MCEN90008 Fluid Dynamics 2024 Unit 2 - Compressible flow assignment - Part IIR

Java Python MCEN90008 Fluid Dynamics 2024

Unit 2 - Compressible flow assignment - Part II

Supersonic wing design

Instructions:

• Assignment is due 11:59 pm Sunday 20th October 2024

• This assignment should be done in groups of 2 students. Both students in the group will get the same mark for this assignment. If you choose to do the assignment alone, no concession will be given (your assignment will be marked the same as an assignment done by two students). Groups of three are not permitted.

• Please choose your group partner carefully.

• Signs of collusion between groups in the submitted report will be treated as plagarism

• Only hand in one assignment per group.

• Question 1 (shock-expansion theory) is quite tedious - and well suited to computational solution. I suggest that you write a Matlab code, since this will save you time later for question 4 & 6. You may use the tabulated data and graphical information provided at the back if you prefer to get some practise using tables (as you would in the exam).

• You should at every step show which equations and assumptions you are using to arrive at results.

• The airfoil that you are asked to analyze is very thin, thus the flow deflec tions are relatively small (and therefore the shocks relatively weak). Your solutions will be more accurate if you use the functions provided to solve the various equations. If you do use the tabulated data given at the back of this assignment (not really recommended), it is important where possible that you adequately interpolate between values in tables to maintain accuracy. If your solutions require tabulated data beyond the range provided here, full tables are available under the resources tab of LMS

• Where you are asked to include graphs, you must adequately label key values and include all axis labels etc to receive full marks.

• Append your Matlab codes to your submissions.

Supersonic wing design

Figure 1: Supersonic airfoil cross-section

A new supersonic passenger aircraft with a cruise Mach number M = 2.2 is designed to employ the above wing profile.

1. Use shock-expansion theory to calculate the Mach number and the ratio p/p∞ in all six flow regions labeled 1 − 6 on the figure, for angle-of-attack α = 4. Fill out the first two columns of table 1. Show all workings in your report for this angle of attack and also include your codes. For the latter parts of this question, it is strongly suggested that you write a Matlab code to compute these. (12 Marks)

Table 1: Tabulated results for questions 1 and 2

2. Now also compute the Mach number and the ratio p/p∞ in all six flow regions using Ackeret’s linearized Mach-line approximation (which dropped out from Ackeret’s clever Taylor’s series expansion that we covered in lectures)

and

where η is the angle that each facet makes with the oncoming flow, and again ± and ∓ refer to the upper and lower aerodynamic surface (upper sign is upper surface, lower sign is lower surface). M∞, ρ∞ and V∞ are the conditions of the oncoming flow (upstream of the airfoil) and δp and δM are changes about these reference conditions. Fill the answers for regions 1 − 6 in the last two columns in Table 1. Show workings in your report for facets 2 and 5 and also include your codes. (6 Marks)

3. Hence calculate the lift and drag coefficient at this angle-of-attack for the two different meth-ods (shock-expansion, linearized Mach-Line) for the results tabulated in Table 1 and also compare these with the solution from Ackeret’s approximate solutions for CD MCEN90008 Fluid Dynamics 2024 Unit 2 - Compressible flow assignment - Part IIR and CL given below. Fill answers n the Table 2 shown below. Show workings in your report for the shock-expansion calculation of CL and CD and also the calculation using Ackeret theory given in equations (1) and (2). It will be easier for subsequent questions (4 & 6) if you write a Matlab code for the shock-expansion method that modifies your code from question 1 to return CL and CD for a given angle-of-attack α.

Table 2: Tabulated results for question 3

(12 Marks)

4. Consider an aircraft of mass m in straight and level flight using the wing cross-section shown in figure 1. Use your shock-expansion code to seek a numerical solution for the angle-of-attack α that will give the smallest drag due to the wing. (5 Marks)

5. Compare this to the minimum angle predicted from Ackeret’s equations for CL and CD. Prove that the angle for the minimum drag is given by,

(6 Marks)

6. Plot mg/D against angle-of-attack α (for 0 < α < 8 ◦ ) for the shock-expansion and Ackeret solution, where m is the aircraft mass, D is the drag of the wing and g is acceleration due to gravity. Mark the minimum drag angles on this figure. Comment on the agreement / disagreement as a function of α. (2 Marks)

7. Comment on the advantages and disadvantages for an aircraft designer employing the mini-mum drag angle in cruise? (1 Marks)

θ-β-M relationship

Relationship between shock angle β and flow deflection angle θ for (top) several Mach numbers M = 1.2, 1.5, 2, 2.5, 3, 10, (bottom) zoomed view on inset with curves for M = 1.85 to 2.40 in steps of 0.01.

Rather than using the θ −β −M chart, it would be more accurate to make use of the Matlab script. file getbeta_betathetaM.m which solves the equation:

getbeta_betathetaM.m will return the shock angle β for a given θ and M. For example for an upstream Mach number M = 2 and a flow deflection of 5◦ ,

>>beta = getbeta_betathetaM(2,5)

>>beta = 34.3016

Normal shock relations

It will be more accurate to use the provided Matlab scripts to calculate the pressure and Mach number change across a normal shock. The function M2_nshock will return the downstream (normal component of) Mach Number for a given upstream (normal component of) Mach number. This script. solves the following relationship,

M2_nshock.m will return M2N for a given M1N . For example for an upstream Mach number M1N = 1.5,

>>M2 = M2_nshock(1.5)

>>M2 = 0.7011

For the pressure change across a normal shock, the Matlab script. p2_over_p1_nshock can be used, which uses the following relationship,

p2_over_p1_nshock will return p1/p2 for a given M1N . For example for an upstream Mach number M1N = 1.5,

>>p2onp1 = p2_over_p1_nshock(1.5)

>>p2onp1 = 2.4583

Prandtl-Meyer Relationship

It will be more accurate to use the provided Matlab scripts to calculate the Prandtl angle from the Mach number (and vice-verse). The function M_given_prandtl_angle will return the Mach Number for a given Prandtl angle. The function prandtl_angle_given_M will return the Prandtl angle ω for a given Mach number. These scripts solve the following relationship,

prandtl_angle_given_M will return ω for a given M. For example for an upstream Mach number M1 = 2,

>>omega = prandl_angle_given_M(2)

>>omega = 26.3798

M_given_prandtl_angle will return M for a given ω. For example for a Prandtl angle ω = 26.3798,

>>M = M_given_prandtl_angle(26