Turbulent flow over NACA0012 airfoil (2D)

Table of Contents

Overview

Flow physics:

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

Solver:

Tutorial case:

Keywords: Reynolds-averaged Navier-Stokes, simpleFoam

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: \( \nu_\text{fluid} = 8.58 \times 10^{-6} \) [m2⋅s-1]
  • 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):
Numerical domain

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):
Mesh

Equation discretisation:

Spatial derivatives and variables:

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

Temporal derivatives and variables:

Numerical boundary conditions:

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

α

U α
\( \alpha = 0^o \) (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 [m2⋅s-2]
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 [m2⋅s-1]
Inlet freestream \( 8.58e^{-6} \approx \nu_\text{fluid} \) [82]
Outlet freestream \( 8.58e^{-6}\approx \nu_\text{fluid} \) [82]
Sides \(\text{(}y \)-dir) empty -
Aerofoil fixedValue 0.0 [82]
  • Spalart-Allmaras model modified viscosity, nuTilda (i.e. \(\tilde{\nu}\))
Patch Condition Value [m2⋅s-1]
Inlet freestream \( 3.432e^{-5} \approx 4 \nu_\text{fluid} \) [82]
Outlet freestream \( 3.432e^{-5}\approx 4 \nu_\text{fluid} \) [82]
Sides \(\text{(}y \)-dir) empty -
Aerofoil fixedValue 0.0 [82]

Solution algorithms and solvers:

  • Pressure-velocity: SIMPLE algorithm
  • Parallel decomposition of spatial domain and fields: Not applicable
  • Linear solvers:
Field Linear Solver Smoother Tolerance (rel)
U Smooth solvers Gauss Seidel Smoother 0.01
p GAMG Solver Gauss Seidel Smoother 0.01
nuTilda Smooth solvers Gauss Seidel Smoother 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
Lift coefficient vs. Angle of attack
Drag coefficient vs. Angle of attack
Drag coefficient vs. Lift coefficient


Surface pressure coefficient vs. Normalised chord length at α=0 [degree]
Surface pressure coefficient vs. Normalised chord length at α=10 [degree]
Surface pressure coefficient vs. Normalised chord length at α=15 [degree]


Surface skin friction coefficient vs. Normalised chord length at α=0 [degree]
Surface skin friction coefficient vs. Normalised chord length at α=10 [degree]
Surface skin friction coefficient vs. Normalised chord length at α=15 [degree]


Resources

Note: Links will take you to the NASA website

Mesh

Datasets for verifications (plain text)

Lift and drag coefficients vs angle of attack

Pressure distribution vs local chord length

Lift coefficient vs angle of attack

Skin friction coefficient vs local chord length