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External Aerodynamics: Parametric modeling of of an airfoil using blockMeshDict in OpenFOAM

Flow over Aerofoil

The construction of OpenFOAM cases for flow over an aerofoil can be done in many ways. For example, the tutorial section demonstrates two methods. The first method uses simpleFoam to demonstrate steady state and incompressible flow over the aerofoil. This method focuses on the overall methodology and a mesh generated in other application has been used. The second method as described in section compressible / sonicFoam / ras which uses a mesh created in STAR and utility star3ToFoam to convert the mesh to the OpenFOAM format. Yet another method using the data points of the airfoil and GMSH is explained in this tutorial.

Mesh Generation using blockMesh

Parametric Model

This page demonstrates a method which can be used to generate mesh for flow over an aerofoil when the points defining the cross-section of the aerofoil is known. The aerofoil shape is created using arc and spline utility of blockMesh. The entire domain will look like as shown below.

Aerofoil Blocking

Mesh: Flow over Aerofoil

Mesh: Flow over Aerofoil

Using the boundary conditions defined in tutorial case incompressible / simpleFoam / airFoil2D, the pressure contour generated out simulation is as per screeshot shown below.

Aerofoil Blocking

The blockmeshDict file is here. The complete case data is here.
Check the effect of parameters on shape of the NACA airfoil:

Shapes of NACA Airfoils

Aerodynamics: 2D Wing Theory

  • R = Resultant force acting on the wing
  • N = Component of R acting perpendicular to the chord
  • A = Component of R acting parallel to the chord
  • Lift L = Component of R acting perpendicular to the relative wind
  • Drag D = Component of R acting parallel to the relative wind direction
  • Angle of Attack, AoA: α = angle between the chord line and relative wind direction.
  • L = N × cosα - A × sinα, D = N × sinα + A × cosα
  • AWING (also designated as 'S' in many textbooks), Wing (projected) area: area of the wing responsible for lift
  • Freestream density: ρ = density of air flow over wings, Freestream velocity: V = velocity of air flow over wings
  • CL = Lift Coefficient,     CD = Drag coefficient
  • L = CL × ½ρV2AWING,   D = CD × ½ρV2AWING
  • At level flight: lift = weigth of aircraft (W), drag = thrust of the engine
  • Wing loading, q = W / AWING
  • Drag coefficient = [CD_i] induced drag (drag caused by shape responsible for generation of lift, includes that due to trailing edge vortex downstream the lifting surface) + [CD_o] zero-lift drag (drag due to shapes not contributing to generation of lift such as fuselage, cockpit...)
  • Drag polar equation: CD = CD_o + K.CL2 where K = induced drag correcton factor and lift-dependent drag coefficient CD_i = K.CL2
  • Steady level flight: Thrust (T) = Drag (D), Weight (W) = Lift (L). Thus: W = L = L/D × D = L/D × T
  • Similarly, D = L × D/L = W × D/L = W × CD/CL. Thus, drag D would be minimum at maximum value of the ratio of CD/CL
  • m: chordwise location for maximum ordinate of airfoil on camber line
  • p: maximum ordinate of 2-digit camber line
  • ζ = x/c, M = m/c, P = p/c, a NACA airfoil can be either of the form y/c = a0 ζ1/2 + a1 ζ + a2 ζ2 + a3 ζ3 + a4 ζ4 or y/c = M/P2 (2Pζ - ζ2) for 0≤ζ≤P, y/c = M/(1-P)2 (1-2P+2Pζ-ζ2) for P≤ζ≤1.
  • CMLE: Pitching moment about leading edge
  • CM1/4: Pitching moment about quarter-chord point

The following integration formula will be helpful in calculation of lift coefficient of airfoils.

Integration of cosine functions

Integration of sine functions

Integration of higher exponent of sin-cos

Integration useful for Fourier Transforms

Integrations useful for Fourier Transforms

Calculation of Angle of Attack at Zero Lift

For small angle of attacks where MCL = Mean Camber Line: αL0

Definition of lift coefficient

Integrating each terms of the equation:

Calculation of lift coefficient

Formula: Lift Coefficient and Pitching Moment

Calculation of lift coefficient

Lift Coefficient and Pitching Moment for a cubic Airfoil

The airfoil profile is given by cubic polynomial described below.

Cubic airfoil profile

Coefficients A1, A2 and A3 are calculated as:

Cubic airfoil zero lift angle of attach

Cubic airfoil list coefficient and pitching moments

Pitching moment
  • Pitching moment is positive when it tends to raise the nose of the aircraft upward. If Ma = pitching moment about a known distance 'a' from the leading edge 'LE' then: MLE = Ma - L × a × cosα - D × a × sinα ...[I] Similarly, pitching moment about a point away from LE by distance 'x': MLE = Mx - L · x · cosα - D · x · sinα ...[II]
  • Subtracting equation [I] from [II]: Ma = Mx + (L × cosα + D × sinα) (a - x)
  • CMx = CMa - (CL × cosα + CD × sinα) (a - x)/c ... [III]
  • CMLE = CMa - (CL × cosα + CD × sinα) (a - 0)/c ... [IV]
  • [III] - [IV]: CMx = CMLE + (CL × cosα + CD × sinα) x/c

Aerodynamic Centre: This is the point on chord at which CM is independent of CL. For angle of incidence up to 10 °, A.C. falls 20~25% behind the leading edge.

CMAC = CMa + (CL × cosα + CD × sinα) (xAC - a)/c

For small value of α, cos(α) ≈ 1 and sin(α) ≈ 0. Thus:

CMAC = CMa + CL × cosα (xAC - a)/c

If a is chosen such that CL = 0, CMAC = CMa and hence pitching moment coefficient about an axis at ZERO LIFT equals the pitching moment coefficient about Aerodynamic Centre. Thus, CMAC is also designated as CM0, the moment coefficient at ZERO LIFT.
Centre of Pressure: From equation [II], MLE = MAC - [L · cos(α) - D · sin(α)] · xAC
Converging-Diverging Nozzle:

Converging-Diverging Nozzle

Converging-Diverging - CD Nozzle Area Ratio and Mach Number

Problem Types:
  • Type-a: Flow is isentropic thoughout, given Mach number at exit and area ratio calculate mass flow rate, temperature...
  • Type-b: Given area ratio and operating pressure ratio, find out location of shock.
  • Type-c: Area ratio and Mach number at exit is given, calculate operating pressure ratio.
  • Type-d: Given location of shock and Ae/A*, find operating pressure ratio.
  • Type-e: Design exit Mach number and reservoir pressure specified, calculate critical pressure ratios.
  • Type-f: Given inlet conditions, exit//throat area ratio and Mach number before shock, find exit pressure, mass flow rate.

Type-a Problem:

Converging-Diverging - CD Nozzle Type-a problem

Type-b Problem:

Converging-Diverging - CD Nozzle Type-b problem

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