1MAE 5540 – Propulsion Systems II

Homework 1.2 SolutionAssignment 1.2 Date Assigned: Wednesday September 4, 2019Date Due: Friday September 20, 2019Title: 2-D Method of Characteristics (M.O.C.) Grid Solver Development Num. of Points: 10 Ø Code and Verify subroutines or scripts for

• Initial Data Line along Expansion Section Wall • Internal Flow Unit Process• Centerline Intercept Unit Process (C- characteristic line)• Wall Intercept Unit Process (C+ characteristic line)• Minimum Length Nozzle Maximum Turning Angle

Ø Link and Sequence Unit Process Modules to Calculate M.O.C Grid

2MAE 5540 – Propulsion Systems II

Homework 1.2 Solution (2)Solve Problem 11.1 in Anderson, page 429. (See Section 1.1 Notes for Example)

... Minimum Length Nozzle with Maximum Turning Angle -- infinitesimal expansion section... Mexit = 2.0... D* = 2.0 cm

... Assume g = 1.4

... Repeat with g = 1.2Solve Problem 11.2 in Anderson, Page 430 but with ...

... Finite expansion section radius of Curvature equal to 1.5 x throat radius

... Mexit = 2.0

... D* = 2.0 cm ... Assume g = 1.4... Repeat with g = 1.2

for all parts Plot nozzle half-contoursPlot nozzle Mach number profile along upper wall and along centerlineCompare to Mach number profile calculated using A/A* equation

3MAE 5540 – Propulsion Systems II

Homework 1.2 Solution (3)For each of the 4 Nozzle Contours above à• Use the approximate mapping technique to solve for P, Q, S, and T of the Parabolic

Contour• Plot Contours against the derived M.O.C contours, compare shapes• Use M.O.C. values for qmax, Nozzle length LN, and Radius of Curvature Rc of the

expansion section for these calculations• Be sure to show Calculations for Xn, Yn, P, Q, S, T, etc. • Assume Nozzle exit angle is zero for each case

4MAE 5540 – Propulsion Systems II

Solution1.2, part 1• Minimum Length Nozzle … g = 1.4• Let N = 6

180π

1.4 1+1.4 1−# $% &

0.5 1.4 1−1.4 1+# $% & 2.02 1−( )# $% &

0.5

# $' (% &

atan 2.02 1−( )( ) 0.5( )atan−# $' (% &

Mexit = 2.0→ν(Mexit ) =γ +1γ −1

tan−1 γ −1γ +1

2.02 −1( )%&'('

)*'

+'− tan−1 2.02 −1 =

= 26.3798o

θw Max =νexit2

== 13.1899o

5MAE 5540 – Propulsion Systems II

Solution 1.2, part 1 (2)Minimum Length à Rc expansion = 0

θw Max =νexit2

== 13.1899o

6MAE 5540 – Propulsion Systems II

Solution 1.2, part 1 (3)

7MAE 5540 – Propulsion Systems II

Solution 1.2, part 1 (4)• Minimum Length Nozzle … g = 1.2• Let N=6

Mexit = 2.0→ν(Mexit ) =γ +1γ −1

tan−1 γ −1γ +1

2.02 −1( )%&'('

)*'

+'− tan−1 2.02 −1 =

= 31.46o

θw Max =νexit2

= 15.731oMaximium Turning Angle changes!

8MAE 5540 – Propulsion Systems II

Solution 1.2, part 1 (5)Minimum Length à Rc expansion = 0

θw Max =νexit2

== 15.731o

9MAE 5540 – Propulsion Systems II

Solution 1.2, part 1 (6)

10MAE 5540 – Propulsion Systems II

Solution 1.2, Part 1 comparisonsθw Max =

νexit2

== 13.1899o

θw Max =νexit2

=15.731o

11MAE 5540 – Propulsion Systems II

Solution 1.2, part 2 (1)Rc expansion = 1.5 x Rthroat = 1.5 cm

θw Max =νexit2

== 13.189o

12MAE 5540 – Propulsion Systems II

Solution 1.2, part 2 (2)

13MAE 5540 – Propulsion Systems II

Solution 1.2, part 2 (3)Rc expansion = 1.5 x Rthroat = 1.5 cm

θw Max =νexit2

== 15.732o

14MAE 5540 – Propulsion Systems II

Solution 1.2, part 2 (4)

15MAE 5540 – Propulsion Systems II

Solution 1.2, Part 2 comparisons

θw Max =νexit2

=

θw Max =νexit2

== 15.731o

= 13.189o

16MAE 5540 – Propulsion Systems II

Part 3, Approximate Contour Mappinga) Minimum Length à Rc expansion = 0, g=1.4, A/A* = 1.6875

M.O.C Nozzle Spline Fit Nozzle

θw Max =νexit2

== 13.1899o

17MAE 5540 – Propulsion Systems II

Part 3, Approximate Contour Mapping (2)b) Non-Minimum Length à Rc expansion = 1.5 cm, g=1.4, A/A* = 1.6875

M.O.C Nozzle Spline Fit Nozzle

θw Max =νexit2

== 13.1899o

18MAE 5540 – Propulsion Systems II

Part 2, Approximate Contour Mapping (3)c) Minimum Length à Rc expansion = 0, g=1.2, A/A* = 1.884

M.O.C Nozzle Spline Fit Nozzle

θw Max =νexit2

== 15.732o

19MAE 5540 – Propulsion Systems II

Part 2, Approximate Contour Mapping (3)b) Non-Minimum Length à Rc expansion = 1.5 cm, g=1.2, A/A* = 1.884

M.O.C Nozzle Spline Fit Nozzle

θw Max =νexit2

== 15.732o