Flow Around an Inclined NACA 0012 Airfoil

This example simulates the flow around an inclined NACA 0012 airfoil using the SST turbulence model and compares the results with the experimental lift data of Ladson (Ref. 1) and pressure data of Gregory and O’Reilly (Ref. 2). The SST model combines the near-wall capabilities of the k-ω model with the superior free-stream behavior of the k-ε model to enable accurate simulations of a wide variety of internal and external flow problems. See the theory for the SST turbulence model in the CFD Module User’s Guide for further information.

Model Definition

Consider the flow relative to a reference frame fixed on a NACA 0012 airfoil with chord-length c = 1.8 m. The temperature of the ambient air is 20 °C and the relative free-stream velocity is U∞ = 50 m/s resulting in a Mach number of 0.15. The Reynolds number based on the chord length is roughly 6·106, so you can assume that the boundary layers are turbulent over practically the entire airfoil. The airfoil is inclined at an angle α to the oncoming stream,

To obtain a sharp trailing edge, the airfoil is slightly altered from its original shape (Ref. 3),

The upstream, top, and bottom edges of the computational domain are located 100 chord-lengths away from the trailing edge of the airfoil and the downstream edge is located 200 chord-lengths away. This is to minimize the effect of the applied boundary conditions.

Figure 1 shows the flow domain and the applied far-field boundary conditions,

Figure 1: Flow domain and far-field boundary conditions.

Ref. 4 provides far-field values for the turbulence variables,

where the free-stream value of the turbulence kinetic energy is given by

and L is the appropriate length of the computational domain. The current model applies the upper limit of the provided free-stream turbulence values,

Figure 2 shows a close-up of the airfoil section. A no-slip condition is applied on the surface of the airfoil.

Figure 2: Close-up of the airfoil section.

The computations employ a structured mesh with a high size ratio between the outermost and wall-adjacent elements.

Potential Flow Solution

The simplest option when setting the initial velocity field is to use a constant velocity, which does not satisfy the wall boundary conditions. A more accurate and robust initial guess can be obtained solving the potential flow equation.

Assuming irrotational, inviscid flow, the velocity potential

is defined as

The velocity potential

must satisfy the continuity equation for incompressible flow,

. The continuity equation can be expressed as a Laplace equation

which is the potential flow equation.

Once the velocity potential

is computed, the pressure can be approximated using Bernoulli’s equation for steady flows:

Results and Discussion

The study performs a Parametric Sweep with the angle of attack α taking the values,

Figure 3 shows the velocity magnitude and the streamlines for the steady flow around the NACA 0012 profile at α = 14°.

Figure 3: Velocity magnitude and streamlines for the flow around a NACA 0012 airfoil.

A small separation bubble appears at the trailing edge for higher values of α, and the flow is unlikely to remain steady and two-dimensional hereon. Ref. 1 provides experimental data for the lift coefficient versus the angle of attack,

where the pressure coefficient is defined as,

and c is the chord length. Note that the normal is directed outward from the flow domain (into the airfoil). Figure 4 shows computational and experimental results for the lift coefficient versus angle of attack.

Figure 4: Computational (solid) and experimental (dots) results for the lift coefficient vs. angle of attack.

No discernible discrepancy between the computational and experimental results occurs within the range of α values used in the computations. The experimental results continue through the parameter regime where the airfoil stalls. Figure 5 shows a comparison between the computed pressure coefficient at α = 10° and the experimental results in Ref. 2.

Figure 5: Computational (solid) and experimental (dots) results for the pressure coefficient along the airfoil.

Experimental data is only available on the low-pressure side of the airfoil. The agreement between the computational and experimental results is very good.

Notes About the COMSOL Implementation

The model uses the SST turbulence model together with a Parametric Sweep for the angle of attack to compute the different flows on a mapped mesh.

The initial values for the velocity components are obtained by solving a potential flow equation, which is set up using a PDE interface.

References

1. C.L. Ladson, “Effects of Independent Variation of Mach and Reynolds Numbers on the Low-Speed Aerodynamic Characteristics of the NACA 0012 Airfoil Section,” NASA TM 4074, 1988.

2. N. Gregory and C. L. O’Reilly, “Low-Speed Aerodynamic Characteristics of NACA 0012 Aerofoil Section, including the Effects of Upper-Surface Roughness Simulating Hoar Frost,” A.R.C., R. & M. no. 3726, 1970.

3. NASA Langley Research Center, Turbulence Modeling Resource, “2D NACA 0012 Airfoil Validation Case,” http://turbmodels.larc.nasa.gov/naca0012_val.html

4. F.R. Menter, “Two-Equation Eddy-Viscosity Models for Engineering Applications,” AIAA Journal, vol. 32, no. 8, pp. 1598–1605, 1994.

CFD_Module/Verification_Examples/naca0012_airfoil

Modeling Instructions

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Laplace Equation (lpeq)

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Global Definitions

Parameters 1

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In the table, enter the following settings:


Name	Expression	Value	Description
U_inf	50[m*s^-1]	50 m/s	Free-stream velocity
rho_inf	1.2043[kg*m^-3]	1.2043 kg/m³	Free-stream density
mu_inf	1.81397e-5[kgm^-1s^-1]	1.814E-5 kg/(m·s)	Free-stream dynamic viscosity
L	180[m]	180 m	Domain reference length
c	1.8[m]	1.8 m	Chord length
k_inf	0.1mu_infU_inf/(rho_inf*L)	4.184E-7 m²/s²	Free-stream turbulent kinetic energy
om_inf	10*U_inf/L	2.7778 1/s	Free-stream specific dissipation rate
alpha	0	0	Angle of attack

Geometry 1

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Dependent variable quantity	Unit
Custom unit	m^2/s

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Source term quantity	Unit
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Dirichlet Boundary Condition 1

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Turbulent Flow, SST (spf)

Since the density variation is not small, the flow cannot be regarded as incompressible. Therefore set the flow to be compressible.

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Fluid Properties 1

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Inlet 1

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