Question Utah State
UNIVERSITY
Mechanical & Aerospace
Engineering
Initial Data Line
• Unit Processes must start somewhere .. Need a
datum from which too start process
Example nozzle flow ... Throat
Expansion section
шах
M<I
Sonic line (generally curved)-
MAE 5540 - Propulsion Systems I
50
50 Utah State
UNIVERSITY
Initial Data Line
Mechanical & Aerospace
Engineering
Using Method of Characteristics to Design a Bell Nozzle
Ө
Δθ=
O MAX
N
0₁₁ = 0
Ꮎ
M₁ =1→ v(1)=0
0
-
θ. = ί· Δθ
X₁ = R · sinė
•
Y
1
= D
i
2
throat
+ R · (1— cose,
(2)
v ( M ) −
1.5
0₁
=
v ( M₁) −0
v (M²) =
Ꮎ
Δθ
Extract M,
+1
γ-1
tan
0₁ = i· ≤ 0
i
v(M):
γ
-1
γ+
∙1
=
-1
-(M² - 1) – tan¯ √(M² −1)
= {1,...N}
- -
√(M²-1) 0.5
Mi
=
sin
M
MAE 5540 – Propulsion Systems I
-
0
0:04 / 0:32
sonic line.
V
S
F
ම
66
"A0" increments
51 Utah State
UNIVERSITY
Mechanical & Aerospace
Using Method of Characteristics to Design a Bell Nozzle
1) Wall point: 0 Ꮴ →M
wall
wall
wall
→ μ wall
0 V₁₁ = 0 wall + V wall
cl → Vcl
cl
=
2) Centerline point: 0a
V cl
Ma
3) Characteristic line slope: C_ =
0.5
sonic line
→ μcl
11.25
O wall - Mwall + (0 - μd)
136
11.54
2
Centerline Intercept
calculation (see next
slide)
X₁ = R ·sine
Y.
=
1
2
i
Engineering
(3)
D + R. (1― cos 0₂)
throat
с
0
0:04 / 0:32
MAE 5540 - P Utah State
Mechanical & Aerospace
Engineering
UNIVERSITY
Centerline Intercept Solution
V₁
1) Initial Point: {x₁,y₁,0,,M₁}→
2) Centerline Intercept: 0₁ =0
cl
right running characteristic line⇒→0₁+V₁ = 0₁+v₁
→V cl
= 0₁₂ + V₁
[
M cl
μει
y₁ = 0
0-y₁
Xcl-X₁
cl
cl
→ Slope(C_)=0₁-14-12
-=tan (Slope (C)) → Xc =
'cl
2
y₁
cl
tan (Slope (C_))
MAE 5540 – Propulsion Systems I
-
+x₁
53 Utah State
UNIVERSITY
Ꮎ
03
=
Ꮴ
3
Mechanical & Aerospace
Using Method of Characteristics to Design a Bell Nozzle
(0₁ + V₁ ) + ( 02 −√2) (K_), +(K),
2
-
V₂) _
2
(0, + V₁ ) - (02 - V₂) (K), -(K),
2
2
Engineering
(4)
0.5
0
1
0:04 / 0:32
MAE 554
66
"A" increments
Internal Flow unit
sonic line
11.26
process calculation
(slides 24-26)
54✡/n The figure assumes a throat height of 10 cm (arbitrary guess).
% Assume throat height is 0.1 m (10 cm);
Assume throat circular arc is 20 cm (was 5 cm).
clear
close all
r = 2; center = [0 2.5]; % (r was 5,)
x = 0.02:1.35;
y = sqrt (r^2
x.^2) + center (1,2);
plot (x, y)
x = [-1 8];
y =
[00];
line (x,y)
xlim ([58])
ylim ([.5 3])
hold on
x1 = .2:.05:3;
x2 = .2:.2:2;
%y_circ
=
@(x) (r^2-x^2)^.5 + center (1,2);
nu E =
PrandtlMeyer(6,1.4);
delta max = nu E/2;
n = 8;
delta inc = delta max/n;
for i = 1:n
✗circ (i) = r*sin(delta_inc*i); y_circ(i) = -r*cos(delta_inc⭑i) + center (1,2);
응 plot (✗_circ (i),y_circ(i),'*')
end
M = 1.1: (6-1.1)/7:6; % Create the 'polydown' lines (C-)
for i = 1:n
= asin (1/M(i));
mu (i)
delta_x =
y_circ(i)/tan (mu (i));
x mu (i) =
✗_circ (i) +delta_x;
✗ vec(i,:)
=
(✗_circ (i) x_mu(i)];
y_vec (i, :)
=
[y_circ(i) 0];
end
line (✗_vec (i, :),y_vec (i,:))
axis equal
Must save in folder to run code
function v=PrandtlMeyer(M,gamma)
v=sqrt((gamma+1)/(gamma-1))*atan(sqrt((gamma-1)/(gamma+1)*(M^2-1)))-atan(sqrt(M^2-1));
end
3
2
1
0
-1
-2
AQQ
Exit mach of 6
0
1
2
3
4
5
6
7
8 N lines deped on the angle of nozzle
m>1 to max mah to get radius
1/M = mu/n AME 323 Project 2 v2.3
AME 323: Project 2: Wind Tunnel Design
Spring 2024
Assigned: 4/5/2024
Due: 5/6/2024
High-pressure air tank
Pressure
regulator
Test section
Settling chamber
Nozzle
ㄸ
Figure 1: Schematic of a blowdown wind tunnel.
A
1 Introduction
You are tasked with producing the rough design of a Mach 6 supersonic blowdown wind tunnel with a planar
2D nozzle. This will consist of several related tasks:
•
production of a nozzle design code using the method of characteristics;
•
design of a nozzle contour and characterization of its flow field;
• estimation of tunnel run envelope and characteristics; and
•
compile your results into a technical report.
2 Tasks
2.1
Task 1: Nozzle design code
Design a computer code employing the method of characteristics to calculate the wall contour of the expansion
section of a 2-D, uniform flow, supersonic nozzle with an exit Mach number of 6. Use the following parameters
in the design of your nozzle:
• Assume the throat is a circular arc with radius of curvature equal to the throat half-height (such a
shape can help avoid boundary-layer separation). Note: Do not compute a minimum-length nozzle
with sharp corners at the throat as shown in several textbooks.
• Start by approximating the expansion through your throat region with N equally-spaced Prandtl-
Meyer waves. Start with small N for simplicity but design your code to take any value of N. Note:
You cannot start exactly at the throat since μ 90° at that point.
=
• Make sure your program runs fine on any machine. Any work required in order to make it run while
grading will cause points to be docked.
•
Tip: Above all else, this is primarily a bookkeeping problem, so it may be helpful to do the problem
first in Microsoft Excel due to its visual nature, then translate it into MatlabⓇ or Python or another
language of your choice.
© 2024, S. A. Craig, All rights reserved. AME 323 Project 2 v2.3
Settling chamber
M
h = ?
sc
SC
Nozzle
Test section
h=
= 350 mm
↓
A
Figure 2: Close-up view of the settling chamber, nozzle, and test section.
2.2 Task 2: Nozzle contour
1. Perform a grid convergence study on the nozzle design problem using your code. In other words,
identify a suitable known parameter to be used to measure the error in your calculations and vary N
to determine the dependence of the error on how many initial waves are used. Make a plot of the error
as a function of N.
2. Plot the nozzle expansion contour and the contour map of Mach number inside the nozzle from throat
to exit. Make sure you have used enough N for a reasonably accurate nozzle contour. Justify what
constitutes a suitable accuracy.
3. Plot the centerline distributions of M, p/po, and T/To along the centerline of the nozzle from the
throat to the exit. Determine the pressure ratio required to operate the nozzle with no shocks or waves
at the exit (i.e. Per3).
4. Determine the length of the triangle of uniform flow that extends inside the nozzle.
2.3 Task 3: Estimate tunnel characteristics
Use your design to estimate the run characteristics for your tunnel. Figure 1 shows a schematic of the
tunnel system including a pressure tank, variable pressure regulator, and the tunnel. Assume that the nozzle
exhausts as a free jet into the test section, which is held at constant vacuum level by a vacuum system not
shown. A closer view of the tunnel is shown in Figure 2.
Minimum total temperature: In order to avoid liquefaction, the free-stream temperature must be greater
than 55 K. Determine the minimum total temperature required to operate the tunnel without lique-
faction.
Operating pressure and mass flow range: A regulator supplies a constant pressure that can range from
0 psia to 500 psia to your tunnel system. Suppose your vacuum system is capable of maintaining a
back pressure of p
0.1 psia regardless of the tunnel settings.
-
1. What is the minimum total pressure required for steady operation of the tunnel with no shocks
at the exit?
2. If your nozzle exit has a height of 350 mm and the entire nozzle has a width of 500 mm, what is
the mass flow rate across your operating pressure range? Plot the relationship.
3. You have a pre-existing compressed air system available to provide air for your tunnel. The tank
is 40 m³ in volume and air is stored at 300 K and held approximately constant by a thermal
matrix in the tanks. Determine how long your tunnel can run as a function of total pressure. Plot
the relationship.
4. Given the range of total pressure and minimum total temperature previously calculated, determine
the free-stream unit Reynolds number as a function of total pressure. Plot the relationship. Note:
© 2024, S. A. Craig, All rights reserved.
2 AME 323 Project 2 v2.3
M
∞
7°
w=200 mm
into page
A
500 mm
Figure 3: Half-wedge model.
Unit Reynolds number is the Reynolds number without a length scale, Re' = pU/μ and has units
of 1/m. Viscosity can be estimated using Sutherland's law of viscosity (Equation 1).
μ = μref
3/2
T
Tref
Tref+S
T+S
-5
Href = 1.716 x 10 Pa S, Tref = 273.15 K,
Additional geometric characteristics: How much space do you need?
1. What is your nozzle expansion length?
(la)
S = 110.4 K
(1b)
2. What cross-sectional area is required in your settling chamber such that M≤ 0.05? (A settling
chamber is the section just upstream of the converging section that often contains flow conditioning
devices.)
Model forces: You wish to place a half-wedge model (bottom flat, top angled, see Figure 3) in your tunnel
with an angle of 7°, a width of 200 mm, and a length of 500 mm. Ignoring the force on the back side
of the model, determine the horizontal and vertical forces on the model (drag and lift) that must be
withstood by the model mount system during steady operation at maximum pressure.
2.4
Task 4: Report
Prepare a design report (maximum 10 pages of text, 20 pages including figures) detailing your code, nozzle,
and tunnel. The report should adhere to the following guidelines:
• Include in-text citations and a references section containing any sources you use to produce your report
other than the course notes.
•
Explain the basic operation of your design code.
• Include and discuss all plots requested in Task 2.
• Include and discuss all calculations and plots requested in Task 3.
• Add any additional plots where appropriate.
• You are free to do all calculations using a computer, but the process for calculating each requested
quantity should be explained using equations as appropriate.
© 2024, S. A. Craig, All rights reserved.
3
