ENAE 488C – Computational Fluid Dynamics

NACA 0012 Airfoil Validation

Mario Nirman Mondal

UID# 113269066

Department of Aerospace Engineering

University of Maryland, College Park, MD

ENAE 488C – Computational Fluid Dynamics

NACA 0012 Airfoil Validation

Mario Nirman Mondal

UID# 113269066

Abstract

In this assignment, a validation study (i.e. comparison to experimental data) of the NACA

0012 airfoil was conducted at various angles of attack (). The study was done using three

different grid topologies: 1) Structured O-grid, 2) Structured C-H grid, 3) Unstructured T-Rex

grid. First of all, we created the grids using grid generation software named Pointwise. In this

case, the provided NACA 0012 airfoil geometry was used, and the chord length was assumed to

be 1 meter. Far field boundary was set approximately 50 chord lengths away from the airfoil in

all directions. In order to minimize the differences between each grid topology, the wall spacing

was kept same at s=1.0e-4, and the grid point distribution was kept same at d=0.01 at the both

leading and the trailing edge. After grid generation, CFD++ solver was used to get the solutions

at different angles of attack. After CFD++ ran the simulations and generated the data, the data

was analyzed in Microsoft Excel. Coefficient of lift (Cl) was plotted against angle of attack (),

and was compared with the experimental data provided. Also, the theoretical incompressible Cl

was calculated using the thin airfoil theory and plotted within the same graph to compare.

Coefficient of lift (Cl) was also plotted against coefficient of drag (Cd) and compared with the

experimental data. After all the analysis, the conclusion was drawn that the structured grids give

better results than unstructured grids if the grid resolution is done well. However, structured grid

generation takes longer time than unstructured grid. On the other hand, if we keep the wall

spacing and grid point distribution same for both types of grid, then the unstructured grids give

better solution accuracy than the structured grids. Though grid generation of the unstructured

topology takes lesser time than structured topology, computation and execution of the codes of

the unstructured topology takes higher time.

I. Introduction

In computational fluid dynamics, solution of a flow field greatly depends on the

generation of grid topology and setting up the solver. There are two main types of grid:

1) Structured grid

2) Unstructured grid

In this assignment, three different topologies of grid are made. They are: 1) Structured O-grid, 2)

Structured C-H grid, 3) Unstructured T-Rex grid. After doing the grid generation, CFD++ solver

was set up to solve the flow field. After solving the flow field at different angles of attack, data

analysis was done, and conclusion and observation was made.

II. Theory

If an airfoil is placed in air flow of free stream velocity of V∞, there are aerodynamic

forces excreted on the airfoil that creates lift (L) and drag (D). Lift (L) is in the perpendicular

direction of V∞, and Drag (D) is in the parallel to the V∞. The force that acts normal to the airfoil

body is called Normal Force (N), and the force that acts parallel to the airfoil body is called

Axial force (A).

If the airfoil is at an angle of attack, with the free stream velocity V∞, then the relation between

lift-drag and normal-axial is as follows:

L = Ncos – Asin

D = Nsin +Acos

lso, the relations between the coefficients and the forces are as follows:

;

Where, Cl = Coefficient of Lift

Cd = Coeffiient of Drag

q∞ = Dynamic Pressure

= Density of the fluid (in this case, Air)

S = Surface area of the airfoil = Chord length (C) X Span (b)

(Since the chord length is 1 m, and the problem is 2-D, so S=1)

In the assignment, it is assumed that,

N = Fy

A=Fx

So the force coefficients are,

The relation between Cx – Cy and Cl – Cd are as follows:

Cl = Cycos – Cx sin

Cd = Cysin + Cx cos

All the equations are found from “Introduction to Flight” by John D. Anderson Jr.)

III. Methods

1. Grid Generation:

The provided geometry of NACA 0012 airfoil was imported in Pointwise as it was.

The geometry of the airfoil was symmetric.

Wall spacing of s=1.0e-4 was chosen for all grids.

Farfield boundary was placed approximately 50 chord lengths away from the airfoil in all

directions.

The grid point distribution was kept same for all the grid topology at d=0.01 at the both

leading and the trailing edge of the airfoil.

Topologies, Points and Number of 2-D cells:

1) Structured O-Grid : Points – 201X66; Number of cells – 13,000

2) Structured C-H Grid : Points – 653X301; Number of cells – 40,000

3) Unstructured T-Rex Grid: Points – 20,405; Number of cells – 39,217

In order to keep the s and d same, the number of cell points in structured O-grid has

become lesser than the other ones.

2. CFD++ Solver Setup:

The Wizards was used to set the equation type, initialize the domain with given

quantities, set up turbulence model with k - , set up fluid properties of air and set up time

integration quantities.

The solver was run at steady-state condition.

The dimension and units were used in S.I system.

Freestream Mach number was set at 0.15.

Initial condition of the domain was set at sea level condition (Altitude 0 Km).

Freestream turbulence intensity was set at 3%.

Simple turbulence model (2-equation model) was used.

Turbulent/laminar viscosity ratio was used as 50.0.

Boundary conditions were set adiabetic viscous wall function for airfoil wall, and

characteristics-based inflow/outflow for the farfield.

Simulations were run at =-5, 0, 5, 10, 20

3. Flow visualization, Data collection and Plotting in Microsoft Excel:

After CFD++ ran the simulations, residual plots and the flow visualization plots were

observed and analyzed.

Data was recorded for approximately last 50 iterations and average was calculated.

Entries for dimensional forces and coefficients were created in Microsoft Excel.

Plots of Cl vs. and Cl vs. Cd were created.

Plots were compared with the experimental values and theoretical incompressible values.

4. Hand Calculation:

In order to verify the CFD++ results, Fx and Fy were used from the data, and Cx, Cy, Cl and Cd

were calculated manually for = 10°. Hand calculations are included in the appendix.

-1.00E+00

-5.00E-01

0.00E+00

5.00E-01

1.00E+00

1.50E+00

2.00E+00

2.50E+00

-10 -5 0 5 10 15 20 25

Cl

Angle of Attack, α

O-Grid: Cl vs. α

Cl from CFD

Experiemental Cl

Theoritical Cl

-1.00E+00

-5.00E-01

0.00E+00

5.00E-01

1.00E+00

1.50E+00

2.00E+00

2.50E+00

-10 -5 0 5 10 15 20 25

Cl

Angle of Attack, α

C-H Grid: Cl vs α

Cl from CFD

Experimental Cl

Theoritical

IV. Results and Discussion

From the plots Coefficient of lift (Cl) vs. angle of attack ( analysis can be done. We plotted

CFD results, theoretical result using the thin airfoil theory and the experimental results.

Computational fluid dynamics can predict fairly accurate results of the actual flow field provided

that the grid generation is decent and the

solver setup was accurate.

From figure 1, we can see that at low

angles of attack, CFD was successfully

predicting the coefficient of lift (Cl).

However, the CFD results predicted the

stall (where Cl starts to fall down as we

increase ) the before the actual

experimental value. The result is still

acceptable and did a pretty good job at

lower Figure is made from structured

O-grid, and the highest lift is produced at

°.

Similar result was found also found for

C-H grid, though the numbers of cells

are different. (Figure 2)

There are two main reasons that might

cause the poor result: 1) the structured

grid setup was not descent to capture all

the results at the boundary layers of the

airfoil, 2) the turbulence model was not

tuned enough to solve the flow field at

higher angles of attack.

In order to get more accuracy with

structured grid, we need to resolve the

grid and include more cells to capture

the flow behavior. Also we need to

redefine our turbulence as close as

possible to the flow behavior. Because of

the simple 2-equations turbulence, our

Figure 1

Figure 2

-1.00E+00

-5.00E-01

0.00E+00

5.00E-01

1.00E+00

1.50E+00

2.00E+00

0.00E+00 1.00E-01 2.00E-01 3.00E-01 4.00E-01 5.00E-01

Co

eff

of

Lift

, Cl

Coeff of Drag, Cd

O-grid: Cd vs Cl

Cd vs Cl Experimental

-1.00E+00

-5.00E-01

0.00E+00

5.00E-01

1.00E+00

1.50E+00

2.00E+00

0.00E+00 1.00E-01 2.00E-01 3.00E-01 4.00E-01 5.00E-01C

oef

f o

f Li

ft, C

l

Coeff of Drag, Cd

C-H grid: Cd vs Cl

CFD Cl vs. Cd Experimental

-1.00E+00

-5.00E-01

0.00E+00

5.00E-01

1.00E+00

1.50E+00

2.00E+00

2.50E+00

-10 -5 0 5 10 15 20 25

Co

eff

of

Lift

, Cl

Angle of Attack, α

Unstructured Grid: Cl vs α

Cl from CFD

Experiemental Cl

Theoritical Cl

drag prediction was higher than the experimental value.

The plots of Cl vs. Cd of the structured grids are provided below.

As it indicates, we got lower lift than the experimental values at higher drag. That is because of

over estimation of drag by the CFD++ solver.

On the other hand, at the same wall

spacing, s and at the same points

distribution d over airfoil, the

unstructured grids predicted better

solution than the structured grids.

It predicted stall about at =15°

(Figure 5). We could push it further

by resolving the grid points at the

farfield, where we can reduce the

unnecessary grid cells.

Unstructured grid also overestimated

the drag coefficients (Figure 6). As

we told before, this is the because of

turbulence model in the CFD solver.

Figure 3 Figure 4

Figure 5

-1.00E+00

-5.00E-01

0.00E+00

5.00E-01

1.00E+00

1.50E+00

2.00E+00

0.00E+00 1.00E-01 2.00E-01 3.00E-01 4.00E-01 5.00E-01C

oe

ff o

f Li

ft, C

l

Coeff of Drag, Cd

Unstructured grid: Cd vs Cl

Cl vs. Cd

Experimental

-8.00E-01

-6.00E-01

-4.00E-01

-2.00E-01

0.00E+00

2.00E-01

4.00E-01

6.00E-01

8.00E-01

1.00E+00

1.20E+00

1.40E+00

-10 -5 0 5 10 15 20 25

Co

eff

of

lift,

Cl

Angle of Attack, α

Different Grid: Cl vs. α O-Grid

C-H Grid

Unstructured

From the analysis, it can be

understood that at lower number of

cells unstructured grid do a good

job predicting the flow behavior.

However, it takes longer time than

the structured grid. Conversely,

structured is much faster. In order

to get more accuracy from

structured grid, the grid need to be

resolved until it capture most of the

parts of the flow behavior,

especially at the boundary of the

airfoil.

Comparison of the three types of

grid topologies at same wall

spacing and same grid point

distribution are shown in the graph

below (Figure 7).

Figure 6

Figure 7

From figure 1, 2 and 5, it can be seen that the „green‟ straight line corresponds to the theoretical

value of coefficient of lift Cl for incompressible flow. According to incompressible thin airfoil

theory,

Coefficient of lift, Cl = 2π

Incompressible theory of thin of airfoil does account for turbulence, and thus it does not predict

stall at all. For this reason, according to the equation, as we increase α, Cl should also

continuously increase. However, this does not happen in real world. Actually, at lower angles of

attack, approximately at α<5°-6°, the theoretical results match up nicely with the experimental

and CFD data.

From the observation of the 2 contour plots of Mach number at =5° and =20° from C-H grid,

we can say that at low angle of attack (Figure 8) the air velocity does not change much when

leaves the airfoil. However, the Mach number decreases at leading edge and decreased Mach

number leaves at the trailing edge. Airflow is smooth around the airfoil, and there is no flow

separation.

As the angle of attack increases, flow separation begins. As shown in figure 9, massive flow

separation occurs as the flow leaves the airfoil body at angle of attack α=20°. The flow separates

and creates turbulent wakes when it leaves the trailing edge. At some points Mach number

approach to a value of 0. The separated flow and the turbulent wakes together cause stalling of

the airfoil.

Figure 8: Mach Number Contour Plot using C-H Grid at α=5°

After analyzing the convergence of different grid topologies, it is observed that the structured

grids converge quickly (takes less iterations) than the unstructured grid. (Figure 10 and 11)

Figure 9: Mach Number Contour Plot using C-H Grid at α=20°

Figure 10: Residual plot of C-H grid at α=5°

V. Conclusion

After all the analysis, the conclusion was drawn that the structured grids give better

results than unstructured grids if the grid resolution is done well. However, structured grid

generation takes longer time than unstructured grid. On the other hand, if we keep the wall

spacing and grid point distribution same for both types of grid, then the unstructured grids give

better solution accuracy than the structured grids. Though grid generation of the unstructured

topology takes lesser time than structured topology, computation and execution of the codes of

takes much longer time. Also, the structured grid converges to solution quickly than the

unstructured grid. There are some trade-offs between structured and unstructured grids. The

answer to the question “Which grid is better?” is problem specific, and it totally depends on the

time, accuracy and solution constraints.

Figure 11: Residual plot of Unstructured grid at α=5°

Appendix A

(Hand Calculation)

Appendix B

(Tables)

Structured O – Grid:

Angle of Attack(in Deg) -5 0 5 10 15 20

Fx -47.19473935 2.08E+01 -47.24664765 -207.63028 -9.68E+01 2.74E+00

Fy -853.5053069 0.203864 853.500409 1540.84688 1.19E+03 1.07E+03

Cx -2.96E-02 1.30E-02 -2.96E-02 -1.30E-01 -6.07E-02 1.72E-03

Cy -5.35E-01 1.28E-04 5.35E-01 9.66E-01 7.43E-01 6.73E-01

Cl -5.35E-01 1.28E-04 5.35E-01 9.73E-01 7.34E-01 6.32E-01

Cd 1.72E-02 1.30E-02 1.71E-02 3.95E-02 1.34E-01 2.32E-01

Structured C-H Grid:

Angle of Attack (in deg) -5 0 5 10 15 20

Fx -4.57E+01 1.95E+01 -4.56E+01 -1.90E+02 -9.61E+01 -1.05E+02

Fy -8.19E+02 -5.12E-02 8.18E+02 1.40E+03 1.09E+03 1.10E+03

Cx -2.86E-02 1.22E-02 -2.86E-02 -1.19E-01 -6.02E-02 -6.57E-02

Cy -5.13E-01 -3.21E-05 5.12E-01 8.78E-01 6.83E-01 6.89E-01

Cl -5.14E-01 -3.21E-05 5.13E-01 8.85E-01 6.76E-01 6.70E-01

Cd 1.62E-02 1.22E-02 1.62E-02 3.53E-02 1.19E-01 1.74E-01

Unstructured T-Rex Grid:

Angle of Attack (in deg) -5 0 5 10 15 20

Fx -5.39E+01 1.83E+01 -5.44E+01 -2.45E+02 -3.49E+02 6.26E+01

Fy -8.64E+02 -1.42E-01 8.65E+02 1.61E+03 1.74E+03 1.01E+03

Cx -3.38E-02 1.15E-02 -3.41E-02 -1.53E-01 -2.19E-01 3.96E-02

Cy -5.41E-01 -8.87E-05 5.42E-01 1.01E+00 1.09E+00 6.29E-01

Cl -5.42E-01 -8.87E-05 5.43E-01 1.02E+00 1.11E+00 5.78E-01

Cd 1.35E-02 1.15E-02 1.33E-02 2.38E-02 7.04E-02 2.52E-01

Experimental Value Alpha, α (in deg)

Experimental Cl

Experimental Cd

t grit "180"

-3.99 -0.4363 0.00871

-1.98 -0.2213 0.00792

-0.03 -0.0115 0.00803

0.04 -0.0013 0.00811

2 0.2213 0.00814

4.06 0.4365 0.00814

6.09 0.6558 0.00851

8.09 0.8689 0.00985

10.18 1.0809 0.01165

11.13 1.1731 0.01247

12.1 1.2644 0.01299

13.31 1.3676 0.01408

14.08 1.4316 0.01533

15.24 1.5169 0.0187

16.33 1.5855 0.02186

17.13 1.6219 0.02513

18.21 1.0104 0.25899

19.27 1.0664 0.43446

Thin Airfoil Theory: Incompressible Theoretical Lift Coefficient

Alpha, α (in deg)

Coeff of Lift, Cl

-5 -0.54831392

0 0.000000

5 0.54831392

10 1.09662784

15 1.64494176

20 2.19325568