Turbulent flow over NACA0012 airfoil (2D)

Overview

• Based on https://turbmodels.larc.nasa.gov/naca0012_val.html 53
• References: Gregory-O'Reilly (1970) 21, McCroskey (1987) 47, Ladson (1988) 32, Spalart-Allmaras (1994) 69, and Krist et al. (1997) 29
• See the resources section for additional data files

Flow physics:

• External flow
• Steady
• High Reynolds number
• Low Mach number, subsonic
• Newtonian, single-phase, incompressible, non-reacting

Solver:

• simpleFoam

Tutorial case:

• airFoil2D

Physics and Numerics

Physical domain:

• The case is a two-dimensional airfoil located around the centre of a computational domain whose dimensions are considerably larger than the chord-length of the airfoil.
  • $x$: Longitudinal direction (mean flow direction)
  • $y$: Spanwise direction (statistically homogeneous direction)
  • $z$: Vertical direction (wall-normal direction)
  • $O$: Origin at the leading edge of the airfoil

Physical modelling:

• Reynolds number based on local chord length: $\text{Re}_c = U_x \, c \, \nu^{-1} \approx 6\times10^6$
  • Streamwise far-field flow speed: $U_x = 51.4815$ [m⋅s^-1]
  • Characteristic length (Local chord length of the airfoil): $c = 1.0$ [m]
  • Kinematic viscosity of fluid: $\nu_\text{fluid} = 8.58 \times 10^{-6}$

\[m<sup>2</sup>⋅s<sup>-1</sup>\]

• Mach number: $\text{Ma} = U_x / U_s \approx 0.15$
  • Speed of sound: $U_s = 343.21$ [m⋅s^-1]
• Turbulence model: Spalart Allmaras

Numerical domain modelling:

• Shape: extruded C-grid
• Dimensions: $(x, y, z) \approx (985.5, 1.0, 1015.6)$ [m]
• Sketch (View direction to $y$-positive):

Spatial domain discretisation:

• Mesh type: hexahedral cells in plot3d format
• Mesh converter: plot3dToFoam
• Number of cells, $N$ : $(N_x, N_y, N_z) = (257, 1, 897)$
• First wall-normal cell centre height: $\Delta_y^+ < 1$
• Mesh detail (View direction to $y$-positive):

Equation discretisation:

Spatial derivatives and variables:

• Gradient: Gauss linear
• Divergence:
  • default: Gauss linear
  • div(phi,U): bounded Gauss

[linearUpwind](https://doc.openfoam.com/2606/tools/processing/numerics/schemes/divergence/rtm/linearUpwind/) `grad(U)`

• div(phi,nuTilda): bounded Gauss linearUpwind grad(nuTilda)
• Laplacian: Gaussian linear corrected
• Surface-normal gradient: corrected

Temporal derivatives and variables:

• ddtSchemes: steadyState

Numerical boundary conditions:

• Velocity, $\mathbf{U}$

Patch	Condition	Value [m⋅s<sup>-1</sup>]
Inlet	freestreamVelocity	$\mathbf{U}_\alpha$
Outlet	freestreamVelocity	$\mathbf{U}_\alpha$
Sides $\text{(}y$-dir)	empty	-
Aerofoil	fixedValue	(0.0, 0.0, 0.0)

<b> α </b>	<b> U </b><sub>α</sub>
$\alpha = 0^o$	(51.4815, 0.00, 0.0000)
$\alpha = 10^o$	(50.6994, 0.00, 8.9397)
$\alpha = 15^o$	(49.7273, 0.00, 13.3244)

• Kinematic pressure, p

Patch	Condition	Value [m<sup>2</sup>⋅s<sup>-2</sup>]
Inlet	freestreamPressure	0.0
Outlet	freestreamPressure	0.0
Sides $\text{(}y$-dir)	empty	-
Aerofoil	zeroGradient	-

• Turbulent kinematic viscosity, nut (i.e. $\nu_t$)

Patch	Condition	Value [m<sup>2</sup>⋅s<sup>-1</sup>]
Inlet	freestream	$8.58e^{-6} \approx \nu_\text{fluid}$ 54
Outlet	freestream	$8.58e^{-6}\approx \nu_\text{fluid}$ 54
Sides $\text{(}y$-dir)	empty	-
Aerofoil	fixedValue	0.0 54

• Spalart-Allmaras model modified viscosity, nuTilda (i.e. $\tilde{\nu}$)

Patch	Condition	Value [m<sup>2</sup>⋅s<sup>-1</sup>]
Inlet	freestream	$3.432e^{-5} \approx 4 \nu_\text{fluid}$ 54
Outlet	freestream	$3.432e^{-5}\approx 4 \nu_\text{fluid}$ 54
Sides $\text{(}y$-dir)	empty	-
Aerofoil	fixedValue	0.0 54

Solution algorithms and solvers:

• Pressure-velocity: SIMPLE
• Parallel decomposition of spatial domain and fields: Not applicable

• Linear solvers:

Field	Linear Solver	Smoother	Tolerance (rel)
U	smooth	GaussSeidel	0.01
p	GAMG	GaussSeidel	0.01
nuTilda	smooth	GaussSeidel	0.01

Results

List of metrics:

• Lift coefficient $\mathrm{C}_\mathrm{L}$ vs. Angle of attack $\alpha$
• Drag coefficient $\mathrm{C}_\mathrm{D}$ vs. Angle of attack $\alpha$
• Drag coefficient $\mathrm{C}_\mathrm{D}$ vs. Lift coefficient $\mathrm{C}_\mathrm{L}$
• Surface pressure coefficient $\mathrm{C}_p$ vs. Normalised chord length $x/c$
• Surface skin friction coefficient $\mathrm{C}_f$ vs. Normalised chord length $x/c$
• ${\overline{\cdot}}$ is the time-averaging operator

Resources

Note: Links will take you to the NASA website

Mesh

• Mesh description
• Mesh: (3-D) 2 x 897 x 257 with 2 x 513 points on airfoil surface (gzipped, 5.1 MB)

Datasets for verifications (plain text)

Lift and drag coefficients vs angle of attack

• Experiment 32
• Numerical (CFL3D software) 29

Pressure distribution vs local chord length

• Experiment 32
• Numerical (CFL3D software) 29

Lift coefficient vs angle of attack

• Theoretical 47

Skin friction coefficient vs local chord length

• Numerical (CFL3D software) 29