1. Consider the flow of an incompressible fluid. If the flow is incompressible, then the speed of sound in
the fluid will be infinite. Thus, the governing equation for the irrotational flow of an incompressible
fluid becomes:
2 0
(12.282)
Governin
18. Small Perturbation Theory
Recall that the equation of motion for an irrotational flow where body and viscous forces are negligible is:
1
2 2 F t
(12.284)
c t t
where the velocity is given in terms of a potential function, u=, and c is the speed of
Substituting Eqn. (12.288) into Eqn. (12.287) and expanding (refer to the Appendix):
2
2
2
1 Ma x y z
2
2
'
ux
U
+
1
1
2
2
u 'y
2 2
2
2
2
Ma 1 2 1 2 2 2 Ma
y
U
x
z
1
2
Ma
'
u x
U
2 u 'y
x 2 U
2 1
2
Ma
'
u x
U
2
u'
2
2Ma x
U
u
Notes:
1. Note that the assumptions (Eqn. (12.289) used in deriving Eqn. (12.292) also indicate that the Mach
number cannot be too large (e.g. we cant use Eqn. (12.292) to model hypersonic flows).
2.
Equation (12.292) is a linear PDE. This means that we c
4.
Since we linearized the governing PDE using the small perturbation assumption, we should also
linearize the boundary conditions. The appropriate boundary condition at a solid boundary for an
inviscid flow is not the no-slip condition since the fluid ca
5.
An additional quantity that is often helpful when analyzing external flows is the pressure coefficient,
Cp, which is defined as:
p p
Cp
1 U2
2
This can be written for a perfect gas as:
p
1
p
(12.295)
Cp
1 Ma 2
2
The pressure ratio can be written in
Using the assumption that the perturbation velocities are small in comparison to the free stream
velocity, the remaining terms in the series can be neglected. Substituting this result into Eqn. (12.296)
and then substituting this result back into the defi
6.
Equation (12.293) for a subsonic flow may be reduced to Laplaces equation using an appropriate
transformation of variables. This is useful since there has been a large body of work in determining
the solution to Laplaces equation for various boundary c