Aerodynamics Computational Assignment #1: Computation

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Matlab Code Policy:

Computational codes must be written individually and are expected to be written in MAT-LAB. If you have collaborated with others while writing your code be sure to acknowledge them in the header of your code, otherwise you may receive a zero for plagiarism. All code les required to successfully run the computational assignment driver or main script should be submitted via the course website by 11:59pm on the due date. Code les will not be accepted after the given due date.

Re ection Questions:

In this assignment, there are multiple re ection questions. These re ection questions are provided to help you review the functionality of your code, help you analyze and understand your results, and to test your understanding of the concepts being studied.

Learning Outcomes:

1. Understand how to integrate pressure values to calculate lift and drag.

2. Understand di erent numerical integration techniques and when to use them.

3. Understand the e ect of the number of pressure sensors and their placement on the accuracy of calculated lift and drag.

4. Understand the e ects of di erent models on lift and drag.

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ASEN 3111 Aero Computational Assignment # 1

Problem #1:

Consider ideal (incompressible and inviscid) ow over a rotating cylinder as depicted in the gure below:

For this ow problem, the coe cient of pressure, de ned as:

Cp
p
p1

q1
where p1 is the freestream pressure and q1 is the dynamic pressure, is known to be:
" #
2
Cp( ) = 1 4 sin2( ) + 2 sin( ) +

RV1 2 RV1

where represents the circulation about the cylinder. The sectional coe cients of lift and drag are de ned as:
cl

L0
and cd
D0

2q1R

2q1R
where L0 and D0 are the lift and drag per unit span, are known to be
cl =
2
Z0
2
and cd =
2
Z0
2

Cp( ) sin( )d

Cp( ) cos( )d

1

1

For = 2 RV1, complete the following tasks:

Analytically determine the sectional lift and drag coe cients and print these values to the command window.

Produce plots of the sectional lift and drag coe cients predicted by the composite trapezoidal rule versus the number of panels, N, used to discretize the surface of the cylinder.

Produce plots of the sectional lift and drag coe cients predicted by the composite Simpson’s rule versus the number of panels, N, used to discretize the surface of the cylinder.

Print to the command window the number of panels, N, required to achieve a pre-dicted sectional lift coe cient with 1/10 percent relative error using the composite trapezoidal rule.

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ASEN 3111 Aero Computational Assignment # 1

Print to the command window the number of panels, N, required to achieve a predicted sectional lift coe cient with 1/10 percent relative error using the composite Simpson’s rule.

Re ection: How do the two methods compare? Would they still perform the same way if you were given experimental data in the form of a discrete array of pressure coe cient values instead of being given an exact solution/equation?

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ASEN 3111 Aero Computational Assignment # 1

in the

y

x
α
c
V∞

For this ow problem, the coe cient of pressure cannot be determined analytically. Nev-ertheless, the vortex panel method has been employed to approximate the coe cient of pressure along both the upper and lower surfaces of the NACA 0012 airfoil, yielding the results displayed below:

6

Lower Surface

5

Upper Surface

4

3

p
2

C

-

1

0

-1

-2
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1

0

x/c

The vortex panel results have been further interpolated using splines and the results are stored within a MATLAB .mat le Cp.mat located in the Lab directory on the course web-site. To open the MATLAB le, type load Cp into the Command Window. This will load two spline variables, Cp_upper and Cp_lower, into the Workspace. Then, to evaluate the coe cient of pressure along some location x=c along the upper surface, simply type fnval(Cp_upper, x/c). Similarly, to evaluate the coe cient of pressure then along some location x=c along the lower surface, type fnval(Cp_lower, x/c).

4
ASEN 3111 Aero Computational Assignment # 1

Using the MATLAB spline variables Cp_upper and Cp_lower and the composite trapezoidal rule, determine and print to the command window the lift and drag (per unit span) on a stationary NACA 0012 airfoil with chord length c = 5 m at 9 angle of attack in an ideal airfoil with freestream airspeed V1 = 20 m/s, air density 1 = 1:225 kg/m3, and pressure p1 = 101:3 103 Pa.

In addition to the above, complete the following tasks:

Determine and print to the command window the number of equispaced (with respect to chord line distance, x) integration points, n, required to obtain a lift solution with ve percent relative error.

Determine and print to the command window the number of equispaced (with respect to chord line distance, x) integration points, n, required to obtain a lift solution with one percent relative error.

Determine and print to the command window the number of equispaced (with respect to chord line distance, x) integration points, n, required to obtain a lift solution with 1=10 percent relative error.

Above n refers to the total number of integration points along the airfoil surface, not just the points on either the lower or upper surfaces.

Re ection: Given the number of required equispaced integration points required to obtain an accurate lift solution, how should one go about measuring pressure in the wind tunnel to experimentally determine coe cient of lift? If the number of experimental pressure ports is limited how can you best locate these ports along to surface to improve the accuracy in the lift estimate?

Note: The formula for the shape of a NACA 00xx airfoil, with \xx" being replaced by the percentage of thickness to chord, is:

yt = 0t:2c

0:2969
r

c
0:1260
c
0:3516
c

+ 0:2843
c

0:1036
c

x

x

x

2

x

3

x
4

where c is the chord length, x is the position along the chord from 0 to c, yt is the half thickness at a given value of x (centerline to surface), and t is the maximum thickness as a fraction of the chord (i.e., t = xx=100).

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