FEATool Multiphysics
v1.10 Finite Element Analysis Toolbox |

Potential Flow Over an Airfoil

Calculation of the inviscid flow field around a NACA airfoil using the potential equation. The potential or Laplace equation is equivalent to Poisson equation with zero source term. Boundary conditions are set as zero normal flow on the airfoil body, and unit velocity magnitude at the external boundaries of the domain.

This model is available as an automated tutorial by selecting **Model Examples and Tutorials...** > **Fluid Dynamics** > **Potential Flow Over an Airfoil** from the **File** menu. Or alternatively, follow the step-by-step instructions below.

- To start a new model click the
**New Model**toolbar button, or select*New Model...*from the*File*menu. - Select the
**Poisson Equation**physics mode from the*Select Physics*drop-down menu. - Enter
`phi`

into the*Dependent Variable Names*edit field. - Press
**OK**to finish the physics mode selection. - Press the
**Create NACA airfoil***Toolbar*button. - Enter
`0012`

into the*series*edit field. - Enter
`0`

into the*angle*edit field. - Enter
`100`

into the*resolution*edit field. - Press
**OK**to finish and close the dialog box. - To create a circle or ellipse, first click on the
**Create circle/ellipse***Toolbar*button. Then left click in the main plot axes window, and hold down the mouse button. Move the mouse pointer to draw the shape outline, and release the button to finalize the shape. - Select
**E1**in the geometry object*Selection*list box. - To modify and edit the selected ellipse, click on the
**Inspect/edit selected geometry object***Toolbar*button to open the*Edit Geometry Object*dialog box. - Enter
`0.5 0`

into the*center*edit field. - Enter
`1.5`

into the*x*edit field._{radius} - Enter
`1.5`

into the*y*edit field._{radius} - Press
**OK**to finish and close the dialog box. - Select
**E1**and**N1**in the geometry object*Selection*list box. - Press the
**- / Subtract geometry objects***Toolbar*button. - Switch to
**Grid**mode by clicking on the corresponding*Mode Toolbar*button. - The default grid may be too coarse ensure an accurate solution. Press the
**Settings**button to open the*Grid Settings*dialog box and select the**Gridgen2D**grid generation algorithm. - To fine tune the settings, enter
`0.3`

in the*Subdomain Grid Size*edit field and`0.3 0.3 0.3 0.3 0.05 0.05`

for the*Boundary Grid Size*. This will ensure that the airfoil boundaries are resolved with a small grid size, while the rest of the domain uses a coarse grid. - Also select the
**Boundary layers**checkbox to create higher quality grids near the airfoil boundaries. - Press the
**Generate**button to call the grid generation algorithm. - Press
**OK**to finish and close the dialog box. - Switch to
**Equation**mode by clicking on the corresponding*Mode Toolbar*button. - The Poisson physics mode is used to model the potential flow equation. In the
*Equation Settings*dialog box, set the source term coefficient*f*to`0`

and also select**(P2/Q2) second order conforming**for the*FEM Discretization*order to ensure that the velocities which are derivatives of the potential is represented with high accuracy.

A convenient way to to define and store coefficients, variables, and expressions is using the *Model Constants and Expressions* functionality. The defined expressions can then be used in point, equation, boundary coefficients, as well as postprocessing expressions, and can easily be changed and updated in a single place.

- Define expressions for the velocities, angle of attack, as well as pressure coefficient by pressing the
**Constants***Toolbar*button, or selecting the corresponding entry from the*Equation*menu, and entering the following variables in the*Model Constants and Expressions*dialog box. Press*Enter*after the last expression or use the**Add Row**button to expand the expression list.

Name | Expression |
---|---|

u | phix |

v | phiy |

U | sqrt(u^2+v^2) |

alfa | 0 |

uinf | cos(alfa*pi/180) |

vinf | sin(alfa*pi/180) |

cp | 1-U^2/(uinf^2+vinf^2) |

- Switch to
**Boundary**mode by clicking on the corresponding*Mode Toolbar*button.

For potential flow normal velocities can naturally be prescribed as *Neumann* boundary conditions. Set the flow at the exterior boundaries to *nx*uinf+ny*uinf* and airfoil boundaries to zero (Where *nx* and *ny* will be evaluated as the unit normal vectors of the boundaries).

- Select
**1**,**2**,**3**, and**4**in the*Boundaries*list box. - Select
**Neumann boundary condition**from the*Poisson Equation*drop-down menu. - Enter
`nx*uinf+ny*vinf`

into the*Neumann coefficient*edit field. - Select
**5**and**6**in the*Boundaries*list box. - Select
**Neumann boundary condition**from the*Poisson Equation*drop-down menu. - Enter
`0`

into the*Neumann coefficient*edit field. - Press
**OK**to finish the boundary condition specification.

To ensure a unique solution for stationary problems without any Dirichlet boundary prescribed value conditions, set a reference level for the potential *phi* at one of the points.

- Select
**Add Point Constraints...**from the*Boundary*menu. - Enter
`0`

into the edit field. - Press
**OK**to finish and close the dialog box. - Switch to
**Solve**mode by clicking on the corresponding*Mode Toolbar*button. - Press the
**=***Toolbar*button to call the solver. After the problem has been solved FEATool will automatically switch to postprocessing mode and plot the computed solution.

After the problem has been solved FEATool will automatically switch to postprocessing mode and here display the potential function. Open the *Postprocessing* settings dialog box and visualize the velocity field *U* as surface, contour, and arrow plots.

- Press the
**Plot Options***Toolbar*button. - Enter
`U`

into the*User defined surface plot expression*edit field. - Select the
**Contour Plot**check box. - Enter
`U`

into the*User defined contour plot expression*edit field. - Enter
`20`

into the*Number or specified vector of contour levels to plot*edit field. - Select the
**Arrow Plot**check box. - Enter
`u`

into the*User defined arrow plot expression, x-direction*edit field. - Enter
`v`

into the*User defined arrow plot expression, y-direction*edit field. - Press
**OK**to plot and visualize the selected postprocessing options.

Use the *Point/Line Evaluation* functionality to plot the pressure coefficient *cp* along the upper wing boundary. At the stagnation point at the left edge the pressure coefficient should be close to *1*, it then rapidly jumps towards *-0.5* as the flow quickly accelerates, after which it slowly increases towards the trailing edge.

- Select
**Point/Line Evaluation...**from the*Post*menu. - Select
**6**in the*Boundaries*list box. - Enter
`cp`

into the edit field. - Press
**OK**to finish and close the dialog box.

To see how a higher angle of attack effects the flow field, change the constant *alfa* and solve the model again.

- Select
**Model Constants and Expressions...**from the*Equation*menu. - Enter
`6`

into the*Expression_4*edit field. - Press
**OK**to finish and close the dialog box. - Switch to
**Solve**mode by clicking on the corresponding*Mode Toolbar*button. - Press the
**=***Toolbar*button to call the solver. After the problem has been solved FEATool will automatically switch to postprocessing mode and plot the computed solution.

Note that the flow field now is unsymmetric with two stagnation points. As the viscosity and the *Kutta* condition at the trailing edge is not accounted for in this model the second stagnation point is found at the rear top boundary of the airfoil instead of at the trailing edge as would be expected.

The *potential flow over an airfoil* fluid dynamics model has now been completed and can be saved as a binary (.fea) model file, or exported as a programmable MATLAB m-script text file, or GUI script (.fes) file.